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Modular curves and the eisenstein ideal

Modular curves and the eisenstein ideal by B. MAZUR (1) INTRODUCTION Much current and past work on elliptic curves over number fields fits into this general program: Given a number field K and a subgroup H of GL2(Z)----IIGL2(Zp) classify all elliptic curves E/K whose associated Galois representation on torsion points maps Gal(K/K) into HC GL2(Z ). By a theorem of Serre [67] , if we ignore elliptic curves with complex multiplication, we may take H to be a subgroup of finite index. This program includes the problems of classifying elliptic curves over K with a point of given order N in its Mordell-Weil group over K, or with a cyclic subgroup of order N rational over K (equivalently: possessing a K-rational N-isogeny). These last two problems may be rephrased in diophantine terms: Find the K-rational points of the modular curves NI(N ) and X0(N ) (cf. I, w I). In this paper we study these diophantine questions mainly for K=Q. In par- ticular we shall determine the (QT)rational points of XI(N ) for all N. The precise nature of our results (which require close control of a certain part of the Mordell-Weil group of j=j0(N), the jacobian of X0(N ) when N is a prime number) may indeed be peculiar to the ground field O. There are other reasons why this ground field may be a reasonable one on which to focus. For example, the recent conjecture of Weil would have every elliptic curve over Q. obtainable as a quotient ofJ0(N ) for some N. Thus, a detailed analysis of the Mordell-Weil groups of these jacobians may be relevant to a systematic diophantine theory for elliptic curves over Q. We shall now describe the main arithmetic results of this paper (2). Theorem (x) (conjecture 2 of Ogg [49])- -- Let N> 5 be a prime number, and [~N-- I n----numerator( -2 ). The torsion subgroup of the Mordell-Weil group of J is a cyclic group \] of order n, generated by the linear equivalence class of the difference of the two cusps (o)--(oo) (chap. III, (1.2)). (1) Some of the work for this paper was done at the Institut des Hautes l~tudes scientifiques, whose warm hospitality I greatly appreciate. It was also partially supported by a grant from the National Science Foundation. (3) See also [36], [38] which give surveys of these results. 33 34 B. MAZUR Control of the 2-torsion part of this Mordell-Weil group presents special difficulties. Ogg has made use of theorem I to establish, by an elegant argument, that for prime numbers N such that the genus of X0(N ) is >2 (i.e. N~23), the only automorphisms of the curve X0(N ) (defined over t3) are the identity and the canonical involution w, except when N--37 [5i]. Ogg had also conjectured the precise structure of the maximal torsion sub-Galois module of J which is isomorphic to a sub-Galois module of Gin: Theorem (2) (conjecture 2 (twisted) of Ogg [49])- -- The maximal ~-type group (chap. I, w 3) is the Shimura subgroup (chap. II, w iI) which is cyclic of order n. Despite their " dual " appearance, theorem 2 lies somewhat deeper than theorem I. Decomposing the jacobian J by means of the canonical involution w, we may consider the exact sequence o-+J+-+J--~J---~o where J+=(~+w).J. One finds a markedly different behavior in the Mordell-Weil groups of j+ and J- (as is predictable by the Birch-Swinnerton-Dyer conjectures). Theorem (3). -- The Mordell-Weil group of J+ is a free abelian group of positive rank, provided g+=dimJ+>o (i.e. N>73 or N=37 , 43, 53, 6I, 67) (chap. III, (2.8)). As for the minus part of the jacobian, a quotient JofJ is constructed (chap. II, (IO.4)), the Eisenstein quotient. It is shown that J'is actually a quotient of J- (chap. II, (i 7. io)), and its Mordell-Weil group is computed: Theorem (4)- -- The natural map J-+J induces an isomorphism of the cyclic group of order n generated by the linear equivalence class of (o)--(oo) onto the Mordell-Weil group of J. We have: J(Q)=Z/n (chap. III, (3.I)). Since n> I whenever the genus of X0(N ) is >o, it follows from theorem 4 that 3" is nontrivial whenever J is, and one can obtain bounds on the dimension of simple factors of J (chap. III, (5.2)). Here is a consequence, which is stated explicitly only because one has, at present, no other way of producing such examples: Theorem (5)- -- There are absolutely simple abelian varieties of arbitrarily high dimension, defined over O, whose Mordell-Weil groups are finite (chap. III, (5.3)). Using theorem 4, one obtains: Theorem (g). -- Let N be a prime number such that X0(N ) has positive genus (i.e. N 4: 2, 3, 5, 7, and I3). Then X0(N ) has only a finite number of rational points over Q, (chap. III, (4. I)). One obtains theorem 6 from theorem 4 as follows: since the image of X0(N ) in j generates the nontrivial group variety ~, it follows that X0(N ) maps in a finite-to-one 34 MODULAR CURVES AND THE EISENSTEIN IDEAL 35 manner to J. Finiteness of the Mordell-Weil group of J then implies finiteness of the set of rational points of X0(N ). The purely qualitative result (finiteness of J(Q)) is comparatively easy to obtain. It uses extremely little modular information, and in an earlier write-up I collected the necessary input to its proof in a few simple axioms. To follow the proof of theorem 6, one need only read these sections: chap. I, w i; chap. II, w167 6, 8. to, prop. (14. I) and chap. III, w 3- See also the outline given in [39]. To be sure, the assertion of mere finiteness is not all that is wanted. One expects, in fact, that the known list of rational points on X0(N ) (all N) exhausts the totality of rational points, and in particular that the only rational points of X0(N ) for N any integer>I63 are the two cusps (o) and (~) [49]- In this direction, we prove the following result, conjectured by Ogg (1): Theorem (7) (conjecture I of Ogg [49])- -- Let m be an integer such that the genus 0fXl(m ) is greater than zero (i.e. m=iI or m>~ 13). Then the only rational points ofXl(m ) are the rational cusps (III, (5-3))- This uses results of Kubert concerning the rational points of Xl(m ) for low values of composite numbers m [27]. Equivalently: Theorem (7')- -- Let an elliptic curve over Q, possess a point of order m< +oo, rational over Q. Then m~io or m=I2. This result may be used to provide a complete determination of the possible torsion subgroups of Mordell-Weil groups of elliptic curves over Q,. Namely: Theorem (8). -- Let rb be the torsion subgroup of the MordeU-Weil group of an elliptic curve defined over Q. Then q~ is isomorphic to one of the following 15 groups: Z/m.Z for m~<io or m= i2 or: (Z/2.Z)� for v<~ 4. (III, (5-i). By [27] theorems 7 and 8 are implied by theorem 7 for prime values of m~> 23. See also the discussion of this problem in [49]-) Since theorems 7 and 8 may be of interest to readers who do not wish to enter into the detailed study of the Eisenstein quotient 3~, I have tried to present the proof of these theorems in as self-contained a manner as possible. For their complete proof one needs to know: a) J(Q,) is a torsion group (see discussion after theorem 6 above) and b) the cusp (o)--(oo) does not project to zero in J if the genus of X0(N ) is greater than zero (which is easy). (1) Demjanenko has published [19] a proof of the following assertion: (?) For any number field k (and, in particular, for k = Q), there is an integer m(k) such that Xl(m ) has no noncuspidal points rational over h, if m >i m(h). However, the proof does not seem to be complete. See the discussion of this in (Math. Reviews, 44, 2755) and in [27]- 85 36 B. MAZUR One then need only read w 5 of chapter III. If ogeC GL~(Fs) is any subgroup such that detYf=F~ there is a projective curve Xav over Qparametrizing elliptic curves with "level W-structures " [9] (chap. IV). The determination of the rational points of Xae amounts to a classification of elliptic curves over O~ satisfying the property that the associated representation of Gal(Q/O) on N-division points factors through a conjugate of 3(f. If N~> 5 is a prime number, any proper subgroup ~4 ~ C GL2(F~) is contained in one of the following four types of subgroups ([67] , w 2): (i) o~= a Borel subgroup. Then Xa~ = X0(N ). (ii) ~= the normalizer of a split Cartan (" ddploy6 " [67] ) subgroup. In this case, denote Xa~ o = Xsplit(N ). It is an elementary exercise to obtain a natural isomorphism between Xsplit(N ) and X0(N2)/wN~ as projective curves over Q, where wN~ is the canonical involution induced from z~--~/N 2 z on the upper half-plane. (iii) ,gt ~ the normalizer of a nonsplit Cartan subgroup. In this case write X~t'= Xn0nsplit(N). (iv) Yr an exceptional subgroup (or to keep to the terminology of [67] (2.5)), ~ is the inverse image in GL~(FN) of an exceptional subgroup of PGLg(FIq). An exceptional subgroup of PGL2(FN) is a subgroup isomor- phic to the symmetric group ~4, or alternating groups ~[4 or ~5" The further requirement det,~f'= F~r insures that the image of ~ in PGLz(FN) be isomorphic to ~. Moreover, if such an ~ (with surjective determinant) exists when K = Q, then N ~. + 3 mod 8. For such N write Xar X~,(N). We do not treat cases (iii) and (iv) in this paper. Of the four types of subgroups of GL~(FI,r listed above, the normalizer of a nonsplit Caftan subgroup seems the least approachable by known methods. In particular (to my knowledge) there is no value of N for which Xnons!0iit(N ) has been shown to have a finite number of rational points. As for case (ii) Serre remarked recently that for any fixed number field K there are very few N/> 5 such that X gt~(K) is nonempty when,~ is an exceptional subgroup of GL2(FN). Firstly, if the image of,~ff in PGL2(FN) is ~4 or ~5, then det Jr (F~) 2. Using the r of Weil, one sees that if Xaf'(N) has a K-rational point, then K contains the quadratic subfield of Q(~N). This can happen for only finitely many values of N for a given K, and not at all when K=Q. Secondly, Serre proves the following local result: Let ,T" be a finite extension of QsI , of ramification index e. Let E be an elliptic curve over ~{" w~l~ a semi-stable N~ron model over the ring of integers ~,~. Let r : Gal(,;~/uT') ~ PGL~(Fz~) denote the projective representation associated to the action of Galois on N-divlsion points of E. Then: if 2e < N-- I, the image of the inertia subgroup under r contains an element of order >1 (N-- I)/e. Using this local result one sees that there is a bound c(.ft") such that if N > c(~U) then Xa~(,,T') is empty for all exceptional subgroups ~o. In the case of ,~T'= Q, X| has no points ratienal over QN if N > 13. Hence it has no points rational over Q for N > 13. Serre constructs, however, a rational point on X~,(I I) and on X~,(I3) corresponding to elliptic curves with complex multiplication by "k/~3. Concerning case (ii) (elliptic curves over O such that the associated Gal(Q/Q)- representation on N-division points factors through the normalizer of a split Cartan subgroup of GL2(Fs) ) we obtain the following result: Theorem (9)- --/f N=II or N~> 17 (i.e./fX~pnt(N ) is ofpositive genus and N+ I3) then X~p~t(N ) has only a finite number of rational points (chap. III,w 6). 36 MODULAR CURVES AND THE EISENSTEIN IDEAL Remarks. -- Since Xsplit(I3) is of genus 3, one expects it to have only a finite number of rational points as well. The proof of theorem 9 is given in Chapter II, w 9. It uses the following two facts: a) J(Q) is finite (see the discussion after theorem 6 above) and b) "J factors through J-. It is interesting to note that when N-I mod 8 fact b) seems to depend on the detailed study of J (chap. II, (I7.Io)). It is often an interesting problem to apply theorem 4 to obtain an ef[~etive deter- mination of the rational points on X0(N ) for a given (even relatively low) value of N, and, to that end, somewhat sharper results are useful. Theorem (xo). -- Let p :X0(N)(Q)~Z/n denote the map obtained by projecting the linear equivalence class of x--(oe) to the MordeU-Weil group of "J (cf. theorem 4)- Then O(x) is equal to one of these five values in Z/n :o, i, i/2 (possible only /f N-----i rood 4), I/3 or 2]3 (the latter two being possible only /f N- --I mod 3) (chap. III, (4-2)). Using these results, tables of Wada [70], Atkin, and Tingley, and work of Ogg [49], Brumer and Kramer [4], and Parry [73], one obtains the chart given at the end of this introduction where the rational points of X0(N ) for N<25 o, N4= I5I, I99, and 223 are determined explicitly (also see note added in proof (end of chap. III)). The main technique of this paper involves a close study of the Hecke algebra T (chap. II, w 6) which we prove to be isomorphic to the full ring of C-endomorphisms of J (mildly sharpening a result of Ribet). We establish a dictionary between maximal ideals 9Jr in T and finite sub-Galois representations of J which are two-dimensional over the residue field of 9J~ (cf. chap. II, (i4.2), for a precise statement and precise hypotheses). The prime ideals that distinguish themselves as corresponding to reducible representations are the primes in the support of a certain ideal which we call the Eisenstein ideal ~ (chap. II, w 9), and which is the central object of our investigation. We like to view the Eisenstein ideal geometrically as follows: Let T* denote the algebra generated by the action of the Hecke operators T t (t~eN) and by w, on the space of holomorphic modular forms of weight 2 for F0(N ). The algebra T is the image of T* in the ring of endomorphisms of parabolic forms. We envision the spectra of these rings schematically as follows: Eisenstein prime ~3 T~ ~ ~ Eisenstein li ne = Spec spec L _ }spec T where the extra irreducible component belonging to T* (the Eisenstein line) corresponds to the action of T* on the Eisenstein series of weight 2. The Eisenstein ideal is the ideal 87 38 B. MAZUR defining the scheme-theoretic intersection of Spec T and the Eisenstein line. The Eisenstein quotient ff is the quotient of J associated to (chap. II, w i o) the union of irreducible components of Spec T which meet the Eisenstein line. One may think of the " geometric descent " argument of chapter III,w 3, as a technique of passing from knowledge of the arithmetic of the Eisenstein line (i.e. of Eisenstein series, and of Gin) to knowledge of the arithmetic of irreducible components meeting the Eisenstein line (i.e. of J) by a " descent " performed at a common prime ideal. One might hope that for other prime ideals common to distinct irreducible components (primes of fusion) one might make an analogous passage (cf. [39], w 5, Prop. 4)- Control of the local structure of T is necessary for the more detailed work. For example, it is easily seen that the kernel of the ideal 9Jl _~T in J(O) is 2-dimensional as a vector space over the residue field of g3l if and only if T~ (the completion at ~J~) is a Gorenstein ring (chap. II, (15. I)). We prove that T~ is a Gorenstein ring, at least if 93l is an Eisenstein prime, or if its residual characteristic is + 2, or if it is super- singular (chap. II, w I4). When ~ is not an Eisenstein prime this is relatively easy to prove. When gJt is an Eisenstein prime, it involves the structure theory of admissible group schemes developed in chapter I and a close study of modular forms mod p (chap. II). Using this work we prove: Theorem (ix). -- The Eisenstein ideal ~ is locally principal in T. If ~3=(t,p) is an Eisenstein prime of residual characteristic p, then the element ~qt = 1 q- g--T t is a local generator of the ideal ~ at ~3 if and only if: (i) t is not a p-th power modulo N J (ii) g- i 9 o mod P I (if not both ~ and p are equal to 2) or (when g=p= 2) 2 is not a quartic residue modulo N (chap. II, (I8. Io)). Most of this analysis of T~ is crucial for the proof of Ogg's conjectures 2 and 2 (twisted) (theorems i and 2) and for the more delicate descent needed to establish: Theorem (I2). -- If ~3 is an Eisenstein prime whose residue field is of odd characteristic, then the ~3-primary component of the Shafarevich-Tate group of J vanishes (chap. III, (3.6)). As described in the survey [36], theorems 4 and i o may be used to prove a version of the Birch-Swinnerton-Dyer conjecture relative to the prime ideal ~3. In the last two sections of chapter III we pursue this theme obliquely by studying the ~-adic L series (1). Guided by formulas, and by conjectures, we are led to the following result, which we prove, independent of any conjectures: Theorem (I3). -- Let ~3 be an Eisenstein prime whose residual characteristic is an odd prime p. Let l~ v) be the unique Zp-extension of Q. Let "~cv) be the p-Eisenstein quotient (the (1) Ref. [34]. 88 MODULAR CURVES AND THE EISENSTEIN IDEAL 39 abelian variety quotient of "J corresponding to the union of all irreducible components of Spec T containing ~3 = (~, p), chap. II, w i o). Then J(p)(Q~p)), the group of points of J(p) rational over O~ p), is a finitely generated group, and is finite if p is not a p-th power modulo N. We also obtain asymptotic control of the ~3-primary component of the Shafarevich- Tare group of J in the finite layers of O~ p). It would be interesting to develop the theory of the Eisenstein ideal in broader contexts (i.e. wherever there are Eisenstein series). Five special settings suggest them- selves, with evident applications to arithmetic: One might study X0(N ) for N not necessarily prime (e.g., N square-free). Although much will carry over (cf. Appendix) there is, as yet, no suitable analogue of the Shimura subgroup, and the " geometric descent" is bound not to be decisive without new ideas: (but see forthcoming work of Berkovich). One might attempt the same with XI(N ) (cf. [7I]) and here one interesting new difficulty is that the endomorphism ring is nonabelian. One might work with modular forms of higher weight k for SL2(Z), where a major problem will be to understand the action of inertia at p on the p-adic Galois repre- sentation associated to the modular form. One might stretch the analogy somewhat and consider some important non- congruence modular curves (e.g., the Fermat curves, following the forthcoming Ph d. Thesis of D. Rohrlich) where the " Eisenstein ideal " has no other definition than the annihilator, in the endomorphism ring, of the group generated by the cuspidal divisors in the jacobian of the curve. One might also work over function fields in the context of Drinfeld's new theory [I3]. Throughout this project, I have been in continual communication with A. Ogg and J.-P. Serre. It would be hard to completely document all the suggestions, conjec- tures and calculations that are theirs, or all that I learned from them in the course of things. I look back with pleasure on conversations and correspondence I had with them, and with Atkin, Brumer, Deligne, Katz, Kramer, Kubert, Lenstra, and Van Emde Boas, Ligozat, Rapoport, Ribet, and Tate. We conclude this introduction with a chart describing the numerical situation for prime numbers N<25o. The columns are as follows: N: ranges through all primes less than 25o such that g=genus(X0(N))>o. g+ = number of=L eigenvalues of w acting on parabolic modular forms of weight 2 for to(N). Also g+=dimJ+=genus(X0(N))+; g_=dimJ-. We write g+ as a sum of the dimensions of the simple factors comprising J=~. When a simple factor is a quotient of the Eisenstein factor, it is boldface. When the 39 B. MAZUR 4 ~ TABLE Values of N n g+ g_ v p(x) ep ii 5 o 1 3 o, _+ I/3 I7 4 o 1 2 4- I/3 19 3 o 1 i o" 23 x~ o 2 o 29 7 o 2 o 3 x 5 o 2 o e 5 = 2 37 3 i 1 2 _+ i 4 t to o 3 o e2 = 3 43 7 I 2 I O 47 23 o 4 o 53 I3 I 3 o 59 29 o 5 o 6I 5 I 8 O 67 xI 2 I+2 I o 71 35 o 35+3 ~ o 73 6 2 1~+28 o 79 ~3 ~ 5 o 8 3 41 I 6 O 89 22 I 1~@511 O 97 8 3 4 o I0I 2 5 I 7 o IO 3 X 7 '~ 6 o Io7 53 ':' 7 o ex7 = 2 IO9 9 3 I-}-4 o I I 3 08 3 1~+22+3 ~ o % = 3 x~ 7 2I 3 7 o e 7 = 2 131 65 i 10 o e 5 = I37 34 4 7 o e2 = 3 I39 ~ 3 I+7 o I49 37 3 9 o I5I ~5 3 3+6 ?? ?? I57 I3 5 7 o I63 27 I +5 7 I o I67 83 2 12 o 173 43 4 10 o 179 89 3 I -~ 0 I8I 15 5 9 O es = 3 I91 95 2 14 o i93 I6 2+5 8 o I97 49 I -}-5 10 o I99 33 4 2+10S,la ?? e s = 2 2I I 35 3+3 25+97 0 e 5 = 2 223 37 2+4 12 o 227 I 13 2+ 3 2+2+10 ?? ?? 229 19 x +6 11 o 233 58 7 1~+1129 o 239 I 19 3 17 o 241 20 7 12 o 40 MODULAR CURVES AND THE EISENSTEIN IDEAL Eisenstein factor is not simple, there is a subscript p to each boldface simple factor. If a factor has p as subscript, then it is a quotient of the p-Eisenstein quotient ~(p/. An asterisk * signals a Neumann-Setzer curve (chap. III,w 7). ,~ = the number of noncuspidal rational points on X0(N ). For a see theorem 7 above. ep=rankzp T~, where ~ is the Eisenstein prime associate to p and T~ is the completion at ~. For the range of the table it is the case that T~ is a discrete valuation ring except when N=II 3 and p=i (see chap. III, remark after (5.5) where this case is shown to be " forced " by the existence of a Neumann-Setzer factor). Thus, for all entries of the table except N = I I3, p = 2, ep = absolute ramification index of T~. In the majority of cases T~=Zp (equivalently: ep=I); therefore we only give ep when it is greater than I. By forthcoming work of Brumer and Kramer [4] all the non-boldface elliptic curve factors of J- (resp. J+) on our table have Mordell-Weil rank o (resp. ~I). The factorization of j:L into simple components comes from tables of Atkin, Wada, and Tingley. The fact that T 0 is integrally closed (N+ I I3) comes in part from a theorem (chap. II, (i 9. I)) and from Wada's tables, as do the calculations of ep. The fact that the values of a given are the only possible involves work of Ogg, completed by Brumer and Parry (also see note added in proof at the end of chap. III,w 9). 6 TABLE OF CONTENTS ................................................................................ I. -- Admissible groups ................................................................. I. Generalities ...................................................................... 2. Extensions of ~p by Z/p over S ................................................... 3. Etale admissible groups ........................................................... 4. Pure admissible groups ........................................................... 5. A special calculation for p = 2 ....................... ~ ............................ 6o II. -- The modular curve Xo(N ) ......................................................... I. Generalities ...................................................................... 62 2. Ramification structure of Xl(N ) --> Xo(N ) .......................................... 3- Regular differentials .............................................................. 4. Parabolic modular forms .......................................................... 5. Nonparabolic modular forms ....................................................... 6. Hecke operators ................................................................. 7- Quotients and completions of the Hecke algebra ..................................... 9 ~ 8. Modules of rank I ............................................................... 9. Multiplicity one ................................................................. 1 o. The spectrum of T and quotients of J ............................................. I i. The cuspidal and Shimura subgroups ............................................... I2. The subgroup D cJ[P] (p = 2; n even) ........................................... IO 3 13. The dihedral action on XI(N ) .................................................... lO 9 I4. The action of Galois on the torsion points of J ..................................... ii2 15. The Gorenstein condition ......................................................... I6. Eisenstein primes (mainly p ~ 2) .................................................. I24 17. Eisenstein primes (p = 2) ........................................................ 129 I8. Winding homomorphisms ......................................................... I35 19. The structure of the algebra T~ ................................................... 14o III. -- Arithmetic applications ............................................................ I4I I. Torsion points ................................................................... I41 2. Points of complex multiplication ................................................... I43 3. The Mordell-Weil group of J ..................................................... I48 4. Rational points on X0(N ) ......................................................... I5I 5. A complete description of torsion in the Mordell-Weil groups of elliptic curves over Q. .... I56 6. Rational points on Xsplit(N ) ...................................................... 16o 7. Factors of the Eisenstein quotient .................................................. 16i 8. The ~-adic L series .............................................................. I64 9. Behavior in the cyclotomic tower .................................................. t66 APPENDIX (by B. MAZUR and M. Behavior of the N6ron model of the jacobian of X0(N ) at bad primes ...................................................................... I73 RAPOPORT). CONVENTIONS CONVENTIONS N : a fixed prime number> 5 (the level) (1). N--I n : numerator (~2---2)" S : Spec Z. S' : Spec Z[i/N]. Ifp is a prime number we write pr[In when pr is the highest power ofp dividing n. If X is a scheme over the base S and T---~S is any base change, X/, r wiU denote the pullback of X to T. If T=Spec A, we may also denote this scheme by X/~. By X(T) we mean the T-rational points of the S-scheme X, and again, if T=Spec A, we may also denote this set by X(A). the Ndron model of the jacobian of X0(N ). J/s : T : the Hecke algebra acting on Jls (chap. II, w 6). the Eisenstein ideal in T (chap. II, w 9)- the Eisenstein prime associated to a prime number p (chap. II, w 9). 92: a general maximal prime ideal in T (not necessarily Eisenstein). If a C T is an ideal, then T, denotes completion with respect to a. If m is an integer and A an object of an abelian category, then A[m] denotes the kernel of multiplication by m. I. -- ADMISSIBLE GROUPS x. Generalities. Consider quasi-finite separated commutative group schemes of finite presentation over the base S = Spec Z which are finite flat group schemes over S'= Spec Z[I/N]. In this chapter we refer to such an object as a group scheme or (if there is no possible confusion) a group over S (or over whatever restriction of the base S concerns us). If G/s is such a group scheme, its associated Galois module is the (finite) Gal(Q/Q)- module G(O) (of ~3-rational points of G, where ~. is some fixed algebraic closure of Q). By the order of G/s we mean the order of the finite abelian group G(Q), or, equivalently. the rank over Z[I/N] of the affine algebra of the scheme G/s, (rank meaning its rank as locally free Z[I/N]-module). For the general properties of group schemes the reader (1) In the appendix we consider more general N. 43 44 B. MAZUR may consult [4o], [9]. We now fix a prime number p different from N, and suppose that the order of G/s is a power ofp. In this case, if S"= Spec Z [I/p], G/s,, is an dtale quasi- finite group ([41], lemma 5) and consequently it is an ~tale finite group over: S'n S"= Spec Z[i/p.N], determined up to isomorphism by its associated Galois module, which is a representation of Gal(Q/Q,) on G(Q)--unramified except possibly at p and N. (a) The structure of G away from p: Let us fix a choice of compatible algebraic closures: Let G/s,, be a group scheme as above. It is given, over S", by the following diagram of compatible Galois modules: (i2) 0(6.) where G(Q) is the Galois module (the Gal(()~/O)-module) associated to G/Q, G(FN) is the Gal(FN/FN)-module associated to G/F ~ and the homomorphism j maps G(F~) into the part of G(Q) which is fixed under the action of the inertia subgroup of Gal(Q~/Q,N). The homomorphism j is compatible with Galois action (compatibility being defined in an evident manner using (i. I)) and it is injective since G is assumed to be separated. (b) Extensions of group schemes from S' to S: Let G~s, be a group scheme (as at the beginning). To give an extension, G/s of G' to the base S (up to canonical isomorphism) amounts to giving a sub-Gal(QN/Q,N)- module H C G(Q) whose elements are fixed under the inertia group at N. For the sub-Gal(Qx/Qs)-module H then inherits a Gal(FN/FN)-structure and H C G(Q) may be viewed as a compatible diagram of Galois modules of the form (i. 2). This compatible diagram gives us an ~tale quasi-finite group scheme G~,, and an isomorphism: G;~ ~ G;s "flS' ~- 'nS", which, by patching, gives our extension G/s. For a given G~s, there is a minimal and a maximal extension to the base S, which we shall denote G ~ and G ~ respectively. The minimal extension (extension-by-zero; compare [33]) G~ is defined by taking H={o}C G((~,,). The maximal extension G~ is defined by taking H to be the subgroup of G(Q) consisting in all elements invariant under the inertia group at N. If G/s is any extension of G~s, , we have G ~ C G C G ~. 44 MODULAR CURVES AND THE EISENSTEIN IDEAL From this discussion we have: Proposition (I.3). -- These are equivalent: (i) G/s, admits an extension G/s which is a finite flat group scheme. (ii) The inertia group at N operates trivially in the Gal(Q/Q,)-module associated to G/s,. (iii) G/~s is a finite flat group scheme. (c) Subgroup scheme extensions: Let G/s be a group scheme as above, and let H(Q) be any sub-Gal(Q/Q)-module of G(Q,). By the subgroup scheme extension to S, H/s , of H(Q) we mean the scheme- theoretic closure of H(Q) in G/s. To understand this, it is perhaps best to consider it over the bases S' and S" separately. Over S', we are taking the scheme-theoretic closure of H(Q) in the finite flat group scheme G/s, [55], which is a finite flat subgroup scheme H/s,C G/s, whose associated Galois module is our H(Q.) [55]- To describe it over S" we must give its " diagram (~ .2)" H(FN)CH(Q); one sees easily that H(FN)=H(Q)nG(FN) , the intersection taking place in G(Q). It follows that the subgroup scheme extension of H(Q) in G/s is a quasi-finite closed subgroup scheme H/s C G/s which is finite and flat over S', and whose associated Galois module is our original H(Q). Moreover, this construction provides a one-one correspondence between sub-Gal(Q/O)-modules in G(Q) and closed subgroup schemes in G/s. If H/sC G/s is a closed subgroup scheme, we may consider the quotient (G/H)/s ([SGA 3], exp. V, VIB) first as sheaf for the fppf topology. This quotient is representable by a group scheme (of the type we are considering) as can be seen, again, by working separately over the bases S' and S": Over S', H/s is a finite flat subgroup scheme of the finite flat group scheme G/s , and the quotient is representable, by [57] theorem i. Over S", one easily constructs the "diagram (1.2)" of the quotient and one finds: o o H(~N ) J" - > H(Q) G(FN ) J~ (G/H)(FN) fi~ (G/H)(Q) o o where JC~/H)is injective because H(FN)=G(F~)c~H(Q ). 45 4 6 B. MAZUR It follows from this discussion that there is a one-one correspondence between filtrations of G/s by closed subgroup schemes, and filtrations of G(Q) by sub-Gal(Q/Q)- modules. Moreover the successive quotients of any filtration of G/s by closed subgroup schemes are again group schemes (of the type we are considering), and their associated Galois modules are canonically isomorphic to the successive quotients of the corresponding filtration of G(~). (d) Determining G/s, by its associated Galois module: Here is a consequence of the work of Fontaine. Theorem (x .4) (Fontaine). -- Let G}~s!, G}~s ~, be two finite flat p-primary group schemes with isomorphic associated Galois modules. If either: (a) p+2 or (b) G}~, are both unipotent finite group schemes. Then r,.cl) is isomorphic to Gl2s ! v/s, Discussion. -- By Fontaine's theorem 2 [I4] and the subsequent remark (p. I424) , the isomorphism between the associated Galois modules extends to an isomorphism: GIll ,,.r~_csl (Fontaine works over the Witt vectors of a perfect field) /z~ = --'/zp A standard patching argument gives the version of Fontaine's result quoted above. (e) Vector group schemes of rank i. If V/s, is a finite flat group scheme killed by p, we may view V/s, in a natural way as admitting an Fp-module structure. If k is any finite field and V/s, is endowed with a k-module structure, we shall call V/s, a k-vector group scheme. The rank of V/s, (as k-vector group scheme) is the dimension of the k-vector space V(O). If V/s, is a kl-vector group scheme and ks/k 1 is a finite field extension, then by V| s the evident construction is meant (one takes the direct sum of as many copies of V as there are elements in a kl-basis of ks, and gives it the natural ks-module structure). Proposition (x .5). -- Let k be a finite field of characteristic p. Let V/s be a finite flat ~ | k-vector group scheme of rank I. Then either V/s~(Z/p)/s| or: V/s=~p/s Fpk. Proof. -- The Gal(Q/Q)-representation of a k-vector group of rank i is given by a character Z :Gal(Q/O_,,)~k* and hence determines a cyclic abelian extension of Q of order dividing pt i (pr = card(k)) unramified except at p. Such an extension must be contained in O~,(~) (~p a primitive p-th root of i) and therefore has order dividing p--I. Consequently the character Z takes values in F~Ck* and it follows that there is a sub-Fp-vector group scheme of rank i, V0/sC V/s whose associated Gal(Q/O) representation is given by the character Z : Gal(Q/Q)-~F~. By the Oort-Tate classi- fication theorem ([54], Cor. to thm, 3) applied to the group scheme of order p, V0/s, 46 MODULAR CURVES AND THE EISENSTEIN IDEAL one has that V0/Fp is either of multiplicative type or 6tale (1). Replacing V/s by its Cartier dual, if necessary, we may suppose that V0/s is Etale, and consequently the character Z is trivial. Moreover, the group scheme (V/F~) ~t has a k-module structure and is nontrivial since it contains V0/Fp. Its order is then >q, and at the same time <q since the order of V/s is q. It follows that V/s is 6tale, and has trivial Gal(Q/O)-action; it follows that as k-vector group scheme, V/s=(Z/p)| For a detailed study of k-vector group schemes, especially of rank I, see Ray- naud's [55]- Corollary (x. 6). -- Let V/s be a group scheme of order p. (i) Let p +-2. If the associated Galois module to V is Z/p, then V/s, ~ (Z/p)/s,. If the associated Galois module to V is ~p, then V/s, ~ ~p/s,- (ii) Let p = 2. Then V/s, is isomorphic either to (Z/2)/s, or to ~2/s'- (f) Admissible p-groups. Definition. -- An admissible (p-)group G over S (or over S') is a group scheme (as usual in this chapter: commutative, quasi-finite, separated, flat, such that G/s, is finite and flat) which is killed by a power of p, and such that G/s, possesses a filtration by finite flat subgroup schemes such that the successive quotients are S'-isomorphic to one of the two group schemes: Z/p or !% (called an admissible filtration). By (i. 6) and (c) G/s, possesses an admissible filtration if and only if its associated Gal(Q/Q)-module possesses a filtration by sub-Gal(Q/Q)-modules whose successive quotients are isomorphic to the Gal(Q/O)-modules Zip or ~p (called an admissible filtration of a Gal(Q/Q,)-module). Clearly a closed subgroup scheme of an admissible p-group is again admissible, as is the quotient group scheme of an admissible p-group by a closed subgroup scheme. We have the notion of short exact sequence of admissible p-groups: 0--> GI-+ G~-+ G3-+ 0 where O 1 is closed in G 2 and the morphism G 2--> G 3 induces an isomorphism of fppf sheaves 03/01-~ 03. To an admissible p-group we may attach the following numerical invariants: t(O) =logp(order of G/s, ) (the length of G) 8(G)=logp(order of G/s,)--logp(order of GIFt) (the defect of G) e(G) = the number of (Z/p)'s occurring as successive quotients in an admissible filtration of G/s,. (a) We may deduce this from the following simple consequence of the theory of Oort-Tate, which may also be checked directly: The group scheme czp over the base Spee (Z/p) admits no extension to a finite flat group of order p over the base Spec (Z/p2). 47 4 8 B. MAZUR h~(G)=logp(order(Hi(S, G))), cohomology being taken for the fppf topology ([SGA 3], Exp IV, w 6). Remarks. -- The invariant d(G)=logp(order G(Q)) depends only on the Galois module associated to G. The invariant 3(G) is detectable from the structure of G/spe~(z~ ). The invariant e(G) is detectable from the structure of G/Fp: ~(G) = logp(order G(Fp)). If p+2, one can also determine ~(G) from the Gal(Q/Q)-module structure of G(Q). This is, of course not the case if p=2. We are mainly interested, in this paper, in h ~ for i=o, 1. Note that: i-i0(s, G)=G(S), while Hz(S, G) may be given an appropriate " geometric " interpretation. (g) Elementary admissible p-groups. By an elementary admissible group G we shall mean an admissible group of length one. Up to isomorphism there are four elementary admissible p-groups: z/p, Z/p , v,,, where (Z/p~)/s is, as in (b), the extension-by-zero of (Z/p)/s, and similarly with t~/s,. The invariants of these elementary groups are given by the following table: z/p z/p 0 I 0 t OC I I 0 0 o(p+2) h ~ I o o I (p=2) o(p+2) h :t o o O if N~ I modp (p odd) where ~ = or N = -- I mod 4 (p = 2) I otherwise. It is straight forward to establish the first three lines of the above table. To compute Hi(S, l~p) use the Kummer sequence (offppf sheaves) o-+[~p-+Gm-+Gm-+o giving: Ha(S, Wp)=(Z*)/(Z*) p since the ideal class group of Z is zero. Also, Hi(S, Z/p)=o because there are no unramified cyclic p-extensions of Q,. The nontrivial class in HI(S, 1~2) is represented by the S-scheme Spec Z[%/~I], regarded as l~2--principal homogeneous space (torseur) over S. Forming the exact sequences offppf sheaves over S: 48 MODULAR CURVES AND THE EISENSTEIN IDEAL 49 o --~- Z/p t' -+Z/p ~ ~ -+o o-+ one computes H~ q~)=Z/p; H~ O?)=t~p(FN). The natural map: Hi(S, ~) -+ Hi(S, t~) is injective if and only if the principal homogeneous space Spec Z[~/~---I] for iz2 over S does not split when restricted to the base Spec F N (i.e. when N=--I mod 4). These facts establish the table. Proposition (x.7). -- Let G/s be an admissible group. Then: hi(G) -- h~ < 3(G) -- :r Proof. -- The right hand side of the above inequality is additive for short exact sequences of admissible groups. The left hand side is sub-additive in the sense that if o-->Gx-+G~-->G2-+o is such a short exact sequence, then: hi (G2) -- h~ (hl(Gi) -- h~ -~ (hi (Ga) -- h0(Ga)). To see this, one simply uses the long exact sequence offppf cohomology coming from our short exact sequence. One clearly has equality if, instead of hX(Ga) one inserts hl(Ga) '~- logp(order(image Hi(G2) in Ha(Ga))) in the displayed line above. The asserted subadditivity follows. Since any admissible group G has a filtration by closed subgroup schemes whose successive quotients are elementarY admissible groups, the discussion above reduces the problem of checking the asserted inequality for any admissible group to the same problem for elementary admissible groups, where it follows from an inspection of the table above. Remark. -- When r I (which will be the case in our applications) the asserted inequality is, in fact, an equality for elementary admissible groups. 2. Extensions of try by Z/p over S. The point of this section is to show that there are no nontrivial such extensions. Proposition (2. x). -- Let p be any prime number. Then: Ext~(l~v, Z/p)= o. Proof (1). __ To begin, we reduce our problem to a calculation in &ale cohomology. Let Sch ~-p/s denote the underlying scheme (ignoring group structure) and let: s : Sch ~, � Sch ls.v --~ Sch ~p, be the group law. (1) An alternate approach to the proof of (2. i) in the case of an odd prime p is to show that an element of Ext~(l/,p, Z/p) must go to zero in Ext~pec Q, using Herbrand's theorem below, and the argument of chapter III, w 5. One then could apply Fontaine's theorem (I .4) to conclude. 49 5 ~ B. MAZUR Since Sch lap/s is connected, and (ZIp)/s is Stale, there are no nontrivial 2-cocycles for lap/s with coefficients in (Z/p)/s, and therefore Ext~(~p, Z/p) is the kernel of: ~*-=~-~ H~(Sch ~, Z/p) > H~(Sch ~x ~, Z/p) where ~i are the first and second projections (i=I, 2) and cohomology is computed for the fppf topology or ([15] , w ii) since Z/p is Stale, for the Stale topology. To see the assertion made, the reader may verify it directly, following [12] and (e.g.) [SGA 3], exp. III, w I. If X, Y are any two schemes equipped with Fp-valued points: /x (i. I ) Spec Fp we allow ourselves to use the symbol XvY to refer to any scheme-theoretic union of X and Y along (subschemes which are nilpotent extensions of) Spec(Fp). Taking Spec(Fp)-+X to be one such scheme, and Y=S=Spec(Z) to be the other, denote by Hi(X) the Stale cohomology group HI(XvY, Z/p). One obtains an exact sequence: o -+ l(x) _+ Hi(X, z/p) -+ Itl(Spec(F ), Z/l,) using: the Mayer-Vietoris exact sequence for Stale cohomology, the fact that Spec(Fp) is connected, and that Hi(S, Z/p)=o. We learn, in particular, that the group Hi(X) is independent of which scheme-theoretic union of X and Y was made (provided that it is subject to the above conditions). A similar calculation gives an additivity formula for ~1 (for any diagram (2. I)): (2.2) HI(X V y) = Hl(X ) (~ Hi(y) o We may write Sch t~p =TvS where T denotes the " cyclotomic scheme "" Spec(Z[x]/(x'-l + x'-~ + . . . + ~ ) ). If M denotes the p-primary component of the Galois group of the Hilbert class field extension of (the field of fractions of) Z Ix]/(xP-l+xP-~+... + 1), then: HI(T, Z/l~ ) = Hom(M, Z/p). Therefore by (2.3), if l~i is the maximal quotient of M such that p splits completely in the field extension classified by l~i, we have: (2.5) HX(T) ----- Hom(~l, Z/p). The automorphism group of I*p/s maps to the automorphism group of the scheme T and we have canonical identifications: Aut(I*p/s) = Aut(T) = F; where asF~ operates by aa=" raising to the a-th power" in the group scheme !*~. The isomorphism (2.5) is compatible with this action in the following sense: 50 MODULAR CURVES AND THE EISENSTEIN IDEAL 5 I If a~F~ and 9etiom(l~i,Z/p), then q~(m)=(aa.9)(aa.m ) where the action of a a on M is the natural action of the morphism % : T-->T on M_cHI(T ) (one- dimensional homology). To decompose our spaces into eigenspaces for the action of F~ we need some terminology: If H is a Zp[F~]-module and jeZ/(p--i)Z, let H(J)={heH t %.h = aJ.h} (where if a eF~ we denote its operation on H by %). Then H= @ H (i), the summation being taken over all j~Z/(p--i)Z. By the compatibility formula above, we get: (2.6) HI(T)(J) = Hom(l~i (-j), Z/p) for all jeZ/(p--I)Z. Note that Sch(t~v� p.p) is a wedge (in the sense of v) ofS withp+i copies ofT; these copies can be considered as the images: TC Sch ~p ~ Sch(~v� ~p) where, to be noncanonical for a moment, we may take the maps x to be given by the set of 2 � I matrices: (a, I) for a=o, I, ...,p--I and (I,O). Using (2.2) we obtain that H~(Sch(~p� is a direct sum of p-~-~ copies of Hi(T). Let us describe this group in a more " choice-free " manner. Consider all imbeddings -c : ~p ~ i~p � ~p. The 2 � i matrices representing all imbeddings z range through the set of nonzero elements of Fp� Let Funct(A, B) denote the set of functions from A to B and form: Hl(Sch(I~p x y.,)) ~ Funct(Fp x Fp, Hi(T)) by sending heH~(Sch(I-tp� to the function (~:~'~*h). Let OF.(Fp� H~(T)) denote those functions which send (o, o) in Fp � Fp to o in Hi(T), and which are compa- tible with the natural action of F~ on domain and range. From our noncanonical description of Hl(Sch(ap� ~p)) it follows that ~ induces an isomorphism between H~(Sch(g.p� and @~(F~� HI(T)). By the analogous but easier construction for Sch I~v we get an isomorphism: Hl(Sch g.p) ~- r HI(T))=HI(T) and one can check the commutative diagram: Hl(Sch ~%) > Ul(Sch(g.p x ~p)) 9 OF;(FpxF p H~(T)) Ol,~ (Fp, H~(T)) a 5l 52 B. MAZUR where 8 is just the obstruction-to-linearity: If fe(PF~(Fp, Hi(T)) then 8f(x,y)=f(x§ We are reduced to analyzing the kernel of 3, the obstruction-to-linearity. Let q)j denote functions which bring o to o and are homogeneous of degree j, under the natural action of F~; on domain and range. Thus: q)Ff(F,, H~(T)) 8 >. OF$(Fp� ~I(T)) @ (Pj(Fp, Fp)| (j) ) @ (I)j(F~� Fp)| ~ where the summation is taken over j~Z/(p--I)Z. We are led to consider the maps: (2.7) %(V,� F,) for each j~Z/(p--i) Z. Clearly (I)j(Fp, Fp) is a one-dimensional vector space over Fp generated by the function x~x ~, and ~ applied to it is the function (x-t-y)~--xS--j. Thus (2.7) is injective if j=t=i. To show that Ext~(~p, Z/p)=o it therefore suffices to show that Hl(T)(t)=o. Equivalently it suffices to show that ~I(-I)=o. In fact, M (-a) vanishes. This is a consequence of a theorem of Herbrand [2o] together with the calculation of the second Bernoulli number. For the convenience of the reader, we shall reprove the theorem of Herbrand, which follows from the theorem of Clausen-von Staudt, Kummer's congruence, a power summation congruence (cf. [72], chap. V, w 8) and Stickelberger's theorem (cf. [23]). To prepare, let the Bernoulli numbers Bi be defined by: t/(et i)=ZBit~/i! (1) and the Bernoulli polynomials: B,(X) ~--~ (n).B,. X"-' (So B 0=I, B 1=-I/2, ...). We have these classical facts: If p is a prime, p. Bm is a p-integer, and B m itself is a p-integer provided m ~ o mod(p - I ) ( Clausen-von Staudt). If p is a prime, and mr o mod(p-- I ), then Bm/m is a p-integer whose residue class mod p depends only on m mod(p--I): gm/m-Bm+p_l/(m-t-p--i ) modp (Kummer). (1) This differs from Iwasawa's choice [22]. 52 MODULAR CURVES AND THE EISENSTEIN IDEAL Let p be an odd prime number. Suppose k is a nonnegative integer such that k -}- I ~- o rood p, and k--I ~o rood(p--I). Then: p--1 a k -=p. B k rood p2. (Power summation congruence [72], chap. V (8. i i) Cor. to theorem 4.) To apply the Stickelberger theorem, we use the class field theory isomorphism to identify the Galois group of the Hilbert class field of K=Q,[x]/(xP-l+... +x) with the ideal class group of K (1). Let Y denote the p-primary component of this ideal class group. Thus: M-+Y. It is important to check that 0 commutes with the natural action of 1~ on domain and range. The action on the domain may be viewed as follows: If L/K is the Hilbert class field extension, then L/Q, is Galois and the natural action of G on itself by inner automorphisms (~g(x)----gxg -1) induces an action of F~ on M, which is equal to the action considered above. The action on Y is induced by the natural action of F; = Gal(K/O) on ideals. The fact that 0 commutes with these actions is, then, VII, theorem (I i. 5) (i) of [5]. Thus, we have: M(J) =y(J). In what follows we suppose that p2>2 and j is odd. This makes sense because j is an integer mod(p--i) and p is odd. For convenience, takej to be an ordinary integer in the range o<j<p--I. Write f=p--~--j (so j=--j mod(p--~), and o<j-<p--I). Let co :F;---~Z; be the Teichmfiller character. We shall now quote (what is, in essence) Stickelberger's theorem (cf. Iwasawa's p-ac L functions [23]. Our " Y " replaces his " So "): Proposition (2.8). -- y(1) = o. If j + i, then the p-adic number: p--1 a~0 is a p-adic integer, and ~j. Y(J) = o. Corollary (2.9) (Herbrand). -- Let j be odd and different from ~. If Bp_j~o modp, then Y(J~ = o. Proof. -- We show, under the hypotheses of the corollary, that ~j is a p-adic unit. For this, we examine: p--1 p--1 Z a.o~-3(a)= Z a.oJ(a) modp ~. a=O a=O (1) For definiteness, take the class field theory isomorphism 0 to be the map induced from +-a as in [5], VII, w 5. 53 54 B. MAZUR p--1 p--1 Since o~(a)--aPmodp 2, Y~ a.eoY(a) = Z akmodp ~, where k:pj+I. a=0 a~0 Since p42, k+I,omodp. Since j4I, k--I,omod(p--i). Therefore the power sum congruence (above) applies, giving: p--1 ak=_p.Bk modp 2. a=0 To prove corollary (2.9) we show that if Bp_~=BT+ t is not congruent to zero modp, then Bk=Bp~+l also is not. But pf+I=f§ and so Kummer's congruence applies; it proves the assertion since p j§ i and j§ i are both p-adic units. Corollary (2.I0).- Y(-t)=o (also: Y(-3)=Y(-5)=Y(-7)=Y(-9)--o) for all p (also: Y(-11)=o for all p4=69I, ...). Proof. -- We may suppose p odd (this is the only place in this paper where p = 2 is significantly easier than its fellow primes). Writing Y(-~)=Y(P-(~+I)) we see (2.9)that Y(-i)----o if B~+l~o modp and i§ or (2.8) if i§ The Corollary then follows from Clausen-von Staudt, and determination of the first few Bernoulli numbers. 3" Etale admissible groups. Fix a prime number p different from N. We consider only p-groups in this section. By a constant group over any base we mean an ~tale finite flat group scheme with trivial (constant) Galois representation. By a [x-type group we mean a finite flat group scheme whose Cartier dual is a constant group. By a pure (admissible) group we mean a finite flat group scheme which is the direct product of a constant group by a ix-type group. Proposition (3- x ). -- Any dtale admissible finite flat group over S is constant. Any admissible finite flat group of multiplicative type over S is a [x-type group. Proof. -- The second assertion follows from the first, by Cartier duality. To see the first, let G be an ~tale, finite flat admissible group over S. Proceed by induction on the length of G, and suppose /(G)_> I. Then, there is a finite flat subgroup G0C G such that G/Go=Z/p , since G is both 6tale and admissible. By induction, G o is constant, and G represents an element in Ext,(Z/p, Go). Now consider the Ext i exact sequence associated to o-+Z-+Z--~Z/p-~o over S. Note that Ext~(Z, Go)=Hi(S, Go), and Hi(S, Go) vanishes since G o is a constant group and there are no nontrivial unramified (abelian) extensions of Z. We obtain an isomorphism: H0(S, G0)/p.H~ Go) -~ Ext,(Z/p, Go). 54 MODULAR CURVES AND THE EISENSTEIN IDEAL Performing the same calculation over (e.g.) Spec(C) rather than S, and comparing (using that S is connected) we get: Ext,(Z/p, Go) -+ Ext~per , Go) which indeed implies that every extension of G o by Z/p over S is constant. Q.E.D. If G is a constant admissible group over S', killed byp ", it is sometimes convenient to write: G = (Z/p') | C where C is an abstract finite group killed by p", and Z/p ~ is, to be sure, the constant S'-group scheme. The | construction is the evident one. We may take: C = Homs,(Z/p', G). Similarly, if G is a [z-type group over S', killed by p", we may write: G = i~:| M where M is the abstract finite group Homs,(btp, , G). Now let G/s, be an 6tale admissible group which is an extension of A/s, by B/s, where both A and B are constant groups over S'. Write A~Z/p~| B~Z/p~| for an appropriate integer e, and abstract finite groups A, B killed by p". We may view G/s, as giving rise to an element: gcExt~, (A, B). To deal with Ext~,(A, B) it is useful to have the following fairly complete des- cription. Let p~[IN--r. Set (Z/N)*=Hom((Z/N)*,Z/p ~) (the Pontrjagin p-dual). Lemma (3.2). -- There is a canonical isomorphism: Ext~,(A, B)= Ext(A, B)| ((Z/N)*| , B)[p~]) (to be described in the course of the proof below). Proof. -- By Ext(A, B), we mean Ext in the category of abelian groups. By Horn(A, B)[p~] we mean the kernel of p~ in Horn(A, B). The map Ext(A, B) -~ Ext,, (A, B) is the one which associates to an extension of abstract groups o ~ B ~ E -+ A-+ o the corresponding extension of constant groups over S'. The map Ext~.(A, B) -~ Ext(A, B) is " passage to underlying abstract group " (or equivalently: restriction of the base from S' to Spec(C)). To establish the isomorphism, resolve A by free abelian groups (of finite rank): o-+R-+F-+A-+o and evaluate the long exact sequence of Ext's to get: o-+ Ext(A, B) -+ Ext~,(A, B) -+ Hom(a, HI(S ', B)) -~o. Since B is a constant group scheme, an element in HI(S ', B) is given by the following data: an abelian extension K/Q unramified outside N, and an injection Gal(K/Q) C B. 55 56 B. MAZUR Since any such extension is isomorphic to a subfield of Q,(~N) (recall: p4~N) we have the canonical isomorphism: Ha(S ', B)= Hom((Z/N)*, B) (using the isomorphism Gal(Q(~N)/O,)~ (Z/N)*) and therefore, we have the canonical isomorphisms : Horn(A, Ha(S ', B))= Hom((Z/N)*, Horn(A, B)) = Hom((Z/N)~, Hom(A, B) [p~])) where the subscript p above means p-primary component. Since (Z/N)~ is a free module of rank i over Zip ~', we have: Hom(A, HI(S ', B))= (Z/N)*| Hom(A, B)[p~]. Remark (3.3)- -- It is sometimes convenient to make a choice of a generator +N : (Z/N)*-+ Zip ~ (1), in which case, an element geExt~,(A, B) gives rise (under projection to the second factor of the formula of (3.2)) to a well-defined element N| where yeHom(A,B)[p~]. We refer to y as the classifying map for g (dependent, of course, on the choice of +~). The associated Galois module to the group scheme G/s, may be neatly described in terms of +N and y, as follows. Fix oeGal(Q/O_.). For xEG({~..) the mapping x~a(x)--x induces a homomorphism from A----A({~..) to B=B(Q) which is simply q~N(~).y(~) where ~ is the image of x in A. If G is an dtale admissible group over S', let the canonical sequence of G denote the filtration of closed (dtale admissible) subgroup schemes over S': o=GoC GIC 9 9 9 C G defincd inductively as follows: Gi+ I is thc inversc image in G of thc group generatcd by the S'-sections (i.e. the Galois invariant sections) of G/G~. Thus, thc succcssivc quotients are constant groups and G = Gm for some integcr m. If m is the Icast such integer, say that G is an dtale (admissible) group of m stages. If G1C G is the " first stage" then, by definition, Gx is the largest constant subgroup of G. If G 2 C G is the "second stage ", then G~ is an extension of the constant group A-----Gs/G1 by the constant group B=G~ and, furthermore, its classifying map y is injective since Gz is the maximal constant subgroup of G~. If G is an admissible group of multiplicative type over S', we may similarly define the canonical sequence for G, as follows: G i + 1 C G is the inverse image in G of the largest ~t-type subgroup of G/G v Note that if G is an admissible multiplicative type group then its canonical sequence (1) Which we also view as a homomorphism from Gal(Q/Q) to Zip a by composition with: Gal(Q/Q) ~ Gal(Q(~N)/Q) ~ (Z]N)*. 56 MODULAR CURVES AND THE EISENSTEIN IDEAL is not necessarily dual to the Cartier dual of the canonical sequence of the dtale admissible group G v. Rather, G 1 is dual to the largest constant quotient group of G v, etc. The natural functor which passes from multiplicative type admissible groups to dtale admissible groups, and which preserves canonical sequences is the functor: G~,gg'oms,(~a It, G) where gh ~t = lim t*:.. Lemma (3-4) (Criterion for constancy). -- Let G be an gtale admissible group over S'. If N ~ i rood p, then G is constant. In general, G is constant if and only if there is a prime number t +- N such that: a) t is not a p-th power modulo N; b) The action of ~t (1) in the Galois representation of G is trivial. Proof. -- Consider the canonical sequence (Gi) for G. We need only show that G~=G1, under the above hypotheses. Thus we may assume G=G 2 is an dtale admissible group of two stages. Let y : O2(O) --+GI(O ) be its classifying homomorphism which is injective by the above discussion. Thus, for any t+N (even for g=p) the endomorphism q~t--I of G(Q) induces a homomorphism +s(t).y :G2(Q)--~GI(Q) where +N is the chosen homomorphism of remark (3.3)- Also p~.y=o, where P~IIN--I" It follows that if N~I modp, y=o, and we are done. Ift is not a p-th power mod N, it is a generator of the p-part of the group (Z/N)*, and therefore +N(t) is a unit in the ring Zip ~. Hypothesis b) then implies that y = o. Lemma (3-5) (A ~-type criterion). -- Let G be an admissible multiplicative type group over S'. If N ~i mod p, then G is a ~-type group. In general, G is ~z-type if and only if there is a prime number t +p, N such that: a) t is not a p-th power rood N. b) The Frobenius element ?t acts as multiplication by t in the Galois representation of G. Pro@ -- Pass to the dtale admissible situation by applying ~oms,($'a ~, --) (or by Cartier duality) and then use lemma (3-4)- Lemma (3- 6). -- Let t +p, N be a prime number not a p-th power mod N. If G is a multiplicative type group, then the Galois module of G 1 (the first stage in its canonical sequence) is the kernel of cpt--t in the Galois module of G. Pro@ -- Passing to the dtale situation by the functor ~oms,(~a ~, --) we may replace G by an gtale admissibie group over S', and we must show that the Galois module of G1 is the kernel of q~t--i. Work by induction on the number of stages of G. Suppose that it is true for groups ofm--i stages and let G have m stages (m>2). Thus G/GIn_ 2 has two stages (1) t-Frobenius. 57 58 B. MAZUR and its "first stage subgroup " is, by construction, Gm_l/Gm_ ~. Using the for- mula (3.3) ?t(x)--x=r where 7 is the classifying homomorphism for the 2-stage group G/GIn_2, xE(G/G,._~)(Q.) and Y is its image in (G/Gm_I)(Q), we see that the kernel of ?t--i in (G/G,._~)(Q) is the subgroup (G,~_I/Gm_2)(~..) (since 3" is injective, and r is a unit in the ring Zips). Consequently, anyelement xeG(Q) which is in the kernel of ~t--i must be contained in Gin_l(0.. ) C G(~=)). But Gm_l is a group of m--I stages, and therefore xeGl(Q), by induction. 4" Pure admissible groups. Proposition (4. x ). -- Let p ~ 2. Let A be a constant group and M a ~-type group. Then: Ext,(A, M)= o. Pro@ -- This reduces to showing Ext,(Z/p, ~p)=o. But applying Ex((--, ~p) to the exact sequence offppf sheaves o~Z-+Z-+Z/p-+o yields a long exact sequence which may be evaluated using the fact that Ext,(Z, lzp)=Ht(S, ~p). One gets the short exact sequence: o H~ + Ext,(Z/p, + n (s, o. From the Kummer sequence o-'~p-~Gm-~Gm-+o offppf sheaves, and the fact that the ideal class group of Z vanishes, one gets: Hi(S, ~p) -= (Z*) /(Z*) p. Thus we have a short exact sequence: o-+ (Z') [p] -+ Ext,(Z/p, ~p) -+ (Z*)/(Z*) P-+ o. Now suppose p + 2, and one sees that the middle group must vanish. If p = 2, we get: Proposition (,t.2). -- There are three nontrivial extensions of Z/2 by ~2 over S: Extension 1: an extension whose associated Galois representation is trivial, and whose underlying abelian group is cyclic of order 4. Extension 2: the unique nontrivial extension over S killed by 2: o-+ ~2~D-+Z/2-+o. Its associated Galois representation factors through Q(X/~-~-I). If we let: r : Gal(Q/Q) -+ Gal(Q(~v/---~)/O.~) -~ Z/2 be the composite where the first map is the natural projection, and the second an isomorphism, then the Galois representation associated to D is given as follows: (4.3) o(x) -- x = r a(O)- V(~) where if xeD(Q), then ~ is its projection to Z/2, and 7 is the only surjective homomorphism D(O_~) :+ ~(Q) with kernel ~(Q). 58 MODULAR CURVES AND THE EISENSTEIN IDEAL As usual, identifying Gal(Q(~v/~ )/O) with (Z/4)* , /f ?e denotes t-Frobenius (in the Galois group of any extension field of Q,(~r which is Galois over Q, and unramified over t) then we also write ~-1(~)for +-l(?e). One has: +_l(t)=I /f and only /f /-=--I mod 4. Extension 8: (the sum in Ext 1 of the above two elements) an extension whose underlying abelian group is cycgc of order 4, and whose Galois representation satisfies the same formula as above. Proof. -- This is evident from the exact sequences in the proof of (4-I) except for the assertions concerning Galois representations. To see those, one must recall that the nontrivial tJ.2-torseur representing the (nontrivial) element in Hi(S, y.~) is the S-scheme Spec Z[~v/~i- ]. Remark. -- The group scheme D/s (Extension 2 of (4- 2) above) will play a central role in our study of the prime 2. Since Fontaine's theorem does not apply to admissible 2-groups in general, the following result is useful: Proposition (4-4)- -- Let D/s, be a finite flat group scheme, and q~ : D/Q-~ D/Q an isomorphism over Q, (equivalently: an isomorphism of associated Galois modules). Then ~ extends to an isomorphism q~ : D/s, ~ Dfs, of group schemes over S'. Pro@ -- Since the associated Galois module to D' is admissible, D~s, is admissible, and since the inertia group at N operates trivially in the Galois representation of D (and hence also of D'), D' extends to a finite flat group scheme over S. Since the Galois representation of D' satisfies (4.3), D' cannot be an extension of F2 by Z/2 (2. i), nor of Z/2 by Z/2 (3-3), nor of Ix2 by p~z (applying (3-3) to its Cartier dual). Therefore it must indeed be isomorphic to D, by (4.2)- Since there is only one nontrivial autommphism of the Galois module associated to D, and this automorphism extends to an automorphism of D/s, our proposition follows. Proposition (t-5) (Criterion for purity: p+2).- Let p+2, and let G/s, be an admissible group. These are equivalent: a) G is pure. b) The associated Galois module to G is pure (i.e. it is the direct sum of a constant Galois module and the Cartier dual of a constant Galois module). c) The action of inertia at N is trivial on the associated Galois module to G. d) G extends to a finite flat group scheme over S. Pro@ -- Clearly a)=>b)=~c). By (~.3), c)=>d). To conclude, we must show that any finite flat admissible group over S is pure. Let G be such a group, and: o=GoC GIC G~C ... C G,=G an admissiblc filtration. Thus thc succcssivc quotients arc isomorphic either to Z/p or to ls.p over S. 59 60 B. MAZUR Step 1: We may suppose all the successive quotients isomorphic to l.tp precede those isomorphic to Z/p. This follows immediately from proposition (2. I), and induction. Therefore, for some s, G 8 C G is an admissible subgroup of multiplicative type, and G/G8 is an admissible dtale group. Step 2: G, is a ~-type group and G/G s is constant. Proof: (3-i). Step 3: G is a trivial extension of the constant group G/G~ by the ~-type group G~. Proof'. (4.1). Remark. -- Demanding that the action of inertia at N be trivial in the Galois representation is clearly not sufficient to insure purity when p =-2 (e.g., consider the nontrivial extension D of (4.2)). Nevertheless, for admissible 2-groups over S killed by 2, purity is equivalent to the requirement that the action of Gal(C/R) be trivial in the associated Galois module. As it turns out in our ultimate applications, however, the notion of purity is not the relevant one when p = 2. The final proposition of this section will be used in studying the cuspidal subgroup chap. II, w I I). Proposition (4-6). -- Let Cjs be a finite flat group whose underlying Galois module zs a finite cyclic group with trivial Galois action. If C is of odd order, then C is a constant group. If C contains a subgroup isomorphic to W2, then the quotient C/W 2 is a constant group. Proof. -- The first assertion of (4.6) follows from (I .6) and (3-4)- As for the second, we may suppose that C is killed by a power of 2 (say 2~). If ~=I, we are done. Now suppose that e----- 2. It suffices to show that C/V~2 is dtale over S. Clearly C/~2 cannot be isomorphic to ~2, for then the Cartier dual of C/s would be 6tale, hence constant, and so the Galois action on C could not be trivial. Thus C/W2=~Z/2. Now let c~>2. We shall show that C/W2 is ~tale as follows: filtering C by the kernels of successive powers of 2, if C/Vq were not ~tale, using the result proved for 0~=2, one could obtain a subquotient of C, whose underlying abelian group is cyclic of order 4, and which is an extension of W2 by Z/2, which is impossible by (2. i). 5. A special calculation for p = 2. Let Ext,_ s'(A, B) denote the subgroup of elements in Ext~,(A, B) which represent extensions of A by B which are killed by multiplication by 2. Consider the (nonflat) surjective homomorphism Z/2--~ll~ (over S'). This induces a homomorphism: Ext,_ s.(ll2, Z/2) % Ext~_s.(Z/2 , Z/2) and we shall show that this map is injective. The full story, however, is the following: 60 MODULAR CURVES AND THE EISENSTEIN IDEAL 61 Proposition (5. x ): a) Ext~_s,(Z/2, Z/2) is of order 2. b) If N~ zk i mod 8, then Ext~_s,(l~2, Z/2)=o. c) If N = + I rood 8, then the homomorphism ~ is an isomorphism of groups of order 2. Pro@ -- a) Follows from HI(S ', Z/2)=Ext~_s,(Z/e, Z/2). As for an analysis of Ext~_s,(p%, Z/2) there are two ways to proceed. We may adapt the general method of (2. i) to the base S', or (since our group schemes have such small orders) we may work directly. We choose the latter course. Consider the composition: : Ext~_s,(l~2, Z/2) ~ Ext~_s,(Z/2, Z/2) -+ HI(S ', Z/2). A " geometric " construction of ~ is the following: If.x is an element in Ext~_s,(tt2, Z/2) represented by an extension: (5.2) o-+Z/2-+E~ tt2-+o, let r : S'--~t~2/s, denote the nontrivial section, and let E,C E denote the fiber-product: E r > E S' '> P~2 Thus E, is the " nontrivial " Z/2-coset. It is a Z/2-torseur over S' and represents the element ~Ix) in HIIS ', Z/21. The scheme-theoretic intersection of E~ and Z/2 in E consists in two points lying over Spec F 2. From this we deduce that the prime 2 splits in the S'-extension E,. If ~(x)=o, E, is a trivial Z/e-torseur. Take the subgroup of E generated by the (unique) S'-section of E, which meets (at Spec(F2)) the zero-section of E. This is a group scheme which projects isomorphically to ~, and therefore gives a splitting of (5.2), showing that x=o. Thus ~ (and hence ~) is injective. Now suppose that x is nontrivial (i.e. (5- 2) does not split). The Galois representation associated to the group scheme E of (5-2) is isomorphic (to be sure) with the Galois representation associated to the pull-back via ~. In particular (3.3): If aeGal(Q/Q) and zeE(O..), we have ~(z)--z=+N(z).y(5) where 5is the image of z in b%(Q) and y : ~2(Q)~Z/2 is an isomorphism. This Galois representation factors through the unique quadratic number field in Q(~N), namely" t Q.(v'-N ) if N~--I mod4 K = [ Q,(V/N) if N - i mod 4. Note that if N ~ =k I mod 8, then 2 does not split in K, whence b). As for c) we need only construct a nontrivial extension (5.2) when N---~-i rood 8. We omit 61 62 B. MAZUR the details (noting that no use of c) is made in this paper) and merely sketch this construction: Since 2 does split in K, when N = :51 mod 8, one can glue the S'-scheme Z/2 and the nontrivial Z/2-torseur over S' transversally at their closed points of characteristic 2, and check that the evider~t group law away from characteristic 2 extends to a group- scheme structure of S'. II. -- THE MODULAR CURVE Xo(N ) x. Generalities. We shall be reading closely in two sources of information concerning moduli stacks, their associated coarse moduli schemes, and the theory of modular forms: [9], [24]- Our ultimate object is to derive as complete a description as possible of j/s, the Ndron model of the jacobian of X0(N ) over S (N~ 5, a prime number; X0(N ) the modular curve associated to P0(N)). Technically, reduction to characteristics 2, 3, and N (in that order) produce the thorniest problems, and we shall spend most of our time dealing with them. We keep to most of the conventions of [9]- Thus, for m any integer, and HC GL~(Z/m) we have the algebraic moduli stack d/a ([9], IV, (3.3)) proper over S, which may be interpreted over Spec Z[I/m] as the fine moduli stack classifying generalized elliptic curves with a level H-structure ([9], IV, (3-i)). Its associated coarse moduli stack ([9], I, (8. i)) may be denoted M H. If H is the trivial subgroup of GL2(Z/m ) we writeJg,~for~ H. If H=F0(N)={( a ~)c-omodN} write: #r R =M/o(N) ; MH = M0(N ) . Given a pair (E/T , h) where E is an elliptic curve (or a generalized elliptic curve [9], chap. II) over the scheme T, and h is a level H-structure of EIT, then the T-valued point of M H determined by this pair will be denoted j(E/T , h). In relating modular forms to differential forms, and in other arguments as well, we shall have use for certain refinements of M0(N), associated to level structures H, where d/~H= M H (i.e. where the fine moduli stack " exists " as an algebraic space). Two notable refinements having this property are ([9], IV, th. (2.7)): a) Take re=N, and H=Po0(N)={( a bd)c=omodN, a=imodN} (recall: N> 5) in which case we write MH=MI(N ). b) Take m=3N and H=P0(N;3)={(: ~)-(; ~ in which case we write M H = Mo(N; 3). The schemes MN-+ MI(N )-+M0(N ) are smooth when restricted to: S'= Spec(Z [i/N]). 69 MODULAR CURVES AND THE EISENSTEIN IDEAL As in [9], the superscript h (d/Z0(N) h, M0(N) h, etc.) refers to the open substack or subscheme obtained by removing the " supersingular points " of characteristic N. The precise geometric structure of M0(N)/s is given by [9], IV, th. (6.9). In particular, Mo(N)/F~ is a union of two copies of P~F~ (the j-line) intersecting transversally at the " supersingular points ", where a point x on the second copy gets glued to the image under N-Frobenius x/NI on the first. One has that M0(N)) s is smooth, and if j is a supersingular point of characteristic N (using [9], IV, (6.9) (iii) and the fact that N>5) then M0(N)/s is regular at j if j+o, I728. In the latter two cases, M0(N ) is formally isomorphic to : W(~N) [x,y]/(x .y-- N 3) if j~o W(F~) [x,y]/(x .y-- N 2) if j=I728. In any case, M0(N )-~S is locally a complete intersection, hence Gorenstein, and hence also Cohen-Macaulay [3]. By suitable blow-up of the points j=o, 1728 in characteristic N, when they are supersingular, we may arrive at the minimal regular resolution of M0(N)/s, which we call X0(N)/s. See the appendix for a study of these minimal regular resolutions in a somewhat broader context. The structure of the " bad fiber " (i.e. over FN) of X0(N ) may be schematized as follows: Z' E Z blow-up of j = 1728 when supersingular (-~N - -- i mod 4) transversal intersection at j+ ~ 728, o, supersingular cusps blow-up of j=o when supersingular (-r ~ --I mod 3) Diagram I The irreducible components E (which occurs if and only if N-- i mod 4) and F, G (which occur if and only if N ~- -- i mod 3) are the " results " of the appropriate blow-ups, and are all isomorphic to P~F,- See appendix for further discussion. The mmphism Xo(N)-+S is a local complete intersection, and, again therefore a Gorenstein morphism, and hence Cohen-Macaulay. Clearly M0(N)/s,=X0(N)/s, , and therefore (over any base extension of S') we have two possible names for the same thing. We try to keep to this usage: it will be called M0(N)/s, when we are interested primarily in questions of modular forms, and X0(N)/s, when we are interested in more 63 64 B. MAZUR geometric questions. Also, for reasons of consistency, and compatibility with other authors, we allow ourselves the same double notation M~(N)/s, = XI(N)/s, in dealing with H=P00(N), and likewise: M0(N; 3)/spec(zEt/3m)----X0(N; The usual names (o and oo) are given to the two cusps of M0(N ). We view these as (nowhere intersecting) sections of M0(N)/s ([9], VII, w 2). They also give rise to sections of X0(N)/s (denoted by the same symbols) and, after arbitrary base change T~S, to T-sections of X0(N)/T. The cuspidal sections o and oo distinguish themselves as follows: The morphism of stacks dt'0(N)--->.~r ) induced by the rule (E/w , H)~E/T is unramified at oo and ramified at o. 2. Ramification structure of XI(N ) ~ X0(N ). As always, let N be a prime number -->5. Let k be a field which is algebraically closed and of characteristic different from N. The map (2. I) XI(N )-+X0(N ) over k is unramified at the cusps, and has precisely these points as ramification points: TABLE i Name of point Occurs if Structure Char k in X0(N ) Value of j and only if of inertia group (i)+, (i)_ 1728 N = I mod 4 cyclic of order 2 ~=2, 3, N (p)+, (p)- o N =- i mod 3 cyclic of order 3 (p)+, (p)- N ~ I mod 3 cyclic of order 3 2 o = x 728 N -=-- I mod 4 cyclic of order 2: "wild 6) ramification of first type " (i)+ , (i)_ N = i mod 4 order 2 o = I728 (P) N ~---- x mod 3 order 3: "wild rami- fication of first type " Definition (2.2). -- A Galois p-cyclic extension of local fields whose residue fields are of characteristic p will be said to be wildly ramified of the ,~-th type if the higher ramification sequence (Gi) (cf. [6o], chap. IV) of subgroups of its Galois group G has the following structure: G = Go = G1 ..... G~ ..... o. We shall establish the facts of the above table. Recall that (since k is of charac- teristic different from N) the cusps are unramified in the mapping (2. I). If (E, C) 3)/sp0~(z~lj3Nj)- MODULAR CURVES AND THE EISENSTEIN IDEAL is a pair representing a point j(E, C) eX0(1V ) (1), then the automorphism group Aut(E, C) denotes the stabilizer of C in Aut(E); since N~5, the natural homomorphism: Aut(E, C) ~ Aut(C)=(Z/N)* is injective (2). Passing to the quotient: Aut(E, C)/(i I) ~ (Z/N)*/( I)=Gal(Xx(N)/X0(N)) the above homomorphism identifies Aut(E, C)/(+~) with the inertia group of the point j(E, C). If j(E)+o, I728 then Aut(E)=(+ i), and therefore j(E, C) is not a point of ramification. Characteristic k+2, 3, N. -- j(E)=I728: The group Aut(E) is cyclic of order 4. It can stabilize no cyclic subgroup of order N, C C E if N ~ i mod 4. On the other hand, if N = I rood 4 there is a 4-th root of unity in F and consequently Aut(E) stabilizes precisely two cyclic subgroups of order N. Call them C* and write (i)  =j(E, C We have (i)++(i) - since no element of Aut(E) interchanges C + and C-. This establishes the first line of the table. j(E) = o: The group Aut(E) is cyclic of order 6 and reasoning similar to the above establishes the second line of the table. Characteristic k-= 2. -- Let E be an elliptic curve with j(E)= 1728 = o. We may take E to be the curve y2-ky=xL The endomorphism ring of E is the ring of Hurwitz quaternions and its automorphism group is of order 24. The quotient Aut(E)/(zki) is isomorphic to 9.I4, the alternating group on 4 letters. The cyclic subgroups of 9/4 have orders I, 2, 3 and any two cyclic subgroups of the same order are conjugate. Fix cyclic subgroups Hz, HaCAut(E)/(zkI) of orders 2 and 3 respectively. Note that the inverse images of these in Aut(E) are cyclic groups of orders 4 and 6 respectively. As above, then, H3 stabilizes precisely two cyclic subgroups of order N (call them C if N--Imod 3 and none if N~Imod 3. Write (p)177 Since H a is its own normalizer in Aut(E)/(~I), (p)+4=(p)- and we have established the third line of the table. The subgroup H 2 stabilizes two cyclic subgroups of order N (call them, again, C +CE) if N=imod4 and none if N*I rood4. But the normalizer of I-I 2 in Aut(E)/(zLI) is isomorphic to the Klein 4-group. Since the entire Klein 4-group cannot stabilize C :~, any element in the normalizer of H 2 which is not in H z must inter- change C + and C-. Consequently j(E, C +) =j(E, C-). Denote this point (i). Clearly, (i) is a point of wild ramification. We shall show it to be offirst type, using an argument (a) Here C is a cyclic subgroup of order N in the elliptic curve E, giving the " level r0(N ) structure ". (3) If a ~ + I is an automorphism of any elliptic curve over any field k, then a is of order 4 or 6, and generates a ring ofendomorphisms isomorphic to the ring ofcyclotomic integers of that order. See the discussion in Appendix I of [29] concerning endomorphism rings, and automorphisms. The assertion concerning injectivity above then follows, for there is no homomorphism of the ring of cyclotomic integers of order 4 or 6 to FN, which sends a to x, provided N~>5. 65 66 B. MAZUR communicated to us by Serre: For any field k of characteristic different from N, one has the short exact sequence: --->- 1"* t~l t~l ~ 1 0 J hZXo(N)[ k ---> XZX,(N)[ k ~Xx(N)/Xo(N ) ---> 0 where the zero on the right comes from the fact that Xo(N ) is smooth, and f: XI(N ) --->Xo(N ) is generically separable. Note that dimkH~ is the degree of the global different ([6o], chap. III, w 7, Prop. I4) giving us the Hurwitz Formula. Namely, the degree of the global different of XI(N)/X0(N ) is: 2 .gl(N) -- 2 -- (~-~) 9 (2g0(N)-- 2) where gi(N) is the genus of the curve X~.(N). It follows that the degree of the global different of XI(N)/X0(N ) is independent of the characteristic of the field h (provided that it is different from N). From the first two lines of our table, choosing k to be of characteristic diffeIent from 2, 3 and N, we compute the degree of the global different to be: where if r is a rational number we let the symbol < r > be r if r is an integer and o if not. On the other hand, if k is of characteristic two and if (i) is wildly ramified of the v-th type, using prop. 4 of [6o] chapter IV, from what we have established concerning the ramification structure of XI(N )/X0(N ) we compute the degree of the global different to be: Consequently, v=i, and the third and fourth lines of our table have been established. Characteristic k = 3. -- Here, again, we take E to be an elliptic curve with j(E)=o=I728; for example: y2=x~--x. The group of automorphisms Aut(E) is of order i2 and has the following structure: it contains a normal subgroup of order 3, 9I 3 C Aut(E) such that the quotient of Aut(E) by ~3 is a cyclic group of order 4, which acts in the unique nontrivial way on ~I3 ([29], App. i, w 2). The center of Aut(E) is (i I) and Aut(E)/(+I) is isomorphic to 63, the symmetric group on 3 letters. Again we have that the cyclic subgroups of ~a have orders i, 2, 3 and any two cyclic subgroups of the same order are conjugate. Fix cyclic subgroups H~, H a C Aut(E)/(~:I) of orders 2 and 3 respectively. It is again true that the inverse images of these in Aut(E) are cyclic groups of orders 4 and 6 respectively. From this point on, to establish the last two lines of our table, we proceed exactly as in the case of characteristic 2, with the one important difference that now it is H2 which is its own normalizer in Aut(E)/(4-I) while H3 is normal in Aut(E)/(4-i). Our table is established. 66 MODULAR CURVES AND THE EISENSTEIN IDEAL N--I) S'= Corollary (2.3).- Let re=numerator ~. Let Spec(Z[I/N]). Let X2(N)/s, -~ Xo(N)/s, denote the unique covering intermediate to XI(N)/s, ~ Xo(N)/s, which is a Galois covering, cyclic of order n. Then X2(N)/s, ~ X0(N)/s, is itale. We shall refer to the above ~tale covering as the Shimura covering. 3" Regular differentials. Deligne and Rapoport [9] work out Grothendieck's duality theory in the case of a Cohen-Macaulay morphism ~ : X-~T (purely of dimension d). We shall recall the contents of [9] in the case d=I, with some change of notation. Definition (3. x ). -- If rc : X~T is a Cohen-Macaulay morphism purely of dimension i, where T is a noetherian scheme, the sheaf of regular differentials is: f~x, =3r (1) ([9], chap. I, (2.~. ~)). The sheaves ~x/, are flat over T, their formation commutes with arbitrary base change T'--~T and with ~tale localization of X. If X/T is smooth, then f2x/" = ~,. If X is a reduced curve over an algebraically closed field k which has only ordinary double point singularities xl, ..., xe and if (x~, x~') denotes the inverse image ofxi in X*, the normalization of X, then the regular differentials on X consist in meromorphic dif- ferential forms on X* regular outside of the x~, x", having at worst a simple pole at the x~ and x~', and verifying: re N =-- res~, (i = I, ..., t). The duality theorem gives an isomorphism. If o~ is a locally free 0T-Module, and if the RJ~.o ~ are locally free 0T-Modules , the duality theorem (loc. cir. (2.2.3)) gives: (3.2) R~-S=,(~ | ~ ) (RJT~,~') ~ where ~ denotes OT-dual, We prepare to apply the duality theorem to the morphisms 7~T, 7~, which are the base changes to T---~S of the morphisms occuring in the diagram: where i is the minimal regular resolution introduced in w i. (x) Deligne-Rapoport call this ~XjT" We often omit the subscript X/:r when no confusion can arise. 67 68 B. MAZUR Let ~M.(s)is(cusps) denote the locally free sheaf, which when restricted to the complement of the cuspidal divisor, is equal to the sheaf of regular differentials and whose sections in a neighborhood of the cuspidal divisor are meromorphic differentials with, at worst, a simple pole along the cuspidal divisor. Let ~9~.(~)~,(cusps) be the subsheaf of functions in 8 which are zero along the cuspidal divisor. An easy computation gives that R~.r ) vanishes when j+ i, and is an extension of 1, ~, R ~T,d)Mo(S)~, by dTT, when j=I. Consequently, the R ~T,d~o(s)(cusps) are locally j , free d)r-Modules, when the R nw, are. Lemma (3.3). -- Let T be a noetherian scheme flat over S = Spec (Z), or over the spectrum of a field. Then: R~ ' T, M0(N)/T j , R z% ) RJ r~T, d)xo(s)l, are locally free OT-Modules. Remark. -- The duality isomorphism (3.2) then applies in these cases. Proof. -- By the preceding discussion we need only prove the assertion for R~f, Oy where f: Y---~T stands for either the morphism 7: or r:'. Formation of R~fT, d)yl ~ commutes with flat base change T'-->T ([EGA], III, (r.4. r5)), which reduces us to considering the unique case T=Spec(Z). Also, j=i is the only nonobvious dimension. Let p be any prime. Since d~y is flat over Z, we have the exact sequence: o P ~ 1 P o-+ R)C.Or-+ R~ ~ R~ Cy,r, R f,~Py-+ Rlf, ~)y. The only global functions on Y/Fp are constant functions. This is evident for p + N, since Y/rp is then smooth and irreducible, and follows for p = N from the explicit description of the fibers X0(N)/1,,, and M0(N)/FN (w I). It follows that Rlf.Or has no nontrivial p-torsion. Proposition (3.3) (commutation with base change). -- Consider the category of rings which are flat over Z/m for some m, or over Z. Let R-+R' be a homomorphism in this category, then: H0(M0(N)/R, f2)| R' -~ H0(Mo(N)m, , ~2) and: H~ f~(cusps))| ~ H~ , f2(cusps)) are isomorphisms. Pro@ -- The assertion holds for R~R' flat, by [EGA], III, (I.4.15). This allows one to reduce the question to the base changes Z~Z/m (for m an arbitrary integer); for these the assertion is true since HI(M0(N)/z, f2) and HI(Mo(N)/z, f2(cusps)) are torsion-free Z-modules, by (3.3) and the duality isomorphism (3.2). d~o(Iq)jT(CUSpS d)M0(r~)j, OM0(N)~ MODULAR CURVES AND THE EISENSTEIN IDEAL Proposition (3.4). -- Let T be a (noetherian) scheme flat over S or over a field. The natural map induces an isomorphism: 0 t R~ ~* -+ R xT. Pro@ -- This is evident ifN is invertible in T. Thus, since formation of RifT,~y/T commutes with flat base change, we are reduced to the cases T=Spec(Z), and T=Spec(F~). For the latter case, we must check that the regular differentials on Mo(N)jF~ and on X0(N)/F~ coincide. But this is elementary, taking account of the explicit description (diagram I of w I) of X0(N)/F~ in terms of Mo(N)/FN and using the fact that a meromorphic differential on p1 with at worst simple poles at two points a, beP 1 is uniquely determined by its residue (at a, say). For T=Spec(Z), we have: i : H~ s176 -+ H~ fl~~ is a morphism of free Z-modules of finite rank (3.3), (3.2) 9 Since i| ] is an isomorphism, it follows that i is injective, with cokernel c~ a finite N-primary abelian group. Since by (3.3), (3.2) HI(Ko(N), g~x0(~)) and HI(Mo(N), O%(i~)) are free Z-modules, we have the diagram: o o >o o > H~ ~x.