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Residually reducible representations and modular forms

Residually reducible representations and modular forms 6 C.M. SKINNER, A d. WIIJ'S 1. Introduction In this paper we give criteria for the modularity of certain two-dimensional Galois representations. Originally conjectural criteria were tbrmulated for compatible systems of ~.-adic representations, but a more suitable formulation for our work was given by Fontaine and Mazur. Throughout this paper p will denote an odd prime. Conjecture (Fontaine-Mazur [FM]). -- Suppose that p 9 Gal(Q,/Q) , GL2(E ) is a continuous representation, irreducible and unrami~ed outside a ~nite set of primes, where E is a/~Tite e~Tension of Q#. Suppose also that (i) PlI# ~' ( ; t )' where Ip is an inertia group at p (ii) det p = ~s ~- 1 for some k >1 2 and is odd, where ~ is the cyclotomic character and ~ is of ~nite order. Then p comes from a modular form. To say that p comes from a modular form is to mean that there exists a modular form /with the property that T(g)/= tracep(Frob~)/for all g at which p is unramified. Here T(g) is the g,h tIecke operator, and an arbitrary embedding of E into C is chosen so that tracep(Frobe) can be viewed in C. Fontaine and Mazur actually state a more general conjecture where condition (i) is replaced by a more general, but more technical, hypothesis. The condition which we use, which we refer to as the condition that p be ordinary, is essential to the methods of this paper. If we pick a stable lattice in E 2, and reduce p modulo a uniformizer )~ of ~:E, the ring of integers of E, we get a representation ~ of Gal(Q,/Q) into GL2(~-' E/~.). If is irreducible, then it is uniquely determined by p. In general we write Oss tbr the semisimplification of ~, and this is uniquely determined by p in all cases. Previous work on this conjecture has mostly focused on the case where ~ is irreducible (cf. [Mill, [D1]). In that case the main theorems prove weakened versions of the conjecture under the important additional hypothesis that ~ has some lifting which is modular. This hypothesis, which is in fact a conjecture of Serre, is as yet unproved. In this paper we consider the case where ~ is reducible, and we prove the following theorem. Theorem. -- Suppose that p 9 Gal(O~/Q) , GL2(E ) is a continuous representation, irreducible and unramilied outside a tinite set of primes, where E is a tinite extension of Qp. Suppose also that ~' ~_ 1 | x and that (i) XlD~ ~: 1, where Dp is a decomposition group at p, (ii) PIt, -~ (0 1)' (iii) det p = ~s k- t for some k >1 2 and is odd, where e is the cyclotomic character and ~r is of linite order. Then p comes from a modular form. RESIDUALLY REI)U(I[BLE REPRESEN'IATI()NS AND MOI)ULAR FORMS We also prove similar but weaker statements when Q is replaced by a general totally real number field: see Theorems A and B of w Ill the irreducible case the proof consists of identifying certain universal deforma- tion rings associated to ~ with certain Hecke rings. However in the reducible case even for a fixed ~s _~ 1 (9 Z we have to consider all the deformation rings corresponding to the possible extensions of Z by 1. These deformation rings are not nearly as well- behaved as in the irreducible case. They are not in general equidimensional. Indeed there is a part corresponding to the reducible representations whose dimension grows with Z, the finite set of primes at which we permit ramification in the deibrmation problem. Just as in the irreducible case, we do not know whether there is an irreducible lifting for each extension of Z by 1, but happily we do not need to assume this. In a previous paper [SW] we examined some special cases where we could identify the deformation rings with Hecke rings. These cases roughly corresponded to the condition that there is a unique extension of 1 by Z- In this paper we proceed quite differently. In particular we do not identify the deformation rings with Hecke rings. As we mentioned earlier, we consider the problem over a general totally real number field. This is not just to extend the theorem but is, in thct, an essential part of the proof. For it allows us by base change to restrict ourselves to situations where the part of the deibrmation ring corresponding to reducible representations has large codimension inside the full deformation ring. It should be noted that the base change we choose depends on Z. We now give an outline of the paper. In w we introduce and give a detailed analysis of certain deformation rings R~. These are associated to an extension c of Z by 1. They are given as the universal deformation ring of the representation where the implied extension is given by c. Here Qz is the maximal extension of Q unramified outside Z and oc although in the main body of the paper Q is replaced by a totally real field F. More precise definitions are given in w In w we give a corresponding detailed analysis of certain nearly ordinary Hecke rings introduced by Hida. We say that a prime of R,/ is pro-modular if the trace of the corresponding representation occurs in a Hecke ring in a sense that is made precise in w If all the primes on an irreducible component of Re/ are pro-modular then we say that the component is pro-modular. If all the irreducible components of R~ are pro-modular then we say that R2/ is pro-modular. The above theorem is deduced from our main result which establishes the pro- modularity of R@ for suitable c,,~. There are three main steps in the proof of this latter result: (I.M. SKINNF, R, AJ. WII,ES (I) We show that if p is a "nice" prime of R~ then every component containing p is pro-modular. (The definition of a nice prime is given in w it includes the requirement that p itself is pro-modular). (II) We show that Rj has a nice prime p. (III) We show that Rj is pro-modular. The proof of step (I) is modelled on that for the residually irreducible case and is given in w The point is that the representation associated to R~/p is irreducible of dimension one and pro-modular. However tile techniques of the irreducible case have to be modified as this representation, which we now view as our residual representation, takes values in an infinite field of characteristic ,0. We should note also that the analog of the patching argument of [TW] is here performed on the detbrmation rings rather than on the Hecke rings. The proof of step (III) is given in Proposition 4.1. Steps (I) and ([I) show that some irreducible component at the minimum level is modular. Then we use a connectivity result of M. Raynaud (see w to show that there is a nice prime in every component at the minimum level. By step (I) again we deduce pro-modularity at the minimum level. A more straightforward argument then shows that there is a nice prime in every component of R~, so that we can again apply step (I) to deduce pro-modularity. For step (II) we proceed as fi)llows. First we show, using the main result of w (which in turn uses techniques for proving the existence of congruences between cusp forms and Eisenstein series), that R(/ has a nice prime tbr some extension Co of Z by 1. Using commutative algebra we show that there are primes in the subring of traces of R~ which correspond to representations with other reduction types, i.e. corresponding to a different extension c (the pair 1, )~ are fixed though). We make a construction to show that we can achieve all extensions in this way, and hence find nice primes for all extensions c. These primes are necessarily primes of the ring of traces which do not extend to R(/ itself. The proof of step (II) is given in Proposition 4.2. At the start of the proof of this proposition is a more detailed outline of how we carry out step (II). We now briefly indicate the extra restriction in the case of a general totally real field F. We need to be able to make large solvable extensions of F0~), the splitting field of )r with prescribed local behavior at a finite number of primes and such that the relative class number is controlled. When F(Z ) is abelian over Q we can do this using a theorem of Washinoon about the behavior of the p-part of the class number of Zt-extensions. In the general case such a result is not known. Finally we note that the ordinary hypothesis which is essential to our method is frequently satisfied in applications. I~br example, suppose that 9 (with ~ reducible) arises as the ~,-adic representation associated to an abelian variety A over Q with a field of endomorphisms K~--~ EndQ(A)| Q such that dim A = [K : Q.]. Then the nearly ordinary hypothesis will hold provided A is semistable at p, or even if A acquires RESIDUALLY REI)UCIBLE REPRESENTATIONS AND MOI)ULAR FORMS 9 semistability over an extension of Qp with ramification degree < p- 1. This can be verified by considering the Zariski closure of ker(~.) in the Neron model of A. 2. Deformation data and deformation rings 2.1. Generators and relations Let F be a totally real number field of degree d. For any finite set of finite places Z, let F~ be the maximal extension of F unramified outside of X and all vle~. For each place v, fix once and for all an embedding of P into ]~v. Doing so fixes a choice of decomposition group Dv and inertia group Iv for each finite place v and a choice of complex conjugation for each infinite place. Let zx, ..., ze be the d complex conjugations so chosen, and let vl,..., vt be the places dividing p. Write Di and Ii for the decomposition group and inertia group chosen for the place vi. Let di be the degree of Fv~ over Qp. Normalize the reciprocity maps of Class Field Theory so that uniformizers correspond to arithmetic Frobenii and write Frob fi)r a Frobenius at a place v. Suppose that k0 is a finite field of characteristic p and that )~ : GaI(F/F) ~ k~ is a character such that 9 )~l~i ~ 1 for i= 1,...,t 9 )~(zi) = - 1 for i = I, ..., d. A deformation datum for F is a 4-tuple ..~ = (~(5 ~ , E, c, ../A~) consisting of the ring of integers C of a local field with finite residue field k containing k0, a finite set of finite places E containing all those at which 3~ is ramified together with ,~ = {vl, ..., vt}, a non-zero cohomology class res t'TN TTI/T'~ (2.1) 0 ~ cE ker {H~(Fx/F, k(~-~))----+ t~l;~n/~i, k(X-~))}, i= 1 and a set of places ~ C X\?) '~ at each of which either c is ramified or ZII,, is non- trivial. For future reference write Hx(F, k) for the kernel of the map in (2.1). A cocycle class c E Hx(F, k) is called admissible. Let F(g) be the splitting field of ?~. There is a canonical isomorphism (via the restriction map) HI(Fx/F, k(~-L))~, HI(Fx/F~), k(~-l)) Gal(F0~)/I'~. Using this identification, one sees that for any cocycle c there is a unique representation 9,:" Gal(Fx/P; ' GL2(k), 9~ = X 10 C.M. SKINNER, A.J. WILES such that 9~(z~) = ( 1 -1) p~(a)= (1 c(1)) for ~E Gal(Fx/F(; 0). If c is admissible, then Pc also satisfies (1) i= 1,...,t. P, IDi -~ Z ' A deformation of p~ is a local complete Noetherian ring A with residue field k and maxima] ideal mA together with a strict equivalence class of continuous representations p 9 Gal(F/F) ~ GL2(A ) satisfying p~ = pmodmA. Such a deformation is oftype-~ if 9 A is an C-algebra, 9 P is unramified outside of Z and the places above 0% v~i) with )~ = ~q mod ma for each i, and 9 ( ") 2 for each w E .//g. Here ~ denotes the Teichmtiller lift of X to A. We usually denote a deformation by a single member of its equivalence class. For any deformation datum ~, there is a universal deformation of type-~ p.g 9 GaI(Fz/F) , GL2(R~,~ ). We omit the precise formulation of the universal property as well as the proof of existence as these are now standard (see [M], [R], [W1]). A totally real finite extension F' of F is permissible for ~ provided 9 o, loal(~/v", is non-split; 9 if v E dg and X]I,, =~ 1, then ZII. ~: 1 for each place w of F' dividing v; 9 if v E./E~ and P,[I~, q= 1 but zlI~ = 1, then P~lI,, =~ 1 for each place w ofF' dividing v ; 9 if w is a place of F' dividing p, then ?~lD~, :~ 1. Remark 2.1. - - If F' is permissible for ~, then ~ determines a deformation datum ~' = (L "t2j , Z', c, ./f/~') for F' with Z' and ~fd' being the sets of places of F' dividing those in X and ,/dd~, respectively. Clearly, if p " Gal(F/F) , GL2(A ) is a deformation of type-~, then Pl(;al~F/v'i is a deformation of type-.~'. In this subsection we give a preliminary analysis of the structure of Ry as an abstract ring. ~Ib start, we analyze the versal deformation rings associated to RESII)UALLY REDUCIBI,E REPRESENTATIONS AND MODULAR FOR-MS representations p : D: ~ GL~(A) satisfying p modmA = )C | 1 and det9 = ~. Such a deformation is a local ~:-deformation if A is an ~'-algebra, and it is nearly ordinary if ( ") in addition p _~ g~' ~r with 1 = tg mod mA. Applying the criteria of Schlessinger as in [M], one sees that there is a versal local ~:-deformation and a versal nearly ordinary deformation 9:': D~ , GL2(R':; ) and 9oi{.a "D,. , GLg(R~'~a) respectively. The representative [ord(~) can be chosen so that Pord = ~u- 1 9 The following lemma gives a ring-theoretic description of Rot d. Lemma 2.2. -- Let co be the character giving the action of D: on the pth roots of unity. There is an isomorphism " ..., = = R(,) , ,~, [[Xl, x24+2]]/(f) if )~ II)i 03 or if m 1, {~ --or~ -- ~i, [[xl,..., xzu/+l]] otherwise. Proof. -- Our proof follows along the lines of that of [M, Proposition 2]. Let V be the representation space of p01Di where P0 = 9c with c = 0. Clearly, V ~ k | k(x ). Let ado() = Hom~(V, V) be the adjoint representation, and let ad~ be the submodule consisting of homomorphisms whose trace is zero. The reduced tangent space of ~(') a~-or d has dimension equal to r = dim k ker { U 1 (Di, ad~ , H [ (Di, Homk(k(~), k)) }. A simple calculation using local class field theory and local Galois duality shows that 2di + 2 if X IE~, = co or if co = 1, r = 2di + 1 otherwise. It follows that R"o';~d is a quotient of the power series ring P = ~" [[x~, ..., x(l] by some ideal I. Consider the exact sequence 1~(') 0 ~ I/mI , P/mI ~ ""ord ~ 0 ~('? for deformations where m is the maximal ideal of P. The universal deformation ring ~-1 of the trivial character satisfies gy,, ...,A]/(h) 12 C.M. SKINNER, A d. WILES where (2.2) s= dimkHl(Di, k) and if co =~ 1 then h= 0. This follows immediately from local class field theory. There is a natural map o/,~ corresponding to the character ~g. Choose a compatible homomorphism R('l ~ ~ ~"ord ~' ~Y~,...,Ys] ' P/mI. This induces a continuous character * : Di ' (P//mI, h))� projecting to q~. Choose a (continuous) set-theoretic map 0:D~ , GL 2 (P/(m], h)) _ (,) projecting to Pond such that * ) Define a 2-cocycle ~: Di - ' I/(mI, h)(z) by 0(~j, a2)0(~2)_, 0(6L)-1 = (1 ~(~1~ r ) and consider its class ['/] in H2(Di, (I/(mI, h))~))-~ H2(Di, k(z))| h). The map (2.3) (I/(mI, h))* , H2(Di, k(Z)) , f~----. (1 | is injective. Here the superscript '*' denotes the k-dual. For if f E (I/(mI, h))* maps to zero, then 7mod(mI, h, ker f) equals d~ fbr some map ~ : Di , (I/(mI, h, ker f))(~), and 0' = (1 ~)0 is a representation into GL~(P/(mI, h, ker f)) that is clearly a nearly ordinary deformation. By the versality of R~ d there is then a homomorphism R(o0rd ~ P/(mI, h, ker f) inducing 0', and its composition with the ~(z? is an isomorphism. Comparing maps on reduced projection P/(mI, h, ker f) --, ~'ord tangent spaces shows that ""ord D(z) -- ~ P/(mI, h, ker f), which is possible only if f= 0. Let g = dim k I/mI. This is the minimal number of generators of the ideal I. By (2.2) and the injectivity of (2.3), 1 if ~=1 g ~< dim k H2(Di, k~)) + 0 otherwise 1 if co=l dim~I[~ k(z-lc~ + 0 otherwise. I if co=l orif )~=o~ 0 otherwise. This proves the lemma. [] RESIDUAI,LY REDUCIBI,E REPRESENTATIONS AND MOI)ULAR FORMS 13 r~('; of R ('~ ideal generated by Corollary 2.3. -- The ring L,or d i,Y a quotient by an 2 if ~ll), = o) = ~-1 ]I), d, + 1 if ZID~ = O) :~ Z-IID~, or o~= 1 zl,,i ,o = Z- ID , 0 otherwise e/ements. Proof. ---The ring R (i) is a quotient of C'~ [[YL,-..,Ye]] with r' = dim kHI(Di, ad~ 3 if ~=0~=Z -1 =3di+ 2 if 0~=1 or ~=0~fik~-I or ~:~o~=)U 1 1 otherwise. Combining this with the previous lemma and the fact that "o~d ~('~ is a quotient of R/'l yields the corollas. [] The above lemma and its corollary, together with minor variations of the methods used to prove them, yield the following ring-theoretic description of Re. Let ~- be the Zp-rank of the Galois group of the maximal abelian pro-p-extension of F unramified away from primes above p. Proposition 2.4. Suppose that ~ = (8, Y., c, J~) is a deformation datum. There exist integers g and r, depending on 5~, such that Re Ix,, ..., ...,frr) and g- r/> d+ fly - 2t- 3 9 #rig. Recall that t is the number of places of F dividing p. Proof. -- First we introduce an auxiliary deformation problem. A deformation P : Gal(ff/F) , GL2(A ) of 9c is of auxiliary O~pe-~ if 9 A is an ~-algebra 9 det p = 9 p is unramified outside of Z and the places above oo. There is a universal deformation of auxiliary type-~.~ RI~X aLIX 9e " Gal (Fz/F) ~ GL2(R e ). 14 C.M. SKINNER, AJ. WILES &tax aux Clearly, there are natural maps q0i " R('; ~ R.~ corresponding to P.~ ID, for i = I, ..., t. Let J2 be the kernel of the projection R(') ' and let J be the ideal generated by ug)i(Ji). It follows from Corollary 2.3 that (2.4) J is generated by ~_~(di + 2) = d + 2t elements. t'= l Now p~modJ is clearly a deformation of type-~.~ ', where c~, = (~:, y., c, ~). Using the versality of the various rings one finds that R~, ~ R~X/J | ~2 [[Gal(L(Y.)/F)]] (2.5) where L(Z) is the maximal abelian pro-p-extension of F unramified away from Z. (One difference between deformations of type-6_~ r' and auxiliary deformations of type-~ is that the former include deformations of the determinant whereas the latter do not.) It is easy to see that there is an isomorphism (2.6) [[Gal (L(X)/F)]] ~_ ~' [[x~ , ..., x~ r , y, , ..., y~]/(g, , ..., g,). Let I be the kernel of R~, ~ R~. For each v G ~ let % E I~ be a generator of the p-part of tame inertia. Choose for each v E ,/tg a basis for 9~' such that P~'(%) m~ = (' *)1 . Write P~'(%) = (a~c~ d~b~) with respect to the basis. Clearly, I is generated by the set { a~ - 1, d~ - 1, q, 9 v E ,.rig }. It follows that (2.7) I is generated by 3. #~/fg elements. Arguing as in [M, Proposition 2] shows that 1Ix,, ..., ...,&) where e' = dimk H' (Fz/F, ad~ r' ~< dim~ H2(Fz/F, ad ~ Combining this with (2.4), (2.5), (2.6), and (2.7) shows that ..., x ll/(j5, ...,f) where g=g'+S+~F and r=r'+s+d+2t+3.#Jg. The desired bound for g- r is a consequence of the global Euler characteristic formula for ad~ [] --o~aP('?, RESIDUALIN REI)UCIBI,E REPRESENTATIONS AND MODULAR FORMS 15 We conclude this subsection with two simple facts about deformation rings. Suppose that ~ is a deformation datum. Fix a basis for p~. With respect to the basis write p~ (~)= (aoco ~) for each (~ E Gal(F/F). Let R' C_ R~ be the ~;-subalgebra generated by { do, bo, co, d~](~ ~ Gal~/F)}. Let m' = m~ 71 R', where m~ is the maximal ideal of R~. Let R~ = R~, and denote by R~ the completion of Rl at its maximal ideal. The inclusion Rt C_ Rg, induces a map i" P--I ~ R~. Lemma 2.5. -- The map i" P,-I , R~ /s surjective. Proof. -- Let m, be the maximal ideal of P,,~. Let P, " Gal(F/F) , GL2(R.~) by PI((~) = (c~ bzz)" Clearly, composing Pl with the homomorphism be defined GL2(13~t) , GL~(R~) induced by i yields p~. It follows from the definitions of RT and Pl that plmodml = p~. Let a = rn~R~. The deformation p~moda is the same as the deformation obtained by composing ptmodmt with the homomorphism GL,(k) = GL.2(P,I/m~ ) , GI,.~(R~/tt) obtained from i. As plmodm~ = Pc, it follows from the universality of R~ that there is a unique map R~ ~ R~/ml whose kernel is necessarily m~. The composition R~ ~ Rt/ml ~ R~/a must be the same as the canonical map R~ ~ R~/a. Therefore a = m~. This proves that dim k (R~/mlR~ ) = 1, from which it follows that R~ is generated as an Rwmodule by one element (cf. [Mat, Theorem 8.4]). [] I~br future retkrence we also record the following fact. i.emma 2.6. -- /fp C_ Ry ~ not the maximal ideal, and zfp~ = Rt 71p, then dimRi/pl /> 1. Proof. -- If" Pl is maximal, then Pl, and hence also p, contains a uniformizer of r ~. In the deformation p~ modplR~ the matrix entries are in k. Therefore the deformation p~2 ~modp is obtained by composing p~ modm~ with the natural inclusion k ~ R~/p. From the universality of R~ it then follows that p = m~. [] 2.2. Reducible deformations A reducible deformation of p~ is a deformation p such that p -~ (zl *) X2 ' In this subsection we analyze the universal reducible deformation of type- ~c.~r where = (~, Z, c, ,~). Write red red p~ 9 6al(Fz/F) , GL:(R~) for the universal reducible deformation. A consequence of our analysis of p~ will Dred be an upper bound for the dimension of l,,~. This bound will be important in our subsequent analysis of R~. 16 C.M. SKINNER, AJ. WILES Choose a basis for p~ such that p~ (zt) = /~1 --1 \~" For a E Gal(Fx/F) write 9~ (a)= (o~ ~ ), and let I be the ideal generated by the c~'s. Clearly, Rred red ~ =R~/I and 9~ =9~ modI. 1Dred Unfortunatel~ this description of ~,~ does not easily yield a non-trivial bound for the we take a more pedestrian approach. dimension. Therefore Let L(Z) be the maximal abelian pro-p-extension of F unramified away from Z. Write g~ G = Gal (L(E)(Z)/N ~-- A x r x Z~ r' where A _~ Gal(F(x)/I"), r is a finite p-group, and L(Z)(X) = L(E). F(X). Let M be the maximal abelian pro-p-extension of L(Z)~) unramified away from Z\.5~ and such that A acts on Gal (MILd)Of,)) via the unique representation over Zp associated to )U 1. Any reducible deformation of type-6.