(~)) > H~ , ~.(m) > ~ >o o > H~ ~xo(s)) > H~ ~0(i~)) > c~ H~ ~Xo(S)a,~) ~> H~ s176 O O giving that W=o. 4" Parabolic modular forms. In this section R will denote a ring flat over Z, or over Z/m for some m. We shall be interested in comparing three different points of view concerning holomorphic modular forms of weight 2 over F0(N), defined over R. i. q-expansions of classical modular forms (Serre [47]). If RC Q, let B(R) C R[[q]] be the R-submodule of q-expansions (at oo) of (classical) modular forms of the above type (1) (whose q-expansion coefficients (at 0o) (1) Holomorphic, of weight 2, over l~0(N). flM.(N)~T" Y~Xo/(N)jT 7 ~ B. MAZUR lie in R). Let B~ B(R) be the subspace generated by parabolic forms. We do not require that the q-expansion coefficients at the other cusp o lie in R. Using ([9], VII, (3.18)) and the discussion of w 6, (1) below one sees, however, that these " other " coefficients lie in N-1.RC (3.. The unspecified term q-expansion will mean: at oo. It follows from the work of Igusa and Deligne or ([69] , p. 85, th. (3.52)) that: B(Z)| and B~ |176 for RC O (formation " commutes with base change ") and we define the g-submodules: B~ C B(R) C R[[q]] for an arbitrary ring R by the above isomorphisms. ~. Sections of the sheaf co | 2 over the moduli sta& (which are holomorphic at the cusps) (Katz [24]; Deligne-Rapoport [9]). Let A(R) (resp. A~ denote the R-module of modular forms (resp. parabolic modular forms) of the above sort, as defined in [24], (I.3) (compare [9], VII, w 3). We also refer to an element of A(R) as a modular form in o~ | Thus, an element 0~A(R) is a rule which assigns to each pair (E/T , H), where E is an elliptic curve over an R-scheme T, and H a finite flat subgroup scheme of E/T of order N, a section e(E/T , H) of co | where is the sheaf of invariant differentials. EIT f~E/T The rule o~ must depend only on the isomorphism class of the pair (E/T , H) and its formation must commute with arbitrary base change T'-+T. Finally, it must satisfy the condition of holomorphy at the two cusps. The q-expansion morphism: q-exp : A(R) -+ R[[q]] ~ defined by: 0~(Tate curve/a(cq)), bt~) = ~. square of canonical differential (1) is injective, if R is flat over Z or if I/NeR (z) and allows us to identify A(R) with an R-submodule of R[[q]] in these cases. We shall be especially interested in A(R) for rings R containing I/N. In this case one has an alternate description of A(R) as the space of holomorphic modular forms of level N, defined over R ([24], (1.2)) which are invariant under the action of the appropriate Borel subgroup. The question of whether formation of A(R) commutes with base change is a difficult one, and may be viewed as the main technical problem of this paragraph (3). (1) Compare [9], VII, (I.I6); [24], A 1.2, p. I6I. (2) This follows from the argument of VII, (3.9) of [9], or, if I/N~R, [~4], (1.6.I). r Compare [24l, (I.7) and (I.8). 70 MODULAR CURVES AND THE EISENSTEIN IDEAL 7i 3" q-expansions of regular differentials. The pair (Tate curve/z[[q]], ~) gives rise to a morphism: -~ : Spec Z[[q]] ~ M0(N)/z as in [9] VII, Th. (2. I) and z identifies Z[[q]] with the formal completion of M0(N)/z, along the section over S = Spec Z corresponding to the cusp oo. For any ring R, induces a morphism: t : Spec R((q)) ~ M0(N)m where R((q))=R[[q]][I/q] is the ring of " finite-tailed" Laurent-series. Suppose U is an open subscheme of Mo(N)/R through which the above morphism t factors, and such that U/Spe.CR/m is contained in the irreducible component of the Hasse domain h M0(N)/sp~0CR/N ) to which the cusp oo belongs. If 7 is a regular differential on U, we refer to y as a meromorphic differential on M0(N)/R. Define the q-expansion of y to be that element ~ of R((q)) such that t*y='~, dq. The q-expansion morphism is an injection of the space of meromorphic differentials over R to R((q)). The reason for this is, briefly, as follows. If 7 is defined on U and ~=o, then y is defined, and vanishes, on a formal neighborhood of the section in M0(N)/R corresponding to the cusp oo. Since ~ is an invertible sheaf, and the support of y inter- sects each geometric fiber of U in a finite number of points, 7 = o (cf. argument of [9], VII, th. (3"9); or of [24], .(I.6.2)). The q-expansion morphism also induces an injection: q-exp : H~ ~(cusps)) -+ R[[q]]. To prove this when R = Z/N use the structure of the fiber in characteristic N and the fact that a differential on P~F~, which possesses at worst simple poles, is known when its poles and (all but one of) its residues are known. It then follows for R= Z]N" (m~1) by an argument using (3.3). If I/NeR, the argument of the preceding paragraph gives injectivity; ifR is flat over Z one must use that M0(N ) is Cohen-Macaulay. By means of the map q-exp, we identify H~ , f2(cusps)) with a sub- R-module of R[[q]]. The relation between A(R) and H~ ~(cusps)) is given by the " Kodaira- Spencer style morphism" of [24], (I.5) and A, (I.3.I7). For our purposes, the fol- lowing statement is convenient. Lemma (4. x ). -- The natural mapping ( [24], A, (i. 3.17)) : a) is an isomorph#m on the complement of the cuspidal sections in XI(N)/R, for any R, as above, which contains I/N; 71 7 2 B. MAZUR b) is defined on the complement of the cuspidal sections and the supersingular points of characteristic N in X0(N; 3),for any R, as above, which contains I /6. Now when R contains I/N, let: Ul(N)/R=the open subscheme of XI(N)/R obtained by removing the discriminant locus of Xl(N) Xo(N). Uo(N)m=the image of UI(N)m in X0(N)/n. The Galois covering UI(N)/R-+ Uo(N)/R is a finite dtale Galois extension with Galois group (Z/N)*/( I) (1). When R contains i]6, let: V0(N; 3)m =the open subscheme of X0(N; 3)m obtained by removing the discriminant locus of X0(N; 3)-+ X0(N) and the " supersingular points " in charac- teristic N. V0(N)/R =the image of V0(N; 3)m in M0(N)m. If G is the covering group of X0(N; 3) -+ X0(N), then V0(N; 3) -+V0(N) is a finite ~tale Galois extension with covering group G. Lemma (4.2). -- The Kodaira-Spencer morphism induces: an imbedding: A(R) ~H~ g~(cusps)) /f I/NeR; a morphism: A(R) -+ H~ f~(cusps)) /f I/6~R. Moreover, these morphisms bring A~ to the subspace 0f regular differentials on the respective bases. Proof. -- Suppose I/NeR. Modular forms for F0(N ) on ~o | ([24] , (i.3)) are modular forms for FI(N) which are invariant under the action of the covering group. Using lemma (4- i), the Kodaira-Spencer morphism associates to an element ~ in A(R) a regular differential al on the complement of the cuspidal sections in UI(N)m , which is invariant under the action of the covering group. Since UI(N)/R-+U0(N)m is dtale, a 1 descends to a regular differential a on the complement of the cuspidal sections in U0(N)/R. By Cor. A, (I.3. I8) of [24], the q-expansions of a coincide with the q-expansions of e. The condition of holomorphy (resp. parabolicity) at the cusps then insures that a have at worst a simple pole (resp. is regular) at the cusps; consequently a is a section of f~(cusps) (resp. f2) on all of U0(N)/R. Similarly, if I/6~R, one constructs a differential on V0(N)/R. Note that both U0(N)I R and V0(N)IR, when defined, are open dense subschemes of Mo(N)m. Also, the construction which associates to ~ differentials on these open subschemes yields the same differential on the intersection (same q-expansion). Consequently, to any o~A(R), and for any ring R as considered in this section, we may associate a meromorph# differential on M0(N)/R, a, with the same q-expansion as a. (I) To avoid confusion with various Galois actions we refer to this group as covering group. 72 MODULAR CURVES AND THE EISENSTEIN IDEAL To compare differentials with elements of B(R), we begin with: Lemma (4- 3) : H~ a) C B~ C R[[q]] f (cusps)) c B(R) C R[[q]]. Proof. -- If R=Z, the first inclusion follows since H~ ~) is a subspace of H~ s having integral q-expansions. Consequently we obtain the desired inclusion for any R of the type considered in this section, since formation of both range and domain commute with base change Z-~R (3.3). The second inclusion follows similarly. Lemma (4.4): (i) If R is a field of characteristic p  N, then: A0(R) =H~ n) if p= 2 and N-I rood 4 and: A(R) =H~ f~(eusps, (i;)) A(R) =H~ n(cusps, (0))) if P= 3 and N-I mod 3 otherwise A(R) =H~ n(cusps)) N---Imod3). (See Table I.) (i.e. p> 5, or p = 2, N---I rood 4 or p = 3, (2) /f R=Z[I/m] for some integer m, then: A(R) c HO(Mo(N)/ , f (eusps)) A~ C H~ g~). Note. ~ By f~(cusps, (i)) is meant the sheaf of meromorphic differentials which have, at worst, simple poles at the cusps and at the point (i) of Table I. Proof. -- Let eeA(R) and let a be its associated meromorphic differential. (i) R a field of characteristic p. N: Here a is a meromorphic differential on M0(N)I R which is regular on U0(N)/R, except for possible simple poles at the cusps, and which lifts to a differential on XI(N)/R regular except at the cusps. We shall make a local calculation to determine when a meromorphic differential can become regular, after finite extension. Explicitly, let k be an (algebraically closed) field of characteristic p and D 1 C D2 a finite extension of k-algebras, which are discrete valuation rings, with residue field k. Let a 1 denote a meromorphic differential on D1 relative to k, and let a 2 denote the induced differential on D~, relative to k. Sublemma. -- If D t C D~ is (gtaIe, or) tamely ramified, then the meromorphic differential a S is a regular differential (resp. has a simple pole) if and only if a 1 is a regular differential (resp. has a simple pole) on D 1. 10 74 B. MAZUR If D1 C D2 is wildly ramified of the first type (2.2), then a2 is a regular differential if and only if a 1 has (at worst) a simple pole on D1. Pro@ -- Since there are no nontrivial 6tale extensions in our situation, we may assume D1CD2 a totally ramified Galois extension of degree r. Write D2=k[[y]] for a choice of uniformizery of D~, and Dl=k[[x]], where x is a uniformizer, chosen so that x=-c?(y), where q~(Y)ek[Y] is a polynomial. Using ([6o], III, 7, Cor. 2) one calculates the different of D1C D~ to be (q~'(y)). If v 2 is the valuation on D~ such that v2(y)=i , then v2(x)=r , and v2(~'(y)) can be calculated in terms of the orders of the higher ramification groups of D1C D2 ([6o], oo IV, w 2, Prop. 4: v~(q~'(y))= 52, (Card(Gi)-I)) and consequently: i=0 v2(~'(y)) = r--I (tamely ramified case) v~(?'(y))=2p--2 (wild ramification of first type). Up to multiplication by a unit in Dr, we may write at as xS.dx for some sEZ. Thus, a s is, up to a unit, of the form xS.~'(y).dy, and: v2(x'.~'(y))=rs+r--I (tame ramification of degree r) =ps + 2(p-- i) (wild ramification of first type). The assertions of the lemma can now be read off from the above formulae (e.g., in the case of wild ramification of first type, s>--I if and only if ps+ 2(p--I)>O). Now return to the case (i) of lemma (4.4), and the meromorphic differential a. By Table I, Xx(N)/R-+ X0(N)m has at most one point of wild ramification, and none if characteristic R:t: 2, 3. Moreover, if there is a point of wild ramification, it is of first type. By the sublemma, the meromorphic differential a is regular with the exception of possible simple poles at o, 0% and (i) and (p), if they occur (see Table I). Conversely, any meromorphic differential which is regular, except for such simple poles will (by the sublemma) lift to a differential on XI(N) with, at worst, simple poles at cusps. This gives us the identification of A(R) with the appropriate space of meromorphic diffe- rentials, as in the statement of (i). The subspace A~ is then identified with the space of differentials on M0(N ) which are regular everywhere with the exception of a possible simple pole at (i) (if p=2, and N-Imod4) or at (p) (if p=3, and N-= I mod 3). Since the sum of the residues of a differential over a complete curve is zero, it follows that A~ is identified with the space of everywhere regular differentials. (2) R =- Z[I/m] : We show A~ C H~ ~) ; the other inclusion is proved in the same way. Recall that M0(N)~ a denotes the complement of the characteristic N supersingular points, in M0(N)m. The meromorphic differential a is regular on an open dense subscheme of Mo(N)~ a. 74 MODULAR CURVES AND THE EISENSTEIN IDEAL Let D~ o (resp. Do) denote the divisor of poles (resp. of zeroes) of a, on M0(N)) a. Recall their definition: ifx is a point of the scheme M0(N)~R, and d~ x the local ring at x, let q~x be a local generator of f~0(N)~R at x. Since 0~ is a unique factorization domain, one can find g~,hxe6) ~ with no common factors such that g~.a-=hx.~. A local equation at x for Do, (resp. for Do) is given by: gz=o (resp. hx=o ). Now let p be a prime number with these properties: a) p~'2.3.N.m; b) D~ o and D O have disjoint support in characteristic p: [D~o| n [D0| =O. It follows from the definition of polar divisor and a), b) that a| is definitely nonholomorphic at D~o| p. Therefore part (I) of our proposition implies that the support of D| is disjoint from the fibre of M0(N)m-+ Spec(R) in characteristic p. Since D~o contains no irreducible component of any fibre of ~, it follows that Do~= o; therefore a is regular on M0(N)) R. To see that a is, in fact, regular on M0(N)/R, use that the supersingular points of characteristic N are of codimension 2 in M0(N)/R, and f~ is an invertible sheaf, and M0(N)/R is Cohen-Macaulay (SGA ~, Exp. III, Cor. (3-5)). Lemma (4.5). -- Let R be flat over Z[I/N]. Then: A(R) ---- HO(Mo(N)~R, n(cusps)) = B(R) A0(R) = H0(Mo(N)/R, f2) = B0(R). Proof. -- We establish the first line above; the second may be obtained by essentially the same argument. First let R=Z[I/N]. By the previous two lemmas, we have inclusions: A(R) C H~ f~(cusps)) C B(R) and so we must prove that A(R)= B(R). But this follows from the q-expansion prin- ciple ([24], Cor. (1.6.2)). To be more precise, using the notation of (I.6.2), take f to be any element in B(R), n=N, K--Q, L=R. Katz's corollary (I.6.2) then gives us that f is a holomorphic modular form (in co | of level N, defined over R. Since f, viewed as a modular form of level N, is invariant under the appropriate Borel subgroup of GL~(FN) , it is in A(R). Now let R be flat over Z[I/N]. Lemma (4.5) will follow from what we have done, provided we show that: A (R) = A (Z [ I/N]) | R. Even this " commutation with base change " is not totally trivial. If one takes the point of view that A(R) is the space of (Z/N)*/(zk i)-invariant differentials (regular, with the possible exception of simple poles at cusps) on Xl(N)/R, however, it is an easy exercise (1). (1) If G is a group and M a Z[x/N][G]-module, flat over Z[I/N], and R a flat Z[I/N]-module, then M G@Z[1/N]R is isomorphic to (M| (The superscript G denotes invariants under G.) 75 B. MAZUR Lemma (4.6): H~ , f2(cusps)) = B(R) ItO(Mo(N)/R, f2) = BO(R). Proof. -- We show the second equality; the first is done similarly. It suffices to prove this equality for R=Z, since formation of both sides of the equation commutes with base change from Z to any of the rings R we consider. By lemmas (4.3) and (4.5), H~ f2) is a subgroup of B~ and the quotient Q is an N-primary finite abelian group. Since: H~ f~) C B~ C FN[[q]] one checks that multiplication by N is an isomorphism on Q. Lemma (4.7). -- Let m be an integer prime to N and R=Z/m. Then: H~ ~2(cusps)) C A(R) HO(Mo(N)/R, f2) C A~ Proof. -- These inclusions follow from (4.5) and the fact that the morphisms: HO(Mo(N)zE1/N], f2(cusps)) -+ H~ ~(cusps)) HO(Mo(N)z[1/~], f~) ---> HO(Mo(N)m , f2) are surjective (3.3). Lemma (4.8). -- Let m be prime to N, and R=Z/m. Then: AO(R) =H~ , ~2) = B~ and: A(R) =H~ f2(cusps))= u(g) if m and N satisfy the following properties: (a) either m ~-o mod 2, or N # I rood 4 and (b) either m ~ o mod 3, or N ~s I mod 3. Proof. -- In the light of (4.6), what must be shown is that the inclusions of (4.7) are equalities, under the hypotheses above. Lemma (4.4) (i) assures us that they are if m is a prime number. We now proceed by induction. Let p be a prime dividing m; m =m'.p. Let R' C R be the sub-R-module consisting in multiples of p (R'~Z/m'). Consider : > A(R) > A(Fp) A(R') , H(R) > H(Fv) > o H(R') 76 MODULAR CURVES AND THE EISENSTEIN IDEAL where I-I(.) stands for YI~ f~(cusps)). The bottom line is exact since formation of H(*) commutes with the type of base change which occurs in that line (3-3)- The top line is exact, by an application of the q-expansion principle ([24], (1.6.2)). The two flanking vertical inclusions are isomorphisms by induction, since if rn and N satisfy (a), (b), then m' and N also satisfy (a), (b). Therefore the central vertical inclusion is an equality, as well. This establishes the assertion of lemma (4.8) concerning A(R); the assertion concerning A~ is established by a similar argument. Summary and convention (4-9)- -- We shall be chiefly concerned with modular forms of weight 2, over I'0(N), for some (usually fixed) prime number N> 5- Except when indicated explicitly to the contrary, a parabolic modular form (over 1-'.(N), defined over R) will mean an element of B~ or, equivalently, a regular differential on Mo(N)/R; or, equivalently (if R is flat over Z or over a field) a regular differential on X0(N)/R; or (if R=Z/m with (re, N)=1 (4.8); or R flat over Z[I/N] (4.5)) an element of A~ For holomorphic (nonparabolic) modular forms it is true that elements of B(R) coincide with differentials defined over R, regular with the possible exception of simple poles at cusps (4.6). Nevertheless, for certain rings R, A(R) may differ from B(R) (e.g., Remark below). Thus we shall always make clear, in what follows, whether we are dealing with an element of A(R) (a modular form in o~ | or of B(R), and both notions will be useful. Remark (concerning the distinction between A(R) and B(R)). -- The Riemann- Roch Theorem and the description given in (4-4) (I) show that B(R) is of codimension I in A(R), if R is a field of characteristic 2 and N'---I mod 4; or of characteristic 3 and N--- I mod 3- In certain cases one can exhibit an element of A(R), not in B(R). For example, if char R=2, and N- 5 mod 8, it follows from the description in (5-I2) below that the power series 3 modulo 2 is (the q-expansion of) such an element; the power series modulo 3 is such an element if charR=3, and N- 4 or 7m~ On the other hand, the Eisenstein series e' (w 5) is in B(Z) but not A(Z), since its q-expansion coefficients at the cusp o can be seen to lie in N -1.Z but not in Z. Proposition (4.xo). -- There are no nonvanishing parabolic modular forms over F0(I ) (in ~| defined over any ring R flat over Z or over Z/m. Remark. -- There are nontrivial holomorphic modular forms over F0(~ ) (in ~o| defined over certain rings R (cf. (5.6)). Pro@ -- If I/5eR, lift to M0(5)/. This is a curve of genus o, and therefore has no nonvanishing regular differentials on it. Since I/5~R there are no parabolic modular forms (in o~ | over P0(5), as well. Therefore there are none over l'0(I ). If I/7eR, lift to X0(7) and use the same argument. The general ring R (as considered in this section) is then treated by patching. 77 78 B. MAZUR 5" Nonparabolic modular forms. Consider the following three power series in Z[[q]]. ~o e = I -- 24Z=t ~(m)q" where a(m) is the sum of the positive divisors of m. oo (5 -x ) e'=I--N-- 24~=1 a'(m)q "~ where C(m) is the sum of the positive divisors of m which are prime to N (as usual, N is a fixed prime number > 5). oo 8= 52 C(m)q". The power series e is the q-expansion of the Eisenstein series of weight 2 of level I (1). It is the logarithmic derivative of the q-expansion of the normalized modular form (of level i) of weight 12 : co A=q II (I--qm) ~' (~), The power series e'(q)=e(q)--N.e(q N) is the q-expansion of the Eisenstein series of weight 2 on P0(N)(3). It may be regarded, as meromorphic differential, as the logarithmic derivative of the function A(q)/A(q N) on M0(N)/Q. Since this functions has zeroes and poles only at the cusps, e' is (the q-expansion of) a differential whose only poles are (simple) poles, occurring at the cusps. Since e' has integral coefficients, we have e'E B(Z). Viewed as modular form in o~ | ~ over Q, the q-expansion of e' at the cusp o may be seen to be (using [9], VII, (3-i8)): I/N. (N--I + 24~ t C(m)q "l~) and therefore e' is not in A(Z). The power series 8 is simply e', deprived of its constant term and conveniently normalized. It will be of interest to consider those rings R over which 8 is a modular form. It is proved in [24], (4.5.4) (also (A.2.4) if P--~5) that e is a p-adic modular form ([24] , (2.2)) for everyp. Thus, if R=Fp, e is the q-expansion ofa meromorphic differential on the Hasse domain X0(1)~ R. This differential may have poles at the (1) It is denoted P It is the q-expansion of /~|.Gdv;o,o, i) in Hecke's terminology ([I9], p. 474). \7~-1 in [24]- (~) Cf. [24], A x .4.4 for a proof of this fact, which does not use the Jacobi identity. (") p- The q-expansion of r in HECg-~ ['9], 78 MODULAR CURVES AND THE EISENSTEIN IDEAL 79 supersingular points. Our first object will be to study e, both as section of co | over the moduli stack, restricted to the Hasse domain, and as meromorphic differential. Lemma (5.x). -- The power series e is the q-expansion of the meromorphic differential: -- dj modulo 24. 32. 5 and of: dj modulo 24. 33. 7 j--I728 on X0(I). Remarks. -- In the above, j is the elliptic modular function, which is a rational parameter for X0(i), and has q-expansion beginning I/q+744+ .... If: e4=I+24o ~ ~(m)q" ra=l co e =I--5o4 Z m=l are the normalized Eisenstein series of weight 4 and 6 respectively, we have: (5.2) j = e, /A = 1728+ e~/A. It would be interesting to study the poles and residues of e at the supersingular points of X0(I)/z/p, for pr any power of a prime. O.A.L. Atkin, M. Ashworth, and (independently) N'. Koblitz have some interesting formulae, algorithms, and machine computations which suggest some precise conjectures in this direction. Proof of lemma (5. I). -- Take logarithmic derivatives of formulas (5.2), regarded as identities in power series in q, noting that: d log(~) = o modulo 3.24 ~ = 24.33. 5 d log(e~) - o modulo 2.5o4 = 24. 33 9 7. O .E.D. To study e as a section of c~ | 2 over the moduli stack, recall the standard formulas giving elliptic curves in " generalized Weierstrass form " over arbitrary bases (we use the notation and conventions of Tate. Cf. Appendix i [29]). Thus, if (E/T , ~) is an elliptic curve over the base scheme T, equipped with an invariant differential, ~, we may represent (E/T , ~) locally for the Zariski topology over T as a curve: y2 + alxy + a~y = x 8 + a2x ~ + a4x + a s (5-3) = dx/(2y + alx + a3) = dy/(3 x~ + 2a~x + a4-- aly) where, if f : E-+T is the structure map, and ~ : T-+E the zero-section, then (I, x) is a basis off,0(2,) and (i, x,y) is a basis off, C(3, ). This representation may be modified by making a different choice (7:', x',y') to 79 8o B. MAZUR obtain a new equation (5.3)'- The relation between the old and new choices is given by the " data "" (u, r, s, t) where u~I~(T, 0~) r, s, t~I'(T, @T) defined by the formulas: 7~ p = UT~ X =U2X'-~r y = u3y'+su2x'+ t and, conversely, any such data gives us a new choice. The new formula (5.3)' is related to the old by: ua~ = a 1 + 2s u~ a; = as-- sal + 3 r- s" u3 a'3 = a 3 + ra 1 + 2t u4 a4, = am_ s a 3 + era 2 -- (t + rs)a 1 + 3 rg- 2st u%'~ = a n + ra m + r2 a~ + r a- ta3-- t z- rtal. Following Tate, define: b2=a~+ra2; b4=ala3+2a4. For the new formula (5-3)' one has: 2 ' (5"4) u b2=b2+ I2r; u4b'~=b4+rb2+6r 9". Lemma (5.5). -- Let R=Z/72 (72=23.33). Let T be an R-scheme. Let (E/T , ~) be a pair consisting in an elliptic curve E/T and an invariant differential ~ such that b 2 is invertible in P(T, @T)" Then the function ~ = b2--I2b4/b2 depends only on the isomorphism class of the pair (E/x, ~z) and not on the representation (5.3) chosen. It defines a section of co | over the open substack of the moduli stack of level I over R obtained by removing the cusps and" inverting b2 " The q-expansion of ~ is e, modulo 72. Proof. -- One checks, using (5.4) and working modulo 72, that the relation between and ~' (under a change of representation given by the " data " (u, r, s, t)) is u~'= which establishes everything but the last sentence of lemma (5.5). For this, we evaluate ~ on the Tate curve whose equation is ([63] , IV, 3o): y2--xy=x3+a~x+a6 (a~ =-- 5 ~, i--q ] na q n ~ ma q m Thus, b2=I and ba-=--I~ I--q"' giving ~= I + 120m=tI_q,,. Since 12ore 3 ~ -- 24m mod 72, we conclude that , = e. Remark. -- This lemma gives the first two terms of an " asymptotic expansion " of e in terms of the parameter b 2 (which cuts out the supersingular locus, 2-adically 80 MODULAR CURVES AND THE EISENSTEIN IDEAL 81 and 3-adically). Using his algorithm and machine computation, N. Koblitz has obtained the first 4 ~ terms. Proposition (5- 6). -- (Holomorphic modular forms of level I) : (a) There are no nontrivial holomorphic modular forms of level I (in o~ | defined over a field R of characteristic 4= 2, 3. (b) The " square of the Hasse invariant" is a holomorphic modular form mod 4, with q-expansion equal to e. The " Hasse invariant " is a holomorphic modular form mod 3, with q-expansion equal to e. (c) If q~ is a holomorphic modular form (of level I; in o~| defined over R=Z/m, with q-expansion beginning with the constant I, then: (i) m divides 12 ; (ii) q~=e. Summary. -- Every holomorphic modular form of level I, defined over R=Z/m has q-expansion equal to a constant. Proof: (a) R a field of characteristic 4: 2, 3 Let ~ be such a holomorphic modular form defined over R, and denote by the same letter the meromorphic differential on X0(I)/l~ associated to % Since the moduli stack associated to P0(I ; 3) " exists " (w i), lifting ~? to X0(I ; 3) yields a meromorphic differential, with at worst, simple poles at the cusps. Since X0(I ; 3) -+ X0(I) is a tamely ramified Galois extension, the sublemma in the proof of (4.4) assures us that has, at worst a simple pole at the cusp ~ of X0(i)/R. Since X0(I)t g is of genus o, q~ must vanish. (b) Going back to (5.5) one sees that, modulo 12, e is given by b2, and is therefore a holomorphic modular form modulo 12. Its q-expansion is the constant I. Modulo p (any p) the Hasse invariant is a holomorphic modular form of weight p--i, and q-expansion equal to I ([24], (2.o)). Thus, modulo 2, the Hasse invariant is of weight I and can be taken to be a 1. By " the square of FIasse invariant mod 4 " we mean a~, which is a section of c0 | mod 4- But, e- ba- a~ mod 4- Working modulo 3, the Hasse invariant is a modular form of weight 2, with the same q-expansion as e. It coincides, therefore, with e. (c) Let q0 be a holomorphic modular form mod m, such that the constant term of its q-expanslon is i. By (a) m=2~.3 b. To show that m divides 12, it suffices to show that no such q~ can exist mod 8 or mod 9. Let q~ be such a modular form mod 8 (resp. mod 9). Note that q~-e mod4 (resp. mod 3), because, by (b) q0--e is a holomorphic modular form mod 4 (resp. mod 3) ; 11 8~ B. MAZUR it is parabolic by our assumptions on % and therefore must be zero by (4.8). Let R=Z/8 (resp. Z/9 ). We may write ~= r(j). e, where r(j) is a rational function in j (viewed as rational parameter of X0(i)) with coefficients in R. If we view both q0 and e as meromorphic dj, differentials, and use (5.I) that e-- and (4.2) that ~ is a regular differential on the open subscheme Spec R[j,j -1] of X0(I)/R, we obtain that r(j) is in R[j,j-t]. By the above, we may write r(j)=iq-4L(j ) (resp. I-t-3L(j)) where L(j) is a " Laurent polynomial " in R[j,j-1]. We now use holomorphicity of q~ about the point j=o, together with the above description of r(j). If R = Z/8, consider the following elliptic curve E over the power series ring R[[t]]. E : y2+txy+y=x3, rc=dx/(2y+tx+I) One computes: b 2 = t 2 e -~ b~-- i ~b4/b~ = t~-----z 2/t E R [ It]] It- t] b 4 = t j =tl~(t"--27)-IER[[t]]. We now compute the value of the section q~ of c0 | on the pair (E, ~) over the ring of finite-tailed Laurent series R[[t]] [t-l]. q~(E, ~z)= (I +4L(j))(t2--I 2/t)eR[[t]] [t -~] =t2--I~/t +4t~.L(j). Since q~ is holomorphic, and (E, ~) is defined over R[[t]], q~(E, ~) must lie in R[[t]]C R[[t]] [t-l]. That is: (5-7) 12/t--4 t~. L(j) e R[[t]]. Let jb be the lowest power of j occuring in the Laurent polynomial L(j) with coefficient a unit mod 8. Writing 4 t2 L(j) as a finite-tailed Laurent series in t, one has that t 12b+2 is the lowest power of t occuring with nonzero coefficient. One reasons now, that if b is nonnegative, the I2/t term in (5.7) cannot be cancelled by any term in 4t~L(j), while if b is negative, t 12b + 2 is the lowest power of t occurring in the expression of (5.7). In either case, one has a contradiction. If R-~Z/9 , it is convenient to work with the elliptic curve E given by the representation: al=a3-=I ; a2=(t--I)/4; a4:o; a6 =--I ]4. Then: b2~--I e = b2--I2b4/b 2 = t--i2/teR[[t]] [t -t] b4z I j =4t6(t ~- 32)- le R [[t]]. 82 MODULAR CURVES AND THE EISENSTEIN IDEAL Again: q)(E, 7:) = (i -t- 3L(j)) (t-- 12/t) (5,8) = t--i2/t+ 3 t. L(j) eR[[t]] [t-~]. Ifj b is the lowest power ofj occurring with unit coefficient in the Laurent poly- nomial L, then t 6b+1 is the lowest power of t occurring with nonzero coefficient in 3t.L(j)~R[[t]][t-t~, and, as above, (5.8) leads to a contradiction. Q.E.D. We now prepare to study the status of the q-expansion " I " as a modular form over P0(N). The following lemma, which is in the spirit of the theory of Atkin-Lehner, and which was suggested to me by J.-P. Serre, will be helpful. Lemma (5-9) (reduction of level). -- Let I/NER. Let ~ be a holomorphic modular form in o~ | over P0(N), defined over R (k> 2). Suppose, further, that the q-expansion (at oo) of ~ is a power series in qN :~,=3~q~), Then ffis the q-expansion of a holomorphk modular form over Po(~) (again in o~ | and defined over R). To obtain an analogue of (5-9) in characteristic N, we return to the setting of interest to us: Lemma (5.IO). -- Let N_>5 be a prime number and ~ a holomorphiv modular form over ~'o(N), in B(FN) (w 4). Suppose, further, that the q-expansion of ~ is a power series in Then ~ = o. Proof of lemma (5.9). -- Let .#" denote the stack ..dt'0(N ) over R and ./V ~ =~'0(N) ~ Thus, if one is given a pair (EjT , H) where T is an R-scheme, E is an elliptic curve over T, and HC E is a subgroup of order N, defined over T, one may associate to (EIT , H) a T-valued section of the stack .A/-. There are maps: where sr and ~,o are the moduli stacks of level N and i respectively, defined over R. These maps are determined by the rules: where y :Z/NxZ/N~E[N] is an isomorphism of group schemes over T, and: E' is taken to be E/.f(oxZ/N); H is taken to be the image of y(Z/Nxo). (L T, H) s (E/H). The map ~3~ :.//~'~-~dt '~ is a Galois, dtale morphism of stacks. The Galois (covering) group may be identified with GL2(F~) acting in the natural way (by compo- sition with y) on d/~ The intermediate stack ~/'0 is fixed under the Borel subgroup 83 84 B. MAZUR .or\ over the stack ~A/'0 or as a section over ~o, invariant under the action of B. A " formal neighborhood of the cusp oo " in A/'0 is induced from the pair (Tate(q)m((q/~, ~) while a " formal neighborhood of the (unique) cusp ~ " in Me' 1 is induced from the Tate curve over R((q)) ([24], (I. 3))- We have the following commu- tative diagram: Spec R((q)) -> X ~ q l-~ qN t f~ Spec R((q)) .go where we can check that the left-hand vertical map is given by q ~ qN as follows: By ([24] , (I.II), p. 9 I) we have Tate(q)/~N=Tate(q~), and Tate(q ~) is induced from Tare(q) by extension of scalars R((q)) ~ R((q)); q~qS. By the above discussion we may give the following geometric interpretation to our hypothesis concerning ?: the restriction ~ of ? to Spec R((q)) descends to a section of eo | over the " formal neighborhood of the cusp " in ~go. We now consider the cusps of J/C'N, and for this we make the base change from R to R 0=R[~N]. Note that the map J/~-~-~A/" is dtale over the cusp oo. Let: The inertia groups in GL2(Fs) of the cusps in -///s consist in the conjugates of the group U (cf. [9] Cor. (2.5) of VII). From the definitior~ of the map e one sees that the inertia groups of those cusps lying above ooe,4/" consist in those N conjugates of U which do not lie in B. Let ~ be a cusp in Jgs, lying over 0% whose inertia group (for the Galois extension ~gN~Jt'l) is U. Since the group A normalizes U, it follows that, for all aeA, a. ~ also has U as inertia group. Viewing ? as a section of o~ | over jgo, the Fourier expansions ~'~.~ descend to a formal neighborhood of the cusp in dr176 and therefore q0"~.~ is invariant under the action of the inertia group of a. ~ (namely U). Thus, for any ueU, q0"--q~ has zero q-expansions at each of the cusps a. ~ for aeA. Since the group A operates transitively on the N--I distinct connected components of ,g~ we have that q0"--? has zero q-expansion at (at least) one cusp belonging to each of the N--I distinct connected components of the geometric fiber of ~g~. Therefore theorem (i .6. I) of [24] applies, giving that ~"-q~ = o. It follows that ~ is it, variant under both B and U. Since B 84 MODULAR CURVES AND THE EISENSTEIN IDEAL and U generate GLe(FI~), q~ descends to a modular form over Po(i), defined over R0=R[~I~ ]. Since its q-expansion has coefficients in R, [9], VII, th. (3-9) (ii) insures that 9 is defined over R. Proof of lemma (5. IO). -- Suppose that 9 is a nonzero holomorphic modular form satisfying the hypotheses of our lemma. Since 9 is the reduction modulo N of a modular form of weight 2 over I'0(N ) With integral q-expansion, we use [6i], th. i I (c), and regard 9 as the reduction modulo N of a modular form over SL2(Z), of weight N-}-I. In the terminology of [61], q~ is of filtration <N + I, as a modular form over SLy(Z). Since the filtration of 9 is congruent to N@I modulo N--I ([66], th. 2) and since it cannot be 2, the filtration of 9 is N+ i. On the other hand, our hypotheses may be interpreted as saying 09=0 , where 0 is the derivation q.~qq. Since N~5, we may apply lemma i (a) of [6I], which gives an absurd equality for the filtration of 09 = o. Conse- quently, there are no nonzero modular forms 9 satisfying the hypotheses of (5-m). Corollary (5.Ix). -- Let "~(q)=I+a~qN-?a2~a2N-r -. .. be a power series in qN, with integral coefficients, beginning with constant term I. Then: (i) ~' reduced modulo N is not a holomorphic modular form (for F0(N)) in B(FN) (w 4). (ii) If m is prime to N, and ~, reduced modulo m, is a holomorphic modular form (in ~| over P0(N)), then m divides I2, and 9-I modulom. Proof. -- (i) is a repetition of (5.IO), while (ii) follows from (5-9) and (5.6) and (4. IO). We now consider the status of tile power series 3(q)----~ e'(m)q ~ (see beginning of w 5), as modular form, when reduced modulo integers m. Proposition (5- x2): (i) The power series ~ is not the q-expansion of a holomorphic modular form of weight 2 over Po(N), modulo N (" holomorphic modular form" in B(Fs) (w 4)). (ii) Let m be prime to N. The power series 3(q) is the q-expansion of a holomorphic modular N--I form over Po(N) modulo m (in o~ | if and only if m divides -- (2). (iii) Let m be any integer. The power series 3(q) ia the q-expansion of a parabolic modular <i) form if and only if m divides n = numerator ~ . Pro@ -- Consider the formula: --e'=(N--i)+~4~ (1) See also KoIK~ [36] when m is a prime ~ 5. 85 86 B. MAZUR from which it follows that if 8 were a modular form modulo N (in B(FI~)) then the constant I would be the q-expansion of such a modular form as well. This is not true by (5.11) (i), whence (i). We shall now prove (ii). But first we need a fact about modular forms (in co | which is not totally obvious: Let ~ be a power series in q with integral coefficients. Let a, b be integers. Then ~ is a holomorphic modular form mod b if and only if a~ is a holomorphic modular form mod ab. To prove this, we invoke the q-expansion principle ([9], VII, (3.9) (ii); [24], (I .6.2)). We view a.Z/b as submodule of Z/ab and note that aq~ has all q-expansion coefficients lying in the above submodule. Now, suppose that 8 is a holomorphic modular form modulo m with (m, N)= I. From the formula quoted above, it follows that N--I is (the q-expansion of) a holo- morphic modular form, modulo 24 m. By (5. i i) (ii) and the fact proved above, if m' is any integer prime to N such that N--I is a holomorphic modular form modulo m', then m' divides I2(N--I). It follows that 24 m divides I2(N--I), or m divides (~). Conversely, e is a holomorphic modular form (in o~ | modulo I2. Therefore (N--i).e is a holomorphic modular form modulo I2(N--I). Moreover: (5.I3) --e'--- (N--I).e+24 8 modulo 24(N--I), from which it follows that 8 is a holomorphic modular form modulo (}-~-~/" This \'1/ proves (ii). /3%T .k As for (iii), it suffices to consider integers m which divide {~--~), by (ii). Consider (5.13) as an equation of meromorphic diffelential forms, and we shall compute the residues of each term appearing in it, at the sections oo and o. To do this, consider the involution w of X0(N ) induced by the rule: (E, H) ~ (E/H, E[N]/H) operating on sections of the moduli stack ,/fro and on modular forms (cf. terminology and discussion in proof of lemma (5.9) above; for a discussion of w cf. w 6) below. If q~ is a (holomorphic) modular form, defined over R, of level I, and if we denote by 9, again lifting to sff ~ defined by the rule ?(E, H)=q~(E), then the q-expansions of q~ and ~0.w are related by: w(q) = as follows from the discussion in the proof of lemma (5.9). Since w interchanges the cuspidal sections oo and o, we have the following Sublemma. ~ Let I/NeR. If ~ is a holomorphic modular form (in o~ | of level I, defined over R, regarded as meromorphie differential, and if the same letter ~ denotes its lifting to Mo(N)m as above, we have the formula for residues: Res0(,) = N. Res~o(~). 86 MODULAR CURVES AND THE EISENSTEIN IDEAL Thus: Reso0((N--I) .e) --N--I mod 24(N--I) (5. x4) (~). Reso((N--I ) .e) -N(N--I) mod 24(N--I ) Since e' is an eigenvector for w with eigenvalue --I, we have: (5" I5) Res~(--e')=N--i (1). Res0(-- e') = I --N Formula (5.15) would also follow from the fact that the only poles of e' occur at o and oo. Combining (5.i3), (5-14) and (5.15) we get: Res~o(~)=o (as it should) (5.I6) I_N ~ Res0(~ ) -= modulo N--I. Assertion (iii) then follows from (ii), (5. I6), and the following elementary fact: n = g.c.d. N~-I, 6. Hecke operators. I) The involution w (induced by (z~--I/Nz) on the upper half-plane). This is defined on M0(N)lzE1/N 1 by the rule (E, H) ~ (E/H, E[N]/H); it extends to an involution of Mo(N)/z (by [9], IV, (3. I9)), and of X0(N)/z. We denote this involution (as well as the involutions induced by it on the moduli schemes M0(N. N'), where N and N' are relatively prime) by w~, or by w, if no confusion can arise. In the terminology of [9], IV, (3- 16), w is induced by conjugation of Po(N) (o Io) It interchangesthecuspidalsections oo and o. by the matrix g= N " By " transport of structure " (i.e. functoriality of the sheaf of regular differentials) the involution w induces an involution on the space of regular differentials (on B~ and also on B(R). Care should be taken to distinguish this involution w (which is indeed the " classical " one) from the mapping on modular forms in o~ | defined by Deligne and Rapoport ([9], VII, (3.18)). Referring to their w by the bold letter w, one can show that for a modular form in co | 2 over Q, w? =N. wq0. Our mapping w does not necessarily " preserve " A(R). If fEI-I~ f~) has q-expansion f=y,%qm, the q-expansion of w.f is given by w.f=--Y,a~.mq m ([61], (2.I), and (3.3), th. II (a)). 2) T t for prime numbers t 4= N. (1) In these formulas e and e' are regarded as meromorphic differentials. 87 88 B. MAZUR These are correspondences determined by the diagram of morphisms: M0(N.?) (,) 2 / M0(N ) , M0(N ) Tt where c, on the moduli stack, is determined by the rule: (E, H~, He) ~ (E, H~). Here H N C E is a subgroup scheme of order N, H t C E of order r Compare [9], VI, (6. i i). The morphisms c, cw t are finite (loc. cir.) (1). If x=j(E/E , HN) is a point on the curve X0(N ) with values in a field K, then Tex is the divisor: (6. x) Y4(E/H, (H~ + H)/H) where the summation is taken over all cyclic subgroups I-I of order t of E, defined over ~x. Define morphisms : (a) c* : Ha(M0(N)/z, dTMo(S)) -+ HI(Mo(N. ~)/z, d~M.(Nt)) (b) c* : H~ f2) -+ H~ f2) as follows: (a) is induced from the natural map: OM~ ) -+ C, OM,(N ' t)" As for (b), let U denote the open subscheme of Mo(N.l ) which is the complement of the supersingular points of characteristics N and I (the smooth locus of M0(N.I) -+ Spec (Z)) and let V be the image of U under c. The restriction of f~ to U (resp. to V) is f~r/~ (resp. f2~/s). One has the natural map: which induces a morphism c* : H~ f2) ---> H0(U, f2). But since f~ is an invertible sheaf on M0(N.d ) and the complement ofUin M0(N.t ) consists in afinite set of points of codimension two, whose local rings are Cohen-Macaulay, we have: H0(U, f2)----HO(M0(N, e), f2) whence the mapping in (b) above. Applying the Grothendieck duality isomorphism v ((3.3) + (3.2)) to (a) and (b), we obtain morphisms: (a ~) c. ~ (c*)v : HO(M0(N. t)/z ' f2) -, H~ ~2) (b v) c.=(c')" : HI(M0(N./)/z, d~~ Ha(M0(N)tz, d~ro(S)). (t) They are not necessarily fiat. To determine the (finite) set of points at which they are nonfiat is an easy exercise, using [62], IV, Prop. 23; [9], V, (6.9). 88 MODULAR CURVES AND THE EISENSTEIN IDEAL We now define the endomorphism T t on H~ ~) and on: HI(Mo(N)/z , r by the formula: T t ---- c.. (cwt)* = (cwt) , . c*. From the definition one sees that the action of T t on HI(M0(N), 0~0(1~) ) and on H~ ~) are adjoint with respect to Grothendieck duality. The correspondence T t also induces endomorphisms of: (i) The Hodge filtration on 1-dimensional de Rham cohomology: o -~ H~ .e~, ~1) ~ H~R(Xo(N)/z[lm .el) -+ Hl(Xo(N)/mm. tl, eXo(N)) ~ o. (This action is hermitian - (Ttx, y)=(x, Try ) - with respect to the cup-product self- duality on H~m and it exhibits the adjointness of the action of T t on the two flanking members of the above exact sequence) (1). (ii) The jacobian of X0(N)/Q; its Ndron model J/z; the " connected component " of the Ndron model JTz; the singular cohomology groups of X0(N)/c with coefficients in Z; the p-divisible (Barsotti-Tate) groups Jp/z[lm]" The endomorphisms T t are hermitian with respect to the cup-product self-duality of i-dimensional singular cohomology of X0(N)/e and the auto-duality of the Barsotti- Tate groups Jp/zE1/N~- The effect of T t on the q-expansions of elements in H~ f2) may be computed over the base Q. (or c) and one finds (applying (6. i)) the classical formula: If the q-expansion of f is given by a~=Za,,q " then: w~ (6.2) Ttf=~bmq , where bm=g.a,~/t+at. ~ (with the convention that am/t = o unless e I m). Consider the action of T e on the Ndron model J/z and restrict to characteristic t. The Eichler-Shimura relation on the level of correspondence, (whose proof in [7] works mutatis mutandis for F0(N)) gives the fmmula: Eichler-Shimura : T t =Frob t +t/Frob t on J/Ft (14=N). Here Frob t is the Frobenius endomorphism of the group scheme J/Ft , and g/Frob t may be regarded as the canonical " Verschiebung " of the group scheme J/Ft. It follows that Frob t satisfies the quadratic Eichler-Shimura equation: X 2-T t . X + g : o in the endomorphism ring of J/vt. (1) Duality for de Rham cohomology is compatible with (indeed: constructed by means of) duality for coherent sheaves (of. [I8]). 12 9 ~ B. MAZUR Definition. -- By the Hecke Algebra T we shall mean the subring of End(J/Q) generated by the Hecke operators T t (g+N) and by w. The algebra T operates, by definition, on J/Q. It also operates (via the previously defined actions of T t and w) on the following list of objects: -J,z; J~z; Pic~ (xo (N)/z) ~ J~z; HI(Xo(N)/z, (9Xo(Ni)=Tan. space Pic~ ; H~ ~2) (which is the dual of the above) ; H~m(Xo(N)IQ) (which is the Lie algebra of the universal extension of J/. [3 7]) ; _Hsing(Xo(N)/c, Z). Clearly, T is a free Z-module of finite rank. It is known that T| is a commutative Q-algebra of rank g = genus(Xo(N)), and that it is isomorphic to a product of totally real algebraic number fields: TQQ,= 1-[k~ (6.3) (1). ~=I~ . . .~ t (6.4) Say that a T-module IV[ (of finite type) is of rank r (as opposed to free or locaUy free of rank r) if, equivalently: (a) M@O is free over TNQ of rank r. (a') For some, or any, field K of characteristic o, M| is free of rank r over T| (b) M| is a vector space of dimension r, over k~, for a=I, . .., t. (c) M contains a free T-module of rank r, of finite index. Note that ifM is a T-module of rank r, then the Z-dual T-module MY=Horn(M, Z) is again a T-module of rank r (2). Since H~ f21) is known to be a free TNE module of rank I (as follows from lemma 27 of [2]), one has: (6.5) H~ a) and Hl(Xo(N)/z, Ox.Im) are T-modules of rank I. H]i~g(Xo(N)/c, Z) is a T-module of rank 2. (x) This follows from lemmas x3, 27 of [2]. (2) It is not at all evident, however, that the operation v preserves the category of locally free T-modules. This latter assertion is equivalent to saying that T is a Gorenstein ring (but see w167 x5-I 7 below). 90 MODULAR CURVES AND THE EISENSTEIN IDEAL 7- Quotients and completions of the Hecke algebra. Let m be an integer. Let J[m]/z denote the scheme-theoretic kernel of multipli- cation by m in the N6ron model J/z. Since J is semi-stable (cf. appendix) J[m]/z is a quasi-finite fiat group scheme, whose restriction to S'= Spec Z[I/N] is finite and flat. Let a C T be an ideal containing m. By J [a]/Q we shall mean the kernel of the ideal a in the jacobian J/Q. That is: J[a]/Q = f'] (kernel of ~ in J/Q) = ['1 (kernel of 0~ in J[m]/Q). From the second description it is clear that J[a]/Q is a finite subgroup scheme of J[m]/Q. Now define J[a]/z to be the Zariski-closure of J[a]/Q in J/z. It is the subgroup scheme extension of J[a]/Q in J[m]/z, as in chapter I, w i; J[a]/z is a quasi- finite fiat group, which is, by constIuction, a closed subgroup scheme of J/z, and killed by a. The quotient T/a operates naturally on J [a]/z. Caution. -- The group scheme J[a]/z is not necessarily the full scheme-theoretic kernel of a in J/z. This kernel is not necessarily flat over Z. Fix a prime p. Let a C T be any ideal containing p. Let T a = lim T/a m denote the completion of T at a. Denote by Tp the completion of T at the ideal generated by p. Thus Tp =T| Since T is a finite Z-module, T, is a direct factor of the semi- local ring Tp. Write: (7.') (a) Tp=T~� (b) i = ~eL -t- ~'a where T~ is our notation for the factor complementary to T~, and (7. I) (b) is the associated idempotent decomposition of I in T,. Form the inductive limits of the quasi-finite group schemes: (7.2) J,,,. = 1~ J [pm]/,. J~ = lim J [d"]/z. Thus, Jp/z is an ind-quasi-finite group scheme, whose restriction to: S'= Spec(Z [I/N]) is a p-divisible (Barsotti-Tate)group admitting a natural continuous action of T~,. We may use the idempotent decomposition (7-i) to write J, as a direct factor of Jp: (7.3) Jp=J,� Restricting to the base S', (7.3) becomes a product decomposition of Barsotti- Tate groups. Moreover, since the action of T is hermitian with respect to the auto- duality of Jp/s,, one obtains an induced auto-duality on J,/s" 91 92 B. MAZUR To pass to pro-p-groups, one uses the Tare construction. We recall this in the category of modules. The functor M~ M| is an equivalence between the categories of free Zp-modules of rank r, and p-divisible torsion Zp-modules of corank r. The Tate construc- tion W ~Hom(Q~/Zp, W)= g~(W) provides an essential inverse to the above functor. There is a perfect Zp-pairing between g'a(W) and the Pontrjagin p-dual of W, W* = Hom(W, Q,/Z,). The isomorphism W*--~g~a(W)V=I-Iom(g~(W),Zp) takes q0eW to: ~-a(q~) : g'a(W) -+ g'a(Q,/Z,)~Z,. Let X0(N)e denote the analytic curve associated to X0(N)/e, andJc the complex Lie group associated to J/e. We may identify the singular homology group HI(X0(N)e , Z) with the kernel of the homomorphism of the universal covering group of Je to Je. By means of this identification, we obtain an isomorphism: (7.4) J,(C) = HI(X0(N ) e, Z) | Q,/Z, = I{l(X0(N ) e, Q,/Z,) where the left-hand group is the group of C-valued points of Jp. Applying the Tate construction: (7" 5) ga(Jp) (C) = gh(Jp(C)) = HI(X0(N)c , Zp) and this isomorphism is compatible with the action of Tp. Applying the idempotent % to (7-5) gives: (7.6) g'a(Jo) (C) = Nh(Jo(C)) = HI(Xo(N)e , Zv) | = HI(X0(N)c , Z) NTT o . The last equality, together with (7-5) gives: Lemma (7-7)- -- Let K be an algebraically close~l fiel~l of characteristic o. Then ~(J~ is of rank 2 over T o. (That is: g~(Jo(K))| is free of rank 2 over To| 8. Modules of rank I~ If M is a To-module of rank 2 (6.4) and there is an exact sequence of To-modules (up to torsion): o-+MI~M-+M2~o where M 1 and M s are Zp-dual (up to torsion), then they are each of rank I. We use this elementary assertion three times in this section. i. The defect sequence. Suppose p+N. Then Jp is an ind-6tale (quasi-finite) group scheme over the base Spec(Z). Consider the natural imbedding: 92 MODULAR CURVES AND THE EISENSTEIN IDEAL which induces an imbedding on Tate constructions. Form the exact sequence: (8. x) o -+ ~'a(Jv(F~)) -+ ~a(Jv(QN)) -+ -+o where A is the cokernel (the module of defect). The sequence (8. I) is compatible with the action of Tp. By the " th~or~me d'orthogonalit~ " (th. (2.4) of exp. IX, SGA 7), the toric part of ~'a(Jv(Q~)) is orthogonal to itself. On the other hand, the fiber J/~, is isomorphic to Ggm where g is the genus of X0(N ) (cf. appendix). It follows by computing ranks over Zp that the self-duality of $'a(Jp(Ox)) induces, up to torsion, a Z:duality between $'a(Jp(F~)) and A. Applying % (7. I) (b) to (8.I) yields an exact sequence: (8.2) o ~ ~'a(J,(fs) ) -+ ~'a(J,(Q~)) -+~, -~o where A~=A| and where Aa is dual to ~(J,(Fz~)), up to torsion. Applying lemma (7.7), we have: Proposition (8.3). -- ~a(J,(f~)) and A a are T,-modules of rank i. 2. Etale and Multiplicative type parts, in the ordinary case. Now suppose that p ~ N, and Ja is an ordinary Barsotti-Tate group. This means that over Spec(Zp) it admits a filtration: (8.4) o ~j~,lt. type ~j~ _+ o where j~ta~e is an 6tale Barsotti-Tate group (the gtale part of J,) and j~lt.wp0 is the connected component of J,, and is a group of multiplicative type (the dual of an 6tale Barsotti-Tate group). The self-duality of Ja induces a duality between j~ult.type and j~a~e. Applying ~'a to (8.4) , and using lemma (7.7) one obtains: Proposition (8.5).- $'a(j~u~t'tYPe(Q,)) and $'a(J~tate(Q,))are T,-modules of rank x. 3. Eigenspaces for complex conjugation. Complex conjugation e on the topological space X0(N)c commutes with cup- product and induces multiplication by --I on H ~. Consequently the cup-product pairing induces (up to torsion) a duality between the +i-eigenspace of e operating on Ha(X0(N)c, Z) and the -- I-eigenspace. Using (6.5) it follows that these eigenspaces are T-modules of rank i. 9. Multiplicity one. Let R be any commutative ring. Consider operators Tt:R[[q] ] --~ R[[q]] (g=t=N) and U : R[[q]] --~ R[[q]] defined purely formally by the appropriate equations: If f= Y, amq m, then : (9.I) Ttf=Y~atmq'~+t. Y~a,,q t'm (t+N), and Uf=~,as.,,q". 93 B. MAZUR Let ~ be any set of prime numbers, and .LP' the set of all positive integers which are not divisible by any member of ~ (so I is always in .s Let : f= alq-k a2q ~ +... eR[[q]] be a power series with no constant term, which is an eigenvector for T t (all le.L#, t:#N) with eigenvalue ct~R , and, if Ns& ~ an eigenvector for U, with eigenvalue cseR. The recursive relations: at. m =ct.am-~-g.am/t t~.L#, f ~ N aN.m = Cr~ . a m if Ne.~ show immediately thatfis determined by the eigenvalues c t for ~.5r and its coefficients a m for m~6~ ~ (1). In particular, given ct~R for all prime numbers ~, there is a unique power series f=~.q+a2q2+.., in R[[q]] such that Tt.f=ct. f for all t+N, and U.f=c~.f Moreover, any eigenvector in R[[q]] possessing the same eigenvalues for all these operators must be a scalar multiple of f. Catl f the generating eigenvector (for the eigenvalues { ct}. ) Proposition (9.2). -- Let R and B~ be as in w 4. Let elements cteR be given, for each prime number L If ~eB~ # a parabolic modular form such that: Tt.~=ct. ~ e+N (*) u. = cN. then the q-expansion of ~3 is a scalar multiple of the generating eigenvector f. The R-submodule Of B~ consisting in all elements which satisfy (.) is a submodule of a free R-module of rank ~. Now let 93~ C T be a maximal ideal, with k~ as residue field, of characteristic p. Let B~ denote the kernel of the ideal 93l. This may be viewed, in a natural way, as a k~-vector space. Proposition (9.3). -- B~ 93l] is of dimension I over k~. Proof. -- Let R = k~, or any field of characteristic p, which is large enough. Let M denote the k~j~-vector space B~ Clearly M+o, since T operates faithfully on B~ Since: M| C B~ (a) A (perhaps too) succinct way of expressing this determination is by the use of formal Diriehlet series with coefficients in R. Once one defines the evident rules of manipulation of these formal Dirichlet series, one has: Zam.m -s= ( Z am,m-S). [I Dg m m~.r ' g~.~ where: D t = (I --c t. t -s -t- tt-zs) -1 if t ak N, and Dbl =(I --c~.N-S) -1. 96 MODULAR CURVES AND THE EISENSTEIN IDEAL the proposition will follow, if we show that B~ is an R-vector space of dimension less than or equal to [k~ : Fp]. The action of T on B~ induces an action of k~n on B~ [gJ~] which commutes with the action of R. Since R contains k~, B~ possesses an R-basis of k~-eigenvectors. To each eigenvector in this basis, we may associate a homomorphism k~n-+R (by passing to eigenvalues). By the previous proposition, no two eigenvectors in this basis are associated to the same homomorphism. The proposition follows. Proposition (9.4)- -- t-Ia(Xo(N)/z, 0) is a locally free T-module, of rank I (1). Pro@ -- Note that if M is a T-module of rank I, it is locaUyfree of rank I provided M/gJt. M is a k~-vector space of dimension i, for all maximal primes 9XCT. Letting M=HI(X0(N)/z, 0), it is of rank i over T, by (6.5). Also: M = I-Ii(Xo(N)/,p, I-II(Xo(N) ,, (9) and the right-hand side of the above equality is isomorphic to the (FFvector space) dual of I-I~ a)[gJ~] = B~ which is of dimension I by (9.3)- Proposition (9.5). -- The Hecke algebra T is the full ring of endomorphisms 0fJ/c. Remark. -- This is a mild sharpening of a result of Ribet: that: T| End(J/,)|174 End(J/c )| [58] which is, in fact, used in the proof below. Pro@ -- Let T'=End(J/c ). By Ribet's result, any element of T' is defined over Q., and therefore acts on the Ndron model of J/Q; hence on the connected component JTz which is Pic~ hence on the tangent space to Pic~ which is IIa(Xo(N)/z, (9). It also follows by Ribet's result that T' is a subring of T| and hence is a commutative ring, and its action commutes with the action of the I-Iecke algebra T. We get, then, a homomorphism: T'-+ EndT(I-II(X0(N)/z, (9))=T" which is injective, since T| acts faithfully on Ha(X0(N)/o, (9). Since HI(X0(N)/z, (9) is a locally free T-module of rank I (9.4), T":T. The proposition is established. Definition. -- The Eisenstein ideal Z C T is the ideal generated by the elements: I +g--T t (all t4:N) and by I +W. If R is any ring, any element in B ~ (R) [~], the kernel of ~ in B ~ (R), is an eigenvector for the Tt's and for U, satisfying equation (,) above, where: c t =i+t if g4:N Cl~ ----- I. (1) It follows that B~ = H~ f~) is the Z-dual of a locally free T-module of rank x. The assertion that B~ is locally free over T is therefore equivalent to the assertion that T is a Gorenstein ring (see w 15 below). 95 96 B. MAZUR In R[[q]], the generating eigenvector for the above package of eigenvalues c t is the power series 8 of (5-I). Consequently, the q-expansion of any element of the R-module B~ must be a scalar multiple of 8. Proposition (9.6).- Let mbe any integer divisible by n=num(~-I]. Then ~ ,z z B~ [3] is a cyclic group of order n, generated by (m/n). 8. Proof. -- This follows from the above discussion and (5. I2). Proposition (9.7). -- T/~=Z/n; the Eisenstein ideal 3 contains the integer n (1). Proof. -- We have a natural map Z-+T/3 which is surjective, since, modulo 3, the operators T t (r 4: N) and w are all congruent to integers. We cannot have T/3 = Z, for then 8 would be the q-expansion of a modular form (of weight 2 for P0(N)) over C, which it is not. Therefore, T/3 = Z/m for some integer m, which must be divisible by n, since 8eB~ is of order n, and is annihilated by 3. We prepare to use the previous proposition. Since: B~ = H~ n) is the Z/m-dual of I-P(Xo(N)/(z/m), 0) (3-e) we have that B~ is the Z/m-dual of: Hl(Xo(N)i(z/m), 0)/3. I-~l(Xo(N)/(z/m), 0) =Ha(Xo(N)/z, 0)/3. HI(Xo(N)/z, 0) where, we have the equality above since me3. By the previous proposition, then, the cokernel of 3.HI(X0(N)/z, 0) in HI(X0(N)/z, (9) is cyclic of order n. Since (9.4) HI(Xo(N)/z, 0) is a locally free T-module of rank I, it follows that T/3 is cyclic of order n. Q.E.D. Definition. -- A prime ideal ~ C T in the support of the Eisenstein ideal is called an Eisenstein prime. The Eisenstein primes ~3 are in one-one correspondence with the prime numbers p which divide n by (9-7)- Ifp is such a prime number, then the Eisenstein prime corre- sponding top (which is the unique Eisenstein prime whose residue field is ofcharacteristicp) is given by: Clearly: T/~ =F,. One checks easily that n> i if and only if the genus of Xo(N ) is greater than o. Thus: Proposition (9.8). -- If the genus of X0(N ) is greater than o, the Eisenstein ideal ~ is a proper ideal in T; there are Eisenstein primes. (1) This vague result is sufficient for our purposes. It appears to be significantly more difficult to give an expression for n in terms of the operators Tt, in T. This would be particularly useful in questions related to w 19 below. 96 MODULAR CURVES AND THE EISENSTEIN IDEAL xo. The spectrum of T and quotients of J. As follows from the result of Ribet [58], there are one-to-one correspondences: isogeny classes of 1 l isogeny classes of 1 C-simple I Q-simple I < .'-. abelian variety 1 abelian variety t factors of J/c l factors of J/Q l IO. I ) t fields k s occurring irreducible I in the product de- ~ components composition (6.3) of Spec T of T| Define j+=(i+w).jcj; j_=(l-W).jcj. These are sub-abelian varieties, defined over Q. Form the quotients indicated in the diagram below: J+ (Io.2) o -+j_ -~J -+J+ -+ o J_ Thus J+, J- are quotients of J on which w acts as + i, and --I respectively. We let J~z denote the N6ron model of J~ over the base Z. By the criterion of N~ron- Ogg-Shafarevitch, J/z[lmj  is an abelian scheme, as are J+/z[1/~]. The abelian variety J+/Q can be identified with the jacobian of the quotient curve X+=X0(N)/w. One sees this as follows: since the map X0(N)-+X + is ramified (w has fixed points), the induced map on Pic ~ is injective and identifies the jacobian of X + with the connected component of the identity in the +-eigenspace of w in J. But the diagram (lO.2) identifies (I+W).J=J+ with this same connected component. To any ideal a C T we may associate an abelian variety Jl~ ) which is a quotient of J/Q, whose C-simple factors are in one to one correspondence under (io. i) with those irreducible components of Spec T which meet the support of the ideal a. To define J ("), let yaCT be the kernel of T-+T,=limT/am; let -~.JCJ be the sub- abelian variety (defined over Q) generated by the images ~.J for eeu Take Jl~ ~ to be the quotient abelian variety: (x o.s) o -+ v..J -"J _+jc.) _+ o. 13 98 B. MAZUR Let J}~) denote the N6ron model ofJ}~ ) over the base Z. By the criterion of N6ron- Ogg-Shafarevitch one has that l(a/ is an abelian scheme. O/Z[I/N] Definitions (xo.4): I) If a =3, the Eisenstein ideal, call J(") the Eisenstein quotient of j, and denote it J. 2) If a --~3, the Eisenstein ideal at p, call J (a) the p-Eisenstein quotient and denote it j(~l. Note that, for any p, the p-Eisenstein quotient is a quotient of the Eisenstein quotient. Conversely, any C-simple factor of J is a factor of J(p) for some prime p dividing n. It is also true (but not at all evident when n is even; cf. (x7.Io) below) that j is a quotient of J-. Definitions (xo 5): g = dim(J/0 ) g+ = dim(J~) = dim(J ~'= dim(J/0 ) g/,J= dlm(j/~t). So g=g++g-, and g+=genus(X+). The tturwitz formula computed for the map X0(N)-+X + yields the well known relation: 2(g---g+)=h--2, where h is the number of fixed points of w. Proposition (xo. 6). -- The scheme Spec T is connected. Pro@ -- Suppose not. It would follow that J/c could be expressed as a nontrivial direct product Jjc = A � B. Let us show that the principal polarization X : J ~J (^ denotes the dual abelian variety and X is the 0-polarization ( [43], chapter 6; [44])) induces principal polarizations XA : A-+A and Z B : B-+B. By Ribet's theorem [58], since J decomposes t~J (up to isogeny) into a product of simple factors, each occurring with multiplicity one, the simple factors of A are non-isogenous to simple factors of B, and consequently there are no nontrivial homomorphisms from A to B and from B to ,~. Our assertion follows. But a jacobian (taken with its natural principal polarization) cannot decompose as a nontrivial direct product of principally polarized abelian varieties. This follows from the irreducibility of its 0-divisor. Remark. -- When g+>o, the above proposition insures the existence of " primes of fusion " (see introduction) relating J+ to J-. It would be interesting to understand these primes. Ix. The cuspidal and Shimura subgroups. Let c be the linear equivalence class of the divisor (o)--(oo) in J(Q). Proposition (xx.I). -- The element ceJ(Q) is annihilated by the Eisenstein ideal 3. It is of order n. 98 MODULAR CURVES AND THE EISENSTEIN IDEAL Proof.- Since the correspondence T t (g#N) takes the cusp (o) to (I+g).(o) and (oo) to (I+r one has: Tt.c=(I+Q.c for all t+N. Since w interchanges the cusps o and 0% one has: (I -4- W) .6 = O. It follows that ~.c==o. From proposition we conclude that the order of c divides n. But since (Appendix A. i) the specialization of c to the Ndron fibre Je~ generates the cyclic group of connected components, which is of order n, it follows that the order of c must also be divisible by n. Q.E.D. Remark. -- The fact that order(c)=n was proved originally by Ogg [36]. He shows that the order of c divides n by exhibiting a function f on X0(N ) whose divisor is n. (o)--n. (~). Namely, if v is the g.c.d, of N--I and 12: ( A(z)~l/~=q. fi (I--qmN)-2'/".(I--qm) 2a/~ (xi.2) f(z)=\A(Nz)] ,,=1 can be shown to be invariant under F0(N), and clearly has the indicated divisor. Let C denote the subgroup of J(O) generated by c. Thus, C is a cyclic group of order n, with a distinguished generator. Denote by C/z the finite flat subgroup scheme of J/z generated by CcJ(Q). Let C=the IN-valued points of C/z ("the specialization " of C to J/F~). By the appendix, one has that C is, again, of order n (the specialization map C-+C is an isomorphism) and: (x I. 3) J/F~ =J/~ � C where J~F~ is the connected component of the identity. The retraction 0fJ(Q) to C. -- If xeJ(Ov) , denote the section over Spec Z induced by x in J/z by the same letter. Let x/F ~ denote the restriction of this section to an F~-valued section of J/F~. Let ~-be the image of x/F~, under projection, to C, using the product decomposition (i I. 3). Let p(x) ~C denote the unique element of C which maps to s under the " special- ization map" described above. If M=J(Q) (the Mordell-Weil group of J), we have just described a retraction p :M--+CC M, giving a product decomposition. (xx.4) M=M~215 where p is projection to the second factor; projection to M~ ker 0 is given by x ~ x-- 9(x). The Shimura subgroup. -- The Shimura covering (2.3): (x x. 5 ) X 2 (N)/s, -+ X0 (N)/s, is the maximal dtale extension intermediate to XI(N ) ~ X0(N ) and is a finite, 6tale, Galois extension, whose covering group U is the (unique) quotient group of (Z/N)* 99 ioo B. MAZUR which is (cyclic) of order n. Applying Pic ~ to the morphism (11.5) , we obtain a morphism J/s' ~ Pic~ X2(N)/s' whose group scheme kernel we denote Z/s,. Definition. -- The Shimura subgroup Y~/s C J/s is the group scheme extension (i.e. Zariski closure) of Z/s, in J/s- Let LrTs =.;g~rns(U, WI~) be the Cartier dual of U (where U is viewed as constant group scheme over S). Proposition (xx.6). -- There is a natural isomorphism UTs~X/s. The Shimura subgroup is a ~-type group (chapter I, w 3) over S; in part#ular it is finite and flat. Proof. -- We establish this first over the base S'. Consider the Hochschild-Serre Spectral sequence (for the 6tale topology ([i], III (4-7))) associated to the (finite 6tale Galois) Shimura covering X2(N)/T-+ X0(N)/T and the sheaf G,, where we have made the base change to an (arbitrary) S'-scheme T. We obtain the exact sequence: o ~ H~(U, GIn(T)) ~ H~(X0(N)/r, Gin) ~ H~(X2(N)/T, G~). . i Passing to associated sheaves, the morphism i induces an isomorphism, U/s,-~ Z/s,. Since U* is a finite dtale group scheme over the base S', this isomorphism extends to a . i homomorphism U~s-~ Z/s (by the universal property of the Ndron model). It follows that X/s is a finite fiat group scheme. Restricting to the base S', one has that the morphism i is a homomorphism of locally constant groups, which is an isomorphism on generic fibers. Hence i is an isomorphism over S'; hence i is an isomorphism over S. Proposition (II.7). -- The Shimura subgroup Z is annihilated by the Eisenstein ideal 3. Proof. -- We must show that w acts as --i on Z, and T t acts as I + t for g 4: N. As for the action of w, note that induces an involution w' on XI(N ) which (; projects to the involution w on X0(N ). If ~Po(N), one computes conjugation by w', and obtains: w' c~w'-0~ -1 mod FI(N), which yields what we wish. The operators Tr " act " as well on XI(N), by the formula: Tt : (z) ~ (~.z)+ =0\-7-1. In the above formula, as in the rest of this proof, we view the modular curves Xi(N) (i=I, 2, o) as analytic manifolds, parametrized by the extended upper half-plane. If oh, ~ ale points in the extended uper half-plane, let {~, ~} denote the (relative) homotopy class of paths in the extended upper half-plane beginning at o~ and ending at 9. Recall Ogg's convenient terminology for the cusps of F(N): Let: (~)={p/q~P~(Q) lp-a mod N, q--b mod N; (p, q)=i }. 100 MODULAR CURVES AND THE EISENSTEIN IDEAL 1OI With this notation, (2) is an equivalence class of pl(Q) mod F(N). Therefore it gives rise to a well-defined cusp of Xi(N) (i = i, 2, o). One shows (~) - (0) mod 1-'I(N), provided (b, N) = i. If r~{e, ~} is a path in the extended upper half-plane, let y(r~)~U be the (unique) element of U which maps the image of 0~ in X2(N) to the image of ~ in X2(N). Let ~b be a path in {(0), (0)}, for b an integer relatively prime to N. Then one checks that y(%) is the image of b -~ in U, while ~((Tt.%) is the (I+g)-th power of this image, as follows from the formula: Tt{(0), (0)}={(0), (0)}_}_ y~ {(j), (t!b)} j~0 The proposition follows. The Shimura subgroup over the base F N. -- Note that Z(FN)=Hom(U , ~,(FN)), and that there is a natural generator of this group. Namely: (Z/N)*= , where the unlabeled horizontal map is raising to the ,~-th power (v~(N--I, I2)). The natural projection J(FN) =j0(F~) � C-+ C induces a homomorphism: .8) :c which sends the canonical generator s to some multiple ~ of the canonical generator ~C. Thus ~ is a well-defined integer modulo n. Question. -- What is ~ ? Proposition (II.9). -- The homomorphism (11.8) is an isomorphism. The scheme- theoretic intersection X/r N c~J~ is the trivial group scheme over F~. The integer (modulo n) is relatively prime to n (1). Proof. -- The three assertions of the proposition are equivalent. We prove them by showing that: (xi. xo) Pic~ ~ Pic~ is injective. For this, we may identify Pic~ as group-scheme over Fs with the Gin-dual of the singular one-dimensional homology group of the topological graph (Appendix, w 3) associated to X0(N)~ . (homology with Z coefficients). (1) In the light of this, it is hard to imagine that ~ is anything other than ___ I. We have not, however, succeeded in answering our question. 101 io2 B. MAZUR By inspecting (diagram i of chap. II, w i) it is clear that this is the same as the G,;dual of Hl(Graph(M0(N)/~),Z ). To prove injectivity of (11.Io) it suffices to show that the map: Graph MI(N)/~ ~ Graph M0(N)/~ induces a surjection on one-dimensional homology. But the above map of graphs is an isomorphism as follows from [9], V, th. (2. I2) and VI, Cor. (6. Io). The relation between C and E. -- By the cuspidal subgroup ClzCJ/z we mean the Zariski closure of CcJ(Q) in the group scheme J/z. By the universal property of N~ron models, the isomorphism Z/n-+C (of group schemes over O; I ~c) extends to a homomorphism Z/n/s~C/s , and shows that C/s is a finite flat group. Proposition (xx. IX ). -- If n is odd, the group scheme C is a constant (gtale) group over S; the scheme-theoretic intersection Of C and Z over S is the trivial group; the natural map C| is an injection. If n is even, the group scheme C/s contains a subgroup scheme isomorphic to It2 (and which we shall call ~z2). The cokernel of ~2 in C is a constant (gtale) group. The scheme-theoretic intersection of C and Z in J!s is ~z 2. The natural map C | z--+J [3] has " the diagonal " It2 as kernel, and induces an isomorphism of (C| with the finite flat subgroup of order n2/2 in J[~] generated by C and E (call it C § Z). Proof. -- (a) We show first that the odd part of Z has trivial intersection with (the odd part of) C. For by consideration of Galois modules, the odd part of Z is a It-type group and the odd part of C is a constant group. (b) If n is even, the group E(O) (the rational points of Z) is of order 2. Lemma. -- Z(Q) C C. Proof. -- Suppose that n is even, or, equivalently, N--i mod 8. Then there is an 6tale double covering X0~(N)-+X0(N ) intermediate to the Shimura covering (2.3). This we shall call the Nebentypus (double) covering. Applying the functor Pic ~ to the Nebentypus covering (over O), we obtain a morphism of jacobians J/Q-+Jac(X0~(N)/Q) whose kernel is the group Z(Q). To prove the lemma, it suffices to show that the image, c ~, of c in Jac(X0~(N)/Q) is of order n/2. For this, it suffices to show that iff is the function (II.2) whose divisor is n. (o)--n. (oe), then ftl2 is a rational function on the Nebentypus curve X0~(N): ~o f112 =q,12 1] (I __q,,,)-121~. (I--qm) 12Iv m=l as follows from Dedekind's transformation formulas for the ~-function. (CA'. discussion of this in [48], w 3-) The Zariski closure of E(Q) in J/s (which is its Zariski closure in Z/s ) is a it-type group of order 2. Thus it is canonically isomorphic to It2ts and we shall denote it ~z21 s. 102 MODULAR CURVES AND THE EISENSTEIN IDEAL Io3 By the lemma, ix 2 C C/s. Since C/s is a finite flat group scheme, whose associated Galois module is a cyclic group with trivial Galois action, by (chap. I (4.6)) we have that the cokernel of ~2 in C is a constant (dtale) group scheme over S. It follows by an easy argument that Z n Cls is the finite flat group scheme ~x 2. There is a canonical auto-duality: II, I2) J[n] ~ ~om(J[n], ~,) (over Q) and the section ceC c J [n] (Q) determines, by (I I. 12), a homomorphism: ca: J[n]-~ r (over Q.). Restricting c ~ to Z, we obtain a homomorphism: ca : U*-+~n which, in turn, may be identified with an element uEU (1). Question. -- What is this element u ? This element has been evaluated in no case where n>I. One can show that ifp is an odd prime dividing n, then u projects to a generator of the p-primary component of U/f and only /f T~ (the completion of T at the Eisenstein prime ~3 associated to p) is isomorphic to Zp (cf. (I 9. 2) below) (2). In the light of the table of the introduction, it then follows that u does project to a generator ofthep-primary component of U(p % 2, p [n) for all N<25o except when N=3I , Io3, I27, I3I, I8I, I99 and 2II. x2. The subgroup D cJ[~] (p= 2 ; n even). Suppose n-omod4 (equivalently: N=imodi6). Choose yeZ(Q(~J--~)) an element of order 4. Let x=(n/4 ).c, which is an element of order 4 in C. Thus x, y are elements of C+E rational over Q(~/'----i-). By (Ii. Ii), 2x=2y. Let D C (C+E)/s be the closed subgroup scheme generated by the points x--y, and 2y. In (C+E)(Q(~r these two points are a basis of an F2-vector space (of dimension two) which is stable under the action of Gal(Q(~v/~---~ )/Q). Ifz is the nontrivial element of Gal(Q(~v/~i-)/Q), then the matrix of'~ computed with respect to the basis x--y, is (o :) Since O(x/'~---~)/O ~ is unramified at N, the group scheme D is finite and flat over S (chap. I (I.3)), and it follows from the above discussion and (chap. I (4.4)) that D is isomorphic to the unique nontrivial extension of Z/2/s by t~2/s killed by 2. (1) Since there are two natural choices of sign of the above autoduality (or equivalently, of the en-pairing), the pair of elements u  has, perhaps, greater significance than the element u. (2) Which explains why we might be interested in some reasonable direct method of computation of u. 103 IO 4 B. MAZUR The purpose of this section is to consider the case where n- 2 mod 4 (equivalently: N- 9 mod 16) and to construct a subgroup scheme of J[~], which is isomorphic to D. In this case, the 2-primary components of C and of ~ coincide with ~z 2. The group scheme D, which we construct, will contain ~2, but (necessarily) will not be contained in C+E. The construction of D. -- Ifn is even, the Nebentypus covering (w I I) X0~(N) --+ X0(N ) is 6tale (over S') with Galois group U/U 2. Let v be the nontrivial element in U/U ~, and J-~J~ the induced morphism on jacobians. Using the Leray Spectral sequence (over the base Q.) for Gm-cohomology of the Nebentypus covering, one has: (I2.I) O ~ Vq(Q) ~J(Q) --~ (J~(Q))~ -+ O where the superscript v means the part fixed under the involution v. To describe the Galois module associated to D, we shall construct a point of order 2 in J~(O..), and D(Q) will be, by definition, the subgroup of J(O..) generated by the inverse image of this point. There are four cusps on X0~(N ). Let o, g denote the cusps lying over o in X0(N), and o% ~ those lying over oe. Thus v interchanges o and g (and oo and ~). The cusps o and ~ are rational over Q., while oo and ~ are conjugate over O and defined over Q(~c/N). Compare [48], w I. Proposition (x2.2) (Ogg, Ligozat). -- Let Z be the Legendre symbol of conductor N, a)). z(a)-----(N), and let B2, z be the generalized second Bernoulli number associated to Z ([22 ] Then the divisor class of (o)--(~) (and of (oo)--(~)) in jr is of order B2,x/4. There is a rational function f on X0(N)/~ having the properties: (a) (f) = (B2, ~I4). ((o) --(~)) (b) ,J .f= -- i/f. The function f, and the proof of the proposition of Ligozat and Ogg are discussed below. We now prepare to apply their proposition in the construction of D. then B2,x-o mod8. Lemma (x2.3). -- If N=I mod 8, where B2(X) is the second Bernoulli poly- Proof. -- B~,x=N. 31Z(u).B ~ N nomial, X 2- X + I/6. Thus: ' \N ~ N! 104 MODULAR CURVES AND THE EISENSTEIN IDEAL xo5 and since N- I mod 8: g~z-- N (u ~'-u) mod8 l<~u<N--2 = Z 2u -2u) -4- Z .u. u odd u odd l'~ l~u(N--2 l~u<~N--2 =-4.  (u--I)/2 mod8 u odd l~u~N--2 Writing u=I+2j (j=o, I, 2, ..., (N--3)/2) we get: (N--3)/2. ((N-- 3)/2 +I) B~, z = 4- mod 8 N--I mod 8. -4" (N--3)" 8 Q.E.D. But N=I modS. (o)--(g) and (oo)--(~) are of even We conclude from (I2.2) and (I2.3) that order (I/4) B2, x = n~ in jr. Set: A = (n~/2). ~t((o)--(6)) B = (n~12). ~e((oo)--(~) 2A = 2B = o. Since A and B are fixed under ,J, they are in the image of J. SO Suppose that N=9modI6. -- The image of c in jt~ is the divisor class of (o)@(o)--(m)--(~) which is of odd order m=n/2. Thus: A+B =m.A+m.B=n~/2.~l(m. ((o) + (~)--(oo)--(~))) = o and therefore A--~B. Denote by DcJ(Q.) the inverse image (inJ(Q)) of the group generated by A. Since A is fixed by w and w~ (acting on jt~), D is stable under the action of w (acting on J). Also, D is stable under Galois. Let e~z~(O~) be the non- trivial element, and let ~D be an element in the inverse image of A. Lemma (x2.4). -- D is a Klein four group. The action of Gal(Q/Q) on D is the action which factors through Gal(Q.(v/~)/Q) where the conjugation "~ acts on the basis ~, It by the Proof. -- Using (I2.2), the above lemma is an exercise in Galois theory. To emphasize this, let K be the function field of X0(N)/Q and L the function field of X0~(N)/Q. 14 m6 B. MAZUR Thus L/K is a quadratic extension with ~ as conjugation. By (12.2)fis not a square in L| The extension L(flI2)/K is a quartic extension. Since ~(f)=--I/f, (12.2) (b), the extension L(f 1/~, %/~---~)/K(@--~I) is Galois. Let G denote its Galois group. Fix ~, a lifting ofv to G, and let -~ denote complex conjugation in L(f u2, %/----I). By (12.2)(b), ~(ft/2)= Therefore 52(fi/2)=ft/2, and consequently ~ = I. It follows that G is a Klein four-group and therefore so is D, for D is the Cartier dual of G. Let p denote the automorphism of L(f 1/~, %/-----T)/L(%/~-i), given by p(fl/2)=_fl/2. One checks: ~'~T -I ~ ~ .'v 'Tp~- i = p which yields the Galois action on D asserted in the lemma. Let D/s denote the group scheme extension (Zariski closure) of D m in J/s. Let D1/s denote the finite flat group which is the unique extension of Z/5 s by ~/s killed by 2 (extension ~ of chapter I (4.2)). Lemma (I2.5).- D/s~Dx/s. Pro@ -- The two groups have isomorphic Galois modules. Therefore, if D/s is a finite flat group over S, then (12.5) follows from chapter I (4.4). Consider an isomorphism DI/Q --~ D/QCJ/Q and extend it to an isomorphism: Dl/z[1/21 ~ D/z[l/~ ] C J/z[1/21 by the universal property of Ndron models. In particular, D/z[l/2] is finite. Since D/s, is clearly finite, it follows that D/s is a finite flat group. Lemma (I2.6). -- D is annihilated by the Eisenstein ideal ~3. Proof. -- By the formulas giving the action of T t on the cusps of X0~(N) one has, as in (11.1), Tt.A=(I-t-t).A (/~N). As already mentioned, A is fixed under w, and since it is of order 2, (I -}- w). A = o. It follows that D is annihilated by ~2. Any element ye~ operates as an upper triangular matrix in terms of the basis a, ~. To show that D is annihilated by ~, we show that 7~I operates semi-simply on the vector space D. For this, we choose a prime lying above N in Z[~r and consider the specialization map D(Z[%/~--i])-+D(F~), which is an isomorphism of T-modules. Let D(F~)~176 Note that D(F~) is canonically a direct sum: D(Fs) = D(F~)~174 ~2(Fs) (for the subgroup [x2CC maps isomorphically to the image of D(F~) in C (II.3)). Since the action of T " preserves j0 ,, it follows that the action of T preserves the above direct sum decomposition. Since each summand is an F2-vector space of dimension i, T does act semi-simply on D(Z[%/-----~]). 106 MODULAR CURVES AND THE EISENSTEIN IDEAL ~o7 Discussion of proof of the proposition of Ligozat and Ogg. -- Ligozat constructs the function f using the " Klein forms " of Kubert and Lang [28], which are essentially Eisenstein series of weight I. Ogg has a different point of view; he works with products of differences of Eisenstein series of weight 2. In the end, from either point of view, one emerges with a function f on X0(N)/Q whose divisor is B2, x/4.((o)--(~)) and which has the property that f(o~).f(~)=--~. Assertion (b) of our proposition follows from this equation since (vf).f must be a constant. It also follows that, up to sign, Ligozat's function and Ogg's function must agree (this identity is nontrivial). Both Ligozat and Ogg check that their function f is " smallest possible " and thus B2,x/4 is indeed the order of the divisor class of ((o)--(g)) in J~. Nevertheless, in the light of the use we make of (o)- (g) it is worth noticing that the equation f(~)f(~)=--I immediately implies that this divisor class is not killed by B2,x/8 (1). For if it were, there would be a function g on X0(N)/Q such that g2 = r.f where r is a rational (nonzero) number. This is impossible, for g(oo).g(~) would then be a rational number whose square is negative. In the remainder of this section, although we do not prove the proposition in full, we present an account of the construction of Ligozat's function and some of its salient properties (2). Ligozat's construction. -- We may take N- = I mod 4, N>5. Let ~ be a primitive N-th root of I; set: Sj: ={ I<a< (N--I) /2 l z(a)=  } ~ (I--~a'qm)(1--~-a'qm) " and : g+(z)= lI " + ,. = 1 ( I _ q~) (I~ - 1)/~ The functions g (q=e z'~i*) are expressible as products of Klein forms of level N [28]. Explicitly, let p be the constant: P+ =(--27d)lN-1)]~'exp(~-~'',,m, ~ 11 (I- ~a) -1 a6S+ then, using the notation of [28]: g (z) = p=~. oH kco, a)(z ). One checks : g+(z).g_(z)= II (i-qN")(~-r -N- ~(Nz) ,.=1 -n(z) ~ (1) An integer, by lemma (I~.3). (2) Here I have simply copied a part of a manuscript that Ligozat provided for me, and for which I am extremely grateful. It is to be hoped that Ligozat will present the full story in his future publications. 107 IO8 B. MAZUR and therefore (by Hecke [i9] , p. 924) g+.g_ is a modular form of weight --(N--I)/2 on F0(N), of Nebentypus, whose associated character is the Legendre character Z. g+(z) Definition.- f(Z)=g_-~. Lemma. -- f(z) is a modular form on Fi(N ). Proof. -- This follows from the transformation laws for the Klein forms k of [28]. If v=(I4-NaNy N~ I+Ns]eF(N) one has: k(0,~)(vz ) = (Nyz + NS). ~(Y, 8).k(0,,)(z ) where -- %(y, 8) = (--i)(ya + 1)(8a + 1) exp (2~i (-- Ya2)] under F(N) if and only if: 2N ] and therefore f(z) is invariant N--1 II 8) (~ 1 (a (N--l)/2 for any choice of y, 8. Since N>5, Y,z(a).a~---o mod N and therefore we may rewrite the condition of invariance off(z) as: (y + 8) (l<~<(~s_i)/2z(a).a) +y(I +NS) l<~<(x-1)/2Z z(a)a~-=o mod 2. Now note that if y is even, so is 8, in which case the above congruence holds. If y is odd, it also holds since l<~<(N~]-l)12z(a)(a 4-a 2) --o rood 2. Therefoie f is invariant under F(N). To see that it is invariant under Pi(N), note that k(o,~)(z+i)=k(o,~)(z ) for i~Z. Definition.- If u=(~ bd)~P0(N), define ~(u)=f(uz).f(z) -x(~). Thus ~ is a character of P0(N), trivial on Pi(N), and takes values in the group of (2N)-th roots of i. Lemma.- ~(u)=z(d ). Proof. -- Clearly ~2= i, since the index of Pi(N ) in F0(N ) is relatively prime to N. To establish the lemma, one must show that ~+ I. If (r a b)=u is in I'o(N), and z(d)=--I then: g (uz) = ~ (u) . (cz 4- d) (N -1)/4. g T (z) where ~=L(u) are 2N-th roots of i and s(u)= s+(u)/,(u). But since g+g_ =~(Nz)/~(z) N is of Nebentypus with character Z, %.s_=z(d)------I. It follows that s(u)=--i. Corollary. -- vf = -- i If. By the properties of Klein forms [28] the zeroes off are concentrated at (o), (~) and an elementary computation (compare [48], w 2) gives their order. 108 MODULAR CURVES AND THE EISENSTEIN IDEAL ~o9 z 3. The dihedral action on Xl(N ). We shall be working with the covering XI(N ) -+X0(N ) of curves over Q, and with certain subcoverings. Abbreviate the notation to Xi-+X0, and set: U=(Z/N)*/( So, U operates on X i with quotient curve X0; it operates freely on the open curve Y1 = X1-- cusps. As in [4 o] form a " dihedral " group A containing U as follows: where the w e are " symbols " indexed by the primitive N-th roots of I, ~eQ, where, by convention, the element w~-I is taken to be equal to the element w~. Impose a group law on A by: (I3.1) (gt)~)2= I ; U. W~ = W~u = ~)~.U -1 for all ueU, and primitive N-th roots of I, ~. Here ~,~_~a for a an integer (rood N) projecting to ueU. The dihedral group A acts in a natural way as a group of automorphisms of X 1 (cf. [4 o] w 2). The compatibility of the action of 2~ and of Gal(Q/Q) on Xl(()~) is most conveniently described as follows: Define an action of Gal(Q/Q) on A by the rules u~=u; (w~)~=w~, for 0~eGal(Q/Q), ueU, and ~ a primitive N-th root of I. Then, for SeA, and xeXi(Q), we have: (~.x)~=8~.x ~ (1). The action of A on X 1 " covers the action of the canonical involution w on X 0 ", in the following sense: If r: :Xi-+X 0 is the projection, then ~(w~.x)=w.r~(x); (u = Let 90C X0(Q) be the fixed point set of the canonical involution w. Using the modular definition of w, one sees that a point in 90 is given by an elliptic curve defined over Q together with an endomorphism whose square is --N (note: N>5). That is, the fixed point set is in one-one correspondence with isomorphism classes of elliptic curves over Q which possess a complex multiplication by ~r Suppose that N--I mod4. Then Z[%/~] is the full ring of integers in Q('v/~N) and % is a principal homogeneous set under the natural action of eel, the ideal class group of the field Q(@~N). Let q)i C XI(OL) be the full inverse image of %, and let q~l(~) C q~i be the fixed point set of w;, for each ~. (1) In [4 o] we call A, with its GaI(Q/Q) action, the twisted dihedral group. 109 IIo B. MAZUR If xxeq01, and ~ is a primitive N-th root of I, there is a unique element of U, which we denote u(.. ~) satisfying: W~. X 1 = U(x~, ~). X 1 . Clearly, for wU, u(~,,~)v=v.u(~,~) from which one gets Lemma (i3.2). -- q)l decomposes into the disjoint union: where ~ runs through the set of primitive N-th roots of r, with the convention that we have identified and ~-1. Let x0 denote the image of xl in X 0. An elementary computation gives, for any element u.xl in the inverse image of x0, that: (x3.3) u(..~,, ~) = u- 2. u(~,, ~) and consequently the question of whether or not u(~,, ~) is a square in U depends on x o and ~ hut not on x 1. Write u(~.,~)EU/U 2 for the image of u(~.~). Lemma (x3.4). -- These are equivalent: a) u(~,, ~) is trivial in U/U 2. b) w e possesses a fixed point in the inverse image of x o. Moreover, if these conditions hold, then w e will have exactly two fixed points in the inverse image of xo, and these fixed points will be multiples of each other by the unique element wU which is of precise order two. Proof. -- This is essentially immediate: If a) holds, choose an x t mapping to xo, and let ucU be such that u~=u(=,~). Then (i3.3) shows that u.x 1 is a fixed point of w e. The other direction is totally trivial. Finally, if x 1 is a fixed point of we, from (I3.3) the action of w e on the inverse image of x 0 is: wdu. xl) = u- 1. xl giving the last assertion of our lemma. Since (w~.x)~=w~.x ~ for ~eGaI(Q/Q), it follows that ~ induces a i : i corre- spondence q%(~)--->q~l(~), giving: Lemma (I3.5). -- Let h be the class number of Q,(~V/~---N). Then, for any primitive N-th root of I, ~, w~ has exactly h fixed points in XI(Q). Proof. -- The cardinality of qh is h. (N--I)/2. By (I 3.5), ~1 is the disjoint union of the (N--I)/2 sets q~l(~), which are put in i :i correspondence, one with another, by the action of Gal(Q/Q). It follows that each of these sets has cardinality h. Comparing lemmas (I 3 . 5) and (i3.4) it follows that, for a given ~, precisely 110 MODULAR CURVES AND THE EISENSTEIN IDEAL III half of the elements of % have the property that w~ has a fixed point in their inverse image. It is reasonable to expect that the elements of % with this property, for a given ~, forms a principal homogeneous space under the action of ~ft~C ~g (squares of ideal classes). Now pass to the Nebentypus curve X~X 0 which fits into a diagram: X1 where v denotes the involution of X ~ such that X#/v=X0, induced from the action of any ueU such that uCU ~. From (13 . i) one sees that the (N--I) involutions w~ induce precisely two distinct involutions of X ~ which we arbitrarily call US and v. uS. These are conjugate over Q and defined over Q(%/N). From (13 . i) we have that v and w # commute. Also, from (13.4) it follows that if w e induces w ~, then q~l(~) projects bijectively to the fixed point set of US. Consequently, both US and ~. US have exactly h fixed points in X#(t:~). Now suppose N-I mod 8, so X~--->X 0 is unramified. Consider the diagram: x* (x3.6) X:/w: x:/v = x0 X + = Xo/w Lemma (x 3. 7). -- Both ~r and ~ are ramified. To compute the number Proof. -- As for ~, this follows since uS has h fixed points. of fixed points of ~, we use the Euler characteristic Z: (since X~-+X 0 is unramified) z(X~) = 2. z(Xo) (US has h fixed points) z(X~) = 2. z(X~/us)- h (w has h fixed points) z(Xo) = 2. z(x +) - h and therefore ~ has h/2 fixed points. which gives: z(X~/US)----- 2. z(X+)--h/2 Lemma (i3.8). -- We continue to suppose N-I mod8. The subgroup DCJ/Q (of. w 12) has trivial intersection with the sub-abelian variety J+ =(I + w).j. Proof. -- We work with group schemes over Q. We frst show that the subgroup ~2 of the Shimura subgroup has trivial intersection with J+. If Y~Z is any double covering of (smooth projective) curves, then the induced map on their jacobians (regarded 111 II '~ B. MAZUR as Pie ~ is injective if and only if the double covering is ramified. Since X0-+X+ is ramified, we may identify the jacobian of X + with the sub-abelian variety J+ cJ. The subgroup a2C D is the kernel of the map J-+J* onjacobians induced by X#~X 0 (I2. i). To show that ~2 is not contained in J+, it suffices to show that the composition J+-+J-+J~ is injective. But the map J+-~J~ is induced from the covering of degree 4, X~-+X +. Returning to diagram (I3.6) we have that this map is the composite ~ where by (~3-7) both ~ and ~ are ramified double coverings. Injectivity of J+-+Jl follows. Since J+ is defined over Q, and Dt~J+ is a subgroup scheme of D (over Q) not containing a,,, it must vanish. Q.E.D. Corollary (x3.9).- The subgroup scheme D/s, CJ/s, maps isomorphically onto a subgroup scheme of J~, under the natural projection of abelian schemes Jj.s,-+J;, (cf. w m). Proof. -- Let D~, c J~, be the subgroup scheme extension of the image of D/Q in J~. Then we have a map D/s,-+D ~, which induces an isomorphism on Galois modules. It must be an isomorphism, by chapter I (4-4)- For later purposes: Corollary (i3.xo). -- The subgroup ~ (D /F,J ~t C J/F, /s not in the image of i ~-w. Pro@ -- The image of J/F, under I-}-w goes to zero in J~,, but (D/F,) ~t does not, by (~3-9)- x 4. The action of Galois on torsion points of J. Let m be an integer 4=o, and consider J[m](Q) as a T/(m.T)[G]-module (the group ring of G with coefficients in T/(m. T) where G is some finite quotient of Gal(Q/Q,) through which the natural action of Gal(Q/Q,) on J[m](Q) factors). Say that the T/(m.T) [G]-module V is a constituent of J[m](Q) if it is a constituent of a T/(m.T) [6]- Jordan-H61der filtration ofJ [m] (Q). Since a constituent Vis irreducible (as T/(m. T) [G]- module), its annihilator in T is a maximal ideal 93l. Say that V belongs to gJl. Thus, V is a h~[G]-module where h~ is the residue field T/9~. By the dimension of V we mean its dimension as h~-vector space. Note that any constituent V belonging to ~ is a constituent of the sub-module J[~r](~)cJ[m](~) for suitable integers r, m. Note also that given a generating set of elements (al, ..., a~) of the k~-vector space 9J~r/~lN ~ + 1, the map x ~ a 1. x| | t. x is an injection of the module J [gJl']/J [9)l "+1] (Q) into the direct sum oft copies of J [92R] (Q), and therefore V is isomorphic to a constituent in j[gJI](Q). Regarding V as a specific subquotient of J[m](Q) we may use (chap. I, w I (6)) to obtain a quasi-finite group scheme subquotient V/s of Jim]/s which is finite and flat over S', and whose associated Galois module is the subquotient V. 112 MODULAR CURVES AND THE EISENSTEIN IDEAL II 3 Note however that the isomorphism type of V/s may depend on the way we view V as subquotient ofJ [m] (Q) and is not necessarily predictable from the isomorphy type of V. By Fontaine's theorem, chapter I (i.4), however, it is determined (over S') by the isomorphy type of V provided the characteristic of k~ is different from 2. Let p be the characteristic of k~ and VtF q the fibre of V/s reduced to characteristic p. Consider the two possibiIities: a) TpegJ~. Then, by the Eichler-'Shimura relations (w 6), both the Frobenius and the Verschiebung satisfy the relation X2--TpX+p=o, and therefore, since 93l annihilates V/F p they satisfy the relation : X ~ = o. That is, both Frobenius and Verschiebung are nilpotent on V/F p. Consequently, V/F p has the property that both it and its Cartier dual are unipotent finite group schemes. Equivalently, it has a Jordan-HSlder filtration by finite subgroup schemes, all constituents being isomorphic to % ([9], IV, w 4 (3-14)). In this case say that 9X is supersingular. b) Tpr Then, as above, Frobenius and Verschiebung satisfy X. (X--Tp)=o, where Tp is an automorphism of V/F p and it follows that: __ VIll.t. ~ V/Fp-- /Fp � V/Fp- (The product decomposition arising, if you wish, from the fact that T~ -1. Frobenius and T~ -1. Verschiebung are orthogonal idempotents whose sum is the identity.) Thus V/F ~ is, as we shall say, an ordinary group scheme over Fp. In this case we say that ~ is ordinary. Proposition (z 4. 9 ). -- Let V be a constituent belonging to 9)l. Then V is of dimension I if and only if gJ~ is an Eisenstein prime, ff g~ is an Eisenstein prime, then Jilt]/s is admissible (cf. chap. I, w I (f)). Proof. -- We first show that if V is of dimension I, then it beiongs to an Eisenstein prime. Consider V/s, which is a finite flat group scheme if and only if the inertia group at N operates trivially on the k~-vector space V (chap. I (I.3)). Since the inertia group operates unipotently (SGA 7, exp. IX (3.5) (crit~re galoisien de rdduction semi-stable) which applies since (appendix) J/s has semi-stable reduction at N) and semi-simply (since V is of dimension i over k~), it does operate trivially (1). Thus Vts is a finite flat one-dimensional k~-vector group scheme. By chapter I (I. 5), either: V/s = ~p| or: V/s =Z/p| and in either case, the Eichler-Shimura relations (w 6) give us the following facts about the image of T t (g+N) in k~ which we can think of as contained in End(V/@: Tt- I -}-t modg~ (g4=N). (1) This was pointed out to me by K. Ribet. 15 iI 4 B. MAZUR As for the image of w in k~, since w is or order 2, this image must be 4- i. If the image of w is --i, then 9J/ is visibly the Eisenstein prime of residual characteristic p. To conclude the first part of this proof, one must show that if p is odd, the case w~ + i cannot occur. We show that the ideal 93/generated by:p, I--W, and i +t--T t (all t+N) is the unit ideal in T. Suppose not; then it is a maximal ideal with residue field Fp. By (9-3) the kernel of its action on I-I~ , ~)is of dimension i over Fp. This kernel is generated by a parabolic modular form mod p, g, whose q-expansion is entirely determined (9-2) by the above package of eigenvalues, and the fact that it begins with the term i .q. Comparing the coefficients ofg with that of the Eisenstein series e' (5- i) one sees that f=e'§ is a modular form modulo 24P whose q-expansion (modulo 24P) is a function of q~: y= ( I -- N) -- 4 8. q~ +... If P>5, such a modular form does not exist (1) by lemma (5. IO) (if p=N), by lemmas (4. io), (5.9) (if piN--I) and corollary (5.II) (if p=t=N, pgN--I). If P=3, and N-=I mod3, then J~3= ~- -t-I6-q~-} -.-- does not exist mod 3, as a holomorphic modular form, by corollary (5-i i) (ii) (if N~ i rood 9) and by (4-io) and (5-9) (if N=I mod9). Finally, if P=3, and N=--I mod3, fdoes not exist mod 9 by corollary (5. i I). To conclude the proof of our proposition, we show that if ~3 is an Eisenstein prime, then J[~3 r] is admissible (any r) and consequently any of its constituents is, indeed, of dimension I. In the light of (chap. I, w I (f)) and remarks made at the beginning of this section, it suffices to show that J[~3] (Q) possesses an admissible filtration by sub- Galois modules. Let W denote the Gal(Q/Q.)-module which is the direct sum ofJ [~3] (Q) and its Cartier dual. Thus W is a self-dual Gal(Q/Q.)-modute, annihilated by ~3, of dimension 2d, say, over Fp. We let G denote a finite quotient of Gal(Q/Q) through which the action on W factors. Since T t acts as I+t on W, (t+N), the Eichler- Shimura relations (w 6) impose the relation: on the action of the Frobenius automorphism ?t (t + N, p) on W. Thus, the only eigen- values possible for the action of q~t on W are: I and g. Since Cartier duality " inter- changes " these eigenvalues, and since W has been devised to be self-dual under Cartier duality, it follows that the characteristic polynomial of q~t acting on W must be (X-- i)~(X--~) ' . Now consider the Gal(Q/O)-module (Z/p)a| a, which we also regard as a G-module (the natural action on this module factors through G, and if it did not, we would have augmented G appropriately). It also has the property that the characteristic (I) This has been proven independently by K. Ribet. 114 MODULAR CURVES AND THE EISENSTEIN IDEAL I15 polynomial of ?e acting on it is: (X--I)e(X--t) a. By the ~ebotarev theorem any element in G is the image of some q~t (t #p, N). Thus, any element geG has the same characteristic polynomial for the represen- tation W as for (Z/p)d| a. By the Brauer-Nesbitt theorem ([6], (3o. I6)), the semi-simplification of the representation W is isomorphic to (the already semi-simple) (Z/p)d| e. Thus W has an admissible filtration and therefore, so does J[~](()~). Proposition (x4.2). -- Let 9J~ be a prime which is not an Eisenstein prime, and which is supersingular if char k~=2. Then J[gJl] is an irreducible two-dimensional Gal(I~/Q)- representation over k~oz (1). Proof. -- By theorem (6.7) (and (3.2)) of [IO], there is a unique semi-simple rep- resentation p : Gal(Q/Q) -~ GL2(k~) such that for every e#p, N, if at=image(Tt) C k~: Trace (?t) = at det(q@ =L Denote by V the associated semi-simple k~[Gal(Q/Q)]-module. Let: d = dimk~j~ (J [9~] (Q)). As in the previous proposition, form the Gal (Q/Q)-module W: the direct sum ofJ [gJ~] (Q) with its Cartier dual Let W'= the direct sum of d copies of V. By the Eichler- Shimura relations, the eigenvalues of q~t are constrained to be solutions of the quadratic equation X2--atX+t-----o, and since Cartier duality " interchanges the roots of the above equation " the characteristic polynomial of q~t operating on the self-dual Gal(Q/Q)- module W is: (X~--atX+t) a. But this is also the characteristic polynomial of q0 t acting on the semi-simple Gal(Q/Q)-module W'. It follows that W' is the semi-simplification of W. By prop- osition (I 4. I) (and the fact that 9J~ is not an Eisenstein prime) it follows that V is an irreducible k~[Gal(Q/Q)]-module. Therefore, W has a Jordan-H61der filtration of sub-k~[Gal(Q/O)]-modutes all of whose successive quotients are isomorphic to V. It follows that J[932](Q) also has such a filtration. In particular, considering the first stage of such a filtration, we have an injection VCJ[gJ~](Q). We must prove that V=J[gJI](Q). We do this by studying V/sCJ[~lJt]/s, the quasi-finite group scheme extension of V. Case 1.- Chark~+N and either: a) 93l supersingular or b) 93t ordinary and char k~ + 2. (1) We also establish (cf. (I6.3) below) that J[~] is a 2-dimensional Gal(Q/Q)-representatlon, when ~ is an Eisenstein prime. 115 II6 ]3. MAZUR Here we make use of the contravariant Dieudonn6 module functor of Oda [47], denoted M(--). Its relation to De Rham eohomology is given by corollary (5.II) of [47]. Namely, if A is an abelian variety over a perfect field k, of characteristic p then there is a functorial isomorphism of Dieudonnd modules: M(A[p]) ~ H~R(A/k ). Moreover, under 4, the Hodge filtration: o ~ H~ f2) ~ Hla(A/k) -~ HI(A, (9A) --, o corresponds to the filtration: o --~ M(A[Frob])' -~ M(A[p]) --~ M(A[Ver]) --~ o where [ ] means, as usual, kernel, Frob means the Frobenius endomorphism, Ver means the Verschiebung, and the prime superscript has the following significance: M (A [Frob]) ' = (k, a- 1) | M(A [Frob]) where (k, a-l) is the abelian group k, regarded as k-algebra by the morphism k~ where a is the p-th power map. Moreover: M(A [p])[Frob] --~Ver. M(A[p])=~ M(A[Frob])' where Ver and the first Frob denote the V and F operators of the Dieudonnd module M(A[p]). If G is a finite group scheme over k equipped with a homomorphism: T /pT --~ End(G/k), we induce a T/pT-module structure on M(G) commuting with its module-structure over the Dieudonn6 ring. Since M(--) is an exact contravariant functor, we have M(G)/0Jr. M(G)- M(G [gJl]). Consequently: M(J [~]/F,) = M(J [P]/F,)/~" M(J [P] A',) = H~)R(J/Fp )/92R. HI)R(J/F,) ~ H~)R(Xo(N)/,,)/g.R. H~)R(X0(N)/@ where the last isomorphism comes by the identification of J with the Albanese of X0(N), and all isomorphisms are isomorphisms of T/pT-Dieudonn6 modules. Make these abbreviations: M(V/Fp)=M; t 1 HDR(X0(N)/Fp) = Hi)a. The inclusion V/l,~cJ[9)l]/Fp induces a surjection of the k~-Dieudonn6 modules: H~/OJ~. H~ -+ M ~ o. 116 MODULAR CURVES AND THE EISENSTEIN IDEAL I17 Passing to the cokernel of Verschiebung, one has a diagram: 1 1 HDa~/9"J~.HIm -> M > o Hl(g)x)/gJt.Hl(@x) ). M/Ver. M > o o O where we have written I-P(Ox) for tP(X0(N)/Fp , d)). By (9.4), I-II(@x)/gJ~.Hl(Ox) is a k~-vector space of dimension i. Thus: (I4" 3) dimk~(M/Ver. M)5 I. We now use the hypothesis that either ~ is supersingular or the characteristic of k~ is different from 2. Lemma (x4.4). -- With the above hypotheses, V/s, is an auto-dual finite flat group scheme with respect to Cartier duality. Neither Frob nor Ver vanish identically, nor are they isomorphisms, on the Dieudonng module M. Proof. -- The Gal(Q/Q)-module V is auto-dual under Cartier duality. Therefore Vls, and its Cartier dual V~, have isomorphic associated Gal(Q/Q)-modules. Under our hypotheses, Fontaine's theorem, chapter I (I.4), applies. Thus V/s, is auto-dual. Consequently, M is a self-dual Dieudonn6 module. Since Frob.Ver=p=o on M, it is clear that not both Frob and Ver can be automorphisms of M, and by self- duality, neither are. Also, by self-duality, if one of the two operators Frob and Ver are identically zero, then both are. In particular, Ver would be zero, which is impossible, since its cokernel is of dimension less than or equal to I by (I4.3). An immediate consequence of Lemma (14.4) and (14.3) is: Lemma (x 4 . 5). -- Hl(@x)/93l. Hl(Ox) -+ 1VI/Ver. M is an isomorphism of i-dimensional k~-vector spaces. Lemma (x 4. 6). -- Let 0 -+ M 1 -+ M 2 -+ M 3 -+ 0 be a short exact sequence of (finite) k~-Dieudonng modules satisfying these properties: a) the cohernel of Ver on M 2 # of dimension I over k~; b) Frob is nonzero on M 3. Then Ver is an isomorphism of M1 onto itself. Proof. -- We show that Ver : M1-)-M 1 is surjective, by showing: (i) M2/Ver. M2 ~ M3/Ver. M 3 is injective, (ii) M2[Ver] ~ Ms[Ver] is suriective, 117 lib B. MAZUR and applying the snake-iemma. Since the morphism (i) is surjective, and Mz/Ver. M~ is of dimension i over kan , it suffices to show that Ver. Ma4:M3 to obtain injectivity of (i). But Ver annihilates Frob. M 3 which is nonzero, by b). Therefore Ver is not an automorphism of M3. To show (ii), note first that since Mi is finite-dimensional over k~ (i=2, 3), dim~x~(M2[Ver])=I (as follows from a)); also dim~(Ma[Ver])=I (as follows from the isomorphism (i)). Therefore all that must be shown is that the morphism (ii) is nonzero. But this follows from the diagram: 1V[ 2 > M 3 > o M2[Ver ] > Ma[Ver ] oi) and hypothesis b). To apply lemma (14.6) to our situation, take iV[2= t i HDR/gX.HIm and Ma=M. Both hypotheses a) and b) hold, by lemmas (14.4) and (14.5). We obtain the following conclusion: If M 1 is the kernel of the homomorphism Htm~/gJt. H~R-+M, then Ver is an isomorphism on M 1. That is, M 1 is the Dieudonn6 module of a group scheme of multiplicative type. Therefore the cokernel of V/s, CJ[gX]/s, has multiplicative type reduction in characteristic p. But, by the discussion at the beginning of this proof, and by Fontaine's theorem, this cokernel has a filtration by finite flat subgroup schemes all of whose nontrivial successive quotients are isomorphic to V/s, , which does not have multiplicative type reduction in characteristic p. We conclude that this cokernel is zero. Case 2. -- (A digression) 9)l ordinary and char kan 4:N. There is another more direct way of putting the above argument, when 9X is ordinary. This alternate method does not use Dieudonn6 modules, but rather depends upon an important isomorphism due to Cartier and Serre ([64], w II, Prop. IO). By means of this isomorphism, one may deduce (14.8) below, which will also be useful to us in the case where 93l is an Eisenstein prime (el. (14.9), (14. IO)). Thus, in the present case we let 93l be any ordinary prime in T, Eisenstein or not, with char k~ 4: N. Recall the canonical isomorphism: 3 : J[p] (F,) --+ H~ f~l)* of [64] , w 11, Prop. IO, where the superscript ~ means fixed elements under the Cartier operator. (Note: this isomorphism is defined for any smooth projective curve and not just Xo(N). ) The definition of ~ is as follows: an element x of the domain is rep- resented by a divisor D on X0(N)~ p such that p.D=(f). One takes ~(x)=df/f. 118 MODULAR. CURVES AND THE EISENSTEIN IDEAL II 9 Proposition (I4.7). -- The isomorphism above induces an injection: 8 : (J[p](F,)) | ,-+ H~ a *) which commutes with the action of T/p.T on domain and range. Proof. -- By ([64], w i I, Prop. io) injectivity follows from injectivity of the natural map: (H~ tll)*) | -+ H~ s which is an elementary exercise, using a- Minearity of c~: Let xl, ..., x, be the smallest number (s~o) of FKlinearly independent elements of H~ t~t) ~ such that xl-}-Xg.x~+...4-X~.x~-~o , where ~jeFp. Applying I--C~ to this equation gives a smaller relation. That 8 commutes with the action of w is evident. To check that it commutes with T t boils down, in the end, to checking commutativity of the (two) squares: L* elo~ ~p K* > t~ep d log where K is a function field in one variable over Fp and L is a finite K-algebra. Corollary (I4.B). -- Let 9J~ be an ordinary prime ideal in T with char k~+N. Let (J[p]/pp) ~t denote the gtale part of the group scheme J[p]/~p, and let (J[p]~)[9:R] denote the kernel of the ideal ~ in this group scheme.. Then (J[p]~p)[9J~] is a k~-vector group scheme of rank i. One has the equality: J[9)l]~p=(J[p]~p)[g31] /f p>2. Proof. -- The rank of (J[p]~p)[93l] is at most i as follows immediately from the previous proposition and proposition (9-3)- To obtain the equality asserted, we must lrgj~l~t is nontrivial. If it were trivial, then J[931]/Fp would be of multipli- show that ak J/Fp cative type. Since p::>2, Fontaine's theorem, and the remarks at the beginning of this section, apply, giving us that any constituent of J[!I~r]~p is a constituent of J[~0l]/Fp. It would then follow that the p-divisible (Barsotti-Tate) group J~l/Fp is of multiplicative type, which is impossible, since it is auto-dual under Cartier duality. Ifp divides n, let Jp[~]/s denote (as usual) the group scheme extension to S of the kernel of the Eisenstein ideal 3 in the Barsotti-Tate group Jp/Q. This group scheme is also J[3,pt]/s where pt[In. Let Gp/s denote the p-primary component of the cuspidal subgroup C (regarded as constant group over S). If p = 2, let Dis denote the subgroup scheme of Jts constructed in w I2. 11 ~2o B. MAZUR Proposition (x4.9).- If p is an odd prime dividing n, then: = (J, [3] e,) t = (J YF,) [3] and, if p = 2, divides n: Remarks and pro@ -- The right-hand group scheme on the first line of our prop- osition is the kernel of 3 in the dtale part of the p-Barsotti-Tate group over Fj, associated to J. All of the asserted equalities of the proposition are known inclusions (reading from left to right). To establish the proposition, note that Corollary (14.8) gives that: ~t is a group scheme of order p. The assertion of the proposition for p = 2 then follows simply by noting that (D/F~) ~t is also a group scheme of order p. To obtain the assertion when p;>2, let prlln. Note that (J~Fp)[3] is annihilated by pr by (9.7), and since the kernel of multiplication by p in this group scheme is of order p, it follows that (J~[l,~)[3] is of order p( So is C~, by (1~.I). The proposition follows. Corollary (x 4. xo). -- If p is an odd prime dividing n, then one has a short exact sequence: (i) o ~ Cp -+Jp[3] -3- M ~ o (over S) where M is an admissible group scheme of muhiplicative type. If p = 2 divides n, then one has a short exact sequence: (ii) o ~ D -+J [~3] --~ M -+ o (over S) where M is an admissible group scheme of multiplicative type. Proof. -- In either of the exact sequences above, the cokernel is admissible by (t 4- i), and of multiplicative type by the previous proposition. Corollary (x 4 . xx ). -- Let p be a prime dividing n. Let W denote the Zp-dual of $-a(Jv(Fp)). The To-module W is free of rank I. Proof. -- By proposition (8.4) it is of rank I (i.e. W| is free of rank I over TvNQ). It suffices to show that W/~3.W is oforderp. But (w 7) W is the Pontrjagin dual of J~(F~) and therefore we must show that J~(Fp)[~3] is of order p, which follows from (14.9). As for the alternate argument: suppose gJl is ordinary, not an Eisenstein prime, and such that char k~l+N , 2. By Fontaine's theorem, and the discussion at the beginning of this section, V~r p cannot be of multiplicative type. Thus, the k~-rank of (V/Fy t is 21, and (by (14.8)) the k~-rank of (J[9)l]~rp) 6t is S~I. It follows that V=J[9/II]. 120 MODULAR CURVES AND THE EISENSTEIN IDEAL I2I Case 3. -- Char kzl = N. This case parallels the " alternate argument " in case 2. Note that if char k~ = N, then ~f)2 is not an Eisenstein prime. Also, J [~JX]/s, is a finite dtale group scheme, admitting a Jordan-H01der filtration by finite dtale subgroup schemes all successive quotients being isomorphic to the finite dtale group scheme V/s,. Consider the first layer in such a filtration VCJ[992], and note that we have exact sequences: o " J[~]~ ' J[~](QN) ' J[~](fN) , o I * U U U o , , v(0 N) - , , o where the superscript 0 may be viewed as denoting either the connected component over Spec Zs, or the intersection of the (appropriate) group scheme over Spec Z N with J~z~- Since V is self-dual, we cannot have V(Qs)=V~ (for then it would be of multiplicative type over FN). Therefore dimk~(V(Fs))>I. As in case 2, we must show dim~J[9~](Fl~)<I. For this, we extend Serre's mapping a to cover our present case. Let D represent a divisor class x in J IN] (Q~)= Pic~ Assume D is an eigenvector for w. Letfbe a rational function on Mo(N)/~N (Zr~=ring of integers in QN) such that (f)=N.D on X0(N)/~,~, and such that f does not vanish identically on M0(N)/~N. Since f satisfies an equation of the type fow = 4-f~:*, and w interchanges the two irreducible components of M0(N)/f~, f vanishes on neither component of M0(N)a ~. Form df/f in H~ ~*) where the superscript h denotes the smooth locus. Note that: H~ f~*)= H~ f~)= H~ f~) (the second equality comes from (3.4); for the first, since Mo(N ) is Cohen-Macaulay, g) is invertible, and the supersingular points of characteristic N are of codimension 2 in M0(N)/z~,). Set 8(x) ----- image of df/f in H~ f~). Since the function f is unique up to a possible multiple u.g s where u is a unit in 7.N and g is a rational function on M0(N)/z~, the mapping x~8(x) is well-defined. Also, ~(x)= o if and only if f, reduced modulo N, is an N-th power (or, equivalently, x goes to zero in J[-N](FN) ). Extending the definition of ~, by linearity, to all divisor classes x~J[N](QN), we have: : J[N](Fs) ~ H0(Xo(N)a., f~). lfi x22 B. MAZUR Since the theory of the Cartier operator ([64] , w io) is local, it applies to the smooth quasi-projective curve X0(N))~ and one has, as before, that the image of 8 is contained in the fixed part: H~ ~21) ~, and by the argument of (14.7) , one deduces an injection: : J[N] (Fs)@x,,~Fz~ r H~ n). At this point one uses (9.3) as in the proof of (I4.8) to conclude that: dimk~ J [~] (Fs)~ i. What is the minimal field of definition of the Gal(Q/Q,) representation determined by J[9:R] ? Proposition (I 4. x2). -- Let ?fit be a prime which is not an Eisenstein prime, and which is supersingular if char k~ = 2. Then k~ is generated over Fp by the images of the operators T t (l@char k~), and J[gJ~] is an irreducible Gal(t2~/Q)-module. Proof. -- Before we begin the proof, let us note that the last assertion is stronger than the assertion of Proposition (I 4. 2). We are saying that the abelian group J[gJ~] (l~)=V is irreducible as Fp[Gal(Q/Q)]- module. Let E be the image of Fp[GaI(Q/Q)] in the endomorphism ring ofJ[g)l] (Q). Let k C k~ be the subfield generated by the T t (for all [ +p, N). Using the Eichler- Shimura relations for J[gJ~]/Ft and the fact that J[9:R] is dtale in characteristic [+p, N, we obtain a natural imbedding of k in the center of the ring E, which we therefore view as k-algebra; in fact we take k systematically as our base field. Note that k~ =k[vp] (zp=imagerlp), if p+N. If p=N then k=k~. Let V x be a two-dimensional Gal(Q/Q)-representation over k such that V~| Such a representation exists by [io], Theorem (6.7). Viewing V1 C V as sub-Galois module, and taking the subgroup scheme extension (Chap. I, w I (c)) ofV 1 in V/s=J[gJ~]/s, we obtain a closed (k-vector space) subgroup scheme VI/sC V/s. Let V~| denote the associated k~-vector group scheme, which one can " construct " simply by taking: 2 d--1 V~/s @ up. V~/s @ up. V~/s @... @ =~ . V~/s where d~ [k~ : k], and giving it the natural k~-structure. We have a homomorphism of k~r group schemes: Vl| --~ V/s which is an isomorphism on associated Galois modules. Since, by our hypothesis, Fontaine's theorem (chap. I (i .4)) applies, this is an isomorphism of group schemes over S', and hence also when restricted to characteristic p. Note that over Fp, the endomorphism 122 MODULAR CURVES AND THE EISENSTEIN IDEAL ~3 Frob +Ver preserves the above direct sum decomposition. By the Eichler-Shimura relations, T~ must also preserve the above direct sum decomposition (over Fp), which is possible only if d= I. The proposition follows. Proposition (i4.i3). -- Let 9X, 9X' be primes such that chark~=chark~,4:2 or: 9X and 9J~' are supersingular. Then the Gal(Q/Q)-modules J[gJt](Q) and J[ 9X' ](Q) are isomorphic if and only Pro@ -- By (1 4. I) we may suppose neither prime is an Eisenstein prime. Suppose J[gX] (()_..) and J[gX'] (()_~) are isomorphic as Gal(~_../Q)-modules. By Fontaine's theorem J[gX]/s, is isomorphic to J[ 9J~' ]/s', and the Eichler-Shimura relations together with Proposition (I4.12) enable us to get an isomorphism k~ -% k~, such that if v t is the image of Te in k~ (resp. -~=image of T t in k~0v) then i(vt)----z ~ for all t+N. To show that 9X = ~', it suffices to show that w has the same image in k~ as it does in k~,. Suppose not (i.e. w goes to + i in k~ and --I in k~v ). Then consider the q-expansions of generating eigenvectors (w 9) in H~ ~)[gX] and in H~ ~)[~']. These q-expansions are the same except for the coefficients of powers of qN. Applying (4. IO), (5.9), (5. IO) to the difference of these generating eigenvectors, we obtain that the generating eigenvectors are equal. Therefore ~R = 9X'. I5. The Gorenstein condition. Let R be a local Zp-algebra, free of finite rank as a module over Zp. Then R is a Gorenstein ring [3] if and only if the Zp-dual to R, R* =Homzp(R , Zp), is free (of rank i) as a module over R. Lemma (x5.I). -- Let 93tCT be a maximal ideal. We have the indicated implications of the assertions below: I) J[gX](()_..) /s of dimension 2 over k~. 2) $'a(Js~(Q)) is free of rank 2 over T~. 3) T~ is a Gorenstein ring. 4) H~ ~)| is free of rank i over T~. Proof. -- I) ~-2): Assuming i) we have that the kernel of 9Jr in J~(Q) is of dimension two over k~. Hence the cokernel of ~J~ in Hom(J~(Q), Qv/Z~)=H~ is also of dimension two. But H~ is the Zp-dual of the Tare group (cf. w 7): $'a(J~(Q)) = H~(Xo(N) ~, Z) | = H~. 123 124 B. MAZUR Since H~ is its own Ziduat , we have that H~| ~ is of dimension two. But since (7.7) H~| is free of rank two over T~| it follows that for any homomorphism T~--->K (where K is any field) H~| is of dimension two, and H~m is therefore free of rank 2 over T~. 2)--3): Write H~=FI| the direct sum of two free T~-modules of rank i. Since H~=H~ (* denotes Zidual ) we have an isomorphism F~| FI| 2. Consider the four projections ~i,~ : F* --> Fj (i,j=I, 2). At least one is a surjection, for if not the image of ~,~ would be contained in the maximal proper submodule 9Yr. Fj for all i, j, contradicting our isomorphism. Suppose ~i,~ : F~F~ is surjective. It is also injective since it induces an iso- morphism after tensoring with Q, and the domain is Z-torsion free. Thus, F~ is a free T=-module of rank I, whose Z-dual is also free and therefore T~ is Gorenstein. 3)<=>4): By (9.4) Hl(X0(N)tzp, d))| is free of rank i over T~, and using the idempotents c~, ~ of (7. I), and the fact that T act in a hermitian manner with respect to the duality (3.2) one sees that H~ f~)| is its Zidual. 2) ~ i): An easy reversal of the argument that I) :~2). Corollary (i5.2). -- Let 9X be a maximal ideal in T which is not an Eisenstein prime, and such that, if char k~= 2, then 9J1 is supersingular. Then all four assertions of (i 5. i) hold and in particular T~ is a Gorenstein ring. Remark. -- In the next two sections, we shall establish this Corollary for Eisenstein primes as well. This is significantly harder. We shall have use for the following (elementary) sufficient condition for Gorenstein-ness. Proposition (x5.3). -- If R=Zp[~] is generated by one element over Z~, then R is Gorenstein [3]. x6. Eisenstein primes (mainly p 4:2). Fix p a prime number dividing n. Definition. -- A prime number 14:N will be called good (relative to the pair (p, N), usually unmentioned and understood) if either: a) not both t and p are equal to 2, and (i) ~ is not a p-th power modulo N and t--i (ii) -- * o rood p or (the somewhat special " degenerate " case) : b) ~ =p = 2, and 2 is not a quartic residue modulo N. 124 MODULAR CURVES AND THE EISENSTEIN IDEAL x25 --2) The set of good primes has Dirichlet density ~ if p>2, and - if p=2. In particular, there are some good primes. 4 For anyg set ~t=I+t--T t. The object of this section and the next is to establish the following proposition and to derive some important consequences: Proposition (x6. x). -- The Eisenstein prime ~.T~ C T~ is generated by the elements p and ~t, where t is any good prime (1). Although some finer consequences of the above proposition will be developed later, note these corollaries. Corollary (x6.~z). -- The Zp-algebra T~ is generated by ~t for ~ any good prime. Therefore, by (I 5.3) : Corollary (x6.3). -- The ring T o is Gorenstein. The Fp-vector group J[~3] is two- dimensional. If p>~, then: J [~] = c [p] | E [p]. If p--2, then J[~]----U. The T~-module H~--~'a(J~(Q)) is free of rank 2. Corollary (x6.4). -- If p>2, Jp[~]=J~[~]=ep| (recall: Cp=p-primary component of C, and the same for Ep). Pro@ -- %OE, is contained in Jv[3] ((ii.i), (iI.7)). But [3] (Q) is the PontIjagin dual of H;/3. H (* means Z,-dual) and therefore, by the previous corollary, it has the same order as Cp(())| We begin by establishing a lemma needed to control the action of inertia. Lemma (x6.5)- -- Let B be a subgroup of either the cuspidal or the Shimura subgroup of J. If the superscript i denotes the module of fixed elements under the action of inertia, we have an exact sequence: o -+ B -+j,(QN)' -~ (J,(QN)/B) ~ o (where Jp is the p-divisible (Barsotti-Tate) group associated to J). Proof. -- What must be shown is that Jp(QN)-+Jp(Q.N)/B induces a surjection on elements fixed under inertia. By the appendix we know: Jp(QN)I=J~215 (J~ group associated to J~F,,)- (1) Carefully stated, our proof even works for g = p, ifp happens to be a good prime. This is hardly relevant for the main corollaries; moreover, our second proof of this proposition (by the theory of modular symbols (of. (x 8. I o) below)) makes no distinction whatsoever between the cases t=p and t4:p. Nevertheless, the fact that our proposition is true when l =p is a good prime has significance for the ~3-adic analytic number theory of J, and for the study of the arithmetic of the p-Eisenstein factor .](P) in the p-cyclotomic tower over Q (cf. chap. III, 9)- 125 ~6 B. MAZUR By SGA 7, exp. IX (3.5) (crit~re gatoisien de r~duction semi-stable) we know that if ~, 7 are in the inertia group at N, then (i --~) (I--y) acts trivially on Jp(QN)- Hence, ify is in the inertia subgroup, (1--7) .Jp(QN) CJp(QN) i. But since J~(QN) is ap-divisible group, (1-y).jp(QN) must be contained in the p-divisible part of jp(Qs) l, which is 0- J~(F~) cjp(QN)' , by the above direct product decomposition. Now, take an element e in Jp(Qs) which maps to b- in (Jp(QN)/B) 1. Let y be any element in the inertia subgroup. Since (1--y).e goes to (I--,()~-=o in j~(QN)/B, (i--7).e~B. Therefore, by the above discussion, (l--y).e is in Bc~J~ which is the trivial group, as is clear from the displayed direct product, if B C C and as follows from (II .9) if BC Z. Thus (i--y).e=o, for all y in the inertia subgroup. Q.E.D. From now on, in this section, let p+ 2. -- In this case, proposition (16. I) will follow from a direct proof of the stronger proposition (16.6) below. When p----2, we shall reverse the order of proofs of these propositions. Proposition (i6.6). -- The ideal 3.T~ is a principal ideal in TO, generated by ~t for g any good prime. Proof. -- We shall be working with subgroup schemes (closed quasi-finite) in J~ (hence admissible by (I4.I)). In particular, consider Jv[~]=Jp[,3]. We make extensive use of the tools developed in chapter I. Lemma (x6.7). -- The admissible group Jp[~] is a pure group (1). Pro@ -- Consider the exact sequence (14. io): over S', where C~ is the p-primary component of the cuspidal subgroup, and M is of multiplicative type. We first show that M is a Vt-type group (1). Since ~ annihilates J[~], for any prime number g+N, T t acts as I-r on J[~]. Thus, by the Eichler- Shimura relations, for any t 4:p, N, the g-Frobenius q~t satisfies ?~--(I -}- t). q0 t + g = o, or : =o. If g~I modp, then q~t acts as multiplication by t on M(Q). The reason for this is as follows. Since the Galois module M(Q) is admissible, of multiplicative type, the only eigenvalue that q~t possesses (when acting on M(Q)) is l. Consequently, (q~t--i) maps M(Q) isomorphically onto itself, and the above formula then implies that (~t--g) annihilates M(Q). If M v denotes the Cartier dual of M, then M v is an 6tale admissible group over S' such that ?t acts trivially in its Galois iepresentation (1) Chapter I, w 3- 126 MODULAR CURVES AND THE EISENSTEIN IDEAL ~27 for every g ~ p, N such that l $ i mod p. An elementary density argument (or chap. I (3.4)) implies that M ~ is constant, and therefore M is a ~t-type group. Since M is a ~-type group, the inertia subgroup at N operates trivially on M(Qs) (M extends to a finite flat subgroup over S, of order prime to N). Applying lemma (i6.5) with B=Gp, we obtain that inertia at N operates trivially on Jp[3] which is therefore a pure group by chapter I (4.5). Thus Jp[3]/s is a finite flat group and, over S: =c,� M. It follows that: J [~] = c [p] � M [p]. Let r be a nonnegative integer. Claim 1. -- The quotient group scheme Jp[3. ~3" +1]/jp[3. ~'] over S' is pure. Pro@ -- Set t=diml,p3.~'/3.~ ~+1. As in the discussion at the beginning of w I4, one may obtain an injection of the associated Galois module to Jp[3. ~r+1]/jp [3. ~,] into the associated Galois module to the direct sum of t copies of J[~]. Consequently, by lemma (I6.7), the inertia group at N operates trivially on the Qs-valued points of Jp[~. ~ + ~] [Jp [3. ~'] and therefore it is pure, by chapter I (4.5). Now fix a good prime number ~. Claim 2. -- The group scheme Jp[~.~', ~t]=G, /s pure for all r, and: Gr = Cp � M (') where M (~) is a ~-type group. Proof. ~ By the above group scheme we mean, as usual, the subgroup scheme extension inJl s of the intersection of the kernels of 3. ~r and ~t in Jp/Q (or, equivalently, in J~/Q). We proceed by induction, the first case r=o being already established (lemma (i 6.7)). Suppose G r is of the desired type: a pure group with dtale part Cp and ~-type part M (r). Since its dtale and ~t-type parts are canonically determined, the operation of the Hecke algebra T must preserve these parts; in particular it preserves M t'). Now we work over the base S'. Since ~t annihilates G~+I, the Eichler-Shimura relations give us the equation: on G,+I, if t4=p and hence also on any subquotient of G,+ 1. If t=p we have the above equation on any subquotient of G~+ 1 which is gtale. 127 i~8 B. MAZUR By claim t, and chapter I (4-5), it follows that Gr+l/G ~ is pure. So we may write: (I6.8) o -+ G~ --> Gr +l --> (Z/p)~ � M ' -+o where ~ is some nonnegative integer, and M' is a ~z-type group. We first show that ~ = o. -- Form the pullback: o > G, ) G,+ 1 > (Z/p)~� ' ) o oT o > G~ , G § (Z/p) ~- ~-o and set G =-G/M (~). Thus we have a short exact sequence: o -~Cp->G -+ (Z/p) ~ -+o. That is, G is an admissible 6tale group. Moreover, since (~pt--I)(q~t--t) =o, and t ~ I mod p, it follows that q~t = I on G. By the " criterion for constancy (chap. I (3.4)), G is a constant group. By the manner in which G was constructed, there is a natural induced action of the Hecke algebra T on G. But the ideal ~ C T annihil- ates G. To see this, use the fact that the action of Frob t, on G is trivial (for any prime number g'+N, including t'=p) since G is a constant group over S'. From the Eichler- Shimura relations one then sees that T t, = i +t' (for all t'+N). By construction of G, (i+w) ~ annihilates G. Since w is an involution, and p+2, it follows that i + w = o on G. Thus the ideal ~ annihilates G. Now reduce to characteristic p. From our exact sequences one sees that G/F p is equal to (Gr+m,p) ~t. It follows from what we have just shown that (Gr+l/Fp) ~t is annihilated by 3. But by (~4-io), Cp equals the kernel of ~ in (Jv~y t. Therefore 0(~O. Return to our exact sequence (i6.8), which now may be written: O -->G r ~-> Gr + 1 ~M' -+o. Also we have the exact sequence: .--> M H (x6.9) o --> Cp--> G~ + 1 -->o where M" is an admissible group of multiplicative type which is an extension of M' by M Crt. Since ~t ~ o, applying the Eichler-Shimura relations to the Cartier dual (M") which is an 6tale admissible group, we have that (gt--I)(gt--g)~ o on (M") ~. Since g is a good prime number, we use, again, the above quadratic equation, and the " Criterion of constancy" (chap. I (3.4)) to deduce that (M") ~ is a constant group; thus M" is of ~t-type. In particular, the inertia group at N operates trivially on M"(QN) , and hence also on G~+I(QN), using the exact sequence (I6.9) and lemma (i6.5) with B=C~. Thus, by the criterion of purity (chap. I (4-5)), Gr+t is a pure group, whose 6tale part is C~. 128 MODULAR CURVES AND THE EISENSTEIN IDEAL ~29 Claim 3.- %=(j~yt[~t]. Proof. -- By Claim 2, the kernel of ~t in J~zs, has the following structure: =% x M where MI~ U M/') is a union of ~-type groups. Consequently C~=((Jv[~t])~y ~. Our claim will follow from a lemma (which we also use later when p= ~): Lemma (x6. xo). -- Let p be any prime dividing n, and t any prime number different from N. Then: [Ht] (f,) = [H,]. One sees easily that He is an isogeny of J onto itself, for if it were not, then, by the Eichler-Shimura relations, q)t would have an eigenvalue equal to I or to g in its represen- tation ofJt,(Q ) (r any prime different from t or N) which is impossible for various reasons. Thus, Ht is a surjective endomorphism on all groups of the exact sequence: o -+J,(Qp) -,J,(Fp) o giving us surjectivity of J~(Qp)[~t]~J~(Fp)[~t]-+o by the snake-lemma. It follows that J~ [Ht] (Fp) =J~(Fp) [~t] (~)- Conclusion of the proof of Proposition (I 6.6) for p # 2. -- Let W denote the Zp-dual of $'a(J~(Fp)) (or, equivalently, the Qp/Zp-dual of J~(F~); cf. w 7). By (I4. II ) W is a free T~-module of rank i. By Claim 3, and (~4.9), W/Ht.W=W/~.W. Therefore ~t. T~ : ~. T~. x 7. Eisensteln primes (p = 2). We now begin to study the case where p = 2 divides n. Our first goal is to prove : Proposition (x 7. I ), -- The Eisenstein prime ~3 . T~ C TV is generated by the elements p and Ht, where I is any good prime different from 2. (a) To help the reader see this, it may be worth discursively reviewing the " brackets " terminology at this point. By definition, the group scheme J~[~t]/s' is the subgroup scheme extension in J/s' of the sub-Gal(Q/Q)- module in J~(Q) consisting in the kernel of~t. Thus, since "0t is an isogeny and J/s, is an abelian scheme, J~['tlt]/s, is a finite flat group scheme (it is, in fact, admissible) whose associated Galois module is J~(Q)[~tl =J~(Qp)[~t]. By J~[~t](Fv) we mean, to be sure, the Fv-valued points of the group scheme J~[~qt]/s'- We have a natural map (a surjectlon in fact) from the Qp-valued points of the finite flat group J~[-tlt ] to the Fp-valued points (reduction to characteristic p) : J~[vlt](l~p) --~ Jg~[~]t] (Fp), the range being naturally contained in J~(Fp)[~t] (the kernel of ~t in J~(Fp)). The asserted equality then follows from the previous discussion. 17 I3o B. MAZUR Discussion. -- The case p = 2 differs from p * 2 in many respects, the major ones being: a) Fontaine's theorem does not apply. b) The equation (q~t--/)(?t--i)=o (for l a good prime) on an 6tale or multi- plicative type admissible group does not imply that the group is constant or of ix-type. c) Where we have dealt in w 16 with the cuspidal subgroup, we must now deal, systematically, with the group D. d) Cp and Zp have a nontrivial intersection (when p = 2) and therefore it will turn out that J~[,~] is larger than C~+Zp. If N=Imod 16 we give no direct construction of J~[~]. We deal with a) by keeping strict control of the 6tale part of our group scheme. We are forced by b) and c) to work with groups which are roughly " twice the size " (in terms of lengths of various filtrations) as in the case p 4= 2. In particular, the pure groups of w 16 are replaced by ,-type groups (see below). We " pay for " d) by not being able to give a complete account of the Galois representation on J~ [~]. Recall the terminology of chapter I, w 3, and especially lemma (3.5): Lemma (i7.2). -- Let M/s, be a multiplicative type admissible group. Let ! be a prime number which is not a quadratic residue mod N (e.g. a good prime) (t4=2) such that (,Ot--g)(r on M. Let MIC M be the "first stage " in the canonical sequence of M (el. chap. I, w 3) (i.e. the largest ~-type subgroup 0fM). Then MI(Q) is the kernel of ~t--t and Ot acts trivially on (M/Mr)(Q,). Proof. -- The first assertion is a repetition of chapter I (3.6). The second assertion is then evident since q~t--I brings M(Q) into the kernel of (~t--t). ,-type groups. We work with a fixed good prime number g 4 = 2, and certain admissible subgroup schemes G/s, c J~[~t]/s, (i.e. in J~ and killed by ~t)- Say that such a group scheme is a ,-type group if it can be expressed as a " push-out " (or " amalgamated direct sum ") of the following form: o > ~ > D ~ Z/2 ~ o o >G O >G >Z/2 >o where: DCJ[~] is the subgroup scheme ofw 12, and G~ is some (admissible) subgroup scheme of multiplicative type, containing the subgroup ixz C D. We also denote the " amalgamated sum" as follows: G=G~ D. 130 MODULAR CURVES AND THE EISENSTEIN IDEAL x3x Since D is fixed, a .-type group is determined by its multiplicative part G O C G, and conversely: G O is the connected component containing the identity of the scheme G/s,. Lemma (x7.3). -- If G is .-type, and: G 0 6t o -+ (/z,) -+ G/z, -~ (G/z,) -+ o is the natural sequence displaying the connected and 6tale parts of Glz,, then: (O/,,) ~ = O z,. Remark. -- In particular, the Q~-rational points of (G/z,) ~ (which is, a priori, only stable under the action of Gal(Q2/Q.~)) is stable under the action of GaI(Q/Q.) as subgroup of G(Q)--G(Q~); here we fix any imbedding QC Q2- Applying (I7.2) to G~ we obtain a subgroup scheme G~176 G o containing ~, such that G~176 is the " first stage " in the canonical sequence for G~ In particular, G~176 2 is a ~-type group and its Galois module is the kernel of q~t--t in the Galois module of G~ Also qo t acts trivially on the Galois module of G~ ~176 Lemma (x7.4). -- G ~176 is a ~-type group. Proof. -- Since the inertia group at N operates trivially on G~176 lemma (i6.5) (where we take B = p~.) assures us that it operates trivially on G ~176 We then apply chapter I (3-1)- The key lemma enabling us to construct .-type groups is the following: Lemma (x7.5). -- Let ~ be a good prime number different from 2. Let GC G'cJ~[~t] be (admissible) subgroups stable under the action of T such that: a) G is a ,-type group. b) G'/G is of order 2. c) 2 kills the gtale part of G~F ,. Then G' is a .-type group. Proof. -- In the calculations of Claims i and 2 below, we deal exclusively with Galois modules. For simplicity we let the symbol of the group-scheme stand for the associated Gal((~/Q.)-module, in the proof of those claims. Thus G' would stand for G'(Q), etc. Claim 1: (?t--I)(G'/D) C (G~176 D) /D ---- G~176 Proof. -- We have a filtration: oC (G~176 D)/DC G/DC G'/D G~176 G~ 131 I32 B. MAZUR Since (q~t--i) annihilates G'/G and leaves D stable, we have that (q~t--Q(G'/D) is in G/D. But note that (q~t--g) : G0/G~176 G~ is injective (i 7.2), and consequently, if (q~t--I)(G'/D) were not contained in G~176 ~ we could not have (~t--t)(?t--i)~--o. Claim 2: (q>t--I) G' C G ~176 Pro@-- By Claim i, (q0t--i) G'CG~176 D. But Vt--g maps G0~ onto ix2, with kernel G ~ (since +_t(g)=I, ~pt--t maps D onto 1~2; el. chap. I (4-3))- Again, since (q~t--t)(q)t--I)=O, Claim 2 follows. Claim 8. -- The extension of group schemes over S': o -+Z[2 -> G'/G ~ -+ G'/G-+o ~plits. Proof. -- By Claim 2, q>t acts trivially on G'/G ~ There are two possibilities: Case I. -- G'/G=Z/2 (as group scheme over S'). Then hypothesis c) insures that the 6tale group scheme G'/G ~ is killed by 2, and since q~e acts trivially on it, it is indeed a product, by chapter I (3-4)- Case H. -- G'/G = g,~. It is also true in this case that G'/G" is killed by 2. The reason is that any exten- sion g of ~ by 7./2 splits over Spec Z~ (the splitting is obtained by showing that d ~~ the connected component of @, must project isomorphically to ~2). Therefore, in particular, 2 kills g(Q~) = d~(Q), and hence it also kills d ~ Again since q~t acts trivially on it, it is a product, by chapter I (5-i). We now show that Case I cannot occur. That is, G'/G ~ cannot be the constant group scheme Z/2 � Z/2. Note first that the Hecke algebra T induces a natural action on G'/G ~ For it leaves G' and G stable by hypothesis. We must show that it leaves G o stable. But by Lemma (i7.3) , the Galois submodule G~ of G(Q) is determinable as the sections which specialize to zero in characteristic 2 (the sections of the connected component (G/z,) ~ and is therefore left stable under the action of T. We follow the proof for p odd, quite closely. For all primes t' # 2, N, the Eichler-Shimura relations assure us that T t, =I-kt' on the constant group scheme G'/G ~ Reducing to F2, one has that T 2 = I ( - i ~- 2) on (G'/G~ again by the Eichler-Shimura relations. We have to check that w+I=O, in order to conclude that :cT' ~:F,: ~,t - (O'/O ~ is in (J/F,)~t[~3]. But since (w-kI)2=o (because w+I is certainly nilpotent on G'CJ~ and (G'/G ~ is an F2-vector group of rank 2) and since w+i annihilates Z/2=G/G ~ it suffices to show that (G:~,)~t=(D/F,) a is not in the image of w+I, which is true by (I3. io). Thus :~, ~v/F,: ~tC (J/s,)a[~] which contradicts (I4.9). Therefore we have: G'/G o ---- Z/2 � ~. 132 MODULAR CURVES AND THE EISENSTEIN IDEAL x33 Defining G '~ to be the kernel of the natural projection of G' onto the first factor Z/2 in the above product, we have that G '~ is a subgroup of G', of multiplicative type, and G'=G'~ is therefore a ,-type group. Q.E.D. Let r~i be an integer. Consider the exact sequences of Gal(Q~/Q2)-modules: o-+J~[2 r, ~t]~ [2', ~qt] (Q2)-+J~ [~t] (F~) where the superscript o denotes the connected component (containing the identity) of a group scheme over Spec Z~. Since Jr[2 r, ~t](Qz)=Jv[2 ~, ~t](Q), this group is (in a fixed way) a Gal(Q/Q,)- module. Let G(r) CJ~[2', ~qt](O_,,z) denote the full inverse image of (Jv[~t](F~))[2] in Jr[ 2~, ~t](Q2). It is clear that G(r) inherits a Gal(Q2/Q2)-module structure. It is not clear that G(r) is stable under Gal(Q/Q). Write G~ ', ~]~ We have: (x7.6) (i) o ~O~ -+G(r) -+ (Jv[~t](F~))[2] and, if r is sufficiently large: (x7.6) (ii) o-+G~ -+G(r) -+ (J~[~qt](F2)) [2] -+o. We formulate two hypotheses: I(r): The subgroup G(r)cJ~[2", ~qt](O_..) is stable under the action of Gal(Q/Q). I~ The subgroup G~ r, ~qt](O_..) is stable under the action of Gal(Q/Q). Lemma (I7.7). -- Hypotheses I(r) and I~ hold for all r>o. The group scheme G(r) is a ,-type group for all r>o. Proof of Lemma (i 7. 7). -- Our inductive proof consists in five steps. Set: G(o)= DC J[~3] (w I2). Step 1. -- For r>o, if I(r) and I(r+I) hold, and if G(r) is a ,-type group, then G(r+I) is a ,-type group. Proof. -- Since the groups G(r)C G(r+i) are both stable under the action of T, we may find a filtration: G(r)=FIoC HtC ... CHjC ... C Ht=G(r+I ) by T,[Gal(Q/O_..)]-submodules I-~ such that the successive quotients Hi/Hi_ I are irre- ducible T~[Gal(Q./Q.)]-modules. Since Hj/I-tj_ 1 is therefore a module over: Tv/ Tv [Gal(O./O__)] and since T~/~3.T~--~F2, t-Ij/Hj_ t is an irreducible Gal(O/Q)-module. Since G(r) is admissible, it follows that HJHy_, is of order two. By upwards induction on j, applying Lemma (i 7.5) to G = I~_ 1, G' = Hi, one obtains that G(r -/I) a ,-type group. 133 134 B. MAZUR Step 2. -- If I(r) holds, and G(r) is a ,-type group, then I~ holds. Proof. -- Apply Lemma (i7.3) with G(r)=G. Step 3. -- I~ =~ I(r+I). Proof. -- If xeJ~ [2 r + 1, ?t] (Q), then (since 2 r. 2x = o) by the defining property of G(r+1), x~G(r+I) if and onlyif 2xeG~ Now, if ~eGal(Q/Q), and xeG(r+I) we must show that ~(x)eG(r4-I). Equivalently, we must show that 2.~(x)eG~ But this is true since (by I~ ~ leaves G~ stable, and a(2x)= 2. ~(x). Step 4. -- If I(r) holds, and G(r) is a ,-type group, then I(r+I) holds and G(r+i) is a ,-type group (r~i). Proof. ~ Combine the first three steps. Step 5. -- Conclusion: Clearly D=G(o) is a ,-type group, and, since: G(I) =J[2, ~t](Q), I(I) holds. By Step I it follows that G(I) is a ,-type group. This allows us to apply Step 4 (inductively) to conclude the proof of Lemma (i7.7). Proposition (i7.8). -- The following groups (of order 2) are equal: D (F~) =J [~3] (F~) =J~ (F~) [~1 =J~ (F2) [2, ~t]. Proof. -- It is only the last equality that is new, but they will all follow if we show that the right-most group is of order 2. By Lemma (I7.7) and the exact sequence (I 7- 6) (ii) for r sufficiently large, we deduce that (J~ [Ht] (F~)) [2] is of order 2 (a). To conclude the proposition, we need that J~[~t](F~)--=J~(F2)[~e] which is true by (i6. io). Proof of Proposition (I 7. I). -- We follow the proof for p odd. By (~4.1 I), the Pontrjagin dual, W, of J~(F,) is free over T~ of rank i. By Proposition (I7.8), 2) w. Therefore ~3 = (~qt, 2). Q.E.D. One has these immediate consequences (I 5. I): (i) T~ is a Gorenstein ring; (x7.9) (ii) J[~]=D; (iii) H~ is free of rank 2 over T~. Proposition (x 7. xo). -- The Eisenstein quotient J-+J factors through J- (cf. w 4)- Proof. -- Let us work over the base Q. (1) The point here is that the 6tale part of a *-type group reduced to characteristic 2 is of order 2. This is all we need. 13d MODULAR CURVES AND THE EISENSTEIN IDEAL I35 The fact that for p odd, the p-Eisenstein quotient factors through J- is fairly evident: the kernel of ~ in J+ is zero, since w acts as -I on J[~3] and as +I on J+ = (i + w) .J = ker(J-+J-). For p=2 (if 2In) we must also show that J+[~3]=o. But by (17.9) J[~]=D, and by (13.1o) Dc~J+=o. x8. Winding homomorphlsms. If R is a commutative ring with unit let R[a] denote the commutative R-algebra I. R| a.R where e is a symbol satisfying the law ~2 = I. If M is a free R[~]-module of rank I, then ~ is an involution on M; forming the (~i)-eigen-subspaces M S C M associated to ~, and the corresponding eigenquotient spaces M +, we have the diagram of exact sequences: M+ (I8. x) o -+ M_ -+ M-+ M + -+o ,,2\ + M- where all four R-modules M~, M + are free of rank I. In fact they may be canonically identified with R and in terms of these canonical identifications, the diagonal homo- morphisms above are " multiplication by 2 ". Lemma (x8.2). -- Let K be a commutative local ring with maximal ideal m. Let M be a free R-module of rank 2 endowed with an (R-linear) involution ~ which is not a scalar modulo m. Then M is free of rank I over R[e]. Proof. -- Let k= R/re. Then M = M/m.M is a 2-dimensional vector space over k on which the involution e does not act as a scalar. In particular, there is an element xeM such that YeM is a generator ofM ask[e]-module. Applying Nakayama's Lemma to M over R, one deduces that x is a generator of M as R [~]-module. Moreover, since the R[~]-homomorphism R[~] 2~ M (e~ e.x) is an isomorphism of R-modules modulo m, and M is a free R-module, it follows that i is an isomorphism. : H-+H be the involution Proposition (x8.3). -- Let H=Ht(X0(N)c , Z) and let induced from complex conjugation of the manifold X0(N)r Then H~ is a free T~[~]- Let ?Ol be any maximal ideal in T such that char k~ 4= 2. module of rank i. 135 x36 B. MAZUR Let ~3 be (any) Eisenstein prime. Then HV is a free T 0 [~]-module of rank 1. Let T~ denote the completion of T with respect to the (full) Eisenstein ideal. Then H~ is a free T~[~]-module of rank 1. Pro@ -- Let 9J~ be a maximal ideal such that p=chark~4=2. By (15.2) H~ = $'a(J~)(13) is free over T~ of rank two. On the other hand, one has a perfect duality: 08.4) (c) � (where V~-- 0 t~p,C Gin). Since ~ acts as --I on $-a(t~)(C ) and since p+2, we have --r=l that H~--~H~m+| ~_ where (18.4) puts H~+ and H~_ in duality. Since (again) p+ 2 it follows that ~ does not act as a scalar modulo 9"J~.T~ and consequently H~ is free over T~[~]. Now let ~ be an Eisenstein prime associated to p. By (16.3) and (I7.9) H~ is free over T 0 of rank 2, and by (16.3) the action ore is evident. Namely, when p4=2 acts as + I on 13 and as --I on Y,. Therefore it does not act as a scalar modulo ~3. T 0 . If p = 2, ~ does not act as a scalar on D (C) (cf. chap. I (4.3)). Therefore Lemma (I 8.2) applies again. Since T~=pHT~, H~=piIIH~, the final assertion follows, as well. Q.E.D. Let J0 denote the complex Lie group underlying the jacobian of Xo(N ), and U its universal covering group. We have an exact sequence of topological groups: o I-I &Jc where the discrete subgroup HC U is identified with H =HI(X0(N)0, Z). Moreover, the tIecke algebra T and complex conjugation cr both operate naturally on the above exact sequence. The Lie group U is isomorphic to H| as real Lie group: The real Lie group J0 is canonically isomorphic to H| (R/Z). Consider the fundamental arc [o, iao] = { ~y ] o <y < ioo } in the extended upper-half plane. We regard the fundamental arc as an oriented topological interval (orientation from ioo to o). The parametrization of X0(N)o by the upper-half plane induces a natural homeomorphic injection: [o, ioo] h-~J0 (h(ioo) = origin). The continuous map h lifts uniquely to a continuous map to the universal covering group: h : [o, ioo]-+U (h(ioo)=origin). Definition. -- Set e=h(o)eU. Call e the winding element. Lemma (I8.6). -- We have 3. e C H+ C U. The winding element e is in H+ | Q.. 136 MODULAR CURVES AND THE EISENSTEIN IDEAL I37 Proof. -- The fundamental arc maps to the real locus of X0(N ) and, from the definition it is clear that n(e) = e = c~/((o) -- (oo)) in Jc- Therefore, since 3. c = o (it.i), it follows that ~.eCH, and since e is fixed under ~, ~.eCH+. Since ne~, ee(I/n).H+. Definition. -- Let: e+ : ~-~H+ be the T-homomorphism ~ ~ ~. e. If a is any ideal in T, let: e+ : 3.T~Ha+ denote, as well, the induced T a homomorphism. If H, is free over T, of rank i, let: e + : ~.Ta-~H + be the Ta-homomorphism defined by: 2. e + (e) = image in H + of e+ (~) (using diagram (i 8. I)). We shall call the homomorphisms e+ and e + winding homomorphisms. The winding homomorphisms are (conveniently normalized) " generalizations " of the winding num- bers of [39]" We shall be especially interested in the winding homomorphism e + for a = 3: e + : .~.T~H +. By means of the theory of modular symbols ([32], [35], [39]) we shall be able to completely determine this homomorphism modulo 3, and deduce a number of impli- cations. As we do this it is of interest to keep track of how little use we shall make of all our previous work. We use only the assertions of Proposition (i8.3) (those having to do with Eisenstein primes). These, in turn, are easy corollaries of the (hard) result: T~ is a Gorenstein ring, for ~ an Eisenstein prime. Lemma (xS. 7)- -- H+/3. H + is a cyclic group of order n. There is a canonical (1) surjection ? : (Z/N)*~ H+/~.H + which identifies H+/~.H + with the Galois group of the Shimura covering ((2.3); cf. w ii): XI(N) (Z/N)* ~" (+i) x (N) t $ H +/3. H + X0(N) , Pro@ -- Since H +/3. H + = H +/~.H + and since H~/~. H + is free of rank I over T~/~.T~ by Proposition (I8.3) and the discussion involving Diagram (i8.i), it follows that H+/~.H + is, indeed, a cyclic abelian group of order n (9-7). (1) To make it canonical, one must make, somewhere, a specific choice of sign. Compare the next footnote and relevant text. 18 i38 B. MAZUR Let 5 p denote the unique quotient of (Z/N)* of order n. Thus 50 is a cyclic group which is canonically the Galois (covering) group of the Shimura covering (2.3)- Since the Shimura covering is unramified (2.3) there is a canonical surjection H-~5:. Since X~(N) --> X0(N ) is defined over Q (and hence over R) this canonical surjection factors, to give a surjection HH+~SC Since (Proposition (ii.7)) the Shimura subgroup is annihilated by 3, this surjection factors to yield a surjection t-I+/3.H+~5 :, which must be an isomorphism, since both domain and range have the same order. Q.E.D. Ifa/b is a fraction where b is an integer relatively prime to N, let {a/b}eH denote the modular symbol [32], [35], ([39], w 6). Proposition (i8.8). -- (Congruence formula for the modular symbol.) Let a, b be integers with b relatively prime to N. Let b denote the image of b in (Z/N)*. Let @(a/b)EH+ /~.H + denote the image of the modular symbol { a/b } in H+/3.H +. Then: r = +/3. H +. (Compare footnote in Section (6. I5) of [32].) Pro@ -- Here we again (as in the Proof of (I I. 7)) make use of Ogg's terminology for the cusps of F(N). ( ~ ) = {P /q~pl( Q,) lp = a mod N ; q = b mod N ; (p, q) = I }. From the definition of the modular symbol, one sees that if (b, a.N)=I, r is that unique element of 5:~ H+/3.H + which sends (the image in Xz(N) of) the cusp (~ to (the image in X2(N) of) thecusp (~). Sincean element c~(Z/N)*acts as the matrix (; o C -1 ) (1) and since (b) = (b) mod FI(N ) provided (b, N)=I, it follows that r is the image ofb -1 in H+/3.H +. Q.E.D. Proposition (18.9). -- (Congruence formula for the winding homomorphism.) Let ~t=-I+t--Tt. Let ~+ :~/~2-->H+/3.H+ be the homomorphism induced from the winding homomorphism e + : 3-+ H + 9 Then: where t is any prime number different from N. Remarks. -- First note that the right-hand side makes sense. For if l = 2, and p=2 divides n, then N- I mod 8. By the quadratic reciprocity theorem ~(Z/N)* (a) We follow Ogg in making this choice. 138 MODULAR CURVES AND THE EISENSTEIN IDEAL [g--i\ is then a quadratic residue (7=x2), and consequently l-T2l?(~-)-----q0(x). In any other case, the 2 in the denominator is harmless. The assertion of (18.9) may be viewed as a congruence formula for numbers of rational points over F t. For example, in the first nontrivial case, N = i i, it was first proved by Serre, and takes the following shape: Let N t denote the number of rational points of the elliptic curve X0(i1 ) over Fe(e,l~ ). Let ~:(Z/~)*-+Z/5 be the homomorphism which sends --3e(Z/II) * to 2 rood 5. Then: Nl~-~--5(e--I) .~(e) mod 25. Proof of (i 8.9)- -- Our proposition follows immediately from the formula: (i +e--Tt).e=-- Y~ {k/e} k m0d l (formula (8), w 6 of [39], compare (5-5) of [32]), together with Proposition (I8.8), (I8.3) , and the definition of e +. Theorem (x8. xo). -- (Local principality of the Eisenstein ideal.) Let p be a prime number dividing n. Let ?(3 be the associated Eisenstein prime. Let E be a prime number different from N. Then ~t is a generator of the ideal ~0=~.T~C T~ if and only if t is a good prime number (with respect to p). The winding homomorphism e + : ~ ~ H~ is an isomorphism of To-modules. Proof. -- Reducing the above winding homomorphism mod ~ one gets the homomorphism ~+ :~o/~-~H~/~o.H ~ and by Proposition (i8.9) , the element ~t maps to a generator of H~/~.H, if and only if (V)is not congruent to o modp and| is not ap-thpowermodN (if we are not in the special case e=p=2). In the case t =p= 2, Proposition (I8.9) assures us that / maps to a generator if and only if e is not a quartic residue mod N. Thus, ~}t maps to a generator if and only if e is a good prime. Since good primes do, indeed, exist, we deduce that e + : ~--->H~ is surjective, by Nakayama's lemma. By counting dimensions over O p, we obtain that: e+| : ~| g;| is an isomorphism. Since ~v is torsion-free as a Zp-module, it follows that e + : 3v--->H~ is an isomorphism. Since H~ is free of rank I over T0, our theorem follows. Remark. -- Except for the " only if" part of the theorem and the assertion concerning the winding homomorphism, the " new " information conveyed by (18. IO) is for p=2. For odd p, it is a curious alternate to the methods of w I 6, for (starting with Corollary 16.3. The Gorenstein property for T0) it enables us to quickly retrieve the results of w 16 in their full strength. IfgJl is a maximal non-Eisenstein prime in T, then the winding element e is naturally 139 ~4 o B. MAZUR contained in I-I~+. Thus if 93l is such that H~+ is free over T~ of rank I (e.g., if char k~ Je 2 ; cf. (I 8.3)) then, choosing some identification between the T~-modules H~+ and T~, e will correspond to some element in T~. The principal ideal e~C T~ generated by this element is independent of the choice made and shall be called the winding ideal associated to 9JL 19. The structure of the algebra T~. Fix p a prime dividing n, and !13 the associated Eisenstein prime. We know ( 18. I o) that if/ is any good prime number, ~t generates the Eisenstein ideal ~C Tin, and ,~=Zp[~qt ]. Let Rt(x)~Zv[x ] be the minimal monic polynomial satisfied by ~t over Zp. Thus T~=Zp[x]/(Rt). Denote by gp the rank of T~ as Zp-module, or equivalently, the degree of Rt(x ). Since T~ is local and ~qt is in ~3, R t is a" distinguished polynomial" (i.e. Rt(x )-x gp modpZp[x]). Since Tm/~t.T ~ ~ Zip f where prlln, the constant term of Re(x ) has p-adic valuation f. Since, if t and t' are two good prime numbers, ~qt and ~t, are associate in the ring T~, the Newton polygons of Rt(x ) and Rr(x ) are equal. One might call the common Newton polygon of Re(x ) for l any good prime number, the Newton polygon of T~ (or, more strictly speaking, of ~). Is there anything general that can be said about the Newton polygon of T~, or even about gp? One has hardly enough numerical data to begin serious speculation about this question. As far as my calculations go (N<25 o) there is only one instance where T~ is not a discrete valuation ring (N = i i3, p = 2) (1). In this case f= 2, gp = 3, and the Newton polygon is the only possible one conforming to this data. There is no practical difficulty in computing the Newton polygon of T~, using (e.g.) the tables of Wada [7o]. Wada gives the characteristic polynomial of T t (call it St(x)) acting on the parabolic modular forms for F0(N ). The most straightforward thing to do is to look for the smallest good prime number g such that Se(i +t) has p-adic valuationf (2). For such a prime number t, Re(x ) is simply the " Weierstrass-prepared part " of St(I +t--x). Proposition (x 9. i ). -- Suppose p [] n (i.e. f= I). Then T~ is a discrete valuation ring, totally ramified over Zp, of ramification index gp. Proof. -- In this case, the maximal ideal ~3=~, and is principal, by (I8. Io). Proposition (I9.2). -- Let p*2, prlIn (f>~). The natural auto-duality 0fJ[p r] restricts to a nondegenerate auto-duality of C~| (the direct sum of the p-primary components (1) As we shall see (chap. III, w 5) if we avail ourselves of certain standard conjectures, this instance is the first of an infinite series of analogous instances (all with p = 2). (3) In practice one does not have to go far to find one, at least when N < ~5 o. 140 MODULAR CURVES AND THE EISENSTEIN IDEAL I4! of the cuspidal and Shimura subgroups) if and only if T~ = Zp (i.e. gp = I). In particular, the element u (end of w 1 I) is a generator of the p-primary component of U if and only if gp = ~. Pro@ -- If T~=Zp, then Cp| and on the latter group the natural auto-duality (I 1. I2) is nondegenerate. Conversely, suppose the natural auto-duality of Jv[p r] restricts to a nondegenerate auto-duality of C~| Then the natural auto- duality ofJ~ [p] would restrict to a non-degenerate pairing of C [p] with E [p]. By (I 8.3) J$[p](Q) is free of rank 2 over T~/p.T~, which by the above discussion is isomorphic to Fp[~t ] where ~t satisfies the relation ~=o (over F:o). One sees immediately that J~[p, ~qt](Q) (which is the kernel of ~t in J~[p](~).)) is the image of ~g~-~. If gp--1>o, the relation (~gp-~x,y)=(x, ~gp-ty) gives us that the natural auto-duality restricts to zero on C[p]| contrary to assumption. Remark. -- The only instances (N<25 o, p:t:2) where gp>i are: N=31, IO3, 127, I31 , 18t, I99 and 211. III. -- ARITHMETIC APPLICATIONS I. Torsion points. Lemma (x. x). -- Let AIQ be any quotient abelian variety of JtQ. Let p be a prime number dividing the order of the torsion subgroup A(Q)tor ~ of the Mordell-Weil group of A/Q. Then p divides n. If AIQ is a quotient abelian variety of J/Q on which T operates in a manner compatible with its action on JIQ, then A(Q)tor s is annihilated by a power of the Eisenstein ideal .~. Let A/Q be a simple (equivalently: C-simple or Q-simple; of. chap. II (IO. i)) quotient abelian variety of J/Q such that the prime number p divides the order of A(Q)tor ~. Then A/Q is a quotient of the p-Eisenstein quotient J(P) (Io.4). Proof. -- Start with the first assertion. Consider the surjective morphism of associated p-divisible groups over Q, Jp-+Ap. If r is large enough, the image of the finite group scheme J[p*] in Ap contains A[p]. Find a Jordan-HSlder filtration of J[pr], as T [Gal(Q/Q.)]-module. Since, by hypothesis, there is a point of A [p], defined over Q, some successive quotient of the Jordan-YI61der filtration must have trivial Gal(Q/Q)- action. By (chap. II (14. i)), this subquotient of J[p r] belongs to (1) an Eisenstein prime ~, necessarily associated to p. Therefore p divides n (chap. II (9-7)). The second assertion is similar, but easier. Every successive quotient of a Jordan- H61der filtration of the T[Gal(Q/Q)] module A(Q)to, ~ must belong to some Eisenstein prime, by chapter II (14. I). By the Mordell-Weil theorem, A(Q)tor s is a finite group, and is therefore annihilated by some (finite) power of .~. (2) In the terminology of w 14. 141 r42 B. MAZUR The third assertion depends upon the one-to-one correspondence of chapter II (lO. I) where isogeny classes of simple abelian variety factors of J are " identified " with irreducible components of Spec T. Since p divides the order of A(Q)tor,, by what we have already shown, the irreducible component of Spec T corresponding to the (isogeny class of the) simple abelian variety quotient A must contain the Eisenstein prime ~3eSpecT associated to p. Since jt,/=j/g~.j where y~=0~ ~ (chap. II (lO.4)) it follows that, up to isogeny, ~(~/is a product of those simple factors corresponding to irreducible components of Spec(T) containing ~3. Since J(p) is the quotient of J by a connected subgroup scheme, it follows that J--~A factors through J(P). Theorem (x.at). -- (Conjecture of Ogg): c =J(Q)to,. (Any rational torsion point of J is a multiple of c-~r Proof. -- Set M =J(Q) (the Mordell-Weil group of J) and recall the retraction p : M-+CC M of chapter II, w ii, giving rise to the direct product decomposition M=M~215 (chap. U (11.4)). It follows that C is a direct factor of Mtors=J(O~)t0rs. By (i. i) it suffices to show M ~ [~3] = o for all Eisenstein primes ~. But this follows from the inclusion C[~]�176 and the determination of J[~3] (chap. II (16.3) or (for p4~2) (14.1o)). Theorem (x .3). -- The Shimura subgroup X is the maximal ~-type subgroup in J/s,. Proof. -- The sum of two (finite) ~-type subgroups of J is again a (finite) ~z-type group. It suffices to show that if Y.' is a (finite) ~-type subgroup of J containing N, then Y/= N. We first show that Y~ is a direct summand in Z'. Using the universal property of the N6ron model J/s (and the fact that the inertia group at N operates trivially in the Galois module associated to ~') one has that the subgroup scheme extension NfsCJ/s is a finite flat (~-type) subgroup scheme. Consider specialization to characteristic N, where one obtains a diagram: ~' ..... " J /F. = J /~ � l~ I pr~ where - denotes specialization, and where the bottom horizontal map is an isomorphism, by chapter II (I 1.9). It follows that ~ is a direct summand of ~' and one easily obtains from this that 142 MODULAR CURVES AND THE EISENSTEIN IDEAL I43 5; is a direct summand in E'. Write E'= 5;| where B is a ~z-type group. Applying chapter II (I 4. I), one has that every successive quotient of a Jordan-H61der filtration of B belongs to an Eisenstein prime. It suffices to show that B [~] =o for all Eisenstein primes. But 5;[~]| cJ[~3], and B[~3] must therefore vanish, by chapter II (16.3). Remarks. -- Theorem (i.3) was also conjectured by Ogg [48]. Although (1.2) and (I-3) have the appearance of being of comparable difficulty, there are notable differences between them. Ignoring 2-torsion, Theorem (I.3) is far easier than Theorem (1.2) (it uses only chap. II (I 4. Io), and does not depend on the Gorenstein condition). In dealing with the 2-torsion subgroup of J(Q), however, one must control subgroup schemes of J/s isomorphic to Z/2 as well as subgroup schemes isomorphic to ~ (since either will contribute to a point of order 2 in J(Q)). Consequently, this requires the full strength of chapter II (16.3), e.g., all of chapter II, w 17. Corollary (I. 4)- -- The natural maps induce isomorphisms of torsion subgroups of Mordell- Weil groups: C =J(Q)tor~-+J-(Q),or~ -+J(Q)tors (cf. chap. II, w 1o). Proof. --By (i. i) one has that J-(Q)tor 8 and J(Q)tor8 are annihilated by a power of the Eisenstein ideal 3. We shall show that the natural maps: J~ ~J~ -,J~ are isomorphisms. The map J~-+J~ is an isomorphism since ((i-}-w).J)[~]=o (chap. II (I 7. lO)). The map J~-+J~ is an isomorphism since its kernel is T~.J~ and the supports of T/~ and of T~=rl~ r are disjoint. I* Corollary (I.5). -- The MordeU-Wdl group of J+=(l+W).J is torsionfree. 2. Points of complex multlpHcation. In this section we examine a set of points ofXo(N ) defined over fields of particularly low degree. A somewhat larger class has been studied by Birch and Stephens (called Heegner points). Fix N (a prime number > 5, as usual) and work over the field of complex numbers. If E/c is an elliptic curve, an N-isogeny by complex multiplication r: :E-+E is an endo- morphism such that ker r~ is of order N. Thus, ~ is a complex multiplication of E such that if R is the ring of complex multiplications of E, ~. ~ = u. N where u is a unit in R. Let aE, . =j(E, ker ~)eX0(N ) (C), which we will refer to as a point of complex multiplication. If a = a~.,,, is a point of complex multiplication, set: R(a)=the ring of endomorphisms of E. The ring R(a) is an order in a quadratic imaginary field k(a) which may be viewed as naturally imbedded in C (since 1~3 t44 B. MAZUR End(Tan(E/c))=C and R(a) acts faithfully on Tan(E/c), the tangent space of E/o ). A(a) = a sublattice of C such that C/A(a)~ E. It is well known (cf. [29] ) that A(a) is a locally free R(a)-module of rank I. ~(a)==. It is an element in R(a) of norm N. Given a triple (R, A, 7:) where R C C is an order in a quadratic imaginary field, A is a locally free R-module of rank I, taken up to isomorphism, and ~ is an element of R of norm N, given up to multiplication by a unit in R, then we may construct a unique point of complex multiplication a--=acR.a.~)eX0(N)(C ) such that R=R(a), A=A(a), and n--~n(a). Let dc X0(N)(C) denote the set of all points of complex multiplication. It is easy enough to produce elements of d. Consider equations: N = r 2 + D.s 2 where D is a positive integer not necessarily square-free and r, s are either both positive integers, or both positive half-integers (1). If D= I, suppose r>s. Let r~=r and let R be an order in Q('V/~) containing =. Finally let A be a locally free R-module of rank I (e.g., R itself). The points of complex multiplication are defined over algebraic number fields which are studied in detail by the classical theory of complex multiplication (cf. [29] , chap. IO, w 3, theorem 5 and remarks I, 2 following it). We give a synopsis of this theory below: (2.x) Let R be an order in a quadratic imaginary field k_C and (7~)CR a principal ideal of norm N. Let A 1, ..., Ah(R) run through a system of representatives of isomorphism classes oflocally free R-modules of rank i. Set ai-----a(R, Ai,,~)ed. Then the points al,...,%R)eXo(N)(C) are rational over Q-~k and are a full set of conjugates over k. Let G denote the quotient of Gal(k/k) which acts faithfully on the above set of conjugate points. There is an isomorphism (a~A~) of G onto H(R), the group of isomorphism classes of locally free R-modules of rank I, such that if aeG, then a(al)=a o where A1R(~Ao=A ~. The group G cuts out that ray class field L of k whose conductor is the conductor of R. The field extension L/Q is Galois, with Galois group G. We may write L=k(j(R)) C C in which case the real subfield L + =Lc~R is given by L + =O(j(R)). Let ? denote the nontrivial element of GaI(L/L +) =Gal(k/Q). Then G is a semi-direct product of G and the group { i, ?