~ factors through Gal(M/F). Put A = Zp]]-G]] ~ Zp ][A x 1-]] [[I',, ..., T~v] ]. group H = Gal(M/L(s is a finitely generated A-module generated by The m elements where m = dim e H/mA , } = dimtker{ HI(Fx/F, k(X-1)) i= l = dim~ Hx(F, k). Note that by our hypothesis that g[Di 36 1, tll(Di, k(x -i) --~ H~(Ii, k(X-~)) Di. Here mA is the maximal ideal of A corresponding to X-I and k' is the residue field of Zp[A] associated to )C -1. Fix a presentation rtl a ~ ~ Aei ~ H i= 1 such that em projects to an element ~ of H tbr which pc(bin) = (1 u)t with u :~ 0 and such that if i:~ m then ei projects to an element hi of H for which pc(hi) = (l ) 1 " Choose an element u0 E r reducing to u. Put Al = ~'[[1-]] ~-Wl, ... , TaF]], A2 = ~' ~ [[Sl, ..., S~F]] and fix embeddings q0i 9 A ~ A | ~" ~ Ai RESIDUALIN REDUCIBLE REPRESEN~IATIONS AND MODULAR FORMS 17 where for (DI the map Zp[A] , C ~ is that induced by ~-i and for ~p9 it is that induced by ~. Let J be the ideal of AI ~_x~, ..., Xm-~]] generated by { qIl(al)Xl + q~l(a2)x2 +''" + q)l(am-1)Xm-I + (pl(am)U0 :Zaiei ~ a }. Fix a homomorphism of A-modules : H , B = AI [[Xl, ..., Xm-l~/J; ei I ~ xi, i= l,...,m- 1 era1 ~ UO. Put R = B~ A 2. Observe that Gal(M/F) _~ G ~ H. We may therefore define a reducible deformation of type-_q~', ~' = (65~, Z, c, ~) p 9 Oal(M/F) , GL2(R ) by p(g) = (~P1(g) | q~2(g) ) q~z(g) ' g E G, 9(h) = (1 z(h)) 1 ' hEH. The deformation P is readily seen to be the universal reducible deformation of type- ored 6_~. As easy consequences of this explicit description of ~,~, we obtain the following estimates. Dred Iz, mma 2.7. --- I4~ have dim ,,~ ~< 1 + 23F + dim k H~-(F, k). Lemma 2.8. -- If q C_ R~ is a prime containing p such that p~ mod q is reducible and its determinant has finite order, then dim R~/q ~< 8~, + dim k H~(F, k). A diagonal deformation of the representation 1 G X is a representation P : Gal(F/F) ~ GL2(A), A a complete local Noetherian ring with residue field k, such that P = (,1 ~2) with 1 = 001 modmA and Z = 002modmA" Such a representation P: Gal(F~/F) ~ GL2(A ) is a diagonal deformation of type-(G, 2) if A is a local O ~- algebra (C~ -~' and Y~ as in the definition of a deformation datum). There is a universal diag diagonal deformation of type-(C, s 9(~:~ ~ag I " Gal(Fz/F) , GL~(R/e~,~/). Later we shall need to know an upper bound for the dimension of various primes of R~ g, ~1" The proof of the following lemma is similar to, but much simpler than, the proofs of Lemmas 2.7 and 2.8 and hence is omitted. D diag _ D diag Lemma 2.9. -- (i) dim 1,~6~, ~, ~< 1 + 28r. (ii) If q C ~,~, ~ is a prime containing p such -- diag 1 .. ndiag j that detp(~:,y.)moaq hasfinite ordo;, then aamt~(~ ,z)/q ~< ~v- 18 C.M. SKINNER. AJ. WILES 2.3. Some special deformations In the course of our analysis of the rings R(/ we shall sometimes have to consider some augmented deformation problems as well as deformations of various restricted type. Here we introduce these deformations and, when applicable, their universal deformation rings 9 Let .r = (6~.", Z, c, .//~) be a deformation datum and Q a finite set of finite places, Q = { w~, ..., w~ }, disjoint from X. A deformation p of type-(~ ~' , Z U Q, c, ..~//~) is of type-~ o if 9 detp is unramified at each wicQ. There exists a universal deformation of type-r1(2: p~Q : Gal (FxuQfF) ' GL2(R~,)). For a deformation datum ~ = (~:, E, c, ,///), Zc C Z is the subset of places at which p, is ramified together with the set ,O ~. Similarly, ~/fgc = Z,\ ?)~. Also, we write Z0 for the set of finite places at which )~ is ramified together with ,?~. Given ~ we write .~c for the deformation datum .~c~,. = (~, Z,., c, ,J/gc)- A deformation p " Gal(Fz/F) ----0 GL,)(A) of type-~ is nice if 9 A is a one-dimensional domain of characteristic p, 9 p is a deformation of type-C'~_~, (*) 9 PlDi ~- ~li) ~i) with V( ~/~l/i ~; having infinite order for i= 1, ..., t. For a deformation datum ~ = (~, E, c, ./f~) let Lx/F be the maximal abelian pro-p-extension of F unramified away from 2, and let Nz be the torsion subgroup of Gal(Lx/F). A deformation p 9 6aI(Fx/F) , GL~(A) of type-CJ~o., is .~Q-minimal (6~.'-rninimal if Q= 0) if det P is trivial on Nz. Let min P~a 9 Gal (FxuofF) --, GL2(R~o mi~ ) be the universal ,f~o-minimal deformation. If Q = 0, then we just write p~n and mm rain R~. There is a simple relation between R~z and R~Q. We fix for each s a free Zz-summand Hy, C_ Gal(Lr./F) such that Gal(Lr/F) -~ Hx (9 Nr,. We choose the Hz's to be compatible with varying Z. Let Wz " Gal(Lr/F) ~ Ny denote the character min obtained by projecting modulo Hx. The representation P~)_ | ~z " Gal@'z/F) , GL2(Rff~ ~ | ~:'[Nz]) is easily seen to be a deformation of type-~oc It follows from min the universal properties of R~Q and R~Q that ~-~ min ~ min R~ a _ R~Q | ~' [Nz] and p~Q _ p~za | q%. RESI1)UALIN REDUCIBIA~; REPRESENTATIONS AND MO1)UI.AR FORMS 19 Suppose that ~ is a field and that p : Gal(F/F) , GL)(.~) is a represen- tation. For each place w { p at which p is ramified we distinguish for future reference four possibilities for 0Ix,,: TypeA pllw"( 1 *) _ 1 , ,q=O. TypeB pli,,~"( - ~ I ) , ~ a tinite character. TypeB' PII,-~(* 0-1), ~afinitecharacter. |,'.~ Type C PI,~ = Indv'e~ where F=._, is the unique unramified quadratic extension of F=, and V is a character of Gal(Fw/Fd, ) such that Vl~,,. 3 k ~:r'"'"'lX,,., V[l,. has p-power order, and det P[Dw has order prime to p. Note that Type C can only occur if the characteristic of .-Srg / is zero. 2.4. Pseudo-deformatlons For our purposes, following [W2] a (2-dimensional)pseudo-representation of Gal(F/F) into a topological ring A is a set 9 = {a, d, x} of continuous thnctions a, d : Gal(F/F) , A and x : Gal(F/F) 2 , A such that 9 a(~) = a(<a(~) + x(~, ~), 9 d(o~) : d(o)d(~) + x(~, ~), 9 x(o, ~>(~, ~) = ~(~, [3>(n, ~), 9 x(ox, ~) = a(o)a(13>(x, ~) + a(13)d(x>(o, a) + a(o)d(~>(x, 13) + d(x)d(~)x(o, [3), 9 a(1)= 1 = d(1), 9 a(z.) = 1 - -d(zl), and 9 x(~,g)=0=x(g,o) if g= 1 or zl. The trace and determinant of p are trace 9(0) = a(o) + d(o) and det p(o) = a(o)d(o) - x(o, o). Suppose that P " Gal(F/F) ~ GL2(A ) is a continuous representation such that 9(z,) = (1 _,). Write P(O) = ('~ by). The functions a(o)= a~, d(o)= do, and x(o, z) = boc~ form a pseudo-representation, and the trace and determinant of this pseudo-representation are merely the trace and determinant of the representation p. Let P0 be the pseudo-representation associated to the representation 1 9 X (i.e., a = 1, d = Z, and x = 0). A pseudo-deformation of P0 is a pair (A, 9) consisting of a local complete Noetherian ring A with residue field k (which we assumed finite) and maximal ideal mA and a pseudo-representation p of Gal(F/F) into A such that p mod mA = P0. We often just write p to mean such a pair (A, p). A pseudo-datum for F is a pair ~_~.m = (C, E) consisting of the ring of integers 6 ~ of some local field having 20 C.M. SKINNER, AJ. WILES residue field k and a finite set of finite places X of F containing ~ and those places at which )~ is ramilied. A pseudo-deformation 9 of P0 is of type-~.q~ v~ if 9 A is an c"~:-algebra and 9 p is unramitied outside of Y. (i.e., a, d, and x factor through Gal(Fx/F)). It is relatively straightforward to verify that the functor F~ o~ from the category of local complete Noetherian ~'-algebras with residue field k to the category of sets given by F~p~(A) = {pseudo-deformations into A of type-e_qr vp~} satisfies the criteria of Schlessinger [Sch]. The only non-trivial point is the finiteness of the tangent space, and this is provided by the following lemma. Lemma 2.10. - - Let k[~] be the "dual numbers". Then #F~ps(k[e]) = (#k) T, where r ,< 4(#GaI(F(x)/F)) + 2(dim k HI(Fz/F(x), k))2 + 4. Proof. -- If P = {a, d, x} E F~ps(k[e]), then a=l+~al, d=x+adl, and x=exl. If p' = {a', at, x'} is another such pseudo-deformation, and if cx E k, then {1 + eo~(al + a/l), X + ~cx(d, + ~), ecx(xl + ~l)} is in F~ p~(k[e]). In particular I"~p~(k[e]) is a k-space. Let G x = Gal(Fz/F(x) ). From the relations defining pseudo-representations it follows that Xl[Gn� determines an element of Hom(G;c, Hom(Gx,k)) via xl , , {g, ~ xl(',g)}. If P is in the kernel of the k-linear map FNp~(k[E]) , Hom(Gx, Hom(G~, k))given by p, , XI[Gxx(Ix , then aliGn, dl[Gx E Hom(G x, k). Thus .< #{ p" = = x, = 0 }, where s = dim~ H 1 (Fz/F(~), k). Now let G = Gal(F~:/F) and suppose that p = {a,d,x} E F~p,(k[~]) satisfies allcx = dl[G x = XIIG? x -" 0. Then XI[GxxG determines a 1-cocycle G , Hom(G:~, k(%)) via g~ ~ xl(', g). Moreover, this cocycle vanishes upon restricting to G x. Thus the number of possibilities for Xl [G~� is at most #Hom(Gx, k). A similar argument shows that the number of possibilities for xl IG� is also bounded by the same quantity. Thus #F~ ~(k[E]) ,< (#k) '~+4'. #{ p: a~ IGn = d~ L~ = x, LG~ � = x~ IG� = 0 } <~ (#k)S2 +4s+4"#Gal(F(X)/F). RESIDUALLY REDUCIBLE RI'PRESENTATIONS AND MODULAR FORMS 21 Here we have used that for any pseudo-deformation O = { a, d, x} satisfying aLlGx = dl]c, x = Xl]Gx� = Xl ]G� x = 0 the functions al, dz and xl are constant on cosets of G x in G. [] There is therefore a universal pseudo-deformation (R~ p.,, 9~ p,,) of type-("-/yps. Clearly, any deformation 9 9 Gal(F~/F) , GL2(A ) of some 9c with A an &"-algebra gives rise to a pseudo-deformation of type-(~ ~ , E). Choose a basis for 9 such that p(z,) = (1 -1)-Write p(a)= (~ ~). As we have previously noted, { a~, d~, x~,z = b~cz } is a pseudo-representation, and its reduction modulo ma is P0, so it is a pseudo-deformation. One easily checks that it is also of type-(~", s The entries of p(~) with respect to any other basis for which p(z~) = {1 -l ) are obtained by \ / conjugating the chosen basis by a diagonal matrix. Such a conjugation does not change a~, d~ or b~c,. We call { ~, d~, xa., = b~c, } the pseudo-deformation associated to 9 and sometimes denote it by 9 as well. There is a unique map R~p~ , A ((_~P~ = (r I2)) inducing the pseudo-deformation associated to 9. This argument shows that to any deformation p of p,, where c is some cocycle in H~(F~/F, k(z-~)), one can associate a well-defined pseudo-deformation. In particular, if ~r = (c~, Z, c, ,/~) is a deformation datum, and if ~_9~ "p~ = (r 12), then we obtain a unique map r~ 9 RNp~ ~ R~ min rnin inducing the pseudo-deformation associated to p~. We write r~ : R~ ps ~ R~ for min the composition of r~ with the canonical map R~ ; R~. Let ~P~ = (~, 12) be a pseudo-datum and let Q be a finite set of finite places disjoint from s A pseudo-deformation O of type-(~, Z tO Q) is of type-~-c-~ if 9 det P is unramified at each w E Q. There exists a universal pseudo-deformation of type-!~Q" (R~, p~ ps ). If 6_~ = 6.~Q (~, Z, c, JPg) is a deformation datum, then as in the preceding paragraph there is a unique map r~Q : R~ ps ~ R~ Q inducing the pseudo-deformation associated to P~ ~z" Of course, if Q = 0, then R~ ~_ = R~ p~, p~ ~ = 9~ v~, and r~ Q = r~. Suppose !~ = (~, Z, c, J~g) is a deformation datum and 6~p~ = (G, E). The following proposition reflects the relation of R~ v~ to R~ Q Q" Proposition 2.11. -- If p C_ R~ o - is a one-dimensional prime such that pg Qmodp is __1 A irreducible, and if pps -- rc.c.~ Q(p) C R6c~ ~, then the canonical map R~ ~, pp~ ~ R~ Q, p is surjective. Proof. -- As p~ modl0 is not reducible, pps is not the maximal ideal mps. Therefore mpSR~Q ~ p. Let A = R~Q/p and A p~ = R~ ~/pP~. Let K ps be the field of fractions of A ps. The map r~ : R~ ps ~ R~ o_ induces an inclusion A ps ~ A. Q 22 C.M. SKINNER, AJ. WILES As we have observed, mP~A ~= 0, whence A/mPSA is a zero-dimensional Noetherian local ring with residue field k. It follows that #(A/toP'A) < oc and hence that A is a finite APe-module (cf. [Mat, Theorem 8.4]). Thus A is an integral extension of A I'~ and dim A ps = dim A = 1. Fix a basis of p~Q such that P~(zl) = ,(1 ,) and 9~(o0) = (* u) -I * * u E 6: � for some c0 E Gal(F/F). With respect to this basis write 9~ o(~) = ao tiff It is easily checked that { ao, do, co, bob I o, 1: E GaI(F/F)} is contained in im(r~ ). Let R' C_ R~Q be the subring generated by im(r~) and the set { bo I r E Gal(F/~ }. Let m' = R' M m~ o_, where rn~ ,~ is the maximal ideal of R~ oj Put R~ = R~,. Let p' = R ~ f3 p and P0 = R4~ C'l p. It is a standard tact about localizations that P0 = p'R0. Let A' = R'/p' and ?to = Rq)/p0. Our first claim is that A0 = A and p0R~ o_ = P" "Ib see this, first note that there are inclusions A ps C_ A0 C A. Since A is a finite AP~-module and A vS is Noetherian, A0 is also a finite AP~-module as is any ideal of Ao. It fi)llows that Ao is a Noetherian ring and that it is complete as an AP~-module. Since A0 is local and the radical of rnPSA0 is the maximal ideal of Ao, it follows that Ao is a complete local Noetherian domain. It now follows from I~mma 2.5 that the map from A0 to R~ ~Jp0R~ ~ is surjective, so we have A0 -~ R~ ~t/p0R~ Q -~ R~ ~/pR~ c~ = A. The claim follows. We next claim that the canonical map R~p., p~,~ > R'p, is surjective. As Q, p~ modp is irreducible there exists some % such that % ~ r~ tv :'P'~x j. It follows easily that { bo [ o E Gal(F/F) } C R* = im(R~ p, , R;,). Therefore the image of the canonical map R' ~ R~, is contained in R*. The inverse image in R' of the prime pP~R* is just p', whence localization induces a map Rp, ~ R* whose composition with the inclusion R* ~ R~, is the identity map. It follows that R* = Rp,. As a consequence we have ppsR; = p'R~,. Combining the results of the preceding two paragraphs yields - ' = =pR~ ppSR~ o_, p - P R~ o~, v p0R~ o, v <), p" We also find that A p~, A', A0 and A all have the same field of fractions, namely K p~. It follows that dimKp~(R~Q,p/pP~R~,p ) = I. Therefore the canonical map R~ ~,~ ~ > R;/~ Q,~ is surjective (see [Mat, Theorem 8.4]). Q, PP As a corollary of this we have the following important result. Corollary 2.12. --/fQC R~ is any prime such that p~ modQ is irreducible, and if QpS= r)l (Q) c_ R~ ps, then dim R~p~/QP~ > dimR~/Q with equality holding if Q is a dimension one prime. RESII)UALI,Y Rt-I)UCIBLE REPRESI'NTATIONS AND MOI)UI.AR FORMS 23 Proof. - Equality of dimR~,,,/02 '~ and dim R~/Q when Q is a dimension one prime was shown in the first paragraph of the proof of Proposition 2.1 1. We may therefore assume that Q is a prirrie of dimension at least two. Choose a prime p _D Q of R~ having dimension one and such that P~ modp is also irreducible. Let pps _- r~ 1 (p). We then have dimR~/Q= l+dimR~,p/Q ~< l+dimR~p~p)>~/Q v~ ~< dim R~ p~/Qp~ with the first inequality following from Proposition 2.11. [] We now collect a few results connecting deformations and pseudo-deformations. Suppose that A is a complete DVR with residue field k. Let K be the field of fractions of A, and let ~ be a uniformizer of A. Suppose that 9 : Gal(Fz/F) ---~ GL2(K ) is a continuous representation. As Gal(F~/F) is compact, there exists a Gal(F~/F)- stable A lattice L in the representation space of p. Such a lattice, being a free A-module of rank 2, gives rise to a representation PL" Gal(Fz/F) ~ GL2(A ) such that Pu | K ~-- 9- It is well-known that whereas the reduction 9L = PL rood ~. is not --SS --S~ necessarily independent of L, its semisimplification 9L is. We call 9L the reduction of 9. Ix'mma 2.13. -- Suppose that the reduction of p is 1 | )~ and that 9 is irreducible. (i) There exists a Gal(F~/F)-stable lattice L m the representation space of p such that = ( 1 Z(~jba 1"~ for all ~, and -Or, has scalar centralizer. (ii) For two lattices L, and L2 as in (i), the classes in II~(Fz/F, k(z-~)) of the coqycles (yl , )~(~)-lbi(o), i= 1, 2, are non-zero scalar multiples of one anoth~ Proof - - Choose any Gal(Fz/F)-stable lattice L and pick a basis for PL such that ( ) (~o ~) and let n= min ord~,(bo). As p is irreducible, PI.(Zl)-" 1 --1 " Write P1,(~J) = co ' o n < oo. Let L' be the lattice obtained by scaling L by (~'--'~ 1)" The representation PL' is just = ( )~-" 0, ,) ,) The representation clearly has the properties desired for part (i). To prove part (ii) it suffices to show that the representations 9I.~ and 9L2 are equivalent. Choose bases for the 9Li'S such that PL,(Zl) = (1 -1 )" As 9L, | K "~-- 9, there exists g G GL2(K ) such that -1 (l) (1) g 9L~g = 962. Since g must commute with - l , one may assume that g = a 9 Write 9L1(~)= ~.(a~ ~/b'~] By hypothesis, there exists some 60 such that b,0 is a unit. ClJ As ab~ o E A, it must be that ord~.Ia )/> 0. As the reduction of ab~ is not always zero, it must be that ord~(a) ~< 0. Thus a is a unit and 9L~ and Pb~ are equivalent. [] 24 C.M. SKINNER, AJ. WILES Corollary 2.14. -- IrA is a complete DVR with residue field k as above, and zf(A, q~) is a pseudo-deformation of Po unram~fied away from 2 for which x(a, "c) is not identical~ zero, then there exists a non-zero coqycle c E H'(Fr./F, k0u')) and a deformation Pq~ : Gal(Fy./F) , GL2(A ) of Pc whose associated pseudo-deformation is q). Moreover, c is unique up to multiplication by a non-zero scalar. Proof. --We need only observe that there is some irreducible representation p : Gal(F~/F) ~ GL2(A ) whose associated pseudo-representation is q0, for then the claim follows from the lemma. Fix ~0, ~0 C Gal(Fr./F) such that ord~.x(~0, ~0) is minimal. Define P by = a) " [] Our next result also associates deformations to pseudo-deformations. Suppose that R is a local complete Noetherian domain with residue field k and maximal ideal m. Suppose that P = { a, d, x} is a pseudo-representation of Gal(F~-/F) into R such that P0 = P modm (i.e., (R, P) is a pseudo-deformation). Let p be a prime of R such that the dimension of R/p is one. Let A be the integral closure of" R/p in its field of fractions K. This is a complete DVR with residue field a finite extension of k, say k'. Suppose that xmodp is not identically zero. By Corollary 2.14 there exists a cocycle 0 :~ c in HI(F~:/F, /fix-l)) and a deformation Pp : Gal(Fz/F) , GL2(A ) of Pc such that the pseudo-deformation associated to pp is 9 rood p. We will construct a local complete Noetherian domain R + having the same dimension as R, an injective local homomorphism R ~ R +, and a deformation 9 + : Gal(FyJF) , GL2(R +) of 9c whose associated pseudo-deformation is P. Moreover, R + will have a prime p+ of dimension one such that p = R n p+ and Pp = p+modp +. Let L be the field of fractions of R. Pick c~, ]3 E m, ]3 ~ p, such that or/[3 is a uniformizer of A. Put R' = R[ot/[3] C L. This is a Noetherian domain with maximal ideal m' = (m, c~/]]). To see that m' is in fact a maximal ideal, let q~' : R' -* A be given by ~'(f(ot/~))=f(o~/]3) for any polynomial fwith coefficients in R. Here the "bar" denotes reduction modulo p. This is well-defined, for iff(o~/[3) = 0, then f(o~/~) = 0 as can be seen by first clearing denominators and then reducing. Let p' be the kernel of q0', and let I be the ideal of R' generated by the set { x(r~, ~) }. Let { il, ..., i~ } be a set of generators of I taken from among the x((~, x)'s. As Pp is irreducible, the image of I under q)' is non-trivial. Pick an i E { il, ..., i, } whose image has minimal valuation in A. Define R* by R* = R'[iL/i, ..., ir/t] C_ L. This is a Noetherian integral domain with maximal ideal va* defined as the inverse image of the maximal ideal of A under the homomorphism q0* : R* ~ A given by RESIDUALLY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS f(i,/i, ..., ir/Z)~---~f(-it /-i, ..., i/i) for any polynomial f with coefficients in R'. (Now the "bar" denotes reduction modulo p'.) Let p* be the kernel of the map q)*. Let R + = ~., and let rn + be its maximal ideal and p* the kernel of the induced map R + -~ A. The ring R ~ is clearly a local complete Noetherian ring with residue field k. Moreover, the inclusion R ~-~ R + is a local homomorphism. To see that the dimension of R + is the same as that of R, observe that it follows fi'om the construction of R* that R~. = Rp, whence dimR* =dimR~. =dimR~.+l =dimRp+ 1 =direR. Unfortunately, R § need not be a domain. However, since the "going-down" property holds for the pair R~. and R* (see [Mat, Theorem 9.5]), there is a minimal prime q+ of R + contained in p+ and such that q+ 7/R~. = (0) and dimR+/q + = direR +. We replace R + by the quotient R+/q +. This ring has all of the desired properties. In the ring R 4 the ideal IR + = {i} is principal. As i = x(t~0,"c0) for some t~0, "Co E Gal(Fz/F), one can define a representation 9 + : Gal(Fz/F) ~ GL2(R +) by : o/ " The reduction of p*modm + is non-semisimple as P*(Z,) = (1 -1) and P+(~0) = (* I ). Thus 9+ is a deformation of 9e for some 0 :~ d 9 HI(Fy./F, k'(z -l)). Reducing p+ modulo p+ gives a deformation of Pe into GL2(A ) whose associated pseudo- deformation is p modp. It follows from Corollary 2.14 that c' is a non-zero scalar multiple of c. Thus, after possibly replacing 9+ by a conjugate, we may assume that c = c' and p+modp § = pp. Finally, suppose that A is an U/-algebra with ,~;' the ring of integers of some local field having residue field k' and that ~///~, Q c E are sets of finite places (possibly empty) such that (2.8) (i) Jig C Y~\~ consists of places w such that 9c is ramified at w ; (ii) Q c_ z\?~ u .Ill consists of places w at which 9c is unramified. Let ~9r be the field of fractions of R +. It is easily checked that if (2.9) (i) p+ | [D,"~ ( "~xgl'i *) ~/%i ' %"ira~ = 1, i= 1,...,t, p+ -~llw -~ (1 *'~ for all wEJtg and (ii) \ / (iii) det 9 + II,, = 1 for all w C Q, then c is admissible and 9 + is of type-,~Q where ~ = (r Z, c, ~fg). l%r ease of reference we summarize these results in a proposition. 26 C.M. SKINNER, AJ. WILES Proposition 2.15. -- Suppose (R, 9) is a pseudo-deformation of type-(~;, Z). Suppose also that p C_ R is a prime of dimension one such that x(~, ~)modp /s not identical!y zero (9 = { a, d, x } ). Let k ~ be the residue field of the integral closure of R/p in its field of fractions. There exists a cofycle 0 ~- c E HI(Fx/F, /((~-l)) unique up to multiplication by a scalar, a local complete Noetherian (~-domain R + with residue field k ~ and haviN the same dimension as R, a local homomorphism R~-~ R ~ of ~;-algebras, a dimension one prime p~ C_ R ~ extending p, and a deformation 9 + 9 Gal(Fx/F) , GL2(R +) of p~ whose associated pseudo-deformation is that induced from 9. Moreover, if Q, ,/2,g C_ Z are sets of places satisfying (2.8), and if 9 + @ ~, .~ thefield of fractions of R +, satisfies (2.9), then 9 + is a deformation oftype-~o_ with ~_~ = (C, E\Q., c, JAg), with 6~' = (~;' | W(k'). For a finite field F, W(F) denotes the ring of Witt vectors of F. 2.5. The Iwasawa algebra In this subsection we describe how each of the deformation rings R~ and R~ ~z is an algebra over a certain multivariate "Iwasawa algebra". Let LI be the maximal abelian pro-p-extension of F unramified away from ?~. Let I C Gal(L0/F) be the subgroup generated by the images of the inertia groups Ii, i = 1, ..., t. We fix once and for all a maximal free Zp-summand I0 of I (necessarily of rank 5v). Fix also a free Zp-summand Go of Gal(L0/F) containing I0 (this also has rank 5/). Finally, fix elements 'YI, ..., "/5 F E Gal(F/F) whose images in Gal(L0/F) generate Go and for which there exist integers rl ' ..., rgv such that 3'( ~ '""Ygv /~v generate I0. For each 0~<i~< t fix once and for all y(~ ' , "",Ydi E Ui (the units of F~i) generating a free Zp-summand of rank di. Put A~ = ~'~T,, ..., %F, Y~t', .-., Y~I]I. The rings R~ (and hence the R~(z) are algebras over A~ via 9 Ti . , det p~ (Ti) - 1, i = 1, ..., ~SF; 9 Yf' ' ~g21Y))-- 1, where 9~ I.), - v]~/~ v~/ and U, is identified with the inertia subgroup of D ab via local reciprocity. Suppose that ~ = (6;, E, c, rig) and cj, = (~:,Z', c,.~/g') are deformation data with E C_ E' and .//g~ C_ d,g. The natural map R~, ~ R~ is a map of A~: -algebras. Each universal pseudo-deformation ring R~ p,, and R~ ~ is a A~-algebra in a manner compatible with the canonical maps R~ p., ~ R~ and R~ (p~ ~ R~ ~. To see this, for each i = 1,...,t fix gi E Di such that x(gi):~ 1 and for each ,,g j = 1,..., di let ~j!'? C Di be a lift of yj~. By the choice of gi the polynomial RESII)UALLY REI)UCIBI,E REPRFSFNTATIONS AND MODUI2kR FORMS 27 X 2- tracep~p,(gi)X + detp~p~(g/) has distinct roots in Rcj:p~, say ~i and ~i with 0~i reducing to Z(gi) modulo the maximal ideal of R~ p~. (The images of c~i and ~i in R~ are just the eigenvalues of p~ (gi)). We define a map A~ ~ R~ p~ by * Ti~ ~ det Pcj p~('fi)- 1, i= 1, ..., ~iF, ' ' 0) * Yf, , (trace p~ ,,~(g.~}i/) _ a,. trace p~ p,(~) ))/(~i- o~i)- 1. The compatibility with the A~-algebra structure of R~ is clear. Also, if d;~ ps .~I~, = (~.,, Y'I) and -~2 = (~', Z2) are two pseudo-data with Y.,~ D Z1, then the natural map R~!~ ~ R~ ~,~ is a map of A~ -algebras 9 ~ I 3. Nearly ordinary Hecke algebras and Galois representations 3.1. Modular forms and Hecke operators We keep our previous conventions for the field F. We write A and Af for the adeles and the finite adeles of F, respectively. If G is any algebraic group over F, then we identify G(A) with the restricted direct product of the groups G(F~.) with respect to the subgroups G(~F, ,1,) (for finite w), writing .'cw for the w-component of x E G(A), and similarly for G(Aj). For a finite place w, we sometimes write xp for xw with p the prime ideal of F corresponding to w. Let I be the set of infinite places of F (equivalently; the set of embeddings "c 9 F ~ R). This description of G(A) identifies G(F | R) with G(R) I. We also fix an algebraic closure Qr of Qp and an embedding of Q = F into Qp. l~br an ideal n of ~';v we define various standard open compact subgroups of GL2(Aj) as tbllows: Uo(n) ={(: b) EGL2(~;F| moan}, 9 a-- 1 modn}, and {(++ 9 d- 1 modn}. U(n) = c ,1 E U t(n) For k= Zk~ E Z[-l] and x E (31 write x ~ for the product II~. Let t= E~. To each k = Zk~, with each k~/> 2 and having the same parity as the others, we associate quantities m, v E Z[I] and la E Z as follows: m=k-2t and v=Zv~'~, v~)0, somev~=0; m+2v=g.t. 28 C.M. SKINNER, A.J. V~qI,ES Let H denote the complex upper-half plane. Define j" GL2(F | R) x H l ~ C I by , = = (o, C (F | R). j(u~ z) (c~z~ + d~), u~ ~. Definc also an action of GL~(F | R) = GL~(R) I on H I by uoo(z)= + b~'~ Denote by z0 the point (i, ..., z) G H I. We now recall the notion of a (holomorphic) modular form on GL 2. First, for any congruence subgroup F C_ GL2(F), denote by Mr(F) and St(F) the spaces of (classical, Hilbert) modular forms and cusp forms on H I, respectivel)4 of weight k (cf. [Sh]). For a functionf : 6L2(A ) , C and u = uf- uo,z G GL2(,A ) - GL2(Af). GL2(F| ) we define fl . by (/Itu)(g) = j(u~ , Zo) -t det(u~)V+t-t f (gu-'). Write Co~ for the subgroup (R � -SO2(R)) I C_ GL~(F| A functionf 9 GL2(A ) , C satisfying f Itu = f fbr all u E Coo gives rise to a function ~: I-I I > C for each x E GL2(A/): f~(z) =j(Uoo, Zo) t det(uoo)t-t-V f(xuoo), uo~(Zo) = z. Let U C_ GL2(Af) be an open compact subgroup. A function f: GL2(A ) , C is a modular form of weight k and level U if 9 f(ax) =f(x) Va E GL2(F ), 9 f]tu=f VuEU'C~, 9 f~(z) E Mk(F,), Fx = GL2(F) nxU-GL~(F| Vx E GL2(At). Such a function is a cusp form if f~(z) E Sk(Fx) for all x E GL2(Aj). Denote by Mk(U) and Sk(U) the spaces of modular forms and cusp forms of weight k and level U, respectively. For more on such forms see [Sh] and [H1]. If U = Ut(n), then Mk(U) and Sk(U) are just the spaces Mk(n) and Sk(n) defined in [Sh]. For each n choose once and for all representatives t (') E A x of the ideal classes of F (i = 1, ..., h) such that t~ ) = 1 for each place w lNm(np).oo. Put xi = ( t(O 1) and write Fi for the subgroup Fx i and for each f E Mr(n) write f for f,. There are h h isomorphisms Mk(n) --~ I-[ Mk(F;) and Sk(n) -~ I-[ Sk(F;) given by f~--~ (f). Each f(z) has i= 1 i= I a Fourier expansion of the form f(z) = ai(O)+ ~+ai(g)e(g" z) where (P)) is the ideal go(#)) of F associated to the idele t/'l, the sum is over totally positive elements of (t(')), and (a~z.___5~ RESIDUALLY REI)UCIBI,E REPRESENTATIONS AND MOI)UI2kR FORMS 29 g'z = Xz(g)-z,. For a ring A _C C, let Me(n, A) be the space of modular formsfE M~(n) such that each f has Fourier coefficients in A. Define Sk(n, A) similarly. Shimura has shown that Mk(n, A) = Mk(n, Z) | A and Sk(n, A) = Sa(n, Z) | A. For a ring R C Qr define Mk(n, R) and Sk(n, R) by Mk(n, R) = Ma(n, Z)| R and Sk(n, R) = Sk(n, Z)| R. If R C_ Q as well, then this agrees with the earlier definitions, as Shimura's result shows. From now on we require each U to satisfy U = II Uw, U~ C_ GL2(~F,w). ?/,' f OC We also require that U(n) C U C U0(n) for some n. Next we recall the connection between modular forms on GL~ and automorphic representations of GL 2. For details and definitions the reader should consult [De], [Ge], and [J-L]. Let ,_~0 be the space of all cusp forms on GL 2 (over F, of course) of weight k. The group GL2(Af) acts on ,.d~ via (~)(x) =f(xg). Under this action ,~ is an admissible representation of GL2(Af). Moreover, ~4~ decomposes into a direct sum ~ = (~VTr where, for each g, V~ is an irreducible admissible representation ]I of GL2(Af) (which we often denote just by re), and the Vn are all non-isomorphic. For an open subgroup U C_ GL2(O,; | 7.)let Ilk(U)= {~ I V~ ~( 0 }. Clearly the space (~ V~ is just Sk(U). We recall that each rc E FIt(U) can be written as a restricted tensor product ~ = I~)lt~ where v runs over the finite places of F and each ~ is an irreducible admissible representation of GL2(F~ ). Let V~ - | ~ be the corresponding tensor product decomposition of V,~. Clearly V~ ~ = @ V~U,~ It follows from the theory /; of newforms that dim V L'~ -- 1 for each place v for which U~ = GL2(~F ' ,). l%r each g E GL2(Aj) define a Hecke operator [UgU'] 9 Mk(U) , Mk(U') by X -1 (3.1) [UgU']f(x) = ~_.,f( "gi ), UgU' = I lUg,. gi Of course, [UgU'] maps St(U) to Sk(U'). For each prime ideal g of F choose an element ~.(g)E r | such that )~{)is a uniformizer of (~v,t and ~t)= 1 for p :~ g. If pL0 then we require that ;~P) also be an element of ~F such that X(~P)) ~: 1 and that )~) E (J~.~,,, for each p'~ but p' =~ p. (For y E F~ we define )~(y) to be the value obtained from composing )~ with the local reciprocity map F~ , Gal(F~pb/Fp).) Denote by T(g)and S(g)the operators [U( 1 ~(~))U] and [U(~'(t) ~(t))U], respectively. These operators commute one with another. Moreover, it is easy to see that T(g) and S(g) are independent of the choice of ~(t/ if Ut = GL2(~"v,t). Also, if V C_ U and if 30 C.M. SKINNER, AJ. WILES GL.~(~ F, C) = Ve = Ue, then the inclusion M~(U)C_ Mk(V) respects the actions of T(g) and S(g). If U = U1 (n), then T(g) and S(g) (for g r n) are just the Hecke operators defined and so denoted in [Sh]. These operators stabilize each M~(n, R) and S~(n, R). As S~(U)= (~ V~ we find that the Hecke operator [UgU] stabilizes each V~ ~, ~EFIk(U) the action being given by (3,1 t) [UgU]x -- E -1 U ~(~i )x, x 9 UgU=UUg,. gz l:br each place v let g,. be the v-component ofg. Under the tensor product decomposition gl7 u,. u. = | lUgU] decomposes as [-UgU] = @ [U,.g,.UJ with [ld, g,U,] 9 End(V~,~) being given by = x E V~'~, U~g~ U~. = [IU~hi. hi 3.2. Nearly ordinary Hecke algebras Keeping the conventions for U introduced in the preceding subsection, for each positive integer a define U~, U~, and Uo by "e'~ = U r] Uo(pa), U~ = U ["] UlOOa), and U~ = U N U(p~'), respectively. Suppose that U~ = GL2(~::v,~ ) for each v[p. There is an action of the group G~U~) = U a" ~ C ~ r/Ua" � ~V � on M~(Ua) with x.y acting via the operator (UaxUa) = (~r -~) [IS~xUa] where x = (i b) E U~ andy E d~-. Here co" Gal(F"b/F) ~ F� '12" is the Teichmaller character, and c0(a,.) is defined by composing co with the global ,>: X reciprocity map taking d; v.l. to the inertia group of v in Gal(F~U/F). Recall that we -- -- X have fixed an embedding of F into Q~, so co, which a priori takes values in Qp, can be considered as taking values in F � e t el Let (p) = p,, ...,p, be the prime factorization of (p) in F with Pi the prime corresponding to the place vi. For each i = 1, ..., t define an operator T(~(pi) on Mk(U,) by = (~P'~)-VT(pi). Define an operator T0(p) similarly by T.(p)= p-v[U~(~ ~)U~] where p, = p if v[p and p,, = 1 otherwise. Note that T0(p) differs from FI T.(pi)" by i= 1 multiplication by some ~,. [U~ (e) 1 Ua] where ordp,(~,) = 0 and ep, E (?'� F, p, for each pi. RESIDUALLY REDUCIBI,E REPRESENTATIONS AND MOI)UI.AR FORMS 31 As discussed in the previous subsection, these operators act on each V~, rc E Ht(U,). If V _C U and if V~, = GL2(dv,,, ) for each vlp, then the inclusion Mt(U~) C Mt(V~)(b /> a) is compatible with the natural homomorphism G(Vb) , G(U,) and with the actions of T~ (p) and the T0(Pi)'s. Let k = Y~kr with each k~ >/ 2. Let n E II~(U~). Suppose that vlP and that p~ is the corresponding prime ideal. It is an easy consequence of the classification of local automorphic representations that if there exists a line in V~('I/ on which T0(p,) acts via an element of F that is a unit in the ring of integers of/~r via the fixed embedding F~--~ O-~0, then the line is unique. Call such a line v-good. A v-good line exists if and I I only if n~ is either a principal series representation ~(n~l 9 , 9 or a special 1 I representation rc({~ I 9 I[ ~ , {~l" I~)such that in either case ~.~-v{~()~-I)is a unit in the ring of integers of O~ (cf. [H3, Corollary 2.2]). Here ~.~. is the uniformizer of ~' r',~ chosen in the definition of T0(p~). The representation n is said to be nearly ordinary if a v-good line exists in V~,~ for each v dividing p. Similarly, a newformfE Sx(U~) is called nearly ordinary if the corresponding automorphic representation nj is nearly ordinary. Let H""a/ll ~ C rlk(U~) be the subset of nearly ordinary representations. A representation k k ''~a] -- ord n E 11 k (U~) is said to be ordinary if ~ is unramified at v for each v~. Similarly, a newform f E Sk(U~) is ordinary if the corresponding automorphic representation is ordinary. Fix an identification ofC with Q~ extending the fixed embedding of 0 into O-~0" For each rc E II~rd(ua)let w(rc)= | v) E @V;)'I ~ be a vector such that l' w(x, v) spans a v-good line for each v~b. Each w(n) is an eigenw:ctor for the Hecke operators T0(Pi) for i = 1, ..., t, T0(p), T(g) and S(g) for each prime ideal g r tbr which Ue = GL2(tg?v, g), and for each element of G(U~), and the corresponding eigenvalues are integers in Qr Let S~ (Uo) c S~(U,,) be the subspace spanned by the w(n)'s (recall that Sk(ga) = (~ v~Ta). Let Tk(ga) C Endc(S~.(U,) ) be the subalgebra generated over Zp /IE I-lk(l_'a; by the aforementioned Hecke operators. The ring Tk(U~) is a finite, flat, commutative, reduced Zp-algebra. In fact, we have an injection ~C-II k (l a/ Note that the definition of Tk(U~) is independent of the choice of the w(n)'s. Also, if V _D U is another open compact subgroup, then there is a canonical homomorphism Tk(Vb) ' Tk(Ua) (b >/a). 32 C.M. SKINNER, AJ. WII3~;S For ~' the ring of integers of some finite extension K of Q0, put T~(U,, 8) = T~(Ua) | ~:. Put also T~(U, ~') = lim T2(U~, 8:). Suppose k is parallel (i.e., v = 0). We now give an alternate (but equivalent) definition of Tk(Uo, 8'). As we shall see, both definitions will have their uses. Let Tk(Ua) be the subring of Endc(M~(Ua)) generated over Z by T0(Pi) tbr i= 1,..., t, T0(p), the action of G(Ua), and by the operators T(g) and S(g) for each prime ideal g ~ p for which Ue = GL2(~r, e ). Let T~.(Ua) be the quotient ring obtained by restricting the action of the Hecke operators to the space Sk(U,) of cusp forms. These rings are finite, flat, commutative Z-algebras. Put Tk(Ua, ~,~) = 'rk(Ua) @ ~' and T~(Ua, ~') = T~(U~) | C~ with 8' as in the preceding paragraph. For all sufficiently divisible integers rn, the operator e lim F0(p) e v, -, exists in Tk(U,,, ~;) and is independent of m. Moreover, n~cx3 * T e is an idempotent. Put T~(U~, .~,') = eTk(Ua , 8). That this definition of T~(Uo, ~) yields the same ring as did the previous one can be seen as follows. The ring T~(U~, C ~) is the subring of Endc(Sk(Ua)) generated over ~' by T0(p/) for i = 1, ..., t, T0(p) , the action of G(Ua), and by T(g) and S(g) for each g Cp such that Ue = GL~(~'F.e). In particular, e is identified with an idempotent in Endc(Sk(Ua)) and eT~(Ua, ~)is just the image of T~(Uo, ~)in Endc(eSk(Wa)). Write Ua ord eSk(Ua) = ~ eVn . From the definition of e we have that eV~" = 0 if r~ r I1, (Ua) r~El Ik(Ua) Fl~ T ~ Ua and that if g E "'k ~,,J then eV= = { x = | 9 x~ spans a v-good line Vv~0 }. It is now immediate that eT~(Ua, -~;) agrees with the first definition of Tk(U~, ~/). Let G(U) = lira G(Ua), where the transition maps are the maps induced from the inclusions U~ C U~, a >/b. There is a homomorphism ~" [[G(U)~ , T~(U, r Put t/= u a (&)� = uo n (A2� , Z(Ua) = Ur. and Z(U) = lim Z(Ua). The map (~ b), , (~(a_ld)~, a)induces isomorphisms G(Ua) --~ (~'F/pa) � X Z(Ua) and G(U) ~ (.~f,4' F ~) Zp) � x Z(W). RESIDUALIN REDUCIBLE REPRESENTATIONS ANI) MODULAR FORMS For (y, 1) E G(U) we write T~. for the corresponding Hecke operator. Similarly, we write Sx for the operator corresponding to (1, x) E G(U). Let ez = n c_ | zp) � = n v i v i where ~ ~ C ~� is the subgroup of units congruent to one modulo vi. Let y} ~ E '~.~ -- J F, v i ' ' "'"YgF ' respectively, via the be as in w Let x~ ..., x~ v E Z(U) be the images of ~'(~ /~v global reciprocity map (for the definition of Yi and ri see w The xi's generate a maximal Zp-free direct summand of Z(U). The ring T~(U, (9') is an algebra over the ring A' = ()? [[-Xl, ..., X~F:, Y(I 1), Y~ via Xi, , S,,- 1 and ~'3, , T (,: - 1. 9 .., )) The principal goal of this subsection is to show that T~(U, ~) is a finite, torsion- ti'ee A~.-module. We only prove this for F having even degree, although the result is true in general. Our proof involves analyzing modular forms on a twisted-form of GL 2. Suppose that F has even degree. Let D be the unique quaternion algebra over F ramified at every infinite place and unramified at all finite places, and let R be a maximal order of D. Let G D be the unique algebraic group over F such that GD(F) = D x. Let VD " G D ~ Gm be the reduced norm. For each finite place v fix an isomorphism R | ~'F. 0 --~ M2((~F.,). This induces an isomorphism GD(Af) _~ GL2(Af) which we use to identify these two groups. For each open compact subgroup U _C GL~(Af) put c~'~ = {f: D� , 13 }. (Note that D� is a finite set.) We distinguish a subspace ID(u) = {f E c~'D(u): ffactors through GD(Af)/U-~(Af)� }. For any g E GD(Af) -~ GL2(Af) there is a Hecke operator [UgU'] : J3~D(u) , Jfi'~)(U') defined as in (3.1). It is easy to see that [UgU'] maps ID(U) to ID(u'). A theorem of Jacquet, Langlands, and Shimizu [J-L], [Shi] states that there is a system of isomorphisms sD(u) = s2(u) compatible with the action of the tIecke operators [UgU']. Thus T_~(Ua) can be identified with the subring of Endc(SD(Ua)) generated over Z by T0(p), T0(P/) for i= 1,...,t, G(U~), and T(g) and S(Q for all prime ideals ~ for which Ue = GL2(~v,e). Put X(U) = D� and define H~ Z) = {f E .r taking values in Z }. 34 C.M. SKINNER, AJ. WII,ES This is a free Z-module of rank equal to #X(U). I~br any Z-module R, put H~ R) = H~ Z)| R. Note that H~ C) = ,~)(U). The action of [UgU] on H~ (3) stabilizes H~ Z) and hence induces an action of [UgU l on H~ R) for any Z-module R. If R is an ~-module then the operator e= l'ma[U,( 1 ~)U~] f~-l/ exists in Endr176 R)) for sufficiently divisible m. tl Moreover, e annihilates ID(U,, Z)| R, where ID(Ua, Z) = { f E ID(Ua) taking values in Z}. Let T(U~, (~") be the C-subalgebra of End,~ (H~ ~:')) generated over C by T0(p), T0(Pi) for i= 1,...,t, G(U~), and T(g) and S(g) for all prime ideals g {p such that Ue = GL,2(~Yr, e). It follows that T2(U~, ~') can be identified with eT(U~,, ~) (equivalently, with the image of T(U,, ~'~) in End~,, (eH~ ~))). Put H~(U) = lim eH~ K/~). (K is the field of fractions of ~'.) This is a T~(U, ~)-module. For any open subgroup U put U = U/UnF � For each x E GD(Ap) put c (x) = #{u E U = x}. Let R be any ~'-algebra. If each cu(x) is invertible in R, then define a pairing ( , )~:" H~ R) � H~ R) , R by (f, g)u = ~_~ Cu(X)-l f(x)g(x) - xGX(U) This is a non-degenerate pairing, and the map f. , (f, .)t: determines an isomorphism H~ R) -~ HomR(H~ R), R) that is functorial in R. The pairing ( , )u is not Hecke-equivariant, but a straight-forward calculation shows that ([UgU]f, h)t: = (f, [Ug-~U]h}u for any g E GD(Af). It follows that for each t E T(Ua, ~') there exists t § E Endr (H~ ~')) such that (t .f, h)u~ = (f, t +" h)u~ for all f, h E H~ C). Let T+(Ua, ~) C_ End(H~ ~)) be the ~;'-subalgebra generated by {t + 9 t E T(Ua, ~:)}. Clearly the map t ~ ~ t* determines an isomorphism of ~- algebras T(U~, ~') ~-T+(U~, ~). For any C'-module R write H~ R) ~ for the T(U~, ~')-module whose underlying ~-module is just .~s~ ~ 9 ~'~J, R) but the action of t E T(Ua, ~') is via t +. It follows that the pairing ( , )t:, induces a perfect RESIDUAI,LY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 35 Put pairing (,)G: eH~ R)x eH~ R)+ , R of T2(U~, (~)-modtfles. H+ (U~) = limeH~ K/U) +. If V C_ U is any subgroup, then we define a trace map tr(V, U) 9 H~ R) H~ R) by tr(V, U)f(x) = ~_,f(xx~-l), U = kJVxi. If V = Vb and U = U~ (b /> a) then it is easy to see that this is independent of the chosen coset representatives and that it is compatible with the actions of t and t + for t E T2(V, ~'). The pairings ( , )v and ( , )v satisfy the following compatibility whenever they are both defined: the diagram ( , )v " H~ R) x H~ R) , R T tr(V, II ( , )v" H~ R) x H~ R) , R commutes. Since Cuo(X) = 1 for all x if a is sufficiently large, it follows that by putting Mo~(U) = lim eH~ ~),";:' ~ where the transition maps are just the trace maps tr(Ub, U~)(b ) a), we have an identification of T~(U, ~')-modules Ms(U) -~ lim Hom(:~ (eH~ C), r162 _~ lim Hom~. (eH~ K/~::'), K/(Y ') _~ Hom~ (lim eH~ K/~"), K/~ ~) --~ Hom~: (H~(U), K/C'), the Pontryacn dual of H~(U). Putting ML(U) = lim eH~ ~) we obtain a similar identification of M~(U) + with the Pontryagin dual of Ha(U). + The following proposition is due to Hida [H2, Theorem 3.8]. Proposition 3.3. --If the action of every element 0fU/UVIF � on D� f) isfixed-point M~(U) are ' -modules of rank equal to flee, then Mo~(U) and + flee A~, { order of the torsion "~ rank6:~ eH~ ~) x \ subgroup of G(U) 1" 36 C.M. SKINNER, AJ. WII,ES Proof. -- We first claim that the Pontryagin dual of eH~ K/~ ~:) is a free E[[G(U~)~-module of rank equal to the ~'-rank of the Pontryagin dual of eH~ K/~). Clearly; it suffices to prove the claim without having ap- plied the operator e, in which case it is a simple consequence of the fact that H~ K/6 ~') = H~ K/r ~tJ~ and that #X(U~) = #X(U~ #G(Ua), the latter equality a consequence of the assumption that G(U~) acts freely on X(U,). The assertion of the proposition for Mo(U) will follow if we can establish that eH~ K/~") = eH~ K/~), for then we will have that the dual of eH~ K/~ ~) is a free r of rank equal to the ~-rank of the dual of eH~176 K/~ ~') which in turn equals rank6~ H~176 E). Now, if a /> 2, (1) 0 0(1 ) 0 then U~ ~ U a = U a ~ Ua_l, so T0(p). H~ K/#) C_ H~ K/6~), whence eH~ K/eft) = eH~ K/(~ ~) as desired. The same argument applies to the Pontryagin dual of eH~ K/~) + yielding the assertion of the proposition for M~(U). + [] Corollary 3.4. -- For any U, To(U, ~') is a finite, torsion-flee A~-module. In particular To(U, ~) is a semilocal ring complete with respect to its radical. Proof. - Choose a prime g of F for which V = U nU(g) is such that V/V N F � acts freely on D� The existence of such an g is proven in Lemma 3.5 below. The induced map Mo(U)~ Mo(V) is compatible with the action of To(V, ~'~) and hence is a map of A~-modules. As M~(V) is a free A'F-module by Proposition 3.3, Mo(U) is a finite, torsion-free A~-module. As there is an injection To(U, ~)"-~ EndA~.~ (Mo(U)), the same is therefore true of To(U, (Y). [] Lemma 3.5. --/fg ~6 is an unramifiedprime ideal ofF, then U(e)/U(e)NF � o~ts3~ee~ on D � \GD(Az). Proof. -- If~x= xu for some ~ E D � x E GD(Af), and u E U(g), then fi E Fx where Fx = D � n xU(g)x -l. We claim that Fx/Fx n F � has finite order. To see this, note that the canonical injection i : D� � ~ GD(A)/A � identifies D� � with a discrete subgroup of GD(A)/A � . Now let V = Gr~(R | F)/(R | F) � � (xU(e)x-~/U(s n (A/)� This is a compact open subgroup of GD(A)/A � so W = V N im(i) is a finite group, and it is clear from the definitions that Fx/Fx n F � ~ W. This proves the claim. Thus some power of 5 lies in F � and the same is true ofu. By the choice of g, u must itself be in F � . [] Corollary 3.6. -- Let M be the exponent of the torsion subgroup OfD� � . If{g,, ..., e, } is a set of unramfied primes of F such that (i) gi { 6 and (Nm(gi) - 1, M) = 2~i for each i, RESIDUALIN REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 37 (ii) for each y E ~ ~, that is totally positive some g~ does not split in F(v~) , then U 1 (e I ... es) / g I (e l ... e~) ['-'l F x acts freely on D � \GD(A~). Proof. -- If 8x = xu for some 8 E D � , x E GD(Az), and u E [Jl(gl ,-,gs), then 8*x = xu ~ for e = II(Nm(g~) - 1). Since u ~ E U(g~ ...g,) it follows from I~emma 3.5 that i= I u ~ E F � and therefore ~ E F � . It then follows from our hypotheses on gi that 6 2' E F � for some r. If r= 0, then ~, u E F � We may therefore suppose that r/> 1 but that 2 r-I Fx, o2 Fx" 2r-I r F � Put 7 = 62'' and to = u . Then 7, o) ~ but y2, E Let ot and II be the eigenvalues of 7. As ((x/~) ~ = 1 it must be that either a = II or ot = -13. If a = [3 then 7 E F � since 2ct = 0t + 1~ E F. Therefore a = -ll. Note that det(7) = oc13 must be totally positive. Since o~ and [~ are also the eigenvalues of co and since co E U~(gl ... gs) we find that a and II are in Fe~ for each i and that all E ~; F" � Therefore each g, splits in F(x/-2~) = F(v/~), contradicting our hypotheses. [] Suppose k : T~(U, G) , Qr is a homomorphism of" ~'-algebras such that = klz(u 1 and q0 = kl(~,~.