} where the action of ? is given by pgp-l=g-1 for geG. Thus G is a dihedral extension of G. One has: 9 a(R,A, ~ ) ~ a(R,A-~,~ ) 9 (1) H. Lenstra and P. Van Emde Boas have tables of the smallest such D for a given N < 50o,ooo. 144 MODULAR CURVES AND THE EISENSTEIN IDEAL x45 The action of the canonical involution w on d is easily determined: (2.3) w. a(R,A,~ ) ~--- a(R,A, ~ / . Let a~X0(N)(C ) be a fixed point of w. Then a is represented by an isogeny t~ E-~ E' which is isomorphic to E -~ E (its dual). It follows that E': E and consequently the isogeny must be a complex multiplication E-~E and ~:2=u.N where u is a unit in R(a). Multiplying ~ by a unit in R(a), if necessary, we may suppose that ~ =~r N. Consequently, R is either Z['v/~N] or Z[I_] [r ,~-~N / 1_, where the latter case may [ .: j occur only if N - -- I mod 4. Using classical facts concerning the class numbers of the orders Z[A/~] and Z - ([28], chap. 8, w I, th. 7) we may give the following description of the fixed point set of w. Let h(N) be the class number of Q,(X/~-N). If N = I mod 4, then the fixed point set of w consists in one Q:conjugacy class of h(N) points. If N-----I mod 4 it consists in two distinct Q-conjugacy classes, the first containing h(N) points and the second containing h(N) or 3h(N) points according as N -= -- i mod 8 or N-3mod8. Proposition (2.4). -- Let a be a point of complex multiplication and let a+eX0(N)+(C) be its image in X0(N ) * = X0(N )/w. Then a + is defined over Q, if and only if the class number of R(a) is i. Proof. -- This follows immediately from (2. I) and (2.3)- Such points a+eX + are examples of rational noncuspidal points. It is natural to refer to them as points of class number one. One obtains a point a + of class number one for each order R (in a quadratic imaginary field) of class number one, in which N splits or ramifies. Note that if N splits or ramifies in any one of the 9 quadratic imaginary fields of class number one, there are some points of class number one on the associated X +. This is the case, for example, for all prime numbers N<7ooo except for N=3167, as was communicated to me by H. Lenstra and P. Van Emde Boas. The Dirichlet density of primes N whose associated X + possesses no point of class number I is I ]512. What further noncuspidal rational points does the curve X + possess ? This diophantine question (when the genus g+>o) is extremely interesting, since no known method appears to be applicable to it, for any value of N. In the first nontrivial case (N =67) the genus of X + is ~. A. Brumer has obtained its hyperelliptic representation, and has begun a numerical study. When h(N)=i, the description of the fixed point set of w given above shows that there is a (unique) rational point aeX0(N)(Q, ) fixed under w. 19 x46 B. MAZUR Proposition (2.5) (compare [48]). -- Let N----II, i9, 43, 67, or I63. Then Xo(N ) possesses a rational point fixed under the action of w. Moreover (when g=genus X0(N):>o ) these are all the points of complex multiplication in Xo(N ) (C) which are rational. Recall that J+=(i+w).JcJ may be identified with the jacobian of X0(N) + (cf. chap. II, proof of (I3.8)). We shall produce some rational points in J+. If R C C is a fixed order in a quadratic imaginary number field such that the ideal generated by N splits into a product of conjugate principal ideals: (N) be the linear equivalence class containing the divisor: let a+~J+ :c a+ A,--h(R). (oo) A~HCR) ' ' ' where a(R ,+ A) is the common image of a(R,A,r:) and a(R,A,~ ) in X +, ooEX + is the unique cusp and h(R) is the order of H(R). By (2. i-3) a+ is defined over Q, and therefore represents an element in the Mordell- Weil group of J+. To study these points we use a modification of an elegant trick due to Ogg: [49]. Lemma (2.6). -- Let d be an integer. If the dimension of H~ ~(d.oo)) is :>I, then d<N/96. Proof. -- Suppose d is as in the assertion above. Using (chap. II, w IO), J+/s, is an abelian scheme. By ([9], VI, 6.7) one sees easily that X0(N) + has a smooth model over S' (which we call X~s, ). Consider the base change Spec F4-+S'. Using the upper-semi-continuity property (EGA III (7.7.5), I) one obtains that the dimension over F 4 of H~ /F,, 0(d. oo)) is also :>I. Thus there is a morphism f: X~,-+P~p, of degree d, such that the inverse image of the point oo of p1 is the divisor d. oo of X +. Composingfwith the projection X-+X +, we obtain a map g : X/l, ' -+P~F, of degree 2d such that the inverse image of the point oo of p1 consist in the cusps. This gives us the upper bound 8d for the number of rational (noncuspidal) F4-valued points of X0(N ). But, as Ogg remarked [49], all the supersingular points of X0(N)/~, are rational over F 4 and there are more than N/I2 of them. Corollary (2.7). -- If R is an order in a quadratic imaginary field such that the ideal generated by N splits into a product of conjugate principal ideals, and such that h(R)<N/96, then a + is a point of infinite order in the Mordell-Weil group of J+. Proof. -- By (I.5) , the Mordell-Weil group of J+ is torsion-free. Therefore it suffices to prove that a + ~eo. Suppose a + =o. Then there would be a function f 146 MODULAR CURVES AND THE EISENSTEIN IDEAL on X~ whose divisor of poles is h(R).(oo). By (2.6) h(R)>N/96 contrary to assumption. Proposition (2.8). -- Suppose g+>o (which is the case for all N>73 , as well as N=37, 43, 53, 61, 67). Then the MordeU-Weil group 0f J+ is a torsion-free group of infinite order (i.e. of positive rank). Pro@ -- Write 4N=a2+Db ~" with a, b integers, D>o and a s largest possible. One obtains N = ~.~ with ~ in R, the ring of integers of O~(~v/-~ D) and if A is the discriminant of R, then I AI<4v/N. By a standard upper estimate for the class number h(R), we have h(R)~(I/3 ) [AII/~.loN[A[<(I/3)NI/41og(IaN ) if [51> 4. A calculation shows that (I/3).N1/41og(I6N)<N/96 when N1/~>7, or N>24oi. Thus by (2.7) a~ is a point of infinite order when N>24oi. But by the calculations of Lenstra and Van Erode Boas, X0(N) + possesses a point of class number i (hence defined over Q (2.2)) for all N<3167 and therefore (2.8) follows. Remarks. -- I. Using the estimates in the proof above one may show that if N is sufficiently large, each of the points a~eX0(N) + is of infinite order. 2. The above theorem depends on the fact that J+(Q) is torsion-free, which, in turn, depends on the full strength of chapter II, w 17. It is significantly easier to show that 2 .J+ (Q) is torsion:free (for one has far less to do with Eisenstein primes associated to 2). If one wishes to obtain the above proposition using only this weaker fact, one must prove that for same R, 2.a ++o. The estimates give this for R as in the proof above, provided N< 7ooo. We may then use the calculations of Lenstra and Van Emde Boas quoted above to reduce considerations to the one case: N=3167. But, quoting their tables, 3167=562+3I.I ~ and Q(%/~----3 I) has class number 3. For: the estimates above enable us to conclude that 2.a +4:o. 3. Let v+=J+(Q)| which we regard as a T+| module, where: T + =T/(I -- w)T. We have shown that V + is a Q-vector space of positive dimension if g+~o. Let V + Vo.m.) be the sub-T +| of V + module generated by the point a + where R is (resp. + the ring of integers in Q,(V'~) (resp. by all points of complex multiplication). Consideration of Dirichlet L-series and the Birch Swinnerton-Dyer conjectures might lead one to suspect that V + will play a significant role in studies of the MordeU-Weil group of J+. It is tempting to hope that V + is always a free T+| module of rank i. Numerical evidence is too slim to make any conjectures yet, but Atkin has recently produced some interesting tables which bear on the problem. 147 t48 B. MAZUR 3" The MordeU-Weil group of J. The object of this section is to prove Theorem (3. 9 ). -- The natural projection J-+'J induces an isomorphism of the cuspidal subgroup C onto the Mordell-Weil group J(Q). We shall also obtain complementary information concerning a part of the Shafarevich-Tate group of J. Our method will be to use " geometric descent " together with much of the information we have accumulated up to this point. Let ~ be an Eisenstein prime and J~ the connected component containing the identity (which differs fromJ only in its fibre at N). Let J~ m] be the kernel ofp m, and j0[pm]~ its ?(3-component (which is the image of J~ m] under the idempotent a~ for the Eisenstein ideal, as discussed in chap. II, w 7)- Lemma (3.2). --ja[p,~]~ is an admissible group (chap. I, w i(f)) and when m varies, the order of Hl(S,J~ ) remains bounded. Pro@ -- It is admissible as can be easily seen by chapter II (I4. I ). Since H~176 is a subgroup of the torsion part of the Mordell-Weil group of j, it is a finite group which has bounded order as m varies. Thus, to prove the lemma, it suffices to show that h 1- h ~ has bounded order. But by chapter I, Prop. (I. 7) it suffices, then, to prove that ~(j0[pm]~)_e,(j0[p,~]~] has bounded order. This is done by showing: (a) a(J~ =m.g~ + O(I) (b) cr (j0 [pm] ~) = m.g~ + O( I ) where g~ is the rank of T~ over Zp. Proof of (a). -- Letting J~ denote the p-divisible group associated to J over S, and J~ its ~3-component (i.e. the image of the idempotent ,~) then J[p'~]~=J~[p'~] and j0[pm]~=j~[p,~] where the superscript o denotes, as usual, the inverse image of j0. We now make use of the results and terminology of chapter I, w 8. Consider, in particular, the exact sequence chapter I (8.2): o --> Wa(J~(fN) ) ---> g'a(J~(QN)) --> A~ -> o where (8.3) A~ is a Tv-module " of rank I " (i.e. it contains a free T~-module of rank i as a subgroup of finite index). One checks that: S (J~ [p~]) = log, (order (A~/pm A;) ) + 0 ( I ) where v denotes Zp-dua]. Since A~ is also a T~-module of rank 1, we have: S(J~ [p'/]) = logp(order T~/p"T~) + O(I). Proof of (b). -- This follows the same lines as (a) above. One need only note that ~(J~176176 and since ~ is an 148 MODULAR CURVES AND THE EISENSTEIN IDEAL I49 ordinary prime, we have the exact sequence of chapter II, w 4 and chapter II, Prop- osition (8.5). Lemma (3.3). -- Let M=j(Q.) be the Mordell-Weil group of j, regarded as T-module. Then T~ | M and III~ (the ~3,component of the Shafarevich- Tare group III 0f J) are finite groups. Proof. -- Set M~ I-I~ j0) and note that 1V[ = I-I~ J). The quotient M/M ~ is finite. Therefore to show that T~| is finite it suffices to show that T~| M~ is finite. The long exact sequence of cohomology associated to the exact sequence of fppf sheaves over S: pra o j0[pm] JTs J?s o yields: o ~ M~ '~. g ~ ~ IF(S, j0[pm]) _+ IF(S, j0) [pm] ~ o and, by passage to the limit as m tends to m, using the maps: o > j0[p,,] > j0 pm j0 > > 0 o >j0[p,~+~] >j0,m+~j0 > ;) 0 we obtain an exact sequence of Tp-modules: o -~ O_..,/Z,N M ~ ~ lim IF(S, j0[pm]) ~ Hi(S, j0)p __~ o where the subscript p on the right means p-primary component. Passing to U-component (by applying the idempotent ~) we obtain: o ---> T~NTp (Q~/Zp@ g ~ ---> lim H~(S, jo [pm] ~) __> tp(S, jo)v ___> o. rn > Since the middle group is finite, by Lemma (3.2), the two flanking groups are. Since M ~ is a finitely generated group (by the theorem of Mordell-Weil), finiteness of TV|174 ~ implies finiteness of T~QTpM ~ To see that III~ is finite we use that (working modulo the category of groups whose order is a power of two) III may be identified with the image of HI(S, j0) in Hi(S, J) (Appendix of [34]), and the 2-primary component of IlI is a subgroup of the 2-primary component of this image. Finiteness of III~ then follows from finiteness of Hi(S, j0)v. To use (3.3) conveniently, we make a digression and recall the terminology of chapter II, w I o. Let a C T be any ideal of finite index in T, Yta) = N a r, T (at --~T/',((a) (so T ("/maps injectively to the completion T,) and j(,/=J/Y(,t'J, the quotient associated toJ (chap. II, w io). Let v=j(Q)| as T| module, and V("/=J("I(Q,)NQ, as T(")| 149 t5o B. MAZUR Lemma (3-4)-- V~aI=V/Y(,)-V=TI")| V. Proof. -- On the category of abelian varieties over Q, the functor A ~ A(Q)N Q is exact since A(Q) is finitely generated and I-P(Gal(Q/Q), A) is a torsion group, for all A in the category. The 1emma then follows by applying this exact functor to the diagram: o ,~,(,).J ,j ,jc~ ,o (~, ..., ~t) J�215215 where el, 9 9 et is a system of generators of the ideal a. Corollary (3-5). --If TaNTV=o, then the MordeU-Weil group of J(") is finite. Proof. -- Let W be the torsion-free quotient of the Mordell-Weil group J(Q.). Thus W is a free Z-module of finite rank and gives rise to a coherent sheaf over Spec T. By hypothesis, the support of T~| contains no irreducible component of Spec T,. Since the support of a meets every irreducible component of Spec T (al it follows that the support of W contains no irreducible component of Spec T ~al. The support of W then meets Spec T Cal in a finite union of closed points and (3.5) follows from Lemma (3.4). Proof of Theorem (3.1). -- Applying (3.3) for all Eisenstein primes we obtain that T~| T M is finite, where ,~ denotes the Eisenstein ideal. It then follows from (3.5) that J(Q) is finite. The theorem follows from Corollary (I .4). Proposition (3.6). -- Let ~3 be an Eisenstein prime associated to an odd prime number p. Then III~ = o. Pro@ -- We shall perform a more delicate descent to establish this. Let t be a good prime number and ~ =~t (using the terminology of chap. II, w i6). Then is an isogeny (cf. Proof of (I 6. i o)) ; we consider: (3.7) o --~ ker ~q ~-~ J ~ j -~ coker ~ -~ o as an exact sequence offppf sheaves of T-modules over S. Let A denote a finite set of points in Spec T, not containing ~, but containing all other points in the support of the T-modules ker ~(Q) and coker ~(Q). We shall work in the category of T-modules, modulo the category of T-modules whose supports lie in A (modulo A). By chapter II (16.6) and (16.4) it follows that: ker ~-Cp| modulo A 150 MODULAR CURVES AND THE EISENSTEIN IDEAL x5x and cok ~q is a skyscraper sheaf concentrated in characteristic N, whose stalk in charac- teristic N is isomorphic to the T-module Cp, modulo A. Since Gp is a constant group over S and Zp is a ix-type group, we have (e.g., chap. I (I. 7)) that Hi(S, Cp@Y,p)----o, and therefore Ha(S, ker ~q) =o modulo A. One obtains then the following exact sequence modulo A of fppf cohomology from the exact sequence (3.7) : o -+ Cp -+ H~ J) -+ n H 0 (S, J) ~ H~ coker "~). Since t-I~ coker ~q) = Cp modulo A, the above exact sequence shows that i is surjective modulo A. It also shows that ~ is automorphism, modulo A, of the torsion-free quotient of the Mordell-Weil group, which can be used as an alternative to the proof of Theorem (3-i), at least as it concerns odd Eisenstein primes. Reconsidering the exact sequence (3-7), surjectivity (modulo A) of the mapping i, gives that: : Hi(s, j) Hi(S, j) is injective, modulo A. Since III is a submodule of Ha(S, J), multiplication by ~ is also injective modulo A on HI, which establishes our proposition. Combining this with recent results of Brumer and Kramer [4] we may obtain: Proposition (3.8). -- Let N<25o. The natural map J~J- induces an isomorphism of C onto the Mordell- Weil group J- (Q) except possibly in cases N = ~ 5 i, 199 and 227. Proof. -- From the table of the introduction, one sees that for N<25 o, J-=J except for the following values of N: N=67, lO 9 , 139, I5I, I79, I99, 227 and when N = 67, I o9, 139, 179, J- differs from ~ by an elliptic curve factor. Brumer and Kramer have shown (1) that these elliptic curve factors have finite MordeU-Weil groups over Q. It then follows that J-(Q) is finite, using (3. I), for all values of N considered in Proposition (3.8). The assertion then follows from (i.4). Recently, Atkin communicated to me that J- is a simple abelian variety (and hence equal to J) for N=383, 419, 479, 491, and consequently (3.8) holds for these values of N, as well. 4" Rational points on Xo(N ). Theorem (4.i). -- Let N:~(2, 3), 5, 7 and x 3 (i.e. thegenus 0fX0(N ) is >o). Then Xo(N) (Q) isfinite. Pro@ -- Work over Q, and consider the projection X0(N)~ J defined by x~image(x--oo) in J. Since ~" is nontrivial and the image of X= Xo(N ) generates (1) See their forthcoming publication [4]. 151 ~52 B. MAZUR as a group variety, if.~ is the image ofX m J, then X/Q is a curve, and X--~X is a finite morphism. Since .X(Q) Cj(Q)=C (3. I), X(Q) is a finite set and therefore X(Q) is also finite. To be sure, we have little control over this set if we know nothing concerning the structure of the finite morphism ~. What is its degree? What are the singular points of the image? Remark. -- It is a theorem of Manin ([3I], [65] ) that for every number field K and integer re>i, there is an integer e(m, K) such that Xo(m~)(K) is finite for all e~_e(m, K), but no effective bound for e(m, K) is obtained. I understand that the Russian mathematician Berkovich has recently obtained such effective bounds using the techniques of this paper, and in particular the techniques of the proof of (4. I). To analyze the finite set X(Q) we make use of the retraction p :J(O)-+Z/n of chapter II, w I I. Proposition (4.2). -- If xeX0(N)(Q), the element p(x) of Z/n is equal to one of the following values: OOT I (4" 2) i) I/2 (possible only /f N - -- I mod 4) I/3 or 2/3 (possible only /f N-- --I mod 3). Remarks. -- i. If N=--- i mod 3, the integer n is not divisible by 3 and t/3 has a sense in Z/n; similarly I/2 has a sense in Z/n when n is odd, which is the case if N----I mod4. 2. p(o)=i and p(oo)=o. Proof of (4.2). -- The point x extends (by Zariski closure) to a section of X0(N)/s . This section must lie in the smooth locus of X0(N)-~S and hence, if ~is its pullback to Spec(F~), ~ must lie in exactly one irreducible component of the fiber X0(N)y N (see diagram I of chap. II, w I). Thus, Y lies on one of these: Z'or Z (4.2) ii) E (possible only if N- -- i mod 4) G or F (possible only if N----I mod 3). But the natural map X o~,//~ r~s-,o,th ~ J/F~ -+ C = Z/n sends the five components listed in (4.2) ii) to the corresponding values listed in (4.2) i) as follows from the table of the appendix. Let J_=(i-w)JCJ. One obtains a map: r : X0(N)--,J_ by the rule x~cl((x)--(wx)). If xeX0(N)(Q), set X(x)=p(r(x))eZ/n. Corollary (4.3)- -- One has X(x)=~=I, o, or j=I/3. Pro@ -- This follows from (4.2) and the fact that X(x)=2f~(x)--I. 152 MODULAR CURVES AND THE EISENSTEIN IDEAL I53 Corollary (4-4).- Suppose J_(Q.) is finite. Then: a) One has J_(Q.)=C. b) For all x~X0(N)(Q) , one has r(x)=X(x).c with ~(x)=-}-I,o, or:~:x/3. Pro@ -- Assertion a) comes from the fact that C is contained in J_ (Q,) and it is the torsion subgroup of J(Q.) (i .2). Assertion b) follows from (4.3) and the fact that p is the identity on C. Remark.- By (3.8) and the remarks after it, (4-4) applies to at least these values: N<25 o, with the possible exceptions of N=I5I , 199, 227 and N=383, 419, 479, 491. The next proposition is due mainly to Ogg and includes work of Parry and of Brumer. Proposition ('t-5)- -- a) If N>23 and N=37 then the morphism r : X0(N)---> J_ is injective off the locus of fixed points of w. b) IfJ_(Q) is finite, and xeX0(N)(O..), one has: X(x)=:~I/3 ~N=II, 17 X(x)= o =>N=iI, I9,43,67, orr63. Discussion of the proposition. -- The following is Ogg's proof of a). Suppose x,y~X0(N)(C ) such that r(x)=r(y), x4=y, and x is not fixed under w. Write z=w(y) and we have: x + z -- w(x) + w(z) where - denotes linear equivalence on Xo(N ). Since x+z is not invariant under w, it follows that Xo(N ) is a hyperelliptic curve, and moreover, the involution w is not the hyperelliptic involution. But Ogg ([38], theorem i) has determined that N = 37 is the only value of N such that X0(N ) admits such a description. As for b), let xeX0(N )(Q) be a point such that ;~(x)=-E1/3. Then N=--I rood 3, and (replacing x by w(x) if necessary) we may assume: sx + (oo) = sw(x) + (o). By an elegant argument (end of [49]) Ogg shows that n<24 o. We recall the argument and sharpen this upper estimate a bit. Let - denote specialization of a section over S to Spec F 4. Then we have: yielding a function f on Xo(N)/F, such that the inverse image of the point ooEPI(F4) is the divisor 3~+(~) and the inverse image of oePl(F4) is 3w(s Since the points of X0(N)(F4) different from ~, w(~), ~, g must lie in the fibers off above the 3 points of PI(F4) different from oo and o, we have: X0(N ) (F4)< - ~6. 20 154 B. MAZUR But Ogg has constructed [49] at least N/I2 noncuspidal rational points in Xo(N)(F4) , so: N/I2+2<I6 or N<__ I68. Let us now consider an argument which helps to eliminate many low values of N. Findfa function from X0(N)/Q to P~Q whose divisor is 3 x + oo-- 3wx--o. Definef w by f'~ Since f.fw has neither zeroes nor poles, it is a constant e. Define an involution w : I~I-+P 1 by the formula y~e[y. We obtain a commutative square: X ~> X pl w~ p1 and consequently f induces a rational map on quotients by w: X "> X/w = X + pt '~> pt/w = pl+ The double covering pa___>pl+ has precisely 2 ramification points y=+v%. Also, the mapf + is of degree 4- Consequently, the double covering ~ : X-+X + can have at most 2.4=8 ramification points. That is, the number of fixed points of w is ~8. +< This condition is easily shown to be equivalent to the condition g---g _3, by the Hurwitz formula applied to the covering X-+X +, and one checks (e.g., consult the table of introduction) that those N such that a) N~I68, b) N-----I mod3, c) g---g+<__3, are: N=11, 17, 23, 29, 4 I, 53, I13, and 137. When N----I I, I7, Xo(N ) is of genus I, and there is a (unique) point xeX0(N)(Q) such that ),(x) = i/3. When N----23, 29, 41, X0(N) + is of genus o, and Ogg has special arguments to show that there are no rational points such that X(x)=I/3 [49]. The remaining three cases have been ruled out by work of W. Parry and A. Brumer. Finally, note that if xeX0(N)(C), X(x)=o-z~x is a fixed point of w, and so the final assertion of (4- 5) follows from (2.2) and the solution of the class number one problem (Heegner-Baker-Stark). Remarks. -- i. By (3.8) and (4.5) we have determined all the rational points on all curves X0(N ) for N<25o, with the exception of N=I5I , I99 and 227. 154 MODULAR CURVES AND THE EISENSTEIN IDEAL I55 2. (Fields of low odd degree.) If J_(Q) is finite, Ogg's trick has strong implications concerning rational points of Xo(N ) in the totality of fields of a given degree. Brumer has some computations for degree 2, and we shall give a fragmentary result for degree 3- By a point on X0(N ) of degree d we mean a point of Xo(N ) (Q) defined over any extension field of degree d over Q. Proposition (4.6). -- If N=383, 419, 479, or 491, then Xo(N ) has only a finite number of cubic points (points of degree 3)- Proof. -- Let d be any odd positive number. Let K d be the set of all Q;conjugacy classes of points of degree d in X0(N)(Q) not containing a fixed point of w. Define a mapping t :K~--->J_(Q.) by: ,:~cl( Z x- X w(x)). xG~ xGK Lemma ('t-7)- -- If (d is an odd positive number, and) N>i2o.d, then : : K s -~J_(Q) h injective. Pro@ -- Suppose :(K) = ~(K') where K ~:K'. Writing K"= w(K') we get a relation: (4.8) Z x+ Z y- Z w(x)+ Z w(y). x@K y~" xG~ YGK" The above linear equivalence cannot be an identity of divisors, for by hypothesis w does not interchange the conjugacy classes 1< and ~", nor does w stabilize either co~ugacy class. The reason for the latter assertion is that an involution of a finite set of odd cardinality must have a fixed point and the conjugacy classes containing fixed points of w are excluded from K s. Thus there is a nonconstant function f on X0(N ) whose divisor of zeroes is the left-hand side of (4.8) and whose divisor of poles is the right-hand side. Letting D be the Cartier divisor in X0(N)/s, obtained by taking the closure of the right-hand side of (4-8), we have that H~ ~(D)) is of dimension > ~, using (EGA III (7- 7- 5), I) as in the proof of Lemma (2.6). It follows that there is a mapping f: X0(N)/F,---> la~F, of degree 2dand therefore io.d is an upper bound for the cardinality of Xo(N)(Fd). Since this cardinality is greater than N/I2, the lemma follows. The proposition then follows by taking dz3, and using the remark after (4-4)- It is interesting to consider the problem of showing finiteness of X0(m ) (O) when genus(X0(m))~'o , for all integers m. By (4-I) we may restrict attention to composite numbers m, and it is evident that it suffices to treat those composite numbers m such that genus X0(d ) = o for all proper divisors d of m. There are x 7 such values of m, of which 9 are of genus I and have been shown to have finite Mordell-Weil groups, by various people, including Ligozat ([3o], and see discussion in [48]). The case m= 26 155 156 B. MAZUR is treated in [4I]. The cases m=35, 5 ~ have been taken care of by Kubert [27] ; the case m----5 ~ was also done independently by Birch. The case m = 39 has been settled by a descent argument on the elliptic curve quotient of X0(39 ) using an explicit equation given for this curve which can be found in an extensive table compiled by Kiepert. This equation (formula 63ib on page 391 of [25] ) and these useful tables were pointed out to me by Kubert. Re-writing the curve as a quotient defined over Q, its minimal model is y2+xy=xa-l-x~--4x--5 and the descent follows the lines of the case m=35 ([27] , [34], w 9)- Sixteen of the seventeen cases (all m above except m= 125) have been covered by the recent work of Berkovich (cf. remark following (4-I)). 5" A complete description of torsion in the Mordell-Weil group of elliptic curves over Q. In this section we shall prove the following theorem, first conjectured by Ogg [49] : Theorem (5- x ). -- Let a~ be the torsion subgroup of the Mordell- Well group of an elliptic curve E, over Q. Then (I) is isomorphic to one of the following 18 groups: Z/mZ for m<lo or m=I2 or: Z/2.Z� .for ~<_4. Remark. -- All these groups do occur. The fifteen curves: Xl(m ) and X(2) � for m, ~ in the above range are all isomorphic to P~Q. Consequently, the elliptic curves E/Q whose Mordell-Weil group contains a given group (I) (chosen from among the 15 above) occur in an infinite (rationally parametrized) family. These fifteen explicit rational parametrizations are given in the table of [27], chapter IV. Corollary (5- 2). -- Let an elliptic curve, defined over Q, possess a point of order m rational over Q. Then m<Io or m=I2. Equivalently: Corollary (5.3)- -- Let m be an integer such that the genus of Xl(m) /s greater than o (i.e. m = 11 or m >I 3)- Then the only rational points of X~(m) (over Q) are the rational cusps. We shall begin the proof of (5- 1-3) with a series of reduction steps. First reduction. -- To prove (5-1-3) it suffices to prove (5.2) in the special case where re=N, a prime number such that the genus of X0(N ) is 2>o (i.e. N 4:2, 3, 5, 7, and 13). This is so by virtue of the close study of the above conjecture of Ogg, made by Kubert, for low values of composite numbers m. In particular, Kubert has shown ([27] , chap. IV) that it suffices to consider only prime values of m, greater than or equal to 23. For m=i3, see [4o]. For the duration of the proof, let, then, N denote a prime number 4:2, 3, 5, 7 or 13, 156 MODULAR CURVES AND THE EISENSTEIN IDEAL I57 and let Z/N C E be an elliptic curve over Q with a point of order N, rational over Q (generating the subgroup Z/N). The object of the proof will be to show that Z/N C E does not exist. As usual, E/s will denote the Ndron model of E over S and Z/Nrs C E/s is the 6tale constant subgroup scheme over S generated by our point of order N. Let K=Q(~N), where ~N is a primitive N-th root of unity, and let L be the field extension of Q generated by the N-division points of E. By considering the short exact sequence of Gal(Q/Q)-modules: (5.4) o -+ Z/N -+ E[N] ~ ~N -+ O ~ sees that Gal(L/K)has a faithfuI representati~ int~ GL2(FN)~ the f~ (I O ~) where Z : Gal(L/Q) ,, Gal(K/Q) >~ F} is the cyclotomic character. Thus, one has the diagram of field extensions: / I~'~ where L/K is either an N-cyclic extension, or is the trivial extension. Moreover, an elementary computation gives that the natural action of Gal(K/Q,) on Gal(L/K) (conjugation in Gal(L/Q)) is by multiplication by z -1. This computation uses the existence of the faithful representation of Gal(L/Q,) of the form (i .~, and, as Serre \o ] remarked, can be most conveniently seen by noting that the 9 in the upper right corner takes its values, canonically, in the vector space Horn(aN, Z/N). It is clear that the exact sequence (5-4) splits if and only if L = K. Second reduction. -- It suffices to prove that (5.4) splits, or equivalently, that L = K. For we would then have the following result from which we easily derive a contradiction: Given any eUiptic curve ~Q and a sub-Galois module Z/NC d ~, there is a sub-Galois module I~,C g. Let us obtain a contradiction from this. Forming the quotient d~'=N/lxN, we get another elliptic curve over Q and the image of Z/N provides g' with, again, a sub-Galois module Z/NC N'. We may then apply the above result inductively to obtain a chain of such elliptic curves over Q,, related by ~N-isogenies, rational over Q,: 8--~g'--~g~ all containing sub-Galois modules isomorphic to Z/N. This is impossible for various reasons. Firstly, the members of the above chain cannot be all mutually nonisomorphic. For, if they were, they would represent an infinite number of elliptic curves over Q with good reduction outside a given finite set of primes. This would contradict the 157 158 B. MAZUR theorem of Shafarevitch (cf. [63] , IV (I.4)). Alternatively (and more in the spirit of the present work), it would provide an infinite number of distinct rational points on X0(N); and this would contradict Theorem (4-i). We have therefore shown that for suitable i# j, #(~)x #(J), and consequently there is a non-scalar endomorphism of 8 (i) defined over O. In particular, d "(i) possesses a complex multiplication over Q, which is impossible. Third reduction. -- It suffices to show that L/K is unramified (at all places). For suppose that L/K is unramified, and nontrivial. Then it is an N-cyclic (unramified) extension, and consequently N must be an irregular prime. Since L/Q is Galois and the natural action of Gal(K/Q) on Gal(L/K) is ) -1 it would then follow, by Herbrand's theorem (chap. I (2.9)), that the Bernoulli number B 2 must have numerator divisible by N. Since B 2 = I/6, L/K must be the trivial extension. We shall now prove that L/K is unramified. Although this is a local question at each place v of K, it is unlikely that one can prove this by local arguments. Indeed, the essential step 3 below is global. We proceed by 4 steps, analyzing the structure of the putative Z/NC E. Step 1. -- E/s is semi-stable. That is, E has semi-stable (i.e. good or multiplicative) reduction at all points of S. Proof. -- Let q be a (rational) prime of nonsemi-stable (i.e. additive) reduction for E. Thus the connected component of the fibre E/Fq, (E/@ ~ is an additive group, and, as is well known, the index of (E/Fq) ~ in E/F q is 2 a. 3 b for suitable integers a, b. It follows that the specialization Z/NjF q must be contained in (E/Fq) ~ Consequently, q=N. Using ([72], w 2, Cor. 3) one sees that there is a finite extension field ~ff/Oz q such that E/~ has semi-stable reduction at the maximal ideal of the ring of integers 1~ = g)~c, and if e = e(Cf : QN) is the absolute ramification index, we may choose o~# so that e<6. IfE/~ is the Ndron model of E over the base 0, and E/z| is the pullback to ~) of the Ndron model over Z, there is a natural morphism E/zNg~-+ E/~ which is trivial on the connected component of the closed fiber, since there are no nontrivial maps from an additive group over a field to a multiplicative group, or to an elliptic curve. If G/oC E/e is the closed subgroup scheme generated by Z/N/jcC E/~, we have a natural morphism Z/N/o-+ G/o which is an isomorphism on generic fibers, and not an iso- morphism on the special fibers, by the above discussion. From this, one sees that G/o is a finite flat group scheme. But since e<6<N--I, by [55] a finite flat group scheme of order N over 0 is determined by its generic fiber. In other words, G/o must be isomorphic to Z/N/~, which is a contradiction. Step 2. -- If q = 2, or 3, then E has bad (hence multiplicative) reduction at q, and the specialization Z/N/F q is not contained in the connected component of the identity (E/Fq) ~ 158 MODULAR CURVES AND THE EISENSTEIN IDEAL Pro@- If E/rq were an elliptic curve, and Z/NC E/rq, then by the " Riemann hypothesis" N<_I+q+2%/~t, which is impossible for q=~, 3. Therefore, E has bad reduction at q (= 2, 3) necessarily of multiplicative type, by step i. But, by Tate's theory ([63] , IV, A.I. i), (E/Fr ~ is isomorphic to G,,/F r which has qZ--I points. Again, we cannot have Z/NC Gmfpq, , for q=2, 3, by virtue of our hypotheses on N. Step 3. -- If q is any prime of bad (hence multiplicative) reduction for E, then the special- ization Z/N/Fq is not contained in the connected component of the identity, (E/@ ~ Proof. -- Let q be a prime of multiplicative reduction such that Z/N/rqC (E/rq) ~ By steps I and 2 we may assume q + ~, 3 or N. Consider the base T= Spec Z[I/2.N] and let x be the T-valued point of X0(N)/r determined by the couple (E/T , Z/N/r). That is, x=j(E/r , Z/NIT). It is illuminating to draw the scheme-theoretic diagram: 9 9 ,~ 2 3 q where oo and o are the cuspidal sections over T. We are justified in drawing the intersections: x/F= ~/F,, X/F~=O/F~, because, by ([7], VI, w 5), the modular interpret- ation of c~fr t is the " generalized elliptic curve ": (Gin � Z/N, Z/N)F t (i.e. the cyclic subgroup of order N which gives the F0(N)-structure is not contained in the connected component containing the identity) while the interpretation of o/F t is the " generalized elliptic curve ": (Gin x Z/N, ~N)~F, (i.e. the cyclic subgroup of order N which gives the P0(N)-structure is contained in the connected component containing the identity). Now consider the natural projection to the Eisenstein quotient X0(N)/r-+J/~. By (3. I) we know that J(O~)= C. For the present proof, however, it suffices to know that ~(O) is a torsion group. Thus j(T):j(o,) is a torsion group. Let '~ denote the image of sections of X0(N ) in J. Since T is an open subscheme of Spec Z over which 2 is invertible, if A is any abelian scheme over T, and g any rational prime representing 159 i6o B. MAZUR a closed point of T, the specialization map A(T)tor s --> A(Ft) is injective (1). Applying thisfact to 1=7, one sees that J(T)-+j(Ft) isinjective. But the equations ~F =~/r,, and x/Fq=O/Fq, then imply that ~=oo. Since ~--~ is of order n in j, this can only be true if n = I, or equivalently, if N < 7 or N = i3. Since N is constrained to be :>7 and + i3, we obtain the contradiction that we seek. Step 4. -- L/K is unramified. Pro@ -- (i) q a rational prime of good reduction for E; q+N: Since E[N']/Zq is an 6tale, finite flat group scheme, L/K is unramified over all places of K lying over q. (ii) q=N; E has good reduction at N: Again E [N]/z~ is a finite flat group scheme. Applying the " connected component of the identity" functor to (4.4) one sees that (E/zN) 0 [.LN ' and therefore we get a splitting: E [N']/z, , = Z/N/z ~ � ~wz~,, which again shows that L/K is unramified at all places of K lying above N. (iii) q a rational prime of bad reduction for E: Since Z/NlFqd; (E/@ ~ by step 3, one obtains, as in (ii), E[N]/zq~Z/N/zq� giving us the same conclusion: that all places of K above q are unramified in L/K. 6. Rational points on X~plit(N ). Keeping to the terminology of the Introduction (cf. discussion preceding Theorem 9) elliptic curves with a normalizer-of-split-Cartan structure on their N-division points are classified by noncuspidal rational points on X0pnt(N)=X0(Ng")/wN,. Theorem (6.x). -- If N+2, 3, 5, 7 and I3, then Xsplit(N ) has only a finite number of rational points. Note. -- If N~7, then Xsplit(N ) is isomorphic to P~0 and therefore its set of rational points form a rationally parametrized infinite set. The curve Xsplit(i3) is of genus 3. It is to be expected that Xspnt(I3) has only a finite number of rational points, but my methods have not been able to establish this. Proof of theorem (6. I). m Consider the two natural morphisms: f, g : X0(N ~) -> X0(N ). The mapfis defined by the prescription f: (E, CN, ) ~ (E, CN) where, ifE is an elliptic curve, and C N, is a subgroup of E of order N 2, then C N-~N.CN,CE. It induces a map from parabolic modular forms (of weight 2) on Fo(N ) to parabolic modular forms on F0(N ~) with the same q-expansion. The map g is defined by the prescription g : (E, C~,) ~ (E', C~) where E'= E/C N and C~=C~,/C N. (1) This is a standard application of the Oort-Tate classification theorem [54] to the group scheme over T generated by an element of order p in the kemel of the above specialization map. If A is an elliptic curve, then this result is due to Nagel-Lutz. 160 MODULAR CURVES AND THE EISENSTEIN IDEAL x6I If r is a parabolic modular form (of weight 2) and ~(q) denotes its q-expansion at 0% then (~-'~)(q)=~(q~). We denote the canonical involution (chap. II, w 6) of X0(N ) by w N to distinguish it from the canonical involution of X0(N~), denoted wN,. As usual, J is the jacobian of X0(N ). Let h : X0(N *) --->J be the map which associates to x the divisor class of f(x)--g(x). A straightforward calculation yields the formula h. w N, =--ws.h (and the minus sign will be of importance to our proof). It follows from this formula that the composition X0(N ~) --->Jh ~J-=J/(I+wN)" J factors through X0(N ~) ---> X0(N ~)/w N, = Xsplit(N ) and thereby induces a map: h-- : Xsplit(N ) --->J-. The map X0(N ~) -->J induces a surjection on the jacobian Jo(N ~) --->J as can be seen as follows. The induced map from parabolic modular forms of weight 2 under F0(N ) to parabolic modular forms under F0(N 2) is injective. This latter assertion is true since a modular form of weight 2 under P0(N) which is sent to zero by the map in question must have its first N q-expansion coefficients equal to zero. Hence it is zero. It follows that the map h- : Xspa(N)-~ J- induces a surjection from the jacobian of Xspn~(N ) to J-. Let h:X~plit(N)-->J denote the composition of h- with the projection map to the Eisenstein quotient (chap. II (I 7. IO)). Since N=I I or N>_I7, it follows that X0(N ) is of positive genus, and that J is nontrivial. Letting X~p~it(N) cJ denote the image of Xsplit(N ) under h, one sees that Xsput(N) must be a curve, and Xspiit(N ) --> :~sput(N) a finite morphism. Since J(Q)=C is a finite group, the proof of Theorem (6. I) is completed. Remark. -- We have made essential use of the fact that J factors through J- (chap. II (i 7. io)). This fact (when N- I rood 8) seems to depend on some of the more delicate aspects of the theory developed in chapter II. 7. Factors of the Eisenstein quotient. Consider a surjective morphism defined over Q, J-->A where A/Q is a Q-simple (equivalently: C-simple) abelian variety. Let p [ n be a prime number such that this morphism factors through the p-Eisenstein quotient (such a prime number p must exist, but may net be unique) and let a = dim A. Replacing A by an abelian variety isogenous to it, if necessary, we may suppose that the Hecke algebra T leaves the kernel of J-+A stable, and consequently that we can induce a natural action of T on A. Since the Eisenstein prime ~ associated to p is contained in the irreducible component of Spec T which corresponds to A (chap. II (io. I)) it follows that A[~3] (the kernel of f~ in A) is nontrivial. Consequently, by admissibility of the kernel of ~3 (chap. II (14. i)) it follows that there is an abelian variety A;Q isogenous to A over Qsuch that A' possesses a point oforderp in its MordeU- Weil group. (More precisely, we may take A;s to contain a subgroup scheme isomorphic to Z/p/s 0 21 16e B. MAZUR Using the criterion of N~ron-Ogg-ghafarevich, one sees that A-~A' extends to an isogeny of abelian schemes over S'. Reduce to characteristic 2 and obtain an isogeny A/F -+A~ ' of abelian varieties where A'(F~) contains a point of order p. Standard estimates for the number of rational point of an abelian variety over a finite field (the Well conjectures) give: p< #A,(F2)< (i +~r or: logp (7. i) a2 2 .log(i -? %/2)" We obtain : Proposition (7.2). -- Every simple factor of the p-Eisenstein quotient ~Cp) has dimension log p -- 2. log(i +%/2)" Corollary (7.3). -- There are absolutely simple abelian varieties of arbitrarily high dimension, defined over Q,, whose Mordell-Weil group is finite. Proof. -- For any positive integer a0, find a prime number P-->5 such that logp>2a0.1og(i+V/2) and, by Dirichlet's theorem, choose a prime number N such that N--=I modp. Then (7.2) every simple factor of the p-Eisenstein quotient of j=j0(N) has dimension >a 0 and (4. ~) has finite Mordell-Weil group. What are the elliptic curve factors of the Eisenstein quotient J? If E/Q is a quotient elliptic curve of jC~), then E has (prime) conductor N, and by the above discussion, after modification of E by Q-isogeny if necessary, we may suppose that the Mordell-Weil group of E possesses a point of order p. There has been some recent work ([68], [46], [42], [I6]) on elliptic curves of prime conductor N possessing a torsion point of order p over Q. In particular, one has that p< 5 (using the Well estimates to the reduction of E in characteristic 2) and by [42] one has, further, if p = 5, then N =II and E is isogenous to X0(II); if P=3, then N=I9, or37 and E is isogenous to X0(I9) or to the Eisenstein quotient of J0(37). Thus we are reduced to the case p=2. In this case, either N=I7 and E is isogenous to X0(I7), or it is a Neumann-getzer curve (which, by definition, is an elfiptic curve over Q, of prime conductor N + 17 possessing a point of order 2 in its Mordell- Well group). The facts concerning Neumann-Setzer curves are these ([68], [46]): A Neumann-Setzer curve of conductor N exists if and only if N is of the form 64 q-u 2 (u an integer). If N is of the above form there are precisely two isomorphism classes of Neumann-Setzer curves of conductor N, given by the equations: y2 = x 3 + ux 2_ 16x yZ = x ~-- 2ux 2 + Nx. 162 MODULAR CURVES AND THE EISENSTEIN IDEAL I63 One may pass from one curve to the other by the 2-isogeny obtained by division by the rational point of order 2. Proposition (7.4). -- i) Let p > 2. The p-Eisenstein quotient has no elliptic curve factor unless P=5, N=II or P=3 and N=I90r37. ii) The 2-Eisenstein quotient 7 (21 has no elliptic curve factor unless N = 17, or N = 64 + u S with u an integer. If the 2-Eisenstein quotient has an elliptic curve factor, then this factor is unique up to isogeny and if N + 17 its isogeny class is that of the Neumann-Setzer curves of conductor N. If the (two) Neumann-Setzer curves of conductor N are parametrized by modular functions for P0 (N) (i.e. /f they occur as quotients of J, a special case of the conjecture of Weil) then they are quot#nts of jl ). Proof. -- This combines the work of [68], [46] as in the discussion above, and chapter II (I 4. i). The following gives (granted conjectures of Weil and Hardy-Littlewood) an infinite number of values of N for which the estimate of (7.2) is sharp for the 2-Eisenstein quotient. Proposition (7.5). -- Let N be a prime number of the form 64+u ~ such that N r i mod I6. Suppose that the (two) Neumann-Setzer curves are parametrized by modular functions for P0(N). Then "~c~) is of dimension I, and is a Neumann-Setzer curve of conductor N. Proof. -- Let ~ be the Eisenstein prime associated to 2. By our hypothesis on N, 2 ]In. Therefore, by chapter II (I 9.I), T~ is a discrete valuation ring. Since the irreducible components of Spec T~0 map surjectively to the (isogeny classes of) factors of 7 (2/, it follows that 7 (2/is a simple abelian variety. But, by the hypothesis of (7.5) and by (7.4) ii) the Neumann-Setzer curves are factors ofJ (~/. Our proposition follows. Remark. -- Let N be a prime number of the form 64 + u S such that N -=- 1 mod 16. The 2-Eisenstein quotient contains a point of order 4 (at least). It must have dimension greater than I, for if it were an elliptic curve, it would be a Neumann-Setzer curve and a Neumann-Setzer curve does not possess a point of order 4 [68]. Suppose, further, that the Neumann-Setzer curves of conductor N are parametrized by modular functions for F0(N ). It then follows that ]~(2/is not a simple abelian variety, since it has a Neumann-Setzer curve as a proper factor. Consequently the completion of the ttecke algebra T~ is not an integral domain. Conjectures of Weil and of Hardy- Littlewood would give that this occurs for infinitely many values of N. The only case of a pair (N,p) where N<25 o and T~ is not a discrete valuation ring, for ~3 the Eisenstein prime associated to p, is: N=II3, p----2. This is the first instance of the (conjecturally infinite) family of examples described in the paragraph above. 163 I64 B. MAZUR 8. The ~-adic L-series. Fix ~3 an Eisenstein prime associated to a prime number p4= 2. In this section and the next we shall examine the analytically-defined ~-adic L series [39] and the arithmetically-defined ~3-adic characteristic polynomial [34]. We recall terminology and results from the papers cited. Since both p and ~p = I-+-p--Tp are in the ideal ~3, we have T v-- i mod ~3 (1) and therefore Tp is a unit in T~. The standard recursive process (e.g., p. 47 of [39]) gives two roots of the quadratic equation: X~- Tp. X +p = o in T~. Call the unit root r~ and the other one ~ to be consistent with the terminology of [39]- Let { }:Q/Z'-->I-I=HI(X0(N), Z) denote the modular symbol, where Q/Z' means rationals with denominator prime to N, modulo I ([22], [29] ). Let f: Q/Z'-+H~ be the composition of { } with H~H~=T~| For any fixed choice of integer A 0 prime top, set A,= A0.p n and ZA = l im Z/A, regarded as topological ring. Z~ is then the topological group of its units. We now wish to use the construction of [39], w 8, to obtain an H~-valued measure on Z~x from the function fl This may be done, for f is an eigenfunction for the Hecke operator rip with eigenvalue a unit in T~. One remark, however, must be made: in the terminology of [39], w 8, we take T~=D, H~ =W. Note, however, that in [39] (8. I) the hypothesis on D is that it be the ring of integers in a finite extension of Qp. This is not needed. All that is used is that D is a local ring with maximal ideal m containing the prime p and that D is p-adically complete. Let p~/', then, denote the It~-valued measure associated to the eigenfunction f ([39] (8.1)). Let (chap. II (18.1)) o~H~H~-+H~o be the decomposition of H~ into -- and + eigen (sub- and quotient-) spaces. Let Z be a continuous multi- plicative character on Z~ whose values lie in (and generate) the To-algebra T~[Z]. We consider the Fourier transform of the measure ~xA: Lv(Z) = fz3, Z- ExaeHv [)~] =Tv [Z] | Hr. If the formula )(--I) -(sign x). I defines sign ), L~(Z) lies in the (sign Z)- eigenspace of the complex conjugation involution and, if Z is even, it is natural to let LV()~) take its value in H~[Z] , by projection. We refer to L~ as the ~3-adic L-series, and the general theory of [39] applies to it. In particular we have its various developments as analytic function in the s- and T-planes, keeping the conventions of [39]. (a) We think of this relation as expressing the fact that Eisenstein primes are anomalous, in the spirit of the notion introduced for elliptic curves in [34]. 164 MODULAR CURVES AND THE EISENSTEIN IDEAL Let ~z~, + be the projection of the measure ~/' to I-I~. Then the ~3-adic L-series restricted to even characters is the Fourier transform of tx/'' + Proposition (8. I) (divisibiaty). -- ~a, + takes its values in ~3. I-I~ C H~. Pro@ -- By chapter II (i8.8), f(b/Am) depends only on A~ mod.~.H~, if b is prime to A. By formula (2) of (8.1) of [39], A+ evaluated on the fundamental open set a+A,ZaCZ x (for a prime to A) is given by: lim r~ -m ~ f(b/Am), rain b ~ amod An" from which our proposition is seen to follow. Corollary (8.2). -- If Z is an even character, L~(Z)e3.H~[7. ]. The proposition also implies that if we develop L~ in a power series expansion about an even character )(o in either the s- or the T-plane (cf. [39], w 9) then every coefficient of these power series will lie in ~.K~[Xo ]. To evaluate the constant term L~(z0 ) of the ~3-adic L-series, where Z0 is the principal character of conductor p, we use [39] (8.2). Take A0= I. We work in the ring D=T~. The proposition of [39] (8.2) gives: --~v.S p--1 where S is E {alp} projected to H~. a=0 If e~ denotes the image of the winding dement in H~| (chap. II, w I8), formula (8) of page 35 of [39] yields "~j,.e~=--S, giving: (8.3) L~(Xo) = ~q~.e; ell+ (compare with the formula at top of page 55 of [39]). To analyze this constant term more closely, fix g a good prime number (relative to p, N) as in chapter II, w i6. For convenience, if p itself is good (i.e. ifp is not a p-th power modulo N), take t=p. Let ~=~t, which is a generator in T~ of the ideal Z~ by chapter II (i6.6). Write ~, = 3. ~. Therefore 3eT~ and 8 is a unit (= i) if and only ifp is good. Since ~, ~% are units in the ring T~| (e.g., as in chap. II, proof of (I6. Io)) so is 8. Corollary (8.4). -- There is a suitable generator y of the T~-module H~ such that: L~(z0 ) = 32. ~.y where 8 is a unit in T~ | Q. Furthermore, 8 is a unit in T~ if and only if p is not a p-th power rood N. 165 x66 B. MAZUR Proof. -- This follows from the above discussion, and (8.3), by taking: y = ~ .e;/(~--P) (~--~)~. Now make a choice of a i-unit y~Z~ = Z; and form ([39], w 9) the ~-adic L-series in the T-plane about ~(0: L(T) = L~(Zo , T)(v)E H;| The constant term is just L~(Z0), and by (8.2) each of its coefficients is divisible by ~, and therefore we may write L(T):g(T).~.y, where g(T)cT~[[T]] is a power series whose constant term is 82. Thus Corollary (8.5). -- Identify T v with H~ by the map z~z.y. Then: ~]-I.L~(z0 , T)(v) e T~[iT]] is a power series with constant term 82. It is a unit in Tr if and only if p is not a p-th power modulo N. Remark. -- When p is a p-th power modulo N, we have then a " secondary " analogue to the phenomenon of anomalous primes studied in [34], [39]. Namely, either Lr , T)(v) is divisible by more than ~, or it has at least one zero in the open unit T~-disc (or both). 9" Behavior in cyclotomic towers. Guided by conjectures made in [39], the results concerning the ~3-adic L-series (w 6) suggest that the following proposition is true. We prove it below (independent of any conjectures). We shall also take the opportunity to correct an erroneous assertion made in [34]. Proposition (9. x ). -- Let p ~ 2 be a divisor of n. Let (j~P) /O. denote the unique Galois extension with aalois group isomorphic to Zp (the p-cyclotomic F-extension). The group "J(p)(~')) of rational points of the p-Eisenstein quotient with values in the p-cyclotomic F-extension is a finitely generated group. If p is not a p-th power modulo N, then it is a finite group. One has an accompanying assertion about the ~3-primary component of the Shafarevich-Tate group. Namely, let F= Gal(O~P)/O), and for every positive integer m, let F mC P be the subgroup of index p'~. Set O~C O.~ p) to be the fixed field of pro, and III m the ~3-primary component of the Shafarevich-Tate group of J(p) (or of J: it is the same) over O.~ ). Set A=liI+_nnT~0[I'/Pm] (the projective limit of topological rings, where T~[P/Fm] is given the natural topology) and III~o=lim III m regarded as A-module. Proposition (9.2). -- The kernel and cokernel of the natural map: IIIm (III Jm 166 MODULAR CURVES AND THE EISENSTEIN IDEAL I67 are finite groups whose orders are bounded (independent of n). That is, the above sequence is controlled in the sense of [24], [34]. The A-module III oo is isomorphic, modulo finite groups, to the Pontrjagin dual of the A-module A/.~s~.A~Ts~/~s~[[T]]. There is a constant c0>o such that if fl[n, then: I logp(order IIIm)--f.p"[< c o for all m>o. Remarks. -- Guided by the same conjectures of [39] (6.5), one would expect that if p (42) is a p-th power modulo N, then either J(p)(O~ p)) is a finitely generated group of positive rank, or III m grows more rapidly than the bound of (9-2). The proof of these propositions may be regarded as a " generalization " of the case N=II, treated in [34]- It proceeds closely along the lines of argument used for the case N = i i, but incorporates work we have already done concerning Eisenstein primes, and uses a recent result: Theorem (Imai [2I]). -- Let K be a number field (a finite extension of Q) and L/K the p-cyclotomic extension (L = V K(~pr)). Let AlL be an abelian variety. Then the torsion sub- group A(L)tor s of the group of rational points of A over L is a finite group. Correction. -- I am thankful to Ito for pointing out that an assertion I made in [34] (labelled (6.18)) is incorrect (for abelian varieties of CM-type of dimension greater than I). Therefore, my proof that A(L)t0r s is finite when A is of CM-type ([34] (6. I2 (i))) is incomplete. The theorem of Imai [2I] shows, however, that the result is valid for all abelian varieties. Discussion. -- Imai proves a local result based on Sen's analysis of the structure of the Lie algebra of a Galois group acting on a Hodge-Tate module [59]- Serre has communicated to me a proof along rather different (global) lines by means of which he obtains finiteness of the group of rational torsion points of the abelian variety A with values in many F-extensions over K not only the p-cyclotomic F-extension. We now prepare to prove (9-1) and (9-2) by the method of [34]- Let Ym denote the spectrum of the ring of integers in O~ and let Y be the spectrum of the ring of integers in O~ p). Thus Y0= S = Spec(Z). If j/y,, is the base change of the N~ron model J/s then it is the N~ron model of the jacobian of X0(N )tQ~p) since p is the only ramified prime, and J has good reduction at p. Let ~q (as in w 6) be a generator of the ideal ~s~ C T~ (chap. II (i6.6)). SO: Jv = lim jv [~r]/s is represented in this way as an inductive limit of quasi-finite group schemes over S, and is naturally endowed with the structure of Ts~-module. 167 I68 B. MAZUR We have the analogue of diagram (6.6) of [34], which may be written: o o (9.3) 0 > HI(Y~-p~,J,~) > H.(Y~,p~, J~) ---> H~(Y~, J~) > H~(Y~, J~) CXm 1 , H2:y T ~rm > Hl(y,j~)rm_____> Hl(Y__p~o j~)rm__~ .~ p~o,Jw o O where Pm is the unique closed point of characteristic p in Ym, P~ is the unique closed point of characteristic p in Y; Ym, pm is the completion of Ym at Pro, and Ypoo is the completion of Y at Poo ; the superscript F m means invariants under the action of I~,, and the subscript mean coinvariants; H 2. denotes cohomology with supports at the closed point. We view the above diagram, whose horizontal and vertical lines are exact, as a diagram of To-modules. There are three necessary calculations that must be made, in order to prove (9- I) and (9.2) and we collect them in the following lemma: Lemma (9-4) : i. J~(O~')) is isomorphic to T~/~. 2. Hl(Spec(Z), J~) = o. 3. For any m, the T~-module E,~ is (noncanonically) isomorphic to (T,/~)| (T~/~). Granted the lemma, we shall prove our propositions. Let I-I denote I-P(Y, J~) The lemma enables us to " evaluate " the above diagram regarded as A-module. for m=o: 0 0 T~I~ > (T~I3~) | (T~I~) H~(Yo --Po, J~) > H.(Yo,,o, J~) o > H r , HI(Y--p~,J~) r > W.(Y,~,J~) r o o 168 MODULAR CURVES AND THE EISENSTEIN IDEAL ~69 Note that T~/53 v is a cyclic group of order pr. From the above diagram, it follows that H r is a cyclic abelian group, hence, as Tv-module, a quotient of T~ by some ideal a C T~. It also follows from the above diagram that ~VC a. From this, we obtain the analogous information about H*, the Pontrjagin dual. Namely, H*| ~ is isomorphic to T~/a, as T~-module. Now consider the " descent sequence "" o , C~:~ ---~J~ --*J?~ ~ o 0 -----~ ~ ~ ~0 which is a sequence of sheaves of T~-modules for the fppf topology over Spec(Z)=Y0, or, after base change, over the schemes Y~, and Y. Here Cp~ Zip f is the p-primary component of the cuspidal subgroup and Ep (T t~pl noncanonicalty) is the p-primary component of the Shimura subgroup (chap. II (16.4)). The sheaf (P is representable by a nonseparated but finite &ale group (pre-)scheme whose support is concentrated at the prime of characteristic N, and whose fiber at N is a free Tv/~V-module of rank I. Noting that by (I) of the lemma the group H~ is generated by the appro- priate multiple of the point (o)- (oe), and as T~-module is isomorphic to Tv/~, one obtains that H~ A consequence of the above diagram is, then, that: (9.6) I-It(Y, C,| ---> H is an injection of A-modules. By a result of Iwasawa, the p-primary components of the ideal class group of the fields O~ ) vanish. It follows that: Hi(Y, Zip r) = o and, by " Kummer theory "" HI(Ym, ~pf)=U~/U~ I (all these cohomology groups being H2 (Ym, ~v~) = o fppf-cohomology) where U m is the group of units in the ring of integers of O~ ). By the Dirichlet unit theorem, Um/U~ I is a free (Z/pr)-module of rank pm--I. Replacing A by ~vs in the diagram (7-3) and evaluating (using that l~s(lj~)=o, He(Y,,, l~ps)=o and that H.(Ym, p,,, btpl) is dual to HI(Y,,,Vm, Z/pf)), one finds that: H~(Y~, ~,) -+ H'(,/, ~,~)~ is injective, for each m, with cokernel cyclic of order p( It follows that Hi(Y, ~vt) rm is a free (Z/pr)-module of rank pro. An application of Nakayama's lemma gives that the Pontrjagin dual of Hi(Y, btvl) is a free module of rank one over Z/pf[[P]] = A/~. A. Taking the Pontrjagin dual of (9-6), one gets a surjective map of A-modules: H* ~ ) A/~.A. 22 ~7o B. MAZUR Let R denote the kernel of the above homomorphism. Form the long exact sequence: Tor~(A/~?.A, T~) -+ R| v ~ H*| ~ -+ T~/Z~ ~ o. By the resolution o-+A-+A-+A/~A-~o, one sees that the Tor ~ term in the above sequence vanishes. Since H*| ~ is isomorphic to T~/a where a contains ~, it follows that R| vanishes as well. By Nakayama's Iemma one has R =o, and therefore : H* =~ A/~. A ~ Z/p~[[I']] as A-module. Proposition (9.~) is an immediate consequence of this, and Proposition (9. I) follows from Proposition (6. i I) of [39] and the theorem of Imai and Serre quoted above. Proof of Lerama (9.4). -- Part 1: Consider the filtration of jr over the base Spec(Zp) (chap. II (8.4)): o _+j~l~. t,p~ __>j~ ___> j~ ___> o. We show that the specialization map Jv(O~ p)) --+J~3(Fp) is injective by noting that Jv(O~p))nJ~Ult. type(Qp) vanishes. But the kernel of ~ in the latter intersection is just E(O~ p)) by chapter II (i6.4). It is zero, since E is a ~z-type group, and O~ p) does not contain the p-th roots of I. Since J~(O~ p)) f.~jsult, type(Q,p) is also killed by a power of 3, it must vanish. We shall conclude Part I by noting that J~(Fp)~ Cp, the p-primary component of the cuspidal subgroup. Sincep is not ap-th power rood N (and p ~ 2), ~p is a generator of 3~C T~ (chap. II (I 8. Io)). If r: is the unit root of X 2-TpX +p = o in T~ and 5 is the non-unit root, using the Eichler-Shimura relations and well known arguments (repeated in [39], w 4 d) and e)) one deduces that J~(Fp) is the kernel of I -- X in J~(Fp). But ~p = (I -- 7~) (I -- K) and therefore J~(Fp) is the kernel of ~ in J~(Fp). Part 2: Write out the descent sequence [39] (3-3) for the isogeny ~ on J: o . j [~;] ) j ) j0 ----~ o o ___~j0 .__.~j ~ 9 , o and the related long exact sequences for fppf cohomology. These latter we regard as exact sequences of T-modules and we tensor them with T~, which preserves exactness. We get: o---~ C~-~ M| T~--~ M~ TV --~H ~ (S, J, [~;]) --~H ~ (S, J) ~-~H t (S, jo) *-~W(S, j,~ [~;]) o--~M~174174 )H~ r )HI(S,J~ 170 MODULAR CURVES AND THE EISENSTEIN IDEAL I7i By (3.3), M| and M~174 are finite, and therefore by (I.2) we may evaluate them as follows: M| p and M~174 By chapter II (I6.4) , we have Jo[hp]=(Cp| Using the facts: Hi(S, Cp) = H~(S, Zp) = o for i = I, 2 and: H~ O) is free of rank I over To/~v, we may evaluate the above diagram for r = I and obtain the fact that ~q induces an isomorphism Hi(S, J) o -~ Hi( S, j0) o (from the top line) and the kernel of :qp in Hi(S, J) 0 is zero (from the bottom line). Consequently, Itl(S,J)o =Hl(S,J~ If we now consider the top line for general r, we have that Itl(S, Jo [~]) is flanked by groups which vanish and hence must vanish itself. Had [39] (5.7) been written in appropriate generality we would apply it directly to obtain what we wish. Part 3: As it is, we reconsider its proof. Let ofp denote the formal completion of J/sp~Izp)- Since Jv is naturally a T~=T| we have the decomposition (chap. II (7.1)) ~=Jo�162 using the idempotent decomposition I=r ~. We now prepare to copy the exact sequence of [39], Corollary (4.6). To convert to the notation of that Corollary, set A=J, L m = the completion of O~ / at the prime p,,, Din=the ring of integers in Lm, and, for some fixed m 0 set K=Lmo, D=Dm.. Then, for m=mo4-h (h_>o), Corollary (4.6) of [39] reads: ofp (D)/NLm/K ~(Dm) ~ J(K)/Ntr,/KJ(Lm) -+ J(Fp)/J(Fp) ph -+ o which is an exact sequence of T-modules. Tensoring with T o gives: (9.7) or (D)/NL,~/K oil o (D,~) ~ J (K)/NL,,/~: J (L~) | TO -+ J(F,)/J(Fp)~hGTTo --~ o But fV is a formal group of multiplicative type to which Corollary (4-33) of [39] applies, giving: The subgroups NL,,/~fo(Dm) C ~0(D) stabilize for large m, and: (9.8) ~o (D)/NL,,m tO (D,~)~ [r/P,,] | T0/(I -- ~). [F/P,] | where Fm=Gal(O.~P)/O~,P2) and rc is the unit root (which is the twist matrix [39], w 4 for J0)" Since the ideal in Tv generated by (1--r~) is just ~o, the above isomorphism yields that the left-hand T~-module of (9.8) is free of rank i over To/~ O for large enough m. Also, by the discussion of Part I, J(Fp)/J(Fp)Ph| is a free TV/3 v- module of rank i, if h is large enough. We now check that the left-hand map of exact sequence (9-7) is injective. This is as in Proposition (4.42) of [39]. Form the short exact sequence of Fmo/Pm-modules: (9.9) o ~ Jo(Dm) -+ J(L~)| -+ J(Fp)| -~ o 171 i7~ B. MAZUR and note that J(Fp)| is generated by the specialization of C~o which is contained in J(K)| ~. It follows that (9-9) splits as an exact sequence of Tv[Fm0/Fm]-modules. But the left-hand map of the exact sequence (9.7) is the map induced on o-dimensional Tate cohomology by the map of F,~./Pm-modules Jv(Dm)| appearing in the split exact sequence (9.9). Putting all the information we now have into the exact sequence (9.7) we obtain the following split exact sequence of Tv-modules : o -+ T~/~ v ->J(K)/NLm/KJ(Lm)QTT v -+ TV/~ v --> o for m large. We now apply Corollary (5.4) (P. 225 of [39]) and the discussion on page 226 to conclude that the kernel of: H.2(Ym, J~) -+ H2.(Y, J~) is a free module over T~/~ of rank 2. Added in proof (August 1977): I. Using results of the present paper, and some new techniques, the (Q-) rational points of X0(N ) can be completely determined for all prime numbers N. One finds that there are no noncuspidal rational points on X0(N), and hence no Q-rational N-isogenies, when N is a prime number ~>23, such that N+ 37, 43, 67, and I63. In particular the question-marks occuring in the TABLE of the introduction have been resolved. See: Rational isogenies of prime degree to appear in Invent. math. 2. An incorrect entry in a previous table of mine ([38], w 4) is corrected in the TABLE of the introduction to the present paper. Namely, when N= i99 , the data for g_ (in the table at the end of [38]) should read: 2 -t-10 and not: 2-1-t0. In particular, when N=I99 , J is not equal to J-. Therefore, remark 2 of [38], 2. 5 should be amended to read: J--J- for N<25 o, except when N=67, Io9, I39, I5I, I79, 22I and I99. 172 APPENDIX Behavior of the N@ron model of the jacobian of X0(N ) at bad primes by B. MAZUR and M. RAPOPORT Throughout this appendix we depart from the convention of the rest of this paper and let N denote a square free number not divisible by ~ or 3, and p a prime divisor of N. The connected component of the fibre at p of the Ndron model J of the jacobian of the modular curve Mo(N ) (chap. II, w I) was determined in [9]- Our purpose here is to get somewhat finer information about J, in particular about the finite abelian group : of the connected components of the fibre at p of J. The following theorem, which is the main result of this appendix, is due to P. Deligne: Theorem (A. x). -- Let N=p be a prime number. a) The connected component jo of the fibre at Fp of J is a group of multiplicative type; considering it over Fv, the Frobenius endomorphism acts on the p-adic Tare module: as: F* ~ --p. w, where w is induced from the canonical involution (z~--i/pz). b) We have a canonical decomposition of the fibre at p o f J: J,=J~ where C is a cyclic group of order num((p--i)/I 2) generated by the class of the divisor (o)- ( oo). More generally, write: N=p.qi, ..., q~ (allowing for v = o to include the case N =p). The connected component of the fibre at p of J is an extension of j0(ql, ..., q~)p �176 ..., q~)p by a group of multiplicative type (cf. [9] and section I below). As for the group @ =q)p of connected components of the fibre at p of J one has table 2 below: 173 B. MAZUR TABLE 2 Order of (o)--(oo) Structure of Order Relations satisfied by "standard " elements of (I) (u, v) in ~ q)/(o)--(~) ofO Structure of O (O, 0) o~.P~2 1 trivial Q.. P~2 I Z/(Q..~2 I) z = (o)-- (oo) is a generator (I,0) Q .p-~- i Z/2~`0_1 z Q..p-- I 22`0 q) is generated by the Ei 6 12 (i = I,.,., 2'~); (~ Z/2 2~-2 Z relations: = 2E i (i = I ..... 2 `0 ) p--I (o, I) Q.p-- i Z/a~_iZ Q. ~ .32~ is generated by the Gi (i= I, ..., 2`0); | Z/3~- i Z relations: E~ =--s'.g ~ =.g~l . } (~ = i .... ,2`0) 2 ! (I, I) Q.p--I Z]62`0_1 z O.P--I .6~` o Z/(O, p-- 1)Z (I) is generated by the E i ,~Gj 2 I2 (i, j = I, ..., 2"r | Z/2Z`0- 2 Z relations: @ Z/3 2"- 1 Z i 3 = ~i (i = I .... ,0`0) Yj = oui/= 1 ~ ,J = i .... ,2`0) Notation: a) Set u = I if: P-7 or Ii(mod I2) and : all q~--- I (mod 4) i = I, . .., v otherwise set u =o. b) Set v=I if: P-5 or II(mod I2) and: all qi~I(mod 3) i=I, ...,v; otherwise set v=O. 174 MODULAR CURVES AND THE EISENSTEIN IDEAL I75 c) Set Q= __IIl(qi+I ) (=I if v=o). d) The last column gives information about "standard elements " in q) (cf. section 2). In particular Z=(o)--(oo). Remarks. -- I) In table 2, Z is an element (but not necessarily a generator) of the first cyclic group occurring in the column labelled " structure of @ " Z is a generator of this cyclic group if v = o or if (u, v) is (o, o) or (o, I). In all other cases Z is twice a generator. 2) The table shows that, ignoring 2- and 3-primary components, ~bp is a cyclic group generated by the image of the divisor class (o)--(oo). Its order (again ignoring products of powers of 2 and 3) equals Q. (p-l). The order of (o) --(oe) in J is divisible by the 1.c.m. of the orders of qbp, for all p dividing N. G. Ligozat has computed this order (as yet unpublished). The plan of exposition is the following. In section I we recall relevant results from [9] about the moduli schemes of interest. After recalling results of Raynaud [5 6] about the relation between the jacobian of the minimal model of a smooth curve over a discretely valued field and the N6ron model of its jacobian, we reduce our problem to a computation. This computation is outlined in section 2. The final section 3 proves a) of Theorem i. x. Relation between Tnlnimal model and N6ron model. The following is a somewhat simplified version of ([9], VI (5-9))- Set N'=N/p. Theorem (I.I). -- a) M0(N ) is smooth over Z[I/N'] outside the supersingular points in characteristic p. b) M0(N ) | is the union of two copies of M0(N' ) | crossing transversally at the supersingular points. If x ----j(E, H) is a supersingular point of M0(N') | (i.e. E=super- singular elliptic curve and HC E[N'] a cyclic subgroup of order precisely N'), then x on the second copy is glued to the point x (p) of the first copy of M0(N' ) | c) Let x =j(E, H) be a supersingular point of M0(N' ) | and set: k = I [ aut(E, H) I. At the corresponding point of M0(N)| p the scheme M0(N ) has a singularity whose strict localization is isomorphic to: W(F,) [ IX, Y]]/(X. Y--f) (i.e. is of type Ak_l). 175 r7 6 B. M AZUR d) In particular, the reduction modulo p of the minimal model X0(N ) 0fM0(N ) (over Z[I/N']) is obtained by glueing two copies of Mo(N') | at corresponding supersingular points, and then replacing a crossing point by a chain of k-- I projective lines. If p # 2, 3 (which we will always assume), then: k>I implies either: j(x) =o, and then k = 3 or: j(x) = 1728, and then k = 2. Those projective lines, considered as divisors on the minimal model, have self-intersection --2. Our next task is to determine the number of supersingular points explicitly. Let: S'=the number of supersingular curves E over Fp with j(E)+o, 1728. if there exists a supersingular curve E over Fp with j(E)---- 1728. , ={'o otherwise. if there exists a supersingular curve E over Fp with j(E) =o. otherwise. Recall [1, VI (4.9)] that: S'+~.I+ I.R= p-I . 3 12 Recall from the introduction that Q= i II=1 (qi + I). M0(N ) | lying above a supersingular Proposition (i .2). -- (i) The number of points in point xzMo(P)| p is: Q if j(x)#o, I728 Q if j(x) = 1728 but not all qi- I (mod 4) I_ (Q_ 2~ ) iof j(x) =1728 and all qi - i (mod 4) but not all q~- i (mod 3) I_Q if j(x) =o and all q~- i (mod 3). if j(x)=o Hence: (ii) S'= number of supersingular points x in Mo(N ) | ~'p with j(x)+-o, ~ 728 _Qp;i ~[u (for u, v consult the introduction to this appendix). 176 MODULAR CURVES AND THE EISENSTEIN IDEAL t77 Pro@ -- (i) is a consequence of the following facts: a) The morphism M0(N)| p --~ M0(t)| p is a covering of degree Q. b) Let j(E) 1728 and let (E, I-I) correspond to a point in X0(N')| p. If Aut(E, H)4={ }, there is a primitive 4-th root of unity in (Z/qi)* (the automorphism group of the qi-primary component of H) for each i: I, ..., v, i.e.: qi= I (mod 4) i=I, ..., v. c) Similarly, if (E, H) corresponds to a point in M0(N' ) | with j(E)= o, and if Aut(E, H) 4={-4-I}, then there is a primitive 6-th root of unity in (Z/q~)* for each i=I, . .., v, i.e.: q~i (mod3) i=i, ...,v. (ii) follows from (i) by taking into account the formula recalled shortly before the statement of the proposition and the fact that: j= o is supersingular if p--I (mod 6) j=~728 is supersingular if p-I (mod 4). Q.E.D. We obtain the following picture for the reduction modulo p of Xo(N): Z,~, r ! Zt X i -- = Xo(N ) | 0(N ) E1 i 9 these are present 9 ~ if and only if u= I E2'~ ) these present 9 ~ 9 these are present L li, -~v=I 9 ~ if and only if v= F~ J ~ G~ We now recall results of Raynaud [5 6] which will allow us to pass from The- orem (i. i) to Theorem (A. I) and its variants. 23 I78 B. MAZUR Let: K= discretely valued field, complete for the valuation. R = ring of integers in K, k = R/(rc)= residue field (assumed algebraically closed). S =Spec(R), ~q and s its generic and closed points respectively. C = a curve, smooth, geometrically irreducible and proper over K. f : W---~ S = minimal model of C over R. (Recall that q~ is the (unique) regular scheme, proper and flat over R, with generic fibre ~ =C such that for any other regular scheme 5' flat over R with generic fibre rg~=C, the birational map ~'--~W is a morphism.) of=jacobian variety of C=Pic~ J =Ndron model of of. (Recall that J is the (unique) group scheme smooth over R such that for every other smooth group scheme J' smooth over R, any K-morphism J'~J~ comes from a unique R-morphism J'~J.) The following result of Raynaud gives the connection between ~ and J: Theorem (1.3)- -- With the above notations, assume that d=g.c.d, of all multiplicities d~ of the irreducible components Ci of ~ is equal to I. Then: J ~ Pic~]s/E, where: P;c E~ =kernel of the morphism " degree " deg : Pice/s-+Z ~r and: E = scheme-theoretic closure of the unit section in Pic~e/s. (This result is not stated in this form in [56]; it is a consequence of the results in that paper (we adhere to the terminology of [56]): a) f verifies condition (N) and we have: f. (d)~e)= ~)s, hence f is cohomologically flat in dimension o ([56] (7.2. i)). b) Pic~/s is representable by a formally smooth algebraic space in groups; and Pic~/s is represented by a separated smooth group scheme ([54] (8.2. i)). The quotient: Q= Pic~e/s/E is representable by a separated smooth group scheme over S ([56] (8.o.1)). c) The group scheme: Q*=inverse image in Q of the torsion part of Q/Q0 is the Ndron model of of= Pic~ d) The morphism: 9 [o1 elc~/s~Q2 178 MODULAR CURVES AND THE EISENSTEIN IDEAL is surjective, with kernel E (cf. [5 6] (8. I. 2)); hence Pi@)s/E_~Q" is the Ndron model of J.) We extract from [5 6] (8. i. 2) the following additional information: Proposition (I.4). -- Let D_~Z" be the free abelian group generated by the irreducible Let D*=Hom(D, Z) be the dual group. Define: componen~ C~ of~. o~ : D-+D* and: : D*-+Z E I-. deg(~[ q). C~, identifies: Then ~oe-=o; and, sending ~q~ePic(~') to i d i Js/J~ To apply these results in our case we note that we may pass to the algebraic closure F~ of Fp since formation of N~ron models (respectively of minimal models of curves) commutes with ~tale base change (Fp is perfect). 2. Calculation of the table. We use the Proposition (I.4) of section I. The irreducible components of the reduction modulo p of X0(N), the minimal model of M0(N ), are Z, Z', El, Fi, G i (i=I, ..., 2 ~) (with the convention that Ei, respectively F~ and G~, are missing if u = o, respectively v = o). They all have multi- plicities equal to one. Hence: (2.i) D=free abelian group generated by Z, Z', El, Fi, Gi. Let D0=ker(~)=elements in D of degree o (cf. section I, Proposition (1.4)). Let D* and D O be their respective dual groups. Then: (2.2) A basis of D; is given by: Z = Z*--Z'* = z'* V;-- Z'" G~= G~-- Z'*. 179 I8O B. MAZUR The intersection products, as read off from the configuration of divisors given in section I, determine the self-intersection numbers: (2.3) Z.Z =--(S'+ 2~.u+2~.v) Z'. Z'=-- (S'+ 2~. u + 2~. v). Hence, since 9 = Do/Im(0~ ) : (2.4) (I) = D0/modulo the relations m. EEi+n. ZFi-(S'+ 2~m + 2~n).Z-=o i i m. EEl+ n. ZC,~ + S'.2- o i i 2-2.g~-o n. 2- 2n~ + n~ ---- o 2nG,-- nF, =- o. (2.5) The order of (I) equals the absolute value of the determinant of the intersection matrix of Z', Ei, Fi, G i. To fill in the table we distinguish cases: 1st case: (u, v) = (o, o) Here (I) =Z.Z/S - '- Z, hence its order is S'=Q. p- I ; (I) is generated by Z. 2nd case: (u, v)=(o, I) The order of (I) equals the absolute value of the determinant of the following intersection matrix: Z' F 1 Gj F 2 G 2 F2~ ' G~v Z t --(S' + 2 ~) o i o i o I O --2 I O O 0 o F1 I I --2 O O o o G1 0 o O O O --2 I F2 G~ I O 0 I --2 o o o 0 0 0 F2v O O O O 0 --2 1 O O 0 0 I --2 G2~, 180 MODULAR CURVES AND THE EISENSTEIN IDEAL x81 Adding to the Z'-row: 2 (sum over Gcrows ) (sum over F,-rows) -t- gives as new Z'-row" --S'--2".-, O, O, . .., O; hence the determinant equals: act =--(S' + 3.2~)3~ =- 32~-1(3. S ' +2~). The relations (2.4) allow us to eliminate Fi, and the Ei are absent: ( )/ (I)/(cyclic subgroup generated by 2)~Z/32~-lZ. The order of Z in @ is thus: Hence 3.Q. (p--I)/I2; since this number is prime to 3 (because, if /)=i, then P---5 or 7 (mod I2) and qi-I (mod3) , i=I,..., 2~), the cyclic subgroup of 9 generated by Z is a direct summand. 3rd case: (u, v) = (I, o) The order of 9 equals the absolute value of the determinant of the following intersection matrix: Z" E 1 E~ E2v Z p --(S' + 2 ~) i I i I --2 0 0 E1 E~ I 0 --2 0 E~v I 0 0 --2 Adding the Z'-row to I. (sum of the Ei-rows ) one obtains as new Z'-row: --S '-~-.2 ~, O, O, ..., O, hence: et= 181 x82 B. MAZUR The relations (2.4) become in this case: E g~(S'+ 2 ~) .Z=o EE~+S'.2 -o --2E~+Z =o. Hence ~/(cyclic subgroup generated by Z)=~Z/2~-tZ. The order of Z is thus 2.Q.(p-I)/I2. If v>I, then the cyclic subgroup generated by Z is not a direct summand of @ but is of index 2 in a direct summand. 4th case: (u, v) = (I, i) The order of 9 equals the absolute value of the determinant of the following intersection matrix: Z' E 1 . . . E2'~ 171 G 1 . . . F2v G2~ Z t --'~t~' --~]~ 2~ .xl), I . . . I 0 I . . . 0 I l --2 0 ... 0 0 0 . . . 0 0 E1 0 --9 0 0 0 . . . 0 0 I O . . . --~2 0 0 . 9 . 0 0 E2v Fa 0 0 . . . 0 --9 I . . . 0 0 131 I 0 0 I --2 0 0 0 0 0 0 0 --9 ! F2,0 I 0 0 0 0 I --2 G2v Add to the Z'-row: I (sum of Ei-rows ) +3I (sum of Fcrows ) + 3 (sum of G~-rows) to get as new Z'-row: _8,+52~=_S, 2~+ ~ i 2~ 2 6 +~' +~ "2~' o,o,.. ,o. 182 MODULAR CURVES AND THE EISENSTEIN IDEAL i83 Hence: det =--62~(S'+ 5 . 2~). The relations (2.4) become: 7,--2-g~-3-Gs_ _ '. ) ZE~+ZGj-=--S 2 (i,j=i,...,2~). Hence q)/(cyclic subgroup generated by ]~)-~Z/62~-lZ. The order of 2 in is thus 6.Q.(p--I)/I2. If v>I, the cyclic subgroup of 9 generated by Z is not a direct summand of 9 but is of index 2 in a direct summand. In conclusion, we have filled in all entries of the table; sections I and 2 also prove Theorem (A. i) except for the statement about the action of Frobenius on J~ 3" The Frobenlus action. Let N=p be a prime number. Denote by F the following graph: vertices = components Z, Z' edges-= supersingular points (joining Z and Z') F: Z~ * 9 Z~ Z' There is a canonical isomorphism (cf. [9]): J~174 ~ Hi(F, Z) | The action of the Frobenius endomorphism of J~174 may be identified with: ~| : Hi(P, Z)| z)| where ~ : F-+ I" is the map which fixes the vertices and which sends a supersingular point x (corresponding to an " edge " of F) to the unique supersingular point x'=o~(x) such that j(x')=j(x) p. But the map a induces the endomorphism --w on t-P(P, Z), because ~ is the composition of w with the automorphism of F which interchanges the vertices and keeps the edges fixed. Hence: F----- ~| =--p.w. This proves part a) of Theorem (A. I). 183 MODULAR CURVES AND THE EISENSTEIN IDEAL I85 [27] KUBERT (D.), Universal bounds on the torsion of elliptic curves, Proe. London Math. Soc. (3), 33 (1976), i93-237. [28] KImERT (D.), LANa (S.), Units in the modular function field, I, II, III, Math. Ann., 218 (I975), 67-96, 175-189, 273-285. [29] I~NG (S.), Elliptic Functions, Addison Wesley, Reading, I974. [3 o] LmOZAT (G.), Fonctions L des courbes modulaires, Sgminaire Delange-Pisot-Poitou, Jan. 197o. Thesis: Courbes modulaires de genre I, Bull. Sor math. France, mSmoire 43, 1975. [3 i] MAmN (Y.), A uniform bound for p-torsion in elliptic curves [in Russian], Izv. Akad. Nauk. CCCP, 38 (t969) , 459-465 9 [32] Ma-~xN (Y.), Parabolic points and zeta functions of modular forms [in Russian], Izv. Akad. Nauk. GGCP, 36 (I972), I9-65 . [33] MAZVR (B.), Notes on 6tale cohomology of number fields, Ann. Seient. l~,c. Norm. Sup., 4 e sSrie, t. 6 (1973), 521-556 . [34] YIAZUR (B.), Rational points on abelian varieties with values in towers of number fields, Inventiones Math., 18 (I972), I83-266. [35] MAZUR (B.), Courbes elliptiques et symboles modulaires. S6minaire Bourbaki, No. 414, Lecture Notes in Mathematics, No. 31"/, Berlin-Heidelberg-New York, Springer, I973. [36] MAZUR (B.), p-adic analytic number theory of elliptic curves and abelian varieties over Q, Proc. of Imernational Congress of Mathematidans at Vancouver, 1974, vol. I, 369-377, Canadian Math. Soc. (i975). [37] MAZUR (B.), M~SSlNO (W.), Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, No. 370, Berlin-HeideIberg-New York, Springer, 1974. [38] MAZVR (B.), S~RR~. (J.-P.), Points rationnels des courbes modulaires X0(N ). S6minaire Bourbaki, No. 469, Lecture Notes in Mathematics, No. 514, Berlin-Heidelberg-New York, Springer, I976. SWlNNERTON-D~R (P.), Arithmetic of Well curves, Inventiones math., 25 (I974), I-6L [39] MAZUR (B.), [40] MAzuR (B.), TAT~ (J.), Points of order 13 on elliptic curves, Inventiones math., 22 (I973), 4t-49 . [4 I] MAZUR (B.), V~LV (J.), Courbes de Weil de conducteur 26, C. R. Acad. Sc. Paris, t. 275 (I972), s~rie A, 743-745" [4 2] MIYAWAKA (I.), Elliptic curves of prime power conductor with Q-rational points of finite order, Osaka J Math., 10 (1973), 309-323 . [43] MUMFORD (D.), Geometric invariant theory, Ergebnisse der Math., 34, Berlin-Heidelberg-New York, Springer, x 965 . [44] MUMFORD (D.), Curves and their jacobians, Ann Arbor, The University of Michigan Press, 1975. [45] N~RON (A.), ModUles minimaux des vari6tds abdliennes sur les corps locaux et globaux, Publ. Math. LH.E.S., 21 (i964), 361-483 [MR 3~, 3424] 9 [46] NEUMANN (O.), Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten, I, II, Math. Naehr., 49 (x97I), io7-123; Math. Nachr., 56 (I973), 269-280. [47] OI)A (T.), The first De Rham cohomology group and Dieudonn6 modules, Ann. sdent. ~e. Norm. Sup., 4 e s~rie, t. 2 (I969), 63-I35. [48] OoG (A.), Rational points on certain elliptic modular curves, Proc. Syrup. Pare Math., 24 (1973) , 221-231, A.M.S., Providence. [49] Ooo (A.), Diophantine equations and modular forms, Bull. A.M.S., 81 (I975), 14-27. [5o] Ooo (A.), Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), 449-462- [5 I] Ooa (A.), Automorphismes des courbes modulaires, S~minaire Delange-Pisot-Poitou, d~c. I974 (mimeo. notes distributed by Secrfitariat mathfimatique, 1I, rue Pierre-et-Marie-Cnrie, 75231 Paris, Cedex o5). [52] OHTA (M.), On reductions and zeta functions of varieties obtained from F0(N ) (to appear). [53] OORT (F.), Commutative group schemes, Lecture Notes in Mathematics, No. 15, Berlin-Heidelberg-New York, Springer, 1966. [54] OORT (F.), TAT~ (J.), Group schemes of prime order, Ann. Scient. 12c. Norm. Sup., s6rie 4, 8 (I97o), I-2x. [55] RA'e~AUD (M.), Schfimas en groupes de type (p, ...,p), Bull. Soc. Math. France, 102 (I974), 241-28o. [56] RAYNAUD (M.), Sp~cialisation du foncteur de Picard, Publ. Math. LH.E.S., 38 (I97O), 27-76. [57] I~YNAVB (M.), Passage au quotient par une relation d'dquivalence plate, Proc. of a Conference on Local Fields, NUFFIC Summer School held at Driebergen in z966 , I33-I57, Berlin-Heidelberg-New York, Springer, I967. [58] RmET (K.), Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555-562. 24 MODULAR CURVES AND THE EISENSTEIN IDEAL I85 [27] KUBERT (D.), Universal bounds on the torsion of elliptic curves, Proe. London Math. Soc. (3), 33 (1976), i93-237. [28] KImERT (D.), LANa (S.), Units in the modular function field, I, II, III, Math. Ann., 218 (I975), 67-96, 175-189, 273-285. [29] I~NG (S.), Elliptic Functions, Addison Wesley, Reading, I974. [3 o] LmOZAT (G.), Fonctions L des courbes modulaires, Sgminaire Delange-Pisot-Poitou, Jan. 197o. Thesis: Courbes modulaires de genre I, Bull. Sor math. France, mSmoire 43, 1975. [3 i] MAmN (Y.), A uniform bound for p-torsion in elliptic curves [in Russian], Izv. Akad. Nauk. CCCP, 38 (t969) , 459-465 9 [32] Ma-~xN (Y.), Parabolic points and zeta functions of modular forms [in Russian], Izv. Akad. Nauk. GGCP, 36 (I972), I9-65 . [33] MAZVR (B.), Notes on 6tale cohomology of number fields, Ann. Seient. l~,c. Norm. Sup., 4 e sSrie, t. 6 (1973), 521-556 . [34] YIAZUR (B.), Rational points on abelian varieties with values in towers of number fields, Inventiones Math., 18 (I972), I83-266. [35] MAZUR (B.), Courbes elliptiques et symboles modulaires. S6minaire Bourbaki, No. 414, Lecture Notes in Mathematics, No. 31"/, Berlin-Heidelberg-New York, Springer, I973. [36] MAZUR (B.), p-adic analytic number theory of elliptic curves and abelian varieties over Q, Proc. of Imernational Congress of Mathematidans at Vancouver, 1974, vol. I, 369-377, Canadian Math. Soc. (i975). [37] MAZUR (B.), M~SSlNO (W.), Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, No. 370, Berlin-HeideIberg-New York, Springer, 1974. [38] MAZVR (B.), S~RR~. (J.-P.), Points rationnels des courbes modulaires X0(N ). S6minaire Bourbaki, No. 469, Lecture Notes in Mathematics, No. 514, Berlin-Heidelberg-New York, Springer, I976. SWlNNERTON-D~R (P.), Arithmetic of Well curves, Inventiones math., 25 (I974), I-6L [39] MAZUR (B.), [40] MAzuR (B.), TAT~ (J.), Points of order 13 on elliptic curves, Inventiones math., 22 (I973), 4t-49 . [4 I] MAZUR (B.), V~LV (J.), Courbes de Weil de conducteur 26, C. R. Acad. Sc. Paris, t. 275 (I972), s~rie A, 743-745" [4 2] MIYAWAKA (I.), Elliptic curves of prime power conductor with Q-rational points of finite order, Osaka J Math., 10 (1973), 309-323 . [43] MUMFORD (D.), Geometric invariant theory, Ergebnisse der Math., 34, Berlin-Heidelberg-New York, Springer, x 965 . [44] MUMFORD (D.), Curves and their jacobians, Ann Arbor, The University of Michigan Press, 1975. [45] N~RON (A.), ModUles minimaux des vari6tds abdliennes sur les corps locaux et globaux, Publ. Math. LH.E.S., 21 (i964), 361-483 [MR 3~, 3424] 9 [46] NEUMANN (O.), Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten, I, II, Math. Naehr., 49 (x97I), io7-123; Math. Nachr., 56 (I973), 269-280. [47] OI)A (T.), The first De Rham cohomology group and Dieudonn6 modules, Ann. sdent. ~e. Norm. Sup., 4 e s~rie, t. 2 (I969), 63-I35. [48] OoG (A.), Rational points on certain elliptic modular curves, Proc. Syrup. Pare Math., 24 (1973) , 221-231, A.M.S., Providence. [49] Ooo (A.), Diophantine equations and modular forms, Bull. A.M.S., 81 (I975), 14-27. [5o] Ooo (A.), Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), 449-462- [5 I] Ooa (A.), Automorphismes des courbes modulaires, S~minaire Delange-Pisot-Poitou, d~c. I974 (mimeo. notes distributed by Secrfitariat mathfimatique, 1I, rue Pierre-et-Marie-Cnrie, 75231 Paris, Cedex o5). [52] OHTA (M.), On reductions and zeta functions of varieties obtained from F0(N ) (to appear). [53] OORT (F.), Commutative group schemes, Lecture Notes in Mathematics, No. 15, Berlin-Heidelberg-New York, Springer, 1966. [54] OORT (F.), TAT~ (J.), Group schemes of prime order, Ann. Scient. 12c. Norm. Sup., s6rie 4, 8 (I97o), I-2x. [55] RA'e~AUD (M.), Schfimas en groupes de type (p, ...,p), Bull. Soc. Math. France, 102 (I974), 241-28o. [56] RAYNAUD (M.), Sp~cialisation du foncteur de Picard, Publ. Math. LH.E.S., 38 (I97O), 27-76. [57] I~YNAVB (M.), Passage au quotient par une relation d'dquivalence plate, Proc. of a Conference on Local Fields, NUFFIC Summer School held at Driebergen in z966 , I33-I57, Berlin-Heidelberg-New York, Springer, I967. [58] RmET (K.), Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555-562. 24 x86 B. MAZUR [59] SEn (S.), Lie algebras of Galois groups arising from Hodge-Tate modules, Ann. of Math., 97 (i973) , i6o-x7o. [60] S~RP.~ (J.-P.), Corps locaux, Paris, Hermann, 1962. [61] SERP~ (J.-P.), Formes modulaires et fonctions zdta p-adiques, vol. III of the Proceedings of the International Summer School on Modular Functions, Antwerp (~972), Lecture Notes in Mathematics, 350, I9X-268, Berlin- Heidelberg-New York, Springer, i973. [62] S~mp.e (J.-P.), Alg~bre locale. Multiplicit&, Lecture Notes in Mathematics, No. 11 (3rd edition), Berlin-Heidelberg- New York, Springer, 1975. [63] S~m~a~(J.-P.),Abelianl-adicrepresentationsandelliptlccurves, Lectures at McGill University, NewYork-Amsterdam, W. A. Benjamin Inc., I968. [64] S~RR~ (J.-P.), Q uelques propri6t6s des vari6tfis abdliennes en caract6ristique p, Amer. J. Math., 80 (I958), 7x5-739 9 [65] SERRE (J.-P.), p-torsion des courbes elliptiques (d'apr~s Y. Manin). S6minaire Bourbaki, 69-7o , No. 38o, Lecture Notes in Mathematics, No. 180, Berlin-Heidelberg-New York, Springer, x97 I. [66] S~m~ (J.-P.), Congruences et formes modulaires (d'apr~s H. P. F. Swinnerton-Dyer). S6minaire Bourbaki, 7~-72, No. 416, Lecture Notes in Mathematics, No. 317, Berlin-Heidelberg-New York, Springer, I973. [67] SEm~ (J.-P.), Propri~tds galoisiennes des points d'ordre fini des courbes elliptiques, Inventiones math., 15 (I972), 259-33 I. [68] SETZEa (B.), Elliptic curves of prime conductor, J. Lond. Math. Soe. (2), 10 (1975), 367-378. [69] SHXMURA (G.), Introduction to the arithmetic theory of automorphlc functions, Publ. Math. Soc. Japan, No. 11, Tokyo-Prlnceton, I97x. [7 o] WADA (H.), A table of Hecke operators, Proc. Japan Acad., 49 (I973) , 38o-384 . [71 ] Y~AUCHI (M.), On the fields generated by certain points of finite order on Shimura's elliptic curves, J. Math. Kyoto Univ., 14 (2) (I974), 243-255. [72] BOREVICH (Z. I.l, SHA~'~VmH (I. R.), Number theory, London-New York, Academic Press, 1966. [73] PARRY (W. 1~.), A determination of the points which are rational over Q of three modular curves (unpu- blished). [74] SERR~ (J.-P.), TATE (J.), Good reduction of abelian varieties, Ann. of Math., 88 (I968), 492-5x7 . [75] TATE (J.), Algorithm for determining the type of a singular fiber in an elliptic pencil, vol. IV of The Proceedings of the International Summer School on Modular Functions, Antwerp (i972), Lecture Notes in Mathematics, 4'/6, Berlin-Heidelberg-New York, Springer, I975. [SGA 3] Sdminaire de Gdomdtrie algdbrique du Bois-Marie, 62-64. Directed by M. DEMAZURE and A. GROTHENDIECK, Lecture Notes in Mathematics, Nos. 151, 152, 153, Berlin-Heidelberg-New York, Springer, I97o. [SGA 7] Sdminaire de G6om~trie alg6brique du Bois-Marie, 67-69. P. D~LmNE and N. KATZ, Lecture Notes in Mathematics, Nos. 288, 340, Berlin-Heidelberg-New York, Springer, ~972, I973. Manuscrit refule 16 avril 1976. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Modular curves and the eisenstein ideal