|215 are finite characters. (Recall that we have identified G(U) with Z(U)x (-~v | Z~) � It is not difficult to deduce from the definition of T~(U, C?) that ~ factors through some T~(U~, (~) and hence corresponds to some rl~ 1 rr~ ~ That is to say; there exists a unique n E ..~ ~,~ and an eigenvector ~ *'2 \"~a]" Ua v E V~ for T~(U, 62P.) such that the eigenvalue of each t E T~:(U, C) acting on v, viewed as an element of Qr is just ~(t). The existence of such a n follows from the ord definition of T~(U~, ~(2v). It is also easy to see that any ~ E I-I~. (Ua) determines such a homomorphism ~.. Therefore there is a correspondence between homomorphisms as at the start of this paragraph and nearly ordinary autornorphic representations ord n E U I-I~ (U,). This correspondence generalizes to other weights k as summarized in the following remarkable result of Hida [H2, Corollary 2.5]. Proposition 3.7. -- /f X " T~(U, ('~) , Qr an ~'-a~ebra homomorphism such that ~.[z~/ = 9 e~, g >/ 0, with ~ and g~ = ~.[(~v~zp~� finite characters, then there exists a nearly ordinary automorphic representation n of weight k = (g + 2). t for which ~.~F(g) ) and k(S(g)) equal, respectively, the eigenvalues of T(g) and S(g) acting on the newform associated to n for all prime ideals g r p for which Ue = GL2(~'F.t). Here, as in the preceding section, e denotes the cyclotomic character giving the action of Gal(F~b/F) on the Zp-module lim~p., ~p, being the group of p"th roots of unity. The character ~ factors through Gal(F~)/F) where F~) is the maximal abelian extension of F unramified outside of those places dividing p and oc. Global reciprocity determines a homomorphism Z(U) ~ GaI(F~//F) via which we view ~ as a character on Z(U). 38 C.M. SKINNER, AJ. WILES We continue to assume that the degree of F is even. A prime P of Toe(U, C) that is the kernel of a homomorphism as in Proposition 3.7 is called an algebraic prime. The associated element k = m + 2t E Z[I] is called the wei~t of P. For an algebraic prime P of T~(U, ~') there are finitely many homomorphisms ~. : T..,o(U, C') > whose kernel is P since T~(U, 65')/P is a finite extension of ~5 ~:. Let .~(P) denote the set of such homomorphisms. The set of algebraic primes of T~(U, ~') is Zariski dense, as the following lemma shows. Lemma 3.8. -- Let Q be a minimal prime of T~(U, ~) and let ,2g'(Q) be the set of algebraic primes of weight 2 containing Q. The set 32/(Q) is Zariski dense in spec(T~(U, 6':)/0,). Proof. -By Corollary 3.4, T~(U, ~?)/Q is an integral extension of A~:. Call a prime IJ C_ A} algebraic (of weight 2) if it is of the form p = A} NP for some algebraic prime P of T~(U, U'.')/Q of weight 2. The algebraic primes of A~.: are just those corresponding to kernels of homomorphisms A' , ~ sending 1 +YJf, , q)~vj") and 1 + Xi, , ~(xi) for finite characters q0 and V of (~?v | Zp) � and Z(U), respectively. That such primes are Zariski-dense in spec(A~. ) is immediate. [] Corollary 3.9. -- Let Q be a minimal prime 0fToo(U, ~). /fV D U is such that some )~ E ,~ (P)factors through the map T~(U, ~>') , T~(V, ~')for all P in a subset of .3g'(O~ that is Zariski-dense in spec(Too(U, ~')/Q), then e ~ the inverse image of a minimal prime of T~(V, r 3.3. Hecke algebras, representations, and pseudo-representatlons In this subsection we assume that F has even degree. Let U C_ GL2.(6~: v | Z) be as in the preceding subsection. Write n for the product of those prime ideals g . Vl~ fl " T for which Ut ~= GLo(Cv.e). Suppose that k = ~:- t with  /> 2. Let r~ E --h ~,oa) and let )v 9 Tk(Ua, .~) > ~ be the corresponding homomorphism. Suppose that 7t is ordinary. In [W2] it was shown that there exists a continuous, irreducible representation p,~" Gal(F/F) , GLg(O_.# ) such that (3.{~) 9 Dn(Zl) = (1 --1) 9 p~ is unramified at all primes g {np 9 trace o~(Frobe) = ~.(T(g)) for all g ~np 9 det p~(Frobt) = X~S(e))Nm(e) for all g ~np 9 det On(X) = ~.(Sx)e(x) tbr all x E Z(U) 9 Pro[l), ~ t~{i) V~) with ~'?(y)= )v(Ty) for all y E ~"~., ~,~ and v9 v"~, ) = ~.(F0(Pi) ) for all i = 1, ..., t. RESIDUALLY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS ord Now suppose that n is any element of rI~ (U~). Given any finite set S of finite places of F distinct from those dividing p, there exists a finite character ~ unramified at S and such that n | ~ is ordinary. The representation p~ = (p~)| ~ is independent of ~, and by varying S one finds that (3.2) also holds for this p~. Now, for any representation p 9 Gal(F/F) ~ GL2(()~0 ) and tbr any finite place v ~p let n~(p) be the automorphic representation of GL2(F~, ) corresponding to Pit), via the local Langlands' correspondence (see [Ca] and [Ku]) normalized as in [C]. So, I 1 in particular, if OID~ ~ (gl g2* ) then n,.(p)= n(gl' ] 9 If ~ , ~j~211 9 Iv2). For any ordinary ord representation n E FI t (U) it was shown in [W2, Theorem 2.1.3] that n~(prc) ~ n~. Now 1-;otd/! -~ suppose that n is any element in H k ~uj and that qt is a finite character unramified at v such that n | ~ is ordinary. It follows from the preceding observations that n~, | qt~. = (n | qt)~ _~ n,(Pn| = n,(pn | qr -1) = n,.(pn) | qty. We therelbre have that vI~ ,'~ (3.3) n~.--~ n,,(pTt), n E "'k t'-~J, vCpoo. ]'he representation pn can be generalized as follows. Suppose that Q is a prime of T~(U, ~'). Let R = T~(U, ~')/Q and let L be the field of fractions of R. Note that R is a complete local domain. Hida has shown that there is a continuous, semi-simple representation pQ" Gal(F/F) , GL2(L ) such that (3.4) (i) PQ(Zl) = (1 -1) (ii) Po is unramified at all primes g ~ np (iii) trace pQ(Frobg) = T(g) mod Q for all g ~ np (iv) det po(I"robe) = S(e) Nm(g) mod Q for all e ~ np (v) det pQ(x) = S~ ~(x) mod Q for all x E Z(U) V~) with ~r = ~Iy mod Q for all y E C,"v ~, ,, (vi) p~,D%( v~'' " ) ' and qt 2 (~i ') = T0(p/) mod Q for all i = 1,..., t. By pQ being continuous we mean that there is a finitely generated Gal(F/F)-stable R-module ,////~ in the underlying representation space of PQ. such that Gal(F/F) acts continuously on J/-g. We give a proof of the existence of Pe. in the next few paragraphs. If Q is an algebraic prime of weight 2, then the desired representation follows immediately from the existence and properties of the representations pn. For let ~. E .5gr (Q) and let n be the automorphic representation corresponding to ~.. The homomorphism ~." T~(U, ~:~) , O p determines an embedding R~--* Qp which extends to an identifcation of L with Qp. Under this identification we may take 94 = P,t. Properties (3.4i-vi) follow from (3.2). Suppose now that Q is a minimal prime. Using Lemma 3.8 one then deduces the existence of PQ as in the proof of [W2, Theorem 2.2.1]. Of the properties of PQ 40 C.M. SKINNER, AJ. WILES listed in (3.4) the only one that is not immediate from the construction is the final one concerning the restrictions 9e[Di. This can be deduced from the corresponding properties of the 9p's, P an algebraic prime of weight 2, as follows. First, arguing as in the proof of [W2, Lemma 2.2.4] shows that the semisimplification of 9QID, is the sum of two characters gtl '~ and " (') ~ � V2 with ~)(y) = Ty for each y E ~F.v,- Moreover, (,?, we may assume that gt 1 II,,i ~ for otherwise there would be nothing to prove. Choose ~0 E Iv, such that gtl')(~0):~ gt~)(~0). Let ~" C f(Q) be the Zariski-dense subset of primes P for which ~i~('c0) ~: q~)('t0)mod P. Choose a basis of 9Q such that pQ('c0) = (~ [3 ) and Po[,~, = (x~ ~2) with either Z~ = ~]~? or Z, = ~?. If 9QIDi is split, then property (vi) is immediate, so assume otherwise. Let ~0 E Di be such that 9Q(~j0 ) = (* bo ) with b0 * 0, and let go E Gal(F/F) be such that PQ(g0) = (co *) with co :~ 0. Let R C L be a finite integral extension of R containing a and 13. Let ~- be the set of primes in R consisting of those primes P such that P M R is in ~'. Let C ~- be the subset consisting of primes P such that boco E R{. The set ~ is non-empty. For each P E o~; it is not difficult to see that pQ = b0 pQ bo t ) takes values in GL2(I~). As 9}l]DmodP is nonsplit for each prime P E ~; it follows that for P E o~;, )~lmodP = vl0modP. Since V~ ) ~ ~')modP and either ;(l = ~i '~ or gt~ ~ we conclude that ;~ = gt/~ '). Note that arguing again as in [W2, Lemma 2.2.4] (z)/,~ (Pi),~ _. = shows that either V2,'~,, T0(p/)modQ or gt?()~"/)) T0(Pi)modQ. Arguing as before shows that the former must hold. This proves property (vi). Note that the representation pQ gives rise to a pseudo-representation into R = T.~(U, ~)/Q. This is just the pseudo-representation associated to pQ (cf. w For a non-minimal O~ C__ Q the representation pe, can be constructed in the usual way (cf. end of the proof of [W2, Lemma 2.2.3]) from the pseudo-representation into T~(U, (~)/Q' obtained by reducing modulo O~ the pseudo-representation associated to pQ. The only property that is not immediate is (3.4vi). For this we note that if pQ, is reducible then there is nothing more to prove (as one of the characters has the desired property), so assume that pa is irreducible. Let R be a finite integral extension of R containing the values of ~t~"' and gt~ ), and let (~ be an extension of O~ to R. It is easy to see that the semisimplication of Ooj I~i is the sum of the characters ~t:i ') and ~ " (') modulo (~. If ~:2 = gt~)m~ then there is nothing more to prove. If gi'~ ~,)modQ~, then for a suitable choice of basis oo takes values in R~, and satisfies OoJi~ = (~r * ) ~r - Reducing modulo Q~ yields the representation 9Q'. Property (3.4vi) is now immediate. Suppose now that m is a maximal ideal of T~(U, ~). Patching together the pseudo-representations for the various minimal primes Q contained in rn yields a ~l/~?[I~.~ RESIDUALIN REDUCIBLF REPRFSENTATIONS AND MODUI,AR I,'ORMS 41 pseudo-representation p'~'"~ into T~(U, ~"),, (here we have used the fact that T~(U, ~) is reduced) satisfying (3.5) (i) pmod is unramified at primes g ~ np, m~ ~ = T(g) for all g ~ np, (ii) trace p~ ~ .... tJ mod r (iii) detpm (F ob~) = s(e)Nm(e) for all g r np. Let ~ and k be as in w Henceforth #' is the ring of integers of a finite extension of O~ having residue field k. A maximal ideal of T~(U, #:~) is permissible if m A #:: JIG(U)]] is the maximal Zo ~ - i ideal corresponding to the character G(U) ~ Z(U) , k, if m contains T0(p~)- 1 tbr each i = 1, ..., t, and if p~ -~ X | 1. Such a maximal ideal, if it exists, is unique. For this reason we will drop the subscript m from the notation for p,,od whenever m is permissible 9 Suppose that m is a permissible maximal ideal of T~(U, 6~'). The ring T~(U, ~')m is an algebra over A~ "~4a 1 + Y!'~ ~ ~ T (i: and 1 + T i D , det prnod(~). The - .I Yj 9 r. r. homomorphism A' C ~ A,< determined by 1 + Xi~ . (1 + Tj)P~e(~ 7#) is compatible with the A~.-algebra structure of To~(U, ~)m and makes A~ a free A~-module of rank r = 2r i. Consequently, we obtain the Ibllowing lemma. Lemma 3.10. -- If m is a permissible maximal ideal OfT~(U, ~'), then (i) T~(U, G)m is a torsion-flee, finite A~ -algebra, (ii) Jbr U satisfying the hypotheses of Proposition 3.3, M~(U)m and M~(U)m are .free Ar -modules of equal rank 9 Let Z be the places of F for which U,: 3(GLT(~F.,.) together with vl,..., v~. If m is a permissible maximal ideal of T~(U, ~'), then it is easy to see that pmoa is a pseudo-deformation of type-6.~ p~ = ( ;G., Z). Consequently, there is a map (3.6) R~ p~ , T~(U, ~:~)m inducing p m~ Lemma 3.11. --- Suppose that m is a permissible maximal ideal of T~(U, c<~'). /f S is any finite set of primes of F containing all those for which Ut ~ GL.,(~'v,e), then the ring T~(U, -#~)m is generated over he': by the operators { T(e), S(e)" e r S }. -- Let T s C T~(U, ~)m be the subring generated over A~ by Proof. 9 g)ES}. Note that T s is a local, complete A~-algebra 9 Let p m~ = { T(e), s(e) { x(o, t) } and let Z = S U { v~, ..., v, }. The pseudo-representation p m~ factors GaI(Fx/F). Since Gal(Fx/I:) is topologically generated by through Z} and since trace p m~ and detp rand are continuous maps, it follows { Frobt : 42 C.M. SKINNER, AJ. WILES that T s contains trace prnod(ty) and detpm~ for every ~ E Gal(F/F). It remains to show that T s contains T~' tbr eachy E (~'~,QZp) � as well as T0(p/) for each i = 1, ..., t. Let gi E Di be such that )~(gi) :~ i. Let 0~i and 13i E T s be the roots of the polynomial X 2- trace p'n~ + detpm~ with a~ reducing to 1 modulo m. One then has Ty = ([~i- oti) ([~i trace pm~ -- trace pm~ E T S ~ x ab fory E .~ v, ,~,, where % E Di is anv lift of the element of Gal(Fo/F~) corresponding to y via local reciprocity. Similarly, one also has T0(Pi) = ([3i- 0~i) (~i trace ~3nl~ -- trace 9m~ E T s where ~i E Di is any lift of an element of Gal(~ b/F,:) corresponding to (X~P'~). These expressions for 'F~, and T0(pi) can be checked for each Po., Q a minimal prime of T~(U, ~)m, using (3.4) 9 [] , T~,(U, ~)~ /s Corollary 3.12. - /f V C_ U, then the natural map T~], ~')., surjective. Corollary 3.13. The map (3.6) is surjective. We conclude this subsection with a t~w results about the "level" of a prime of T~(U, ~'). The first of these is a generalization of Carayol's n~, _~ n(~,:) result (see [C]). Indeed, its proof boils down to Carayol's result as generalized in [W2]. For w a finite place of F write gw for the prime ideal of ~v associated to w and write A~,, for the Sylow p-subgroup of (.~:' F/gw) � . Proposition 3.14. -Let w ~ p be a finite place of F. Suppose that U C_ GL2(~ v Q Z) /s such that U~, _D {(a b) E GL2(~v ~)" c, a- 1 E g~. } for some s. Given a minimal prime Q c_ T~(U, U) there exists a subgroup V D_ U such that Q is the inverse image of a prime of To~(V, C) and V satisfies (i) /foe ir unramfied at w, then V _D GLe(6"F,w); 9 c E ~,~ ] (ii) f pe /s type a at w, then V D_ I ( ci b) E GL2(6-:' F 9 "') a rood g ~,, E Aw f; is type B at w, then V D I (~i b) (iii) E GL2(r - 1,c E wh e e2 I,', w) 9 a if Pe is the conductor 0f~ = detpoli,,; -t 5- . ~. 9 (iv) if 9(2 is type C at w, then V D_ ,. ,1 EGL2(C~I,',w) a- 1, c E g~ S RESII)UALLY REDU(JIBI,E RI'PRESFNTATIONS AND MOI)UI,AR FORMS 43 Proof. -- Recall that types A, B, and C were defined in w We first claim that for P E .:~'(Q) the representation pp is of the same type at w as 9o. and that if , with condw(,) = gw then PPlI~ -~ ,, with condw(q~') = g~, as well. In light of Corollary 3.9 it then suffices to show that some ~, E ,~f (P) factors through T~(V, ~') for some V as in the statement of the proposition. To prove the claim, first assume that OQ is unramified at w. In this case it is obvious that each 9P, P E .~'(Q), is also unramified at w. This can be seen, for instance, by observing that the pseudo-representation associated to 9Q. is trivial on Iw and hence the same is true of the pseudo-representation associated to Pc- As pp is irreducible, this tbrces pp to be trivial on Iw. Next assume that pe, is type A at w. If P E J~)'(Q), then it is easily deduced that the semisimplification of 9vi~,. is just 9 egt for some character ~ unramified at w. If PPID, were unramified then this would contradict (3.3). Therefore, it must be that OI~ID,,. is ramified, and it follows from the description of its semisimplification that it must be of type A. Now suppose that Po is type B at w. Write 9oJl~ = ~lg?| 1 with ~l of order prime to p and gt2 of p-power order. Note that cond~,,(gtl~?)= max(cond (~), cond (~2)) and that both It./1 and g2 take values in T~(U, C)/Q. It follows that PPI~= -~ (~L~r mod P) | 1. As p ~ P one sees that cond,,(gtl~t? modP) = max(cond (gtl), cond (~)), proving the claim in this case. The remaining case (i.e., po being of type C at w) is proved similarly. Now let P E ~,q~/(Q) and choose ~, E .~,7r Let n be the automorphic representation corresponding to ~.. rib prove that ~, factors through T~(V, ~") for Fl~ some V as in the statement of the proposition we need only show that rc E ..,) t,:. In other words, we need to show that if W;~ is the underlying representation space for =, then W v ~: 0. Let x = | E W~; and let x' w E W~,= be the new vector at w. It Uzr follows from our hypotheses on U~ and the theory of newforms that x' E W=, w. Put y = @ x~ | ~,,. We claim that y is fixed by a subgroup of the desired type. For this we v:t: t0 note that it follows easily from (3.3) that x' w is fixed by a subgroup of GL~(U"v,w) of the desired type necessarily containing Uw. [] As a variant of the above we have the following result. For a place w of F, let (~)',,/gw) � = Aw x A'. (Recall that A w is the p-Sylow subgroup of (~v/gw) � .) Proposition 3.15. -- Let w ~ p be a finite place of F and let U ~,, = { ~c E GL2(~ F,w) " c E g~, ad-~modg~, E A'w}. Suppose that Q c__ To~(U, ~) is a minimal prime such that oczl,,~ (* ,-1 ) with, of p-power ord . Put U' = rI x U;. There exists a minimal z~ w prime Q~ c_ T=(U', ~:) such that 9c2 ~- 9Q' and such that Q and O~ have the same inverse image in T~(U M U', ~). Proof. -- We prove the existence of a minimal prime Q~ c T~(U', ~)') such that Q~ and Q have the same inverse image in T~(U M U', ~,~). Clearly the assertion that 44 C.M. SKINNFR, AJ. WILES 9o~ -~ 9Q will follow from this. Upon replacing U by U N U' and Q by its inverse image in T~(U M U', ~:') we need only show that Q is the inverse image of some minimal prime in T.~(U', ~). By Corollary 3.9 it then suffices to show for each P C .~'(Q) and ll'~ T '~ ~. C ,~_~gS (P) that )~ factors through T~(U', ~'). Fix such a P and ~. Let u C ..,~ ~,~j be the automorphic representation associated to )~. To know that ~ factors through ord ,I l-'a T~(U', ~) it suffices to know that rc C 11 2 (U~). Now suppose that x = | E V~ . U' ;t We will establish the existence of a non-zero vector x' w C V m ~,.. The non-zero vector U' U' y = @ x~ | ~,. will then lie in V~ ~ showing that V~ ~ :~ 0, from which it follows that l: z~ [U ]lOrd/| It '~ g ~ ""2 \"~a]" ,,_ ~ with We now establish the existence of ~,. First we note that ppll ~ (*' ) 00' a character of p-power order. To see this observe that by hypothesis PqlI,, factors through a quotient of Iw of p-power order and detp@i~ = 1. Hence the same is true of the pseudo-representation associated to 9% As the pseudo-representation associated to 9P is obtained by reducing modulo P the one associated to pQ, it follows easily that PPlI,~ factors through a finite quotient of p-power order and that detpvll,, = 1. That P~'I~, has the form asserted is now immediate. It follows from (3.3) that ~,,, is a S: X principal series representation n(gl, g'2) with ILlg2 trivial on ~' v, ~, and each gi trivial on a subgroup of ~'~ x ~,, (~ v, of index a power of p. Let v0 E V~, ~, be the vector corresponding to a new vector of ~w | 00 where 00 is a finite character such that 00]~ v. ~ w _ ~ g~-l I~- � 9 I:, It' h\ It follows that {: ~t)vo = g,(a)g'2(d) for all [c )) E U~,. Thus x' w = v0 is the desired .e / \ 1_,'. vector in V~, w. [] Next we record for later reference the following relations between PQII~, and the subgroups U~. Lemma 3.16. -- Suppose that w ~ p is a finite place of F. Suppose also that QC_ T~(U, ~:) is a minimalprime and that UwD_ {(~ ~):cE g~, amodg'w E Aw} J'br some r >1 1. (i)/f Pal',, ~- ( *~ **2 ) with 001 and 002 of p-power order and 00 non-trivial of order prime to p, and if cond (00) = g~,, then 001 and 002 are trivial. (ii)/f OQII),,, -~ ( E0  ) and if (Nm(w) - 1, p) = 1, then 001ia. is a finite character of order prime to p, and if r = 1, then 001Iz,, = 1. Proof. -- Let R = T~(U, ~')/Q. We first prove (i). The characters 00~, 002 and 00 take values in R, so they may be reduced modulo P for any P E ,~'(Q). We denote these reductions by 001,P,002, p, and 00P, respectively. The reduced characters have the same orders as the corresponding non-reduced characters. Now fax a choice RESII)L'ALIN REDUCIBI,E REPRESENTATIONS AND MOI)UI2kR FORMS 45 of P E ,2Z'(Q) and ~. E .~ (P). Let ~ be the nearly ordinary automorphic representation c~176 t~ )~" It is easy t~ see that PI'[', "~ %'p- ( *P*2, v) from. It follows this description of PPIL, and from (3.3) that cond(n,,,) = condw(qbt, Cp,) p)cond,.(Ol,p ) = (b) gw" cond(Ol,p). However, since by hypothesis x~,. has a vector fixed by { ~ ,1 " c E g,~,, amodgi~ E A~,} it follows that cond(xw)l e2 and that the restriction to Iw of the central character of x~: has order prime to p. From this we deduce that qbl, p and r are both trivial. As 0h,~ and ~2, P have the same order as ~1 and ~2, respectively, the latter are trivial as well. We now prove (ii). Our hypothesis on Nm(w) ensures that ~ takes values in R. As in the proof of (i) we write r for the reduction of ~ modulo P. The character 0Op[i~. has the same order as does {~[I~,. Again, fix a P E ,542'(Q) and a )~ E ,~ (P). Let n be the automorphic representation corresponding to ~.. From the hypotheses on U we find that cond(nw)]g~,, and that the restriction to I~. of the central character of n~ has order prime to p. It is easy to see that PPID,--~ (eOr, *t,* )" It follows from (3.3)that cond(n,,.) = max(g~,,, cond(qb~,) ~) and that the restriction to I~. of the central character of xw is just @ L,." From this, one deduces that Cvilw, and hence OO]L, , has order prime to p. And moreover, if r = 1, then cond(r [g .... hence 0Ov]]~ (and so q~[l~) is trivial. [] We conclude this subsection with a simple observation about twists of the representations pQ. /emma 3.17. - - Suppose that Q c_ p c_ T~(U, ~:) are primes with Q minimal. Let L be the field of fractions OfT~(U, ~)/Q and suppose that R C L is a finite integral extension of T~(U, &~')/Q./f ho " Gal(F/F) , R � is a character of finite order, then there exists primes O~ C p' _C T~(U 71 U1 (cond'~/(qJ)2), (5 :') with O~ minimal and such that PQ' ~- PQ | ho and pp, ~ pp | u?. Here cond~~ denotes the prime-to-p part of the conductor of q~. Proof. -- Let V = U 71Ul(cond(q~) 2) and let R0 = To~(U, u~,')/Q. Let .3~ '(Q) be the set of primes of R extending those in ,~'(Q). For each P E .3~"(Q) we write ,Sref (P) for ,~ (P 71 Ro). Now, for each P E ,~"(Q), let hop = ho mod P. Let n be the product of those primes g such that either glcond*)('e) or Ue :1: GL2(6"'F,g). We claim that tbr each P E ,3~"(Q) there exists a homomorphism zp : T~(V, ~') , R/P such that (3.7) 9 zp(T(g)) = ~I'(g) rood P). ~Fp(Frobt) for all g r np; ,p(S(g)) = (S(g) mod P) 9 Udp(l~robt) 2 . for all g { np ; r = (Sx mod P). ho~,(x) for all x E Z(U); t. (Pi)\- 1 zi,(T0(Pi)) = (T0(Pi)modP) 9 ~FpCLpi ) for i = 1,..., t; "~P('I~r) = (T~' mod P)- hop[y) for all y E r and each i = 1,..., t. 46 C.M. SKINNER, AJ. WILES ord We construct zp as follows. Let )L E ~ (P) and let zc E I12 (U~) be the corresponding automorphic representation. We fix an embedding R/P ~ O~ extending the embedding C L coming from We thus view as taking values in CL � hence in ~x (via the fixed embedding F~--~ ~). Clearly 7t | q~p E II~"d(vb) for some b /> a. Therefore there exists an algebraic prime P~ of Too(V, O) whose corresponding representation is just PP,v -~ Pp | ~Pp- Let zp E o~(P,v) be the homomorphism corresponding to rc | ~Fp. Viewing R/P as an O-subalgebra of O~ as above, we see from the fact that 9Pv ~- PP | ~Fp that zp takes values in R/P and satisfies (3.7). Now consider the map "c 9 Too(V, G) > I-I R/P given by z(t) = Ilzp(t). PE.5~"(O0 It is easily deduced from (3.7) that the image of z is contained in the image of the diagonal embedding R~-+ II R/P. In particular, x determines a homomorphism pc.