Publications mathématiques de l'IHÉS , Volume 47 (1) – Aug 4, 2007

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Springer Journals
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Copyright © 1977 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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1618-1913
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10.1007/BF02684339
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Abstract

by B. MAZUR (1) INTRODUCTION Much current and past work on elliptic curves over number fields fits into this general program: Given a number field K and a subgroup H of GL2(Z)----IIGL2(Zp) classify all elliptic curves E/K whose associated Galois representation on torsion points maps Gal(K/K) into HC GL2(Z ). By a theorem of Serre [67] , if we ignore elliptic curves with complex multiplication, we may take H to be a subgroup of finite index. This program includes the problems of classifying elliptic curves over K with a point of given order N in its Mordell-Weil group over K, or with a cyclic subgroup of order N rational over K (equivalently: possessing a K-rational N-isogeny). These last two problems may be rephrased in diophantine terms: Find the K-rational points of the modular curves NI(N ) and X0(N ) (cf. I, w I). In this paper we study these diophantine questions mainly for K=Q. In par- ticular we shall determine the (QT)rational points of XI(N ) for all N. The precise nature of our results (which require close control of a certain part of the Mordell-Weil group of j=j0(N), the jacobian of X0(N ) when N is a prime number) may indeed be peculiar to the ground field O. There are other reasons why this ground field may be a reasonable one on which to focus. For example, the recent conjecture of Weil would have every elliptic curve over Q. obtainable as a quotient ofJ0(N ) for some N. Thus, a detailed analysis of the Mordell-Weil groups of these jacobians may be relevant to a systematic diophantine theory for elliptic curves over Q. We shall now describe the main arithmetic results of this paper (2). Theorem (x) (conjecture 2 of Ogg [49])- -- Let N> 5 be a prime number, and [~N-- I n----numerator( -2 ). The torsion subgroup of the Mordell-Weil group of J is a cyclic group \] of order n, generated by the linear equivalence class of the difference of the two cusps (o)--(oo) (chap. III, (1.2)). (1) Some of the work for this paper was done at the Institut des Hautes l~tudes scientifiques, whose warm hospitality I greatly appreciate. It was also partially supported by a grant from the National Science Foundation. (3) See also [36], [38] which give surveys of these results. 33 34 B. MAZUR Control of the 2-torsion part of this Mordell-Weil group presents special difficulties. Ogg has made use of theorem I to establish, by an elegant argument, that for prime numbers N such that the genus of X0(N ) is >2 (i.e. N~23), the only automorphisms of the curve X0(N ) (defined over t3) are the identity and the canonical involution w, except when N--37 [5i]. Ogg had also conjectured the precise structure of the maximal torsion sub-Galois module of J which is isomorphic to a sub-Galois module of Gin: Theorem (2) (conjecture 2 (twisted) of Ogg [49])- -- The maximal ~-type group (chap. I, w 3) is the Shimura subgroup (chap. II, w iI) which is cyclic of order n. Despite their " dual " appearance, theorem 2 lies somewhat deeper than theorem I. Decomposing the jacobian J by means of the canonical involution w, we may consider the exact sequence o-+J+-+J--~J---~o where J+=(~+w).J. One finds a markedly different behavior in the Mordell-Weil groups of j+ and J- (as is predictable by the Birch-Swinnerton-Dyer conjectures). Theorem (3). -- The Mordell-Weil group of J+ is a free abelian group of positive rank, provided g+=dimJ+>o (i.e. N>73 or N=37 , 43, 53, 6I, 67) (chap. III, (2.8)). As for the minus part of the jacobian, a quotient JofJ is constructed (chap. II, (IO.4)), the Eisenstein quotient. It is shown that J'is actually a quotient of J- (chap. II, (i 7. io)), and its Mordell-Weil group is computed: Theorem (4)- -- The natural map J-+J induces an isomorphism of the cyclic group of order n generated by the linear equivalence class of (o)--(oo) onto the Mordell-Weil group of J. We have: J(Q)=Z/n (chap. III, (3.I)). Since n> I whenever the genus of X0(N ) is >o, it follows from theorem 4 that 3" is nontrivial whenever J is, and one can obtain bounds on the dimension of simple factors of J (chap. III, (5.2)). Here is a consequence, which is stated explicitly only because one has, at present, no other way of producing such examples: Theorem (5)- -- There are absolutely simple abelian varieties of arbitrarily high dimension, defined over O, whose Mordell-Weil groups are finite (chap. III, (5.3)). Using theorem 4, one obtains: Theorem (g). -- Let N be a prime number such that X0(N ) has positive genus (i.e. N 4: 2, 3, 5, 7, and I3). Then X0(N ) has only a finite number of rational points over Q, (chap. III, (4. I)). One obtains theorem 6 from theorem 4 as follows: since the image of X0(N ) in j generates the nontrivial group variety ~, it follows that X0(N ) maps in a finite-to-one 34 MODULAR CURVES AND THE EISENSTEIN IDEAL 35 manner to J. Finiteness of the Mordell-Weil group of J then implies finiteness of the set of rational points of X0(N ). The purely qualitative result (finiteness of J(Q)) is comparatively easy to obtain. It uses extremely little modular information, and in an earlier write-up I collected the necessary input to its proof in a few simple axioms. To follow the proof of theorem 6, one need only read these sections: chap. I, w i; chap. II, w167 6, 8. to, prop. (14. I) and chap. III, w 3- See also the outline given in [39]. To be sure, the assertion of mere finiteness is not all that is wanted. One expects, in fact, that the known list of rational points on X0(N ) (all N) exhausts the totality of rational points, and in particular that the only rational points of X0(N ) for N any integer>I63 are the two cusps (o) and (~) [49]- In this direction, we prove the following result, conjectured by Ogg (1): Theorem (7) (conjecture I of Ogg [49])- -- Let m be an integer such that the genus 0fXl(m ) is greater than zero (i.e. m=iI or m>~ 13). Then the only rational points ofXl(m ) are the rational cusps (III, (5-3))- This uses results of Kubert concerning the rational points of Xl(m ) for low values of composite numbers m [27]. Equivalently: Theorem (7')- -- Let an elliptic curve over Q, possess a point of order m< +oo, rational over Q. Then m~io or m=I2. This result may be used to provide a complete determination of the possible torsion subgroups of Mordell-Weil groups of elliptic curves over Q,. Namely: Theorem (8). -- Let rb be the torsion subgroup of the MordeU-Weil group of an elliptic curve defined over Q. Then q~ is isomorphic to one of the following 15 groups: Z/m.Z for m~<io or m= i2 or: (Z/2.Z)� for v<~ 4. (III, (5-i). By [27] theorems 7 and 8 are implied by theorem 7 for prime values of m~> 23. See also the discussion of this problem in [49]-) Since theorems 7 and 8 may be of interest to readers who do not wish to enter into the detailed study of the Eisenstein quotient 3~, I have tried to present the proof of these theorems in as self-contained a manner as possible. For their complete proof one needs to know: a) J(Q,) is a torsion group (see discussion after theorem 6 above) and b) the cusp (o)--(oo) does not project to zero in J if the genus of X0(N ) is greater than zero (which is easy). (1) Demjanenko has published [19] a proof of the following assertion: (?) For any number field k (and, in particular, for k = Q), there is an integer m(k) such that Xl(m ) has no noncuspidal points rational over h, if m >i m(h). However, the proof does not seem to be complete. See the discussion of this in (Math. Reviews, 44, 2755) and in [27]- 85 36 B. MAZUR One then need only read w 5 of chapter III. If ogeC GL~(Fs) is any subgroup such that detYf=F~ there is a projective curve Xav over Qparametrizing elliptic curves with "level W-structures " [9] (chap. IV). The determination of the rational points of Xae amounts to a classification of elliptic curves over O~ satisfying the property that the associated representation of Gal(Q/O) on N-division points factors through a conjugate of 3(f. If N~> 5 is a prime number, any proper subgroup ~4 ~ C GL2(F~) is contained in one of the following four types of subgroups ([67] , w 2): (i) o~= a Borel subgroup. Then Xa~ = X0(N ). (ii) ~= the normalizer of a split Cartan (" ddploy6 " [67] ) subgroup. In this case, denote Xa~ o = Xsplit(N ). It is an elementary exercise to obtain a natural isomorphism between Xsplit(N ) and X0(N2)/wN~ as projective curves over Q, where wN~ is the canonical involution induced from z~--~/N 2 z on the upper half-plane. (iii) ,gt ~ the normalizer of a nonsplit Cartan subgroup. In this case write X~t'= Xn0nsplit(N). (iv) Yr an exceptional subgroup (or to keep to the terminology of [67] (2.5)), ~ is the inverse image in GL~(FN) of an exceptional subgroup of PGLg(FIq). An exceptional subgroup of PGL2(FN) is a subgroup isomor- phic to the symmetric group ~4, or alternating groups ~[4 or ~5" The further requirement det,~f'= F~r insures that the image of ~ in PGLz(FN) be isomorphic to ~. Moreover, if such an ~ (with surjective determinant) exists when K = Q, then N ~. + 3 mod 8. For such N write Xar X~,(N). We do not treat cases (iii) and (iv) in this paper. Of the four types of subgroups of GL~(FI,r listed above, the normalizer of a nonsplit Caftan subgroup seems the least approachable by known methods. In particular (to my knowledge) there is no value of N for which Xnons!0iit(N ) has been shown to have a finite number of rational points. As for case (ii) Serre remarked recently that for any fixed number field K there are very few N/> 5 such that X gt~(K) is nonempty when,~ is an exceptional subgroup of GL2(FN). Firstly, if the image of,~ff in PGL2(FN) is ~4 or ~5, then det Jr (F~) 2. Using the r of Weil, one sees that if Xaf'(N) has a K-rational point, then K contains the quadratic subfield of Q(~N). This can happen for only finitely many values of N for a given K, and not at all when K=Q. Secondly, Serre proves the following local result: Let ,T" be a finite extension of QsI , of ramification index e. Let E be an elliptic curve over ~{" w~l~ a semi-stable N~ron model over the ring of integers ~,~. Let r : Gal(,;~/uT') ~ PGL~(Fz~) denote the projective representation associated to the action of Galois on N-divlsion points of E. Then: if 2e < N-- I, the image of the inertia subgroup under r contains an element of order >1 (N-- I)/e. Using this local result one sees that there is a bound c(.ft") such that if N > c(~U) then Xa~(,,T') is empty for all exceptional subgroups ~o. In the case of ,~T'= Q, X| has no points ratienal over QN if N > 13. Hence it has no points rational over Q for N > 13. Serre constructs, however, a rational point on X~,(I I) and on X~,(I3) corresponding to elliptic curves with complex multiplication by "k/~3. Concerning case (ii) (elliptic curves over O such that the associated Gal(Q/Q)- representation on N-division points factors through the normalizer of a split Cartan subgroup of GL2(Fs) ) we obtain the following result: Theorem (9)- --/f N=II or N~> 17 (i.e./fX~pnt(N ) is ofpositive genus and N+ I3) then X~p~t(N ) has only a finite number of rational points (chap. III,w 6). 36 MODULAR CURVES AND THE EISENSTEIN IDEAL Remarks. -- Since Xsplit(I3) is of genus 3, one expects it to have only a finite number of rational points as well. The proof of theorem 9 is given in Chapter II, w 9. It uses the following two facts: a) J(Q) is finite (see the discussion after theorem 6 above) and b) "J factors through J-. It is interesting to note that when N-I mod 8 fact b) seems to depend on the detailed study of J (chap. II, (I7.Io)). It is often an interesting problem to apply theorem 4 to obtain an ef[~etive deter- mination of the rational points on X0(N ) for a given (even relatively low) value of N, and, to that end, somewhat sharper results are useful. Theorem (xo). -- Let p :X0(N)(Q)~Z/n denote the map obtained by projecting the linear equivalence class of x--(oe) to the MordeU-Weil group of "J (cf. theorem 4)- Then O(x) is equal to one of these five values in Z/n :o, i, i/2 (possible only /f N-----i rood 4), I/3 or 2]3 (the latter two being possible only /f N- --I mod 3) (chap. III, (4-2)). Using these results, tables of Wada [70], Atkin, and Tingley, and work of Ogg [49], Brumer and Kramer [4], and Parry [73], one obtains the chart given at the end of this introduction where the rational points of X0(N ) for N<25 o, N4= I5I, I99, and 223 are determined explicitly (also see note added in proof (end of chap. III)). The main technique of this paper involves a close study of the Hecke algebra T (chap. II, w 6) which we prove to be isomorphic to the full ring of C-endomorphisms of J (mildly sharpening a result of Ribet). We establish a dictionary between maximal ideals 9Jr in T and finite sub-Galois representations of J which are two-dimensional over the residue field of 9J~ (cf. chap. II, (i4.2), for a precise statement and precise hypotheses). The prime ideals that distinguish themselves as corresponding to reducible representations are the primes in the support of a certain ideal which we call the Eisenstein ideal ~ (chap. II, w 9), and which is the central object of our investigation. We like to view the Eisenstein ideal geometrically as follows: Let T* denote the algebra generated by the action of the Hecke operators T t (t~eN) and by w, on the space of holomorphic modular forms of weight 2 for F0(N ). The algebra T is the image of T* in the ring of endomorphisms of parabolic forms. We envision the spectra of these rings schematically as follows: Eisenstein prime ~3 T~ ~ ~ Eisenstein li ne = Spec spec L _ }spec T where the extra irreducible component belonging to T* (the Eisenstein line) corresponds to the action of T* on the Eisenstein series of weight 2. The Eisenstein ideal is the ideal 87 38 B. MAZUR defining the scheme-theoretic intersection of Spec T and the Eisenstein line. The Eisenstein quotient ff is the quotient of J associated to (chap. II, w i o) the union of irreducible components of Spec T which meet the Eisenstein line. One may think of the " geometric descent " argument of chapter III,w 3, as a technique of passing from knowledge of the arithmetic of the Eisenstein line (i.e. of Eisenstein series, and of Gin) to knowledge of the arithmetic of irreducible components meeting the Eisenstein line (i.e. of J) by a " descent " performed at a common prime ideal. One might hope that for other prime ideals common to distinct irreducible components (primes of fusion) one might make an analogous passage (cf. [39], w 5, Prop. 4)- Control of the local structure of T is necessary for the more detailed work. For example, it is easily seen that the kernel of the ideal 9Jl _~T in J(O) is 2-dimensional as a vector space over the residue field of g3l if and only if T~ (the completion at ~J~) is a Gorenstein ring (chap. II, (15. I)). We prove that T~ is a Gorenstein ring, at least if 93l is an Eisenstein prime, or if its residual characteristic is + 2, or if it is super- singular (chap. II, w I4). When ~ is not an Eisenstein prime this is relatively easy to prove. When gJt is an Eisenstein prime, it involves the structure theory of admissible group schemes developed in chapter I and a close study of modular forms mod p (chap. II). Using this work we prove: Theorem (ix). -- The Eisenstein ideal ~ is locally principal in T. If ~3=(t,p) is an Eisenstein prime of residual characteristic p, then the element ~qt = 1 q- g--T t is a local generator of the ideal ~ at ~3 if and only if: (i) t is not a p-th power modulo N J (ii) g- i 9 o mod P I (if not both ~ and p are equal to 2) or (when g=p= 2) 2 is not a quartic residue modulo N (chap. II, (I8. Io)). Most of this analysis of T~ is crucial for the proof of Ogg's conjectures 2 and 2 (twisted) (theorems i and 2) and for the more delicate descent needed to establish: Theorem (I2). -- If ~3 is an Eisenstein prime whose residue field is of odd characteristic, then the ~3-primary component of the Shafarevich-Tate group of J vanishes (chap. III, (3.6)). As described in the survey [36], theorems 4 and i o may be used to prove a version of the Birch-Swinnerton-Dyer conjecture relative to the prime ideal ~3. In the last two sections of chapter III we pursue this theme obliquely by studying the ~-adic L series (1). Guided by formulas, and by conjectures, we are led to the following result, which we prove, independent of any conjectures: Theorem (I3). -- Let ~3 be an Eisenstein prime whose residual characteristic is an odd prime p. Let l~ v) be the unique Zp-extension of Q. Let "~cv) be the p-Eisenstein quotient (the (1) Ref. [34]. 88 MODULAR CURVES AND THE EISENSTEIN IDEAL 39 abelian variety quotient of "J corresponding to the union of all irreducible components of Spec T containing ~3 = (~, p), chap. II, w i o). Then J(p)(Q~p)), the group of points of J(p) rational over O~ p), is a finitely generated group, and is finite if p is not a p-th power modulo N. We also obtain asymptotic control of the ~3-primary component of the Shafarevich- Tare group of J in the finite layers of O~ p). It would be interesting to develop the theory of the Eisenstein ideal in broader contexts (i.e. wherever there are Eisenstein series). Five special settings suggest them- selves, with evident applications to arithmetic: One might study X0(N ) for N not necessarily prime (e.g., N square-free). Although much will carry over (cf. Appendix) there is, as yet, no suitable analogue of the Shimura subgroup, and the " geometric descent" is bound not to be decisive without new ideas: (but see forthcoming work of Berkovich). One might attempt the same with XI(N ) (cf. [7I]) and here one interesting new difficulty is that the endomorphism ring is nonabelian. One might work with modular forms of higher weight k for SL2(Z), where a major problem will be to understand the action of inertia at p on the p-adic Galois repre- sentation associated to the modular form. One might stretch the analogy somewhat and consider some important non- congruence modular curves (e.g., the Fermat curves, following the forthcoming Ph d. Thesis of D. Rohrlich) where the " Eisenstein ideal " has no other definition than the annihilator, in the endomorphism ring, of the group generated by the cuspidal divisors in the jacobian of the curve. One might also work over function fields in the context of Drinfeld's new theory [I3]. Throughout this project, I have been in continual communication with A. Ogg and J.-P. Serre. It would be hard to completely document all the suggestions, conjec- tures and calculations that are theirs, or all that I learned from them in the course of things. I look back with pleasure on conversations and correspondence I had with them, and with Atkin, Brumer, Deligne, Katz, Kramer, Kubert, Lenstra, and Van Emde Boas, Ligozat, Rapoport, Ribet, and Tate. We conclude this introduction with a chart describing the numerical situation for prime numbers N<25o. The columns are as follows: N: ranges through all primes less than 25o such that g=genus(X0(N))>o. g+ = number of=L eigenvalues of w acting on parabolic modular forms of weight 2 for to(N). Also g+=dimJ+=genus(X0(N))+; g_=dimJ-. We write g+ as a sum of the dimensions of the simple factors comprising J=~. When a simple factor is a quotient of the Eisenstein factor, it is boldface. When the 39 B. MAZUR 4 ~ TABLE Values of N n g+ g_ v p(x) ep ii 5 o 1 3 o, _+ I/3 I7 4 o 1 2 4- I/3 19 3 o 1 i o" 23 x~ o 2 o 29 7 o 2 o 3 x 5 o 2 o e 5 = 2 37 3 i 1 2 _+ i 4 t to o 3 o e2 = 3 43 7 I 2 I O 47 23 o 4 o 53 I3 I 3 o 59 29 o 5 o 6I 5 I 8 O 67 xI 2 I+2 I o 71 35 o 35+3 ~ o 73 6 2 1~+28 o 79 ~3 ~ 5 o 8 3 41 I 6 O 89 22 I 1~@511 O 97 8 3 4 o I0I 2 5 I 7 o IO 3 X 7 '~ 6 o Io7 53 ':' 7 o ex7 = 2 IO9 9 3 I-}-4 o I I 3 08 3 1~+22+3 ~ o % = 3 x~ 7 2I 3 7 o e 7 = 2 131 65 i 10 o e 5 = I37 34 4 7 o e2 = 3 I39 ~ 3 I+7 o I49 37 3 9 o I5I ~5 3 3+6 ?? ?? I57 I3 5 7 o I63 27 I +5 7 I o I67 83 2 12 o 173 43 4 10 o 179 89 3 I -~ 0 I8I 15 5 9 O es = 3 I91 95 2 14 o i93 I6 2+5 8 o I97 49 I -}-5 10 o I99 33 4 2+10S,la ?? e s = 2 2I I 35 3+3 25+97 0 e 5 = 2 223 37 2+4 12 o 227 I 13 2+ 3 2+2+10 ?? ?? 229 19 x +6 11 o 233 58 7 1~+1129 o 239 I 19 3 17 o 241 20 7 12 o 40 MODULAR CURVES AND THE EISENSTEIN IDEAL Eisenstein factor is not simple, there is a subscript p to each boldface simple factor. If a factor has p as subscript, then it is a quotient of the p-Eisenstein quotient ~(p/. An asterisk * signals a Neumann-Setzer curve (chap. III,w 7). ,~ = the number of noncuspidal rational points on X0(N ). For a see theorem 7 above. ep=rankzp T~, where ~ is the Eisenstein prime associate to p and T~ is the completion at ~. For the range of the table it is the case that T~ is a discrete valuation ring except when N=II 3 and p=i (see chap. III, remark after (5.5) where this case is shown to be " forced " by the existence of a Neumann-Setzer factor). Thus, for all entries of the table except N = I I3, p = 2, ep = absolute ramification index of T~. In the majority of cases T~=Zp (equivalently: ep=I); therefore we only give ep when it is greater than I. By forthcoming work of Brumer and Kramer [4] all the non-boldface elliptic curve factors of J- (resp. J+) on our table have Mordell-Weil rank o (resp. ~I). The factorization of j:L into simple components comes from tables of Atkin, Wada, and Tingley. The fact that T 0 is integrally closed (N+ I I3) comes in part from a theorem (chap. II, (i 9. I)) and from Wada's tables, as do the calculations of ep. The fact that the values of a given are the only possible involves work of Ogg, completed by Brumer and Parry (also see note added in proof at the end of chap. III,w 9). 6 TABLE OF CONTENTS ................................................................................ I. -- Admissible groups ................................................................. I. Generalities ...................................................................... 2. Extensions of ~p by Z/p over S ................................................... 3. Etale admissible groups ........................................................... 4. Pure admissible groups ........................................................... 5. A special calculation for p = 2 ....................... ~ ............................ 6o II. -- The modular curve Xo(N ) ......................................................... I. Generalities ...................................................................... 62 2. Ramification structure of Xl(N ) --> Xo(N ) .......................................... 3- Regular differentials .............................................................. 4. Parabolic modular forms .......................................................... 5. Nonparabolic modular forms ....................................................... 6. Hecke operators ................................................................. 7- Quotients and completions of the Hecke algebra ..................................... 9 ~ 8. Modules of rank I ............................................................... 9. Multiplicity one ................................................................. 1 o. The spectrum of T and quotients of J ............................................. I i. The cuspidal and Shimura subgroups ............................................... I2. The subgroup D cJ[P] (p = 2; n even) ........................................... IO 3 13. The dihedral action on XI(N ) .................................................... lO 9 I4. The action of Galois on the torsion points of J ..................................... ii2 15. The Gorenstein condition ......................................................... I6. Eisenstein primes (mainly p ~ 2) .................................................. I24 17. Eisenstein primes (p = 2) ........................................................ 129 I8. Winding homomorphisms ......................................................... I35 19. The structure of the algebra T~ ................................................... 14o III. -- Arithmetic applications ............................................................ I4I I. Torsion points ................................................................... I41 2. Points of complex multiplication ................................................... I43 3. The Mordell-Weil group of J ..................................................... I48 4. Rational points on X0(N ) ......................................................... I5I 5. A complete description of torsion in the Mordell-Weil groups of elliptic curves over Q. .... I56 6. Rational points on Xsplit(N ) ...................................................... 16o 7. Factors of the Eisenstein quotient .................................................. 16i 8. The ~-adic L series .............................................................. I64 9. Behavior in the cyclotomic tower .................................................. t66 APPENDIX (by B. MAZUR and M. Behavior of the N6ron model of the jacobian of X0(N ) at bad primes ...................................................................... I73 RAPOPORT). CONVENTIONS CONVENTIONS N : a fixed prime number> 5 (the level) (1). N--I n : numerator (~2---2)" S : Spec Z. S' : Spec Z[i/N]. Ifp is a prime number we write pr[In when pr is the highest power ofp dividing n. If X is a scheme over the base S and T---~S is any base change, X/, r wiU denote the pullback of X to T. If T=Spec A, we may also denote this scheme by X/~. By X(T) we mean the T-rational points of the S-scheme X, and again, if T=Spec A, we may also denote this set by X(A). the Ndron model of the jacobian of X0(N ). J/s : T : the Hecke algebra acting on Jls (chap. II, w 6). the Eisenstein ideal in T (chap. II, w 9)- the Eisenstein prime associated to a prime number p (chap. II, w 9). 92: a general maximal prime ideal in T (not necessarily Eisenstein). If a C T is an ideal, then T, denotes completion with respect to a. If m is an integer and A an object of an abelian category, then A[m] denotes the kernel of multiplication by m. I. -- ADMISSIBLE GROUPS x. Generalities. Consider quasi-finite separated commutative group schemes of finite presentation over the base S = Spec Z which are finite flat group schemes over S'= Spec Z[I/N]. In this chapter we refer to such an object as a group scheme or (if there is no possible confusion) a group over S (or over whatever restriction of the base S concerns us). If G/s is such a group scheme, its associated Galois module is the (finite) Gal(Q/Q)- module G(O) (of ~3-rational points of G, where ~. is some fixed algebraic closure of Q). By the order of G/s we mean the order of the finite abelian group G(Q), or, equivalently. the rank over Z[I/N] of the affine algebra of the scheme G/s, (rank meaning its rank as locally free Z[I/N]-module). For the general properties of group schemes the reader (1) In the appendix we consider more general N. 43 44 B. MAZUR may consult [4o], [9]. We now fix a prime number p different from N, and suppose that the order of G/s is a power ofp. In this case, if S"= Spec Z [I/p], G/s,, is an dtale quasi- finite group ([41], lemma 5) and consequently it is an ~tale finite group over: S'n S"= Spec Z[i/p.N], determined up to isomorphism by its associated Galois module, which is a representation of Gal(Q/Q,) on G(Q)--unramified except possibly at p and N. (a) The structure of G away from p: Let us fix a choice of compatible algebraic closures: Let G/s,, be a group scheme as above. It is given, over S", by the following diagram of compatible Galois modules: (i2) 0(6.) where G(Q) is the Galois module (the Gal(()~/O)-module) associated to G/Q, G(FN) is the Gal(FN/FN)-module associated to G/F ~ and the homomorphism j maps G(F~) into the part of G(Q) which is fixed under the action of the inertia subgroup of Gal(Q~/Q,N). The homomorphism j is compatible with Galois action (compatibility being defined in an evident manner using (i. I)) and it is injective since G is assumed to be separated. (b) Extensions of group schemes from S' to S: Let G~s, be a group scheme (as at the beginning). To give an extension, G/s of G' to the base S (up to canonical isomorphism) amounts to giving a sub-Gal(QN/Q,N)- module H C G(Q) whose elements are fixed under the inertia group at N. For the sub-Gal(Qx/Qs)-module H then inherits a Gal(FN/FN)-structure and H C G(Q) may be viewed as a compatible diagram of Galois modules of the form (i. 2). This compatible diagram gives us an ~tale quasi-finite group scheme G~,, and an isomorphism: G;~ ~ G;s "flS' ~- 'nS", which, by patching, gives our extension G/s. For a given G~s, there is a minimal and a maximal extension to the base S, which we shall denote G ~ and G ~ respectively. The minimal extension (extension-by-zero; compare [33]) G~ is defined by taking H={o}C G((~,,). The maximal extension G~ is defined by taking H to be the subgroup of G(Q) consisting in all elements invariant under the inertia group at N. If G/s is any extension of G~s, , we have G ~ C G C G ~. 44 MODULAR CURVES AND THE EISENSTEIN IDEAL From this discussion we have: Proposition (I.3). -- These are equivalent: (i) G/s, admits an extension G/s which is a finite flat group scheme. (ii) The inertia group at N operates trivially in the Gal(Q/Q,)-module associated to G/s,. (iii) G/~s is a finite flat group scheme. (c) Subgroup scheme extensions: Let G/s be a group scheme as above, and let H(Q) be any sub-Gal(Q/Q)-module of G(Q,). By the subgroup scheme extension to S, H/s , of H(Q) we mean the scheme- theoretic closure of H(Q) in G/s. To understand this, it is perhaps best to consider it over the bases S' and S" separately. Over S', we are taking the scheme-theoretic closure of H(Q) in the finite flat group scheme G/s, [55], which is a finite flat subgroup scheme H/s,C G/s, whose associated Galois module is our H(Q.) [55]- To describe it over S" we must give its " diagram (~ .2)" H(FN)CH(Q); one sees easily that H(FN)=H(Q)nG(FN) , the intersection taking place in G(Q). It follows that the subgroup scheme extension of H(Q) in G/s is a quasi-finite closed subgroup scheme H/s C G/s which is finite and flat over S', and whose associated Galois module is our original H(Q). Moreover, this construction provides a one-one correspondence between sub-Gal(Q/O)-modules in G(Q) and closed subgroup schemes in G/s. If H/sC G/s is a closed subgroup scheme, we may consider the quotient (G/H)/s ([SGA 3], exp. V, VIB) first as sheaf for the fppf topology. This quotient is representable by a group scheme (of the type we are considering) as can be seen, again, by working separately over the bases S' and S": Over S', H/s is a finite flat subgroup scheme of the finite flat group scheme G/s , and the quotient is representable, by [57] theorem i. Over S", one easily constructs the "diagram (1.2)" of the quotient and one finds: o o H(~N ) J" - > H(Q) G(FN ) J~ (G/H)(FN) fi~ (G/H)(Q) o o where JC~/H)is injective because H(FN)=G(F~)c~H(Q ). 45 4 6 B. MAZUR It follows from this discussion that there is a one-one correspondence between filtrations of G/s by closed subgroup schemes, and filtrations of G(Q) by sub-Gal(Q/Q)- modules. Moreover the successive quotients of any filtration of G/s by closed subgroup schemes are again group schemes (of the type we are considering), and their associated Galois modules are canonically isomorphic to the successive quotients of the corresponding filtration of G(~). (d) Determining G/s, by its associated Galois module: Here is a consequence of the work of Fontaine. Theorem (x .4) (Fontaine). -- Let G}~s!, G}~s ~, be two finite flat p-primary group schemes with isomorphic associated Galois modules. If either: (a) p+2 or (b) G}~, are both unipotent finite group schemes. Then r,.cl) is isomorphic to Gl2s ! v/s, Discussion. -- By Fontaine's theorem 2 [I4] and the subsequent remark (p. I424) , the isomorphism between the associated Galois modules extends to an isomorphism: GIll ,,.r~_csl (Fontaine works over the Witt vectors of a perfect field) /z~ = --'/zp A standard patching argument gives the version of Fontaine's result quoted above. (e) Vector group schemes of rank i. If V/s, is a finite flat group scheme killed by p, we may view V/s, in a natural way as admitting an Fp-module structure. If k is any finite field and V/s, is endowed with a k-module structure, we shall call V/s, a k-vector group scheme. The rank of V/s, (as k-vector group scheme) is the dimension of the k-vector space V(O). If V/s, is a kl-vector group scheme and ks/k 1 is a finite field extension, then by V| s the evident construction is meant (one takes the direct sum of as many copies of V as there are elements in a kl-basis of ks, and gives it the natural ks-module structure). Proposition (x .5). -- Let k be a finite field of characteristic p. Let V/s be a finite flat ~ | k-vector group scheme of rank I. Then either V/s~(Z/p)/s| or: V/s=~p/s Fpk. Proof. -- The Gal(Q/Q)-representation of a k-vector group of rank i is given by a character Z :Gal(Q/O_,,)~k* and hence determines a cyclic abelian extension of Q of order dividing pt i (pr = card(k)) unramified except at p. Such an extension must be contained in O~,(~) (~p a primitive p-th root of i) and therefore has order dividing p--I. Consequently the character Z takes values in F~Ck* and it follows that there is a sub-Fp-vector group scheme of rank i, V0/sC V/s whose associated Gal(Q/O) representation is given by the character Z : Gal(Q/Q)-~F~. By the Oort-Tate classi- fication theorem ([54], Cor. to thm, 3) applied to the group scheme of order p, V0/s, 46 MODULAR CURVES AND THE EISENSTEIN IDEAL one has that V0/Fp is either of multiplicative type or 6tale (1). Replacing V/s by its Cartier dual, if necessary, we may suppose that V0/s is Etale, and consequently the character Z is trivial. Moreover, the group scheme (V/F~) ~t has a k-module structure and is nontrivial since it contains V0/Fp. Its order is then >q, and at the same time <q since the order of V/s is q. It follows that V/s is 6tale, and has trivial Gal(Q/O)-action; it follows that as k-vector group scheme, V/s=(Z/p)| For a detailed study of k-vector group schemes, especially of rank I, see Ray- naud's [55]- Corollary (x. 6). -- Let V/s be a group scheme of order p. (i) Let p +-2. If the associated Galois module to V is Z/p, then V/s, ~ (Z/p)/s,. If the associated Galois module to V is ~p, then V/s, ~ ~p/s,- (ii) Let p = 2. Then V/s, is isomorphic either to (Z/2)/s, or to ~2/s'- (f) Admissible p-groups. Definition. -- An admissible (p-)group G over S (or over S') is a group scheme (as usual in this chapter: commutative, quasi-finite, separated, flat, such that G/s, is finite and flat) which is killed by a power of p, and such that G/s, possesses a filtration by finite flat subgroup schemes such that the successive quotients are S'-isomorphic to one of the two group schemes: Z/p or !% (called an admissible filtration). By (i. 6) and (c) G/s, possesses an admissible filtration if and only if its associated Gal(Q/Q)-module possesses a filtration by sub-Gal(Q/Q)-modules whose successive quotients are isomorphic to the Gal(Q/O)-modules Zip or ~p (called an admissible filtration of a Gal(Q/Q,)-module). Clearly a closed subgroup scheme of an admissible p-group is again admissible, as is the quotient group scheme of an admissible p-group by a closed subgroup scheme. We have the notion of short exact sequence of admissible p-groups: 0--> GI-+ G~-+ G3-+ 0 where O 1 is closed in G 2 and the morphism G 2--> G 3 induces an isomorphism of fppf sheaves 03/01-~ 03. To an admissible p-group we may attach the following numerical invariants: t(O) =logp(order of G/s, ) (the length of G) 8(G)=logp(order of G/s,)--logp(order of GIFt) (the defect of G) e(G) = the number of (Z/p)'s occurring as successive quotients in an admissible filtration of G/s,. (a) We may deduce this from the following simple consequence of the theory of Oort-Tate, which may also be checked directly: The group scheme czp over the base Spee (Z/p) admits no extension to a finite flat group of order p over the base Spec (Z/p2). 47 4 8 B. MAZUR h~(G)=logp(order(Hi(S, G))), cohomology being taken for the fppf topology ([SGA 3], Exp IV, w 6). Remarks. -- The invariant d(G)=logp(order G(Q)) depends only on the Galois module associated to G. The invariant 3(G) is detectable from the structure of G/spe~(z~ ). The invariant e(G) is detectable from the structure of G/Fp: ~(G) = logp(order G(Fp)). If p+2, one can also determine ~(G) from the Gal(Q/Q)-module structure of G(Q). This is, of course not the case if p=2. We are mainly interested, in this paper, in h ~ for i=o, 1. Note that: i-i0(s, G)=G(S), while Hz(S, G) may be given an appropriate " geometric " interpretation. (g) Elementary admissible p-groups. By an elementary admissible group G we shall mean an admissible group of length one. Up to isomorphism there are four elementary admissible p-groups: z/p, Z/p , v,,, where (Z/p~)/s is, as in (b), the extension-by-zero of (Z/p)/s, and similarly with t~/s,. The invariants of these elementary groups are given by the following table: z/p z/p 0 I 0 t OC I I 0 0 o(p+2) h ~ I o o I (p=2) o(p+2) h :t o o O if N~ I modp (p odd) where ~ = or N = -- I mod 4 (p = 2) I otherwise. It is straight forward to establish the first three lines of the above table. To compute Hi(S, l~p) use the Kummer sequence (offppf sheaves) o-+[~p-+Gm-+Gm-+o giving: Ha(S, Wp)=(Z*)/(Z*) p since the ideal class group of Z is zero. Also, Hi(S, Z/p)=o because there are no unramified cyclic p-extensions of Q,. The nontrivial class in HI(S, 1~2) is represented by the S-scheme Spec Z[%/~I], regarded as l~2--principal homogeneous space (torseur) over S. Forming the exact sequences offppf sheaves over S: 48 MODULAR CURVES AND THE EISENSTEIN IDEAL 49 o --~- Z/p t' -+Z/p ~ ~ -+o o-+ one computes H~ q~)=Z/p; H~ O?)=t~p(FN). The natural map: Hi(S, ~) -+ Hi(S, t~) is injective if and only if the principal homogeneous space Spec Z[~/~---I] for iz2 over S does not split when restricted to the base Spec F N (i.e. when N=--I mod 4). These facts establish the table. Proposition (x.7). -- Let G/s be an admissible group. Then: hi(G) -- h~ < 3(G) -- :r Proof. -- The right hand side of the above inequality is additive for short exact sequences of admissible groups. The left hand side is sub-additive in the sense that if o-->Gx-+G~-->G2-+o is such a short exact sequence, then: hi (G2) -- h~ (hl(Gi) -- h~ -~ (hi (Ga) -- h0(Ga)). To see this, one simply uses the long exact sequence offppf cohomology coming from our short exact sequence. One clearly has equality if, instead of hX(Ga) one inserts hl(Ga) '~- logp(order(image Hi(G2) in Ha(Ga))) in the displayed line above. The asserted subadditivity follows. Since any admissible group G has a filtration by closed subgroup schemes whose successive quotients are elementarY admissible groups, the discussion above reduces the problem of checking the asserted inequality for any admissible group to the same problem for elementary admissible groups, where it follows from an inspection of the table above. Remark. -- When r I (which will be the case in our applications) the asserted inequality is, in fact, an equality for elementary admissible groups. 2. Extensions of try by Z/p over S. The point of this section is to show that there are no nontrivial such extensions. Proposition (2. x). -- Let p be any prime number. Then: Ext~(l~v, Z/p)= o. Proof (1). __ To begin, we reduce our problem to a calculation in &ale cohomology. Let Sch ~-p/s denote the underlying scheme (ignoring group structure) and let: s : Sch ~, � Sch ls.v --~ Sch ~p, be the group law. (1) An alternate approach to the proof of (2. i) in the case of an odd prime p is to show that an element of Ext~(l/,p, Z/p) must go to zero in Ext~pec Q, using Herbrand's theorem below, and the argument of chapter III, w 5. One then could apply Fontaine's theorem (I .4) to conclude. 49 5 ~ B. MAZUR Since Sch lap/s is connected, and (ZIp)/s is Stale, there are no nontrivial 2-cocycles for lap/s with coefficients in (Z/p)/s, and therefore Ext~(~p, Z/p) is the kernel of: ~*-=~-~ H~(Sch ~, Z/p) > H~(Sch ~x ~, Z/p) where ~i are the first and second projections (i=I, 2) and cohomology is computed for the fppf topology or ([15] , w ii) since Z/p is Stale, for the Stale topology. To see the assertion made, the reader may verify it directly, following [12] and (e.g.) [SGA 3], exp. III, w I. If X, Y are any two schemes equipped with Fp-valued points: /x (i. I ) Spec Fp we allow ourselves to use the symbol XvY to refer to any scheme-theoretic union of X and Y along (subschemes which are nilpotent extensions of) Spec(Fp). Taking Spec(Fp)-+X to be one such scheme, and Y=S=Spec(Z) to be the other, denote by Hi(X) the Stale cohomology group HI(XvY, Z/p). One obtains an exact sequence: o -+ l(x) _+ Hi(X, z/p) -+ Itl(Spec(F ), Z/l,) using: the Mayer-Vietoris exact sequence for Stale cohomology, the fact that Spec(Fp) is connected, and that Hi(S, Z/p)=o. We learn, in particular, that the group Hi(X) is independent of which scheme-theoretic union of X and Y was made (provided that it is subject to the above conditions). A similar calculation gives an additivity formula for ~1 (for any diagram (2. I)): (2.2) HI(X V y) = Hl(X ) (~ Hi(y) o We may write Sch t~p =TvS where T denotes the " cyclotomic scheme "" Spec(Z[x]/(x'-l + x'-~ + . . . + ~ ) ). If M denotes the p-primary component of the Galois group of the Hilbert class field extension of (the field of fractions of) Z Ix]/(xP-l+xP-~+... + 1), then: HI(T, Z/l~ ) = Hom(M, Z/p). Therefore by (2.3), if l~i is the maximal quotient of M such that p splits completely in the field extension classified by l~i, we have: (2.5) HX(T) ----- Hom(~l, Z/p). The automorphism group of I*p/s maps to the automorphism group of the scheme T and we have canonical identifications: Aut(I*p/s) = Aut(T) = F; where asF~ operates by aa=" raising to the a-th power" in the group scheme !*~. The isomorphism (2.5) is compatible with this action in the following sense: 50 MODULAR CURVES AND THE EISENSTEIN IDEAL 5 I If a~F~ and 9etiom(l~i,Z/p), then q~(m)=(aa.9)(aa.m ) where the action of a a on M is the natural action of the morphism % : T-->T on M_cHI(T ) (one- dimensional homology). To decompose our spaces into eigenspaces for the action of F~ we need some terminology: If H is a Zp[F~]-module and jeZ/(p--i)Z, let H(J)={heH t %.h = aJ.h} (where if a eF~ we denote its operation on H by %). Then H= @ H (i), the summation being taken over all j~Z/(p--i)Z. By the compatibility formula above, we get: (2.6) HI(T)(J) = Hom(l~i (-j), Z/p) for all jeZ/(p--I)Z. Note that Sch(t~v� p.p) is a wedge (in the sense of v) ofS withp+i copies ofT; these copies can be considered as the images: TC Sch ~p ~ Sch(~v� ~p) where, to be noncanonical for a moment, we may take the maps x to be given by the set of 2 � I matrices: (a, I) for a=o, I, ...,p--I and (I,O). Using (2.2) we obtain that H~(Sch(~p� is a direct sum of p-~-~ copies of Hi(T). Let us describe this group in a more " choice-free " manner. Consider all imbeddings -c : ~p ~ i~p � ~p. The 2 � i matrices representing all imbeddings z range through the set of nonzero elements of Fp� Let Funct(A, B) denote the set of functions from A to B and form: Hl(Sch(I~p x y.,)) ~ Funct(Fp x Fp, Hi(T)) by sending heH~(Sch(I-tp� to the function (~:~'~*h). Let OF.(Fp� H~(T)) denote those functions which send (o, o) in Fp � Fp to o in Hi(T), and which are compa- tible with the natural action of F~ on domain and range. From our noncanonical description of Hl(Sch(ap� ~p)) it follows that ~ induces an isomorphism between H~(Sch(g.p� and @~(F~� HI(T)). By the analogous but easier construction for Sch I~v we get an isomorphism: Hl(Sch g.p) ~- r HI(T))=HI(T) and one can check the commutative diagram: Hl(Sch ~%) > Ul(Sch(g.p x ~p)) 9 OF;(FpxF p H~(T)) Ol,~ (Fp, H~(T)) a 5l 52 B. MAZUR where 8 is just the obstruction-to-linearity: If fe(PF~(Fp, Hi(T)) then 8f(x,y)=f(x§ We are reduced to analyzing the kernel of 3, the obstruction-to-linearity. Let q)j denote functions which bring o to o and are homogeneous of degree j, under the natural action of F~; on domain and range. Thus: q)Ff(F,, H~(T)) 8 >. OF$(Fp� ~I(T)) @ (Pj(Fp, Fp)| (j) ) @ (I)j(F~� Fp)| ~ where the summation is taken over j~Z/(p--I)Z. We are led to consider the maps: (2.7) %(V,� F,) for each j~Z/(p--i) Z. Clearly (I)j(Fp, Fp) is a one-dimensional vector space over Fp generated by the function x~x ~, and ~ applied to it is the function (x-t-y)~--xS--j. Thus (2.7) is injective if j=t=i. To show that Ext~(~p, Z/p)=o it therefore suffices to show that Hl(T)(t)=o. Equivalently it suffices to show that ~I(-I)=o. In fact, M (-a) vanishes. This is a consequence of a theorem of Herbrand [2o] together with the calculation of the second Bernoulli number. For the convenience of the reader, we shall reprove the theorem of Herbrand, which follows from the theorem of Clausen-von Staudt, Kummer's congruence, a power summation congruence (cf. [72], chap. V, w 8) and Stickelberger's theorem (cf. [23]). To prepare, let the Bernoulli numbers Bi be defined by: t/(et i)=ZBit~/i! (1) and the Bernoulli polynomials: B,(X) ~--~ (n).B,. X"-' (So B 0=I, B 1=-I/2, ...). We have these classical facts: If p is a prime, p. Bm is a p-integer, and B m itself is a p-integer provided m ~ o mod(p - I ) ( Clausen-von Staudt). If p is a prime, and mr o mod(p-- I ), then Bm/m is a p-integer whose residue class mod p depends only on m mod(p--I): gm/m-Bm+p_l/(m-t-p--i ) modp (Kummer). (1) This differs from Iwasawa's choice [22]. 52 MODULAR CURVES AND THE EISENSTEIN IDEAL Let p be an odd prime number. Suppose k is a nonnegative integer such that k -}- I ~- o rood p, and k--I ~o rood(p--I). Then: p--1 a k -=p. B k rood p2. (Power summation congruence [72], chap. V (8. i i) Cor. to theorem 4.) To apply the Stickelberger theorem, we use the class field theory isomorphism to identify the Galois group of the Hilbert class field of K=Q,[x]/(xP-l+... +x) with the ideal class group of K (1). Let Y denote the p-primary component of this ideal class group. Thus: M-+Y. It is important to check that 0 commutes with the natural action of 1~ on domain and range. The action on the domain may be viewed as follows: If L/K is the Hilbert class field extension, then L/Q, is Galois and the natural action of G on itself by inner automorphisms (~g(x)----gxg -1) induces an action of F~ on M, which is equal to the action considered above. The action on Y is induced by the natural action of F; = Gal(K/O) on ideals. The fact that 0 commutes with these actions is, then, VII, theorem (I i. 5) (i) of [5]. Thus, we have: M(J) =y(J). In what follows we suppose that p2>2 and j is odd. This makes sense because j is an integer mod(p--i) and p is odd. For convenience, takej to be an ordinary integer in the range o<j<p--I. Write f=p--~--j (so j=--j mod(p--~), and o<j-<p--I). Let co :F;---~Z; be the Teichmfiller character. We shall now quote (what is, in essence) Stickelberger's theorem (cf. Iwasawa's p-ac L functions [23]. Our " Y " replaces his " So "): Proposition (2.8). -- y(1) = o. If j + i, then the p-adic number: p--1 a~0 is a p-adic integer, and ~j. Y(J) = o. Corollary (2.9) (Herbrand). -- Let j be odd and different from ~. If Bp_j~o modp, then Y(J~ = o. Proof. -- We show, under the hypotheses of the corollary, that ~j is a p-adic unit. For this, we examine: p--1 p--1 Z a.o~-3(a)= Z a.oJ(a) modp ~. a=O a=O (1) For definiteness, take the class field theory isomorphism 0 to be the map induced from +-a as in [5], VII, w 5. 53 54 B. MAZUR p--1 p--1 Since o~(a)--aPmodp 2, Y~ a.eoY(a) = Z akmodp ~, where k:pj+I. a=0 a~0 Since p42, k+I,omodp. Since j4I, k--I,omod(p--i). Therefore the power sum congruence (above) applies, giving: p--1 ak=_p.Bk modp 2. a=0 To prove corollary (2.9) we show that if Bp_~=BT+ t is not congruent to zero modp, then Bk=Bp~+l also is not. But pf+I=f§ and so Kummer's congruence applies; it proves the assertion since p j§ i and j§ i are both p-adic units. Corollary (2.I0).- Y(-t)=o (also: Y(-3)=Y(-5)=Y(-7)=Y(-9)--o) for all p (also: Y(-11)=o for all p4=69I, ...). Proof. -- We may suppose p odd (this is the only place in this paper where p = 2 is significantly easier than its fellow primes). Writing Y(-~)=Y(P-(~+I)) we see (2.9)that Y(-i)----o if B~+l~o modp and i§ or (2.8) if i§ The Corollary then follows from Clausen-von Staudt, and determination of the first few Bernoulli numbers. 3" Etale admissible groups. Fix a prime number p different from N. We consider only p-groups in this section. By a constant group over any base we mean an ~tale finite flat group scheme with trivial (constant) Galois representation. By a [x-type group we mean a finite flat group scheme whose Cartier dual is a constant group. By a pure (admissible) group we mean a finite flat group scheme which is the direct product of a constant group by a ix-type group. Proposition (3- x ). -- Any dtale admissible finite flat group over S is constant. Any admissible finite flat group of multiplicative type over S is a [x-type group. Proof. -- The second assertion follows from the first, by Cartier duality. To see the first, let G be an ~tale, finite flat admissible group over S. Proceed by induction on the length of G, and suppose /(G)_> I. Then, there is a finite flat subgroup G0C G such that G/Go=Z/p , since G is both 6tale and admissible. By induction, G o is constant, and G represents an element in Ext,(Z/p, Go). Now consider the Ext i exact sequence associated to o-+Z-+Z--~Z/p-~o over S. Note that Ext~(Z, Go)=Hi(S, Go), and Hi(S, Go) vanishes since G o is a constant group and there are no nontrivial unramified (abelian) extensions of Z. We obtain an isomorphism: H0(S, G0)/p.H~ Go) -~ Ext,(Z/p, Go). 54 MODULAR CURVES AND THE EISENSTEIN IDEAL Performing the same calculation over (e.g.) Spec(C) rather than S, and comparing (using that S is connected) we get: Ext,(Z/p, Go) -+ Ext~per , Go) which indeed implies that every extension of G o by Z/p over S is constant. Q.E.D. If G is a constant admissible group over S', killed byp ", it is sometimes convenient to write: G = (Z/p') | C where C is an abstract finite group killed by p", and Z/p ~ is, to be sure, the constant S'-group scheme. The | construction is the evident one. We may take: C = Homs,(Z/p', G). Similarly, if G is a [z-type group over S', killed by p", we may write: G = i~:| M where M is the abstract finite group Homs,(btp, , G). Now let G/s, be an 6tale admissible group which is an extension of A/s, by B/s, where both A and B are constant groups over S'. Write A~Z/p~| B~Z/p~| for an appropriate integer e, and abstract finite groups A, B killed by p". We may view G/s, as giving rise to an element: gcExt~, (A, B). To deal with Ext~,(A, B) it is useful to have the following fairly complete des- cription. Let p~[IN--r. Set (Z/N)*=Hom((Z/N)*,Z/p ~) (the Pontrjagin p-dual). Lemma (3.2). -- There is a canonical isomorphism: Ext~,(A, B)= Ext(A, B)| ((Z/N)*| , B)[p~]) (to be described in the course of the proof below). Proof. -- By Ext(A, B), we mean Ext in the category of abelian groups. By Horn(A, B)[p~] we mean the kernel of p~ in Horn(A, B). The map Ext(A, B) -~ Ext,, (A, B) is the one which associates to an extension of abstract groups o ~ B ~ E -+ A-+ o the corresponding extension of constant groups over S'. The map Ext~.(A, B) -~ Ext(A, B) is " passage to underlying abstract group " (or equivalently: restriction of the base from S' to Spec(C)). To establish the isomorphism, resolve A by free abelian groups (of finite rank): o-+R-+F-+A-+o and evaluate the long exact sequence of Ext's to get: o-+ Ext(A, B) -+ Ext~,(A, B) -+ Hom(a, HI(S ', B)) -~o. Since B is a constant group scheme, an element in HI(S ', B) is given by the following data: an abelian extension K/Q unramified outside N, and an injection Gal(K/Q) C B. 55 56 B. MAZUR Since any such extension is isomorphic to a subfield of Q,(~N) (recall: p4~N) we have the canonical isomorphism: Ha(S ', B)= Hom((Z/N)*, B) (using the isomorphism Gal(Q(~N)/O,)~ (Z/N)*) and therefore, we have the canonical isomorphisms : Horn(A, Ha(S ', B))= Hom((Z/N)*, Horn(A, B)) = Hom((Z/N)~, Hom(A, B) [p~])) where the subscript p above means p-primary component. Since (Z/N)~ is a free module of rank i over Zip ~', we have: Hom(A, HI(S ', B))= (Z/N)*| Hom(A, B)[p~]. Remark (3.3)- -- It is sometimes convenient to make a choice of a generator +N : (Z/N)*-+ Zip ~ (1), in which case, an element geExt~,(A, B) gives rise (under projection to the second factor of the formula of (3.2)) to a well-defined element N| where yeHom(A,B)[p~]. We refer to y as the classifying map for g (dependent, of course, on the choice of +~). The associated Galois module to the group scheme G/s, may be neatly described in terms of +N and y, as follows. Fix oeGal(Q/O_.). For xEG({~..) the mapping x~a(x)--x induces a homomorphism from A----A({~..) to B=B(Q) which is simply q~N(~).y(~) where ~ is the image of x in A. If G is an dtale admissible group over S', let the canonical sequence of G denote the filtration of closed (dtale admissible) subgroup schemes over S': o=GoC GIC 9 9 9 C G defincd inductively as follows: Gi+ I is thc inversc image in G of thc group generatcd by the S'-sections (i.e. the Galois invariant sections) of G/G~. Thus, thc succcssivc quotients are constant groups and G = Gm for some integcr m. If m is the Icast such integer, say that G is an dtale (admissible) group of m stages. If G1C G is the " first stage" then, by definition, Gx is the largest constant subgroup of G. If G 2 C G is the "second stage ", then G~ is an extension of the constant group A-----Gs/G1 by the constant group B=G~ and, furthermore, its classifying map y is injective since Gz is the maximal constant subgroup of G~. If G is an admissible group of multiplicative type over S', we may similarly define the canonical sequence for G, as follows: G i + 1 C G is the inverse image in G of the largest ~t-type subgroup of G/G v Note that if G is an admissible multiplicative type group then its canonical sequence (1) Which we also view as a homomorphism from Gal(Q/Q) to Zip a by composition with: Gal(Q/Q) ~ Gal(Q(~N)/Q) ~ (Z]N)*. 56 MODULAR CURVES AND THE EISENSTEIN IDEAL is not necessarily dual to the Cartier dual of the canonical sequence of the dtale admissible group G v. Rather, G 1 is dual to the largest constant quotient group of G v, etc. The natural functor which passes from multiplicative type admissible groups to dtale admissible groups, and which preserves canonical sequences is the functor: G~,gg'oms,(~a It, G) where gh ~t = lim t*:.. Lemma (3-4) (Criterion for constancy). -- Let G be an gtale admissible group over S'. If N ~ i rood p, then G is constant. In general, G is constant if and only if there is a prime number t +- N such that: a) t is not a p-th power modulo N; b) The action of ~t (1) in the Galois representation of G is trivial. Proof. -- Consider the canonical sequence (Gi) for G. We need only show that G~=G1, under the above hypotheses. Thus we may assume G=G 2 is an dtale admissible group of two stages. Let y : O2(O) --+GI(O ) be its classifying homomorphism which is injective by the above discussion. Thus, for any t+N (even for g=p) the endomorphism q~t--I of G(Q) induces a homomorphism +s(t).y :G2(Q)--~GI(Q) where +N is the chosen homomorphism of remark (3.3)- Also p~.y=o, where P~IIN--I" It follows that if N~I modp, y=o, and we are done. Ift is not a p-th power mod N, it is a generator of the p-part of the group (Z/N)*, and therefore +N(t) is a unit in the ring Zip ~. Hypothesis b) then implies that y = o. Lemma (3-5) (A ~-type criterion). -- Let G be an admissible multiplicative type group over S'. If N ~i mod p, then G is a ~-type group. In general, G is ~z-type if and only if there is a prime number t +p, N such that: a) t is not a p-th power rood N. b) The Frobenius element ?t acts as multiplication by t in the Galois representation of G. Pro@ -- Pass to the dtale admissible situation by applying ~oms,($'a ~, --) (or by Cartier duality) and then use lemma (3-4)- Lemma (3- 6). -- Let t +p, N be a prime number not a p-th power mod N. If G is a multiplicative type group, then the Galois module of G 1 (the first stage in its canonical sequence) is the kernel of cpt--t in the Galois module of G. Pro@ -- Passing to the dtale situation by the functor ~oms,(~a ~, --) we may replace G by an gtale admissibie group over S', and we must show that the Galois module of G1 is the kernel of q~t--i. Work by induction on the number of stages of G. Suppose that it is true for groups ofm--i stages and let G have m stages (m>2). Thus G/GIn_ 2 has two stages (1) t-Frobenius. 57 58 B. MAZUR and its "first stage subgroup " is, by construction, Gm_l/Gm_ ~. Using the for- mula (3.3) ?t(x)--x=r where 7 is the classifying homomorphism for the 2-stage group G/GIn_2, xE(G/G,._~)(Q.) and Y is its image in (G/Gm_I)(Q), we see that the kernel of ?t--i in (G/G,._~)(Q) is the subgroup (G,~_I/Gm_2)(~..) (since 3" is injective, and r is a unit in the ring Zips). Consequently, anyelement xeG(Q) which is in the kernel of ~t--i must be contained in Gin_l(0.. ) C G(~=)). But Gm_l is a group of m--I stages, and therefore xeGl(Q), by induction. 4" Pure admissible groups. Proposition (4. x ). -- Let p ~ 2. Let A be a constant group and M a ~-type group. Then: Ext,(A, M)= o. Pro@ -- This reduces to showing Ext,(Z/p, ~p)=o. But applying Ex((--, ~p) to the exact sequence offppf sheaves o~Z-+Z-+Z/p-+o yields a long exact sequence which may be evaluated using the fact that Ext,(Z, lzp)=Ht(S, ~p). One gets the short exact sequence: o H~ + Ext,(Z/p, + n (s, o. From the Kummer sequence o-'~p-~Gm-~Gm-+o offppf sheaves, and the fact that the ideal class group of Z vanishes, one gets: Hi(S, ~p) -= (Z*) /(Z*) p. Thus we have a short exact sequence: o-+ (Z') [p] -+ Ext,(Z/p, ~p) -+ (Z*)/(Z*) P-+ o. Now suppose p + 2, and one sees that the middle group must vanish. If p = 2, we get: Proposition (,t.2). -- There are three nontrivial extensions of Z/2 by ~2 over S: Extension 1: an extension whose associated Galois representation is trivial, and whose underlying abelian group is cyclic of order 4. Extension 2: the unique nontrivial extension over S killed by 2: o-+ ~2~D-+Z/2-+o. Its associated Galois representation factors through Q(X/~-~-I). If we let: r : Gal(Q/Q) -+ Gal(Q(~v/---~)/O.~) -~ Z/2 be the composite where the first map is the natural projection, and the second an isomorphism, then the Galois representation associated to D is given as follows: (4.3) o(x) -- x = r a(O)- V(~) where if xeD(Q), then ~ is its projection to Z/2, and 7 is the only surjective homomorphism D(O_~) :+ ~(Q) with kernel ~(Q). 58 MODULAR CURVES AND THE EISENSTEIN IDEAL As usual, identifying Gal(Q(~v/~ )/O) with (Z/4)* , /f ?e denotes t-Frobenius (in the Galois group of any extension field of Q,(~r which is Galois over Q, and unramified over t) then we also write ~-1(~)for +-l(?e). One has: +_l(t)=I /f and only /f /-=--I mod 4. Extension 8: (the sum in Ext 1 of the above two elements) an extension whose underlying abelian group is cycgc of order 4, and whose Galois representation satisfies the same formula as above. Proof. -- This is evident from the exact sequences in the proof of (4-I) except for the assertions concerning Galois representations. To see those, one must recall that the nontrivial tJ.2-torseur representing the (nontrivial) element in Hi(S, y.~) is the S-scheme Spec Z[~v/~i- ]. Remark. -- The group scheme D/s (Extension 2 of (4- 2) above) will play a central role in our study of the prime 2. Since Fontaine's theorem does not apply to admissible 2-groups in general, the following result is useful: Proposition (4-4)- -- Let D/s, be a finite flat group scheme, and q~ : D/Q-~ D/Q an isomorphism over Q, (equivalently: an isomorphism of associated Galois modules). Then ~ extends to an isomorphism q~ : D/s, ~ Dfs, of group schemes over S'. Pro@ -- Since the associated Galois module to D' is admissible, D~s, is admissible, and since the inertia group at N operates trivially in the Galois representation of D (and hence also of D'), D' extends to a finite flat group scheme over S. Since the Galois representation of D' satisfies (4.3), D' cannot be an extension of F2 by Z/2 (2. i), nor of Z/2 by Z/2 (3-3), nor of Ix2 by p~z (applying (3-3) to its Cartier dual). Therefore it must indeed be isomorphic to D, by (4.2)- Since there is only one nontrivial autommphism of the Galois module associated to D, and this automorphism extends to an automorphism of D/s, our proposition follows. Proposition (t-5) (Criterion for purity: p+2).- Let p+2, and let G/s, be an admissible group. These are equivalent: a) G is pure. b) The associated Galois module to G is pure (i.e. it is the direct sum of a constant Galois module and the Cartier dual of a constant Galois module). c) The action of inertia at N is trivial on the associated Galois module to G. d) G extends to a finite flat group scheme over S. Pro@ -- Clearly a)=>b)=~c). By (~.3), c)=>d). To conclude, we must show that any finite flat admissible group over S is pure. Let G be such a group, and: o=GoC GIC G~C ... C G,=G an admissiblc filtration. Thus thc succcssivc quotients arc isomorphic either to Z/p or to ls.p over S. 59 60 B. MAZUR Step 1: We may suppose all the successive quotients isomorphic to l.tp precede those isomorphic to Z/p. This follows immediately from proposition (2. I), and induction. Therefore, for some s, G 8 C G is an admissible subgroup of multiplicative type, and G/G8 is an admissible dtale group. Step 2: G, is a ~-type group and G/G s is constant. Proof: (3-i). Step 3: G is a trivial extension of the constant group G/G~ by the ~-type group G~. Proof'. (4.1). Remark. -- Demanding that the action of inertia at N be trivial in the Galois representation is clearly not sufficient to insure purity when p =-2 (e.g., consider the nontrivial extension D of (4.2)). Nevertheless, for admissible 2-groups over S killed by 2, purity is equivalent to the requirement that the action of Gal(C/R) be trivial in the associated Galois module. As it turns out in our ultimate applications, however, the notion of purity is not the relevant one when p = 2. The final proposition of this section will be used in studying the cuspidal subgroup chap. II, w I I). Proposition (4-6). -- Let Cjs be a finite flat group whose underlying Galois module zs a finite cyclic group with trivial Galois action. If C is of odd order, then C is a constant group. If C contains a subgroup isomorphic to W2, then the quotient C/W 2 is a constant group. Proof. -- The first assertion of (4.6) follows from (I .6) and (3-4)- As for the second, we may suppose that C is killed by a power of 2 (say 2~). If ~=I, we are done. Now suppose that e----- 2. It suffices to show that C/V~2 is dtale over S. Clearly C/~2 cannot be isomorphic to ~2, for then the Cartier dual of C/s would be 6tale, hence constant, and so the Galois action on C could not be trivial. Thus C/W2=~Z/2. Now let c~>2. We shall show that C/W2 is ~tale as follows: filtering C by the kernels of successive powers of 2, if C/Vq were not ~tale, using the result proved for 0~=2, one could obtain a subquotient of C, whose underlying abelian group is cyclic of order 4, and which is an extension of W2 by Z/2, which is impossible by (2. i). 5. A special calculation for p = 2. Let Ext,_ s'(A, B) denote the subgroup of elements in Ext~,(A, B) which represent extensions of A by B which are killed by multiplication by 2. Consider the (nonflat) surjective homomorphism Z/2--~ll~ (over S'). This induces a homomorphism: Ext,_ s.(ll2, Z/2) % Ext~_s.(Z/2 , Z/2) and we shall show that this map is injective. The full story, however, is the following: 60 MODULAR CURVES AND THE EISENSTEIN IDEAL 61 Proposition (5. x ): a) Ext~_s,(Z/2, Z/2) is of order 2. b) If N~ zk i mod 8, then Ext~_s,(l~2, Z/2)=o. c) If N = + I rood 8, then the homomorphism ~ is an isomorphism of groups of order 2. Pro@ -- a) Follows from HI(S ', Z/2)=Ext~_s,(Z/e, Z/2). As for an analysis of Ext~_s,(p%, Z/2) there are two ways to proceed. We may adapt the general method of (2. i) to the base S', or (since our group schemes have such small orders) we may work directly. We choose the latter course. Consider the composition: : Ext~_s,(l~2, Z/2) ~ Ext~_s,(Z/2, Z/2) -+ HI(S ', Z/2). A " geometric " construction of ~ is the following: If.x is an element in Ext~_s,(tt2, Z/2) represented by an extension: (5.2) o-+Z/2-+E~ tt2-+o, let r : S'--~t~2/s, denote the nontrivial section, and let E,C E denote the fiber-product: E r > E S' '> P~2 Thus E, is the " nontrivial " Z/2-coset. It is a Z/2-torseur over S' and represents the element ~Ix) in HIIS ', Z/21. The scheme-theoretic intersection of E~ and Z/2 in E consists in two points lying over Spec F 2. From this we deduce that the prime 2 splits in the S'-extension E,. If ~(x)=o, E, is a trivial Z/e-torseur. Take the subgroup of E generated by the (unique) S'-section of E, which meets (at Spec(F2)) the zero-section of E. This is a group scheme which projects isomorphically to ~, and therefore gives a splitting of (5.2), showing that x=o. Thus ~ (and hence ~) is injective. Now suppose that x is nontrivial (i.e. (5- 2) does not split). The Galois representation associated to the group scheme E of (5-2) is isomorphic (to be sure) with the Galois representation associated to the pull-back via ~. In particular (3.3): If aeGal(Q/Q) and zeE(O..), we have ~(z)--z=+N(z).y(5) where 5is the image of z in b%(Q) and y : ~2(Q)~Z/2 is an isomorphism. This Galois representation factors through the unique quadratic number field in Q(~N), namely" t Q.(v'-N ) if N~--I mod4 K = [ Q,(V/N) if N - i mod 4. Note that if N ~ =k I mod 8, then 2 does not split in K, whence b). As for c) we need only construct a nontrivial extension (5.2) when N---~-i rood 8. We omit 61 62 B. MAZUR the details (noting that no use of c) is made in this paper) and merely sketch this construction: Since 2 does split in K, when N = :51 mod 8, one can glue the S'-scheme Z/2 and the nontrivial Z/2-torseur over S' transversally at their closed points of characteristic 2, and check that the evider~t group law away from characteristic 2 extends to a group- scheme structure of S'. II. -- THE MODULAR CURVE Xo(N ) x. Generalities. We shall be reading closely in two sources of information concerning moduli stacks, their associated coarse moduli schemes, and the theory of modular forms: [9], [24]- Our ultimate object is to derive as complete a description as possible of j/s, the Ndron model of the jacobian of X0(N ) over S (N~ 5, a prime number; X0(N ) the modular curve associated to P0(N)). Technically, reduction to characteristics 2, 3, and N (in that order) produce the thorniest problems, and we shall spend most of our time dealing with them. We keep to most of the conventions of [9]- Thus, for m any integer, and HC GL~(Z/m) we have the algebraic moduli stack d/a ([9], IV, (3.3)) proper over S, which may be interpreted over Spec Z[I/m] as the fine moduli stack classifying generalized elliptic curves with a level H-structure ([9], IV, (3-i)). Its associated coarse moduli stack ([9], I, (8. i)) may be denoted M H. If H is the trivial subgroup of GL2(Z/m ) we writeJg,~for~ H. If H=F0(N)={( a ~)c-omodN} write: #r R =M/o(N) ; MH = M0(N ) . Given a pair (E/T , h) where E is an elliptic curve (or a generalized elliptic curve [9], chap. II) over the scheme T, and h is a level H-structure of EIT, then the T-valued point of M H determined by this pair will be denoted j(E/T , h). In relating modular forms to differential forms, and in other arguments as well, we shall have use for certain refinements of M0(N), associated to level structures H, where d/~H= M H (i.e. where the fine moduli stack " exists " as an algebraic space). Two notable refinements having this property are ([9], IV, th. (2.7)): a) Take re=N, and H=Po0(N)={( a bd)c=omodN, a=imodN} (recall: N> 5) in which case we write MH=MI(N ). b) Take m=3N and H=P0(N;3)={(: ~)-(; ~ in which case we write M H = Mo(N; 3). The schemes MN-+ MI(N )-+M0(N ) are smooth when restricted to: S'= Spec(Z [i/N]). 69 MODULAR CURVES AND THE EISENSTEIN IDEAL As in [9], the superscript h (d/Z0(N) h, M0(N) h, etc.) refers to the open substack or subscheme obtained by removing the " supersingular points " of characteristic N. The precise geometric structure of M0(N)/s is given by [9], IV, th. (6.9). In particular, Mo(N)/F~ is a union of two copies of P~F~ (the j-line) intersecting transversally at the " supersingular points ", where a point x on the second copy gets glued to the image under N-Frobenius x/NI on the first. One has that M0(N)) s is smooth, and if j is a supersingular point of characteristic N (using [9], IV, (6.9) (iii) and the fact that N>5) then M0(N)/s is regular at j if j+o, I728. In the latter two cases, M0(N ) is formally isomorphic to : W(~N) [x,y]/(x .y-- N 3) if j~o W(F~) [x,y]/(x .y-- N 2) if j=I728. In any case, M0(N )-~S is locally a complete intersection, hence Gorenstein, and hence also Cohen-Macaulay [3]. By suitable blow-up of the points j=o, 1728 in characteristic N, when they are supersingular, we may arrive at the minimal regular resolution of M0(N)/s, which we call X0(N)/s. See the appendix for a study of these minimal regular resolutions in a somewhat broader context. The structure of the " bad fiber " (i.e. over FN) of X0(N ) may be schematized as follows: Z' E Z blow-up of j = 1728 when supersingular (-~N - -- i mod 4) transversal intersection at j+ ~ 728, o, supersingular cusps blow-up of j=o when supersingular (-r ~ --I mod 3) Diagram I The irreducible components E (which occurs if and only if N-- i mod 4) and F, G (which occur if and only if N ~- -- i mod 3) are the " results " of the appropriate blow-ups, and are all isomorphic to P~F,- See appendix for further discussion. The mmphism Xo(N)-+S is a local complete intersection, and, again therefore a Gorenstein morphism, and hence Cohen-Macaulay. Clearly M0(N)/s,=X0(N)/s, , and therefore (over any base extension of S') we have two possible names for the same thing. We try to keep to this usage: it will be called M0(N)/s, when we are interested primarily in questions of modular forms, and X0(N)/s, when we are interested in more 63 64 B. MAZUR geometric questions. Also, for reasons of consistency, and compatibility with other authors, we allow ourselves the same double notation M~(N)/s, = XI(N)/s, in dealing with H=P00(N), and likewise: M0(N; 3)/spec(zEt/3m)----X0(N; The usual names (o and oo) are given to the two cusps of M0(N ). We view these as (nowhere intersecting) sections of M0(N)/s ([9], VII, w 2). They also give rise to sections of X0(N)/s (denoted by the same symbols) and, after arbitrary base change T~S, to T-sections of X0(N)/T. The cuspidal sections o and oo distinguish themselves as follows: The morphism of stacks dt'0(N)--->.~r ) induced by the rule (E/w , H)~E/T is unramified at oo and ramified at o. 2. Ramification structure of XI(N ) ~ X0(N ). As always, let N be a prime number -->5. Let k be a field which is algebraically closed and of characteristic different from N. The map (2. I) XI(N )-+X0(N ) over k is unramified at the cusps, and has precisely these points as ramification points: TABLE i Name of point Occurs if Structure Char k in X0(N ) Value of j and only if of inertia group (i)+, (i)_ 1728 N = I mod 4 cyclic of order 2 ~=2, 3, N (p)+, (p)- o N =- i mod 3 cyclic of order 3 (p)+, (p)- N ~ I mod 3 cyclic of order 3 2 o = x 728 N -=-- I mod 4 cyclic of order 2: "wild 6) ramification of first type " (i)+ , (i)_ N = i mod 4 order 2 o = I728 (P) N ~---- x mod 3 order 3: "wild rami- fication of first type " Definition (2.2). -- A Galois p-cyclic extension of local fields whose residue fields are of characteristic p will be said to be wildly ramified of the ,~-th type if the higher ramification sequence (Gi) (cf. [6o], chap. IV) of subgroups of its Galois group G has the following structure: G = Go = G1 ..... G~ ..... o. We shall establish the facts of the above table. Recall that (since k is of charac- teristic different from N) the cusps are unramified in the mapping (2. I). If (E, C) 3)/sp0~(z~lj3Nj)- MODULAR CURVES AND THE EISENSTEIN IDEAL is a pair representing a point j(E, C) eX0(1V ) (1), then the automorphism group Aut(E, C) denotes the stabilizer of C in Aut(E); since N~5, the natural homomorphism: Aut(E, C) ~ Aut(C)=(Z/N)* is injective (2). Passing to the quotient: Aut(E, C)/(i I) ~ (Z/N)*/( I)=Gal(Xx(N)/X0(N)) the above homomorphism identifies Aut(E, C)/(+~) with the inertia group of the point j(E, C). If j(E)+o, I728 then Aut(E)=(+ i), and therefore j(E, C) is not a point of ramification. Characteristic k+2, 3, N. -- j(E)=I728: The group Aut(E) is cyclic of order 4. It can stabilize no cyclic subgroup of order N, C C E if N ~ i mod 4. On the other hand, if N = I rood 4 there is a 4-th root of unity in F and consequently Aut(E) stabilizes precisely two cyclic subgroups of order N. Call them C* and write (i)  =j(E, C We have (i)++(i) - since no element of Aut(E) interchanges C + and C-. This establishes the first line of the table. j(E) = o: The group Aut(E) is cyclic of order 6 and reasoning similar to the above establishes the second line of the table. Characteristic k-= 2. -- Let E be an elliptic curve with j(E)= 1728 = o. We may take E to be the curve y2-ky=xL The endomorphism ring of E is the ring of Hurwitz quaternions and its automorphism group is of order 24. The quotient Aut(E)/(zki) is isomorphic to 9.I4, the alternating group on 4 letters. The cyclic subgroups of 9/4 have orders I, 2, 3 and any two cyclic subgroups of the same order are conjugate. Fix cyclic subgroups Hz, HaCAut(E)/(zkI) of orders 2 and 3 respectively. Note that the inverse images of these in Aut(E) are cyclic groups of orders 4 and 6 respectively. As above, then, H3 stabilizes precisely two cyclic subgroups of order N (call them C if N--Imod 3 and none if N~Imod 3. Write (p)177 Since H a is its own normalizer in Aut(E)/(~I), (p)+4=(p)- and we have established the third line of the table. The subgroup H 2 stabilizes two cyclic subgroups of order N (call them, again, C +CE) if N=imod4 and none if N*I rood4. But the normalizer of I-I 2 in Aut(E)/(zLI) is isomorphic to the Klein 4-group. Since the entire Klein 4-group cannot stabilize C :~, any element in the normalizer of H 2 which is not in H z must inter- change C + and C-. Consequently j(E, C +) =j(E, C-). Denote this point (i). Clearly, (i) is a point of wild ramification. We shall show it to be offirst type, using an argument (a) Here C is a cyclic subgroup of order N in the elliptic curve E, giving the " level r0(N ) structure ". (3) If a ~ + I is an automorphism of any elliptic curve over any field k, then a is of order 4 or 6, and generates a ring ofendomorphisms isomorphic to the ring ofcyclotomic integers of that order. See the discussion in Appendix I of [29] concerning endomorphism rings, and automorphisms. The assertion concerning injectivity above then follows, for there is no homomorphism of the ring of cyclotomic integers of order 4 or 6 to FN, which sends a to x, provided N~>5. 65 66 B. MAZUR communicated to us by Serre: For any field k of characteristic different from N, one has the short exact sequence: --->- 1"* t~l t~l ~ 1 0 J hZXo(N)[ k ---> XZX,(N)[ k ~Xx(N)/Xo(N ) ---> 0 where the zero on the right comes from the fact that Xo(N ) is smooth, and f: XI(N ) --->Xo(N ) is generically separable. Note that dimkH~ is the degree of the global different ([6o], chap. III, w 7, Prop. I4) giving us the Hurwitz Formula. Namely, the degree of the global different of XI(N)/X0(N ) is: 2 .gl(N) -- 2 -- (~-~) 9 (2g0(N)-- 2) where gi(N) is the genus of the curve X~.(N). It follows that the degree of the global different of XI(N)/X0(N ) is independent of the characteristic of the field h (provided that it is different from N). From the first two lines of our table, choosing k to be of characteristic diffeIent from 2, 3 and N, we compute the degree of the global different to be: where if r is a rational number we let the symbol < r > be r if r is an integer and o if not. On the other hand, if k is of characteristic two and if (i) is wildly ramified of the v-th type, using prop. 4 of [6o] chapter IV, from what we have established concerning the ramification structure of XI(N )/X0(N ) we compute the degree of the global different to be: Consequently, v=i, and the third and fourth lines of our table have been established. Characteristic k = 3. -- Here, again, we take E to be an elliptic curve with j(E)=o=I728; for example: y2=x~--x. The group of automorphisms Aut(E) is of order i2 and has the following structure: it contains a normal subgroup of order 3, 9I 3 C Aut(E) such that the quotient of Aut(E) by ~3 is a cyclic group of order 4, which acts in the unique nontrivial way on ~I3 ([29], App. i, w 2). The center of Aut(E) is (i I) and Aut(E)/(+I) is isomorphic to 63, the symmetric group on 3 letters. Again we have that the cyclic subgroups of ~a have orders i, 2, 3 and any two cyclic subgroups of the same order are conjugate. Fix cyclic subgroups H~, H a C Aut(E)/(~:I) of orders 2 and 3 respectively. It is again true that the inverse images of these in Aut(E) are cyclic groups of orders 4 and 6 respectively. From this point on, to establish the last two lines of our table, we proceed exactly as in the case of characteristic 2, with the one important difference that now it is H2 which is its own normalizer in Aut(E)/(4-I) while H3 is normal in Aut(E)/(4-i). Our table is established. 