~2,(Q) z 9 Too(V, ~) , R such that "c(T(g)) = (T(g) mod Q). ~(Frobe) (3.8) for all g ( np "c(S(g)) = (S(g) mod Q). ~2(Frobe). Let O~ = ker('c). By (3.8) we have 9o2 "~ Pc)_ | u?. Moreover, by comparing dimensions one sees that O~ is minimal. Let P l be any prime of R extending p. Let p' be the kernel of the composition Too(V, ~) ~> R , R/pl. Obviously p' D Q~. Also, it follows from (3.8) that Pp' ~- Pp | tF as well. [] 3.4. Eisenstein maximal ideals (existence) In this subsection we establish sufficient conditions for the existence of a permissible ideal of Too(U, ~). We continue to assume that the degree of F is even. An equivalent definition of permissible maximal ideal is a maximal ideal m of Too(U, G) such that (3.9) 9 m V/& [[G(U)]] is the maximal ideal corresponding to the character ~r l a(u) ,z(u) ,k, 9 m contains T0(p/)- 1 for i = 1, ..., t, and 9 m contains T(g) - 1 - ~(Frobe) for each g {p for which Ut = GL2(~F,e). We will also call a maximal ideal m of any T2(Ua, ~) satisfying (3.9) a permissible maximal ideal since any such ideal determines a permissible maximal ideal of Too(U, ~). Clearly, to conclude that Too(U, 8) has a permissible maximal ideal it suffices to show that T2(U~, ~) does for some a (and hence for all sufficiently large a). This we do, provided a certain p-adic L-function is not a unit. Let rt be the prime-to-p part of the conductor of ;(o) -1. For each prime gin let t4t)lrrt and write Ae for the Sylow p-subgroup of (~v/g) X, which we think of as a RESII)UALLY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 47 subgroup of (~s:v/g~e") � . Define an open compact subgroup U x = 1-I U~ C GL2(~)v@Z) as follows: if gin, U~= {{(:GL,)(~'v,t) :) E GL2(~;v't) c E g~e',, amodg~g)GAg} otherwise. Let Lp(F, s, ~to) be the p-adic L-function associated to ~to (cf. [Co], [D-R]). Let n denote a uniformizer of ~. Proposition 3.I8. -- /f ordn(Lp(F, - 1, Xm)) > 0, then some T2(U~, ~') (and hence T~(U x, ~*)) has a permissible maximal ideal. Proof. -- Let ~r = )~t0 -1. For each integer n /> 2 let Mn(pn, gt) C Mn(pn) be the subspace of modular forms having nebentypus character gt. Define S,(pn, ~) similarly. For a pair of characters ~l and r for which ~)1r = ~ and cond(~l)cond(r let E,(q~l, ~2) E M,(pn, gt) be the Eisenstein series whose associated Dirichlet series is L(F, s, q),)L(F, s-n+ 1, *2) (cf. [Sh]). A complement for the space S,(n, ~) in M,(n, ~) is spanned by the set { E,(r r It is well-known that (3.10) ai(E,(1, V), 0) = 2-dL(F, 1 - n, ~) Nm(ti) '~ where the ai(E,(1, ~/), 0) are the constant terms of the Fourier expansions of E,(1, ~) described in w Let "/be a generator of the Galois group of the cyclotomic Zp-extension of F, and let f= ordp(e(~/) - 1). Now let n = 2 +pf(p - 1)m E Z be so big that there exists a modular form g E M,(np, gt) M M,(np, C) such that (3.11) ai(g, 0)= 2-dNm(ti) ~ , i= 1, ...,h. The existence of such a g for large n is proven in [Ch, w Let E0 = E,(1, ~). It follows from our choice of n and standard facts about p-adic L-functions that ord~(L(F, 1 - n, V)) > 0 as well. Thus E0 E M,(np, ~'). Consider the form f= L(F, 1 -n, gt)g-Eo. By (3.10) and (3.11) this form satisfies ai(f , O) = O, i= 1,...,h. We also have f E M,(np, ~). Let e E ~',(U, (n)71 U(p), ~) be the operator defined in w It is easily checked that E~ = eE0 is the modular form whose associated Dirichlet series is just ~v(s)L:P)(s -- n + 1, ~), where for an ideal a we write L"( 9 ) to mean that the Euler factors at places dividing ct have been removed. In particular, El is an eigenform for the ring T,(U~(n) M U(p), ~':). Let e0 E T,,(U~(n) N U(p), ~:) be the 48 C.M. SKINNER, AJ. WII,ES idempotent associated to the corresponding maximal ideal. The tbrmfcan be expressed as f= F + G with F E Sn(np, ~) and G a linear combination of Eisenstein series, say G = Zc(,)E,,(qb, ,-Iv) , where the sum is over those ~b such that cond(~b), cond(qb--Iv)lnp. It follows that e0G = Zd(~.)E,,(~., ~.-1V) where the sum is now over the unramified characters of p-power order. Since the constant terms off are zero, the same is true of those of e~ Thus e0G must have all constant terms zero (i.e. a~(e0G, 0) = 0 for i = 1, ..., h). Arguing as in the proof of [W3, Proposition 1.6] shows that e0G = 0. It follows that FI = e~)ef is a cusp form. Moreover, since ~IL(F, 1 - n, V) by hypothesis, we have Fl = El modrc. As El ~ 0mode, FI ~= 0. As El is an ordinary eigenform (in the sense of [W2]) and Fl = El ~ 0, it is easily seen that there must be an ordinary newform f such that ~0 f :~ 0. The tbrm ef is a p-stabilized newform in the sense of [W2]. Let 99I be the maximal ideal of the ring of integers of Qr The non-vanishing of e0 f means that the coefficients of the Dirichlet series Ln(ef, s) associated to ef are congruent modulo 9)I to those of n np ~v(s)L (F, s- n + 1 V). By the theory of "A-adic forms" developed in [W2] (see especially [W2, Theorem 1.4.1]) there is some p-stabilized newform~ E S2(np ", V), for some large a, such that the coefficients of L"(~, s) are congruent to those of L"(ef, s) modulo 9N, and hence are congruent to the coefficients of ~v(s)L n ,p (F,s-n+ , 1,V). Being a p-stabilized newform, ~ spans a v-good line in the associated local automorphic representation for each place viP. Thus ~ is an eigenform for the ring of operators T2(Ua, ~) (U = Ut(n)). Let m be the corresponding maximal ideal. We claim that m satisfies (3.9). The second and third properties listed in (3.9) are consequences of the connection between the eigenvalues of the Hecke operators and the coefficients of the Dirichlet series L"(J~, s). The first property listed in (3.9) follows from the t~ct that f, is ordinary (so the operators T~, Y E (~ F @ Zp)x, act trivially) and that J~ E S,,(np a, V) (so S~, x E Z(U), acts via )~0~-t(x)). It remains to show that this maximal ideal occurs in T2(U~, ~). This follows from the fact that n/,, the automorphic representation associated to the p-stabilized vl~ I)C\ newformJ~, is in "-2 ~. This last fact can be seen by considering the possibilities for pp,,lD e at primes g I n (P,) being the algebraic prime of T~(UI(n), ~') corresponding toJ~) and invoking [W2, Theorem 2.1.31 or (3.3). [] 3.5. Some miscellaneous results We keep the conventions of the previous sections. Suppose that U = IIU~, C_ -1 5)~ GL2(C~ v | Z) is a compact open subgroup as usual. In this subsection we consider the effect of altering U at one selected place w. RESIDUALIN REDUCIBLE REPRESEN'FATIONS AND MODULAR FORMS 49 Let w r p be a place of F such that U~ = GL2(~F,w). Let A~ be the Sylow p-subgroup of (~'v/g~,) x and let A 2 be a complementary subgroup (so (~F/G) � ~_ Aw x A'~,,). Put 9 c E f~,} U' w = {(~ ~) E GL2(~r,,) and : ad-lmodg~ E A'}. Put also U'=U'-IIu~ and U"=U'~',.I-IU~;. There is a natural isomorphism (3.12) U'o/U~ ~ , a~ given by (~ ~) ~-* (image of (ad-l)~, in a~). Recall that U'~ = U' N U(p ~) and similarly for U'a. The group U'~ acts on S~(U~') with g E U'~ acting via thc Hccke operator '" "" - 1, Till I It [tGg tJ~l. This action clearly factors through the quotient U~/U~ and hence determines via the isomorphism (3.12) an action of A~ on Sk(U;'). Under the Jacquet- Langlands correspondence (see w this action is compatible with usual action of Uta/Uta , on {f: Dx\GD(Ad)/U'~ ' , (3 } = H~ (3) given by (gf)(x) =f(xg) for , H 0 'q g E Ua. This action clearly stabilizes (X(U~,, Z) and hence we obtain an action of" H~ r"~ R) for any Z-module R. It is straight-forward to check that the action Aw on ~- -\ ~ a/, of Aw commutes with that of G(U~') and the Hecke operators T0(p/), i= 1, 9 t, and T(g) and S(g) for primes g Cpw tbr which U, = GL2(CF, e ). Moreover, the action of A~ is compatible with varying a. We also have that (3.13) 0 -, 0 ,, H (X(U~), R) A~ = U (X(U~), R). If every element of U'/F x ffl U' acts without fixed points on D� then much more is true, as the following lemma shows 9 Lemma 3.19. -- If each element of U'/F � n U' acts without fixed points on D � \G D (At) then (i) ModU") and M*~(U") are fiee A~ ~&,,]l-modules, (ii) M~(U )a,,; = M~(U') and + " ,, Moo( U )a,~ = Moo(U + ' ). Proof. --By (3.13) we have 0 it #H (X(Ua) , 4"~/rQG = #H0(X(UI{), (~)/gn)G = #H~ ,,, 4"47 /r('). 50 C.M. SKINNER, AJ. WILES On the other hand, it follows from the hypothesis on U'/F � N U' that #X(U~') = #Aw #XCU'~). Combining these observations we find that H~ .~.'/n") is a free C/g"[[Aw]]-module. The lemma follows from this, the definitions of M,,~(U") and + II M~(U ), and Proposition 3.3. [] It is a consequence of the lemma that many characteristic p primes of T~(U", (2:) (i.e., primes containing p) come from primes of T~(U', (2:'). We state this more precisely in the next proposition. Note in particular that we are not assuming anything about U'/U' N F � . Suppose that p _C T~(U", ~') is a prime such that 9 pep, 9 det 9p = Z, 9 Pplr~,.--~ (~z ~2" ) with (~l/~t2)l,, having infinite order for some vlp, 9 9p is irreducible but not dihedral (i.e., not induced from a one-dimensional representation over a quadratic extension). Proposition 3.20. -- The prime p is the inverse image of a prime 0fT~(U', ~"). Proof. - Choose o E Iw such that co(o) = 1, detpp(~ ) = 1, and pp((y) has infinite order. Such a ~ exists by the hypothesis on 9p]l~.. We claim that there exists r E Gal(F/F) for which c0(r) :~ 1 and an n such that 9p(gnr) has infinite order. To see ,r this, choose a basis of 9p for which 9p(o) = (a u-1 ). If pp(r) E ~(* *)} for each "c such that c0(r) ~: 1, then it would follow that 9p is dihedral. Therefore there exists some (o('c0) 1, for which pp(r0) = ( ~ b) with either a ~: 0 or d ~: 0. Suppose now that "go, 9p(o"~0) always has finite order. In this case, the roots of X 2- (oCa + a-"d)X + )C('c0) are roots of unity lying in some quadratic extension of the field of fractions of To.(U", ~')/p. As there are only finitely many such roots of unity, there are only finitely many possible values for o~"a + a-"d, which is easily seen to be absurd. Let s be the set of finite places v such that U'/ :~ GL,)((2~F, ,:). Fix now an no for which pp(c"~'c0) has infinite order. Let Frob e E Gal(F~-/F) be a Frobenius element such that g { 6 is unramified in F, det pp(Frobe) = )~(r0), m(Frobg) = (o(r0) :~ 1, and pp(Frobg) has infinite order. Such a prime g can be found by choosing a Frob t sufficiently close to o'er0 in GaI(Fz/F). Put V' = U' 71 U(g) and V" = U" N U(g). The prime p of To~(U", ~:') determines a prime of T~(V", ~:') (the inverse image of p) which we also denote by p. We now claim that p comes from T~(V', 6:::). "lb see this, note that by Lemma 3.5 and Lemma 3.19, M~(V") is a free A~,. [lAw]l-module. Let T C EndA) ~ (M~(V")) be the ring generated by As. and T~(V", 6'). This is a finite RESIDLTALLY REI)UCIBI,E REPRESEN'IATIONS ANI) MOI)UI,AR FORMS 51 integral extension of T~(V", ~J'). Let Pl be an extension of p to T. As p 9 p, it is clear that { 8 - 1 : 8 E A~: } C Pl. As Mec(V", ~9) is a thithlhl T-module, we have that Fit~/p, (M~(V", ~)/p,) = 0. Since T/pl is a domain, it follows that it acts faithfully on M,,c(V", C)/pl. Put B = T.~(V", C)/p C T/p~. It follows that B acts faithfully on Moo0g", ~:')/P~. On the other hand, M~(V", ~r is a quotient of M~(V", ~;)a,, = M~(V', ~') by Lemma 3.19(ii). As the action of T~(V", eg) on M~(V', 6"') is via the natural map T~(V", ~") -+ TooOg', ~:) we have T~(V", ~)/p = T~(V', ~')/im(p) which proves the claim. Write P2 for the corresponding prime of T~0g' , ~'~) (so P2 = im(p)). Our final claim, which proves the proposition, is that P2 is the inverse image of a prime of To~(U', ~'). Let Q c p~ be a minimal prime of T~(V', ~). [t suffices to prove that Q. comes from a prime of T~o(U', ~'~). Consider po_[D e . As f does not divide p, p { (Nm(g) - 1), and pp., --~ pp is unramified at g, there are three possibilities for POJDe : (i) pQID e is unramified at f, (ii) polD e is of type A, (iii) polD e is of type C. If the first possibility holds, then the desired claim is a consequence of Proposition 3.14. We will now show that the second and third possibilities cannot occur. If P@Dg were of type A, then the eigenvalues of pQ(cg) (~e a lift of Frob~), say ~ and 13, would 0~ satisfy ~ = e(g) or -- = e(g) -~ The same would then be true of Pp~(~t). However, since P9 contains p and detpp., =)~ it would follow that the eigenvalues of ppe(~) would have finite order, contradicting our choice of g. Similarly, if Po were of type C at s then trace Po(~e) = 0, but we have chosen f so that trace Pp(~e) ~: 0. ]'his final contradiction completes the proof of the proposition. [] We now assume that T~(U, e'~) has a permissible maximal ideal. The same is then true of T~(U', eg:) and T~(U", ~5: ). We also assume that the place w satisfies z(Frob ) = 1 as well as (~(Frobw) = 1. Let ~,,. ~ I,,. be a generator of the p-part of tame inertia. We identify (~w with ~,~.� (m,,) an element of ~ F, ~ via local reciproci~. The element 8w = , generates &,, via (3.12). Recall that both &,. and T~(U", ~'~)m act on the module M~(U")m. Lemma 3.21. - The element trace pm~ E T~(U", ~')~, acts on M~.(U")~ via 8w+8~ ~. Remark 3.22. -- Since both Toc(U", ~'~)m and A~ are contained in Ende 7 (Mo~(U")m), the lemma identifies &, + 8~,i I with an element of T~(U", ~s')m. Moreover, this identi- fication behaves well with respect to varying U. 52 C.M. SKINNER, AJ. WILES Proof. -- Let V _C eH~ ~i:)m | ~ be an eigenspaee for the action of T~nt", U). The space V is stable under zXw. Under the Jacquet-Langlands correspondence V is identified with a subspace of S2(U~'). Let ~." T~(U", ~;) , O.~ be the homomorphism gi~fing the action of T~(U", C) on V. Clearly ~ factors through l-lOrd ~ ttr Tg(U~', ~i) and hence is an algebraic homomorphism of weight 2. Let 7t E ,) (U~) be the corresponding automorphic representation. The space V is identified with a subspace of V~ 9 We now determine the action of 8~ on V, which is via ~ ( ~' ] ) 9 First we note that ~, cannot be supercuspidal. To see this, let P = ker0v ). The prime P is clearly contained in m. It then follows from (3.3) that if ~. is supercuspidal then Ppll),,, is type C, but clearly this can only occur if x(Frobw) = -1 contradicting our assumptions on w. Now suppose that ~. = 9(its, It~l J~ -1) is a special representation. It follows from the definition of U~. that gl is unramified. From this we find that n~,(~w 1 ) = 1. If ~,. "~_ ~(gl, It,.,)is a principal series representation then it must be \ / that It! and It7 are tamely ramified and ItlIt~ is unramified. Moreover, the action of (oo) 1 on V~.~, is by either gt(~,) or It2(cw) = gTl(gw). Now it follows from (3.3) that if n is either a principle series representation or a special representation then pl,(~,) = (gl(~,) ~]-~(~w) * ) . Thus we find that trace pp(Gu, ) - Itl (cu,) + ItTl(G,r) -= 8,,~ + 821 [] 3.6. The rings T~ and T~ ~ In this subsection we associate Hecke rings to various deformation data. Essentially this is done by first defining a suitable open compact subgroup of GI,,(6~v | Z) and then localizing the corresponding Hecke ring at a permissible maximal ideal. To ensure the existence of such a maximal ideal we henceforth assume that Lp(F, - 1, Xm) is integral but not a unit (see Proposition 3.18). We are, of course, also assuming that the degree of F is even. Suppose that ~ Q = (~)~, Z, c, .//g)e is an (augmented) deformation datum 9 As in the previous subsections, for each finite place w we write gw for the prime ideal of F corresponding to w and we write A~, for the Sylow p-subgroup of (~;'v/gw) � which we identify with a subgroup of ((J ~" v/g,,) r � for any r/> I. We also write A t for ~ x __ t a complementary subgroup of (8 v/g~,) (so (~'F/g~) � ~ &L, X ZX~,). We define r(w) by RESIDUALIN RE1)UCIBI,E REPRESENqTkTIONS AND MODULAR FORMS 53 r(w) g~. ][cond(z~,'). 'Ore define a subgroup U~,~ = 1-I U~ C GL.~(~v | Z') by putting if w ~(Z\~') U Q if w ~ Z\(.~, 7~ U ,/gg) ]ii'" i ,> > U~ q,. = 'l[ ~ ~ m~ ew". max(l , <w) ~ )/~w~ Aw'} , if w ~ .~g. ifw6 Q. {(c a :)~GL~(~',:,w):C~g~., ad-'modgw~A:} Let m be a permissible maximal ideal of To,(U~ ,.t' ~)" Put T~ o_ = T~(U~ a, r mod We define T~ to be T~ 0. We write 9~Q for the pseudo-representation into T~Q described in (3.5). This is in fact a pseudo-deformation of type-C_Z~, 6.~Z~. = (~, Z)Q, and we write n~ : R~ p., ~ T~ and n~ct " R~ ~ ~ T~o_ for the corresponding maps Suppose now that 6_~Q. = (~, Z, c, .-.tg)O - is an augmented deformation datum. rain of T~a. This quotient At times it will be necessary to work with a quotient T~Q is defined as follows. As in w let L~/F be the maximal abelian p-extension of F unramified away from the places in Z\M. Let Gal(L~/F) _~ H~ (9 NN be the decomposition fixed in w (Ny is the torsion subgroup). It is a consequence of our definition of U~ t that if q C_ T~ z is a minimal prime, then ~-~ detgq factors through Gal(L~/F). Let ,.//g(~-~e) be the set of minimal primes q of T~o and let ,_/~min(~..@yQ) min be the subset of those q for which (%- ~ detpq)lNz is trivial. Define T~o " by rnim __ T~Q T~Q/ A q" For this defmition to make sense we must show that dgmi~(6.~f@ :~ 0. TO this end, fix another decomposition Gal(L~/F) ~ M~ x N~ with M~ the free Zp-summand 1 mod __ generated by Tl,-.., 78F" (For the definition of the Ti's see w Write aet 9~ Q O-W. with W trivial on N~ and O trivial on M~. Let ~ be a square root of O (i.e., qi-' = O). It follows from Lemma 3.17 and from the definition of U~q that given any q C ,//~ (6_~e) there exists some qo E ~r (6~e) such that 9q. ~- On | ~-~ Clearly ~-~-detpq . is trivial on Nx, and so q,~ C ./~min(~@. This proves that/~min((~Q) is non-empty. GL2(~TF'w):Ca 54 C.M. SKINNER, AJ. WILES rain We now relate T~o - to T~ a more directly. Let I~/F be the splitting field of O. A priori, Gal(Le/F) is a quotient of N~. We claim that Gal(Le/F) ~ N~. To see this, let { 9 N~ , ~� be any character. Extend { to a character of Gal(F/F) by first setting it to be trivial on M~ and then composing with the projection of Gal(F/F) onto Gal(L~/F). Choose q E .~,gmm(~_~g, @ and P E '2g~'(q). By Lemma 3.17 there is an algebraic prime P{ of T~ Q such that Pp~ "~ PP | {. By the choice of q, ~-l det Pe is trivial on N~Q, whence ~-l -detpe~ -~ ~2. It follows that {2 = OmodPg. As ~2 can be any character of N~, | has trivial kernel. This proves the claim. Now let X(~'.~) be the group of O_,r215 characters of Ng, which we view as characters of Gal(F/F) that factor through Gal(Le/~. For each q E ./Ig(~J)let R(q) = T~o/q and let L(q) be the field of fractions of R(q). We identify Qp with an ~'-subalgebra of L(q). In this way we may view each ~ E as taking values in L(q). }'or each q E ~//~(66ge) and ~ E X(~) let R(q, ~) be the subring of L(q) generated by R(q) and the values of {. This is again a complete local Noetherian domain. By Lemma 3.17 there is a prime q~ E ./;/g)(~e) such that pq~ --~ pq | ~. We next claim that the set ,//~g' = { qg'q E ,//gmin(6-~./Q), ~ E X(~ r) } is just ,/l,g(~Q). For let q E ,/r (~@, and let { E X(~) be the unique character such that 9 = {mod q. Let q' = q~ ~. Clearly q' E ./~g ram(D@. Also, q~ = q since pq~ ~ pq, (x) ~ '~ pq @ ~-1 @ ~ = pq. This proves the claim. Given a prime q E J~gmi~((-g2ge) and a character { E X(~) we have used that Lemma 3.17 ensures that there is a prime q~ E ./r such that 9% -~ Pq | {. However, more is true. It was shown in the proof of Lemma 3.17 that there is a homomorphism x(q, ~) : T~ a ~ R(q, {) whose kernel is qg and such that (T(e)) = (T(g) mod q). ~(Frobe) (3.14) z(q, (S(e)) = (S(g)mod q). {2(Frobt) for an primes g ~ ZuQ. There is also a homomorphism ,(q, ~) 9 T~e~m~ | ~::[N~] , R(q, ~) such that 0(q, ~) (T(g) | Frobr) = (T(e) mod q)- ~(Frobr) (3.15) O(q, ~) (S(g) @ Frobr) = (S(g)rood q). ~9(Frob,). Now define "c "T~/Q ~ I-I R(q, ~) qE,IL mi.(~ @ X(C_2~7,@ RESIDUAl,IN REI)UCIBLE REPRESENTATIONS AND MODULAR F()RMS 55 and 9 T rnin a(q, , II by "c : 1-I "c(q, ~) and * = II *(q, {), q,~ q,~ respectively. It follows from (3.14) and (3.15) that im(,) = im(r from which one deduces the following proposition. (Note that O(Frob) and ~(Frob~) are mapped by each "~(q, ~) to ~9(Frob~) and ~(Frobr) , respectively.) Proposition 3.23. There is an isomorphism 0fA~ min -- -algebras T~ Q| ~; [N~ ] ~ T~ Q such that T(g) | n, , T(g). r 1 9 n) and S(g) | n, , S(g)- 19(Frob~ -l 9 n). nin Corollary 3.24. -- T~Q is a finite, torsion-flee A~;-algebra. rain Lernma 3.25. -- Under the isomorphism in Proposition 3.23 the element 5~, + 5-w I E Tg Q maps to 8~ + 8~-~ E T ~ Q. We now define a T~ Q-module M~Q for each deformation datum c.5~Q. The obvious choice for M~Q is M.~(U~ Q)m, where m is the permissible maximal ideal of T~(U~ ~). Itowever, for technical reasons we find it better to define M~ Q to be min Mg (~ = Mo~(U~ o)m, min ~ umin where U~ Q I-[ ~ o_. w C_ GL2(~' v | 7,) is such that wtec or if w ~ X a max(l r(w)) if w E mi. U~q,w. 1 "amodgw '' GA~ otherwise. c g E GL2(~F,~) " c = 0modg2~, min The module M~o_ is a T~ o-module (and hence a T~ Q-module by Proposition 3.23) min via the natural map T~ Q -~ T~(U~ o)m. We write ~" R~ ps ~ T~" for the composition of ~ with the canonical surjection T~ -~ ~c9 n. 56 C.M. SKINNER, AJ. WILES 3.7. Duality again We now nmke some important observations concerning the modules introduced in w Fix an open subgroup U C_ GL2(.r F | Z) as in the preceding subsections. We assume that U/U OF � acts freely on DX\GI~(Af). Thus by Proposition 3.3 Mo~(U) and M~(U) are free ~:[[G(U)]]-modules of (the same) finite rank. Let tr(a): C [[G(U~)]] , (9 ~ be the "trace map" given by Zxgg, , X~d (where "id" is the identity element in G(U~)). We have an identification of T~(U, ~')-modules Hom~ ~c,(u,n(ML(U), ~' [[G(Ua)]] ) = Homm ~G(Ua)](eH~ ("~), 6? [[G(Ua)]]) tr(a)o(.) Hom~ (eH~ r 6) ( , )U a = eH~ C) +. Denote by Ka this identification of Home Cw,~.)~(M+(U), ~' [[G(U.)]]) with eH~ 6) +. For b/> a we have a commutative diagram of T~(U, ~]-modules Itom,~ ~c,(bsB(M+~(U), #~ ~G(Ub)~) ~ eH~ ~;)+ 1 ~tr(U,, Ua) ltomr~ IG(v:H(M~(U), ~'llG(Ua)]]) ' eH~ ~?)§ where the left vertical arrow is induced from the natural projection CI[G(Ub) ll ' 6"~ I[G(Ua)]]. We obtain therefore an identification )~o~ 9 Homr~ IG(U)u(M+~(U), ~:[[G(U)~) --~ M~(U) satisfying L~(tm) = t~(m) for all t 9 To~(U, ~"). Recall that there is an isomorphism G(U) ___ (Cv | Zp) � x Z(U) inducing an identification C [[G(U)]] = A~ [[Z0]] with Z0 a finite group 9 Composing K~ with the isomorphism (NI~o(U), A~) ~_ Hom~;, Uc(u)~(M~(U), (~[[G(U) 1 ) I IomA~ + ' + coming from the trace from ~ [[G(U)]] to A~ induces an isomorphism Boo(U) : Homn~ " (M~(U), + A n ) _~ Moo(b). 9 Mo~(V ) is any map compatible with the canonical map Moreover, if q0 M+(U) , + C_flU) ~ CffV) then q~ can be written as q) = lim q)a with % " eH~ 6) RESI1)UALIN REDUCIBI.E REPRESENTATIONS AND MOI)UI,AR FOICMS eH~ 6~), and there is a commutative diagram 13~(U) HomA~: (M+~(U), A~ ) T~ = lim ~a ~ , M~o(V) HOmA/~ (NIL[v), A'~. ) where ~ is the adjoint of % with respect to the pairings ( , )t:~ and ( , )v~- Now suppose that p C_ Too(U, ~) is a prime. Let P = A~: 71 p. It is easily deduced from the above that ~loo(U) induces an identification (3.16) Moo(U)p ~- Horn&, ~ ^' (Moo(U), , A~,p) e<~, 1' of q'~(U, G)p-modules. Recall that we defined in w an injection A~:. ~ AC which identifies A~: with A~ [Z1] for some finite group Z1. Suppose that m is a permissible maximal ideal of T~(U, C). By Lemma 3.10 both Moo(U),, and M+~(IJ)"` are free Ar so composing with the trace map from A~ to A~ induces an isomorphism HomA~: (Moo(U)"`,+ A(;, ) _~ HOmA~ (M~(U)"` ,+ A~, ) of T~(U, ~;~)"`-modules. Combining this with (3.16) yields an isomorphism (M~(U)"`, A#::) (3.17) Mo~(U)"` 2 Homhr + of To~(U, ~;)"`-modules. This will be important in our later computation of various congruences. 3.8. Congruence maps In this subsection we prove a number of results that will be helpful in our analysis of "congruences" between Hecke rings in w As always, U C_ GL~(6~F | Z.) is a compact open subgroup such that U = FIUw and Uo(n) D U D U(n) for some n. Let w~,p be such that U~, = GL2(C~?~v,~), let g = g~, and let X= ~(t) be as in the definition of T(f). For any f: GD(Af) ,R (R an .r (Ixf)(g)=f (g(l ~.)). Let V = U n U0(e). Consider the map ~1 " H~ R) 2 ) H~ R) given by ~l(f,g) =f+ txg. The following is the analog of Ihara's Lemma (cf [Ri]) in our setting. ~)q) - 1 - Nm(q) for an), prime ideal q of F that splits complet@ in the ray class field of conductor n . oo. C.M. SKINNER, AJ. WILES Proof. Our proof of this lemma is a straight-forward generalization of [DT, Lemma 2, p. 445]. Put 8 = (1 x-l). Suppose that (f,J?) E ker(~l). We first claim that f(gu) = f(g) for all u E &~GLg(d;v.~.)& This is an easy calculation: if /2 ---- 8-1Z/l" 8 E 8-1GL2(6"F,w)8, then f(g.) = -k(gu8 = -s *.') = -s =A(g). As SL2((~'~ F, w) and &ISL2(~rF, w)8 generate SL2(Fw) it follows that (3.18) S(gu) =f(g) for ail u E U. SL2(F~,,). Now let q be a prime that splits in the ray class field of conductor n. (x~. It follows from class field theory that such a prime has a uniformizer n E F that is totally positive and satisfies x = 1 mod n. Suppose now that 7 E GD(Aj) is any element such that Vl)(y) = x-l. For any g E GD(Af), &)g,/g-1 E GIl)(Af), where GI] C_ G D is the kernel of the reduced norm vi) and 80 E D � is such that VD(80) = X. Such a 8o exists as is totally positive (cf. [We, XI, w Proposition 3]). As G D is a twisted form of SL2 for D .- 8tglAg- 1 which G l (F~) = SL2(Fw), it follows from strong approximation that 80gyg -1 for some 8' E D � and u C U. SL2(F,,,). We have then by (3.18) that (3.19) f (g),) =f(8o'8'~) =f(gu) =f(g). Nm(q)+ 1 VD(gi) = ~, it follows from (3.19) and the definition X(q) i__[Jl Ug~ of [U(' k(q))Ul that [U(1 k(q) )Ulf =(1 + Nm(q))f. The lemma follows. [] ,A Now put U @ = U 71UI(U), r)0. Lemma 3.27. -- For r >>. 1 the sequence , H~ R) 2 ~, H~ R), H~ ('- ')), R) = o~f +f2, is exact. with 8(f) = (f, - co f) and 7(f ,J)) Proof. - To establish exactness, it suffices to prove that if (f ,~) is in the kernel of 7 thenf E H~ R). For any functionf : GD(Aj)~ R put c~-lf(g)=f (g(1 ~-1 )). Suppose that , = = -0~ ~. Now observe (f j~) is in the kernel of 7. As czf -J~ we also have f -1 RFSII)UALIN RI"DUCIBLE REPRESENTATIONS AND MODUI2kR FOILMS 59 t'_ that (x-IJ~(gu) = (z-lf)(g) for all uE U'= ((: :) E U:a-1 E g',cEs beg, It follows that f E H~ R) where W is the subgroup of U generated by U (~) and U'. This subgroup is just U (~-1~. [] Now consider the map {: H~ R) 3 , H~ R) given by {(f, J~, A) =f + (zJ~ + a2j~. As a consequence of Lemmas 3.26 and 3.27 we obtain the following. )~(,.) U - 1 - Nrn(r)for any Lemma 3.28.- The kernel of { is annihilated by IU( 1 ) ] prime ideal r of F that splits completely in the ray class field of conductor n . oo. Proof. - We can write ~ as the composite H~ R) ~ ~) H~ R)4~'~ ~ H~ R) 2 ~)H~ R) where ~(f, ~, .~) = (0, .)~, f, if2). It follows from Lemma 3.27 that {f, O, O, -f} C H~ R) 4 surjects onto the kernel of 7. If ~(f, J~, J~) E ker(7 o (~, if) ~t), then there exists somef E H~ R) such that (-f, J~, f, ff_9+f) E ker(~l O~i). Therefore by Lemma 3.26, f, J~, f, J~ +f are annihilated by the operators in question. This proves the lemma. [] We conclude our discussion of }~ongruence maps with an important application of Lemma 3.26. Let U C GL,)(~r | Z) be as at the start of this subsection, only now we assume that (n, p) = 1. Suppose that p C T~(U, ~:') is a prime such that *pEp 9 9p is irreducible and not dihedral. For simplicity we shall also assume that 9 p is contained in a permissible maximal ideal. Suppose that g is a prime ideal of F such that * gr 9 p ~ (Nm(g) -- 1) 9 the ratio of the eigenvalues of pp(Frobe) does not equal Nm(g) or Nm(g) -l. Put U (~ = U N U0(g) and U (t) = U N Lrj (~). Lemma 3.29. (i) Too(U (l), ~")p --~ Too(U (~ (9~)p --~ To~(U, ~)p. To~(U 0), (r Proof. -- I,et m be the permissible maximal ideal of T~(U, (~) containing p. Write m for the inverse image of this maximal ideal in _~T n~l(0,, ~*) and Too(U (1) , C). 60 C.M. SKINNI:R, AJ. WILES By Corollary 3.12 we have surjections T~(U ''), ~)m -" To~(U '~ 6~)m --~ To~(U, ~').,. To prove part (i) it suffices to show that every minimal prime of T~(U (l' , ~)m contained in p is the inverse image of a prime of T~(U, ~")~. Let Q c_ T~(U (1), ~i~),, be such a prime. An analysis of the possibilities for pQ.[D~ shows that po must be unramified at g. It then follows from Proposition 3.14 that Q is the inverse image of a prime of 0 ~1~ We now prove part (ii). For each a > 0, let ~ 9 eH~ C) 2 ,eH CX(U'~ '), &?) be given by ~a(f,g) =f+ O~g where cz is as in Lemma 3.26. Let ~ = lim{~ 9 + 2 + (V M~(U) ~ M~(U '). Also for each a > 0, let I~ = ker{ --2/~'1" r,.(o), ~) > T2(Ua, ~) }. Then I = lim I, C T~(U ~~ , ~) is just ker{ T~(U '~ , ~') > T~(U, ~'~) }. We claim that (3.20) + 2 Mo~(U)p = ML(U(~ For this we note that M+~(U(~ = limeH~ C)[I~]. Recall that we have fixed an identification of O~ with C (see w The map ~ extends to a map ~@e~J 13 : eH~ 13)2 ~ eH~ 13). By the Jacquet-Langlands correspondence (see w we have T2(Ua, ~')-equivariant isomorphisms U '0) Ua eH~ 13) _~ (~ V~ ~ and eH~ 13) ~_ (~ V~ . ~ord _ (0) ~ordf... ~CII 2 (tJ a ) ~tell 2 ,to a) u(O) a 1-l~ T It is easy to see that V~ [Ia] ~: 0 if and only ifnE..2 ~'~). On the other hand, if l-I~ I " U(0) [Ja lJa nE 2 ~'~), thenV~ ~ =V~ +a(V~),whence eH~ C) [I~] = im({~ | C). Let K be the field of fractions of ~. It follows that (3.21) eH~ K) [I~] = ilTl(~ a @ K). Now consider the commutative diagram lim ~a lirneHO(X(Ua), ~),, a lim eHO(X(U],)) ' C)[Ia] , C , 0 1 1 0 , lim im({a @ K) ~, lim eH~ K)[Ia] , 0 , 0 RESIDUALLY REDUCIBI.E REPRESEN'Ea.TIONS ANI) MOI)UIAR FORMS 61 where the vertical arrows are the natural ones. Applying the snake lemma we find that C embeds into a quotient of limker(~a Q K/~?). Now, for each a > 0 let F~ be the ray class field of conductor gnp a 9 oo. Let F~ = UFa. Let F' be the maximal extension of F unramified away from places dividing gnp. ec. Any element ~ E Gal(F'/F~) is the limit of a sequence of Frobenii { Frob,, } with r~ splitting completely in F~. It follows that trace p,,(o) is the limit of the sequence { T(r~)} and ~(~) is the limit of { Nm(r~)}. It then follows from Lemma 3.26 that trace pm(o )- 1-~(o) annihilates Cm for all ~ C Gal(F'/F~). Thus if Cp ~: 0, then it must be that trace Pm(~)- 1 - e(o) is in p for all o E Gal(F/F~). It is easily deduced from this that PlGaI(-r is reducible, and hence 9p is either reducible or dihedral, contradicting our assumptions on pp. Therefore Cp = 0. The same argument shows that ker({)p = 0. This proves (3.20). It follows from part (i) that M+oo~U%Jp = M+ (U0)p[i] = Moo( U+ 0)[i]p, whence (3.22) + (0) + 2 Mo~(U )p ~- M~(U)p. We next prove that (3.23) + (l) Mo~(W )0 "~ M~(W(~ 9 For this we note that M+(U (~ = M+(U('~)[S(g) - 1]. By part (i), 1~/I + tl T(1)~ + (1) + -(1) ...~,._. ,p = M~(U )p[S(g)- 1] = M~o(U )[S(g)- 1]p, [] from which (3.23) follows. A similar argument shows that M~(U(~)p -~ M~(U)p. 4. The Theorems 4.1. Pro-modularity and primes of R~ We assume throughout w that F, ?~, and k are as in w and that the degree of F is even unless indicated otherwise. In this subsection we also assume that Lp(F, - 1, ~o) is not a unit (so T.~ exists for any 5~r). Suppose that ~ = (~, Z, c, J/g) is a deformation datum for F. Let 6_c_~m = (G, Y.). Let q be a prime of R~. There is a map % : R.~ p~ ~ R~/q corresponding to the pseudo-deformation associated to P~ mod q. The prime q is pro-modular if % factors through r~ : R~ps ~ T~. That is, q is pro-modular if there is a homomorphism 0q :T~ ~ R~/q such that (4.1) (pq = Oq 0 /1~_~ . 62 C.M. SKINNER, AJ. WII,ES (Throughout this section, if a deformation datum ~ = (eS', Z, c, ,l/g) is ,given, then ~vs will denote the pseudo-datum c~_ZJv~ = (~,Z).) Note that in (4.1) 0q(T(g)) = trace P~ (Frobt) mod q for all g 9~ Y.. Similarly, a detbrmation p : Gal (Fx/F) , GL9(A ) of type-~ is a pro-modular deformation if the kernel of the corresponding map R~ , A is a pro-modular prime. It is immediate from the above definition that if q is a pro-modular prime of R~ and if p D q is another prime ideal, then p is also pro-modular. In particular, if a minimal prime of R~ is pro-modular, then so is every prime ideal on the corresponding irreducible component of spec(R~ ). In this case we say that the component is pro-modular. 4.2. Good data and properties (P1) and (P2) Our primary goal is to show that for certain "good" deformation data ~ the components of spec(R~ ) are all pro-modular provided the data have certain properties (labeled (P1) and (P2) below). In this subsection we describe these "good" data and the relevant properties. Let ~ = (~;, Z, c, ,~g) be a deformation datum for F. The pair (F, ~) is good if 9 the degree d of F is even 9 L~(F, - 1, )~t0) E ~' but I,p(F, - 1, )~co) ~ C � 9 d > 2 + 6v + 8. (#E + dim k H~)(F, k) ) 9 for each vilp the degree d~. i of F~. i over Qp satisfies d~ > 2 + 2t + 7 9 (#Z + dimx HL~(F, k) ) 9 if PclI., ~ 1 and w ~p, then either )~[i w z~ 1 or )~[t),~ = 1. As before, t = #?J), where ;~ = {vi} is the set of places of F over p, X~ is the set of finite places at which Z is ramified together with .~/~, and I'(~S I H~,(F, k)= ker{H~(F~,/F, k(X-t))---+ t~H (D,,,, k(z-')) }. i=1 Note that if ~ is good and if ~_~' = (~", E', c, ,//g') is another datum with s C_ Y~, then (F, ~') is also good. However, being good does not behave well with respect to change of fields, meaning that if (F, c~) is good and if L/F is permissible for ~ (as defined before Remark 2.1), then it can happen that (L, c-,~l,) is not good. On the other hand, it can also happen that (F, .~) is not good but (L, ~.~SL) is. This will be a key ingredient in our reduction in w of Theorems A and B to the Main Theorem. Let 6_~ = (t~,~', y~, c, .~d/g) be a deformation datum for F. Let p be a dimension one prime of T~. Let Pp be the representation described in w Let A be the integral closure of T~/p in its field of fractions K. If 9p is irreducible, then Lemma 2.13 RESIDUAI,LY REDU('IBLE RFPRESENTATIONS ANI) MODULAR FORMS 63 associates to pp a representation p : Gal (Fz/F) , GLg(A) such that 9 | K --~ 9p and 9 is a deformation of some 9c' for some cocycle 0 ~: c'C HI(Fz/F, k'(z-l)), k ~ some finite extension of k. We claim that c' is admissible and that 9 is a deformation of type- (~', E, c', 0), where ~' has residue field k ~. To see this, let vi be one of the places over p. Choose c~i E Di such that Z(ai) ~: 1 and choose a basis for 9 such that 9(~i) = ( c~ ) with ~ modmA = )~(Ci). /ks p | K "~ 9p it follows from (3.4) that with respect to this basis either 9[Di is split, 9[Di is non-split and 9[Di = ('1 ,2 * ) , or 9[Di is non-split and ,~) If 91I~, is split then clearly 0 resz.(c' ) E H~(Di, k'(~-l)) and 9[D~ satisfies the desired criteria. If PlDi = ( ,1 02" )' then ~,~()~i)modm,_ . = ?~(~'~i) ~: 1 . (Here ~'~i is the uniformizer of ~i,, ~i chosen for the definition of T0(p/)-- see w However, as 9 | K _~ pp, it follows from (3.4) that T0(pi)modp = ~20~:~) and by the permissibility of the maximal ideal of T~, T0(Pi)modmA = 1. This contradiction shows that if PIp, is non-split, then 91,)~ = (**l ,2) with *~modmA = Z. One sees immediately that res (c') = 0 and that PlDs satisfies the desired hypotheses. Therefore c' is admissible and p is a deformation of type-((9 ~', Y,, c', l~). We say that the prime p is nice for ~:~ if 9 p is a dimension one prime of T~, 9 9p is irreducible, 9 p is the inverse image of a prime of T~i (where c~ is the deformation datum defined in w 9 d is a scalar multiple of c, 9 some conjugate of 9 is a nice deformation of type-(~', 2, c, ,~fg) in the sense of w A prime p of R~ is good if 9~ modp is nice in the sense of w Such a prime is nice if it is also the inverse image of a pro-modular prime of R~. If p is nice for c,~, then the universality of R~ yields a unique map R~ -----* A inducing a conjugate of p. We denote by p~ the kernel of this map. This is a nice prime. The first of the aforementioned properties of ~ is that if p C_ T~ is any prime that is nice for r (P l) then any prime Q c p~ c R~ is pro-modular. The second important property of ~ is that there exists a pro-modular prime of R~ whose I c (P2) corresponding detbrmation is nice in the sense of w 4.3. The key proposition The following proposition is the key ingredient in our proof of the Main Theorem. 64 (3.M. SKINNER, AJ. WILES Proposition 4.1. -- Let ~ be a deformation datum Jbr F. /f (F, (~) is good, and if (P l) and (P2) hold for ~.~ and ~c, then every prime of R~ is pro-modular. Proof. Let L'~ be the set of irreducible components of spec (R~) and let ~mod C_ W'~ be the subset consisting of pro-modular components. The assertion of ff.e~l rood the proposition is equivalent to U~ = t~-~ . We be~n by proving the proposition for the case ~ = ~. (Note that since (F, c~) is good, so is (F, c~,).) The proof consists of two steps. In the first, we show that any component of spec(R~,) containing a nice prime is itself pro-modular. As a consequence of this and of rp9x ~c.,mod ~...; we have that ,:, ~, ~: (~. In the second step we combine step one with our analysis of the structure of the ring R~ to conclude that -lnod Suppose that p is a nice prime of R~. By the definition of pro-modularity of p there is a unique map 0p : T~ c ~ R~c/10 inducing the pseudo-deformation associated to p~,modp. Call the kernel of this map P l. Clearly, P l is nice for e.G'Yc. It follows from (P1) that if Q c_ p is any prime of Rs[, then Q is pro-modular. In particular, any minimal prime of R~, contained in p is pro-modular. This completes step one. Combining this with (P2), which asserts the existence of a nice prime of R~c, yields .~mod ~mod .~t c.~'-~, \ ~m~� The next step is to prove that ~'~ = ~_,~, . Put ~ ~, = r~ ~,\~ . If ~ = 0, then there is nothing to prove, so assume otherwise. It follows from Proposition 2.4 and <~moa U'~, such that C 1 NC 2 Corollary A.2 that there are components Ct G ,~ (/, and C,_, E contains a prime Q of dimension d-2t+ 8v- 3-#,~#g,.. Let I~ be the ideal generated by the set {p; det 9~0(Y~)- 1 I i = 1, ..., 8v}. Let Q~ be a minimal prime of R~/.J(Q, Il). The dimension of Q1 is at least d- 2t - 3. #J/g, - 1 > 1 + ~v + (#Z + dim k H~;c~(F, k)), the inequality by (G). It follows from Lemma 2.6 that pgmod Q1 is irreducible. Since Q~ E C~, O~ is pro-modular. The prime Ql determines a prime Q~Od of T~. The prime Q~Od is the kernel of 0Ol :T~, ~ R~/QI. Moreover, since P~ modQi is irreducible as remarked in the preceding paragraph, it follows from rl-, /g-~rnod Proposition 2.12 that dim ~,/td. 1 >/ dim R~/QI. Recall that T~, is an integral ~n-v!l~ y/t~ Tt, ...,T~F]I (cf. Corollary 3.4). By construction extension of Ae~: = ~ tt ~l ,..., dr, Qmoa ~_ . '~ Y('~ it would follow I I,~,~e contains T~,...,Ta v. IfOm~ also contained Y'~',..., ai that the dimension of O~ ~ would be at most d- di. Comparing this with the lower bound for the dimension of Oi obtained earlier and recalling that the dimension of QI is at most that of O~ TM, one finds that d, ~< 21 + 3 9 #~,,Pg~, + 1 which contradicts (G). Thus, after possibly reordering the Yf's we may assume that Y'( ~ Q~ for each i= 1,...,t. RESIDUALIN RI'~I)UCIBI~E REPRESI';NTATIONS AND MO1)UI_AR I"ORMS 65 ( ) t'o~ bo~ Fix now a basis for p~ for which P~(zl) = 1 -1 9 Write pM(o) = \ca do/. As p~ mod Q~ is irreducible, there is some o0 for which co,, ~ Q~. Let p _D Q~ be a prime of dimension one not containing coo, Y:i 1), ---, Y;f. Such a p always exists. As p E C1 it is pro-modular. We claim that it is also good. By construction p contains p, and it is, of course, a prime of R~, so it remains to check the conditions at each Di. I,et A = R~/p and let p : Gal(F/F) , GL2(A ) be the deformation p~ modp. Consider pjl)i~ (Vli) ~) * ) . By definition V~'~/[ .... ') equals 1 + Y:r which has infinite order in A. Thus gt~ ) is a character of infinite order. On the other hand, det p(~.) = 1 for j = 1, ..., 5F, SO, as char A = p, det P = )~- It follows that ~il'? = )~ 9 gt~ ~-1, whence (~) !i: g4/~:2 has infinite order. Therefore p is a nice prime of R~. As p E C2 it follows ~,mod from step one that C2 E ~r contradicting the assumption that C2 E ~~. This ~r rood proves that 5W~z~ = ,J ~ . "1 We now prove the proposition in its full generality. We first show that any component of spec (R~) containing a good prime is pro-modular. For this we use the proposition in the case ~ = CJc. We then combine this with our previous analysis of rood Rj to conclude that ~'~ = ,~ . Suppose that p is a good prime of R~. It follows that p is the inverse image of a prime ~1 of R~ under the canonical map R~ -~ R~,. By the proposition in the case 6~ = ~,., P l is a pro-modular prime. Thus there is a map 0p~ : T~ ~ R~/pl = R~/p inducing the pseudo-deformation associated to p~,mOdpl = p~modp. Composing Op~ with the canonical map T~ ---T~. yields a map 0p : T~ ~ R~/p inducing the pseudo-deformation associated to P~ modp. Let P2 be the kernel of O~. It follows from the definition of P2 that it is nice for ~, whence by (P1) any prime Q c p~, .~ c R~ is pro-modular. As p = P2, ~, it follows that any component of spec (R~) containing p is also pro-modular. In our final step we complete the proof of the proposition in its full generality. Let Q be a minimal prime of R~. Let I~ C R~ be the ideal defined as follows. Choose a basis for P~ such that P~(z~) = (l _~). Write P~(o)= (~ ~) and p~(o) = (~ ~(o)"~ ) . For each place v E Z\,+:~ fix a generator ~, E I,, of the pro-p-part of tame inertia at v. Let I~ be the ideal generated by the set {p; c% - 1, bz,: - u.r,~, %, d~,,; det p~ (~) - 1 I v E E\,~, j = 1, ..., gF}. Let Q~ be a minimal prime of R~/(Q, I.~). By Proposition 2.4 the dimension of Q~ is at least d- 7. #Z - 1. It follows ti-om this and from (G) that the dimension of Q~ is at least ~iv + #Z+ dimk H:~(F ) + 1 from which it follows by Lemma 2.6 that p~ mod Q.2 is not reducible. Moreover, it is clear from the fact that Q2 _D I~ that P~ mod Q., is a deformation of type-~:. It follows from the proposition in the case ~ = ~_~ that Q.2 is 66 C.M. SKINNER, AJ. WILES pro-modular. Arguing as in step two of the proof in the case ~ = ~ shows that Q2 is contained in a good prime. As Q c_ Q2, the same is true of Q. The conclusion of the preceding paragraph now implies that Q is pro-modular. Therefore, every minimal prime of R~ is pro-modular. This completes the proof of the proposition. [] 4.4. Conditions under which (P2) holds In this subsection we establish the following criteria for (P2) to hold for a .given deformation datum G~. Proposition 4.2. --Let ~ = ((r Z, c, .//~) be a deformation datum. If (F, ~) /s a good pair, and if (P 1) holds for each datum (C', s c', ,/W~ ') with E' C_ E and C' D_ ~', then (p2) holds for Proof. -- The proof of this proposition consists roughly of three steps. In the first we prove that (P2) holds for some deformation datum ~J0 -- (8", Z0, Co, Jg0) with ~'P _D (r From this, together with the hypotheses of the proposition and Proposition 4.1, we obtain that if .