66 MODULAR CURVES AND THE EISENSTEIN IDEAL N--I) S'= Corollary (2.3).- Let re=numerator ~. Let Spec(Z[I/N]). Let X2(N)/s, -~ Xo(N)/s, denote the unique covering intermediate to XI(N)/s, ~ Xo(N)/s, which is a Galois covering, cyclic of order n. Then X2(N)/s, ~ X0(N)/s, is itale. We shall refer to the above ~tale covering as the Shimura covering. 3" Regular differentials. Deligne and Rapoport [9] work out Grothendieck's duality theory in the case of a Cohen-Macaulay morphism ~ : X-~T (purely of dimension d). We shall recall the contents of [9] in the case d=I, with some change of notation. Definition (3. x ). -- If rc : X~T is a Cohen-Macaulay morphism purely of dimension i, where T is a noetherian scheme, the sheaf of regular differentials is: f~x, =3r (1) ([9], chap. I, (2.~. ~)). The sheaves ~x/, are flat over T, their formation commutes with arbitrary base change T'--~T and with ~tale localization of X. If X/T is smooth, then f2x/" = ~,. If X is a reduced curve over an algebraically closed field k which has only ordinary double point singularities xl, ..., xe and if (x~, x~') denotes the inverse image ofxi in X*, the normalization of X, then the regular differentials on X consist in meromorphic dif- ferential forms on X* regular outside of the x~, x", having at worst a simple pole at the x~ and x~', and verifying: re N =-- res~, (i = I, ..., t). The duality theorem gives an isomorphism. If o~ is a locally free 0T-Module, and if the RJ~.o ~ are locally free 0T-Modules , the duality theorem (loc. cir. (2.2.3)) gives: (3.2) R~-S=,(~ | ~ ) (RJT~,~') ~ where ~ denotes OT-dual, We prepare to apply the duality theorem to the morphisms 7~T, 7~, which are the base changes to T---~S of the morphisms occuring in the diagram: where i is the minimal regular resolution introduced in w i. (x) Deligne-Rapoport call this ~XjT" We often omit the subscript X/:r when no confusion can arise. 67 68 B. MAZUR Let ~M.(s)is(cusps) denote the locally free sheaf, which when restricted to the complement of the cuspidal divisor, is equal to the sheaf of regular differentials and whose sections in a neighborhood of the cuspidal divisor are meromorphic differentials with, at worst, a simple pole along the cuspidal divisor. Let ~9~.(~)~,(cusps) be the subsheaf of functions in 8 which are zero along the cuspidal divisor. An easy computation gives that R~.r ) vanishes when j+ i, and is an extension of 1, ~, R ~T,d)Mo(S)~, by dTT, when j=I. Consequently, the R ~T,d~o(s)(cusps) are locally j , free d)r-Modules, when the R nw, are. Lemma (3.3). -- Let T be a noetherian scheme flat over S = Spec (Z), or over the spectrum of a field. Then: R~ ' T, M0(N)/T j , R z% ) RJ r~T, d)xo(s)l, are locally free OT-Modules. Remark. -- The duality isomorphism (3.2) then applies in these cases. Proof. -- By the preceding discussion we need only prove the assertion for R~f, Oy where f: Y---~T stands for either the morphism 7: or r:'. Formation of R~fT, d)yl ~ commutes with flat base change T'-->T ([EGA], III, (r.4. r5)), which reduces us to considering the unique case T=Spec(Z). Also, j=i is the only nonobvious dimension. Let p be any prime. Since d~y is flat over Z, we have the exact sequence: o P ~ 1 P o-+ R)C.Or-+ R~ ~ R~ Cy,r, R f,~Py-+ Rlf, ~)y. The only global functions on Y/Fp are constant functions. This is evident for p + N, since Y/rp is then smooth and irreducible, and follows for p = N from the explicit description of the fibers X0(N)/1,,, and M0(N)/FN (w I). It follows that Rlf.Or has no nontrivial p-torsion. Proposition (3.3) (commutation with base change). -- Consider the category of rings which are flat over Z/m for some m, or over Z. Let R-+R' be a homomorphism in this category, then: H0(M0(N)/R, f2)| R' -~ H0(Mo(N)m, , ~2) and: H~ f~(cusps))| ~ H~ , f2(cusps)) are isomorphisms. Pro@ -- The assertion holds for R~R' flat, by [EGA], III, (I.4.15). This allows one to reduce the question to the base changes Z~Z/m (for m an arbitrary integer); for these the assertion is true since HI(M0(N)/z, f2) and HI(Mo(N)/z, f2(cusps)) are torsion-free Z-modules, by (3.3) and the duality isomorphism (3.2). d~o(Iq)jT(CUSpS d)M0(r~)j, OM0(N)~ MODULAR CURVES AND THE EISENSTEIN IDEAL Proposition (3.4). -- Let T be a (noetherian) scheme flat over S or over a field. The natural map induces an isomorphism: 0 t R~ ~* -+ R xT. Pro@ -- This is evident ifN is invertible in T. Thus, since formation of RifT,~y/T commutes with flat base change, we are reduced to the cases T=Spec(Z), and T=Spec(F~). For the latter case, we must check that the regular differentials on Mo(N)jF~ and on X0(N)/F~ coincide. But this is elementary, taking account of the explicit description (diagram I of w I) of X0(N)/F~ in terms of Mo(N)/FN and using the fact that a meromorphic differential on p1 with at worst simple poles at two points a, beP 1 is uniquely determined by its residue (at a, say). For T=Spec(Z), we have: i : H~ s176 -+ H~ fl~~ is a morphism of free Z-modules of finite rank (3.3), (3.2) 9 Since i| ] is an isomorphism, it follows that i is injective, with cokernel c~ a finite N-primary abelian group. Since by (3.3), (3.2) HI(Ko(N), g~x0(~)) and HI(Mo(N), O%(i~)) are free Z-modules, we have the diagram: o o >o o > H~ ~x.(~)) > H~ , ~.(m) > ~ >o o > H~ ~xo(s)) > H~ ~0(i~)) > c~ H~ ~Xo(S)a,~) ~> H~ s176 O O giving that W=o. 4" Parabolic modular forms. In this section R will denote a ring flat over Z, or over Z/m for some m. We shall be interested in comparing three different points of view concerning holomorphic modular forms of weight 2 over F0(N), defined over R. i. q-expansions of classical modular forms (Serre [47]). If RC Q, let B(R) C R[[q]] be the R-submodule of q-expansions (at oo) of (classical) modular forms of the above type (1) (whose q-expansion coefficients (at 0o) (1) Holomorphic, of weight 2, over l~0(N). flM.(N)~T" Y~Xo/(N)jT 7 ~ B. MAZUR lie in R). Let B~ B(R) be the subspace generated by parabolic forms. We do not require that the q-expansion coefficients at the other cusp o lie in R. Using ([9], VII, (3.18)) and the discussion of w 6, (1) below one sees, however, that these " other " coefficients lie in N-1.RC (3.. The unspecified term q-expansion will mean: at oo. It follows from the work of Igusa and Deligne or ([69] , p. 85, th. (3.52)) that: B(Z)| and B~ |176 for RC O (formation " commutes with base change ") and we define the g-submodules: B~ C B(R) C R[[q]] for an arbitrary ring R by the above isomorphisms. ~. Sections of the sheaf co | 2 over the moduli sta& (which are holomorphic at the cusps) (Katz [24]; Deligne-Rapoport [9]). Let A(R) (resp. A~ denote the R-module of modular forms (resp. parabolic modular forms) of the above sort, as defined in [24], (I.3) (compare [9], VII, w 3). We also refer to an element of A(R) as a modular form in o~ | Thus, an element 0~A(R) is a rule which assigns to each pair (E/T , H), where E is an elliptic curve over an R-scheme T, and H a finite flat subgroup scheme of E/T of order N, a section e(E/T , H) of co | where is the sheaf of invariant differentials. EIT f~E/T The rule o~ must depend only on the isomorphism class of the pair (E/T , H) and its formation must commute with arbitrary base change T'-+T. Finally, it must satisfy the condition of holomorphy at the two cusps. The q-expansion morphism: q-exp : A(R) -+ R[[q]] ~ defined by: 0~(Tate curve/a(cq)), bt~) = ~. square of canonical differential (1) is injective, if R is flat over Z or if I/NeR (z) and allows us to identify A(R) with an R-submodule of R[[q]] in these cases. We shall be especially interested in A(R) for rings R containing I/N. In this case one has an alternate description of A(R) as the space of holomorphic modular forms of level N, defined over R ([24], (1.2)) which are invariant under the action of the appropriate Borel subgroup. The question of whether formation of A(R) commutes with base change is a difficult one, and may be viewed as the main technical problem of this paragraph (3). (1) Compare [9], VII, (I.I6); [24], A 1.2, p. I6I. (2) This follows from the argument of VII, (3.9) of [9], or, if I/N~R, [~4], (1.6.I). r Compare [24l, (I.7) and (I.8). 70 MODULAR CURVES AND THE EISENSTEIN IDEAL 7i 3" q-expansions of regular differentials. The pair (Tate curve/z[[q]], ~) gives rise to a morphism: -~ : Spec Z[[q]] ~ M0(N)/z as in [9] VII, Th. (2. I) and z identifies Z[[q]] with the formal completion of M0(N)/z, along the section over S = Spec Z corresponding to the cusp oo. For any ring R, induces a morphism: t : Spec R((q)) ~ M0(N)m where R((q))=R[[q]][I/q] is the ring of " finite-tailed" Laurent-series. Suppose U is an open subscheme of Mo(N)/R through which the above morphism t factors, and such that U/Spe.CR/m is contained in the irreducible component of the Hasse domain h M0(N)/sp~0CR/N ) to which the cusp oo belongs. If 7 is a regular differential on U, we refer to y as a meromorphic differential on M0(N)/R. Define the q-expansion of y to be that element ~ of R((q)) such that t*y='~, dq. The q-expansion morphism is an injection of the space of meromorphic differentials over R to R((q)). The reason for this is, briefly, as follows. If 7 is defined on U and ~=o, then y is defined, and vanishes, on a formal neighborhood of the section in M0(N)/R corresponding to the cusp oo. Since ~ is an invertible sheaf, and the support of y inter- sects each geometric fiber of U in a finite number of points, 7 = o (cf. argument of [9], VII, th. (3"9); or of [24], .(I.6.2)). The q-expansion morphism also induces an injection: q-exp : H~ ~(cusps)) -+ R[[q]]. To prove this when R = Z/N use the structure of the fiber in characteristic N and the fact that a differential on P~F~, which possesses at worst simple poles, is known when its poles and (all but one of) its residues are known. It then follows for R= Z]N" (m~1) by an argument using (3.3). If I/NeR, the argument of the preceding paragraph gives injectivity; ifR is flat over Z one must use that M0(N ) is Cohen-Macaulay. By means of the map q-exp, we identify H~ , f2(cusps)) with a sub- R-module of R[[q]]. The relation between A(R) and H~ ~(cusps)) is given by the " Kodaira- Spencer style morphism" of [24], (I.5) and A, (I.3.I7). For our purposes, the fol- lowing statement is convenient. Lemma (4. x ). -- The natural mapping ( [24], A, (i. 3.17)) : a) is an isomorph#m on the complement of the cuspidal sections in XI(N)/R, for any R, as above, which contains I/N; 71 7 2 B. MAZUR b) is defined on the complement of the cuspidal sections and the supersingular points of characteristic N in X0(N; 3),for any R, as above, which contains I /6. Now when R contains I/N, let: Ul(N)/R=the open subscheme of XI(N)/R obtained by removing the discriminant locus of Xl(N) Xo(N). Uo(N)m=the image of UI(N)m in X0(N)/n. The Galois covering UI(N)/R-+ Uo(N)/R is a finite dtale Galois extension with Galois group (Z/N)*/( I) (1). When R contains i]6, let: V0(N; 3)m =the open subscheme of X0(N; 3)m obtained by removing the discriminant locus of X0(N; 3)-+ X0(N) and the " supersingular points " in charac- teristic N. V0(N)/R =the image of V0(N; 3)m in M0(N)m. If G is the covering group of X0(N; 3) -+ X0(N), then V0(N; 3) -+V0(N) is a finite ~tale Galois extension with covering group G. Lemma (4.2). -- The Kodaira-Spencer morphism induces: an imbedding: A(R) ~H~ g~(cusps)) /f I/NeR; a morphism: A(R) -+ H~ f~(cusps)) /f I/6~R. Moreover, these morphisms bring A~ to the subspace 0f regular differentials on the respective bases. Proof. -- Suppose I/NeR. Modular forms for F0(N ) on ~o | ([24] , (i.3)) are modular forms for FI(N) which are invariant under the action of the covering group. Using lemma (4- i), the Kodaira-Spencer morphism associates to an element ~ in A(R) a regular differential al on the complement of the cuspidal sections in UI(N)m , which is invariant under the action of the covering group. Since UI(N)/R-+U0(N)m is dtale, a 1 descends to a regular differential a on the complement of the cuspidal sections in U0(N)/R. By Cor. A, (I.3. I8) of [24], the q-expansions of a coincide with the q-expansions of e. The condition of holomorphy (resp. parabolicity) at the cusps then insures that a have at worst a simple pole (resp. is regular) at the cusps; consequently a is a section of f~(cusps) (resp. f2) on all of U0(N)/R. Similarly, if I/6~R, one constructs a differential on V0(N)/R. Note that both U0(N)I R and V0(N)IR, when defined, are open dense subschemes of Mo(N)m. Also, the construction which associates to ~ differentials on these open subschemes yields the same differential on the intersection (same q-expansion). Consequently, to any o~A(R), and for any ring R as considered in this section, we may associate a meromorph# differential on M0(N)/R, a, with the same q-expansion as a. (I) To avoid confusion with various Galois actions we refer to this group as covering group. 72 MODULAR CURVES AND THE EISENSTEIN IDEAL To compare differentials with elements of B(R), we begin with: Lemma (4- 3) : H~ a) C B~ C R[[q]] f (cusps)) c B(R) C R[[q]]. Proof. -- If R=Z, the first inclusion follows since H~ ~) is a subspace of H~ s having integral q-expansions. Consequently we obtain the desired inclusion for any R of the type considered in this section, since formation of both range and domain commute with base change Z-~R (3.3). The second inclusion follows similarly. Lemma (4.4): (i) If R is a field of characteristic p  N, then: A0(R) =H~ n) if p= 2 and N-I rood 4 and: A(R) =H~ f~(eusps, (i;)) A(R) =H~ n(cusps, (0))) if P= 3 and N-I mod 3 otherwise A(R) =H~ n(cusps)) N---Imod3). (See Table I.) (i.e. p> 5, or p = 2, N---I rood 4 or p = 3, (2) /f R=Z[I/m] for some integer m, then: A(R) c HO(Mo(N)/ , f (eusps)) A~ C H~ g~). Note. ~ By f~(cusps, (i)) is meant the sheaf of meromorphic differentials which have, at worst, simple poles at the cusps and at the point (i) of Table I. Proof. -- Let eeA(R) and let a be its associated meromorphic differential. (i) R a field of characteristic p. N: Here a is a meromorphic differential on M0(N)I R which is regular on U0(N)/R, except for possible simple poles at the cusps, and which lifts to a differential on XI(N)/R regular except at the cusps. We shall make a local calculation to determine when a meromorphic differential can become regular, after finite extension. Explicitly, let k be an (algebraically closed) field of characteristic p and D 1 C D2 a finite extension of k-algebras, which are discrete valuation rings, with residue field k. Let a 1 denote a meromorphic differential on D1 relative to k, and let a 2 denote the induced differential on D~, relative to k. Sublemma. -- If D t C D~ is (gtaIe, or) tamely ramified, then the meromorphic differential a S is a regular differential (resp. has a simple pole) if and only if a 1 is a regular differential (resp. has a simple pole) on D 1. 10 74 B. MAZUR If D1 C D2 is wildly ramified of the first type (2.2), then a2 is a regular differential if and only if a 1 has (at worst) a simple pole on D1. Pro@ -- Since there are no nontrivial 6tale extensions in our situation, we may assume D1CD2 a totally ramified Galois extension of degree r. Write D2=k[[y]] for a choice of uniformizery of D~, and Dl=k[[x]], where x is a uniformizer, chosen so that x=-c?(y), where q~(Y)ek[Y] is a polynomial. Using ([6o], III, 7, Cor. 2) one calculates the different of D1C D~ to be (q~'(y)). If v 2 is the valuation on D~ such that v2(y)=i , then v2(x)=r , and v2(~'(y)) can be calculated in terms of the orders of the higher ramification groups of D1C D2 ([6o], oo IV, w 2, Prop. 4: v~(q~'(y))= 52, (Card(Gi)-I)) and consequently: i=0 v2(~'(y)) = r--I (tamely ramified case) v~(?'(y))=2p--2 (wild ramification of first type). Up to multiplication by a unit in Dr, we may write at as xS.dx for some sEZ. Thus, a s is, up to a unit, of the form xS.~'(y).dy, and: v2(x'.~'(y))=rs+r--I (tame ramification of degree r) =ps + 2(p-- i) (wild ramification of first type). The assertions of the lemma can now be read off from the above formulae (e.g., in the case of wild ramification of first type, s>--I if and only if ps+ 2(p--I)>O). Now return to the case (i) of lemma (4.4), and the meromorphic differential a. By Table I, Xx(N)/R-+ X0(N)m has at most one point of wild ramification, and none if characteristic R:t: 2, 3. Moreover, if there is a point of wild ramification, it is of first type. By the sublemma, the meromorphic differential a is regular with the exception of possible simple poles at o, 0% and (i) and (p), if they occur (see Table I). Conversely, any meromorphic differential which is regular, except for such simple poles will (by the sublemma) lift to a differential on XI(N) with, at worst, simple poles at cusps. This gives us the identification of A(R) with the appropriate space of meromorphic diffe- rentials, as in the statement of (i). The subspace A~ is then identified with the space of differentials on M0(N ) which are regular everywhere with the exception of a possible simple pole at (i) (if p=2, and N-Imod4) or at (p) (if p=3, and N-= I mod 3). Since the sum of the residues of a differential over a complete curve is zero, it follows that A~ is identified with the space of everywhere regular differentials. (2) R =- Z[I/m] : We show A~ C H~ ~) ; the other inclusion is proved in the same way. Recall that M0(N)~ a denotes the complement of the characteristic N supersingular points, in M0(N)m. The meromorphic differential a is regular on an open dense subscheme of Mo(N)~ a. 74 MODULAR CURVES AND THE EISENSTEIN IDEAL Let D~ o (resp. Do) denote the divisor of poles (resp. of zeroes) of a, on M0(N)) a. Recall their definition: ifx is a point of the scheme M0(N)~R, and d~ x the local ring at x, let q~x be a local generator of f~0(N)~R at x. Since 0~ is a unique factorization domain, one can find g~,hxe6) ~ with no common factors such that g~.a-=hx.~. A local equation at x for Do, (resp. for Do) is given by: gz=o (resp. hx=o ). Now let p be a prime number with these properties: a) p~'2.3.N.m; b) D~ o and D O have disjoint support in characteristic p: [D~o| n [D0| =O. It follows from the definition of polar divisor and a), b) that a| is definitely nonholomorphic at D~o| p. Therefore part (I) of our proposition implies that the support of D| is disjoint from the fibre of M0(N)m-+ Spec(R) in characteristic p. Since D~o contains no irreducible component of any fibre of ~, it follows that Do~= o; therefore a is regular on M0(N)) R. To see that a is, in fact, regular on M0(N)/R, use that the supersingular points of characteristic N are of codimension 2 in M0(N)/R, and f~ is an invertible sheaf, and M0(N)/R is Cohen-Macaulay (SGA ~, Exp. III, Cor. (3-5)). Lemma (4.5). -- Let R be flat over Z[I/N]. Then: A(R) ---- HO(Mo(N)~R, n(cusps)) = B(R) A0(R) = H0(Mo(N)/R, f2) = B0(R). Proof. -- We establish the first line above; the second may be obtained by essentially the same argument. First let R=Z[I/N]. By the previous two lemmas, we have inclusions: A(R) C H~ f~(cusps)) C B(R) and so we must prove that A(R)= B(R). But this follows from the q-expansion prin- ciple ([24], Cor. (1.6.2)). To be more precise, using the notation of (I.6.2), take f to be any element in B(R), n=N, K--Q, L=R. Katz's corollary (I.6.2) then gives us that f is a holomorphic modular form (in co | of level N, defined over R. Since f, viewed as a modular form of level N, is invariant under the appropriate Borel subgroup of GL~(FN) , it is in A(R). Now let R be flat over Z[I/N]. Lemma (4.5) will follow from what we have done, provided we show that: A (R) = A (Z [ I/N]) | R. Even this " commutation with base change " is not totally trivial. If one takes the point of view that A(R) is the space of (Z/N)*/(zk i)-invariant differentials (regular, with the possible exception of simple poles at cusps) on Xl(N)/R, however, it is an easy exercise (1). (1) If G is a group and M a Z[x/N][G]-module, flat over Z[I/N], and R a flat Z[I/N]-module, then M G@Z[1/N]R is isomorphic to (M| (The superscript G denotes invariants under G.) 75 B. MAZUR Lemma (4.6): H~ , f2(cusps)) = B(R) ItO(Mo(N)/R, f2) = BO(R). Proof. -- We show the second equality; the first is done similarly. It suffices to prove this equality for R=Z, since formation of both sides of the equation commutes with base change from Z to any of the rings R we consider. By lemmas (4.3) and (4.5), H~ f2) is a subgroup of B~ and the quotient Q is an N-primary finite abelian group. Since: H~ f~) C B~ C FN[[q]] one checks that multiplication by N is an isomorphism on Q. Lemma (4.7). -- Let m be an integer prime to N and R=Z/m. Then: H~ ~2(cusps)) C A(R) HO(Mo(N)/R, f2) C A~ Proof. -- These inclusions follow from (4.5) and the fact that the morphisms: HO(Mo(N)zE1/N], f2(cusps)) -+ H~ ~(cusps)) HO(Mo(N)z[1/~], f~) ---> HO(Mo(N)m , f2) are surjective (3.3). Lemma (4.8). -- Let m be prime to N, and R=Z/m. Then: AO(R) =H~ , ~2) = B~ and: A(R) =H~ f2(cusps))= u(g) if m and N satisfy the following properties: (a) either m ~-o mod 2, or N # I rood 4 and (b) either m ~ o mod 3, or N ~s I mod 3. Proof. -- In the light of (4.6), what must be shown is that the inclusions of (4.7) are equalities, under the hypotheses above. Lemma (4.4) (i) assures us that they are if m is a prime number. We now proceed by induction. Let p be a prime dividing m; m =m'.p. Let R' C R be the sub-R-module consisting in multiples of p (R'~Z/m'). Consider : > A(R) > A(Fp) A(R') , H(R) > H(Fv) > o H(R') 76 MODULAR CURVES AND THE EISENSTEIN IDEAL where I-I(.) stands for YI~ f~(cusps)). The bottom line is exact since formation of H(*) commutes with the type of base change which occurs in that line (3-3)- The top line is exact, by an application of the q-expansion principle ([24], (1.6.2)). The two flanking vertical inclusions are isomorphisms by induction, since if rn and N satisfy (a), (b), then m' and N also satisfy (a), (b). Therefore the central vertical inclusion is an equality, as well. This establishes the assertion of lemma (4.8) concerning A(R); the assertion concerning A~ is established by a similar argument. Summary and convention (4-9)- -- We shall be chiefly concerned with modular forms of weight 2, over I'0(N), for some (usually fixed) prime number N> 5- Except when indicated explicitly to the contrary, a parabolic modular form (over 1-'.(N), defined over R) will mean an element of B~ or, equivalently, a regular differential on Mo(N)/R; or, equivalently (if R is flat over Z or over a field) a regular differential on X0(N)/R; or (if R=Z/m with (re, N)=1 (4.8); or R flat over Z[I/N] (4.5)) an element of A~ For holomorphic (nonparabolic) modular forms it is true that elements of B(R) coincide with differentials defined over R, regular with the possible exception of simple poles at cusps (4.6). Nevertheless, for certain rings R, A(R) may differ from B(R) (e.g., Remark below). Thus we shall always make clear, in what follows, whether we are dealing with an element of A(R) (a modular form in o~ | or of B(R), and both notions will be useful. Remark (concerning the distinction between A(R) and B(R)). -- The Riemann- Roch Theorem and the description given in (4-4) (I) show that B(R) is of codimension I in A(R), if R is a field of characteristic 2 and N'---I mod 4; or of characteristic 3 and N--- I mod 3- In certain cases one can exhibit an element of A(R), not in B(R). For example, if char R=2, and N- 5 mod 8, it follows from the description in (5-I2) below that the power series 3 modulo 2 is (the q-expansion of) such an element; the power series modulo 3 is such an element if charR=3, and N- 4 or 7m~ On the other hand, the Eisenstein series e' (w 5) is in B(Z) but not A(Z), since its q-expansion coefficients at the cusp o can be seen to lie in N -1.Z but not in Z. Proposition (4.xo). -- There are no nonvanishing parabolic modular forms over F0(I ) (in ~| defined over any ring R flat over Z or over Z/m. Remark. -- There are nontrivial holomorphic modular forms over F0(~ ) (in ~o| defined over certain rings R (cf. (5.6)). Pro@ -- If I/5eR, lift to M0(5)/. This is a curve of genus o, and therefore has no nonvanishing regular differentials on it. Since I/5~R there are no parabolic modular forms (in o~ | over P0(5), as well. Therefore there are none over l'0(I ). If I/7eR, lift to X0(7) and use the same argument. The general ring R (as considered in this section) is then treated by patching. 77 78 B. MAZUR 5" Nonparabolic modular forms. Consider the following three power series in Z[[q]]. ~o e = I -- 24Z=t ~(m)q" where a(m) is the sum of the positive divisors of m. oo (5 -x ) e'=I--N-- 24~=1 a'(m)q "~ where C(m) is the sum of the positive divisors of m which are prime to N (as usual, N is a fixed prime number > 5). oo 8= 52 C(m)q". The power series e is the q-expansion of the Eisenstein series of weight 2 of level I (1). It is the logarithmic derivative of the q-expansion of the normalized modular form (of level i) of weight 12 : co A=q II (I--qm) ~' (~), The power series e'(q)=e(q)--N.e(q N) is the q-expansion of the Eisenstein series of weight 2 on P0(N)(3). It may be regarded, as meromorphic differential, as the logarithmic derivative of the function A(q)/A(q N) on M0(N)/Q. Since this functions has zeroes and poles only at the cusps, e' is (the q-expansion of) a differential whose only poles are (simple) poles, occurring at the cusps. Since e' has integral coefficients, we have e'E B(Z). Viewed as modular form in o~ | ~ over Q, the q-expansion of e' at the cusp o may be seen to be (using [9], VII, (3-i8)): I/N. (N--I + 24~ t C(m)q "l~) and therefore e' is not in A(Z). The power series 8 is simply e', deprived of its constant term and conveniently normalized. It will be of interest to consider those rings R over which 8 is a modular form. It is proved in [24], (4.5.4) (also (A.2.4) if P--~5) that e is a p-adic modular form ([24] , (2.2)) for everyp. Thus, if R=Fp, e is the q-expansion ofa meromorphic differential on the Hasse domain X0(1)~ R. This differential may have poles at the (1) It is denoted P It is the q-expansion of /~|.Gdv;o,o, i) in Hecke's terminology ([I9], p. 474). \7~-1 in [24]- (~) Cf. [24], A x .4.4 for a proof of this fact, which does not use the Jacobi identity. (") p- The q-expansion of r in HECg-~ ['9], 78 MODULAR CURVES AND THE EISENSTEIN IDEAL 79 supersingular points. Our first object will be to study e, both as section of co | over the moduli stack, restricted to the Hasse domain, and as meromorphic differential. Lemma (5.x). -- The power series e is the q-expansion of the meromorphic differential: -- dj modulo 24. 32. 5 and of: dj modulo 24. 33. 7 j--I728 on X0(I). Remarks. -- In the above, j is the elliptic modular function, which is a rational parameter for X0(i), and has q-expansion beginning I/q+744+ .... If: e4=I+24o ~ ~(m)q" ra=l co e =I--5o4 Z m=l are the normalized Eisenstein series of weight 4 and 6 respectively, we have: (5.2) j = e, /A = 1728+ e~/A. It would be interesting to study the poles and residues of e at the supersingular points of X0(I)/z/p, for pr any power of a prime. O.A.L. Atkin, M. Ashworth, and (independently) N'. Koblitz have some interesting formulae, algorithms, and machine computations which suggest some precise conjectures in this direction. Proof of lemma (5. I). -- Take logarithmic derivatives of formulas (5.2), regarded as identities in power series in q, noting that: d log(~) = o modulo 3.24 ~ = 24.33. 5 d log(e~) - o modulo 2.5o4 = 24. 33 9 7. O .E.D. To study e as a section of c~ | 2 over the moduli stack, recall the standard formulas giving elliptic curves in " generalized Weierstrass form " over arbitrary bases (we use the notation and conventions of Tate. Cf. Appendix i [29]). Thus, if (E/T , ~) is an elliptic curve over the base scheme T, equipped with an invariant differential, ~, we may represent (E/T , ~) locally for the Zariski topology over T as a curve: y2 + alxy + a~y = x 8 + a2x ~ + a4x + a s (5-3) = dx/(2y + alx + a3) = dy/(3 x~ + 2a~x + a4-- aly) where, if f : E-+T is the structure map, and ~ : T-+E the zero-section, then (I, x) is a basis off,0(2,) and (i, x,y) is a basis off, C(3, ). This representation may be modified by making a different choice (7:', x',y') to 79 8o B. MAZUR obtain a new equation (5.3)'- The relation between the old and new choices is given by the " data "" (u, r, s, t) where u~I~(T, 0~) r, s, t~I'(T, @T) defined by the formulas: 7~ p = UT~ X =U2X'-~r y = u3y'+su2x'+ t and, conversely, any such data gives us a new choice. The new formula (5.3)' is related to the old by: ua~ = a 1 + 2s u~ a; = as-- sal + 3 r- s" u3 a'3 = a 3 + ra 1 + 2t u4 a4, = am_ s a 3 + era 2 -- (t + rs)a 1 + 3 rg- 2st u%'~ = a n + ra m + r2 a~ + r a- ta3-- t z- rtal. Following Tate, define: b2=a~+ra2; b4=ala3+2a4. For the new formula (5-3)' one has: 2 ' (5"4) u b2=b2+ I2r; u4b'~=b4+rb2+6r 9". Lemma (5.5). -- Let R=Z/72 (72=23.33). Let T be an R-scheme. Let (E/T , ~) be a pair consisting in an elliptic curve E/T and an invariant differential ~ such that b 2 is invertible in P(T, @T)" Then the function ~ = b2--I2b4/b2 depends only on the isomorphism class of the pair (E/x, ~z) and not on the representation (5.3) chosen. It defines a section of co | over the open substack of the moduli stack of level I over R obtained by removing the cusps and" inverting b2 " The q-expansion of ~ is e, modulo 72. Proof. -- One checks, using (5.4) and working modulo 72, that the relation between and ~' (under a change of representation given by the " data " (u, r, s, t)) is u~'= which establishes everything but the last sentence of lemma (5.5). For this, we evaluate ~ on the Tate curve whose equation is ([63] , IV, 3o): y2--xy=x3+a~x+a6 (a~ =-- 5 ~, i--q ] na q n ~ ma q m Thus, b2=I and ba-=--I~ I--q"' giving ~= I + 120m=tI_q,,. Since 12ore 3 ~ -- 24m mod 72, we conclude that , = e. Remark. -- This lemma gives the first two terms of an " asymptotic expansion " of e in terms of the parameter b 2 (which cuts out the supersingular locus, 2-adically 80 MODULAR CURVES AND THE EISENSTEIN IDEAL 81 and 3-adically). Using his algorithm and machine computation, N. Koblitz has obtained the first 4 ~ terms. Proposition (5- 6). -- (Holomorphic modular forms of level I) : (a) There are no nontrivial holomorphic modular forms of level I (in o~ | defined over a field R of characteristic 4= 2, 3. (b) The " square of the Hasse invariant" is a holomorphic modular form mod 4, with q-expansion equal to e. The " Hasse invariant " is a holomorphic modular form mod 3, with q-expansion equal to e. (c) If q~ is a holomorphic modular form (of level I; in o~| defined over R=Z/m, with q-expansion beginning with the constant I, then: (i) m divides 12 ; (ii) q~=e. Summary. -- Every holomorphic modular form of level I, defined over R=Z/m has q-expansion equal to a constant. Proof: (a) R a field of characteristic 4: 2, 3 Let ~ be such a holomorphic modular form defined over R, and denote by the same letter the meromorphic differential on X0(I)/l~ associated to % Since the moduli stack associated to P0(I ; 3) " exists " (w i), lifting ~? to X0(I ; 3) yields a meromorphic differential, with at worst, simple poles at the cusps. Since X0(I ; 3) -+ X0(I) is a tamely ramified Galois extension, the sublemma in the proof of (4.4) assures us that has, at worst a simple pole at the cusp ~ of X0(i)/R. Since X0(I)t g is of genus o, q~ must vanish. (b) Going back to (5.5) one sees that, modulo 12, e is given by b2, and is therefore a holomorphic modular form modulo 12. Its q-expansion is the constant I. Modulo p (any p) the Hasse invariant is a holomorphic modular form of weight p--i, and q-expansion equal to I ([24], (2.o)). Thus, modulo 2, the Hasse invariant is of weight I and can be taken to be a 1. By " the square of FIasse invariant mod 4 " we mean a~, which is a section of c0 | mod 4- But, e- ba- a~ mod 4- Working modulo 3, the Hasse invariant is a modular form of weight 2, with the same q-expansion as e. It coincides, therefore, with e. (c) Let q0 be a holomorphic modular form mod m, such that the constant term of its q-expanslon is i. By (a) m=2~.3 b. To show that m divides 12, it suffices to show that no such q~ can exist mod 8 or mod 9. Let q~ be such a modular form mod 8 (resp. mod 9). Note that q~-e mod4 (resp. mod 3), because, by (b) q0--e is a holomorphic modular form mod 4 (resp. mod 3) ; 11 8~ B. MAZUR it is parabolic by our assumptions on % and therefore must be zero by (4.8). Let R=Z/8 (resp. Z/9 ). We may write ~= r(j). e, where r(j) is a rational function in j (viewed as rational parameter of X0(i)) with coefficients in R. If we view both q0 and e as meromorphic dj, differentials, and use (5.I) that e-- and (4.2) that ~ is a regular differential on the open subscheme Spec R[j,j -1] of X0(I)/R, we obtain that r(j) is in R[j,j-t]. By the above, we may write r(j)=iq-4L(j ) (resp. I-t-3L(j)) where L(j) is a " Laurent polynomial " in R[j,j-1]. We now use holomorphicity of q~ about the point j=o, together with the above description of r(j). If R = Z/8, consider the following elliptic curve E over the power series ring R[[t]]. E : y2+txy+y=x3, rc=dx/(2y+tx+I) One computes: b 2 = t 2 e -~ b~-- i ~b4/b~ = t~-----z 2/t E R [ It]] It- t] b 4 = t j =tl~(t"--27)-IER[[t]]. We now compute the value of the section q~ of c0 | on the pair (E, ~) over the ring of finite-tailed Laurent series R[[t]] [t-l]. q~(E, ~z)= (I +4L(j))(t2--I 2/t)eR[[t]] [t -~] =t2--I~/t +4t~.L(j). Since q~ is holomorphic, and (E, ~) is defined over R[[t]], q~(E, ~) must lie in R[[t]]C R[[t]] [t-l]. That is: (5-7) 12/t--4 t~. L(j) e R[[t]]. Let jb be the lowest power of j occuring in the Laurent polynomial L(j) with coefficient a unit mod 8. Writing 4 t2 L(j) as a finite-tailed Laurent series in t, one has that t 12b+2 is the lowest power of t occuring with nonzero coefficient. One reasons now, that if b is nonnegative, the I2/t term in (5.7) cannot be cancelled by any term in 4t~L(j), while if b is negative, t 12b + 2 is the lowest power of t occurring in the expression of (5.7). In either case, one has a contradiction. If R-~Z/9 , it is convenient to work with the elliptic curve E given by the representation: al=a3-=I ; a2=(t--I)/4; a4:o; a6 =--I ]4. Then: b2~--I e = b2--I2b4/b 2 = t--i2/teR[[t]] [t -t] b4z I j =4t6(t ~- 32)- le R [[t]]. 82 MODULAR CURVES AND THE EISENSTEIN IDEAL Again: q)(E, 7:) = (i -t- 3L(j)) (t-- 12/t) (5,8) = t--i2/t+ 3 t. L(j) eR[[t]] [t-~]. Ifj b is the lowest power ofj occurring with unit coefficient in the Laurent poly- nomial L, then t 6b+1 is the lowest power of t occurring with nonzero coefficient in 3t.L(j)~R[[t]][t-t~, and, as above, (5.8) leads to a contradiction. Q.E.D. We now prepare to study the status of the q-expansion " I " as a modular form over P0(N). The following lemma, which is in the spirit of the theory of Atkin-Lehner, and which was suggested to me by J.-P. Serre, will be helpful. Lemma (5-9) (reduction of level). -- Let I/NER. Let ~ be a holomorphic modular form in o~ | over P0(N), defined over R (k> 2). Suppose, further, that the q-expansion (at oo) of ~ is a power series in qN :~,=3~q~), Then ffis the q-expansion of a holomorphk modular form over Po(~) (again in o~ | and defined over R). To obtain an analogue of (5-9) in characteristic N, we return to the setting of interest to us: Lemma (5.IO). -- Let N_>5 be a prime number and ~ a holomorphiv modular form over ~'o(N), in B(FN) (w 4). Suppose, further, that the q-expansion of ~ is a power series in Then ~ = o. Proof of lemma (5.9). -- Let .#" denote the stack ..dt'0(N ) over R and ./V ~ =~'0(N) ~ Thus, if one is given a pair (EjT , H) where T is an R-scheme, E is an elliptic curve over T, and HC E is a subgroup of order N, defined over T, one may associate to (EIT , H) a T-valued section of the stack .A/-. There are maps: where sr and ~,o are the moduli stacks of level N and i respectively, defined over R. These maps are determined by the rules: where y :Z/NxZ/N~E[N] is an isomorphism of group schemes over T, and: E' is taken to be E/.f(oxZ/N); H is taken to be the image of y(Z/Nxo). (L T, H) s (E/H). The map ~3~ :.//~'~-~dt '~ is a Galois, dtale morphism of stacks. The Galois (covering) group may be identified with GL2(F~) acting in the natural way (by compo- sition with y) on d/~ The intermediate stack ~/'0 is fixed under the Borel subgroup 83 84 B. MAZUR .or\ over the stack ~A/'0 or as a section over ~o, invariant under the action of B. A " formal neighborhood of the cusp oo " in A/'0 is induced from the pair (Tate(q)m((q/~, ~) while a " formal neighborhood of the (unique) cusp ~ " in Me' 1 is induced from the Tate curve over R((q)) ([24], (I. 3))- We have the following commu- tative diagram: Spec R((q)) -> X ~ q l-~ qN t f~ Spec R((q)) .go where we can check that the left-hand vertical map is given by q ~ qN as follows: By ([24] , (I.II), p. 9 I) we have Tate(q)/~N=Tate(q~), and Tate(q ~) is induced from Tare(q) by extension of scalars R((q)) ~ R((q)); q~qS. By the above discussion we may give the following geometric interpretation to our hypothesis concerning ?: the restriction ~ of ? to Spec R((q)) descends to a section of eo | over the " formal neighborhood of the cusp " in ~go. We now consider the cusps of J/C'N, and for this we make the base change from R to R 0=R[~N]. Note that the map J/~-~-~A/" is dtale over the cusp oo. Let: The inertia groups in GL2(Fs) of the cusps in -///s consist in the conjugates of the group U (cf. [9] Cor. (2.5) of VII). From the definitior~ of the map e one sees that the inertia groups of those cusps lying above ooe,4/" consist in those N conjugates of U which do not lie in B. Let ~ be a cusp in Jgs, lying over 0% whose inertia group (for the Galois extension ~gN~Jt'l) is U. Since the group A normalizes U, it follows that, for all aeA, a. ~ also has U as inertia group. Viewing ? as a section of o~ | over jgo, the Fourier expansions ~'~.~ descend to a formal neighborhood of the cusp in dr176 and therefore q0"~.~ is invariant under the action of the inertia group of a. ~ (namely U). Thus, for any ueU, q0"--q~ has zero q-expansions at each of the cusps a. ~ for aeA. Since the group A operates transitively on the N--I distinct connected components of ,g~ we have that q0"--? has zero q-expansion at (at least) one cusp belonging to each of the N--I distinct connected components of the geometric fiber of ~g~. Therefore theorem (i .6. I) of [24] applies, giving that ~"-q~ = o. It follows that ~ is it, variant under both B and U. Since B 84 MODULAR CURVES AND THE EISENSTEIN IDEAL and U generate GLe(FI~), q~ descends to a modular form over Po(i), defined over R0=R[~I~ ]. Since its q-expansion has coefficients in R, [9], VII, th. (3-9) (ii) insures that 9 is defined over R. Proof of lemma (5. IO). -- Suppose that 9 is a nonzero holomorphic modular form satisfying the hypotheses of our lemma. Since 9 is the reduction modulo N of a modular form of weight 2 over I'0(N ) With integral q-expansion, we use [6i], th. i I (c), and regard 9 as the reduction modulo N of a modular form over SL2(Z), of weight N-}-I. In the terminology of [61], q~ is of filtration <N + I, as a modular form over SLy(Z). Since the filtration of 9 is congruent to N@I modulo N--I ([66], th. 2) and since it cannot be 2, the filtration of 9 is N+ i. On the other hand, our hypotheses may be interpreted as saying 09=0 , where 0 is the derivation q.~qq. Since N~5, we may apply lemma i (a) of [6I], which gives an absurd equality for the filtration of 09 = o. Conse- quently, there are no nonzero modular forms 9 satisfying the hypotheses of (5-m). Corollary (5.Ix). -- Let "~(q)=I+a~qN-?a2~a2N-r -. .. be a power series in qN, with integral coefficients, beginning with constant term I. Then: (i) ~' reduced modulo N is not a holomorphic modular form (for F0(N)) in B(FN) (w 4). (ii) If m is prime to N, and ~, reduced modulo m, is a holomorphic modular form (in ~| over P0(N)), then m divides I2, and 9-I modulom. Proof. -- (i) is a repetition of (5.IO), while (ii) follows from (5-9) and (5.6) and (4. IO). We now consider the status of tile power series 3(q)----~ e'(m)q ~ (see beginning of w 5), as modular form, when reduced modulo integers m. Proposition (5- x2): (i) The power series ~ is not the q-expansion of a holomorphic modular form of weight 2 over Po(N), modulo N (" holomorphic modular form" in B(Fs) (w 4)). (ii) Let m be prime to N. The power series 3(q) is the q-expansion of a holomorphic modular N--I form over Po(N) modulo m (in o~ | if and only if m divides -- (2). (iii) Let m be any integer. The power series 3(q) ia the q-expansion of a parabolic modular <i) form if and only if m divides n = numerator ~ . Pro@ -- Consider the formula: --e'=(N--i)+~4~ (1) See also KoIK~ [36] when m is a prime ~ 5. 85 86 B. MAZUR from which it follows that if 8 were a modular form modulo N (in B(FI~)) then the constant I would be the q-expansion of such a modular form as well. This is not true by (5.11) (i), whence (i). We shall now prove (ii). But first we need a fact about modular forms (in co | which is not totally obvious: Let ~ be a power series in q with integral coefficients. Let a, b be integers. Then ~ is a holomorphic modular form mod b if and only if a~ is a holomorphic modular form mod ab. To prove this, we invoke the q-expansion principle ([9], VII, (3.9) (ii); [24], (I .6.2)). We view a.Z/b as submodule of Z/ab and note that aq~ has all q-expansion coefficients lying in the above submodule. Now, suppose that 8 is a holomorphic modular form modulo m with (m, N)= I. From the formula quoted above, it follows that N--I is (the q-expansion of) a holo- morphic modular form, modulo 24 m. By (5. i i) (ii) and the fact proved above, if m' is any integer prime to N such that N--I is a holomorphic modular form modulo m', then m' divides I2(N--I). It follows that 24 m divides I2(N--I), or m divides (~). Conversely, e is a holomorphic modular form (in o~ | modulo I2. Therefore (N--i).e is a holomorphic modular form modulo I2(N--I). Moreover: (5.I3) --e'--- (N--I).e+24 8 modulo 24(N--I), from which it follows that 8 is a holomorphic modular form modulo (}-~-~/" This \'1/ proves (ii). /3%T .k As for (iii), it suffices to consider integers m which divide {~--~), by (ii). Consider (5.13) as an equation of meromorphic diffelential forms, and we shall compute the residues of each term appearing in it, at the sections oo and o. To do this, consider the involution w of X0(N ) induced by the rule: (E, H) ~ (E/H, E[N]/H) operating on sections of the moduli stack ,/fro and on modular forms (cf. terminology and discussion in proof of lemma (5.9) above; for a discussion of w cf. w 6) below. If q~ is a (holomorphic) modular form, defined over R, of level I, and if we denote by 9, again lifting to sff ~ defined by the rule ?(E, H)=q~(E), then the q-expansions of q~ and ~0.w are related by: w(q) = as follows from the discussion in the proof of lemma (5.9). Since w interchanges the cuspidal sections oo and o, we have the following Sublemma. ~ Let I/NeR. If ~ is a holomorphic modular form (in o~ | of level I, defined over R, regarded as meromorphie differential, and if the same letter ~ denotes its lifting to Mo(N)m as above, we have the formula for residues: Res0(,) = N. Res~o(~). 86 MODULAR CURVES AND THE EISENSTEIN IDEAL Thus: Reso0((N--I) .e) --N--I mod 24(N--I) (5. x4) (~). Reso((N--I ) .e) -N(N--I) mod 24(N--I ) Since e' is an eigenvector for w with eigenvalue --I, we have: (5" I5) Res~(--e')=N--i (1). Res0(-- e') = I --N Formula (5.15) would also follow from the fact that the only poles of e' occur at o and oo. Combining (5.i3), (5-14) and (5.15) we get: Res~o(~)=o (as it should) (5.I6) I_N ~ Res0(~ ) -= modulo N--I. Assertion (iii) then follows from (ii), (5. I6), and the following elementary fact: n = g.c.d. N~-I, 6. Hecke operators. I) The involution w (induced by (z~--I/Nz) on the upper half-plane). This is defined on M0(N)lzE1/N 1 by the rule (E, H) ~ (E/H, E[N]/H); it extends to an involution of Mo(N)/z (by [9], IV, (3. I9)), and of X0(N)/z. We denote this involution (as well as the involutions induced by it on the moduli schemes M0(N. N'), where N and N' are relatively prime) by w~, or by w, if no confusion can arise. In the terminology of [9], IV, (3- 16), w is induced by conjugation of Po(N) (o Io) It interchangesthecuspidalsections oo and o. by the matrix g= N " By " transport of structure " (i.e. functoriality of the sheaf of regular differentials) the involution w induces an involution on the space of regular differentials (on B~ and also on B(R). Care should be taken to distinguish this involution w (which is indeed the " classical " one) from the mapping on modular forms in o~ | defined by Deligne and Rapoport ([9], VII, (3.18)). Referring to their w by the bold letter w, one can show that for a modular form in co | 2 over Q, w? =N. wq0. Our mapping w does not necessarily " preserve " A(R). If fEI-I~ f~) has q-expansion f=y,%qm, the q-expansion of w.f is given by w.f=--Y,a~.mq m ([61], (2.I), and (3.3), th. II (a)). 2) T t for prime numbers t 4= N. (1) In these formulas e and e' are regarded as meromorphic differentials. 87 88 B. MAZUR These are correspondences determined by the diagram of morphisms: M0(N.?) (,) 2 / M0(N ) , M0(N ) Tt where c, on the moduli stack, is determined by the rule: (E, H~, He) ~ (E, H~). Here H N C E is a subgroup scheme of order N, H t C E of order r Compare [9], VI, (6. i i). The morphisms c, cw t are finite (loc. cir.) (1). If x=j(E/E , HN) is a point on the curve X0(N ) with values in a field K, then Tex is the divisor: (6. x) Y4(E/H, (H~ + H)/H) where the summation is taken over all cyclic subgroups I-I of order t of E, defined over ~x. Define morphisms : (a) c* : Ha(M0(N)/z, dTMo(S)) -+ HI(Mo(N. ~)/z, d~M.(Nt)) (b) c* : H~ f2) -+ H~ f2) as follows: (a) is induced from the natural map: OM~ ) -+ C, OM,(N ' t)" As for (b), let U denote the open subscheme of Mo(N.l ) which is the complement of the supersingular points of characteristics N and I (the smooth locus of M0(N.I) -+ Spec (Z)) and let V be the image of U under c. The restriction of f~ to U (resp. to V) is f~r/~ (resp. f2~/s). One has the natural map: which induces a morphism c* : H~ f2) ---> H0(U, f2). But since f~ is an invertible sheaf on M0(N.d ) and the complement ofUin M0(N.t ) consists in afinite set of points of codimension two, whose local rings are Cohen-Macaulay, we have: H0(U, f2)----HO(M0(N, e), f2) whence the mapping in (b) above. Applying the Grothendieck duality isomorphism v ((3.3) + (3.2)) to (a) and (b), we obtain morphisms: (a ~) c. ~ (c*)v : HO(M0(N. t)/z ' f2) -, H~ ~2) (b v) c.=(c')" : HI(M0(N./)/z, d~~ Ha(M0(N)tz, d~ro(S)). (t) They are not necessarily fiat. To determine the (finite) set of points at which they are nonfiat is an easy exercise, using [62], IV, Prop. 23; [9], V, (6.9). 88 MODULAR CURVES AND THE EISENSTEIN IDEAL We now define the endomorphism T t on H~ ~) and on: HI(Mo(N)/z , r by the formula: T t ---- c.. (cwt)* = (cwt) , . c*. From the definition one sees that the action of T t on HI(M0(N), 0~0(1~) ) and on H~ ~) are adjoint with respect to Grothendieck duality. The correspondence T t also induces endomorphisms of: (i) The Hodge filtration on 1-dimensional de Rham cohomology: o -~ H~ .e~, ~1) ~ H~R(Xo(N)/z[lm .el) -+ Hl(Xo(N)/mm. tl, eXo(N)) ~ o. (This action is hermitian - (Ttx, y)=(x, Try ) - with respect to the cup-product self- duality on H~m and it exhibits the adjointness of the action of T t on the two flanking members of the above exact sequence) (1). (ii) The jacobian of X0(N)/Q; its Ndron model J/z; the " connected component " of the Ndron model JTz; the singular cohomology groups of X0(N)/c with coefficients in Z; the p-divisible (Barsotti-Tate) groups Jp/z[lm]" The endomorphisms T t are hermitian with respect to the cup-product self-duality of i-dimensional singular cohomology of X0(N)/e and the auto-duality of the Barsotti- Tate groups Jp/zE1/N~- The effect of T t on the q-expansions of elements in H~ f2) may be computed over the base Q. (or c) and one finds (applying (6. i)) the classical formula: If the q-expansion of f is given by a~=Za,,q " then: w~ (6.2) Ttf=~bmq , where bm=g.a,~/t+at. ~ (with the convention that am/t = o unless e I m). Consider the action of T e on the Ndron model J/z and restrict to characteristic t. The Eichler-Shimura relation on the level of correspondence, (whose proof in [7] works mutatis mutandis for F0(N)) gives the fmmula: Eichler-Shimura : T t =Frob t +t/Frob t on J/Ft (14=N). Here Frob t is the Frobenius endomorphism of the group scheme J/Ft , and g/Frob t may be regarded as the canonical " Verschiebung " of the group scheme J/Ft. It follows that Frob t satisfies the quadratic Eichler-Shimura equation: X 2-T t . X + g : o in the endomorphism ring of J/vt. (1) Duality for de Rham cohomology is compatible with (indeed: constructed by means of) duality for coherent sheaves (of. [I8]). 12 9 ~ B. MAZUR Definition. -- By the Hecke Algebra T we shall mean the subring of End(J/Q) generated by the Hecke operators T t (g+N) and by w. The algebra T operates, by definition, on J/Q. It also operates (via the previously defined actions of T t and w) on the following list of objects: -J,z; J~z; Pic~ (xo (N)/z) ~ J~z; HI(Xo(N)/z, (9Xo(Ni)=Tan. space Pic~ ; H~ ~2) (which is the dual of the above) ; H~m(Xo(N)IQ) (which is the Lie algebra of the universal extension of J/. [3 7]) ; _Hsing(Xo(N)/c, Z). Clearly, T is a free Z-module of finite rank. It is known that T| is a commutative Q-algebra of rank g = genus(Xo(N)), and that it is isomorphic to a product of totally real algebraic number fields: TQQ,= 1-[k~ (6.3) (1). ~=I~ . . .~ t (6.4) Say that a T-module IV[ (of finite type) is of rank r (as opposed to free or locaUy free of rank r) if, equivalently: (a) M@O is free over TNQ of rank r. (a') For some, or any, field K of characteristic o, M| is free of rank r over T| (b) M| is a vector space of dimension r, over k~, for a=I, . .., t. (c) M contains a free T-module of rank r, of finite index. Note that ifM is a T-module of rank r, then the Z-dual T-module MY=Horn(M, Z) is again a T-module of rank r (2). Since H~ f21) is known to be a free TNE module of rank I (as follows from lemma 27 of [2]), one has: (6.5) H~ a) and Hl(Xo(N)/z, Ox.Im) are T-modules of rank I. H]i~g(Xo(N)/c, Z) is a T-module of rank 2. (x) This follows from lemmas x3, 27 of [2]. (2) It is not at all evident, however, that the operation v preserves the category of locally free T-modules. This latter assertion is equivalent to saying that T is a Gorenstein ring (but see w167 x5-I 7 below). 90 MODULAR CURVES AND THE EISENSTEIN IDEAL 7- Quotients and completions of the Hecke algebra. Let m be an integer. Let J[m]/z denote the scheme-theoretic kernel of multipli- cation by m in the N6ron model J/z. Since J is semi-stable (cf. appendix) J[m]/z is a quasi-finite fiat group scheme, whose restriction to S'= Spec Z[I/N] is finite and flat. Let a C T be an ideal containing m. By J [a]/Q we shall mean the kernel of the ideal a in the jacobian J/Q. That is: J[a]/Q = f'] (kernel of ~ in J/Q) = ['1 (kernel of 0~ in J[m]/Q). From the second description it is clear that J[a]/Q is a finite subgroup scheme of J[m]/Q. Now define J[a]/z to be the Zariski-closure of J[a]/Q in J/z. It is the subgroup scheme extension of J[a]/Q in J[m]/z, as in chapter I, w i; J[a]/z is a quasi- finite fiat group, which is, by constIuction, a closed subgroup scheme of J/z, and killed by a. The quotient T/a operates naturally on J [a]/z. Caution. -- The group scheme J[a]/z is not necessarily the full scheme-theoretic kernel of a in J/z. This kernel is not necessarily flat over Z. Fix a prime p. Let a C T be any ideal containing p. Let T a = lim T/a m denote the completion of T at a. Denote by Tp the completion of T at the ideal generated by p. Thus Tp =T| Since T is a finite Z-module, T, is a direct factor of the semi- local ring Tp. Write: (7.') (a) Tp=T~� (b) i = ~eL -t- ~'a where T~ is our notation for the factor complementary to T~, and (7. I) (b) is the associated idempotent decomposition of I in T,. Form the inductive limits of the quasi-finite group schemes: (7.2) J,,,. = 1~ J [pm]/,. J~ = lim J [d"]/z. Thus, Jp/z is an ind-quasi-finite group scheme, whose restriction to: S'= Spec(Z [I/N]) is a p-divisible (Barsotti-Tate)group admitting a natural continuous action of T~,. We may use the idempotent decomposition (7-i) to write J, as a direct factor of Jp: (7.3) Jp=J,� Restricting to the base S', (7.3) becomes a product decomposition of Barsotti- Tate groups. Moreover, since the action of T is hermitian with respect to the auto- duality of Jp/s,, one obtains an induced auto-duality on J,/s" 91 92 B. MAZUR To pass to pro-p-groups, one uses the Tare construction. We recall this in the category of modules. The functor M~ M| is an equivalence between the categories of free Zp-modules of rank r, and p-divisible torsion Zp-modules of corank r. The Tate construc- tion W ~Hom(Q~/Zp, W)= g~(W) provides an essential inverse to the above functor. There is a perfect Zp-pairing between g'a(W) and the Pontrjagin p-dual of W, W* = Hom(W, Q,/Z,). The isomorphism W*--~g~a(W)V=I-Iom(g~(W),Zp) takes q0eW to: ~-a(q~) : g'a(W) -+ g'a(Q,/Z,)~Z,. Let X0(N)e denote the analytic curve associated to X0(N)/e, andJc the complex Lie group associated to J/e. We may identify the singular homology group HI(X0(N)e , Z) with the kernel of the homomorphism of the universal covering group of Je to Je. By means of this identification, we obtain an isomorphism: (7.4) J,(C) = HI(X0(N ) e, Z) | Q,/Z, = I{l(X0(N ) e, Q,/Z,) where the left-hand group is the group of C-valued points of Jp. Applying the Tate construction: (7" 5) ga(Jp) (C) = gh(Jp(C)) = HI(X0(N)c , Zp) and this isomorphism is compatible with the action of Tp. Applying the idempotent % to (7-5) gives: (7.6) g'a(Jo) (C) = Nh(Jo(C)) = HI(Xo(N)e , Zv) | = HI(X0(N)c , Z) NTT o . The last equality, together with (7-5) gives: Lemma (7-7)- -- Let K be an algebraically close~l fiel~l of characteristic o. Then ~(J~ is of rank 2 over T o. (That is: g~(Jo(K))| is free of rank 2 over To| 8. Modules of rank I~ If M is a To-module of rank 2 (6.4) and there is an exact sequence of To-modules (up to torsion): o-+MI~M-+M2~o where M 1 and M s are Zp-dual (up to torsion), then they are each of rank I. We use this elementary assertion three times in this section. i. The defect sequence. Suppose p+N. Then Jp is an ind-6tale (quasi-finite) group scheme over the base Spec(Z). Consider the natural imbedding: 92 MODULAR CURVES AND THE EISENSTEIN IDEAL which induces an imbedding on Tate constructions. Form the exact sequence: (8. x) o -+ ~'a(Jv(F~)) -+ ~a(Jv(QN)) -+ -+o where A is the cokernel (the module of defect). The sequence (8. I) is compatible with the action of Tp. By the " th~or~me d'orthogonalit~ " (th. (2.4) of exp. IX, SGA 7), the toric part of ~'a(Jv(Q~)) is orthogonal to itself. On the other hand, the fiber J/~, is isomorphic to Ggm where g is the genus of X0(N ) (cf. appendix). It follows by computing ranks over Zp that the self-duality of $'a(Jp(Ox)) induces, up to torsion, a Z:duality between $'a(Jp(F~)) and A. Applying % (7. I) (b) to (8.I) yields an exact sequence: (8.2) o ~ ~'a(J,(fs) ) -+ ~'a(J,(Q~)) -+~, -~o where A~=A| and where Aa is dual to ~(J,(Fz~)), up to torsion. Applying lemma (7.7), we have: Proposition (8.3). -- ~a(J,(f~)) and A a are T,-modules of rank i. 2. Etale and Multiplicative type parts, in the ordinary case. Now suppose that p ~ N, and Ja is an ordinary Barsotti-Tate group. This means that over Spec(Zp) it admits a filtration: (8.4) o ~j~,lt. type ~j~ _+ o where j~ta~e is an 6tale Barsotti-Tate group (the gtale part of J,) and j~lt.wp0 is the connected component of J,, and is a group of multiplicative type (the dual of an 6tale Barsotti-Tate group). The self-duality of Ja induces a duality between j~ult.type and j~a~e. Applying ~'a to (8.4) , and using lemma (7.7) one obtains: Proposition (8.5).- $'a(j~u~t'tYPe(Q,)) and $'a(J~tate(Q,))are T,-modules of rank x. 3. Eigenspaces for complex conjugation. Complex conjugation e on the topological space X0(N)c commutes with cup- product and induces multiplication by --I on H ~. Consequently the cup-product pairing induces (up to torsion) a duality between the +i-eigenspace of e operating on Ha(X0(N)c, Z) and the -- I-eigenspace. Using (6.5) it follows that these eigenspaces are T-modules of rank i. 9. Multiplicity one. Let R be any commutative ring. Consider operators Tt:R[[q] ] --~ R[[q]] (g=t=N) and U : R[[q]] --~ R[[q]] defined purely formally by the appropriate equations: If f= Y, amq m, then : (9.I) Ttf=Y~atmq'~+t. Y~a,,q t'm (t+N), and Uf=~,as.,,q". 93 B. MAZUR Let ~ be any set of prime numbers, and .LP' the set of all positive integers which are not divisible by any member of ~ (so I is always in .s Let : f= alq-k a2q ~ +... eR[[q]] be a power series with no constant term, which is an eigenvector for T t (all le.L#, t:#N) with eigenvalue ct~R , and, if Ns& ~ an eigenvector for U, with eigenvalue cseR. The recursive relations: at. m =ct.am-~-g.am/t t~.L#, f ~ N aN.m = Cr~ . a m if Ne.~ show immediately thatfis determined by the eigenvalues c t for ~.5r and its coefficients a m for m~6~ ~ (1). In particular, given ct~R for all prime numbers ~, there is a unique power series f=~.q+a2q2+.., in R[[q]] such that Tt.f=ct. f for all t+N, and U.f=c~.f Moreover, any eigenvector in R[[q]] possessing the same eigenvalues for all these operators must be a scalar multiple of f. Catl f the generating eigenvector (for the eigenvalues { ct}. ) Proposition (9.2). -- Let R and B~ be as in w 4. Let elements cteR be given, for each prime number L If ~eB~ # a parabolic modular form such that: Tt.~=ct. ~ e+N (*) u. = cN. then the q-expansion of ~3 is a scalar multiple of the generating eigenvector f. The R-submodule Of B~ consisting in all elements which satisfy (.) is a submodule of a free R-module of rank ~. Now let 93~ C T be a maximal ideal, with k~ as residue field, of characteristic p. Let B~ denote the kernel of the ideal 93l. This may be viewed, in a natural way, as a k~-vector space. Proposition (9.3). -- B~ 93l] is of dimension I over k~. Proof. -- Let R = k~, or any field of characteristic p, which is large enough. Let M denote the k~j~-vector space B~ Clearly M+o, since T operates faithfully on B~ Since: M| C B~ (a) A (perhaps too) succinct way of expressing this determination is by the use of formal Diriehlet series with coefficients in R. Once one defines the evident rules of manipulation of these formal Dirichlet series, one has: Zam.m -s= ( Z am,m-S). [I Dg m m~.r ' g~.~ where: D t = (I --c t. t -s -t- tt-zs) -1 if t ak N, and Dbl =(I --c~.N-S) -1. 96 MODULAR CURVES AND THE EISENSTEIN IDEAL the proposition will follow, if we show that B~ is an R-vector space of dimension less than or equal to [k~ : Fp]. The action of T on B~ induces an action of k~n on B~ [gJ~] which commutes with the action of R. Since R contains k~, B~ possesses an R-basis of k~-eigenvectors. To each eigenvector in this basis, we may associate a homomorphism k~n-+R (by passing to eigenvalues). By the previous proposition, no two eigenvectors in this basis are associated to the same homomorphism. The proposition follows. Proposition (9.4)- -- t-Ia(Xo(N)/z, 0) is a locally free T-module, of rank I (1). Pro@ -- Note that if M is a T-module of rank I, it is locaUyfree of rank I provided M/gJt. M is a k~-vector space of dimension i, for all maximal primes 9XCT. Letting M=HI(X0(N)/z, 0), it is of rank i over T, by (6.5). Also: M = I-Ii(Xo(N)/,p, I-II(Xo(N) ,, (9) and the right-hand side of the above equality is isomorphic to the (FFvector space) dual of I-I~ a)[gJ~] = B~ which is of dimension I by (9.3)- Proposition (9.5). -- The Hecke algebra T is the full ring of endomorphisms 0fJ/c. Remark. -- This is a mild sharpening of a result of Ribet: that: T| End(J/,)|174 End(J/c )| [58] which is, in fact, used in the proof below. Pro@ -- Let T'=End(J/c ). By Ribet's result, any element of T' is defined over Q., and therefore acts on the Ndron model of J/Q; hence on the connected component JTz which is Pic~ hence on the tangent space to Pic~ which is IIa(Xo(N)/z, (9). It also follows by Ribet's result that T' is a subring of T| and hence is a commutative ring, and its action commutes with the action of the I-Iecke algebra T. We get, then, a homomorphism: T'-+ EndT(I-II(X0(N)/z, (9))=T" which is injective, since T| acts faithfully on Ha(X0(N)/o, (9). Since HI(X0(N)/z, (9) is a locally free T-module of rank I (9.4), T":T. The proposition is established. Definition. -- The Eisenstein ideal Z C T is the ideal generated by the elements: I +g--T t (all t4:N) and by I +W. If R is any ring, any element in B ~ (R) [~], the kernel of ~ in B ~ (R), is an eigenvector for the Tt's and for U, satisfying equation (,) above, where: c t =i+t if g4:N Cl~ ----- I. (1) It follows that B~ = H~ f~) is the Z-dual of a locally free T-module of rank x. The assertion that B~ is locally free over T is therefore equivalent to the assertion that T is a Gorenstein ring (see w 15 below). 95 96 B. MAZUR In R[[q]], the generating eigenvector for the above package of eigenvalues c t is the power series 8 of (5-I). Consequently, the q-expansion of any element of the R-module B~ must be a scalar multiple of 8. Proposition (9.6).- Let mbe any integer divisible by n=num(~-I]. Then ~ ,z z B~ [3] is a cyclic group of order n, generated by (m/n). 8. Proof. -- This follows from the above discussion and (5. I2). Proposition (9.7). -- T/~=Z/n; the Eisenstein ideal 3 contains the integer n (1). Proof. -- We have a natural map Z-+T/3 which is surjective, since, modulo 3, the operators T t (r 4: N) and w are all congruent to integers. We cannot have T/3 = Z, for then 8 would be the q-expansion of a modular form (of weight 2 for P0(N)) over C, which it is not. Therefore, T/3 = Z/m for some integer m, which must be divisible by n, since 8eB~ is of order n, and is annihilated by 3. We prepare to use the previous proposition. Since: B~ = H~ n) is the Z/m-dual of I-P(Xo(N)/(z/m), 0) (3-e) we have that B~ is the Z/m-dual of: Hl(Xo(N)i(z/m), 0)/3. I-~l(Xo(N)/(z/m), 0) =Ha(Xo(N)/z, 0)/3. HI(Xo(N)/z, 0) where, we have the equality above since me3. By the previous proposition, then, the cokernel of 3.HI(X0(N)/z, 0) in HI(X0(N)/z, (9) is cyclic of order n. Since (9.4) HI(Xo(N)/z, 0) is a locally free T-module of rank I, it follows that T/3 is cyclic of order n. Q.E.D. Definition. -- A prime ideal ~ C T in the support of the Eisenstein ideal is called an Eisenstein prime. The Eisenstein primes ~3 are in one-one correspondence with the prime numbers p which divide n by (9-7)- Ifp is such a prime number, then the Eisenstein prime corre- sponding top (which is the unique Eisenstein prime whose residue field is ofcharacteristicp) is given by: Clearly: T/~ =F,. One checks easily that n> i if and only if the genus of Xo(N ) is greater than o. Thus: Proposition (9.8). -- If the genus of X0(N ) is greater than o, the Eisenstein ideal ~ is a proper ideal in T; there are Eisenstein primes. (1) This vague result is sufficient for our purposes. It appears to be significantly more difficult to give an expression for n in terms of the operators Tt, in T. This would be particularly useful in questions related to w 19 below. 96 MODULAR CURVES AND THE EISENSTEIN IDEAL xo. The spectrum of T and quotients of J. As follows from the result of Ribet [58], there are one-to-one correspondences: isogeny classes of 1 l isogeny classes of 1 C-simple I Q-simple I < .'-. abelian variety 1 abelian variety t factors of J/c l factors of J/Q l IO. I ) t fields k s occurring irreducible I in the product de- ~ components composition (6.3) of Spec T of T| Define j+=(i+w).jcj; j_=(l-W).jcj. These are sub-abelian varieties, defined over Q. Form the quotients indicated in the diagram below: J+ (Io.2) o -+j_ -~J -+J+ -+ o J_ Thus J+, J- are quotients of J on which w acts as + i, and --I respectively. We let J~z denote the N6ron model of J~ over the base Z. By the criterion of N~ron- Ogg-Shafarevitch, J/z[lmj  is an abelian scheme, as are J+/z[1/~]. The abelian variety J+/Q can be identified with the jacobian of the quotient curve X+=X0(N)/w. One sees this as follows: since the map X0(N)-+X + is ramified (w has fixed points), the induced map on Pic ~ is injective and identifies the jacobian of X + with the connected component of the identity in the +-eigenspace of w in J. But the diagram (lO.2) identifies (I+W).J=J+ with this same connected component. To any ideal a C T we may associate an abelian variety Jl~ ) which is a quotient of J/Q, whose C-simple factors are in one to one correspondence under (io. i) with those irreducible components of Spec T which meet the support of the ideal a. To define J ("), let yaCT be the kernel of T-+T,=limT/am; let -~.JCJ be the sub- abelian variety (defined over Q) generated by the images ~.J for eeu Take Jl~ ~ to be the quotient abelian variety: (x o.s) o -+ v..J -"J _+jc.) _+ o. 13 98 B. MAZUR Let J}~) denote the N6ron model ofJ}~ ) over the base Z. By the criterion of N6ron- Ogg-Shafarevitch one has that l(a/ is an abelian scheme. O/Z[I/N] Definitions (xo.4): I) If a =3, the Eisenstein ideal, call J(") the Eisenstein quotient of j, and denote it J. 2) If a --~3, the Eisenstein ideal at p, call J (a) the p-Eisenstein quotient and denote it j(~l. Note that, for any p, the p-Eisenstein quotient is a quotient of the Eisenstein quotient. Conversely, any C-simple factor of J is a factor of J(p) for some prime p dividing n. It is also true (but not at all evident when n is even; cf. (x7.Io) below) that j is a quotient of J-. Definitions (xo 5): g = dim(J/0 ) g+ = dim(J~) = dim(J ~'= dim(J/0 ) g/,J= dlm(j/~t). So g=g++g-, and g+=genus(X+). The tturwitz formula computed for the map X0(N)-+X + yields the well known relation: 2(g---g+)=h--2, where h is the number of fixed points of w. Proposition (xo. 6). -- The scheme Spec T is connected. Pro@ -- Suppose not. It would follow that J/c could be expressed as a nontrivial direct product Jjc = A � B. Let us show that the principal polarization X : J ~J (^ denotes the dual abelian variety and X is the 0-polarization ( [43], chapter 6; [44])) induces principal polarizations XA : A-+A and Z B : B-+B. By Ribet's theorem [58], since J decomposes t~J (up to isogeny) into a product of simple factors, each occurring with multiplicity one, the simple factors of A are non-isogenous to simple factors of B, and consequently there are no nontrivial homomorphisms from A to B and from B to ,~. Our assertion follows. But a jacobian (taken with its natural principal polarization) cannot decompose as a nontrivial direct product of principally polarized abelian varieties. This follows from the irreducibility of its 0-divisor. Remark. -- When g+>o, the above proposition insures the existence of " primes of fusion " (see introduction) relating J+ to J-. It would be interesting to understand these primes. Ix. The cuspidal and Shimura subgroups. Let c be the linear equivalence class of the divisor (o)--(oo) in J(Q). Proposition (xx.I). -- The element ceJ(Q) is annihilated by the Eisenstein ideal 3. It is of order n. 98 MODULAR CURVES AND THE EISENSTEIN IDEAL Proof.- Since the correspondence T t (g#N) takes the cusp (o) to (I+g).(o) and (oo) to (I+r one has: Tt.c=(I+Q.c for all t+N. Since w interchanges the cusps o and 0% one has: (I -4- W) .6 = O. It follows that ~.c==o. From proposition we conclude that the order of c divides n. But since (Appendix A. i) the specialization of c to the Ndron fibre Je~ generates the cyclic group of connected components, which is of order n, it follows that the order of c must also be divisible by n. Q.E.D. Remark. -- The fact that order(c)=n was proved originally by Ogg [36]. He shows that the order of c divides n by exhibiting a function f on X0(N ) whose divisor is n. (o)--n. (~). Namely, if v is the g.c.d, of N--I and 12: ( A(z)~l/~=q. fi (I--qmN)-2'/".(I--qm) 2a/~ (xi.2) f(z)=\A(Nz)] ,,=1 can be shown to be invariant under F0(N), and clearly has the indicated divisor. Let C denote the subgroup of J(O) generated by c. Thus, C is a cyclic group of order n, with a distinguished generator. Denote by C/z the finite flat subgroup scheme of J/z generated by CcJ(Q). Let C=the IN-valued points of C/z ("the specialization " of C to J/F~). By the appendix, one has that C is, again, of order n (the specialization map C-+C is an isomorphism) and: (x I. 3) J/F~ =J/~ � C where J~F~ is the connected component of the identity. The retraction 0fJ(Q) to C. -- If xeJ(Ov) , denote the section over Spec Z induced by x in J/z by the same letter. Let x/F ~ denote the restriction of this section to an F~-valued section of J/F~. Let ~-be the image of x/F~, under projection, to C, using the product decomposition (i I. 3). Let p(x) ~C denote the unique element of C which maps to s under the " special- ization map" described above. If M=J(Q) (the Mordell-Weil group of J), we have just described a retraction p :M--+CC M, giving a product decomposition. (xx.4) M=M~215 where p is projection to the second factor; projection to M~ ker 0 is given by x ~ x-- 9(x). The Shimura subgroup. -- The Shimura covering (2.3): (x x. 5 ) X 2 (N)/s, -+ X0 (N)/s, is the maximal dtale extension intermediate to XI(N ) ~ X0(N ) and is a finite, 6tale, Galois extension, whose covering group U is the (unique) quotient group of (Z/N)* 99 ioo B. MAZUR which is (cyclic) of order n. Applying Pic ~ to the morphism (11.5) , we obtain a morphism J/s' ~ Pic~ X2(N)/s' whose group scheme kernel we denote Z/s,. Definition. -- The Shimura subgroup Y~/s C J/s is the group scheme extension (i.e. Zariski closure) of Z/s, in J/s- Let LrTs =.;g~rns(U, WI~) be the Cartier dual of U (where U is viewed as constant group scheme over S). Proposition (xx.6). -- There is a natural isomorphism UTs~X/s. The Shimura subgroup is a ~-type group (chapter I, w 3) over S; in part#ular it is finite and flat. Proof. -- We establish this first over the base S'. Consider the Hochschild-Serre Spectral sequence (for the 6tale topology ([i], III (4-7))) associated to the (finite 6tale Galois) Shimura covering X2(N)/T-+ X0(N)/T and the sheaf G,, where we have made the base change to an (arbitrary) S'-scheme T. We obtain the exact sequence: o ~ H~(U, GIn(T)) ~ H~(X0(N)/r, Gin) ~ H~(X2(N)/T, G~). . i Passing to associated sheaves, the morphism i induces an isomorphism, U/s,-~ Z/s,. Since U* is a finite dtale group scheme over the base S', this isomorphism extends to a . i homomorphism U~s-~ Z/s (by the universal property of the Ndron model). It follows that X/s is a finite fiat group scheme. Restricting to the base S', one has that the morphism i is a homomorphism of locally constant groups, which is an isomorphism on generic fibers. Hence i is an isomorphism over S'; hence i is an isomorphism over S. Proposition (II.7). -- The Shimura subgroup Z is annihilated by the Eisenstein ideal 3. Proof. -- We must show that w acts as --i on Z, and T t acts as I + t for g 4: N. As for the action of w, note that induces an involution w' on XI(N ) which (; projects to the involution w on X0(N ). If ~Po(N), one computes conjugation by w', and obtains: w' c~w'-0~ -1 mod FI(N), which yields what we wish. The operators Tr " act " as well on XI(N), by the formula: Tt : (z) ~ (~.z)+ =0\-7-1. In the above formula, as in the rest of this proof, we view the modular curves Xi(N) (i=I, 2, o) as analytic manifolds, parametrized by the extended upper half-plane. If oh, ~ ale points in the extended uper half-plane, let {~, ~} denote the (relative) homotopy class of paths in the extended upper half-plane beginning at o~ and ending at 9. Recall Ogg's convenient terminology for the cusps of F(N): Let: (~)={p/q~P~(Q) lp-a mod N, q--b mod N; (p, q)=i }. 100 MODULAR CURVES AND THE EISENSTEIN IDEAL 1OI With this notation, (2) is an equivalence class of pl(Q) mod F(N). Therefore it gives rise to a well-defined cusp of Xi(N) (i = i, 2, o). One shows (~) - (0) mod 1-'I(N), provided (b, N) = i. If r~{e, ~} is a path in the extended upper half-plane, let y(r~)~U be the (unique) element of U which maps the image of 0~ in X2(N) to the image of ~ in X2(N). Let ~b be a path in {(0), (0)}, for b an integer relatively prime to N. Then one checks that y(%) is the image of b -~ in U, while ~((Tt.%) is the (I+g)-th power of this image, as follows from the formula: Tt{(0), (0)}={(0), (0)}_}_ y~ {(j), (t!b)} j~0 The proposition follows. The Shimura subgroup over the base F N. -- Note that Z(FN)=Hom(U , ~,(FN)), and that there is a natural generator of this group. Namely: (Z/N)*= , where the unlabeled horizontal map is raising to the ,~-th power (v~(N--I, I2)). The natural projection J(FN) =j0(F~) � C-+ C induces a homomorphism: .8) :c which sends the canonical generator s to some multiple ~ of the canonical generator ~C. Thus ~ is a well-defined integer modulo n. Question. -- What is ~ ? Proposition (II.9). -- The homomorphism (11.8) is an isomorphism. The scheme- theoretic intersection X/r N c~J~ is the trivial group scheme over F~. The integer (modulo n) is relatively prime to n (1). Proof. -- The three assertions of the proposition are equivalent. We prove them by showing that: (xi. xo) Pic~ ~ Pic~ is injective. For this, we may identify Pic~ as group-scheme over Fs with the Gin-dual of the singular one-dimensional homology group of the topological graph (Appendix, w 3) associated to X0(N)~ . (homology with Z coefficients). (1) In the light of this, it is hard to imagine that ~ is anything other than ___ I. We have not, however, succeeded in answering our question. 101 io2 B. MAZUR By inspecting (diagram i of chap. II, w i) it is clear that this is the same as the G,;dual of Hl(Graph(M0(N)/~),Z ). To prove injectivity of (11.Io) it suffices to show that the map: Graph MI(N)/~ ~ Graph M0(N)/~ induces a surjection on one-dimensional homology. But the above map of graphs is an isomorphism as follows from [9], V, th. (2. I2) and VI, Cor. (6. Io). The relation between C and E. -- By the cuspidal subgroup ClzCJ/z we mean the Zariski closure of CcJ(Q) in the group scheme J/z. By the universal property of N~ron models, the isomorphism Z/n-+C (of group schemes over O; I ~c) extends to a homomorphism Z/n/s~C/s , and shows that C/s is a finite flat group. Proposition (xx. IX ). -- If n is odd, the group scheme C is a constant (gtale) group over S; the scheme-theoretic intersection Of C and Z over S is the trivial group; the natural map C| is an injection. If n is even, the group scheme C/s contains a subgroup scheme isomorphic to It2 (and which we shall call ~z2). The cokernel of ~2 in C is a constant (gtale) group. The scheme-theoretic intersection of C and Z in J!s is ~z 2. The natural map C | z--+J [3] has " the diagonal " It2 as kernel, and induces an isomorphism of (C| with the finite flat subgroup of order n2/2 in J[~] generated by C and E (call it C § Z). Proof. -- (a) We show first that the odd part of Z has trivial intersection with (the odd part of) C. For by consideration of Galois modules, the odd part of Z is a It-type group and the odd part of C is a constant group. (b) If n is even, the group E(O) (the rational points of Z) is of order 2. Lemma. -- Z(Q) C C. Proof. -- Suppose that n is even, or, equivalently, N--i mod 8. Then there is an 6tale double covering X0~(N)-+X0(N ) intermediate to the Shimura covering (2.3). This we shall call the Nebentypus (double) covering. Applying the functor Pic ~ to the Nebentypus covering (over O), we obtain a morphism of jacobians J/Q-+Jac(X0~(N)/Q) whose kernel is the group Z(Q). To prove the lemma, it suffices to show that the image, c ~, of c in Jac(X0~(N)/Q) is of order n/2. For this, it suffices to show that iff is the function (II.2) whose divisor is n. (o)--n. (oe), then ftl2 is a rational function on the Nebentypus curve X0~(N): ~o f112 =q,12 1] (I __q,,,)-121~. (I--qm) 12Iv m=l as follows from Dedekind's transformation formulas for the ~-function. (CA'. discussion of this in [48], w 3-) The Zariski closure of E(Q) in J/s (which is its Zariski closure in Z/s ) is a it-type group of order 2. Thus it is canonically isomorphic to It2ts and we shall denote it ~z21 s. 102 MODULAR CURVES AND THE EISENSTEIN IDEAL Io3 By the lemma, ix 2 C C/s. Since C/s is a finite flat group scheme, whose associated Galois module is a cyclic group with trivial Galois action, by (chap. I (4.6)) we have that the cokernel of ~2 in C is a constant (dtale) group scheme over S. It follows by an easy argument that Z n Cls is the finite flat group scheme ~x 2. There is a canonical auto-duality: II, I2) J[n] ~ ~om(J[n], ~,) (over Q) and the section ceC c J [n] (Q) determines, by (I I. 12), a homomorphism: ca: J[n]-~ r (over Q.). Restricting c ~ to Z, we obtain a homomorphism: ca : U*-+~n which, in turn, may be identified with an element uEU (1). Question. -- What is this element u ? This element has been evaluated in no case where n>I. One can show that ifp is an odd prime dividing n, then u projects to a generator of the p-primary component of U/f and only /f T~ (the completion of T at the Eisenstein prime ~3 associated to p) is isomorphic to Zp (cf. (I 9. 2) below) (2). In the light of the table of the introduction, it then follows that u does project to a generator ofthep-primary component of U(p % 2, p [n) for all N<25o except when N=3I , Io3, I27, I3I, I8I, I99 and 2II. x2. The subgroup D cJ[~] (p= 2 ; n even). Suppose n-omod4 (equivalently: N=imodi6). Choose yeZ(Q(~J--~)) an element of order 4. Let x=(n/4 ).c, which is an element of order 4 in C. Thus x, y are elements of C+E rational over Q(~/'----i-). By (Ii. Ii), 2x=2y. Let D C (C+E)/s be the closed subgroup scheme generated by the points x--y, and 2y. In (C+E)(Q(~r these two points are a basis of an F2-vector space (of dimension two) which is stable under the action of Gal(Q(~v/~---~ )/Q). Ifz is the nontrivial element of Gal(Q(~v/~i-)/Q), then the matrix of'~ computed with respect to the basis x--y, is (o :) Since O(x/'~---~)/O ~ is unramified at N, the group scheme D is finite and flat over S (chap. I (I.3)), and it follows from the above discussion and (chap. I (4.4)) that D is isomorphic to the unique nontrivial extension of Z/2/s by t~2/s killed by 2. (1) Since there are two natural choices of sign of the above autoduality (or equivalently, of the en-pairing), the pair of elements u  has, perhaps, greater significance than the element u. (2) Which explains why we might be interested in some reasonable direct method of computation of u. 103 IO 4 B. MAZUR The purpose of this section is to consider the case where n- 2 mod 4 (equivalently: N- 9 mod 16) and to construct a subgroup scheme of J[~], which is isomorphic to D. In this case, the 2-primary components of C and of ~ coincide with ~z 2. The group scheme D, which we construct, will contain ~2, but (necessarily) will not be contained in C+E. The construction of D. -- Ifn is even, the Nebentypus covering (w I I) X0~(N) --+ X0(N ) is 6tale (over S') with Galois group U/U 2. Let v be the nontrivial element in U/U ~, and J-~J~ the induced morphism on jacobians. Using the Leray Spectral sequence (over the base Q.) for Gm-cohomology of the Nebentypus covering, one has: (I2.I) O ~ Vq(Q) ~J(Q) --~ (J~(Q))~ -+ O where the superscript v means the part fixed under the involution v. To describe the Galois module associated to D, we shall construct a point of order 2 in J~(O..), and D(Q) will be, by definition, the subgroup of J(O..) generated by the inverse image of this point. There are four cusps on X0~(N ). Let o, g denote the cusps lying over o in X0(N), and o% ~ those lying over oe. Thus v interchanges o and g (and oo and ~). The cusps o and ~ are rational over Q., while oo and ~ are conjugate over O and defined over Q(~c/N). Compare [48], w I. Proposition (x2.2) (Ogg, Ligozat). -- Let Z be the Legendre symbol of conductor N, a)). z(a)-----(N), and let B2, z be the generalized second Bernoulli number associated to Z ([22 ] Then the divisor class of (o)--(~) (and of (oo)--(~)) in jr is of order B2,x/4. There is a rational function f on X0(N)/~ having the properties: (a) (f) = (B2, ~I4). ((o) --(~)) (b) ,J .f= -- i/f. The function f, and the proof of the proposition of Ligozat and Ogg are discussed below. We now prepare to apply their proposition in the construction of D. then B2,x-o mod8. Lemma (x2.3). -- If N=I mod 8, where B2(X) is the second Bernoulli poly- Proof. -- B~,x=N. 31Z(u).B ~ N nomial, X 2- X + I/6. Thus: ' \N ~ N! 104 MODULAR CURVES AND THE EISENSTEIN IDEAL xo5 and since N- I mod 8: g~z-- N (u ~'-u) mod8 l<~u<N--2 = Z 2u -2u) -4- Z .u. u odd u odd l'~ l~u(N--2 l~u<~N--2 =-4.  (u--I)/2 mod8 u odd l~u~N--2 Writing u=I+2j (j=o, I, 2, ..., (N--3)/2) we get: (N--3)/2. ((N-- 3)/2 +I) B~, z = 4- mod 8 N--I mod 8. -4" (N--3)" 8 Q.E.D. But N=I modS. (o)--(g) and (oo)--(~) are of even We conclude from (I2.2) and (I2.3) that order (I/4) B2, x = n~ in jr. Set: A = (n~/2). ~t((o)--(6)) B = (n~12). ~e((oo)--(~) 2A = 2B = o. Since A and B are fixed under ,J, they are in the image of J. SO Suppose that N=9modI6. -- The image of c in jt~ is the divisor class of (o)@(o)--(m)--(~) which is of odd order m=n/2. Thus: A+B =m.A+m.B=n~/2.~l(m. ((o) + (~)--(oo)--(~))) = o and therefore A--~B. Denote by DcJ(Q.) the inverse image (inJ(Q)) of the group generated by A. Since A is fixed by w and w~ (acting on jt~), D is stable under the action of w (acting on J). Also, D is stable under Galois. Let e~z~(O~) be the non- trivial element, and let ~D be an element in the inverse image of A. Lemma (x2.4). -- D is a Klein four group. The action of Gal(Q/Q) on D is the action which factors through Gal(Q.