~2;' = (~-~r Z', c0,,//g') with Z' C_ E then every prime of R~, is pro-modular. In the second step we combine step one with the existence of suitable reducible deformations to show that there exists a prime P l of Tcj~ 1 (where 6-~1 = ((2~:', Ec, co, ~'~gl) for a suitable JW~ 1) such that the pseudo-deformation associated to P l comes from the pseudo-deformation associated to a deformation 131 of type- (J(~, Ec, c, 0). In the third step we prove that 91 is actually of type-~'_~c and that Pl is essentially the inverse image of a prime of T~, thereby proving that (P2) holds for c~. We now prove that (P2) holds for some deformation datum c_Z0 = (~', Z0, co, ,/IN 0). (Recall that Z0 is the set of finite places at which )~ is ramified together with the places Vl,...,o t over p and that d/~-0 = Z0\{Vl, ..., ut}. ) Let g x C_ GLy(~/'v | 7..) be as in w Since the pair (F,~) is good, L#(F, - 1,Zr is not a unit in ~', so it follows from Proposition 3.14 that T~(U x , 8') has a permissible maximal ideal m. Recall that by Corollary 3.4, T z = T~(U ~, r is an integral extension of A~,. = ~ [[Y(I I), y',0 T1,..., TSF~. Let Q c Tx/(p , "1"~, Ta~.) be a minimal prime. By 9 ", d t ' -- ..., its choice, the dimension of Q is at least d. Let R = Tx/Q. The pseudo-representation associated to pQ. determines a pseudo-deformation into R of type-(~ ;~ , Z0). We denote this pseudo-deformation by 9-~ = {a(~), d(~), x(c, z)}. We claim that x(~, z) is not iden- tically zero. If it were then p" Gal (F~,/F) , GL,)(R) defined by 9(~) = (~(~) ~1(~)) would be a diagonal deformation of type-(~:', Z0) (see w Therefore, there would Ddiag be a map ~/: ~,(~, x) ~ R inducing p. Since it follows from Lemma 3.11 that R is generated (pro-finitely) by the set {trace p(~)} = {trace pQ(o)}, T must be surjective. Ddiag Thus the kernel of 7 would be a prime q of ,,(~-~, z/ of dimension at least d. However , diag 9 , by the choice of Q, det9 (and hence fletp(~iz/moclq) has finite order. Lemma 2.9 would now imply that d ~< 1 + gF, but this contradicts (G). This contradiction implies RESII)UALIN REDUCIBLE REPRFSENTATIONS AND MODUI2kR FORMS 67 that there exists some c0 and z0 such that x((s0, z0) ~: 0. Now let p _D Q be a dimen- sion one prime of T)~ not containing x(60, "Co), y,l),..., y(~. Let A be the normalization of Tx/p (this is a complete DVR with residue field k' a finite extension of k). Let ~7' = ~' | W(k~) 9 I,et q0 = p-'Qmodp be the induced pseudo-deformation into A of type-(~ ~:', E0). This is nothing more titan the pseudo-representation associated to pp. By Corollary 2.14 there exists a cocycle 0 ~= Co E HI(Fr~,/I ', k'(3(-1)) and a deformation p`0 9 Gal(Fx0/F) ~ GL2(A ) of P~0 whose associated pseudo-deformation is q0. Let K be the field of fractions of A (equivalently, the field of fractions of Tz/p). Comparing traces we find that P`0 | K -~ pp. Arguing as in the second full paragraph of w (the paragraph describing primes of T~ that are nice for r shows that co is admissible and that P`0 is a deformation of type-(~r v', Y--o, Co, 0). We claim that it is in fact a nice deformation of type-C:'.~0, where 6_~0 = (~", Y--0, Co, ~/g~o). Recall that dg,~ is nothing more than the set of finite places other than vl, ..., v, at which ~ is ramified. Therefore if w E J/~0, then one sees easily that p`0[,,, _~ (r ~002" ) with r and r finite characters of p-power order. However, since the characteristic of A is p it must be that r = 1 = r This shows that P`0 is of type-C*~0. Moreover, since 9`0@K_~pp it fbllows from (3.4) that p~0,t) _~ (01 i) * ) (,~, ~1 9 ~i) with r " a character of finite order and ,~('~l,.('~x = 1 + Y?, which is an element of infinite order in A � To "f2~l s conclude that 9`0 is a nice defbrmation it remains to check that the corresponding prime of R~0 is of dimension one. This follows from Lemma 2.12. By construction p`0 is a pro-modular deformation of type-6_~0 (since T~, = T:t | ~'~'). It follows that the prime of R~ 0 corresponding to 9~0 is a nice prime. This completes step one. If the cocycle c is a scalar multiple of co, then (P2) holding for 6.~0 easily implies that (P2) holds for 6Ad~ and hence also for ~_Z), as was to be proved. For let P,0 be the deformation of type-CJ0 described in the preceding paragraph. There exists a conjugate 9`0 of 9`0 that takes values in GL2(B) with B C A an r with residue field k and that is a deformation of type-(~ v , ,To, c, JeSt0). Since (~~, 120, c, '//g~0) = 6-~c, this shows that P~ is a nice, pro-modular deformation of type-.~ (since T~ | C' = T~0). Suppose from now on that c is not a scalar multiple of Co. Let ~-~'1 = (~', I?~, Co, d-g ~) with .//{~ the set of finite places w G Z~ other than vl, ..., vt such that )~l[w ~: 1. Since (F, is good so is (F, ~r). Having shown that (P2) holds for c2c_~0 we see that (P2) also holds for .~ZI. Combining this with the hypothesis that (P1) holds for ~0 and 6.~1, and with Proposition 4.1, yields that every prime of R~ is pro-modular. As both c and co are classes in Hz~(F, k') they can both be viewed as Gal (F(,z)/F)- equivariant homomorphisms GaI(F00~)/F~)) ~ k'(g-l), where F0~) is the minimal field over which every cocycle c' G Hz:(F) becomes trivial. FLx Gal(F~)/F)-generators (~l,...,(~, of Gal(F0(z)/F(z)). Then any cocycle in Hz(F, k') is determined completely by its values on the 6i's. Let {(oh,j,...,ot.,.j) ~ k 's,1 ~<j ~< s-2} be s-2 linearly ~.~z;'Y) 68 CM. SKINNER, AJ. WILES s k independent vectors such that ~ o~i, jCo(~,) = 0 and o~i, jc(~i) = 0. Note that i= I i: 1 (4.2) s - 2 = dim e Hz,.(F, k') - 2 ~< #Z~ + dim~ Hz~,(F, k). Fix a lift ~i,j of each oti, a to (S 3''. Fix now a basis of 13~, such that 13~l(Zl) = (t ) Write 13~,(~)= (('~ b~) -- 1 " ca d~ " Let I C_ R~t be the ideal generated by {p; detp ,(' e) lJ = i= 1 g = 1, ..., 8~: }. It follows from (G), (4.2), and Proposition 2.4 that any minimal prime of R~/I has dimension at least dim R~, -(dim k Hz(F, k) - 2) - 8v - 1 > d+ 7 - 3 9 #,//g~ - 4-dimk HL(F , k) - 2t (4.3) > d + 7 - 7 9 (#E~ + dim k H~,(F, k)) > 8v + dimk Hz,(F, k). Comparing this estimate with that in Lemma 9.6 shows that any minimal prime of R~I/I corresponds to an irreducible (pro-modular) deformation. Now, there exists a reducible deformation 13 : Gal(FzJF) , GLg(k'[[x~) of 13q~ given as follows. Let c and Co be cocycle representatives of c and co such that c(za) = 0 = c0(zl). Define 13 by p(o)=(1 X(o).(~0(o)+'~o)X)).X(o) Clearly, 9 is a deformation of type-~l, so 13 corresponds to a dimension one prime p of R~/I. Let Q be a minimal prime of R~/I contained in p. As we observed in the preceding paragraph, 13~lmod Q is irreducible. Let Qr be the inverse image of Q under r~ 9 Ro-zp., ) R~. Let Atr = R~,~/O~ r and let A be the integral closure of --1 N r in its field of fractions L. The ring A is a KruU domain [N, (33.10)]. Let K be the field of fractions of R~/Q. Choose 131,...,13~ E k' such that ~[3ic(~i) = 0 but ~ico(~i)~: 0. Fix a lift i= 1 i= 1 ~i of each ~i to (~-@t. Choose a basis for 13~, such that 13y,(zl) = (1 -1) and 2 ~ibrj i = z/~ E ~("~,,x,~ where 13~I (~): (aaco ~ )" Put x((~, "c)= bocz. Suppose that P is a height one prime of A for which ~ "~ix(,~i, (~p) (/ P for some gl, E Gal(FzJF). Then, since ~'~ix((~i, (~r) = u0%,, %, ~/ P. It follows that b~ E Ap for all ~ E Gal(Fz/F). In particular, the matrix entries of each 13~t(~) modQ are in Ap. Thus, if such a oF RESIDUAI~LY REDUCIBLE RI,;PRESI{NTATIONS r MODUI.*~R FORMS existed for each height one prime P of A, then p~ ]mod Q would have matrix entries in MAp = A. It would then follow from Lemma 2.6 that any dimension one prime of R~I/Q pulls back to a prime of A, and hence to one of Atr, of dimension at least one. However, this is impossible as the non-maximal prime p of R~,/Q pulls back to the maximal ideal of Atr. Therefore, there must exist a height one prime P0 of A" for which { ~ ~Jix((yi, T.)" T, E Gal (Fx,/~ } C_ P0. i = 1 Suppose that x(o, z) E P0 for all o and z. It would follow that the representation Pro defined by Pv0(o) = ( a~ do) EGL9(N~/Po) would be a diagonal deformation of P0 = (1 x) of type-(~;', E,) having determinant equal to X. As Nr/p0 is (pro-finitely) generated by the traces of PP0 it follows that the l~diag ) NVP0 would be a surjection, whence by Lemma 2.9 (ii) the natural map -'-re, , z,) dimension of N"/P0 would be at most 1 + 3v. However dim A'~/P0 = dim Atr -- 1 ) dim R~r l/Q- 1 > 8r" + dime Hz~(F, k') >By+ 1, the first inequality coming from Proposition 2.12, the second from (4.3), and the last from the fact that c and co span a two-dimensional k'-subspace of Hzc(F,/4) by hypothesis. This contradiction implies that there is some o' and z' for which x(o', r P0. We next claim that after possibly renumbering ';, 4 we can assume that Y~I ') ~ P0. l~br this we recall that since Q is a prime of R~ it is pro-modular, so m~ m A ('~mod Atr = T~/Qmod for some prime Q~,od C_ T~ such that p~ ou ,.~ is the pseudo- representation associated to p~]modQ. Recall also that T~, is an integral extension of A~. By the choice of Q, Qmod contains (T1,...,T~v,p). Hence so does P0- If P0 also contained (Y(~'), ..., Y~) then the dimension of P0 would be at most d-d~- 1. Hence the dimension of Qmod (and hence of N r) would be at most d- di. However, as the dimension of Q is at least d + 7 - 3 9 #,M/N 1 -- 4" dim k Hz~(F) - 2t, it would then follow from Proposition 2.12 that di ~< 7. (#Z~ + dimkHz0(F))+ 2t, contradicting (G). This proves the claim. Now let P l be a dimension one prime of N' containing P0 but not containing Yl ,...,Y';, or x(o', "(). Let B be the integral closure of A'~/p] in its field of fractions L. Let k" be the residue field of B. By Corollary 2.14 there is a representation P l " 70 C.M. SKINNER, AJ. WII,ES Gal (Fz~/F) , GL2(B ) whose associated pseudo-deformation comes from P~ ii~ mod P l and for which plmodmg = 9,1 for some cocycle 0 3( cl E HI(Fz:/F, k"(x-I)). Recall that since Q is a pro-modular prime of R~I there is a map T~ ~ A t'. inducing the pseudo-deformation p~p~modO~ r. Thus Pl corresponds to a prime of T~, which we also denote by Pl. It is clear that 91 | L _~ Ppj. Arguing as in the paragraph describing primes of T~ that are nice for ~ shows that Cl is admissible and that Pl is a deformation of type-(.(.; ?'', E:, cl, 0), where C" = ~" | W(k"). We next claim that Cl and c differ by a scalar, or, in other words, Pc "" Pq. Recall that { 2~ix(oi, ~), E ai, jx(oi, z) lz E Gal(F/F); j = 1, ...,s- 2 } is contained in p,. Suppose that 2 oti,./Cl(Oi) 3(_ 0 for some j. Fix a basis for Pl for which Pl(Zl) = (I ) --1 " Write Pl(o) = ('~co ~) . From our supposition it follows that bj = ~ oti, jbo, is a unit in B. But we also have (~ ai, jbo,)co = F, c~i, jx(o~, o) = 0 in B. It follows that co = 0 tbr all o and hence that x(o, ~) = 0 in A for all o and ~, contradicting the assumption that x(o-', ~') ~ Pl. A similar argxlment shows that ~ ~icl(oi) = 0. It follows that cL restricted to Gal(Fz,/F(x) ) is a scalar multiple of c. This proves the claim since restriction determines an isomorphism H' (Fz,/F, k~- 1) ) ,.~ Hom (Gal (Fz,/F0~) ), /doZ -1))c~1 (vz:/v~. Therefore, after possibly replacing Pl by a conjugate, we may assume that Cl = c and that Pl takes values in GL,)(B') with B' a ~::-subalgebra with residue field k and hence that Pl is a deformation of type-(~", E~, c, 0). This completes step two. We now prove that Pl is a nice deformation of type-6.~Y:. The only thing needing proof is that Pl is actually of type-~c, for the desired properties of p~[D~ follow from the isomorphism Pl @ L ~ PpI' and that the corresponding prime of R~: will have dimension one will follow from Lemma 2.12. As .//gc consists only of those finite places other than vl, ..., vt at which Pc is ramified one finds that each w C ,//g~ satisfies exactly one of the following possibilities: (4.4) (ii))~lI~ = 1, XID~, = c0-1= 1. If w E Jgc satisfies (i) then it is easily seen that P l 1I w e~ (| ~) (here we have used that char (B') = p). If w E J/g,. satisfies (ii) then pc[D. is type A. We want to show that the same is true for Pl. Since det Pl = X there are two possibilities for Pl | LID,: it is either type A or type B'. If it were type B' then Pl | I]lI u, ~ (1~ ,-1 ) with a finite character of p-power order. However, since char (L) =p any such ~ must in fact be trivial, from which it would follow that P l is unramified at w, contradicting the assumption that Pc (and hence DI) is ramified at w. Thus it must be that 131 @ L[I),~ is type A. It is now straightforward to show that since [clI)~ = (1 .] is non-split, I,,] RESII)UALLY REDUCIBLI'~ REPRESENTATIONS AND MOI)UI~R FORMS 71 as well. For let u ~ I~: be a generator of the pro-p-part of tame inertia. It follows that p~(u) = (1 b{)) for some 0 :~ b0 ~ k. Let V be the underlying representation space tbr p~ (so V is a free B'-module of rank 2). Let U C_ V be the B'-submodule annihilated by u- 1. That U =~ 0 follows easily from Pl | L being type A at w. That U =~ V follows from the fact that p,.(u) 5 k 1. It follows that U -~ B' and V/U --~ Bq Let et, e2 E V be such that el generates U as a B-module and e2 generates V/U as a B'-module. Thus el and e2 form a basis of V as a B'-module. With respect this basis u acts via a matrix of the form (1 ~). Since the reduction el and to of el and e2 modmB form a basis for V = VmodmB,, the k-space underlying Pc, and since u does not act trivially on V, b is a unit in B ~. After possibily scaling el and e2 by units in B ~ we may assume that b reduces to b0. In this case el and e2 form a basis for follows from the deformation Pl with respect to which pl(u) = (1 ~) (b ~: 0). It now the well-known action of Dw on tame inertia that p,lDw = -(1 *).- (Here we have used that char(B') =p and that det f)l = Z)- This completes the proof that P l is of type-,~c. Let P2 _C R~, be the prime corresponding to Pl. We have shown that p~ modp,2 is nice. We now show that P2 is pro-modular. Put ~ = (r X,, c, .~r Clearly T~, 2 Qnj ~" = T~. From the choice of p~ and the definitions of pe and p~ we have a commutative diagram R~ p~ ~ T~ ,2 ~ T~ ~A~"~A~/p l (4.5) J. r~ ,, B' ~ B R~ 2 ~ R~, where the map T~ ---A tr induces the pseudo-detbrmation associated to p~ l mOdQ and R~ ~ B' corresponds to Pl. Denote by p.~ the kernel of the map T~ ~ ~ At"/pl in (4.5). We need to show that P3 is the inverse image of a prime of T~ c under the canonical surjection Tc~ ~ ~ T~. Let Ql C_ p:~ be a minimal prime of T~ 2" We describe the possibilities for p% ID,, when w C ~/]g,.. First, note that PQI is ramified at every place in .liNe since pp:~ --~ pp~ is. Second, recall that by (G) every place w E J/g~ satisfies one of the two possibilities listed in (4.4). If w satisfies (4.4i) then w E ~//Ni (by the definition of ,~tg 1), and it is easy to see thai 9% [I~ ~- (001 ~02 ) with q~t and ~2 finite characters of p-power order. As w E ,M~Y~Yl, Ur,[c,2, w = c C E g~, ' , amodgw E AL,. especially for notation.) It follows from Lemma 3.16 that q~l and 0~ are in fact trivial. This shows that (4.6a) if ZII~ ~: 1 then pQ~li~ ~ ( 1 ) ' ~ " 72 C.M. SKINNER, AJ. WILES Finally, suppose that w C ,//g~ satisfies (4.4ii). In particular m(Frob) = 1. A straightforward analysis of the possibilities for pe.~ II),,. using that p% I~, factors through the pro-p-part of tame inertia at w shows that there exists a finite character ~ of Dw of p-power order such that Po_a ID,, | is either type A, type B, or type C. As type C can only occur if m(Frob ) = -1, this case is impossible. It follows that if ZIP,, = m--i = 1, then either (1) PQII~, ~- (r ,.~), 01 and r finite characters of p-power order, or (4.6b) ,) O , ~ a finite character of p-power order. Now write det9% = ~" qb.~ where q~ is finite of p-power order and ~ has infinite order and factors through a free Zp-extension ofF (hence gt is ramified only at places in ?/~). It follows from (4.6 a, b) that 0 is ramified only at places in .~/g,.\-tg i and in .~/~. Fix a character 0~ : Gal(Fx,./F) , (T~.,/Qt) � ramified only at places in E~\.~g~ and such that ~b-~1 = ~-1. By Lemma 3.17 there are primes Q:, _c p~ c_ Tj., with Q,~ minimal such that Pe.., ~- 9% ~ qbl and 9p~ ~- Pp:~ | 01. As 9p:~ | ~1 = Dp3, it follows that Pt = P3. Thus Q,2 is contained in P3- It follows from (4.6 a, b) and the definition of ~t that (i) if X!i, + 1, then pQ.,lI,~.~ ( 1 ~), (4.7) (ii) if ZID= = m-1 = 1, then either _ (' "" , OF b) 9Q21Dw ~ (* ,-1 ) with ~)a finite, character of p-power order. Next we introduce some new subgroups of GL2(~v | Z). Write U~ = II U~. ~.. t,' Let ~f" be the set of places w C ./lg,.\,~gl tbr which (4.7iib) holds, l?br w E ~/" define c E g~,, ad-lmodgw E A~w} (see w for the definition of A'w). Put (4.8) U' = II u~,,~, x II u~,, and u" = U~, flU'. z,,4 7/" ~eT/" Let m' and m" be the permissible maximal ideals of T~(U', ~) and T~(U", t~), respectively, obtained by pulling back the permissible maximal ideal of T~(U~c, ~') RESIDUALIN REI)UCIBI,E REPRESENTATIONS AND MOI)UI,AR FORMS 73 via the canonical projections. There is a commutative diagram (4.9) T / S T~(U', 6n~).,, T" = T~(U", d~j.,,, where all the maps are the canonical ones and are surjective. Let ~ C_ p.~' C_ T" be the inverse images of Q2 and P3. It follows from (4.7), (4.8), and Lemma 3.15 that there are primes Q~2 c_ p~ c_ T' whose inverse images in T" are just O~'~ and p~', respectively. It now follows from Proposition 3.20 that p.~ is the inverse image of a prime Pc of T~ c By the commutativity of (4.9) and the fact that P"3 is the inverse image of both P3 and p.~ it follows that P3 is the inverse image of p~.. This proves that the map T~., ---, Atr/p I in (4.5) factors through the canonical surjection T~ 2 ~ T~,, completing the proof that the deformation 91 is nice and pro-modular of type-r This completes the third and final step in the proof of the proposition. [] 4.5. The Main Theorem We now state and prove our Main Theorem. In this subsection and the next, we forego the convention that F has even degree. We will, however, assume property (P1) for certain fields. That this property holds is proven in w (see Proposition 8.4) which are independent of w Main Theorem. -- Suppose that F is a totally real field and that ~.~ = (~ , Z, c, ./Z/g) is a deformation datum for F. Suppose also that 9 9 GaI(Fr./F) , GL2(~' ) is a deformation of type-~ such that 9 p is irreducible 9 det p = gt~ ~t with g >, 1 an integer and V a finite character @) with V'2 II~ of finite order for each i= 1, ..., t. If there exists an extension L/F of totally real fields such that (i) the Galois closure of L over F is solvable (ii) L has even degree over Q (iii) L is permissible for (iv) (L, ~ r,) it a good pair in the sense of w then p | Qp is a representation associated to a Hilbert modular newform. 74 C.M. SKINNER, AJ. WILES Proof. -- Let P~ = [~lGal(L/L)" As L is permissible for F'~, P~ is a deformation of type-~t. Since L has even degree over O~ it follows from Proposition 8.4 that (P1) holds for any deformation datum for L. As the pair (L, ~-~1.) is good it then follows from Propositions 4.1 and 4.2 that 9~ is a pro-modular detbrmation. In particular, there is a map ~. : TCz L > ~> inducing the pseudo-representation associated to P l. By the conditions imposed on p in the statement of the theorem, the map ~. satisfies the hypotheses of Proposition 3.7. Thus there is a Hilbert modular newformf (over L) such that (4.10) trace pt(Vrobt) = (eigenvalue ofT(g) acting on f) for g f~EL. Let PJi " Gal(L/I~) ) GL2(Qp ) be the representation associated to f. (If is the automorphic representation associated to f, then 9Ji is just Pn, the latter being the representation described in (3.2)). This representation satisfies (4.11) trace 9j~(Frobt)= (eigenvalue ofT(g) acting on f) for g f~Ei.. As P is irreducible by assumption and odd, PI (and hence 91 Q Qp) is also irreducible. It therefore follows from (4.10) and (4.11) that Pl Q Q~ ~- 9fL- Now, as the Galois closure of L/F is solvable, it follows from the known cases of base change for 0aolomorphic) Hilbert modular forms (cti [GI,]) that there is a newform f over F such that PJ~GaI',~/L', #.o Pfl (here pj" Gal(F/F) > GL.>(Op) is the representation associated to f). Since 9f]~.~l~/i.l -~ P | Qp ](;~J',~/g) and these are irreducible, it is easy to see that -- --X ~ -- there is some finite character ~ : Gal (F/F) , Qp such that P/| ~) = P | Q/,- As 9f| is the representation associated to the newform corresponding to the twist off by ~), this proves the theorem. [] In the next subsection we will deduce the following theorems from this one. Theorem A. -- Let F be a total~ real abelian extension of Q. Suppose that p is an odd prime and that P " Gal (F/F) ~ GL,,(/Dp) is a continuous, irreducible representation unramified away from a finite number of places ofF. Suppose also that the reduction of P satisfies ~ ~- )~, | If 9 the splittingfield F(~l/~2 ) of)~l/)~ 2 ~ abelian over Q, 9 ~1/)@ (z) = --1 .)Co, each complex conjugation z, 9 ()~1/%'2)Ir),, ~- If or each v[p, ( ") 9 o li) with factoring through a pro-p-group of and of 2 2 finite order for each v~, 9 det p = g~k-1 with k >>. 2 an integer and ~r a character of finite order, then p is a representation associated to a Hilbert modular newform. RI"SIDUALI,Y REI)UCIBI,E REPRESENTATIONS AND M()DUI~'~,R FORMS 75 A critical ingredient in the proof of Theorem A is a result of Washin~on on the boundedness of the p-part of the class group of a cyclotomic Z:-extension of an abelian number field (cs [Wa]). A similar result for any totally real field would yield the same theorem but with the omission of the hypotheses that F and F()~I/Z2 ) be abelian. l~br our next theorem, we make the following hypothesis, which plays a role similar to that of Washington's theorem in the proof of Theorem A. We believe that this hypothesis will be easier to establish than the analog of Washing-ton's theorem, though the latter would yield a stronger result. Hypothesis H. --- There exists 0 < ~. < ~ and a constant c(e) > 0 such that given a totally real field K and a finite set S of finite places of K there is an imaginary quadratic extension L of K having prescribed behavior at each place in S and such that the relative class group of L/K has p-rank at most c(e)deg(K/Q) l-e Theorem B. -- Let F be a total{y real extension of Q. Assume Hypothesis H for all solvable, total~ real extensions ofF. Suppose that p is an odd prime and that 9 : Gal (F/F) , GL2(Qp ) is a continuous, irreducible representation unramified away from a finite number of places of F. Suppose alto that the reduction of 9 satisfies ~ ~- )tl | )t'2. If " (~i/)t~,)(z) = -l for each complex conjugation z, 9 O~,/Z2)[D. has even order for each v~).. 9 911~, ~- ( ~g~)Z~ ~g~v)~ *) 2 with W~ ') factoring through a pro-p-extension ofF, and W.~:[I, of finite order for each 9 det 9 = V ek- ~ with k >1 2 an integer and ~ a character of finite order, then 9 is a representation associated to a Hilbert modular newform. 4.6. Proofs of Theorems A and B We now prove Theorems A and B. In both cases this is done by reducing to a situation to which the Main Theorem applies. Proof of Theorem A. -- Put Pl = 9@Z2 and )C = )~l/X2. I,et Z be the set of finite places at which 91 is ramified together with the places over p. There exists a finite extension K of Q~ such that for some choice of basis 9~ takes values in GL2(~" ) with ~"' the ring of integers of K. Such a basis can be chosen so that the reduction ~ P' m~176 the maximal ideal (~')~ ~, Pl = P'm~ satisfies 91 = ( 1 ~) and is non-split. Let k be the residue field of ~'. It follows that 91 ~- 9c for some cocycle 0 :~ c E H~(Fx/F, LOUt)). The hypotheses on Pll), ensure that c is an admissible cocycle. Thus after possibly replacing Pl by a conjugate we may assume that Pl is a 76 C.M. SKINNER, AJ. WILES deformation of type-C~ ~ , where ~ = (~', Z, c, ~). Clearly 91 satisfies the hypotheses of the Main Theorem. The conclusion of Theorem A will thus follow from that of the Main Theorem if we exhibit an extension L/F of totally real fields that (i) has solvable Galois closure over F, (ii) has even degree over Q, (iii) is permissible for ~, and (iv) is such that (L, ~I~) is a good pair. We will construct such an L. Let E/F be any even extension that is permissible for .