(v/~)/Q) where the conjugation "~ acts on the basis ~, It by the Proof. -- Using (I2.2), the above lemma is an exercise in Galois theory. To emphasize this, let K be the function field of X0(N)/Q and L the function field of X0~(N)/Q. 14 m6 B. MAZUR Thus L/K is a quadratic extension with ~ as conjugation. By (12.2)fis not a square in L| The extension L(flI2)/K is a quartic extension. Since ~(f)=--I/f, (12.2) (b), the extension L(f 1/~, %/~---~)/K(@--~I) is Galois. Let G denote its Galois group. Fix ~, a lifting ofv to G, and let -~ denote complex conjugation in L(f u2, %/----I). By (12.2)(b), ~(ft/2)= Therefore 52(fi/2)=ft/2, and consequently ~ = I. It follows that G is a Klein four-group and therefore so is D, for D is the Cartier dual of G. Let p denote the automorphism of L(f 1/~, %/-----T)/L(%/~-i), given by p(fl/2)=_fl/2. One checks: ~'~T -I ~ ~ .'v 'Tp~- i = p which yields the Galois action on D asserted in the lemma. Let D/s denote the group scheme extension (Zariski closure) of D m in J/s. Let D1/s denote the finite flat group which is the unique extension of Z/5 s by ~/s killed by 2 (extension ~ of chapter I (4.2)). Lemma (I2.5).- D/s~Dx/s. Pro@ -- The two groups have isomorphic Galois modules. Therefore, if D/s is a finite flat group over S, then (12.5) follows from chapter I (4.4). Consider an isomorphism DI/Q --~ D/QCJ/Q and extend it to an isomorphism: Dl/z[1/21 ~ D/z[l/~ ] C J/z[1/21 by the universal property of Ndron models. In particular, D/z[l/2] is finite. Since D/s, is clearly finite, it follows that D/s is a finite flat group. Lemma (I2.6). -- D is annihilated by the Eisenstein ideal ~3. Proof. -- By the formulas giving the action of T t on the cusps of X0~(N) one has, as in (11.1), Tt.A=(I-t-t).A (/~N). As already mentioned, A is fixed under w, and since it is of order 2, (I -}- w). A = o. It follows that D is annihilated by ~2. Any element ye~ operates as an upper triangular matrix in terms of the basis a, ~. To show that D is annihilated by ~, we show that 7~I operates semi-simply on the vector space D. For this, we choose a prime lying above N in Z[~r and consider the specialization map D(Z[%/~--i])-+D(F~), which is an isomorphism of T-modules. Let D(F~)~176 Note that D(F~) is canonically a direct sum: D(Fs) = D(F~)~174 ~2(Fs) (for the subgroup [x2CC maps isomorphically to the image of D(F~) in C (II.3)). Since the action of T " preserves j0 ,, it follows that the action of T preserves the above direct sum decomposition. Since each summand is an F2-vector space of dimension i, T does act semi-simply on D(Z[%/-----~]). 106 MODULAR CURVES AND THE EISENSTEIN IDEAL ~o7 Discussion of proof of the proposition of Ligozat and Ogg. -- Ligozat constructs the function f using the " Klein forms " of Kubert and Lang [28], which are essentially Eisenstein series of weight I. Ogg has a different point of view; he works with products of differences of Eisenstein series of weight 2. In the end, from either point of view, one emerges with a function f on X0(N)/Q whose divisor is B2, x/4.((o)--(~)) and which has the property that f(o~).f(~)=--~. Assertion (b) of our proposition follows from this equation since (vf).f must be a constant. It also follows that, up to sign, Ligozat's function and Ogg's function must agree (this identity is nontrivial). Both Ligozat and Ogg check that their function f is " smallest possible " and thus B2,x/4 is indeed the order of the divisor class of ((o)--(g)) in J~. Nevertheless, in the light of the use we make of (o)- (g) it is worth noticing that the equation f(~)f(~)=--I immediately implies that this divisor class is not killed by B2,x/8 (1). For if it were, there would be a function g on X0(N)/Q such that g2 = r.f where r is a rational (nonzero) number. This is impossible, for g(oo).g(~) would then be a rational number whose square is negative. In the remainder of this section, although we do not prove the proposition in full, we present an account of the construction of Ligozat's function and some of its salient properties (2). Ligozat's construction. -- We may take N- = I mod 4, N>5. Let ~ be a primitive N-th root of I; set: Sj: ={ I<a< (N--I) /2 l z(a)=  } ~ (I--~a'qm)(1--~-a'qm) " and : g+(z)= lI " + ,. = 1 ( I _ q~) (I~ - 1)/~ The functions g (q=e z'~i*) are expressible as products of Klein forms of level N [28]. Explicitly, let p be the constant: P+ =(--27d)lN-1)]~'exp(~-~'',,m, ~ 11 (I- ~a) -1 a6S+ then, using the notation of [28]: g (z) = p=~. oH kco, a)(z ). One checks : g+(z).g_(z)= II (i-qN")(~-r -N- ~(Nz) ,.=1 -n(z) ~ (1) An integer, by lemma (I~.3). (2) Here I have simply copied a part of a manuscript that Ligozat provided for me, and for which I am extremely grateful. It is to be hoped that Ligozat will present the full story in his future publications. 107 IO8 B. MAZUR and therefore (by Hecke [i9] , p. 924) g+.g_ is a modular form of weight --(N--I)/2 on F0(N), of Nebentypus, whose associated character is the Legendre character Z. g+(z) Definition.- f(Z)=g_-~. Lemma. -- f(z) is a modular form on Fi(N ). Proof. -- This follows from the transformation laws for the Klein forms k of [28]. If v=(I4-NaNy N~ I+Ns]eF(N) one has: k(0,~)(vz ) = (Nyz + NS). ~(Y, 8).k(0,,)(z ) where -- %(y, 8) = (--i)(ya + 1)(8a + 1) exp (2~i (-- Ya2)] under F(N) if and only if: 2N ] and therefore f(z) is invariant N--1 II 8) (~ 1 (a (N--l)/2 for any choice of y, 8. Since N>5, Y,z(a).a~---o mod N and therefore we may rewrite the condition of invariance off(z) as: (y + 8) (l<~<(~s_i)/2z(a).a) +y(I +NS) l<~<(x-1)/2Z z(a)a~-=o mod 2. Now note that if y is even, so is 8, in which case the above congruence holds. If y is odd, it also holds since l<~<(N~]-l)12z(a)(a 4-a 2) --o rood 2. Therefoie f is invariant under F(N). To see that it is invariant under Pi(N), note that k(o,~)(z+i)=k(o,~)(z ) for i~Z. Definition.- If u=(~ bd)~P0(N), define ~(u)=f(uz).f(z) -x(~). Thus ~ is a character of P0(N), trivial on Pi(N), and takes values in the group of (2N)-th roots of i. Lemma.- ~(u)=z(d ). Proof. -- Clearly ~2= i, since the index of Pi(N ) in F0(N ) is relatively prime to N. To establish the lemma, one must show that ~+ I. If (r a b)=u is in I'o(N), and z(d)=--I then: g (uz) = ~ (u) . (cz 4- d) (N -1)/4. g T (z) where ~=L(u) are 2N-th roots of i and s(u)= s+(u)/,(u). But since g+g_ =~(Nz)/~(z) N is of Nebentypus with character Z, %.s_=z(d)------I. It follows that s(u)=--i. Corollary. -- vf = -- i If. By the properties of Klein forms [28] the zeroes off are concentrated at (o), (~) and an elementary computation (compare [48], w 2) gives their order. 108 MODULAR CURVES AND THE EISENSTEIN IDEAL ~o9 z 3. The dihedral action on Xl(N ). We shall be working with the covering XI(N ) -+X0(N ) of curves over Q, and with certain subcoverings. Abbreviate the notation to Xi-+X0, and set: U=(Z/N)*/( So, U operates on X i with quotient curve X0; it operates freely on the open curve Y1 = X1-- cusps. As in [4 o] form a " dihedral " group A containing U as follows: where the w e are " symbols " indexed by the primitive N-th roots of I, ~eQ, where, by convention, the element w~-I is taken to be equal to the element w~. Impose a group law on A by: (I3.1) (gt)~)2= I ; U. W~ = W~u = ~)~.U -1 for all ueU, and primitive N-th roots of I, ~. Here ~,~_~a for a an integer (rood N) projecting to ueU. The dihedral group A acts in a natural way as a group of automorphisms of X 1 (cf. [4 o] w 2). The compatibility of the action of 2~ and of Gal(Q/Q) on Xl(()~) is most conveniently described as follows: Define an action of Gal(Q/Q) on A by the rules u~=u; (w~)~=w~, for 0~eGal(Q/Q), ueU, and ~ a primitive N-th root of I. Then, for SeA, and xeXi(Q), we have: (~.x)~=8~.x ~ (1). The action of A on X 1 " covers the action of the canonical involution w on X 0 ", in the following sense: If r: :Xi-+X 0 is the projection, then ~(w~.x)=w.r~(x); (u = Let 90C X0(Q) be the fixed point set of the canonical involution w. Using the modular definition of w, one sees that a point in 90 is given by an elliptic curve defined over Q together with an endomorphism whose square is --N (note: N>5). That is, the fixed point set is in one-one correspondence with isomorphism classes of elliptic curves over Q which possess a complex multiplication by ~r Suppose that N--I mod4. Then Z[%/~] is the full ring of integers in Q('v/~N) and % is a principal homogeneous set under the natural action of eel, the ideal class group of the field Q(@~N). Let q)i C XI(OL) be the full inverse image of %, and let q~l(~) C q~i be the fixed point set of w;, for each ~. (1) In [4 o] we call A, with its GaI(Q/Q) action, the twisted dihedral group. 109 IIo B. MAZUR If xxeq01, and ~ is a primitive N-th root of I, there is a unique element of U, which we denote u(.. ~) satisfying: W~. X 1 = U(x~, ~). X 1 . Clearly, for wU, u(~,,~)v=v.u(~,~) from which one gets Lemma (i3.2). -- q)l decomposes into the disjoint union: where ~ runs through the set of primitive N-th roots of r, with the convention that we have identified and ~-1. Let x0 denote the image of xl in X 0. An elementary computation gives, for any element u.xl in the inverse image of x0, that: (x3.3) u(..~,, ~) = u- 2. u(~,, ~) and consequently the question of whether or not u(~,, ~) is a square in U depends on x o and ~ hut not on x 1. Write u(~.,~)EU/U 2 for the image of u(~.~). Lemma (x3.4). -- These are equivalent: a) u(~,, ~) is trivial in U/U 2. b) w e possesses a fixed point in the inverse image of x o. Moreover, if these conditions hold, then w e will have exactly two fixed points in the inverse image of xo, and these fixed points will be multiples of each other by the unique element wU which is of precise order two. Proof. -- This is essentially immediate: If a) holds, choose an x t mapping to xo, and let ucU be such that u~=u(=,~). Then (i3.3) shows that u.x 1 is a fixed point of w e. The other direction is totally trivial. Finally, if x 1 is a fixed point of we, from (I3.3) the action of w e on the inverse image of x 0 is: wdu. xl) = u- 1. xl giving the last assertion of our lemma. Since (w~.x)~=w~.x ~ for ~eGaI(Q/Q), it follows that ~ induces a i : i corre- spondence q%(~)--->q~l(~), giving: Lemma (I3.5). -- Let h be the class number of Q,(~V/~---N). Then, for any primitive N-th root of I, ~, w~ has exactly h fixed points in XI(Q). Proof. -- The cardinality of qh is h. (N--I)/2. By (I 3.5), ~1 is the disjoint union of the (N--I)/2 sets q~l(~), which are put in i :i correspondence, one with another, by the action of Gal(Q/Q). It follows that each of these sets has cardinality h. Comparing lemmas (I 3 . 5) and (i3.4) it follows that, for a given ~, precisely 110 MODULAR CURVES AND THE EISENSTEIN IDEAL III half of the elements of % have the property that w~ has a fixed point in their inverse image. It is reasonable to expect that the elements of % with this property, for a given ~, forms a principal homogeneous space under the action of ~ft~C ~g (squares of ideal classes). Now pass to the Nebentypus curve X~X 0 which fits into a diagram: X1 where v denotes the involution of X ~ such that X#/v=X0, induced from the action of any ueU such that uCU ~. From (13 . i) one sees that the (N--I) involutions w~ induce precisely two distinct involutions of X ~ which we arbitrarily call US and v. uS. These are conjugate over Q and defined over Q(%/N). From (13 . i) we have that v and w # commute. Also, from (13.4) it follows that if w e induces w ~, then q~l(~) projects bijectively to the fixed point set of US. Consequently, both US and ~. US have exactly h fixed points in X#(t:~). Now suppose N-I mod 8, so X~--->X 0 is unramified. Consider the diagram: x* (x3.6) X:/w: x:/v = x0 X + = Xo/w Lemma (x 3. 7). -- Both ~r and ~ are ramified. To compute the number Proof. -- As for ~, this follows since uS has h fixed points. of fixed points of ~, we use the Euler characteristic Z: (since X~-+X 0 is unramified) z(X~) = 2. z(Xo) (US has h fixed points) z(X~) = 2. z(X~/us)- h (w has h fixed points) z(Xo) = 2. z(x +) - h and therefore ~ has h/2 fixed points. which gives: z(X~/US)----- 2. z(X+)--h/2 Lemma (i3.8). -- We continue to suppose N-I mod8. The subgroup DCJ/Q (of. w 12) has trivial intersection with the sub-abelian variety J+ =(I + w).j. Proof. -- We work with group schemes over Q. We frst show that the subgroup ~2 of the Shimura subgroup has trivial intersection with J+. If Y~Z is any double covering of (smooth projective) curves, then the induced map on their jacobians (regarded 111 II '~ B. MAZUR as Pie ~ is injective if and only if the double covering is ramified. Since X0-+X+ is ramified, we may identify the jacobian of X + with the sub-abelian variety J+ cJ. The subgroup a2C D is the kernel of the map J-+J* onjacobians induced by X#~X 0 (I2. i). To show that ~2 is not contained in J+, it suffices to show that the composition J+-+J-+J~ is injective. But the map J+-~J~ is induced from the covering of degree 4, X~-+X +. Returning to diagram (I3.6) we have that this map is the composite ~ where by (~3-7) both ~ and ~ are ramified double coverings. Injectivity of J+-+Jl follows. Since J+ is defined over Q, and Dt~J+ is a subgroup scheme of D (over Q) not containing a,,, it must vanish. Q.E.D. Corollary (x3.9).- The subgroup scheme D/s, CJ/s, maps isomorphically onto a subgroup scheme of J~, under the natural projection of abelian schemes Jj.s,-+J;, (cf. w m). Proof. -- Let D~, c J~, be the subgroup scheme extension of the image of D/Q in J~. Then we have a map D/s,-+D ~, which induces an isomorphism on Galois modules. It must be an isomorphism, by chapter I (4-4)- For later purposes: Corollary (i3.xo). -- The subgroup ~ (D /F,J ~t C J/F, /s not in the image of i ~-w. Pro@ -- The image of J/F, under I-}-w goes to zero in J~,, but (D/F,) ~t does not, by (~3-9)- x 4. The action of Galois on torsion points of J. Let m be an integer 4=o, and consider J[m](Q) as a T/(m.T)[G]-module (the group ring of G with coefficients in T/(m. T) where G is some finite quotient of Gal(Q/Q,) through which the natural action of Gal(Q/Q,) on J[m](Q) factors). Say that the T/(m.T) [G]-module V is a constituent of J[m](Q) if it is a constituent of a T/(m.T) [6]- Jordan-H61der filtration ofJ [m] (Q). Since a constituent Vis irreducible (as T/(m. T) [G]- module), its annihilator in T is a maximal ideal 93l. Say that V belongs to gJl. Thus, V is a h~[G]-module where h~ is the residue field T/9~. By the dimension of V we mean its dimension as h~-vector space. Note that any constituent V belonging to ~ is a constituent of the sub-module J[~r](~)cJ[m](~) for suitable integers r, m. Note also that given a generating set of elements (al, ..., a~) of the k~-vector space 9J~r/~lN ~ + 1, the map x ~ a 1. x| | t. x is an injection of the module J [gJl']/J [9)l "+1] (Q) into the direct sum oft copies of J [92R] (Q), and therefore V is isomorphic to a constituent in j[gJI](Q). Regarding V as a specific subquotient of J[m](Q) we may use (chap. I, w I (6)) to obtain a quasi-finite group scheme subquotient V/s of Jim]/s which is finite and flat over S', and whose associated Galois module is the subquotient V. 112 MODULAR CURVES AND THE EISENSTEIN IDEAL II 3 Note however that the isomorphism type of V/s may depend on the way we view V as subquotient ofJ [m] (Q) and is not necessarily predictable from the isomorphy type of V. By Fontaine's theorem, chapter I (i.4), however, it is determined (over S') by the isomorphy type of V provided the characteristic of k~ is different from 2. Let p be the characteristic of k~ and VtF q the fibre of V/s reduced to characteristic p. Consider the two possibiIities: a) TpegJ~. Then, by the Eichler-'Shimura relations (w 6), both the Frobenius and the Verschiebung satisfy the relation X2--TpX+p=o, and therefore, since 93l annihilates V/F p they satisfy the relation : X ~ = o. That is, both Frobenius and Verschiebung are nilpotent on V/F p. Consequently, V/F p has the property that both it and its Cartier dual are unipotent finite group schemes. Equivalently, it has a Jordan-HSlder filtration by finite subgroup schemes, all constituents being isomorphic to % ([9], IV, w 4 (3-14)). In this case say that 9X is supersingular. b) Tpr Then, as above, Frobenius and Verschiebung satisfy X. (X--Tp)=o, where Tp is an automorphism of V/F p and it follows that: __ VIll.t. ~ V/Fp-- /Fp � V/Fp- (The product decomposition arising, if you wish, from the fact that T~ -1. Frobenius and T~ -1. Verschiebung are orthogonal idempotents whose sum is the identity.) Thus V/F ~ is, as we shall say, an ordinary group scheme over Fp. In this case we say that ~ is ordinary. Proposition (z 4. 9 ). -- Let V be a constituent belonging to 9)l. Then V is of dimension I if and only if gJ~ is an Eisenstein prime, ff g~ is an Eisenstein prime, then Jilt]/s is admissible (cf. chap. I, w I (f)). Proof. -- We first show that if V is of dimension I, then it beiongs to an Eisenstein prime. Consider V/s, which is a finite flat group scheme if and only if the inertia group at N operates trivially on the k~-vector space V (chap. I (I.3)). Since the inertia group operates unipotently (SGA 7, exp. IX (3.5) (crit~re galoisien de rdduction semi-stable) which applies since (appendix) J/s has semi-stable reduction at N) and semi-simply (since V is of dimension i over k~), it does operate trivially (1). Thus Vts is a finite flat one-dimensional k~-vector group scheme. By chapter I (I. 5), either: V/s = ~p| or: V/s =Z/p| and in either case, the Eichler-Shimura relations (w 6) give us the following facts about the image of T t (g+N) in k~ which we can think of as contained in End(V/@: Tt- I -}-t modg~ (g4=N). (1) This was pointed out to me by K. Ribet. 15 iI 4 B. MAZUR As for the image of w in k~, since w is or order 2, this image must be 4- i. If the image of w is --i, then 9J/ is visibly the Eisenstein prime of residual characteristic p. To conclude the first part of this proof, one must show that if p is odd, the case w~ + i cannot occur. We show that the ideal 93/generated by:p, I--W, and i +t--T t (all t+N) is the unit ideal in T. Suppose not; then it is a maximal ideal with residue field Fp. By (9-3) the kernel of its action on I-I~ , ~)is of dimension i over Fp. This kernel is generated by a parabolic modular form mod p, g, whose q-expansion is entirely determined (9-2) by the above package of eigenvalues, and the fact that it begins with the term i .q. Comparing the coefficients ofg with that of the Eisenstein series e' (5- i) one sees that f=e'§ is a modular form modulo 24P whose q-expansion (modulo 24P) is a function of q~: y= ( I -- N) -- 4 8. q~ +... If P>5, such a modular form does not exist (1) by lemma (5. IO) (if p=N), by lemmas (4. io), (5.9) (if piN--I) and corollary (5.II) (if p=t=N, pgN--I). If P=3, and N-=I mod3, then J~3= ~- -t-I6-q~-} -.-- does not exist mod 3, as a holomorphic modular form, by corollary (5-i i) (ii) (if N~ i rood 9) and by (4-io) and (5-9) (if N=I mod9). Finally, if P=3, and N=--I mod3, fdoes not exist mod 9 by corollary (5. i I). To conclude the proof of our proposition, we show that if ~3 is an Eisenstein prime, then J[~3 r] is admissible (any r) and consequently any of its constituents is, indeed, of dimension I. In the light of (chap. I, w I (f)) and remarks made at the beginning of this section, it suffices to show that J[~3] (Q) possesses an admissible filtration by sub- Galois modules. Let W denote the Gal(Q/Q.)-module which is the direct sum ofJ [~3] (Q) and its Cartier dual. Thus W is a self-dual Gal(Q/Q.)-modute, annihilated by ~3, of dimension 2d, say, over Fp. We let G denote a finite quotient of Gal(Q/Q) through which the action on W factors. Since T t acts as I+t on W, (t+N), the Eichler- Shimura relations (w 6) impose the relation: on the action of the Frobenius automorphism ?t (t + N, p) on W. Thus, the only eigen- values possible for the action of q~t on W are: I and g. Since Cartier duality " inter- changes " these eigenvalues, and since W has been devised to be self-dual under Cartier duality, it follows that the characteristic polynomial of q~t acting on W must be (X-- i)~(X--~) ' . Now consider the Gal(Q/O)-module (Z/p)a| a, which we also regard as a G-module (the natural action on this module factors through G, and if it did not, we would have augmented G appropriately). It also has the property that the characteristic (I) This has been proven independently by K. Ribet. 114 MODULAR CURVES AND THE EISENSTEIN IDEAL I15 polynomial of ?e acting on it is: (X--I)e(X--t) a. By the ~ebotarev theorem any element in G is the image of some q~t (t #p, N). Thus, any element geG has the same characteristic polynomial for the represen- tation W as for (Z/p)d| a. By the Brauer-Nesbitt theorem ([6], (3o. I6)), the semi-simplification of the representation W is isomorphic to (the already semi-simple) (Z/p)d| e. Thus W has an admissible filtration and therefore, so does J[~](()~). Proposition (x4.2). -- Let 9J~ be a prime which is not an Eisenstein prime, and which is supersingular if char k~=2. Then J[gJl] is an irreducible two-dimensional Gal(I~/Q)- representation over k~oz (1). Proof. -- By theorem (6.7) (and (3.2)) of [IO], there is a unique semi-simple rep- resentation p : Gal(Q/Q) -~ GL2(k~) such that for every e#p, N, if at=image(Tt) C k~: Trace (?t) = at det(q@ =L Denote by V the associated semi-simple k~[Gal(Q/Q)]-module. Let: d = dimk~j~ (J [9~] (Q)). As in the previous proposition, form the Gal (Q/Q)-module W: the direct sum ofJ [gJ~] (Q) with its Cartier dual Let W'= the direct sum of d copies of V. By the Eichler- Shimura relations, the eigenvalues of q~t are constrained to be solutions of the quadratic equation X2--atX+t-----o, and since Cartier duality " interchanges the roots of the above equation " the characteristic polynomial of q~t operating on the self-dual Gal(Q/Q)- module W is: (X~--atX+t) a. But this is also the characteristic polynomial of q0 t acting on the semi-simple Gal(Q/Q)-module W'. It follows that W' is the semi-simplification of W. By prop- osition (I 4. I) (and the fact that 9J~ is not an Eisenstein prime) it follows that V is an irreducible k~[Gal(Q/Q)]-module. Therefore, W has a Jordan-H61der filtration of sub-k~[Gal(Q/O)]-modutes all of whose successive quotients are isomorphic to V. It follows that J[932](Q) also has such a filtration. In particular, considering the first stage of such a filtration, we have an injection VCJ[gJ~](Q). We must prove that V=J[gJI](Q). We do this by studying V/sCJ[~lJt]/s, the quasi-finite group scheme extension of V. Case 1.- Chark~+N and either: a) 93l supersingular or b) 93t ordinary and char k~ + 2. (1) We also establish (cf. (I6.3) below) that J[~] is a 2-dimensional Gal(Q/Q)-representatlon, when ~ is an Eisenstein prime. 115 II6 ]3. MAZUR Here we make use of the contravariant Dieudonn6 module functor of Oda [47], denoted M(--). Its relation to De Rham eohomology is given by corollary (5.II) of [47]. Namely, if A is an abelian variety over a perfect field k, of characteristic p then there is a functorial isomorphism of Dieudonnd modules: M(A[p]) ~ H~R(A/k ). Moreover, under 4, the Hodge filtration: o ~ H~ f2) ~ Hla(A/k) -~ HI(A, (9A) --, o corresponds to the filtration: o --~ M(A[Frob])' -~ M(A[p]) --~ M(A[Ver]) --~ o where [ ] means, as usual, kernel, Frob means the Frobenius endomorphism, Ver means the Verschiebung, and the prime superscript has the following significance: M (A [Frob]) ' = (k, a- 1) | M(A [Frob]) where (k, a-l) is the abelian group k, regarded as k-algebra by the morphism k~ where a is the p-th power map. Moreover: M(A [p])[Frob] --~Ver. M(A[p])=~ M(A[Frob])' where Ver and the first Frob denote the V and F operators of the Dieudonnd module M(A[p]). If G is a finite group scheme over k equipped with a homomorphism: T /pT --~ End(G/k), we induce a T/pT-module structure on M(G) commuting with its module-structure over the Dieudonn6 ring. Since M(--) is an exact contravariant functor, we have M(G)/0Jr. M(G)- M(G [gJl]). Consequently: M(J [~]/F,) = M(J [P]/F,)/~" M(J [P] A',) = H~)R(J/Fp )/92R. HI)R(J/F,) ~ H~)R(Xo(N)/,,)/g.R. H~)R(X0(N)/@ where the last isomorphism comes by the identification of J with the Albanese of X0(N), and all isomorphisms are isomorphisms of T/pT-Dieudonn6 modules. Make these abbreviations: M(V/Fp)=M; t 1 HDR(X0(N)/Fp) = Hi)a. The inclusion V/l,~cJ[9)l]/Fp induces a surjection of the k~-Dieudonn6 modules: H~/OJ~. H~ -+ M ~ o. 116 MODULAR CURVES AND THE EISENSTEIN IDEAL I17 Passing to the cokernel of Verschiebung, one has a diagram: 1 1 HDa~/9"J~.HIm -> M > o Hl(g)x)/gJt.Hl(@x) ). M/Ver. M > o o O where we have written I-P(Ox) for tP(X0(N)/Fp , d)). By (9.4), I-II(@x)/gJ~.Hl(Ox) is a k~-vector space of dimension i. Thus: (I4" 3) dimk~(M/Ver. M)5 I. We now use the hypothesis that either ~ is supersingular or the characteristic of k~ is different from 2. Lemma (x4.4). -- With the above hypotheses, V/s, is an auto-dual finite flat group scheme with respect to Cartier duality. Neither Frob nor Ver vanish identically, nor are they isomorphisms, on the Dieudonng module M. Proof. -- The Gal(Q/Q)-module V is auto-dual under Cartier duality. Therefore Vls, and its Cartier dual V~, have isomorphic associated Gal(Q/Q)-modules. Under our hypotheses, Fontaine's theorem, chapter I (I.4), applies. Thus V/s, is auto-dual. Consequently, M is a self-dual Dieudonn6 module. Since Frob.Ver=p=o on M, it is clear that not both Frob and Ver can be automorphisms of M, and by self- duality, neither are. Also, by self-duality, if one of the two operators Frob and Ver are identically zero, then both are. In particular, Ver would be zero, which is impossible, since its cokernel is of dimension less than or equal to I by (I4.3). An immediate consequence of Lemma (14.4) and (14.3) is: Lemma (x 4 . 5). -- Hl(@x)/93l. Hl(Ox) -+ 1VI/Ver. M is an isomorphism of i-dimensional k~-vector spaces. Lemma (x 4. 6). -- Let 0 -+ M 1 -+ M 2 -+ M 3 -+ 0 be a short exact sequence of (finite) k~-Dieudonng modules satisfying these properties: a) the cohernel of Ver on M 2 # of dimension I over k~; b) Frob is nonzero on M 3. Then Ver is an isomorphism of M1 onto itself. Proof. -- We show that Ver : M1-)-M 1 is surjective, by showing: (i) M2/Ver. M2 ~ M3/Ver. M 3 is injective, (ii) M2[Ver] ~ Ms[Ver] is suriective, 117 lib B. MAZUR and applying the snake-iemma. Since the morphism (i) is surjective, and Mz/Ver. M~ is of dimension i over kan , it suffices to show that Ver. Ma4:M3 to obtain injectivity of (i). But Ver annihilates Frob. M 3 which is nonzero, by b). Therefore Ver is not an automorphism of M3. To show (ii), note first that since Mi is finite-dimensional over k~ (i=2, 3), dim~x~(M2[Ver])=I (as follows from a)); also dim~(Ma[Ver])=I (as follows from the isomorphism (i)). Therefore all that must be shown is that the morphism (ii) is nonzero. But this follows from the diagram: 1V[ 2 > M 3 > o M2[Ver ] > Ma[Ver ] oi) and hypothesis b). To apply lemma (14.6) to our situation, take iV[2= t i HDR/gX.HIm and Ma=M. Both hypotheses a) and b) hold, by lemmas (14.4) and (14.5). We obtain the following conclusion: If M 1 is the kernel of the homomorphism Htm~/gJt. H~R-+M, then Ver is an isomorphism on M 1. That is, M 1 is the Dieudonn6 module of a group scheme of multiplicative type. Therefore the cokernel of V/s, CJ[gX]/s, has multiplicative type reduction in characteristic p. But, by the discussion at the beginning of this proof, and by Fontaine's theorem, this cokernel has a filtration by finite flat subgroup schemes all of whose nontrivial successive quotients are isomorphic to V/s, , which does not have multiplicative type reduction in characteristic p. We conclude that this cokernel is zero. Case 2. -- (A digression) 9)l ordinary and char kan 4:N. There is another more direct way of putting the above argument, when 9X is ordinary. This alternate method does not use Dieudonn6 modules, but rather depends upon an important isomorphism due to Cartier and Serre ([64], w II, Prop. IO). By means of this isomorphism, one may deduce (14.8) below, which will also be useful to us in the case where 93l is an Eisenstein prime (el. (14.9), (14. IO)). Thus, in the present case we let 93l be any ordinary prime in T, Eisenstein or not, with char k~ 4: N. Recall the canonical isomorphism: 3 : J[p] (F,) --+ H~ f~l)* of [64] , w 11, Prop. IO, where the superscript ~ means fixed elements under the Cartier operator. (Note: this isomorphism is defined for any smooth projective curve and not just Xo(N). ) The definition of ~ is as follows: an element x of the domain is rep- resented by a divisor D on X0(N)~ p such that p.D=(f). One takes ~(x)=df/f. 118 MODULAR. CURVES AND THE EISENSTEIN IDEAL II 9 Proposition (I4.7). -- The isomorphism above induces an injection: 8 : (J[p](F,)) | ,-+ H~ a *) which commutes with the action of T/p.T on domain and range. Proof. -- By ([64], w i I, Prop. io) injectivity follows from injectivity of the natural map: (H~ tll)*) | -+ H~ s which is an elementary exercise, using a- Minearity of c~: Let xl, ..., x, be the smallest number (s~o) of FKlinearly independent elements of H~ t~t) ~ such that xl-}-Xg.x~+...4-X~.x~-~o , where ~jeFp. Applying I--C~ to this equation gives a smaller relation. That 8 commutes with the action of w is evident. To check that it commutes with T t boils down, in the end, to checking commutativity of the (two) squares: L* elo~ ~p K* > t~ep d log where K is a function field in one variable over Fp and L is a finite K-algebra. Corollary (I4.B). -- Let 9J~ be an ordinary prime ideal in T with char k~+N. Let (J[p]/pp) ~t denote the gtale part of the group scheme J[p]/~p, and let (J[p]~)[9:R] denote the kernel of the ideal ~ in this group scheme.. Then (J[p]~p)[9J~] is a k~-vector group scheme of rank i. One has the equality: J[9)l]~p=(J[p]~p)[g31] /f p>2. Proof. -- The rank of (J[p]~p)[93l] is at most i as follows immediately from the previous proposition and proposition (9-3)- To obtain the equality asserted, we must lrgj~l~t is nontrivial. If it were trivial, then J[931]/Fp would be of multipli- show that ak J/Fp cative type. Since p::>2, Fontaine's theorem, and the remarks at the beginning of this section, apply, giving us that any constituent of J[!I~r]~p is a constituent of J[~0l]/Fp. It would then follow that the p-divisible (Barsotti-Tate) group J~l/Fp is of multiplicative type, which is impossible, since it is auto-dual under Cartier duality. Ifp divides n, let Jp[~]/s denote (as usual) the group scheme extension to S of the kernel of the Eisenstein ideal 3 in the Barsotti-Tate group Jp/Q. This group scheme is also J[3,pt]/s where pt[In. Let Gp/s denote the p-primary component of the cuspidal subgroup C (regarded as constant group over S). If p = 2, let Dis denote the subgroup scheme of Jts constructed in w I2. 11 ~2o B. MAZUR Proposition (x4.9).- If p is an odd prime dividing n, then: = (J, [3] e,) t = (J YF,) [3] and, if p = 2, divides n: Remarks and pro@ -- The right-hand group scheme on the first line of our prop- osition is the kernel of 3 in the dtale part of the p-Barsotti-Tate group over Fj, associated to J. All of the asserted equalities of the proposition are known inclusions (reading from left to right). To establish the proposition, note that Corollary (14.8) gives that: ~t is a group scheme of order p. The assertion of the proposition for p = 2 then follows simply by noting that (D/F~) ~t is also a group scheme of order p. To obtain the assertion when p;>2, let prlln. Note that (J~Fp)[3] is annihilated by pr by (9.7), and since the kernel of multiplication by p in this group scheme is of order p, it follows that (J~[l,~)[3] is of order p( So is C~, by (1~.I). The proposition follows. Corollary (x 4. xo). -- If p is an odd prime dividing n, then one has a short exact sequence: (i) o ~ Cp -+Jp[3] -3- M ~ o (over S) where M is an admissible group scheme of muhiplicative type. If p = 2 divides n, then one has a short exact sequence: (ii) o ~ D -+J [~3] --~ M -+ o (over S) where M is an admissible group scheme of multiplicative type. Proof. -- In either of the exact sequences above, the cokernel is admissible by (t 4- i), and of multiplicative type by the previous proposition. Corollary (x 4 . xx ). -- Let p be a prime dividing n. Let W denote the Zp-dual of $-a(Jv(Fp)). The To-module W is free of rank I. Proof. -- By proposition (8.4) it is of rank I (i.e. W| is free of rank I over TvNQ). It suffices to show that W/~3.W is oforderp. But (w 7) W is the Pontrjagin dual of J~(F~) and therefore we must show that J~(Fp)[~3] is of order p, which follows from (14.9). As for the alternate argument: suppose gJl is ordinary, not an Eisenstein prime, and such that char k~l+N , 2. By Fontaine's theorem, and the discussion at the beginning of this section, V~r p cannot be of multiplicative type. Thus, the k~-rank of (V/Fy t is 21, and (by (14.8)) the k~-rank of (J[9)l]~rp) 6t is S~I. It follows that V=J[9/II]. 120 MODULAR CURVES AND THE EISENSTEIN IDEAL I2I Case 3. -- Char kzl = N. This case parallels the " alternate argument " in case 2. Note that if char k~ = N, then ~f)2 is not an Eisenstein prime. Also, J [~JX]/s, is a finite dtale group scheme, admitting a Jordan-H01der filtration by finite dtale subgroup schemes all successive quotients being isomorphic to the finite dtale group scheme V/s,. Consider the first layer in such a filtration VCJ[992], and note that we have exact sequences: o " J[~]~ ' J[~](QN) ' J[~](fN) , o I * U U U o , , v(0 N) - , , o where the superscript 0 may be viewed as denoting either the connected component over Spec Zs, or the intersection of the (appropriate) group scheme over Spec Z N with J~z~- Since V is self-dual, we cannot have V(Qs)=V~ (for then it would be of multiplicative type over FN). Therefore dimk~(V(Fs))>I. As in case 2, we must show dim~J[9~](Fl~)<I. For this, we extend Serre's mapping a to cover our present case. Let D represent a divisor class x in J IN] (Q~)= Pic~ Assume D is an eigenvector for w. Letfbe a rational function on Mo(N)/~N (Zr~=ring of integers in QN) such that (f)=N.D on X0(N)/~,~, and such that f does not vanish identically on M0(N)/~N. Since f satisfies an equation of the type fow = 4-f~:*, and w interchanges the two irreducible components of M0(N)/f~, f vanishes on neither component of M0(N)a ~. Form df/f in H~ ~*) where the superscript h denotes the smooth locus. Note that: H~ f~*)= H~ f~)= H~ f~) (the second equality comes from (3.4); for the first, since Mo(N ) is Cohen-Macaulay, g) is invertible, and the supersingular points of characteristic N are of codimension 2 in M0(N)/z~,). Set 8(x) ----- image of df/f in H~ f~). Since the function f is unique up to a possible multiple u.g s where u is a unit in 7.N and g is a rational function on M0(N)/z~, the mapping x~8(x) is well-defined. Also, ~(x)= o if and only if f, reduced modulo N, is an N-th power (or, equivalently, x goes to zero in J[-N](FN) ). Extending the definition of ~, by linearity, to all divisor classes x~J[N](QN), we have: : J[N](Fs) ~ H0(Xo(N)a., f~). lfi x22 B. MAZUR Since the theory of the Cartier operator ([64] , w io) is local, it applies to the smooth quasi-projective curve X0(N))~ and one has, as before, that the image of 8 is contained in the fixed part: H~ ~21) ~, and by the argument of (14.7) , one deduces an injection: : J[N] (Fs)@x,,~Fz~ r H~ n). At this point one uses (9.3) as in the proof of (I4.8) to conclude that: dimk~ J [~] (Fs)~ i. What is the minimal field of definition of the Gal(Q/Q,) representation determined by J[9:R] ? Proposition (I 4. x2). -- Let ?fit be a prime which is not an Eisenstein prime, and which is supersingular if char k~ = 2. Then k~ is generated over Fp by the images of the operators T t (l@char k~), and J[gJ~] is an irreducible Gal(t2~/Q)-module. Proof. -- Before we begin the proof, let us note that the last assertion is stronger than the assertion of Proposition (I 4. 2). We are saying that the abelian group J[gJ~] (l~)=V is irreducible as Fp[Gal(Q/Q)]- module. Let E be the image of Fp[GaI(Q/Q)] in the endomorphism ring ofJ[g)l] (Q). Let k C k~ be the subfield generated by the T t (for all [ +p, N). Using the Eichler- Shimura relations for J[gJ~]/Ft and the fact that J[9:R] is dtale in characteristic [+p, N, we obtain a natural imbedding of k in the center of the ring E, which we therefore view as k-algebra; in fact we take k systematically as our base field. Note that k~ =k[vp] (zp=imagerlp), if p+N. If p=N then k=k~. Let V x be a two-dimensional Gal(Q/Q)-representation over k such that V~| Such a representation exists by [io], Theorem (6.7). Viewing V1 C V as sub-Galois module, and taking the subgroup scheme extension (Chap. I, w I (c)) ofV 1 in V/s=J[gJ~]/s, we obtain a closed (k-vector space) subgroup scheme VI/sC V/s. Let V~| denote the associated k~-vector group scheme, which one can " construct " simply by taking: 2 d--1 V~/s @ up. V~/s @ up. V~/s @... @ =~ . V~/s where d~ [k~ : k], and giving it the natural k~-structure. We have a homomorphism of k~r group schemes: Vl| --~ V/s which is an isomorphism on associated Galois modules. Since, by our hypothesis, Fontaine's theorem (chap. I (i .4)) applies, this is an isomorphism of group schemes over S', and hence also when restricted to characteristic p. Note that over Fp, the endomorphism 122 MODULAR CURVES AND THE EISENSTEIN IDEAL ~3 Frob +Ver preserves the above direct sum decomposition. By the Eichler-Shimura relations, T~ must also preserve the above direct sum decomposition (over Fp), which is possible only if d= I. The proposition follows. Proposition (i4.i3). -- Let 9X, 9X' be primes such that chark~=chark~,4:2 or: 9X and 9J~' are supersingular. Then the Gal(Q/Q)-modules J[gJt](Q) and J[ 9X' ](Q) are isomorphic if and only Pro@ -- By (1 4. I) we may suppose neither prime is an Eisenstein prime. Suppose J[gX] (()_..) and J[gX'] (()_~) are isomorphic as Gal(~_../Q)-modules. By Fontaine's theorem J[gX]/s, is isomorphic to J[ 9J~' ]/s', and the Eichler-Shimura relations together with Proposition (I4.12) enable us to get an isomorphism k~ -% k~, such that if v t is the image of Te in k~ (resp. -~=image of T t in k~0v) then i(vt)----z ~ for all t+N. To show that 9X = ~', it suffices to show that w has the same image in k~ as it does in k~,. Suppose not (i.e. w goes to + i in k~ and --I in k~v ). Then consider the q-expansions of generating eigenvectors (w 9) in H~ ~)[gX] and in H~ ~)[~']. These q-expansions are the same except for the coefficients of powers of qN. Applying (4. IO), (5.9), (5. IO) to the difference of these generating eigenvectors, we obtain that the generating eigenvectors are equal. Therefore ~R = 9X'. I5. The Gorenstein condition. Let R be a local Zp-algebra, free of finite rank as a module over Zp. Then R is a Gorenstein ring [3] if and only if the Zp-dual to R, R* =Homzp(R , Zp), is free (of rank i) as a module over R. Lemma (x5.I). -- Let 93tCT be a maximal ideal. We have the indicated implications of the assertions below: I) J[gX](()_..) /s of dimension 2 over k~. 2) $'a(Js~(Q)) is free of rank 2 over T~. 3) T~ is a Gorenstein ring. 4) H~ ~)| is free of rank i over T~. Proof. -- I) ~-2): Assuming i) we have that the kernel of 9Jr in J~(Q) is of dimension two over k~. Hence the cokernel of ~J~ in Hom(J~(Q), Qv/Z~)=H~ is also of dimension two. But H~ is the Zp-dual of the Tare group (cf. w 7): $'a(J~(Q)) = H~(Xo(N) ~, Z) | = H~. 123 124 B. MAZUR Since H~ is its own Ziduat , we have that H~| ~ is of dimension two. But since (7.7) H~| is free of rank two over T~| it follows that for any homomorphism T~--->K (where K is any field) H~| is of dimension two, and H~m is therefore free of rank 2 over T~. 2)--3): Write H~=FI| the direct sum of two free T~-modules of rank i. Since H~=H~ (* denotes Zidual ) we have an isomorphism F~| FI| 2. Consider the four projections ~i,~ : F* --> Fj (i,j=I, 2). At least one is a surjection, for if not the image of ~,~ would be contained in the maximal proper submodule 9Yr. Fj for all i, j, contradicting our isomorphism. Suppose ~i,~ : F~F~ is surjective. It is also injective since it induces an iso- morphism after tensoring with Q, and the domain is Z-torsion free. Thus, F~ is a free T=-module of rank I, whose Z-dual is also free and therefore T~ is Gorenstein. 3)<=>4): By (9.4) Hl(X0(N)tzp, d))| is free of rank i over T~, and using the idempotents c~, ~ of (7. I), and the fact that T act in a hermitian manner with respect to the duality (3.2) one sees that H~ f~)| is its Zidual. 2) ~ i): An easy reversal of the argument that I) :~2). Corollary (i5.2). -- Let 9X be a maximal ideal in T which is not an Eisenstein prime, and such that, if char k~= 2, then 9J1 is supersingular. Then all four assertions of (i 5. i) hold and in particular T~ is a Gorenstein ring. Remark. -- In the next two sections, we shall establish this Corollary for Eisenstein primes as well. This is significantly harder. We shall have use for the following (elementary) sufficient condition for Gorenstein-ness. Proposition (x5.3). -- If R=Zp[~] is generated by one element over Z~, then R is Gorenstein [3]. x6. Eisenstein primes (mainly p 4:2). Fix p a prime number dividing n. Definition. -- A prime number 14:N will be called good (relative to the pair (p, N), usually unmentioned and understood) if either: a) not both t and p are equal to 2, and (i) ~ is not a p-th power modulo N and t--i (ii) -- * o rood p or (the somewhat special " degenerate " case) : b) ~ =p = 2, and 2 is not a quartic residue modulo N. 124 MODULAR CURVES AND THE EISENSTEIN IDEAL x25 --2) The set of good primes has Dirichlet density ~ if p>2, and - if p=2. In particular, there are some good primes. 4 For anyg set ~t=I+t--T t. The object of this section and the next is to establish the following proposition and to derive some important consequences: Proposition (x6. x). -- The Eisenstein prime ~.T~ C T~ is generated by the elements p and ~t, where t is any good prime (1). Although some finer consequences of the above proposition will be developed later, note these corollaries. Corollary (x6.~z). -- The Zp-algebra T~ is generated by ~t for ~ any good prime. Therefore, by (I 5.3) : Corollary (x6.3). -- The ring T o is Gorenstein. The Fp-vector group J[~3] is two- dimensional. If p>~, then: J [~] = c [p] | E [p]. If p--2, then J[~]----U. The T~-module H~--~'a(J~(Q)) is free of rank 2. Corollary (x6.4). -- If p>2, Jp[~]=J~[~]=ep| (recall: Cp=p-primary component of C, and the same for Ep). Pro@ -- %OE, is contained in Jv[3] ((ii.i), (iI.7)). But [3] (Q) is the PontIjagin dual of H;/3. H (* means Z,-dual) and therefore, by the previous corollary, it has the same order as Cp(())| We begin by establishing a lemma needed to control the action of inertia. Lemma (x6.5)- -- Let B be a subgroup of either the cuspidal or the Shimura subgroup of J. If the superscript i denotes the module of fixed elements under the action of inertia, we have an exact sequence: o -+ B -+j,(QN)' -~ (J,(QN)/B) ~ o (where Jp is the p-divisible (Barsotti-Tate) group associated to J). Proof. -- What must be shown is that Jp(QN)-+Jp(Q.N)/B induces a surjection on elements fixed under inertia. By the appendix we know: Jp(QN)I=J~215 (J~ group associated to J~F,,)- (1) Carefully stated, our proof even works for g = p, ifp happens to be a good prime. This is hardly relevant for the main corollaries; moreover, our second proof of this proposition (by the theory of modular symbols (of. (x 8. I o) below)) makes no distinction whatsoever between the cases t=p and t4:p. Nevertheless, the fact that our proposition is true when l =p is a good prime has significance for the ~3-adic analytic number theory of J, and for the study of the arithmetic of the p-Eisenstein factor .](P) in the p-cyclotomic tower over Q (cf. chap. III, 9)- 125 ~6 B. MAZUR By SGA 7, exp. IX (3.5) (crit~re gatoisien de r~duction semi-stable) we know that if ~, 7 are in the inertia group at N, then (i --~) (I--y) acts trivially on Jp(QN)- Hence, ify is in the inertia subgroup, (1--7) .Jp(QN) CJp(QN) i. But since J~(QN) is ap-divisible group, (1-y).jp(QN) must be contained in the p-divisible part of jp(Qs) l, which is 0- J~(F~) cjp(QN)' , by the above direct product decomposition. Now, take an element e in Jp(Qs) which maps to b- in (Jp(QN)/B) 1. Let y be any element in the inertia subgroup. Since (1--y).e goes to (I--,()~-=o in j~(QN)/B, (i--7).e~B. Therefore, by the above discussion, (l--y).e is in Bc~J~ which is the trivial group, as is clear from the displayed direct product, if B C C and as follows from (II .9) if BC Z. Thus (i--y).e=o, for all y in the inertia subgroup. Q.E.D. From now on, in this section, let p+ 2. -- In this case, proposition (16. I) will follow from a direct proof of the stronger proposition (16.6) below. When p----2, we shall reverse the order of proofs of these propositions. Proposition (i6.6). -- The ideal 3.T~ is a principal ideal in TO, generated by ~t for g any good prime. Proof. -- We shall be working with subgroup schemes (closed quasi-finite) in J~ (hence admissible by (I4.I)). In particular, consider Jv[~]=Jp[,3]. We make extensive use of the tools developed in chapter I. Lemma (x6.7). -- The admissible group Jp[~] is a pure group (1). Pro@ -- Consider the exact sequence (14. io): over S', where C~ is the p-primary component of the cuspidal subgroup, and M is of multiplicative type. We first show that M is a Vt-type group (1). Since ~ annihilates J[~], for any prime number g+N, T t acts as I-r on J[~]. Thus, by the Eichler- Shimura relations, for any t 4:p, N, the g-Frobenius q~t satisfies ?~--(I -}- t). q0 t + g = o, or : =o. If g~I modp, then q~t acts as multiplication by t on M(Q). The reason for this is as follows. Since the Galois module M(Q) is admissible, of multiplicative type, the only eigenvalue that q~t possesses (when acting on M(Q)) is l. Consequently, (q~t--i) maps M(Q) isomorphically onto itself, and the above formula then implies that (~t--g) annihilates M(Q). If M v denotes the Cartier dual of M, then M v is an 6tale admissible group over S' such that ?t acts trivially in its Galois iepresentation (1) Chapter I, w 3- 126 MODULAR CURVES AND THE EISENSTEIN IDEAL ~27 for every g ~ p, N such that l $ i mod p. An elementary density argument (or chap. I (3.4)) implies that M ~ is constant, and therefore M is a ~t-type group. Since M is a ~-type group, the inertia subgroup at N operates trivially on M(Qs) (M extends to a finite flat subgroup over S, of order prime to N). Applying lemma (i6.5) with B=Gp, we obtain that inertia at N operates trivially on Jp[3] which is therefore a pure group by chapter I (4.5). Thus Jp[3]/s is a finite flat group and, over S: =c,� M. It follows that: J [~] = c [p] � M [p]. Let r be a nonnegative integer. Claim 1. -- The quotient group scheme Jp[3. ~3" +1]/jp[3. ~'] over S' is pure. Pro@ -- Set t=diml,p3.~'/3.~ ~+1. As in the discussion at the beginning of w I4, one may obtain an injection of the associated Galois module to Jp[3. ~r+1]/jp [3. ~,] into the associated Galois module to the direct sum of t copies of J[~]. Consequently, by lemma (I6.7), the inertia group at N operates trivially on the Qs-valued points of Jp[~. ~ + ~] [Jp [3. ~'] and therefore it is pure, by chapter I (4.5). Now fix a good prime number ~. Claim 2. -- The group scheme Jp[~.~', ~t]=G, /s pure for all r, and: Gr = Cp � M (') where M (~) is a ~-type group. Proof. ~ By the above group scheme we mean, as usual, the subgroup scheme extension inJl s of the intersection of the kernels of 3. ~r and ~t in Jp/Q (or, equivalently, in J~/Q). We proceed by induction, the first case r=o being already established (lemma (i 6.7)). Suppose G r is of the desired type: a pure group with dtale part Cp and ~-type part M (r). Since its dtale and ~t-type parts are canonically determined, the operation of the Hecke algebra T must preserve these parts; in particular it preserves M t'). Now we work over the base S'. Since ~t annihilates G~+I, the Eichler-Shimura relations give us the equation: on G,+I, if t4=p and hence also on any subquotient of G,+ 1. If t=p we have the above equation on any subquotient of G~+ 1 which is gtale. 127 i~8 B. MAZUR By claim t, and chapter I (4-5), it follows that Gr+l/G ~ is pure. So we may write: (I6.8) o -+ G~ --> Gr +l --> (Z/p)~ � M ' -+o where ~ is some nonnegative integer, and M' is a ~z-type group. We first show that ~ = o. -- Form the pullback: o > G, ) G,+ 1 > (Z/p)~� ' ) o oT o > G~ , G § (Z/p) ~- ~-o and set G =-G/M (~). Thus we have a short exact sequence: o -~Cp->G -+ (Z/p) ~ -+o. That is, G is an admissible 6tale group. Moreover, since (~pt--I)(q~t--t) =o, and t ~ I mod p, it follows that q~t = I on G. By the " criterion for constancy (chap. I (3.4)), G is a constant group. By the manner in which G was constructed, there is a natural induced action of the Hecke algebra T on G. But the ideal ~ C T annihil- ates G. To see this, use the fact that the action of Frob t, on G is trivial (for any prime number g'+N, including t'=p) since G is a constant group over S'. From the Eichler- Shimura relations one then sees that T t, = i +t' (for all t'+N). By construction of G, (i+w) ~ annihilates G. Since w is an involution, and p+2, it follows that i + w = o on G. Thus the ideal ~ annihilates G. Now reduce to characteristic p. From our exact sequences one sees that G/F p is equal to (Gr+m,p) ~t. It follows from what we have just shown that (Gr+l/Fp) ~t is annihilated by 3. But by (~4-io), Cp equals the kernel of ~ in (Jv~y t. Therefore 0(~O. Return to our exact sequence (i6.8), which now may be written: O -->G r ~-> Gr + 1 ~M' -+o. Also we have the exact sequence: .--> M H (x6.9) o --> Cp--> G~ + 1 -->o where M" is an admissible group of multiplicative type which is an extension of M' by M Crt. Since ~t ~ o, applying the Eichler-Shimura relations to the Cartier dual (M") which is an 6tale admissible group, we have that (gt--I)(gt--g)~ o on (M") ~. Since g is a good prime number, we use, again, the above quadratic equation, and the " Criterion of constancy" (chap. I (3.4)) to deduce that (M") ~ is a constant group; thus M" is of ~t-type. In particular, the inertia group at N operates trivially on M"(QN) , and hence also on G~+I(QN), using the exact sequence (I6.9) and lemma (i6.5) with B=C~. Thus, by the criterion of purity (chap. I (4-5)), Gr+t is a pure group, whose 6tale part is C~. 128 MODULAR CURVES AND THE EISENSTEIN IDEAL ~29 Claim 3.- %=(j~yt[~t]. Proof. -- By Claim 2, the kernel of ~t in J~zs, has the following structure: =% x M where MI~ U M/') is a union of ~-type groups. Consequently C~=((Jv[~t])~y ~. Our claim will follow from a lemma (which we also use later when p= ~): Lemma (x6. xo). -- Let p be any prime dividing n, and t any prime number different from N. Then: [Ht] (f,) = [H,]. One sees easily that He is an isogeny of J onto itself, for if it were not, then, by the Eichler-Shimura relations, q)t would have an eigenvalue equal to I or to g in its represen- tation ofJt,(Q ) (r any prime different from t or N) which is impossible for various reasons. Thus, Ht is a surjective endomorphism on all groups of the exact sequence: o -+J,(Qp) -,J,(Fp) o giving us surjectivity of J~(Qp)[~t]~J~(Fp)[~t]-+o by the snake-lemma. It follows that J~ [Ht] (Fp) =J~(Fp) [~t] (~)- Conclusion of the proof of Proposition (I 6.6) for p # 2. -- Let W denote the Zp-dual of $'a(J~(Fp)) (or, equivalently, the Qp/Zp-dual of J~(F~); cf. w 7). By (I4. II ) W is a free T~-module of rank i. By Claim 3, and (~4.9), W/Ht.W=W/~.W. Therefore ~t. T~ : ~. T~. x 7. Eisensteln primes (p = 2). We now begin to study the case where p = 2 divides n. Our first goal is to prove : Proposition (x 7. I ), -- The Eisenstein prime ~3 . T~ C TV is generated by the elements p and Ht, where I is any good prime different from 2. (a) To help the reader see this, it may be worth discursively reviewing the " brackets " terminology at this point. By definition, the group scheme J~[~t]/s' is the subgroup scheme extension in J/s' of the sub-Gal(Q/Q)- module in J~(Q) consisting in the kernel of~t. Thus, since "0t is an isogeny and J/s, is an abelian scheme, J~['tlt]/s, is a finite flat group scheme (it is, in fact, admissible) whose associated Galois module is J~(Q)[~tl =J~(Qp)[~t]. By J~[~t](Fv) we mean, to be sure, the Fv-valued points of the group scheme J~[~qt]/s'- We have a natural map (a surjectlon in fact) from the Qp-valued points of the finite flat group J~[-tlt ] to the Fp-valued points (reduction to characteristic p) : J~[vlt](l~p) --~ Jg~[~]t] (Fp), the range being naturally contained in J~(Fp)[~t] (the kernel of ~t in J~(Fp)). The asserted equality then follows from the previous discussion. 17 I3o B. MAZUR Discussion. -- The case p = 2 differs from p * 2 in many respects, the major ones being: a) Fontaine's theorem does not apply. b) The equation (q~t--/)(?t--i)=o (for l a good prime) on an 6tale or multi- plicative type admissible group does not imply that the group is constant or of ix-type. c) Where we have dealt in w 16 with the cuspidal subgroup, we must now deal, systematically, with the group D. d) Cp and Zp have a nontrivial intersection (when p = 2) and therefore it will turn out that J~[,~] is larger than C~+Zp. If N=Imod 16 we give no direct construction of J~[~]. We deal with a) by keeping strict control of the 6tale part of our group scheme. We are forced by b) and c) to work with groups which are roughly " twice the size " (in terms of lengths of various filtrations) as in the case p 4= 2. In particular, the pure groups of w 16 are replaced by ,-type groups (see below). We " pay for " d) by not being able to give a complete account of the Galois representation on J~ [~]. Recall the terminology of chapter I, w 3, and especially lemma (3.5): Lemma (i7.2). -- Let M/s, be a multiplicative type admissible group. Let ! be a prime number which is not a quadratic residue mod N (e.g. a good prime) (t4=2) such that (,Ot--g)(r on M. Let MIC M be the "first stage " in the canonical sequence of M (el. chap. I, w 3) (i.e. the largest ~-type subgroup 0fM). Then MI(Q) is the kernel of ~t--t and Ot acts trivially on (M/Mr)(Q,). Proof. -- The first assertion is a repetition of chapter I (3.6). The second assertion is then evident since q~t--I brings M(Q) into the kernel of (~t--t). ,-type groups. We work with a fixed good prime number g 4 = 2, and certain admissible subgroup schemes G/s, c J~[~t]/s, (i.e. in J~ and killed by ~t)- Say that such a group scheme is a ,-type group if it can be expressed as a " push-out " (or " amalgamated direct sum ") of the following form: o > ~ > D ~ Z/2 ~ o o >G O >G >Z/2 >o where: DCJ[~] is the subgroup scheme ofw 12, and G~ is some (admissible) subgroup scheme of multiplicative type, containing the subgroup ixz C D. We also denote the " amalgamated sum" as follows: G=G~ D. 130 MODULAR CURVES AND THE EISENSTEIN IDEAL x3x Since D is fixed, a .-type group is determined by its multiplicative part G O C G, and conversely: G O is the connected component containing the identity of the scheme G/s,. Lemma (x7.3). -- If G is .-type, and: G 0 6t o -+ (/z,) -+ G/z, -~ (G/z,) -+ o is the natural sequence displaying the connected and 6tale parts of Glz,, then: (O/,,) ~ = O z,. Remark. -- In particular, the Q~-rational points of (G/z,) ~ (which is, a priori, only stable under the action of Gal(Q2/Q.~)) is stable under the action of GaI(Q/Q.) as subgroup of G(Q)--G(Q~); here we fix any imbedding QC Q2- Applying (I7.2) to G~ we obtain a subgroup scheme G~176 G o containing ~, such that G~176 is the " first stage " in the canonical sequence for G~ In particular, G~176 2 is a ~-type group and its Galois module is the kernel of q~t--t in the Galois module of G~ Also qo t acts trivially on the Galois module of G~ ~176 Lemma (x7.4). -- G ~176 is a ~-type group. Proof. -- Since the inertia group at N operates trivially on G~176 lemma (i6.5) (where we take B = p~.) assures us that it operates trivially on G ~176 We then apply chapter I (3-1)- The key lemma enabling us to construct .-type groups is the following: Lemma (x7.5). -- Let ~ be a good prime number different from 2. Let GC G'cJ~[~t] be (admissible) subgroups stable under the action of T such that: a) G is a ,-type group. b) G'/G is of order 2. c) 2 kills the gtale part of G~F ,. Then G' is a .-type group. Proof. -- In the calculations of Claims i and 2 below, we deal exclusively with Galois modules. For simplicity we let the symbol of the group-scheme stand for the associated Gal((~/Q.)-module, in the proof of those claims. Thus G' would stand for G'(Q), etc. Claim 1: (?t--I)(G'/D) C (G~176 D) /D ---- G~176 Proof. -- We have a filtration: oC (G~176 D)/DC G/DC G'/D G~176 G~ 131 I32 B. MAZUR Since (q~t--i) annihilates G'/G and leaves D stable, we have that (q~t--Q(G'/D) is in G/D. But note that (q~t--g) : G0/G~176 G~ is injective (i 7.2), and consequently, if (q~t--I)(G'/D) were not contained in G~176 ~ we could not have (~t--t)(?t--i)~--o. Claim 2: (q>t--I) G' C G ~176 Pro@-- By Claim i, (q0t--i) G'CG~176 D. But Vt--g maps G0~ onto ix2, with kernel G ~ (since +_t(g)=I, ~pt--t maps D onto 1~2; el. chap. I (4-3))- Again, since (q~t--t)(q)t--I)=O, Claim 2 follows. Claim 8. -- The extension of group schemes over S': o -+Z[2 -> G'/G ~ -+ G'/G-+o ~plits. Proof. -- By Claim 2, q>t acts trivially on G'/G ~ There are two possibilities: Case I. -- G'/G=Z/2 (as group scheme over S'). Then hypothesis c) insures that the 6tale group scheme G'/G ~ is killed by 2, and since q~e acts trivially on it, it is indeed a product, by chapter I (3-4)- Case H. -- G'/G = g,~. It is also true in this case that G'/G" is killed by 2. The reason is that any exten- sion g of ~ by 7./2 splits over Spec Z~ (the splitting is obtained by showing that d ~~ the connected component of @, must project isomorphically to ~2). Therefore, in particular, 2 kills g(Q~) = d~(Q), and hence it also kills d ~ Again since q~t acts trivially on it, it is a product, by chapter I (5-i). We now show that Case I cannot occur. That is, G'/G ~ cannot be the constant group scheme Z/2 � Z/2. Note first that the Hecke algebra T induces a natural action on G'/G ~ For it leaves G' and G stable by hypothesis. We must show that it leaves G o stable. But by Lemma (i7.3) , the Galois submodule G~ of G(Q) is determinable as the sections which specialize to zero in characteristic 2 (the sections of the connected component (G/z,) ~ and is therefore left stable under the action of T. We follow the proof for p odd, quite closely. For all primes t' # 2, N, the Eichler-Shimura relations assure us that T t, =I-kt' on the constant group scheme G'/G ~ Reducing to F2, one has that T 2 = I ( - i ~- 2) on (G'/G~ again by the Eichler-Shimura relations. We have to check that w+I=O, in order to conclude that :cT' ~:F,: ~,t - (O'/O ~ is in (J/F,)~t[~3]. But since (w-kI)2=o (because w+I is certainly nilpotent on G'CJ~ and (G'/G ~ is an F2-vector group of rank 2) and since w+i annihilates Z/2=G/G ~ it suffices to show that (G:~,)~t=(D/F,) a is not in the image of w+I, which is true by (I3. io). Thus :~, ~v/F,: ~tC (J/s,)a[~] which contradicts (I4.9). Therefore we have: G'/G o ---- Z/2 � ~. 132 MODULAR CURVES AND THE EISENSTEIN IDEAL x33 Defining G '~ to be the kernel of the natural projection of G' onto the first factor Z/2 in the above product, we have that G '~ is a subgroup of G', of multiplicative type, and G'=G'~ is therefore a ,-type group. Q.E.D. Let r~i be an integer. Consider the exact sequences of Gal(Q~/Q2)-modules: o-+J~[2 r, ~t]~ [2', ~qt] (Q2)-+J~ [~t] (F~) where the superscript o denotes the connected component (containing the identity) of a group scheme over Spec Z~. Since Jr[2 r, ~t](Qz)=Jv[2 ~, ~t](Q), this group is (in a fixed way) a Gal(Q/Q,)- module. Let G(r) CJ~[2', ~qt](O_,,z) denote the full inverse image of (Jv[~t](F~))[2] in Jr[ 2~, ~t](Q2). It is clear that G(r) inherits a Gal(Q2/Q2)-module structure. It is not clear that G(r) is stable under Gal(Q/Q). Write G~ ', ~]~ We have: (x7.6) (i) o ~O~ -+G(r) -+ (Jv[~t](F~))[2] and, if r is sufficiently large: (x7.6) (ii) o-+G~ -+G(r) -+ (J~[~qt](F2)) [2] -+o. We formulate two hypotheses: I(r): The subgroup G(r)cJ~[2", ~qt](O_..) is stable under the action of Gal(Q/Q). I~ The subgroup G~ r, ~qt](O_..) is stable under the action of Gal(Q/Q). Lemma (I7.7). -- Hypotheses I(r) and I~ hold for all r>o. The group scheme G(r) is a ,-type group for all r>o. Proof of Lemma (i 7. 7). -- Our inductive proof consists in five steps. Set: G(o)= DC J[~3] (w I2). Step 1. -- For r>o, if I(r) and I(r+I) hold, and if G(r) is a ,-type group, then G(r+I) is a ,-type group. Proof. -- Since the groups G(r)C G(r+i) are both stable under the action of T, we may find a filtration: G(r)=FIoC HtC ... CHjC ... C Ht=G(r+I ) by T,[Gal(Q/O_..)]-submodules I-~ such that the successive quotients Hi/Hi_ I are irre- ducible T~[Gal(Q./Q.)]-modules. Since Hj/I-tj_ 1 is therefore a module over: Tv/ Tv [Gal(O./O__)] and since T~/~3.T~--~F2, t-Ij/Hj_ t is an irreducible Gal(O/Q)-module. Since G(r) is admissible, it follows that HJHy_, is of order two. By upwards induction on j, applying Lemma (i 7.5) to G = I~_ 1, G' = Hi, one obtains that G(r -/I) a ,-type group. 133 134 B. MAZUR Step 2. -- If I(r) holds, and G(r) is a ,-type group, then I~ holds. Proof. -- Apply Lemma (i7.3) with G(r)=G. Step 3. -- I~ =~ I(r+I). Proof. -- If xeJ~ [2 r + 1, ?t] (Q), then (since 2 r. 2x = o) by the defining property of G(r+1), x~G(r+I) if and onlyif 2xeG~ Now, if ~eGal(Q/Q), and xeG(r+I) we must show that ~(x)eG(r4-I). Equivalently, we must show that 2.~(x)eG~ But this is true since (by I~ ~ leaves G~ stable, and a(2x)= 2. ~(x). Step 4. -- If I(r) holds, and G(r) is a ,-type group, then I(r+I) holds and G(r+i) is a ,-type group (r~i). Proof. ~ Combine the first three steps. Step 5. -- Conclusion: Clearly D=G(o) is a ,-type group, and, since: G(I) =J[2, ~t](Q), I(I) holds. By Step I it follows that G(I) is a ,-type group. This allows us to apply Step 4 (inductively) to conclude the proof of Lemma (i7.7). Proposition (i7.8). -- The following groups (of order 2) are equal: D (F~) =J [~3] (F~) =J~ (F~) [~1 =J~ (F2) [2, ~t]. Proof. -- It is only the last equality that is new, but they will all follow if we show that the right-most group is of order 2. By Lemma (I7.7) and the exact sequence (I 7- 6) (ii) for r sufficiently large, we deduce that (J~ [Ht] (F~)) [2] is of order 2 (a). To conclude the proposition, we need that J~[~t](F~)--=J~(F2)[~e] which is true by (i6. io). Proof of Proposition (I 7. I). -- We follow the proof for p odd. By (~4.1 I), the Pontrjagin dual, W, of J~(F,) is free over T~ of rank i. By Proposition (I7.8), 2) w. Therefore ~3 = (~qt, 2). Q.E.D. One has these immediate consequences (I 5. I): (i) T~ is a Gorenstein ring; (x7.9) (ii) J[~]=D; (iii) H~ is free of rank 2 over T~. Proposition (x 7. xo). -- The Eisenstein quotient J-+J factors through J- (cf. w 4)- Proof. -- Let us work over the base Q. (1) The point here is that the 6tale part of a *-type group reduced to characteristic 2 is of order 2. This is all we need. 13d MODULAR CURVES AND THE EISENSTEIN IDEAL I35 The fact that for p odd, the p-Eisenstein quotient factors through J- is fairly evident: the kernel of ~ in J+ is zero, since w acts as -I on J[~3] and as +I on J+ = (i + w) .J = ker(J-+J-). For p=2 (if 2In) we must also show that J+[~3]=o. But by (17.9) J[~]=D, and by (13.1o) Dc~J+=o. x8. Winding homomorphlsms. If R is a commutative ring with unit let R[a] denote the commutative R-algebra I. R| a.R where e is a symbol satisfying the law ~2 = I. If M is a free R[~]-module of rank I, then ~ is an involution on M; forming the (~i)-eigen-subspaces M S C M associated to ~, and the corresponding eigenquotient spaces M +, we have the diagram of exact sequences: M+ (I8. x) o -+ M_ -+ M-+ M + -+o ,,2\ + M- where all four R-modules M~, M + are free of rank I. In fact they may be canonically identified with R and in terms of these canonical identifications, the diagonal homo- morphisms above are " multiplication by 2 ". Lemma (x8.2). -- Let K be a commutative local ring with maximal ideal m. Let M be a free R-module of rank 2 endowed with an (R-linear) involution ~ which is not a scalar modulo m. Then M is free of rank I over R[e]. Proof. -- Let k= R/re. Then M = M/m.M is a 2-dimensional vector space over k on which the involution e does not act as a scalar. In particular, there is an element xeM such that YeM is a generator ofM ask[e]-module. Applying Nakayama's Lemma to M over R, one deduces that x is a generator of M as R [~]-module. Moreover, since the R[~]-homomorphism R[~] 2~ M (e~ e.x) is an isomorphism of R-modules modulo m, and M is a free R-module, it follows that i is an isomorphism. : H-+H be the involution Proposition (x8.3). -- Let H=Ht(X0(N)c , Z) and let induced from complex conjugation of the manifold X0(N)r Then H~ is a free T~[~]- Let ?Ol be any maximal ideal in T such that char k~ 4= 2. module of rank i. 135 x36 B. MAZUR Let ~3 be (any) Eisenstein prime. Then HV is a free T 0 [~]-module of rank 1. Let T~ denote the completion of T with respect to the (full) Eisenstein ideal. Then H~ is a free T~[~]-module of rank 1. Pro@ -- Let 9J~ be a maximal ideal such that p=chark~4=2. By (15.2) H~ = $'a(J~)(13) is free over T~ of rank two. On the other hand, one has a perfect duality: 08.4) (c) � (where V~-- 0 t~p,C Gin). Since ~ acts as --I on $-a(t~)(C ) and since p+2, we have --r=l that H~--~H~m+| ~_ where (18.4) puts H~+ and H~_ in duality. Since (again) p+ 2 it follows that ~ does not act as a scalar modulo 9"J~.T~ and consequently H~ is free over T~[~]. Now let ~ be an Eisenstein prime associated to p. By (16.3) and (I7.9) H~ is free over T 0 of rank 2, and by (16.3) the action ore is evident. Namely, when p4=2 acts as + I on 13 and as --I on Y,. Therefore it does not act as a scalar modulo ~3. T 0 . If p = 2, ~ does not act as a scalar on D (C) (cf. chap. I (4.3)). Therefore Lemma (I 8.2) applies again. Since T~=pHT~, H~=piIIH~, the final assertion follows, as well. Q.E.D. Let J0 denote the complex Lie group underlying the jacobian of Xo(N ), and U its universal covering group. We have an exact sequence of topological groups: o I-I &Jc where the discrete subgroup HC U is identified with H =HI(X0(N)0, Z). Moreover, the tIecke algebra T and complex conjugation cr both operate naturally on the above exact sequence. The Lie group U is isomorphic to H| as real Lie group: The real Lie group J0 is canonically isomorphic to H| (R/Z). Consider the fundamental arc [o, iao] = { ~y ] o <y < ioo } in the extended upper-half plane. We regard the fundamental arc as an oriented topological interval (orientation from ioo to o). The parametrization of X0(N)o by the upper-half plane induces a natural homeomorphic injection: [o, ioo] h-~J0 (h(ioo) = origin). The continuous map h lifts uniquely to a continuous map to the universal covering group: h : [o, ioo]-+U (h(ioo)=origin). Definition. -- Set e=h(o)eU. Call e the winding element. Lemma (I8.6). -- We have 3. e C H+ C U. The winding element e is in H+ | Q.. 136 MODULAR CURVES AND THE EISENSTEIN IDEAL I37 Proof. -- The fundamental arc maps to the real locus of X0(N ) and, from the definition it is clear that n(e) = e = c~/((o) -- (oo)) in Jc- Therefore, since 3. c = o (it.i), it follows that ~.eCH, and since e is fixed under ~, ~.eCH+. Since ne~, ee(I/n).H+. Definition. -- Let: e+ : ~-~H+ be the T-homomorphism ~ ~ ~. e. If a is any ideal in T, let: e+ : 3.T~Ha+ denote, as well, the induced T a homomorphism. If H, is free over T, of rank i, let: e + : ~.Ta-~H + be the Ta-homomorphism defined by: 2. e + (e) = image in H + of e+ (~) (using diagram (i 8. I)). We shall call the homomorphisms e+ and e + winding homomorphisms. The winding homomorphisms are (conveniently normalized) " generalizations " of the winding num- bers of [39]" We shall be especially interested in the winding homomorphism e + for a = 3: e + : .~.T~H +. By means of the theory of modular symbols ([32], [35], [39]) we shall be able to completely determine this homomorphism modulo 3, and deduce a number of impli- cations. As we do this it is of interest to keep track of how little use we shall make of all our previous work. We use only the assertions of Proposition (i8.3) (those having to do with Eisenstein primes). These, in turn, are easy corollaries of the (hard) result: T~ is a Gorenstein ring, for ~ an Eisenstein prime. Lemma (xS. 7)- -- H+/3. H + is a cyclic group of order n. There is a canonical (1) surjection ? : (Z/N)*~ H+/~.H + which identifies H+/~.H + with the Galois group of the Shimura covering ((2.3); cf. w ii): XI(N) (Z/N)* ~" (+i) x (N) t $ H +/3. H + X0(N) , Pro@ -- Since H +/3. H + = H +/~.H + and since H~/~. H + is free of rank I over T~/~.T~ by Proposition (I8.3) and the discussion involving Diagram (i8.i), it follows that H+/~.H + is, indeed, a cyclic abelian group of order n (9-7). (1) To make it canonical, one must make, somewhere, a specific choice of sign. Compare the next footnote and relevant text. 18 i38 B. MAZUR Let 5 p denote the unique quotient of (Z/N)* of order n. Thus 50 is a cyclic group which is canonically the Galois (covering) group of the Shimura covering (2.3)- Since the Shimura covering is unramified (2.3) there is a canonical surjection H-~5:. Since X~(N) --> X0(N ) is defined over Q (and hence over R) this canonical surjection factors, to give a surjection HH+~SC Since (Proposition (ii.7)) the Shimura subgroup is annihilated by 3, this surjection factors to yield a surjection t-I+/3.H+~5 :, which must be an isomorphism, since both domain and range have the same order. Q.E.D. Ifa/b is a fraction where b is an integer relatively prime to N, let {a/b}eH denote the modular symbol [32], [35], ([39], w 6). Proposition (i8.8). -- (Congruence formula for the modular symbol.) Let a, b be integers with b relatively prime to N. Let b denote the image of b in (Z/N)*. Let @(a/b)EH+ /~.H + denote the image of the modular symbol { a/b } in H+/3.H +. Then: r = +/3. H +. (Compare footnote in Section (6. I5) of [32].) Pro@ -- Here we again (as in the Proof of (I I. 7)) make use of Ogg's terminology for the cusps of F(N). ( ~ ) = {P /q~pl( Q,) lp = a mod N ; q = b mod N ; (p, q) = I }. From the definition of the modular symbol, one sees that if (b, a.N)=I, r is that unique element of 5:~ H+/3.H + which sends (the image in Xz(N) of) the cusp (~ to (the image in X2(N) of) thecusp (~). Sincean element c~(Z/N)*acts as the matrix (; o C -1 ) (1) and since (b) = (b) mod FI(N ) provided (b, N)=I, it follows that r is the image ofb -1 in H+/3.H +. Q.E.D. Proposition (18.9). -- (Congruence formula for the winding homomorphism.) Let ~t=-I+t--Tt. Let ~+ :~/~2-->H+/3.H+ be the homomorphism induced from the winding homomorphism e + : 3-+ H + 9 Then: where t is any prime number different from N. Remarks. -- First note that the right-hand side makes sense. For if l = 2, and p=2 divides n, then N- I mod 8. By the quadratic reciprocity theorem ~(Z/N)* (a) We follow Ogg in making this choice. 138 MODULAR CURVES AND THE EISENSTEIN IDEAL [g--i\ is then a quadratic residue (7=x2), and consequently l-T2l?(~-)-----q0(x). In any other case, the 2 in the denominator is harmless. The assertion of (18.9) may be viewed as a congruence formula for numbers of rational points over F t. For example, in the first nontrivial case, N = i i, it was first proved by Serre, and takes the following shape: Let N t denote the number of rational points of the elliptic curve X0(i1 ) over Fe(e,l~ ). Let ~:(Z/~)*-+Z/5 be the homomorphism which sends --3e(Z/II) * to 2 rood 5. Then: Nl~-~--5(e--I) .~(e) mod 25. Proof of (i 8.9)- -- Our proposition follows immediately from the formula: (i +e--Tt).e=-- Y~ {k/e} k m0d l (formula (8), w 6 of [39], compare (5-5) of [32]), together with Proposition (I8.8), (I8.3) , and the definition of e +. Theorem (x8. xo). -- (Local principality of the Eisenstein ideal.) Let p be a prime number dividing n. Let ?(3 be the associated Eisenstein prime. Let E be a prime number different from N. Then ~t is a generator of the ideal ~0=~.T~C T~ if and only if t is a good prime number (with respect to p). The winding homomorphism e + : ~ ~ H~ is an isomorphism of To-modules. Proof. -- Reducing the above winding homomorphism mod ~ one gets the homomorphism ~+ :~o/~-~H~/~o.H ~ and by Proposition (i8.9) , the element ~t maps to a generator of H~/~.H, if and only if (V)is not congruent to o modp and| is not ap-thpowermodN (if we are not in the special case e=p=2). In the case t =p= 2, Proposition (I8.9) assures us that / maps to a generator if and only if e is not a quartic residue mod N. Thus, ~}t maps to a generator if and only if e is a good prime. Since good primes do, indeed, exist, we deduce that e + : ~--->H~ is surjective, by Nakayama's lemma. By counting dimensions over O p, we obtain that: e+| : ~| g;| is an isomorphism. Since ~v is torsion-free as a Zp-module, it follows that e + : 3v--->H~ is an isomorphism. Since H~ is free of rank I over T0, our theorem follows. Remark. -- Except for the " only if" part of the theorem and the assertion concerning the winding homomorphism, the " new " information conveyed by (18. IO) is for p=2. For odd p, it is a curious alternate to the methods of w I 6, for (starting with Corollary 16.3. The Gorenstein property for T0) it enables us to quickly retrieve the results of w 16 in their full strength. IfgJl is a maximal non-Eisenstein prime in T, then the winding element e is naturally 139 ~4 o B. MAZUR contained in I-I~+. Thus if 93l is such that H~+ is free over T~ of rank I (e.g., if char k~ Je 2 ; cf. (I 8.3)) then, choosing some identification between the T~-modules H~+ and T~, e will correspond to some element in T~. The principal ideal e~C T~ generated by this element is independent of the choice made and shall be called the winding ideal associated to 9JL 19. The structure of the algebra T~. Fix p a prime dividing n, and !13 the associated Eisenstein prime. We know ( 18. I o) that if/ is any good prime number, ~t generates the Eisenstein ideal ~C Tin, and ,~=Zp[~qt ]. Let Rt(x)~Zv[x ] be the minimal monic polynomial satisfied by ~t over Zp. Thus T~=Zp[x]/(Rt). Denote by gp the rank of T~ as Zp-module, or equivalently, the degree of Rt(x ). Since T~ is local and ~qt is in ~3, R t is a" distinguished polynomial" (i.e. Rt(x )-x gp modpZp[x]). Since Tm/~t.T ~ ~ Zip f where prlln, the constant term of Re(x ) has p-adic valuation f. Since, if t and t' are two good prime numbers, ~qt and ~t, are associate in the ring T~, the Newton polygons of Rt(x ) and Rr(x ) are equal. One might call the common Newton polygon of Re(x ) for l any good prime number, the Newton polygon of T~ (or, more strictly speaking, of ~). Is there anything general that can be said about the Newton polygon of T~, or even about gp? One has hardly enough numerical data to begin serious speculation about this question. As far as my calculations go (N<25 o) there is only one instance where T~ is not a discrete valuation ring (N = i i3, p = 2) (1). In this case f= 2, gp = 3, and the Newton polygon is the only possible one conforming to this data. There is no practical difficulty in computing the Newton polygon of T~, using (e.g.) the tables of Wada [7o]. Wada gives the characteristic polynomial of T t (call it St(x)) acting on the parabolic modular forms for F0(N ). The most straightforward thing to do is to look for the smallest good prime number g such that Se(i +t) has p-adic valuationf (2). For such a prime number t, Re(x ) is simply the " Weierstrass-prepared part " of St(I +t--x). Proposition (x 9. i ). -- Suppose p [] n (i.e. f= I). Then T~ is a discrete valuation ring, totally ramified over Zp, of ramification index gp. Proof. -- In this case, the maximal ideal ~3=~, and is principal, by (I8. Io). Proposition (I9.2). -- Let p*2, prlIn (f>~). The natural auto-duality 0fJ[p r] restricts to a nondegenerate auto-duality of C~| (the direct sum of the p-primary components (1) As we shall see (chap. III, w 5) if we avail ourselves of certain standard conjectures, this instance is the first of an infinite series of analogous instances (all with p = 2). (3) In practice one does not have to go far to find one, at least when N < ~5 o. 140 MODULAR CURVES AND THE EISENSTEIN IDEAL I4! of the cuspidal and Shimura subgroups) if and only if T~ = Zp (i.e. gp = I). In particular, the element u (end of w 1 I) is a generator of the p-primary component of U if and only if gp = ~. Pro@ -- If T~=Zp, then Cp| and on the latter group the natural auto-duality (I 1. I2) is nondegenerate. Conversely, suppose the natural auto-duality of Jv[p r] restricts to a nondegenerate auto-duality of C~| Then the natural auto- duality ofJ~ [p] would restrict to a non-degenerate pairing of C [p] with E [p]. By (I 8.3) J$[p](Q) is free of rank 2 over T~/p.T~, which by the above discussion is isomorphic to Fp[~t ] where ~t satisfies the relation ~=o (over F:o). One sees immediately that J~[p, ~qt](Q) (which is the kernel of ~t in J~[p](~).)) is the image of ~g~-~. If gp--1>o, the relation (~gp-~x,y)=(x, ~gp-ty) gives us that the natural auto-duality restricts to zero on C[p]| contrary to assumption. Remark. -- The only instances (N<25 o, p:t:2) where gp>i are: N=31, IO3, 127, I31 , 18t, I99 and 211. III. -- ARITHMETIC APPLICATIONS I. Torsion points. Lemma (x. x). -- Let AIQ be any quotient abelian variety of JtQ. Let p be a prime number dividing the order of the torsion subgroup A(Q)tor ~ of the Mordell-Weil group of A/Q. Then p divides n. If AIQ is a quotient abelian variety of J/Q on which T operates in a manner compatible with its action on JIQ, then A(Q)tor s is annihilated by a power of the Eisenstein ideal .~. Let A/Q be a simple (equivalently: C-simple or Q-simple; of. chap. II (IO. i)) quotient abelian variety of J/Q such that the prime number p divides the order of A(Q)tor ~. Then A/Q is a quotient of the p-Eisenstein quotient J(P) (Io.4). Proof. -- Start with the first assertion. Consider the surjective morphism of associated p-divisible groups over Q, Jp-+Ap. If r is large enough, the image of the finite group scheme J[p*] in Ap contains A[p]. Find a Jordan-HSlder filtration of J[pr], as T [Gal(Q/Q.)]-module. Since, by hypothesis, there is a point of A [p], defined over Q, some successive quotient of the Jordan-YI61der filtration must have trivial Gal(Q/Q)- action. By (chap. II (14. i)), this subquotient of J[p r] belongs to (1) an Eisenstein prime ~, necessarily associated to p. Therefore p divides n (chap. II (9-7)). The second assertion is similar, but easier. Every successive quotient of a Jordan- H61der filtration of the T[Gal(Q/Q)] module A(Q)to, ~ must belong to some Eisenstein prime, by chapter II (14. I). By the Mordell-Weil theorem, A(Q)tor s is a finite group, and is therefore annihilated by some (finite) power of .~. (2) In the terminology of w 14. 141 r42 B. MAZUR The third assertion depends upon the one-to-one correspondence of chapter II (lO. I) where isogeny classes of simple abelian variety factors of J are " identified " with irreducible components of Spec T. Since p divides the order of A(Q)tor,, by what we have already shown, the irreducible component of Spec T corresponding to the (isogeny class of the) simple abelian variety quotient A must contain the Eisenstein prime ~3eSpecT associated to p. Since jt,/=j/g~.j where y~=0~ ~ (chap. II (lO.4)) it follows that, up to isogeny, ~(~/is a product of those simple factors corresponding to irreducible components of Spec(T) containing ~3. Since J(p) is the quotient of J by a connected subgroup scheme, it follows that J--~A factors through J(P). Theorem (x.at). -- (Conjecture of Ogg): c =J(Q)to,. (Any rational torsion point of J is a multiple of c-~r Proof. -- Set M =J(Q) (the Mordell-Weil group of J) and recall the retraction p : M-+CC M of chapter II, w ii, giving rise to the direct product decomposition M=M~215 (chap. U (11.4)). It follows that C is a direct factor of Mtors=J(O~)t0rs. By (i. i) it suffices to show M ~ [~3] = o for all Eisenstein primes ~. But this follows from the inclusion C[~]�176 and the determination of J[~3] (chap. II (16.3) or (for p4~2) (14.1o)). Theorem (x .3). -- The Shimura subgroup X is the maximal ~-type subgroup in J/s,. Proof. -- The sum of two (finite) ~-type subgroups of J is again a (finite) ~z-type group. It suffices to show that if Y.' is a (finite) ~-type subgroup of J containing N, then Y/= N. We first show that Y~ is a direct summand in Z'. Using the universal property of the N6ron model J/s (and the fact that the inertia group at N operates trivially in the Galois module associated to ~') one has that the subgroup scheme extension NfsCJ/s is a finite flat (~-type) subgroup scheme. Consider specialization to characteristic N, where one obtains a diagram: ~' ..... " J /F. = J /~ � l~ I pr~ where - denotes specialization, and where the bottom horizontal map is an isomorphism, by chapter II (I 1.9). It follows that ~ is a direct summand of ~' and one easily obtains from this that 142 MODULAR CURVES AND THE EISENSTEIN IDEAL I43 5; is a direct summand in E'. Write E'= 5;| where B is a ~z-type group. Applying chapter II (I 4. I), one has that every successive quotient of a Jordan-H61der filtration of B belongs to an Eisenstein prime. It suffices to show that B [~] =o for all Eisenstein primes. But 5;[~]| cJ[~3], and B[~3] must therefore vanish, by chapter II (16.3). Remarks. -- Theorem (i.3) was also conjectured by Ogg [48]. Although (1.2) and (I-3) have the appearance of being of comparable difficulty, there are notable differences between them. Ignoring 2-torsion, Theorem (I.3) is far easier than Theorem (1.2) (it uses only chap. II (I 4. Io), and does not depend on the Gorenstein condition). In dealing with the 2-torsion subgroup of J(Q), however, one must control subgroup schemes of J/s isomorphic to Z/2 as well as subgroup schemes isomorphic to ~ (since either will contribute to a point of order 2 in J(Q)). Consequently, this requires the full strength of chapter II (16.3), e.g., all of chapter II, w 17. Corollary (I. 4)- -- The natural maps induce isomorphisms of torsion subgroups of Mordell- Weil groups: C =J(Q)tor~-+J-(Q),or~ -+J(Q)tors (cf. chap. II, w 1o). Proof. --By (i. i) one has that J-(Q)tor 8 and J(Q)tor8 are annihilated by a power of the Eisenstein ideal 3. We shall show that the natural maps: J~ ~J~ -,J~ are isomorphisms. The map J~-+J~ is an isomorphism since ((i-}-w).J)[~]=o (chap. II (I 7. lO)). The map J~-+J~ is an isomorphism since its kernel is T~.J~ and the supports of T/~ and of T~=rl~ r are disjoint. I* Corollary (I.5). -- The MordeU-Wdl group of J+=(l+W).J is torsionfree. 2. Points of complex multlpHcation. In this section we examine a set of points ofXo(N ) defined over fields of particularly low degree. A somewhat larger class has been studied by Birch and Stephens (called Heegner points). Fix N (a prime number > 5, as usual) and work over the field of complex numbers. If E/c is an elliptic curve, an N-isogeny by complex multiplication r: :E-+E is an endo- morphism such that ker r~ is of order N. Thus, ~ is a complex multiplication of E such that if R is the ring of complex multiplications of E, ~. ~ = u. N where u is a unit in R. Let aE, . =j(E, ker ~)eX0(N ) (C), which we will refer to as a point of complex multiplication. If a = a~.,,, is a point of complex multiplication, set: R(a)=the ring of endomorphisms of E. The ring R(a) is an order in a quadratic imaginary field k(a) which may be viewed as naturally imbedded in C (since 1~3 t44 B. MAZUR End(Tan(E/c))=C and R(a) acts faithfully on Tan(E/c), the tangent space of E/o ). A(a) = a sublattice of C such that C/A(a)~ E. It is well known (cf. [29] ) that A(a) is a locally free R(a)-module of rank I. ~(a)==. It is an element in R(a) of norm N. Given a triple (R, A, 7:) where R C C is an order in a quadratic imaginary field, A is a locally free R-module of rank I, taken up to isomorphism, and ~ is an element of R of norm N, given up to multiplication by a unit in R, then we may construct a unique point of complex multiplication a--=acR.a.~)eX0(N)(C ) such that R=R(a), A=A(a), and n--~n(a). Let dc X0(N)(C) denote the set of all points of complex multiplication. It is easy enough to produce elements of d. Consider equations: N = r 2 + D.s 2 where D is a positive integer not necessarily square-free and r, s are either both positive integers, or both positive half-integers (1). If D= I, suppose r>s. Let r~=r and let R be an order in Q('V/~) containing =. Finally let A be a locally free R-module of rank I (e.g., R itself). The points of complex multiplication are defined over algebraic number fields which are studied in detail by the classical theory of complex multiplication (cf. [29] , chap. IO, w 3, theorem 5 and remarks I, 2 following it). We give a synopsis of this theory below: (2.x) Let R be an order in a quadratic imaginary field k_C and (7~)CR a principal ideal of norm N. Let A 1, ..., Ah(R) run through a system of representatives of isomorphism classes oflocally free R-modules of rank i. Set ai-----a(R, Ai,,~)ed. Then the points al,...,%R)eXo(N)(C) are rational over Q-~k and are a full set of conjugates over k. Let G denote the quotient of Gal(k/k) which acts faithfully on the above set of conjugate points. There is an isomorphism (a~A~) of G onto H(R), the group of isomorphism classes of locally free R-modules of rank I, such that if aeG, then a(al)=a o where A1R(~Ao=A ~. The group G cuts out that ray class field L of k whose conductor is the conductor of R. The field extension L/Q is Galois, with Galois group G. We may write L=k(j(R)) C C in which case the real subfield L + =Lc~R is given by L + =O(j(R)). Let ? denote the nontrivial element of GaI(L/L +) =Gal(k/Q). Then G is a semi-direct product of G and the group { i, ?} where the action of ? is given by pgp-l=g-1 for geG. Thus G is a dihedral extension of G. One has: 9 a(R,A, ~ ) ~ a(R,A-~,~ ) 9 (1) H. Lenstra and P. Van Emde Boas have tables of the smallest such D for a given N < 50o,ooo. 144 MODULAR CURVES AND THE EISENSTEIN IDEAL x45 The action of the canonical involution w on d is easily determined: (2.3) w. a(R,A,~ ) ~--- a(R,A, ~ / . Let a~X0(N)(C ) be a fixed point of w. Then a is represented by an isogeny t~ E-~ E' which is isomorphic to E -~ E (its dual). It follows that E': E and consequently the isogeny must be a complex multiplication E-~E and ~:2=u.N where u is a unit in R(a). Multiplying ~ by a unit in R(a), if necessary, we may suppose that ~ =~r N. Consequently, R is either Z['v/~N] or Z[I_] [r ,~-~N / 1_, where the latter case may [ .: j occur only if N - -- I mod 4. Using classical facts concerning the class numbers of the orders Z[A/~] and Z - ([28], chap. 8, w I, th. 7) we may give the following description of the fixed point set of w. Let h(N) be the class number of Q,(X/~-N). If N = I mod 4, then the fixed point set of w consists in one Q:conjugacy class of h(N) points. If N-----I mod 4 it consists in two distinct Q-conjugacy classes, the first containing h(N) points and the second containing h(N) or 3h(N) points according as N -= -- i mod 8 or N-3mod8. Proposition (2.4). -- Let a be a point of complex multiplication and let a+eX0(N)+(C) be its image in X0(N ) * = X0(N )/w. Then a + is defined over Q, if and only if the class number of R(a) is i. Proof. -- This follows immediately from (2. I) and (2.3)- Such points a+eX + are examples of rational noncuspidal points. It is natural to refer to them as points of class number one. One obtains a point a + of class number one for each order R (in a quadratic imaginary field) of class number one, in which N splits or ramifies. Note that if N splits or ramifies in any one of the 9 quadratic imaginary fields of class number one, there are some points of class number one on the associated X +. This is the case, for example, for all prime numbers N<7ooo except for N=3167, as was communicated to me by H. Lenstra and P. Van Emde Boas. The Dirichlet density of primes N whose associated X + possesses no point of class number I is I ]512. What further noncuspidal rational points does the curve X + possess ? This diophantine question (when the genus g+>o) is extremely interesting, since no known method appears to be applicable to it, for any value of N. In the first nontrivial case (N =67) the genus of X + is ~. A. Brumer has obtained its hyperelliptic representation, and has begun a numerical study. When h(N)=i, the description of the fixed point set of w given above shows that there is a (unique) rational point aeX0(N)(Q, ) fixed under w. 19 x46 B. MAZUR Proposition (2.5) (compare [48]). -- Let N----II, i9, 43, 67, or I63. Then Xo(N ) possesses a rational point fixed under the action of w. Moreover (when g=genus X0(N):>o ) these are all the points of complex multiplication in Xo(N ) (C) which are rational. Recall that J+=(i+w).JcJ may be identified with the jacobian of X0(N) + (cf. chap. II, proof of (I3.8)). We shall produce some rational points in J+. If R C C is a fixed order in a quadratic imaginary number field such that the ideal generated by N splits into a product of conjugate principal ideals: (N) be the linear equivalence class containing the divisor: let a+~J+ :c a+ A,--h(R). (oo) A~HCR) ' ' ' where a(R ,+ A) is the common image of a(R,A,r:) and a(R,A,~ ) in X +, ooEX + is the unique cusp and h(R) is the order of H(R). By (2. i-3) a+ is defined over Q, and therefore represents an element in the Mordell- Weil group of J+. To study these points we use a modification of an elegant trick due to Ogg: [49]. Lemma (2.6). -- Let d be an integer. If the dimension of H~ ~(d.oo)) is :>I, then d<N/96. Proof. -- Suppose d is as in the assertion above. Using (chap. II, w IO), J+/s, is an abelian scheme. By ([9], VI, 6.7) one sees easily that X0(N) + has a smooth model over S' (which we call X~s, ). Consider the base change Spec F4-+S'. Using the upper-semi-continuity property (EGA III (7.7.5), I) one obtains that the dimension over F 4 of H~ /F,, 0(d. oo)) is also :>I. Thus there is a morphism f: X~,-+P~p, of degree d, such that the inverse image of the point oo of p1 is the divisor d. oo of X +. Composingfwith the projection X-+X +, we obtain a map g : X/l, ' -+P~F, of degree 2d such that the inverse image of the point oo of p1 consist in the cusps. This gives us the upper bound 8d for the number of rational (noncuspidal) F4-valued points of X0(N ). But, as Ogg remarked [49], all the supersingular points of X0(N)/~, are rational over F 4 and there are more than N/I2 of them. Corollary (2.7). -- If R is an order in a quadratic imaginary field such that the ideal generated by N splits into a product of conjugate principal ideals, and such that h(R)<N/96, then a + is a point of infinite order in the Mordell-Weil group of J+. Proof. -- By (I.5) , the Mordell-Weil group of J+ is torsion-free. Therefore it suffices to prove that a + ~eo. Suppose a + =o. Then there would be a function f 146 MODULAR CURVES AND THE EISENSTEIN IDEAL on X~ whose divisor of poles is h(R).(oo). By (2.6) h(R)>N/96 contrary to assumption. Proposition (2.8). -- Suppose g+>o (which is the case for all N>73 , as well as N=37, 43, 53, 61, 67). Then the MordeU-Weil group 0f J+ is a torsion-free group of infinite order (i.e. of positive rank). Pro@ -- Write 4N=a2+Db ~" with a, b integers, D>o and a s largest possible. One obtains N = ~.~ with ~ in R, the ring of integers of O~(~v/-~ D) and if A is the discriminant of R, then I AI<4v/N. By a standard upper estimate for the class number h(R), we have h(R)~(I/3 ) [AII/~.loN[A[<(I/3)NI/41og(IaN ) if [51> 4. A calculation shows that (I/3).N1/41og(I6N)<N/96 when N1/~>7, or N>24oi. Thus by (2.7) a~ is a point of infinite order when N>24oi. But by the calculations of Lenstra and Van Erode Boas, X0(N) + possesses a point of class number i (hence defined over Q (2.2)) for all N<3167 and therefore (2.8) follows. Remarks. -- I. Using the estimates in the proof above one may show that if N is sufficiently large, each of the points a~eX0(N) + is of infinite order. 2. The above theorem depends on the fact that J+(Q) is torsion-free, which, in turn, depends on the full strength of chapter II, w 17. It is significantly easier to show that 2 .J+ (Q) is torsion:free (for one has far less to do with Eisenstein primes associated to 2). If one wishes to obtain the above proposition using only this weaker fact, one must prove that for same R, 2.a ++o. The estimates give this for R as in the proof above, provided N< 7ooo. We may then use the calculations of Lenstra and Van Emde Boas quoted above to reduce considerations to the one case: N=3167. But, quoting their tables, 3167=562+3I.I ~ and Q(%/~----3 I) has class number 3. For: the estimates above enable us to conclude that 2.a +4:o. 3. Let v+=J+(Q)| which we regard as a T+| module, where: T + =T/(I -- w)T. We have shown that V + is a Q-vector space of positive dimension if g+~o. Let V + Vo.m.) be the sub-T +| of V + module generated by the point a + where R is (resp. + the ring of integers in Q,(V'~) (resp. by all points of complex multiplication). Consideration of Dirichlet L-series and the Birch Swinnerton-Dyer conjectures might lead one to suspect that V + will play a significant role in studies of the MordeU-Weil group of J+. It is tempting to hope that V + is always a free T+| module of rank i. Numerical evidence is too slim to make any conjectures yet, but Atkin has recently produced some interesting tables which bear on the problem. 147 t48 B. MAZUR 3" The MordeU-Weil group of J. The object of this section is to prove Theorem (3. 9 ). -- The natural projection J-+'J induces an isomorphism of the cuspidal subgroup C onto the Mordell-Weil group J(Q). We shall also obtain complementary information concerning a part of the Shafarevich-Tate group of J. Our method will be to use " geometric descent " together with much of the information we have accumulated up to this point. Let ~ be an Eisenstein prime and J~ the connected component containing the identity (which differs fromJ only in its fibre at N). Let J~ m] be the kernel ofp m, and j0[pm]~ its ?(3-component (which is the image of J~ m] under the idempotent a~ for the Eisenstein ideal, as discussed in chap. II, w 7)- Lemma (3.2). --ja[p,~]~ is an admissible group (chap. I, w i(f)) and when m varies, the order of Hl(S,J~ ) remains bounded. Pro@ -- It is admissible as can be easily seen by chapter II (I4. I ). Since H~176 is a subgroup of the torsion part of the Mordell-Weil group of j, it is a finite group which has bounded order as m varies. Thus, to prove the lemma, it suffices to show that h 1- h ~ has bounded order. But by chapter I, Prop. (I. 7) it suffices, then, to prove that ~(j0[pm]~)_e,(j0[p,~]~] has bounded order. This is done by showing: (a) a(J~ =m.g~ + O(I) (b) cr (j0 [pm] ~) = m.g~ + O( I ) where g~ is the rank of T~ over Zp. Proof of (a). -- Letting J~ denote the p-divisible group associated to J over S, and J~ its ~3-component (i.e. the image of the idempotent ,~) then J[p'~]~=J~[p'~] and j0[pm]~=j~[p,~] where the superscript o denotes, as usual, the inverse image of j0. We now make use of the results and terminology of chapter I, w 8. Consider, in particular, the exact sequence chapter I (8.2): o --> Wa(J~(fN) ) ---> g'a(J~(QN)) --> A~ -> o where (8.3) A~ is a Tv-module " of rank I " (i.e. it contains a free T~-module of rank i as a subgroup of finite index). One checks that: S (J~ [p~]) = log, (order (A~/pm A;) ) + 0 ( I ) where v denotes Zp-dua]. Since A~ is also a T~-module of rank 1, we have: S(J~ [p'/]) = logp(order T~/p"T~) + O(I). Proof of (b). -- This follows the same lines as (a) above. One need only note that ~(J~176176 and since ~ is an 148 MODULAR CURVES AND THE EISENSTEIN IDEAL I49 ordinary prime, we have the exact sequence of chapter II, w 4 and chapter II, Prop- osition (8.5). Lemma (3.3). -- Let M=j(Q.) be the Mordell-Weil group of j, regarded as T-module. Then T~ | M and III~ (the ~3,component of the Shafarevich- Tare group III 0f J) are finite groups. Proof. -- Set M~ I-I~ j0) and note that 1V[ = I-I~ J). The quotient M/M ~ is finite. Therefore to show that T~| is finite it suffices to show that T~| M~ is finite. The long exact sequence of cohomology associated to the exact sequence of fppf sheaves over S: pra o j0[pm] JTs J?s o yields: o ~ M~ '~. g ~ ~ IF(S, j0[pm]) _+ IF(S, j0) [pm] ~ o and, by passage to the limit as m tends to m, using the maps: o > j0[p,,] > j0 pm j0 > > 0 o >j0[p,~+~] >j0,m+~j0 > ;) 0 we obtain an exact sequence of Tp-modules: o -~ O_..,/Z,N M ~ ~ lim IF(S, j0[pm]) ~ Hi(S, j0)p __~ o where the subscript p on the right means p-primary component. Passing to U-component (by applying the idempotent ~) we obtain: o ---> T~NTp (Q~/Zp@ g ~ ---> lim H~(S, jo [pm] ~) __> tp(S, jo)v ___> o. rn > Since the middle group is finite, by Lemma (3.2), the two flanking groups are. Since M ~ is a finitely generated group (by the theorem of Mordell-Weil), finiteness of TV|174 ~ implies finiteness of T~QTpM ~ To see that III~ is finite we use that (working modulo the category of groups whose order is a power of two) III may be identified with the image of HI(S, j0) in Hi(S, J) (Appendix of [34]), and the 2-primary component of IlI is a subgroup of the 2-primary component of this image. Finiteness of III~ then follows from finiteness of Hi(S, j0)v. To use (3.3) conveniently, we make a digression and recall the terminology of chapter II, w I o. Let a C T be any ideal of finite index in T, Yta) = N a r, T (at --~T/',((a) (so T ("/maps injectively to the completion T,) and j(,/=J/Y(,t'J, the quotient associated toJ (chap. II, w io). Let v=j(Q)| as T| module, and V("/=J("I(Q,)NQ, as T(")| 149 t5o B. MAZUR Lemma (3-4)-- V~aI=V/Y(,)-V=TI")| V. Proof. -- On the category of abelian varieties over Q, the functor A ~ A(Q)N Q is exact since A(Q) is finitely generated and I-P(Gal(Q/Q), A) is a torsion group, for all A in the category. The 1emma then follows by applying this exact functor to the diagram: o ,~,(,).J ,j ,jc~ ,o (~, ..., ~t) J�215215 where el, 9 9 et is a system of generators of the ideal a. Corollary (3-5). --If TaNTV=o, then the MordeU-Weil group of J(") is finite. Proof. -- Let W be the torsion-free quotient of the Mordell-Weil group J(Q.). Thus W is a free Z-module of finite rank and gives rise to a coherent sheaf over Spec T. By hypothesis, the support of T~| contains no irreducible component of Spec T,. Since the support of a meets every irreducible component of Spec T (al it follows that the support of W contains no irreducible component of Spec T ~al. The support of W then meets Spec T Cal in a finite union of closed points and (3.5) follows from Lemma (3.4). Proof of Theorem (3.1). -- Applying (3.3) for all Eisenstein primes we obtain that T~| T M is finite, where ,~ denotes the Eisenstein ideal. It then follows from (3.5) that J(Q) is finite. The theorem follows from Corollary (I .4). Proposition (3.6). -- Let ~3 be an Eisenstein prime associated to an odd prime number p. Then III~ = o. Pro@ -- We shall perform a more delicate descent to establish this. Let t be a good prime number and ~ =~t (using the terminology of chap. II, w i6). Then is an isogeny (cf. Proof of (I 6. i o)) ; we consider: (3.7) o --~ ker ~q ~-~ J ~ j -~ coker ~ -~ o as an exact sequence offppf sheaves of T-modules over S. Let A denote a finite set of points in Spec T, not containing ~, but containing all other points in the support of the T-modules ker ~(Q) and coker ~(Q). We shall work in the category of T-modules, modulo the category of T-modules whose supports lie in A (modulo A). By chapter II (16.6) and (16.4) it follows that: ker ~-Cp| modulo A 150 MODULAR CURVES AND THE EISENSTEIN IDEAL x5x and cok ~q is a skyscraper sheaf concentrated in characteristic N, whose stalk in charac- teristic N is isomorphic to the T-module Cp, modulo A. Since Gp is a constant group over S and Zp is a ix-type group, we have (e.g., chap. I (I. 7)) that Hi(S, Cp@Y,p)----o, and therefore Ha(S, ker ~q) =o modulo A. One obtains then the following exact sequence modulo A of fppf cohomology from the exact sequence (3.7) : o -+ Cp -+ H~ J) -+ n H 0 (S, J) ~ H~ coker "~). Since t-I~ coker ~q) = Cp modulo A, the above exact sequence shows that i is surjective modulo A. It also shows that ~ is automorphism, modulo A, of the torsion-free quotient of the Mordell-Weil group, which can be used as an alternative to the proof of Theorem (3-i), at least as it concerns odd Eisenstein primes. Reconsidering the exact sequence (3-7), surjectivity (modulo A) of the mapping i, gives that: : Hi(s, j) Hi(S, j) is injective, modulo A. Since III is a submodule of Ha(S, J), multiplication by ~ is also injective modulo A on HI, which establishes our proposition. Combining this with recent results of Brumer and Kramer [4] we may obtain: Proposition (3.8). -- Let N<25o. The natural map J~J- induces an isomorphism of C onto the Mordell- Weil group J- (Q) except possibly in cases N = ~ 5 i, 199 and 227. Proof. -- From the table of the introduction, one sees that for N<25 o, J-=J except for the following values of N: N=67, lO 9 , 139, I5I, I79, I99, 227 and when N = 67, I o9, 139, 179, J- differs from ~ by an elliptic curve factor. Brumer and Kramer have shown (1) that these elliptic curve factors have finite MordeU-Weil groups over Q. It then follows that J-(Q) is finite, using (3. I), for all values of N considered in Proposition (3.8). The assertion then follows from (i.4). Recently, Atkin communicated to me that J- is a simple abelian variety (and hence equal to J) for N=383, 419, 479, 491, and consequently (3.8) holds for these values of N, as well. 4" Rational points on Xo(N ). Theorem (4.i). -- Let N:~(2, 3), 5, 7 and x 3 (i.e. thegenus 0fX0(N ) is >o). Then Xo(N) (Q) isfinite. Pro@ -- Work over Q, and consider the projection X0(N)~ J defined by x~image(x--oo) in J. Since ~" is nontrivial and the image of X= Xo(N ) generates (1) See their forthcoming publication [4]. 151 ~52 B. MAZUR as a group variety, if.~ is the image ofX m J, then X/Q is a curve, and X--~X is a finite morphism. Since .X(Q) Cj(Q)=C (3. I), X(Q) is a finite set and therefore X(Q) is also finite. To be sure, we have little control over this set if we know nothing concerning the structure of the finite morphism ~. What is its degree? What are the singular points of the image? Remark. -- It is a theorem of Manin ([3I], [65] ) that for every number field K and integer re>i, there is an integer e(m, K) such that Xo(m~)(K) is finite for all e~_e(m, K), but no effective bound for e(m, K) is obtained. I understand that the Russian mathematician Berkovich has recently obtained such effective bounds using the techniques of this paper, and in particular the techniques of the proof of (4. I). To analyze the finite set X(Q) we make use of the retraction p :J(O)-+Z/n of chapter II, w I I. Proposition (4.2). -- If xeX0(N)(Q), the element p(x) of Z/n is equal to one of the following values: OOT I (4" 2) i) I/2 (possible only /f N - -- I mod 4) I/3 or 2/3 (possible only /f N-- --I mod 3). Remarks. -- i. If N=--- i mod 3, the integer n is not divisible by 3 and t/3 has a sense in Z/n; similarly I/2 has a sense in Z/n when n is odd, which is the case if N----I mod4. 2. p(o)=i and p(oo)=o. Proof of (4.2). -- The point x extends (by Zariski closure) to a section of X0(N)/s . This section must lie in the smooth locus of X0(N)-~S and hence, if ~is its pullback to Spec(F~), ~ must lie in exactly one irreducible component of the fiber X0(N)y N (see diagram I of chap. II, w I). Thus, Y lies on one of these: Z'or Z (4.2) ii) E (possible only if N- -- i mod 4) G or F (possible only if N----I mod 3). But the natural map X o~,//~ r~s-,o,th ~ J/F~ -+ C = Z/n sends the five components listed in (4.2) ii) to the corresponding values listed in (4.2) i) as follows from the table of the appendix. Let J_=(i-w)JCJ. One obtains a map: r : X0(N)--,J_ by the rule x~cl((x)--(wx)). If xeX0(N)(Q), set X(x)=p(r(x))eZ/n. Corollary (4.3)- -- One has X(x)=~=I, o, or j=I/3. Pro@ -- This follows from (4.2) and the fact that X(x)=2f~(x)--I. 152 MODULAR CURVES AND THE EISENSTEIN IDEAL I53 Corollary (4-4).- Suppose J_(Q.) is finite. Then: a) One has J_(Q.)=C. b) For all x~X0(N)(Q) , one has r(x)=X(x).c with ~(x)=-}-I,o, or:~:x/3. Pro@ -- Assertion a) comes from the fact that C is contained in J_ (Q,) and it is the torsion subgroup of J(Q.) (i .2). Assertion b) follows from (4.3) and the fact that p is the identity on C. Remark.- By (3.8) and the remarks after it, (4-4) applies to at least these values: N<25 o, with the possible exceptions of N=I5I , 199, 227 and N=383, 419, 479, 491. The next proposition is due mainly to Ogg and includes work of Parry and of Brumer. Proposition ('t-5)- -- a) If N>23 and N=37 then the morphism r : X0(N)---> J_ is injective off the locus of fixed points of w. b) IfJ_(Q) is finite, and xeX0(N)(O..), one has: X(x)=:~I/3 ~N=II, 17 X(x)= o =>N=iI, I9,43,67, orr63. Discussion of the proposition. -- The following is Ogg's proof of a). Suppose x,y~X0(N)(C ) such that r(x)=r(y), x4=y, and x is not fixed under w. Write z=w(y) and we have: x + z -- w(x) + w(z) where - denotes linear equivalence on Xo(N ). Since x+z is not invariant under w, it follows that Xo(N ) is a hyperelliptic curve, and moreover, the involution w is not the hyperelliptic involution. But Ogg ([38], theorem i) has determined that N = 37 is the only value of N such that X0(N ) admits such a description. As for b), let xeX0(N )(Q) be a point such that ;~(x)=-E1/3. Then N=--I rood 3, and (replacing x by w(x) if necessary) we may assume: sx + (oo) = sw(x) + (o). By an elegant argument (end of [49]) Ogg shows that n<24 o. We recall the argument and sharpen this upper estimate a bit. Let - denote specialization of a section over S to Spec F 4. Then we have: yielding a function f on Xo(N)/F, such that the inverse image of the point ooEPI(F4) is the divisor 3~+(~) and the inverse image of oePl(F4) is 3w(s Since the points of X0(N)(F4) different from ~, w(~), ~, g must lie in the fibers off above the 3 points of PI(F4) different from oo and o, we have: X0(N ) (F4)< - ~6. 20 154 B. MAZUR But Ogg has constructed [49] at least N/I2 noncuspidal rational points in Xo(N)(F4) , so: N/I2+2<I6 or N<__ I68. Let us now consider an argument which helps to eliminate many low values of N. Findfa function from X0(N)/Q to P~Q whose divisor is 3 x + oo-- 3wx--o. Definef w by f'~ Since f.fw has neither zeroes nor poles, it is a constant e. Define an involution w : I~I-+P 1 by the formula y~e[y. We obtain a commutative square: X ~> X pl w~ p1 and consequently f induces a rational map on quotients by w: X "> X/w = X + pt '~> pt/w = pl+ The double covering pa___>pl+ has precisely 2 ramification points y=+v%. Also, the mapf + is of degree 4- Consequently, the double covering ~ : X-+X + can have at most 2.4=8 ramification points. That is, the number of fixed points of w is ~8. +< This condition is easily shown to be equivalent to the condition g---g _3, by the Hurwitz formula applied to the covering X-+X +, and one checks (e.g., consult the table of introduction) that those N such that a) N~I68, b) N-----I mod3, c) g---g+<__3, are: N=11, 17, 23, 29, 4 I, 53, I13, and 137. When N----I I, I7, Xo(N ) is of genus I, and there is a (unique) point xeX0(N)(Q) such that ),(x) = i/3. When N----23, 29, 41, X0(N) + is of genus o, and Ogg has special arguments to show that there are no rational points such that X(x)=I/3 [49]. The remaining three cases have been ruled out by work of W. Parry and A. Brumer. Finally, note that if xeX0(N)(C), X(x)=o-z~x is a fixed point of w, and so the final assertion of (4- 5) follows from (2.2) and the solution of the class number one problem (Heegner-Baker-Stark). Remarks. -- i. By (3.8) and (4.5) we have determined all the rational points on all curves X0(N ) for N<25o, with the exception of N=I5I , I99 and 227. 154 MODULAR CURVES AND THE EISENSTEIN IDEAL I55 2. (Fields of low odd degree.) If J_(Q) is finite, Ogg's trick has strong implications concerning rational points of Xo(N ) in the totality of fields of a given degree. Brumer has some computations for degree 2, and we shall give a fragmentary result for degree 3- By a point on X0(N ) of degree d we mean a point of Xo(N ) (Q) defined over any extension field of degree d over Q. Proposition (4.6). -- If N=383, 419, 479, or 491, then Xo(N ) has only a finite number of cubic points (points of degree 3)- Proof. -- Let d be any odd positive number. Let K d be the set of all Q;conjugacy classes of points of degree d in X0(N)(Q) not containing a fixed point of w. Define a mapping t :K~--->J_(Q.) by: ,:~cl( Z x- X w(x)). xG~ xGK Lemma ('t-7)- -- If (d is an odd positive number, and) N>i2o.d, then : : K s -~J_(Q) h injective. Pro@ -- Suppose :(K) = ~(K') where K ~:K'. Writing K"= w(K') we get a relation: (4.8) Z x+ Z y- Z w(x)+ Z w(y). x@K y~" xG~ YGK" The above linear equivalence cannot be an identity of divisors, for by hypothesis w does not interchange the conjugacy classes 1< and ~", nor does w stabilize either co~ugacy class. The reason for the latter assertion is that an involution of a finite set of odd cardinality must have a fixed point and the conjugacy classes containing fixed points of w are excluded from K s. Thus there is a nonconstant function f on X0(N ) whose divisor of zeroes is the left-hand side of (4.8) and whose divisor of poles is the right-hand side. Letting D be the Cartier divisor in X0(N)/s, obtained by taking the closure of the right-hand side of (4-8), we have that H~ ~(D)) is of dimension > ~, using (EGA III (7- 7- 5), I) as in the proof of Lemma (2.6). It follows that there is a mapping f: X0(N)/F,---> la~F, of degree 2dand therefore io.d is an upper bound for the cardinality of Xo(N)(Fd). Since this cardinality is greater than N/I2, the lemma follows. The proposition then follows by taking dz3, and using the remark after (4-4)- It is interesting to consider the problem of showing finiteness of X0(m ) (O) when genus(X0(m))~'o , for all integers m. By (4-I) we may restrict attention to composite numbers m, and it is evident that it suffices to treat those composite numbers m such that genus X0(d ) = o for all proper divisors d of m. There are x 7 such values of m, of which 9 are of genus I and have been shown to have finite Mordell-Weil groups, by various people, including Ligozat ([3o], and see discussion in [48]). The case m= 26 155 156 B. MAZUR is treated in [4I]. The cases m=35, 5 ~ have been taken care of by Kubert [27] ; the case m----5 ~ was also done independently by Birch. The case m = 39 has been settled by a descent argument on the elliptic curve quotient of X0(39 ) using an explicit equation given for this curve which can be found in an extensive table compiled by Kiepert. This equation (formula 63ib on page 391 of [25] ) and these useful tables were pointed out to me by Kubert. Re-writing the curve as a quotient defined over Q, its minimal model is y2+xy=xa-l-x~--4x--5 and the descent follows the lines of the case m=35 ([27] , [34], w 9)- Sixteen of the seventeen cases (all m above except m= 125) have been covered by the recent work of Berkovich (cf. remark following (4-I)). 5" A complete description of torsion in the Mordell-Weil group of elliptic curves over Q. In this section we shall prove the following theorem, first conjectured by Ogg [49] : Theorem (5- x ). -- Let a~ be the torsion subgroup of the Mordell- Well group of an elliptic curve E, over Q. Then (I) is isomorphic to one of the following 18 groups: Z/mZ for m<lo or m=I2 or: Z/2.Z� .for ~<_4. Remark. -- All these groups do occur. The fifteen curves: Xl(m ) and X(2) � for m, ~ in the above range are all isomorphic to P~Q. Consequently, the elliptic curves E/Q whose Mordell-Weil group contains a given group (I) (chosen from among the 15 above) occur in an infinite (rationally parametrized) family. These fifteen explicit rational parametrizations are given in the table of [27], chapter IV. Corollary (5- 2). -- Let an elliptic curve, defined over Q, possess a point of order m rational over Q. Then m<Io or m=I2. Equivalently: Corollary (5.3)- -- Let m be an integer such that the genus of Xl(m) /s greater than o (i.e. m = 11 or m >I 3)- Then the only rational points of X~(m) (over Q) are the rational cusps. We shall begin the proof of (5- 1-3) with a series of reduction steps. First reduction. -- To prove (5-1-3) it suffices to prove (5.2) in the special case where re=N, a prime number such that the genus of X0(N ) is 2>o (i.e. N 4:2, 3, 5, 7, and 13). This is so by virtue of the close study of the above conjecture of Ogg, made by Kubert, for low values of composite numbers m. In particular, Kubert has shown ([27] , chap. IV) that it suffices to consider only prime values of m, greater than or equal to 23. For m=i3, see [4o]. For the duration of the proof, let, then, N denote a prime number 4:2, 3, 5, 7 or 13, 156 MODULAR CURVES AND THE EISENSTEIN IDEAL I57 and let Z/N C E be an elliptic curve over Q with a point of order N, rational over Q (generating the subgroup Z/N). The object of the proof will be to show that Z/N C E does not exist. As usual, E/s will denote the Ndron model of E over S and Z/Nrs C E/s is the 6tale constant subgroup scheme over S generated by our point of order N. Let K=Q(~N), where ~N is a primitive N-th root of unity, and let L be the field extension of Q generated by the N-division points of E. By considering the short exact sequence of Gal(Q/Q)-modules: (5.4) o -+ Z/N -+ E[N] ~ ~N -+ O ~ sees that Gal(L/K)has a faithfuI representati~ int~ GL2(FN)~ the f~ (I O ~) where Z : Gal(L/Q) ,, Gal(K/Q) >~ F} is the cyclotomic character. Thus, one has the diagram of field extensions: / I~'~ where L/K is either an N-cyclic extension, or is the trivial extension. Moreover, an elementary computation gives that the natural action of Gal(K/Q,) on Gal(L/K) (conjugation in Gal(L/Q)) is by multiplication by z -1. This computation uses the existence of the faithful representation of Gal(L/Q,) of the form (i .~, and, as Serre \o ] remarked, can be most conveniently seen by noting that the 9 in the upper right corner takes its values, canonically, in the vector space Horn(aN, Z/N). It is clear that the exact sequence (5-4) splits if and only if L = K. Second reduction. -- It suffices to prove that (5.4) splits, or equivalently, that L = K. For we would then have the following result from which we easily derive a contradiction: Given any eUiptic curve ~Q and a sub-Galois module Z/NC d ~, there is a sub-Galois module I~,C g. Let us obtain a contradiction from this. Forming the quotient d~'=N/lxN, we get another elliptic curve over Q and the image of Z/N provides g' with, again, a sub-Galois module Z/NC N'. We may then apply the above result inductively to obtain a chain of such elliptic curves over Q,, related by ~N-isogenies, rational over Q,: 8--~g'--~g~ all containing sub-Galois modules isomorphic to Z/N. This is impossible for various reasons. Firstly, the members of the above chain cannot be all mutually nonisomorphic. For, if they were, they would represent an infinite number of elliptic curves over Q with good reduction outside a given finite set of primes. This would contradict the 157 158 B. MAZUR theorem of Shafarevitch (cf. [63] , IV (I.4)). Alternatively (and more in the spirit of the present work), it would provide an infinite number of distinct rational points on X0(N); and this would contradict Theorem (4-i). We have therefore shown that for suitable i# j, #(~)x #(J), and consequently there is a non-scalar endomorphism of 8 (i) defined over O. In particular, d "(i) possesses a complex multiplication over Q, which is impossible. Third reduction. -- It suffices to show that L/K is unramified (at all places). For suppose that L/K is unramified, and nontrivial. Then it is an N-cyclic (unramified) extension, and consequently N must be an irregular prime. Since L/Q is Galois and the natural action of Gal(K/Q) on Gal(L/K) is ) -1 it would then follow, by Herbrand's theorem (chap. I (2.9)), that the Bernoulli number B 2 must have numerator divisible by N. Since B 2 = I/6, L/K must be the trivial extension. We shall now prove that L/K is unramified. Although this is a local question at each place v of K, it is unlikely that one can prove this by local arguments. Indeed, the essential step 3 below is global. We proceed by 4 steps, analyzing the structure of the putative Z/NC E. Step 1. -- E/s is semi-stable. That is, E has semi-stable (i.e. good or multiplicative) reduction at all points of S. Proof. -- Let q be a (rational) prime of nonsemi-stable (i.e. additive) reduction for E. Thus the connected component of the fibre E/Fq, (E/@ ~ is an additive group, and, as is well known, the index of (E/Fq) ~ in E/F q is 2 a. 3 b for suitable integers a, b. It follows that the specialization Z/NjF q must be contained in (E/Fq) ~ Consequently, q=N. Using ([72], w 2, Cor. 3) one sees that there is a finite extension field ~ff/Oz q such that E/~ has semi-stable reduction at the maximal ideal of the ring of integers 1~ = g)~c, and if e = e(Cf : QN) is the absolute ramification index, we may choose o~# so that e<6. IfE/~ is the Ndron model of E over the base 0, and E/z| is the pullback to ~) of the Ndron model over Z, there is a natural morphism E/zNg~-+ E/~ which is trivial on the connected component of the closed fiber, since there are no nontrivial maps from an additive group over a field to a multiplicative group, or to an elliptic curve. If G/oC E/e is the closed subgroup scheme generated by Z/N/jcC E/~, we have a natural morphism Z/N/o-+ G/o which is an isomorphism on generic fibers, and not an iso- morphism on the special fibers, by the above discussion. From this, one sees that G/o is a finite flat group scheme. But since e<6<N--I, by [55] a finite flat group scheme of order N over 0 is determined by its generic fiber. In other words, G/o must be isomorphic to Z/N/~, which is a contradiction. Step 2. -- If q = 2, or 3, then E has bad (hence multiplicative) reduction at q, and the specialization Z/N/F q is not contained in the connected component of the identity (E/Fq) ~ 158 MODULAR CURVES AND THE EISENSTEIN IDEAL Pro@- If E/rq were an elliptic curve, and Z/NC E/rq, then by the " Riemann hypothesis" N<_I+q+2%/~t, which is impossible for q=~, 3. Therefore, E has bad reduction at q (= 2, 3) necessarily of multiplicative type, by step i. But, by Tate's theory ([63] , IV, A.I. i), (E/Fr ~ is isomorphic to G,,/F r which has qZ--I points. Again, we cannot have Z/NC Gmfpq, , for q=2, 3, by virtue of our hypotheses on N. Step 3. -- If q is any prime of bad (hence multiplicative) reduction for E, then the special- ization Z/N/Fq is not contained in the connected component of the identity, (E/@ ~ Proof. -- Let q be a prime of multiplicative reduction such that Z/N/rqC (E/rq) ~ By steps I and 2 we may assume q + ~, 3 or N. Consider the base T= Spec Z[I/2.N] and let x be the T-valued point of X0(N)/r determined by the couple (E/T , Z/N/r). That is, x=j(E/r , Z/NIT). It is illuminating to draw the scheme-theoretic diagram: 9 9 ,~ 2 3 q where oo and o are the cuspidal sections over T. We are justified in drawing the intersections: x/F= ~/F,, X/F~=O/F~, because, by ([7], VI, w 5), the modular interpret- ation of c~fr t is the " generalized elliptic curve ": (Gin � Z/N, Z/N)F t (i.e. the cyclic subgroup of order N which gives the F0(N)-structure is not contained in the connected component containing the identity) while the interpretation of o/F t is the " generalized elliptic curve ": (Gin x Z/N, ~N)~F, (i.e. the cyclic subgroup of order N which gives the P0(N)-structure is contained in the connected component containing the identity). Now consider the natural projection to the Eisenstein quotient X0(N)/r-+J/~. By (3. I) we know that J(O~)= C. For the present proof, however, it suffices to know that ~(O) is a torsion group. Thus j(T):j(o,) is a torsion group. Let '~ denote the image of sections of X0(N ) in J. Since T is an open subscheme of Spec Z over which 2 is invertible, if A is any abelian scheme over T, and g any rational prime representing 159 i6o B. MAZUR a closed point of T, the specialization map A(T)tor s --> A(Ft) is injective (1). Applying thisfact to 1=7, one sees that J(T)-+j(Ft) isinjective. But the equations ~F =~/r,, and x/Fq=O/Fq, then imply that ~=oo. Since ~--~ is of order n in j, this can only be true if n = I, or equivalently, if N < 7 or N = i3. Since N is constrained to be :>7 and + i3, we obtain the contradiction that we seek. Step 4. -- L/K is unramified. Pro@ -- (i) q a rational prime of good reduction for E; q+N: Since E[N']/Zq is an 6tale, finite flat group scheme, L/K is unramified over all places of K lying over q. (ii) q=N; E has good reduction at N: Again E [N]/z~ is a finite flat group scheme. Applying the " connected component of the identity" functor to (4.4) one sees that (E/zN) 0 [.LN ' and therefore we get a splitting: E [N']/z, , = Z/N/z ~ � ~wz~,, which again shows that L/K is unramified at all places of K lying above N. (iii) q a rational prime of bad reduction for E: Since Z/NlFqd; (E/@ ~ by step 3, one obtains, as in (ii), E[N]/zq~Z/N/zq� giving us the same conclusion: that all places of K above q are unramified in L/K. 6. Rational points on X~plit(N ). Keeping to the terminology of the Introduction (cf. discussion preceding Theorem 9) elliptic curves with a normalizer-of-split-Cartan structure on their N-division points are classified by noncuspidal rational points on X0pnt(N)=X0(Ng")/wN,. Theorem (6.x). -- If N+2, 3, 5, 7 and I3, then Xsplit(N ) has only a finite number of rational points. Note. -- If N~7, then Xsplit(N ) is isomorphic to P~0 and therefore its set of rational points form a rationally parametrized infinite set. The curve Xsplit(i3) is of genus 3. It is to be expected that Xspnt(I3) has only a finite number of rational points, but my methods have not been able to establish this. Proof of theorem (6. I). m Consider the two natural morphisms: f, g : X0(N ~) -> X0(N ). The mapfis defined by the prescription f: (E, CN, ) ~ (E, CN) where, ifE is an elliptic curve, and C N, is a subgroup of E of order N 2, then C N-~N.CN,CE. It induces a map from parabolic modular forms (of weight 2) on Fo(N ) to parabolic modular forms on F0(N ~) with the same q-expansion. The map g is defined by the prescription g : (E, C~,) ~ (E', C~) where E'= E/C N and C~=C~,/C N. (1) This is a standard application of the Oort-Tate classification theorem [54] to the group scheme over T generated by an element of order p in the kemel of the above specialization map. If A is an elliptic curve, then this result is due to Nagel-Lutz. 160 MODULAR CURVES AND THE EISENSTEIN IDEAL x6I If r is a parabolic modular form (of weight 2) and ~(q) denotes its q-expansion at 0% then (~-'~)(q)=~(q~). We denote the canonical involution (chap. II, w 6) of X0(N ) by w N to distinguish it from the canonical involution of X0(N~), denoted wN,. As usual, J is the jacobian of X0(N ). Let h : X0(N *) --->J be the map which associates to x the divisor class of f(x)--g(x). A straightforward calculation yields the formula h. w N, =--ws.h (and the minus sign will be of importance to our proof). It follows from this formula that the composition X0(N ~) --->Jh ~J-=J/(I+wN)" J factors through X0(N ~) ---> X0(N ~)/w N, = Xsplit(N ) and thereby induces a map: h-- : Xsplit(N ) --->J-. The map X0(N ~) -->J induces a surjection on the jacobian Jo(N ~) --->J as can be seen as follows. The induced map from parabolic modular forms of weight 2 under F0(N ) to parabolic modular forms under F0(N 2) is injective. This latter assertion is true since a modular form of weight 2 under P0(N) which is sent to zero by the map in question must have its first N q-expansion coefficients equal to zero. Hence it is zero. It follows that the map h- : Xspa(N)-~ J- induces a surjection from the jacobian of Xspn~(N ) to J-. Let h:X~plit(N)-->J denote the composition of h- with the projection map to the Eisenstein quotient (chap. II (I 7. IO)). Since N=I I or N>_I7, it follows that X0(N ) is of positive genus, and that J is nontrivial. Letting X~p~it(N) cJ denote the image of Xsplit(N ) under h, one sees that Xsput(N) must be a curve, and Xspiit(N ) --> :~sput(N) a finite morphism. Since J(Q)=C is a finite group, the proof of Theorem (6. I) is completed. Remark. -- We have made essential use of the fact that J factors through J- (chap. II (i 7. io)). This fact (when N- I rood 8) seems to depend on some of the more delicate aspects of the theory developed in chapter II. 7. Factors of the Eisenstein quotient. Consider a surjective morphism defined over Q, J-->A where A/Q is a Q-simple (equivalently: C-simple) abelian variety. Let p [ n be a prime number such that this morphism factors through the p-Eisenstein quotient (such a prime number p must exist, but may net be unique) and let a = dim A. Replacing A by an abelian variety isogenous to it, if necessary, we may suppose that the Hecke algebra T leaves the kernel of J-+A stable, and consequently that we can induce a natural action of T on A. Since the Eisenstein prime ~ associated to p is contained in the irreducible component of Spec T which corresponds to A (chap. II (io. I)) it follows that A[~3] (the kernel of f~ in A) is nontrivial. Consequently, by admissibility of the kernel of ~3 (chap. II (14. i)) it follows that there is an abelian variety A;Q isogenous to A over Qsuch that A' possesses a point oforderp in its MordeU- Weil group. (More precisely, we may take A;s to contain a subgroup scheme isomorphic to Z/p/s 0 21 16e B. MAZUR Using the criterion of N~ron-Ogg-ghafarevich, one sees that A-~A' extends to an isogeny of abelian schemes over S'. Reduce to characteristic 2 and obtain an isogeny A/F -+A~ ' of abelian varieties where A'(F~) contains a point of order p. Standard estimates for the number of rational point of an abelian variety over a finite field (the Well conjectures) give: p< #A,(F2)< (i +~r or: logp (7. i) a2 2 .log(i -? %/2)" We obtain : Proposition (7.2). -- Every simple factor of the p-Eisenstein quotient ~Cp) has dimension log p -- 2. log(i +%/2)" Corollary (7.3). -- There are absolutely simple abelian varieties of arbitrarily high dimension, defined over Q,, whose Mordell-Weil group is finite. Proof. -- For any positive integer a0, find a prime number P-->5 such that logp>2a0.1og(i+V/2) and, by Dirichlet's theorem, choose a prime number N such that N--=I modp. Then (7.2) every simple factor of the p-Eisenstein quotient of j=j0(N) has dimension >a 0 and (4. ~) has finite Mordell-Weil group. What are the elliptic curve factors of the Eisenstein quotient J? If E/Q is a quotient elliptic curve of jC~), then E has (prime) conductor N, and by the above discussion, after modification of E by Q-isogeny if necessary, we may suppose that the Mordell-Weil group of E possesses a point of order p. There has been some recent work ([68], [46], [42], [I6]) on elliptic curves of prime conductor N possessing a torsion point of order p over Q. In particular, one has that p< 5 (using the Well estimates to the reduction of E in characteristic 2) and by [42] one has, further, if p = 5, then N =II and E is isogenous to X0(II); if P=3, then N=I9, or37 and E is isogenous to X0(I9) or to the Eisenstein quotient of J0(37). Thus we are reduced to the case p=2. In this case, either N=I7 and E is isogenous to X0(I7), or it is a Neumann-getzer curve (which, by definition, is an elfiptic curve over Q, of prime conductor N + 17 possessing a point of order 2 in its Mordell- Well group). The facts concerning Neumann-Setzer curves are these ([68], [46]): A Neumann-Setzer curve of conductor N exists if and only if N is of the form 64 q-u 2 (u an integer). If N is of the above form there are precisely two isomorphism classes of Neumann-Setzer curves of conductor N, given by the equations: y2 = x 3 + ux 2_ 16x yZ = x ~-- 2ux 2 + Nx. 162 MODULAR CURVES AND THE EISENSTEIN IDEAL I63 One may pass from one curve to the other by the 2-isogeny obtained by division by the rational point of order 2. Proposition (7.4). -- i) Let p > 2. The p-Eisenstein quotient has no elliptic curve factor unless P=5, N=II or P=3 and N=I90r37. ii) The 2-Eisenstein quotient 7 (21 has no elliptic curve factor unless N = 17, or N = 64 + u S with u an integer. If the 2-Eisenstein quotient has an elliptic curve factor, then this factor is unique up to isogeny and if N + 17 its isogeny class is that of the Neumann-Setzer curves of conductor N. If the (two) Neumann-Setzer curves of conductor N are parametrized by modular functions for P0 (N) (i.e. /f they occur as quotients of J, a special case of the conjecture of Weil) then they are quot#nts of jl ). Proof. -- This combines the work of [68], [46] as in the discussion above, and chapter II (I 4. i). The following gives (granted conjectures of Weil and Hardy-Littlewood) an infinite number of values of N for which the estimate of (7.2) is sharp for the 2-Eisenstein quotient. Proposition (7.5). -- Let N be a prime number of the form 64+u ~ such that N r i mod I6. Suppose that the (two) Neumann-Setzer curves are parametrized by modular functions for P0(N). Then "~c~) is of dimension I, and is a Neumann-Setzer curve of conductor N. Proof. -- Let ~ be the Eisenstein prime associated to 2. By our hypothesis on N, 2 ]In. Therefore, by chapter II (I 9.I), T~ is a discrete valuation ring. Since the irreducible components of Spec T~0 map surjectively to the (isogeny classes of) factors of 7 (2/, it follows that 7 (2/is a simple abelian variety. But, by the hypothesis of (7.5) and by (7.4) ii) the Neumann-Setzer curves are factors ofJ (~/. Our proposition follows. Remark. -- Let N be a prime number of the form 64 + u S such that N -=- 1 mod 16. The 2-Eisenstein quotient contains a point of order 4 (at least). It must have dimension greater than I, for if it were an elliptic curve, it would be a Neumann-Setzer curve and a Neumann-Setzer curve does not possess a point of order 4 [68]. Suppose, further, that the Neumann-Setzer curves of conductor N are parametrized by modular functions for F0(N ). It then follows that ]~(2/is not a simple abelian variety, since it has a Neumann-Setzer curve as a proper factor. Consequently the completion of the ttecke algebra T~ is not an integral domain. Conjectures of Weil and of Hardy- Littlewood would give that this occurs for infinitely many values of N. The only case of a pair (N,p) where N<25 o and T~ is not a discrete valuation ring, for ~3 the Eisenstein prime associated to p, is: N=II3, p----2. This is the first instance of the (conjecturally infinite) family of examples described in the paragraph above. 163 I64 B. MAZUR 8. The ~-adic L-series. Fix ~3 an Eisenstein prime associated to a prime number p4= 2. In this section and the next we shall examine the analytically-defined ~-adic L series [39] and the arithmetically-defined ~3-adic characteristic polynomial [34]. We recall terminology and results from the papers cited. Since both p and ~p = I-+-p--Tp are in the ideal ~3, we have T v-- i mod ~3 (1) and therefore Tp is a unit in T~. The standard recursive process (e.g., p. 47 of [39]) gives two roots of the quadratic equation: X~- Tp. X +p = o in T~. Call the unit root r~ and the other one ~ to be consistent with the terminology of [39]- Let { }:Q/Z'-->I-I=HI(X0(N), Z) denote the modular symbol, where Q/Z' means rationals with denominator prime to N, modulo I ([22], [29] ). Let f: Q/Z'-+H~ be the composition of { } with H~H~=T~| For any fixed choice of integer A 0 prime top, set A,= A0.p n and ZA = l im Z/A, regarded as topological ring. Z~ is then the topological group of its units. We now wish to use the construction of [39], w 8, to obtain an H~-valued measure on Z~x from the function fl This may be done, for f is an eigenfunction for the Hecke operator rip with eigenvalue a unit in T~. One remark, however, must be made: in the terminology of [39], w 8, we take T~=D, H~ =W. Note, however, that in [39] (8. I) the hypothesis on D is that it be the ring of integers in a finite extension of Qp. This is not needed. All that is used is that D is a local ring with maximal ideal m containing the prime p and that D is p-adically complete. Let p~/', then, denote the It~-valued measure associated to the eigenfunction f ([39] (8.1)). Let (chap. II (18.1)) o~H~H~-+H~o be the decomposition of H~ into -- and + eigen (sub- and quotient-) spaces. Let Z be a continuous multi- plicative character on Z~ whose values lie in (and generate) the To-algebra T~[Z]. We consider the Fourier transform of the measure ~xA: Lv(Z) = fz3, Z- ExaeHv [)~] =Tv [Z] | Hr. If the formula )(--I) -(sign x). I defines sign ), L~(Z) lies in the (sign Z)- eigenspace of the complex conjugation involution and, if Z is even, it is natural to let LV()~) take its value in H~[Z] , by projection. We refer to L~ as the ~3-adic L-series, and the general theory of [39] applies to it. In particular we have its various developments as analytic function in the s- and T-planes, keeping the conventions of [39]. (a) We think of this relation as expressing the fact that Eisenstein primes are anomalous, in the spirit of the notion introduced for elliptic curves in [34]. 164 MODULAR CURVES AND THE EISENSTEIN IDEAL Let ~z~, + be the projection of the measure ~/' to I-I~. Then the ~3-adic L-series restricted to even characters is the Fourier transform of tx/'' + Proposition (8. I) (divisibiaty). -- ~a, + takes its values in ~3. I-I~ C H~. Pro@ -- By chapter II (i8.8), f(b/Am) depends only on A~ mod.~.H~, if b is prime to A. By formula (2) of (8.1) of [39], A+ evaluated on the fundamental open set a+A,ZaCZ x (for a prime to A) is given by: lim r~ -m ~ f(b/Am), rain b ~ amod An" from which our proposition is seen to follow. Corollary (8.2). -- If Z is an even character, L~(Z)e3.H~[7. ]. The proposition also implies that if we develop L~ in a power series expansion about an even character )(o in either the s- or the T-plane (cf. [39], w 9) then every coefficient of these power series will lie in ~.K~[Xo ]. To evaluate the constant term L~(z0 ) of the ~3-adic L-series, where Z0 is the principal character of conductor p, we use [39] (8.2). Take A0= I. We work in the ring D=T~. The proposition of [39] (8.2) gives: --~v.S p--1 where S is E {alp} projected to H~. a=0 If e~ denotes the image of the winding dement in H~| (chap. II, w I8), formula (8) of page 35 of [39] yields "~j,.e~=--S, giving: (8.3) L~(Xo) = ~q~.e; ell+ (compare with the formula at top of page 55 of [39]). To analyze this constant term more closely, fix g a good prime number (relative to p, N) as in chapter II, w i6. For convenience, if p itself is good (i.e. ifp is not a p-th power modulo N), take t=p. Let ~=~t, which is a generator in T~ of the ideal Z~ by chapter II (i6.6). Write ~, = 3. ~. Therefore 3eT~ and 8 is a unit (= i) if and only ifp is good. Since ~, ~% are units in the ring T~| (e.g., as in chap. II, proof of (I6. Io)) so is 8. Corollary (8.4). -- There is a suitable generator y of the T~-module H~ such that: L~(z0 ) = 32. ~.y where 8 is a unit in T~ | Q. Furthermore, 8 is a unit in T~ if and only if p is not a p-th power rood N. 165 x66 B. MAZUR Proof. -- This follows from the above discussion, and (8.3), by taking: y = ~ .e;/(~--P) (~--~)~. Now make a choice of a i-unit y~Z~ = Z; and form ([39], w 9) the ~-adic L-series in the T-plane about ~(0: L(T) = L~(Zo , T)(v)E H;| The constant term is just L~(Z0), and by (8.2) each of its coefficients is divisible by ~, and therefore we may write L(T):g(T).~.y, where g(T)cT~[[T]] is a power series whose constant term is 82. Thus Corollary (8.5). -- Identify T v with H~ by the map z~z.y. Then: ~]-I.L~(z0 , T)(v) e T~[iT]] is a power series with constant term 82. It is a unit in Tr if and only if p is not a p-th power modulo N. Remark. -- When p is a p-th power modulo N, we have then a " secondary " analogue to the phenomenon of anomalous primes studied in [34], [39]. Namely, either Lr , T)(v) is divisible by more than ~, or it has at least one zero in the open unit T~-disc (or both). 9" Behavior in cyclotomic towers. Guided by conjectures made in [39], the results concerning the ~3-adic L-series (w 6) suggest that the following proposition is true. We prove it below (independent of any conjectures). We shall also take the opportunity to correct an erroneous assertion made in [34]. Proposition (9. x ). -- Let p ~ 2 be a divisor of n. Let (j~P) /O. denote the unique Galois extension with aalois group isomorphic to Zp (the p-cyclotomic F-extension). The group "J(p)(~')) of rational points of the p-Eisenstein quotient with values in the p-cyclotomic F-extension is a finitely generated group. If p is not a p-th power modulo N, then it is a finite group. One has an accompanying assertion about the ~3-primary component of the Shafarevich-Tate group. Namely, let F= Gal(O~P)/O), and for every positive integer m, let F mC P be the subgroup of index p'~. Set O~C O.~ p) to be the fixed field of pro, and III m the ~3-primary component of the Shafarevich-Tate group of J(p) (or of J: it is the same) over O.~ ). Set A=liI+_nnT~0[I'/Pm] (the projective limit of topological rings, where T~[P/Fm] is given the natural topology) and III~o=lim III m regarded as A-module. Proposition (9.2). -- The kernel and cokernel of the natural map: IIIm (III Jm 166 MODULAR CURVES AND THE EISENSTEIN IDEAL I67 are finite groups whose orders are bounded (independent of n). That is, the above sequence is controlled in the sense of [24], [34]. The A-module III oo is isomorphic, modulo finite groups, to the Pontrjagin dual of the A-module A/.~s~.A~Ts~/~s~[[T]]. There is a constant c0>o such that if fl[n, then: I logp(order IIIm)--f.p"[< c o for all m>o. Remarks. -- Guided by the same conjectures of [39] (6.5), one would expect that if p (42) is a p-th power modulo N, then either J(p)(O~ p)) is a finitely generated group of positive rank, or III m grows more rapidly than the bound of (9-2). The proof of these propositions may be regarded as a " generalization " of the case N=II, treated in [34]- It proceeds closely along the lines of argument used for the case N = i i, but incorporates work we have already done concerning Eisenstein primes, and uses a recent result: Theorem (Imai [2I]). -- Let K be a number field (a finite extension of Q) and L/K the p-cyclotomic extension (L = V K(~pr)). Let AlL be an abelian variety. Then the torsion sub- group A(L)tor s of the group of rational points of A over L is a finite group. Correction. -- I am thankful to Ito for pointing out that an assertion I made in [34] (labelled (6.18)) is incorrect (for abelian varieties of CM-type of dimension greater than I). Therefore, my proof that A(L)t0r s is finite when A is of CM-type ([34] (6. I2 (i))) is incomplete. The theorem of Imai [2I] shows, however, that the result is valid for all abelian varieties. Discussion. -- Imai proves a local result based on Sen's analysis of the structure of the Lie algebra of a Galois group acting on a Hodge-Tate module [59]- Serre has communicated to me a proof along rather different (global) lines by means of which he obtains finiteness of the group of rational torsion points of the abelian variety A with values in many F-extensions over K not only the p-cyclotomic F-extension. We now prepare to prove (9-1) and (9-2) by the method of [34]- Let Ym denote the spectrum of the ring of integers in O~ and let Y be the spectrum of the ring of integers in O~ p). Thus Y0= S = Spec(Z). If j/y,, is the base change of the N~ron model J/s then it is the N~ron model of the jacobian of X0(N )tQ~p) since p is the only ramified prime, and J has good reduction at p. Let ~q (as in w 6) be a generator of the ideal ~s~ C T~ (chap. II (i6.6)). SO: Jv = lim jv [~r]/s is represented in this way as an inductive limit of quasi-finite group schemes over S, and is naturally endowed with the structure of Ts~-module. 167 I68 B. MAZUR We have the analogue of diagram (6.6) of [34], which may be written: o o (9.3) 0 > HI(Y~-p~,J,~) > H.(Y~,p~, J~) ---> H~(Y~, J~) > H~(Y~, J~) CXm 1 , H2:y T ~rm > Hl(y,j~)rm_____> Hl(Y__p~o j~)rm__~ .~ p~o,Jw o O where Pm is the unique closed point of characteristic p in Ym, P~ is the unique closed point of characteristic p in Y; Ym, pm is the completion of Ym at Pro, and Ypoo is the completion of Y at Poo ; the superscript F m means invariants under the action of I~,, and the subscript mean coinvariants; H 2. denotes cohomology with supports at the closed point. We view the above diagram, whose horizontal and vertical lines are exact, as a diagram of To-modules. There are three necessary calculations that must be made, in order to prove (9- I) and (9.2) and we collect them in the following lemma: Lemma (9-4) : i. J~(O~')) is isomorphic to T~/~. 2. Hl(Spec(Z), J~) = o. 3. For any m, the T~-module E,~ is (noncanonically) isomorphic to (T,/~)| (T~/~). Granted the lemma, we shall prove our propositions. Let I-I denote I-P(Y, J~) The lemma enables us to " evaluate " the above diagram regarded as A-module. for m=o: 0 0 T~I~ > (T~I3~) | (T~I~) H~(Yo --Po, J~) > H.(Yo,,o, J~) o > H r , HI(Y--p~,J~) r > W.(Y,~,J~) r o o 168 MODULAR CURVES AND THE EISENSTEIN IDEAL ~69 Note that T~/53 v is a cyclic group of order pr. From the above diagram, it follows that H r is a cyclic abelian group, hence, as Tv-module, a quotient of T~ by some ideal a C T~. It also follows from the above diagram that ~VC a. From this, we obtain the analogous information about H*, the Pontrjagin dual. Namely, H*| ~ is isomorphic to T~/a, as T~-module. Now consider the " descent sequence "" o , C~:~ ---~J~ --*J?~ ~ o 0 -----~ ~ ~ ~0 which is a sequence of sheaves of T~-modules for the fppf topology over Spec(Z)=Y0, or, after base change, over the schemes Y~, and Y. Here Cp~ Zip f is the p-primary component of the cuspidal subgroup and Ep (T t~pl noncanonicalty) is the p-primary component of the Shimura subgroup (chap. II (16.4)). The sheaf (P is representable by a nonseparated but finite &ale group (pre-)scheme whose support is concentrated at the prime of characteristic N, and whose fiber at N is a free Tv/~V-module of rank I. Noting that by (I) of the lemma the group H~ is generated by the appro- priate multiple of the point (o)- (oe), and as T~-module is isomorphic to Tv/~, one obtains that H~ A consequence of the above diagram is, then, that: (9.6) I-It(Y, C,| ---> H is an injection of A-modules. By a result of Iwasawa, the p-primary components of the ideal class group of the fields O~ ) vanish. It follows that: Hi(Y, Zip r) = o and, by " Kummer theory "" HI(Ym, ~pf)=U~/U~ I (all these cohomology groups being H2 (Ym, ~v~) = o fppf-cohomology) where U m is the group of units in the ring of integers of O~ ). By the Dirichlet unit theorem, Um/U~ I is a free (Z/pr)-module of rank pm--I. Replacing A by ~vs in the diagram (7-3) and evaluating (using that l~s(lj~)=o, He(Y,,, l~ps)=o and that H.(Ym, p,,, btpl) is dual to HI(Y,,,Vm, Z/pf)), one finds that: H~(Y~, ~,) -+ H'(,/, ~,~)~ is injective, for each m, with cokernel cyclic of order p( It follows that Hi(Y, ~vt) rm is a free (Z/pr)-module of rank pro. An application of Nakayama's lemma gives that the Pontrjagin dual of Hi(Y, btvl) is a free module of rank one over Z/pf[[P]] = A/~. A. Taking the Pontrjagin dual of (9-6), one gets a surjective map of A-modules: H* ~ ) A/~.A. 22 ~7o B. MAZUR Let R denote the kernel of the above homomorphism. Form the long exact sequence: Tor~(A/~?.A, T~) -+ R| v ~ H*| ~ -+ T~/Z~ ~ o. By the resolution o-+A-+A-+A/~A-~o, one sees that the Tor ~ term in the above sequence vanishes. Since H*| ~ is isomorphic to T~/a where a contains ~, it follows that R| vanishes as well. By Nakayama's Iemma one has R =o, and therefore : H* =~ A/~. A ~ Z/p~[[I']] as A-module. Proposition (9.~) is an immediate consequence of this, and Proposition (9. I) follows from Proposition (6. i I) of [39] and the theorem of Imai and Serre quoted above. Proof of Lerama (9.4). -- Part 1: Consider the filtration of jr over the base Spec(Zp) (chap. II (8.4)): o _+j~l~. t,p~ __>j~ ___> j~ ___> o. We show that the specialization map Jv(O~ p)) --+J~3(Fp) is injective by noting that Jv(O~p))nJ~Ult. type(Qp) vanishes. But the kernel of ~ in the latter intersection is just E(O~ p)) by chapter II (i6.4). It is zero, since E is a ~z-type group, and O~ p) does not contain the p-th roots of I. Since J~(O~ p)) f.~jsult, type(Q,p) is also killed by a power of 3, it must vanish. We shall conclude Part I by noting that J~(Fp)~ Cp, the p-primary component of the cuspidal subgroup. Sincep is not ap-th power rood N (and p ~ 2), ~p is a generator of 3~C T~ (chap. II (I 8. Io)). If r: is the unit root of X 2-TpX +p = o in T~ and 5 is the non-unit root, using the Eichler-Shimura relations and well known arguments (repeated in [39], w 4 d) and e)) one deduces that J~(Fp) is the kernel of I -- X in J~(Fp). But ~p = (I -- 7~) (I -- K) and therefore J~(Fp) is the kernel of ~ in J~(Fp). Part 2: Write out the descent sequence [39] (3-3) for the isogeny ~ on J: o . j [~;] ) j ) j0 ----~ o o ___~j0 .__.~j ~ 9 , o and the related long exact sequences for fppf cohomology. These latter we regard as exact sequences of T-modules and we tensor them with T~, which preserves exactness. We get: o---~ C~-~ M| T~--~ M~ TV --~H ~ (S, J, [~;]) --~H ~ (S, J) ~-~H t (S, jo) *-~W(S, j,~ [~;]) o--~M~174174 )H~ r )HI(S,J~ 170 MODULAR CURVES AND THE EISENSTEIN IDEAL I7i By (3.3), M| and M~174 are finite, and therefore by (I.2) we may evaluate them as follows: M| p and M~174 By chapter II (I6.4) , we have Jo[hp]=(Cp| Using the facts: Hi(S, Cp) = H~(S, Zp) = o for i = I, 2 and: H~ O) is free of rank I over To/~v, we may evaluate the above diagram for r = I and obtain the fact that ~q induces an isomorphism Hi(S, J) o -~ Hi( S, j0) o (from the top line) and the kernel of :qp in Hi(S, J) 0 is zero (from the bottom line). Consequently, Itl(S,J)o =Hl(S,J~ If we now consider the top line for general r, we have that Itl(S, Jo [~]) is flanked by groups which vanish and hence must vanish itself. Had [39] (5.7) been written in appropriate generality we would apply it directly to obtain what we wish. Part 3: As it is, we reconsider its proof. Let ofp denote the formal completion of J/sp~Izp)- Since Jv is naturally a T~=T| we have the decomposition (chap. II (7.1)) ~=Jo�162 using the idempotent decomposition I=r ~. We now prepare to copy the exact sequence of [39], Corollary (4.6). To convert to the notation of that Corollary, set A=J, L m = the completion of O~ / at the prime p,,, Din=the ring of integers in Lm, and, for some fixed m 0 set K=Lmo, D=Dm.. Then, for m=mo4-h (h_>o), Corollary (4.6) of [39] reads: ofp (D)/NLm/K ~(Dm) ~ J(K)/Ntr,/KJ(Lm) -+ J(Fp)/J(Fp) ph -+ o which is an exact sequence of T-modules. Tensoring with T o gives: (9.7) or (D)/NL,~/K oil o (D,~) ~ J (K)/NL,,/~: J (L~) | TO -+ J(F,)/J(Fp)~hGTTo --~ o But fV is a formal group of multiplicative type to which Corollary (4-33) of [39] applies, giving: The subgroups NL,,/~fo(Dm) C ~0(D) stabilize for large m, and: (9.8) ~o (D)/NL,,m tO (D,~)~ [r/P,,] | T0/(I -- ~). [F/P,] | where Fm=Gal(O.~P)/O~,P2) and rc is the unit root (which is the twist matrix [39], w 4 for J0)" Since the ideal in Tv generated by (1--r~) is just ~o, the above isomorphism yields that the left-hand T~-module of (9.8) is free of rank i over To/~ O for large enough m. Also, by the discussion of Part I, J(Fp)/J(Fp)Ph| is a free TV/3 v- module of rank i, if h is large enough. We now check that the left-hand map of exact sequence (9-7) is injective. This is as in Proposition (4.42) of [39]. Form the short exact sequence of Fmo/Pm-modules: (9.9) o ~ Jo(Dm) -+ J(L~)| -+ J(Fp)| -~ o 171 i7~ B. MAZUR and note that J(Fp)| is generated by the specialization of C~o which is contained in J(K)| ~. It follows that (9-9) splits as an exact sequence of Tv[Fm0/Fm]-modules. But the left-hand map of the exact sequence (9.7) is the map induced on o-dimensional Tate cohomology by the map of F,~./Pm-modules Jv(Dm)| appearing in the split exact sequence (9.9). Putting all the information we now have into the exact sequence (9.7) we obtain the following split exact sequence of Tv-modules : o -+ T~/~ v ->J(K)/NLm/KJ(Lm)QTT v -+ TV/~ v --> o for m large. We now apply Corollary (5.4) (P. 225 of [39]) and the discussion on page 226 to conclude that the kernel of: H.2(Ym, J~) -+ H2.(Y, J~) is a free module over T~/~ of rank 2. Added in proof (August 1977): I. Using results of the present paper, and some new techniques, the (Q-) rational points of X0(N ) can be completely determined for all prime numbers N. One finds that there are no noncuspidal rational points on X0(N), and hence no Q-rational N-isogenies, when N is a prime number ~>23, such that N+ 37, 43, 67, and I63. In particular the question-marks occuring in the TABLE of the introduction have been resolved. See: Rational isogenies of prime degree to appear in Invent. math. 2. An incorrect entry in a previous table of mine ([38], w 4) is corrected in the TABLE of the introduction to the present paper. Namely, when N= i99 , the data for g_ (in the table at the end of [38]) should read: 2 -t-10 and not: 2-1-t0. In particular, when N=I99 , J is not equal to J-. Therefore, remark 2 of [38], 2. 5 should be amended to read: J--J- for N<25 o, except when N=67, Io9, I39, I5I, I79, 22I and I99. 172 APPENDIX Behavior of the N@ron model of the jacobian of X0(N ) at bad primes by B. MAZUR and M. RAPOPORT Throughout this appendix we depart from the convention of the rest of this paper and let N denote a square free number not divisible by ~ or 3, and p a prime divisor of N. The connected component of the fibre at p of the Ndron model J of the jacobian of the modular curve Mo(N ) (chap. II, w I) was determined in [9]- Our purpose here is to get somewhat finer information about J, in particular about the finite abelian group : of the connected components of the fibre at p of J. The following theorem, which is the main result of this appendix, is due to P. Deligne: Theorem (A. x). -- Let N=p be a prime number. a) The connected component jo of the fibre at Fp of J is a group of multiplicative type; considering it over Fv, the Frobenius endomorphism acts on the p-adic Tare module: as: F* ~ --p. w, where w is induced from the canonical involution (z~--i/pz). b) We have a canonical decomposition of the fibre at p o f J: J,=J~ where C is a cyclic group of order num((p--i)/I 2) generated by the class of the divisor (o)- ( oo). More generally, write: N=p.qi, ..., q~ (allowing for v = o to include the case N =p). The connected component of the fibre at p of J is an extension of j0(ql, ..., q~)p �176 ..., q~)p by a group of multiplicative type (cf. [9] and section I below). As for the group @ =q)p of connected components of the fibre at p of J one has table 2 below: 173 B. MAZUR TABLE 2 Order of (o)--(oo) Structure of Order Relations satisfied by "standard " elements of (I) (u, v) in ~ q)/(o)--(~) ofO Structure of O (O, 0) o~.P~2 1 trivial Q.. P~2 I Z/(Q..~2 I) z = (o)-- (oo) is a generator (I,0) Q .p-~- i Z/2~`0_1 z Q..p-- I 22`0 q) is generated by the Ei 6 12 (i = I,.,., 2'~); (~ Z/2 2~-2 Z relations: = 2E i (i = I ..... 2 `0 ) p--I (o, I) Q.p-- i Z/a~_iZ Q. ~ .32~ is generated by the Gi (i= I, ..., 2`0); | Z/3~- i Z relations: E~ =--s'.g ~ =.g~l . } (~ = i .... ,2`0) 2 ! (I, I) Q.p--I Z]62`0_1 z O.P--I .6~` o Z/(O, p-- 1)Z (I) is generated by the E i ,~Gj 2 I2 (i, j = I, ..., 2"r | Z/2Z`0- 2 Z relations: @ Z/3 2"- 1 Z i 3 = ~i (i = I .... ,0`0) Yj = oui/= 1 ~ ,J = i .... ,2`0) Notation: a) Set u = I if: P-7 or Ii(mod I2) and : all q~--- I (mod 4) i = I, . .., v otherwise set u =o. b) Set v=I if: P-5 or II(mod I2) and: all qi~I(mod 3) i=I, ...,v; otherwise set v=O. 174 MODULAR CURVES AND THE EISENSTEIN IDEAL I75 c) Set Q= __IIl(qi+I ) (=I if v=o). d) The last column gives information about "standard elements " in q) (cf. section 2). In particular Z=(o)--(oo). Remarks. -- I) In table 2, Z is an element (but not necessarily a generator) of the first cyclic group occurring in the column labelled " structure of @ " Z is a generator of this cyclic group if v = o or if (u, v) is (o, o) or (o, I). In all other cases Z is twice a generator. 2) The table shows that, ignoring 2- and 3-primary components, ~bp is a cyclic group generated by the image of the divisor class (o)--(oo). Its order (again ignoring products of powers of 2 and 3) equals Q. (p-l). The order of (o) --(oe) in J is divisible by the 1.c.m. of the orders of qbp, for all p dividing N. G. Ligozat has computed this order (as yet unpublished). The plan of exposition is the following. In section I we recall relevant results from [9] about the moduli schemes of interest. After recalling results of Raynaud [5 6] about the relation between the jacobian of the minimal model of a smooth curve over a discretely valued field and the N6ron model of its jacobian, we reduce our problem to a computation. This computation is outlined in section 2. The final section 3 proves a) of Theorem i. x. Relation between Tnlnimal model and N6ron model. The following is a somewhat simplified version of ([9], VI (5-9))- Set N'=N/p. Theorem (I.I). -- a) M0(N ) is smooth over Z[I/N'] outside the supersingular points in characteristic p. b) M0(N ) | is the union of two copies of M0(N' ) | crossing transversally at the supersingular points. If x ----j(E, H) is a supersingular point of M0(N') | (i.e. E=super- singular elliptic curve and HC E[N'] a cyclic subgroup of order precisely N'), then x on the second copy is glued to the point x (p) of the first copy of M0(N' ) | c) Let x =j(E, H) be a supersingular point of M0(N' ) | and set: k = I [ aut(E, H) I. At the corresponding point of M0(N)| p the scheme M0(N ) has a singularity whose strict localization is isomorphic to: W(F,) [ IX, Y]]/(X. Y--f) (i.e. is of type Ak_l). 175 r7 6 B. M AZUR d) In particular, the reduction modulo p of the minimal model X0(N ) 0fM0(N ) (over Z[I/N']) is obtained by glueing two copies of Mo(N') | at corresponding supersingular points, and then replacing a crossing point by a chain of k-- I projective lines. If p # 2, 3 (which we will always assume), then: k>I implies either: j(x) =o, and then k = 3 or: j(x) = 1728, and then k = 2. Those projective lines, considered as divisors on the minimal model, have self-intersection --2. Our next task is to determine the number of supersingular points explicitly. Let: S'=the number of supersingular curves E over Fp with j(E)+o, 1728. if there exists a supersingular curve E over Fp with j(E)---- 1728. , ={'o otherwise. if there exists a supersingular curve E over Fp with j(E) =o. otherwise. Recall [1, VI (4.9)] that: S'+~.I+ I.R= p-I . 3 12 Recall from the introduction that Q= i II=1 (qi + I). M0(N ) | lying above a supersingular Proposition (i .2). -- (i) The number of points in point xzMo(P)| p is: Q if j(x)#o, I728 Q if j(x) = 1728 but not all qi- I (mod 4) I_ (Q_ 2~ ) iof j(x) =1728 and all qi - i (mod 4) but not all q~- i (mod 3) I_Q if j(x) =o and all q~- i (mod 3). if j(x)=o Hence: (ii) S'= number of supersingular points x in Mo(N ) | ~'p with j(x)+-o, ~ 728 _Qp;i ~[u (for u, v consult the introduction to this appendix). 176 MODULAR CURVES AND THE EISENSTEIN IDEAL t77 Pro@ -- (i) is a consequence of the following facts: a) The morphism M0(N)| p --~ M0(t)| p is a covering of degree Q. b) Let j(E) 1728 and let (E, I-I) correspond to a point in X0(N')| p. If Aut(E, H)4={ }, there is a primitive 4-th root of unity in (Z/qi)* (the automorphism group of the qi-primary component of H) for each i: I, ..., v, i.e.: qi= I (mod 4) i=I, ..., v. c) Similarly, if (E, H) corresponds to a point in M0(N' ) | with j(E)= o, and if Aut(E, H) 4={-4-I}, then there is a primitive 6-th root of unity in (Z/q~)* for each i=I, . .., v, i.e.: q~i (mod3) i=i, ...,v. (ii) follows from (i) by taking into account the formula recalled shortly before the statement of the proposition and the fact that: j= o is supersingular if p--I (mod 6) j=~728 is supersingular if p-I (mod 4). Q.E.D. We obtain the following picture for the reduction modulo p of Xo(N): Z,~, r ! Zt X i -- = Xo(N ) | 0(N ) E1 i 9 these are present 9 ~ if and only if u= I E2'~ ) these present 9 ~ 9 these are present L li, -~v=I 9 ~ if and only if v= F~ J ~ G~ We now recall results of Raynaud [5 6] which will allow us to pass from The- orem (i. i) to Theorem (A. I) and its variants. 23 I78 B. MAZUR Let: K= discretely valued field, complete for the valuation. R = ring of integers in K, k = R/(rc)= residue field (assumed algebraically closed). S =Spec(R), ~q and s its generic and closed points respectively. C = a curve, smooth, geometrically irreducible and proper over K. f : W---~ S = minimal model of C over R. (Recall that q~ is the (unique) regular scheme, proper and flat over R, with generic fibre ~ =C such that for any other regular scheme 5' flat over R with generic fibre rg~=C, the birational map ~'--~W is a morphism.) of=jacobian variety of C=Pic~ J =Ndron model of of. (Recall that J is the (unique) group scheme smooth over R such that for every other smooth group scheme J' smooth over R, any K-morphism J'~J~ comes from a unique R-morphism J'~J.) The following result of Raynaud gives the connection between ~ and J: Theorem (1.3)- -- With the above notations, assume that d=g.c.d, of all multiplicities d~ of the irreducible components Ci of ~ is equal to I. Then: J ~ Pic~]s/E, where: P;c E~ =kernel of the morphism " degree " deg : Pice/s-+Z ~r and: E = scheme-theoretic closure of the unit section in Pic~e/s. (This result is not stated in this form in [56]; it is a consequence of the results in that paper (we adhere to the terminology of [56]): a) f verifies condition (N) and we have: f. (d)~e)= ~)s, hence f is cohomologically flat in dimension o ([56] (7.2. i)). b) Pic~/s is representable by a formally smooth algebraic space in groups; and Pic~/s is represented by a separated smooth group scheme ([54] (8.2. i)). The quotient: Q= Pic~e/s/E is representable by a separated smooth group scheme over S ([56] (8.o.1)). c) The group scheme: Q*=inverse image in Q of the torsion part of Q/Q0 is the Ndron model of of= Pic~ d) The morphism: 9 [o1 elc~/s~Q2 178 MODULAR CURVES AND THE EISENSTEIN IDEAL is surjective, with kernel E (cf. [5 6] (8. I. 2)); hence Pi@)s/E_~Q" is the Ndron model of J.) We extract from [5 6] (8. i. 2) the following additional information: Proposition (I.4). -- Let D_~Z" be the free abelian group generated by the irreducible Let D*=Hom(D, Z) be the dual group. Define: componen~ C~ of~. o~ : D-+D* and: : D*-+Z E I-. deg(~[ q). C~, identifies: Then ~oe-=o; and, sending ~q~ePic(~') to i d i Js/J~ To apply these results in our case we note that we may pass to the algebraic closure F~ of Fp since formation of N~ron models (respectively of minimal models of curves) commutes with ~tale base change (Fp is perfect). 2. Calculation of the table. We use the Proposition (I.4) of section I. The irreducible components of the reduction modulo p of X0(N), the minimal model of M0(N ), are Z, Z', El, Fi, G i (i=I, ..., 2 ~) (with the convention that Ei, respectively F~ and G~, are missing if u = o, respectively v = o). They all have multi- plicities equal to one. Hence: (2.i) D=free abelian group generated by Z, Z', El, Fi, Gi. Let D0=ker(~)=elements in D of degree o (cf. section I, Proposition (1.4)). Let D* and D O be their respective dual groups. Then: (2.2) A basis of D; is given by: Z = Z*--Z'* = z'* V;-- Z'" G~= G~-- Z'*. 179 I8O B. MAZUR The intersection products, as read off from the configuration of divisors given in section I, determine the self-intersection numbers: (2.3) Z.Z =--(S'+ 2~.u+2~.v) Z'. Z'=-- (S'+ 2~. u + 2~. v). Hence, since 9 = Do/Im(0~ ) : (2.4) (I) = D0/modulo the relations m. EEi+n. ZFi-(S'+ 2~m + 2~n).Z-=o i i m. EEl+ n. ZC,~ + S'.2- o i i 2-2.g~-o n. 2- 2n~ + n~ ---- o 2nG,-- nF, =- o. (2.5) The order of (I) equals the absolute value of the determinant of the intersection matrix of Z', Ei, Fi, G i. To fill in the table we distinguish cases: 1st case: (u, v) = (o, o) Here (I) =Z.Z/S - '- Z, hence its order is S'=Q. p- I ; (I) is generated by Z. 2nd case: (u, v)=(o, I) The order of (I) equals the absolute value of the determinant of the following intersection matrix: Z' F 1 Gj F 2 G 2 F2~ ' G~v Z t --(S' + 2 ~) o i o i o I O --2 I O O 0 o F1 I I --2 O O o o G1 0 o O O O --2 I F2 G~ I O 0 I --2 o o o 0 0 0 F2v O O O O 0 --2 1 O O 0 0 I --2 G2~, 180 MODULAR CURVES AND THE EISENSTEIN IDEAL x81 Adding to the Z'-row: 2 (sum over Gcrows ) (sum over F,-rows) -t- gives as new Z'-row" --S'--2".-, O, O, . .., O; hence the determinant equals: act =--(S' + 3.2~)3~ =- 32~-1(3. S ' +2~). The relations (2.4) allow us to eliminate Fi, and the Ei are absent: ( )/ (I)/(cyclic subgroup generated by 2)~Z/32~-lZ. The order of Z in @ is thus: Hence 3.Q. (p--I)/I2; since this number is prime to 3 (because, if /)=i, then P---5 or 7 (mod I2) and qi-I (mod3) , i=I,..., 2~), the cyclic subgroup of 9 generated by Z is a direct summand. 3rd case: (u, v) = (I, o) The order of 9 equals the absolute value of the determinant of the following intersection matrix: Z" E 1 E~ E2v Z p --(S' + 2 ~) i I i I --2 0 0 E1 E~ I 0 --2 0 E~v I 0 0 --2 Adding the Z'-row to I. (sum of the Ei-rows ) one obtains as new Z'-row: --S '-~-.2 ~, O, O, ..., O, hence: et= 181 x82 B. MAZUR The relations (2.4) become in this case: E g~(S'+ 2 ~) .Z=o EE~+S'.2 -o --2E~+Z =o. Hence ~/(cyclic subgroup generated by Z)=~Z/2~-tZ. The order of Z is thus 2.Q.(p-I)/I2. If v>I, then the cyclic subgroup generated by Z is not a direct summand of @ but is of index 2 in a direct summand. 4th case: (u, v) = (I, i) The order of 9 equals the absolute value of the determinant of the following intersection matrix: Z' E 1 . . . E2'~ 171 G 1 . . . F2v G2~ Z t --'~t~' --~]~ 2~ .xl), I . . . I 0 I . . . 0 I l --2 0 ... 0 0 0 . . . 0 0 E1 0 --9 0 0 0 . . . 0 0 I O . . . --~2 0 0 . 9 . 0 0 E2v Fa 0 0 . . . 0 --9 I . . . 0 0 131 I 0 0 I --2 0 0 0 0 0 0 0 --9 ! F2,0 I 0 0 0 0 I --2 G2v Add to the Z'-row: I (sum of Ei-rows ) +3I (sum of Fcrows ) + 3 (sum of G~-rows) to get as new Z'-row: _8,+52~=_S, 2~+ ~ i 2~ 2 6 +~' +~ "2~' o,o,.. ,o. 182 MODULAR CURVES AND THE EISENSTEIN IDEAL i83 Hence: det =--62~(S'+ 5 . 2~). The relations (2.4) become: 7,--2-g~-3-Gs_ _ '. ) ZE~+ZGj-=--S 2 (i,j=i,...,2~). Hence q)/(cyclic subgroup generated by ]~)-~Z/62~-lZ. The order of 2 in is thus 6.Q.(p--I)/I2. If v>I, the cyclic subgroup of 9 generated by Z is not a direct summand of 9 but is of index 2 in a direct summand. In conclusion, we have filled in all entries of the table; sections I and 2 also prove Theorem (A. i) except for the statement about the action of Frobenius on J~ 3" The Frobenlus action. Let N=p be a prime number. Denote by F the following graph: vertices = components Z, Z' edges-= supersingular points (joining Z and Z') F: Z~ * 9 Z~ Z' There is a canonical isomorphism (cf. [9]): J~174 ~ Hi(F, Z) | The action of the Frobenius endomorphism of J~174 may be identified with: ~| : Hi(P, Z)| z)| where ~ : F-+ I" is the map which fixes the vertices and which sends a supersingular point x (corresponding to an " edge " of F) to the unique supersingular point x'=o~(x) such that j(x')=j(x) p. But the map a induces the endomorphism --w on t-P(P, Z), because ~ is the composition of w with the automorphism of F which interchanges the vertices and keeps the edges fixed. Hence: F----- ~| =--p.w. This proves part a) of Theorem (A. I). 183 MODULAR CURVES AND THE EISENSTEIN IDEAL I85 [27] KUBERT (D.), Universal bounds on the torsion of elliptic curves, Proe. London Math. Soc. (3), 33 (1976), i93-237. [28] KImERT (D.), LANa (S.), Units in the modular function field, I, II, III, Math. 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[74] SERR~ (J.-P.), TATE (J.), Good reduction of abelian varieties, Ann. of Math., 88 (I968), 492-5x7 . [75] TATE (J.), Algorithm for determining the type of a singular fiber in an elliptic pencil, vol. IV of The Proceedings of the International Summer School on Modular Functions, Antwerp (i972), Lecture Notes in Mathematics, 4'/6, Berlin-Heidelberg-New York, Springer, I975. [SGA 3] Sdminaire de Gdomdtrie algdbrique du Bois-Marie, 62-64. Directed by M. DEMAZURE and A. GROTHENDIECK, Lecture Notes in Mathematics, Nos. 151, 152, 153, Berlin-Heidelberg-New York, Springer, I97o. [SGA 7] Sdminaire de G6om~trie alg6brique du Bois-Marie, 67-69. P. D~LmNE and N. KATZ, Lecture Notes in Mathematics, Nos. 288, 340, Berlin-Heidelberg-New York, Springer, ~972, I973. Manuscrit refule 16 avril 1976.

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