(/' and is such that E/Q is abelian, each place v[p of F splits in E, and if w is a place of F at which 9c is ramified and )CID~ is unramified then )dDw, = 1 for each place w'lw of E. It is easy to find such fields: take for example, E = F. E' where E' is a real cyclic extension of Q of sufficiently divisible degree in which p splits completely and all primes q :~ p divisible by a place in Z are inert. Choose an odd rational prime g such that g ~ #k � and g is not divisible by any of the places in E. For each positive integer n let E, be the cyclotomic Z/g"-extension of E. It is easily checked that E, is permissible for ~. Let Z, be the set of places of E,, dividing those in Z, and let .~n be the set of places of E,, dividing p. Let rn denote the p-rank of the i~- l-isotypical piece of the p-part of the class group of E,,(X) and let p'" denote the order of the p-part of the class group of E,. From the theory of cyclotomic extensions we know that there exist integers s and t such that (4.12) #Z,=s and #?/,=t for n>>0. Similarly, it follows from [Wa] that there exist r and c such that (4.13) cn =c and r,=r for n>>0. As E,,/F is a Galois extension, we also have that (4.14) deg En, ~./Qp/> g"/t V v[p. Let p~" be the number of p-th power roots of unity in E,,(~p), ~p a primitive p-th root of unity. As the degree of E,/E is a power of g, there is an integer e such that (4.15) e. = e Vn. For each n/> 1 choose a set S,, of en + cn + 1 finite places of E, disjoint from Z, and such that (4.16) * pe,+c,+ t [ (Nm(w) - 1) V w E S,, 9 )c(Frob,~,) = 1 V w E S,, 9 there exists an abelian p-extension L,/En of degree at most pe.+2c.+,, unramified away from Sn, and such that the subgroup of Gal(Ln/E,~) generated by {Iw:w C S~} is isomorphic to (Z/p) e'+c'+l. Note that I~ is necessarily ramified at each place in Sn. The existence of such a set S,, follows easily from Class Field Theory. RESIDUALLY REI)UCIBLE REPRESENTATIONS AND MOI)UI~kR FORMS 77 Let E,, C_ Hn C_ L, be the maximal unramified subextension of L,. It follows that L(L~, - 1, Zro-') = II L(H., - 1, Zeo-~qb) where ~ runs over the characters of GaI(L,JH,,)--~ (Z/p) enfcn+l . Let pC, be the number of p-th power roots of unit)' in H,,(~p). Note that g, ~< e, + c,. It follows from well- known congruences for p-adic L-functions that if Xo-I ~= 1 then L(H,,, - 1, X0~-l~) E Z0 [~c0-1q~] for all , and if Xc0- 1 ~_ 1 then L(H,, - 1, Xm --l*) E Zp [~] for, non-trivial and /"L(H,, - 1, Zc0 -l) E Zp (cf. [Co], [D-R], or [Se]). Here, for any character 0, Zp [0] denotes the ring obtained by adjoining the values of 0 to Zp. We also have by our choice of S~ that if ~ is non-trivial and if n, is a uniformizer of ZpL'~oJ-l~] then L(H., - 1, )~(0--11~)) ~ L(H., - 1, zo-l) I-I (1 - xto- 1,(Frobw)Nm(w)) = 0 mod n,, where S,(~) is the set of places of H, at which ~ is ramified. Combining this with the earlier expression tbr L(L,,, - 1, ;(co -l) we obtain that L(L,, - 1, Xo) -1) E Zp[zr -1] and L(L,, - 1, Xr -1) = 0mode., ~. a uniformizer of Zp[zco-l]. We have thus shown that (4.17) Lp(L~, - 1, Xr E ~\& � Since E, is permissible for 6~, the field L,z is as well. Moreover, it is a simple exercise in p-groups to show that dim~ Hy.o(Ln, k) ~< #Gal (L,/E,) 9 dim k Hr.o(L,, k) Gal(I'n/Fn) (4.18) ~< #Gal (L,/E,). dim k H~,us,(E,, k) <<, pe"+2c"~l(r, + e, + c, + 1). Now choose no so large that g"~ > 2 + 8(f+~c+l(s + r+ e+ c+ 1)) (4.19) t,o > t(2 +pe~2c~I(t+ 7(S+ r+ e+ c+ 1))) and such that the equalities in (4.12) and (4.13) hold. Let L = L, o. By construction the Galois closure of L/F is solvable and the degree of L is even. As noted above, L is permissible for !~. It remains to verify that (L, ~-/Yl.) is a good pair. For this let dl. be the degree of L. By construction dE >/ g,0. Also, by 78 C.M. SKINNER, AJ. WII,ES di It follows that [Wal], ~L ~ --"" dl, ~> 6L + g"/2 > 2 + 8(pe+2'+'(s + e+ lc+ 1 "b?l)'t-~) L (4.20) >/2 + ~ir, + 8(#ZL + dim k HE,(L, k)), the last inequality following ti'om (4.12), (4.13), (4.15), and (4.18). Suppose that v is a place of L dividing p. Let d,, be the degree of L~. The number of such places v is at most pe+')c+lt, so it follows from (4.14) and (4.19) that d~ >1 g'~/t > 2 +p~ 7. ooe+2c+l(s+ r+ e+ lc+ 1)) >1 2 + pe+2~+~t + 7(#EL + dim k Hr~(L, k) ). That (L, C_YYL) is good now follows from this, from (4.17) and (4.20), and from the choice of E. [] Proof of Theorem B. - Let X = Zt/Z2. It follows from base change that it suffices to prove the theorem with F replaced by the maximal totally real subfield F + of F(X ). By the hypotheses in the theorem ZII),, ~: 1 for each place vb0 of F +, and therefore we may assume that X is quadratic. Put Pl = P | ~-1. Let Z be the set of finite places of F at which Pl is ramified together with those dividing p. As in the proof of Theorem A, there exists a finite extension K of Qp with integer ring C~' such that, for a suitable choice of basis, 91 takes values in GL.,(C) and is a deformation of type-C2~ for some ~ = (~, Z, c, 0). The conclusion of the theorem will follow from the Main Theorem if we can find an extension L/F that (i) has solvable Galois closure over F, (ii) has even degree over Q, (iii) is permissible for 6~r and (vi) is such that (L, ~-~I,) is a good pair. Arguing as in the proof of Theorem A shows that we can find a solvable permissible extension E/F that has even degree over Q and is such that (4.21) 9 Lp(E, -1, Zoo) E ~-~' and is not a unit, and wCp, then either X[t,, :~ 1 or X[I),, = 1, for w a place of E. 9 if Pc[I~ :~ 1 Let Z' be the primes of E above those in Z. We now construct a solvable permissible extension L of E such that (L, ~e) is good. By Hypothesis H there is a totally imaginary quadratic character ~g over E such that 9 if w E s and w~p then V is unramified at w and X~g[D~, ~- 1, 9 if vb0 and X[l, :~ 1 then ~ is unramified at v and v(Frob) :~ 1, 9 if v[p and XlI~. = 1 then gt is ramified at v, and 9 the p-rank of the relative class group of" E(~g)/E is at most @)21-~deg(E/Q). Let L, be the splitting field of the character X~g over E. This is a totally real quadratic extension of E and clearly permissible for c~. Let ZI be the set of places RESII)UALIN REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 79 of L1 above those in Y/. It follows from the choice of V that #E~ = #E'. It is also relatively easy to see that dimk HxI(L1, k) ~< dimkHx~(E , k) + c(e)2 I-E deg(E/Q.)+ #Z'. Proceeding inductively, one constructs in the same manner for each n > 1 a totally real quadratic extension L,, of L,_l such that (4.22) 9 L, is permissible, 9 #Zn = #Z I, where s is the set of places of L,, over those in Z', 9 dim k Hx,,(L,,) < dim k Hz,(E, k) + c(e)~,~l 2i(l-e" deg(E/Q)+ #E'. Now choose no so large that (4.23) 2 "~'-~ > 2 + 17-#E' + 8. dim k H~(E, k) + c(g)2 ('~'+1!(1-e)+6 deg(E/Q). dL Put L = Ln0 and ZL = E, 0. Let dE be the degree of L over Q. By [Wal] 5L <~ ~-, so by (4.22) and (4.23) we have that d1. >/~i. + 2 "~ deg(E/Q) (4.24) > gL + 2 + 9. #ZL + 8. dim k HxL(L, k). If v is any place of L dividing p, then d~ = deg (Lv/Qp) is at least 2 '~'. It follows from (4.22) and (4.23) that (4.25) d,, > 2 + 9 9 #ZL + 8. dim~ HzI.(L , k). Since L has even degree by construction, combining (4.25) with (4.24) and (4.21) shows that (L, c~L) is good. [] 5. A formal patching argument In the next four sections we give the proof of property (P1) (see Proposition 8.4). These sections do not make use of any results from w In this section we will describe a formal patching argument which is a variant on the patching argument in [TW] and its refinement in [D2]. The extra complexity in our case is caused by the fact that we are considering the localizations of deformation rings and Hecke rings and not the original rings themselves. In particular the residue fields are not finite. We will, in section w apply our patching argument to localizations of deformation rings (in contrast to [TW] where it is applied to Hecke rings), but in this section we will just axiomatize what is assumed (and later proved) about these rings and consider only the formal aspects of the argument. Let k be a finite field of characteristic p and let A = k[[T]J. Let K be the field of fractions of A. Let .5/' = {N} be a sequence of strictly increasing odd integers together with zero. Let n be a fixed positive integer. 80 C.M. SKINNER, AJ. WILES We introduce rings Ax, BN (for each N E ,5gJ) given by Ax A[Is,, s,]] / (S1 N+I N+I.~ = ..., ,...,S, j, Ao = A Bx = A[[6, ..., t,]] / (~+li/2 ~.X+l~/2. . ,...,t; ' ), B0=A. There is a homomorphism BN ' At,- given by ti, , (1 + si) + (1 + $i)-1 _ 2 which we use to identify BN as a subring of AN. We assume that we are given a ring R ~ for each NE cf~ of the form (5.1) R/'3 = A[x~,..., Xm]l/d a~ with rn independent of N. Furthermore we assume that R Cx) has the following properties: (5.2) (i) R IN) is finite and free as an A-module, (ii) a (x~ G (x,,..., x,,), (iii) 3 a surjective map R (x) -+ R/~ of A-algebras, (iv) R/\3 is a BN-algebra for N > 0. Now letting p(-',3 be the prime of R ~ corresponding to (Xl, ...,Xm) (which we usually abbreviate to p if the N is clear from the context) we assume two further (and less formal) properties of RC'W': (5.3) (i) 3 d(0) > 0 such that pd(0) = 0 in R '~ (ii) p(.X~/(p(X))2 ~ A n | Tor,,~, where the free summand A" is spanned by xl, ..., x, and "For(y) is a finite group whose order is bounded independent of N. For each odd 0 ~< a ~< N together with zero we assume given a ring R~ "4/ which has the following properties: (5.4) (i) R] ~'3 is finite and free as an A-module, (ii) ~(~ R~ -x) :0' "-x =Ri~, =R , (iii) there are surjective maps of Bx-algebras Ri m .., (iv) R(ff 3 is a Ba-algebra (compatible with BN + Ba) such that if" a > 1, then a-I a-I | KI/(tY, ..., tT ) RaT2 | K, (v) R~ ~ @A K is an Aa @A K-algebra satisfying (via the map in (iii)) el N) @A g/(s,, ...,Sn)~ Rff3 ~ @A K, RESII)UALIX REDUCIBLE REPRESEN'IATIONS AND MODULAR FORMS 81 Letting p(~\) denote the prime corresponding to (xl,..., xm) (which we again write as p if a and N are clear from the context) we assume two further properties: (5.5) (i) 3 d(a) > 0 independent of N such that I aJ(~i = 0 in R~ :<, / / -x (ii) '~ ~x;2 A" Pa /(Pa ) ~- {t) TOr(N,a), where the free summand A" is spanned by Xl, ..., x, and Tor:x, ~3 is a finite group whose order is bounded independent of N and a. Associated to the rings R __~ ;x~' are certain subrings R*] ~-'~ (of "traces" in the application) which are assumed to satisfy the following conditions. First we assume given an A-subalgebra R '~(a') C R ix) satisfying (5.6) (i) R '~''~ : A ~Yl, ".,Ym~/[~''N~ (same m as in (.5.1)) where b ~? C Iv,, ...,Y,,,), (ii) R tr(N) is a Bx-algebra for N > 0 compatible with the algebra structure on R i'xi. Setting q~.x, = p(X', 71R t'~'",' (thus the prime corresponding to (Yl, ...,Y,,,) ) we assume in addition the property (5.7) coker" q~r~)/(q(I,0)2 ~ p(N)/(p(N))P, has order bounded independent of N. For 0 ~< a ~< N, a odd or zero, we set R~ 'Ix' =im {R ~''') , R;\'~}. Observe that __~R t'rN~' inherits a B~-algebra structure from R~ "x?. Then we deduce from (5.7) that (N'., (Ng 2 'N'; (N~ 2 coker{% /(q~ ) , p~ /(p~ ) } has order bounded (5.8) independent of N and a, where q]<' = p::.x~ A R~,t~'x'. Associated to the rings we have described we will assume given a set of modules as follows. First we assume given an integer 7, independent of N. Then we assume we are given M i'x~, a finite R<'~)-module, satisfying the hypotheses that (5.9) (i) M $'~ is a free A-module of rank equ~ to the rank of A~, (ii) M (r'~: is an Ax-module compatible with the. Bx-structure via R ''c'; , (iii) there is a map M '''? ~ M '~ of R<a;-modules. For 0 ~< a ~< N (a odd or a = 0) we assume that we are given a Rt'~'NLmodule quotient ',N: of M ix) denoted M a satisfying (5.10) (i) M] v) is a ti'ee A-module of rank equal to the rank of A~, (ii) M::'; (x~ M(O~ x = M(X), M0 C and there exists some z E W '(~ independent of N, ordl.(Z mod q(0)) ~ 0, such that z M (~ C M 'x) 41--- ~ ~ ..., (iii) there are surjective maps of RU','~-modules M~ \) M:i \: M (x) 82 C.M. SKINNER, AJ. WILES (iv) M~ "~) is an R~"N"-module (compatible with the R'r"N"-structure), (Nh (v) M~ is an Aa-module (compatible with the Ba-structure induced in (iv)) in such a way that the maps in (iii) are compatible with A0 ~- Al --- A3..., and the actions of Rtd ('x) and An commute on M~ v', (vi) M2 "~' @A K is a free A~ | K-module and M~ s; | K/(sl, ...,s,) --~ M~ ~ @A K. Furthermore, we assume there exists x ~''; E R trtr~) such that (5.11) (i) ~ annihilates ker{M a /(Sl, s,,) "", *'~0 J, (ii) OrdT(X~'~mod q(r~) = t < oc with t independent of N. We now derive some simple properties of the above rings and modules. Lemma 5.1. - rankAR~ "47 ~< g(a) where e(a) depends only on a. Proof. -- This follows immediately from (5.5i). [] Lemma 5.2. -- There exists an E(a) independent of N such that TE(a)R~ ? C R t'(~3 ---a 9 In particular R tr(~ @A K = R(s "3 | K. Proof. -- Since pd(~) = 0 in R2 ~ by (5.5) it follows that it is enough to check that coker~r, where ~r is the natural map z (N),r// (N)2~r+l (N) r (N) r+l f~r" ~qa ) /~qa ) ' (Pa ) /(Pa ) , is finite and bounded independent of N for r < d(a). For r = 1 this is given by (5.8). A similar bound follows for r = 2 by picking generators for im(~l), lifting them to (N) (r,0, 2 elements, say Zl,...,Zs, in (q~)/(% ) and considering the map (N) ,~) 2 s (qa /(qa )) ' (q~"))2/((q~X))2n(p~'~))3), (al,...,a,), , Zaizi which is surjective. The property for r = 2 can now be deduced from the property for r = 1, and we proceed by induction up to r = d(a) - 1. [] From this lemma we deduce immediately that for any c the kernel and cokernel of (5.12) R~ r('~/T c , R~ "9/T c is annihilated by T E'~). Now let t, x ;~3 be as in (5.11) and let z be as in (5.10ii). Let dl = OrdT(zmodq(~ ). RESIDUALIN REDUCIBLE REPRESEN'IATIONS ANI) MOI)ULAR FORMS Iz, rnma 5.3. --- T annihilates both the kernel and the cokernel of the map ~N~ M,0!/TC M~ / (T c, SI,... , St) --""+ for any c. Proof. --- Let ~ be a lift of z to R t~N~. By (5.11) and the definition of dl, x ~x~- ~- uT t+J~ E q(X) for some unit u E A x . So T t+d' = u Ix(N)~ + v, v E q(N) Hence T d''~)(t+6? = w. x~N)~ + v a~) for some w E R my). By (5.5) ~',~) = 0, so the result follows from the defining properties of x ~'x) and z. [] Next we introduce level structures which we will use to make a patching argument similar to the one in [TW]. A level-(a, c) structure J(N, a, c) is a collection of data comprising (i) Ba-algebras, R~(. ~, = R~(N)/q'c, R~ =, R~<N)/T,' M(N) M(N)/T c ,, mN~ , , (ii) an A~-module _._~ , c = --a /-- that is also an r~o~ -moclme, (iii) a map of B~-algebras R~"~ , R'~"~, tr(~ ~) (iv) a map of Ra' c -modules :.x/ M~O~/TC M~, c/(Sl,..., s,) ' compatible with the actions of A~ and A via A~ ~ A, ..., , R~, c/(xl,..., x,,) ~ A, (v) elements {Xl, Xm} of R]') such that ~.x) n tr(N]// (vi) elements {y,, ...,Ym} of Rt;~) ) such that r%,~ /l.h, ...,Y~)----- A. Let ~o = c~, = {N} as at the be~nning of the section. Let .~'~(0)= .5;f0 and define ~'l(C) C ,~(c- l) inductively as tbllows. We require that 5~l(C ) should be an infinite strictly increasing subsequence of integers from Sr 1) with the property that J(N', 1, c) = J(N", 1, c) for N', N" E .~'l(c). The equality here sig-nifies that the J(N, a, c)-structures for N = N', N" can be identified (non-canonically). Since the total number of such non identifiable structures for fixed a and c is finite a choice of Yg~l (c) can be made. Then define .~', = {N,: N~ E ,Z~L(i)} again with the Ni strictly increasing. Finally we can define (~a for an odd a > 13 inductively by .~(0) = c~'o_~ and defining ~,(c) inductively in the same manner as ~.gf~l(c). We set R~ ~ = lim R~"(c ~ R~ = lim R (N~) , Mo = lim M (m3 p fl, c a,c " 4~)<t+d') 84 C.M. SKINNER, AJ. WILFS Lemma 5.4. a) R~ r | K = R, | K, Ro | K = R '~ | K. b) R, @A K is a quotient ofK[[xl, ...,x,,]]. c) M. | K is a flee An | K-module of rank r and OM. @a K) / (SI, ..., St/) = M a) | K. Proof a) By Lemma 5.2 for any c we have natural maps of A-modules , R;,] , R~, c whose composite is multiplication by T L~a) with E(a) independent of c. Taking projective limits and tensoring with K gives the isomorphism. b) By construction there are elements {xl,..., Xm} of R~ such that R~/(Xl,..., Xm) --~ a. Let pa = (x,, ...,Xm) C Ro. Letting p(~7~ } denote the ideal generated by {Xl,...,Xm} in , we see easily that (No) , ~ , ., 2 a p"N-/(p'X~ F lira P~,c = Pa, lira ~,~ .... c, --- Pa/P 9 +--- Then by (5.5) we deduce that p~/p2 ~_ A n | T, where T is a finite group and Xl, ..., xn span the free summand. Hence (po | K)/(p~ | K) '~ ~ K" by (5.5) which ensures that Ra, p~, p] are all finite A-modules. Part b) follows by Nakayama's lemma. c) By Lemma 5.3, (5.13) (M, @A U)/(Sl, ..., Sn) ~ M (~ | K. By construction, dimK(Ma | K) ) ranKA~lVl ~ ) for large enough c and Nc E oS;~Ja. By (5.10i) the right-hand side has rank equal to rankA(A~). But r is the K-rank of the right-hand side of (5.13). The result follows. [] Lemma 5.5. - l~br odd a > 1, there are surjections a) Ra ~ Ra-2 of A[[xl, ...,Xm]]-algebnz~ and Ba-algebras. b) R~ -+ R~_9_ of A[[yl, ...,ym~-atgebras and Ba-algebras. RESIDUAIAN REDUCIBLE REPRESENTATIONS AND MOI)ULAR FORMS 85 c) M~ ~ M~_z of R~-algebras compatible with b) and of A~-algebras. Here the B~ action via R~ and Aa are the same, and the R~ ~ action commutes with the Aa action. The same holds for a = 1 with a - 2 replaced by O. We have that ,~-2 = {Ni} and ~f~ = {Nj} C_ ~o 2. Note that ji >1 i. Proof -- By our choice of Y~a and ~-2, l(a, - #)~) i --.* R~-2, i = Ra_), (x~ i [] whence taking limits yields R~ -+ R~_2. The same works also for R7 and M~. NOW we set, taking limits over odd integers a, R'. = Ra | K = R~ | K, M'~ = Ma | K, Roo = lim R'., M~ = lim M'~. a a Thus M'~ is an R'~-module and M~ is an R~-module. Lemma 5.6. (i) Roo = K [Ix,,..., x, ifl. (ii) Mor is a free R~-module. Proof. -- By Lemma 5.4 c) there is a map K[[sl, ..., s,]] r --* M~ which is seen to be an isomorphism. Consequently M~ is also a free K[[tt,...,t,]]-module. Thus K[[tl, ..., t,]] ~ R~/AnnR~0VI~ ). On the other hand there is a surjection K[[xl, ..., x,]] --- R.~. By Krull's dimension theorem we deduce part (i). (Note that R~/AnnR~(M~ ) is a finite K[[h, ..., t,]]-module.) Then part (ii) follows from the Auslander-Buchsbaum Theorem ([Mat, Theorem 19.1])since depthR~(M~)>/ n ({h,...,&} is a regular R.-,o- sequence for M~). [] Proposition 5.7. - -/f N >> 1 then dimK(R~ x) | K) >/ 2" dirnK(R~ x) | K). -- ~ N @A K. Choose maps Proof. Fix a projection K[[xl, ..., x,]] R (~ , , such that ( ~---+ (1 + ~)+ (1 + 4) -1- 2 and such that the images of K[[~,...,s',]] and K[I~,..., ~]] in R~ | K are just those of AN @a K and BN | K respectively in the 86 C.M. SKINNER, AJ. WILES sense that the images of ~ and ti are the same, and so are the images of ~ and si. It follows from (5.10 vi) that Bg | K > R~" | K is injective, for by (5.10) the action of Bx | K on M~ ) | K factors through its image in R~ ? | K. So if N+I n--I 9 (N) '' >/n (dXmK(R ~ @A K/(t'~,...,t',)) n = n "-t dimK(R;/'' | K)" then the hypotheses of Lemma 4.1 of [DRS] hold and using (5.4iv) we get that (5.14) K[[Xl, ..., x,]] / (t'l, ... , t',) "" R(~ ~ | K. Applying the Auslander-Buchsbaum Theorem again we find that K[[xl, ..., x,,]] is a free K[[fl,... , ?,]]-module of some rank d. Applying the same theorem yet again we find that K[[x~, ..., x,]] is a free KI[s'~, ..., s',]]-module of rank d/Z". It follows that a = 2 n dimK(K~-Xl, ... , xn]] / (s'l, ..., s',)) 2" dim, (R | ...,s~ ) z mnaK~v, v | K), the last inequality by (5.4v). Combined with (5.14) this proves the proposition. [] Proposition 5.8. -- M ~~ | K ~ (R '~ | K) ~ where e = rkR~(M~ ). Proof. -- We set R := R~/(tl, ..., t,) and := Mo~/(h,...,t~) = M~/(s~,...,s~). Thus M is a free R-module of rank e. Now R --~ R' t since the t~'s arc zero in R't, so (5.15) dimK P-/> dimK R'I ) dimK(R([ "? | K), for any sufficiently large N E 5JJL. (More precisely if T i annihilates the A-torsion submodule of R~l we can take N ) Ni+ l E ,~ (i + 1).) Now let gs(X) for any ring S and S-module X denote the minimum number of generators of X as an S-module. Let ~'~ Ar Then we have the inequality e := rkRoo(Moo ) = g~(M)/> g~(M/(sl, ...,sn)) >>, el. Also we have the inequality (5.16) 2"el dimK~.." /1~(~ | K) /> 2" dimK(M ;0' | K). RESIDUAl,IN REDUCIBI,E REPRESENTATIONS AND MODULAR FORMS 87 Now as remarked at the beginning of the proof of l~emma 5.6, M~ is a free K[[sl,..., s,]]-module of rank r, where r = dimK(M~~ | K), whence (5.17) 2" dimK(M(~ @A K) = dimK(l~ ) = e dimK(P, ). Combining the inequalities (5.15), (5.16), (5.17) with Proposition 5.7 gives el )e. Since also e/> el we have equality and all the inequalities just cited are equalities. In particular el = e, edimK(R(~174 K) = dimK(M (~174 K). It follows that M (~ @A K ~ (R '~ | K) ' as claimed. [] Proposition 5.9. -- R (~ | K is a complete intersection at a K-a~ebra. Proof. -- We recall what we have proved so far. By Lemma 5.6, (5.18) M~ --~ R e and Roo is a power series ring over K of dimension n. By construction we have elements {Sl, ..., s,} acting on M,,~, and (5.19) sl,..., s~ E EndRo~(M~). By Lemma 5.4 c), we have (5.20) M~/(s~,..., s,)M~ --~ M (~ | K. The action of R~ on M (~ | K is via R (~ | K and (5.21) M (~ | K ~ (R~ ,) | K)" by Proposition 5.8. Now let a = ker 9 Roo > R (~ | K. Let N = Z siM~ C Mo~. Then M~/N _"~ M '~ | K _"~ Re/ a by (5.20) and (5.21). Since Mo~ ~ R~ it follows that N _~ a e as R~-modules. (Consider the map q~ : Mo~ --~ R~. of (5.18). Then the above isomorphism easily implies that a e Let wi, ..., we be an R~-basis of Mo~. ]'hen N is generated as an R~-module by the set {siwj : 1 <<. i <~ n, 1 <~ j <<. e}. In particular a set of minimal generators has cardinality ~< en. Let {xl,...,xt} be a minimal set of generators of a. Then {xiwj : 1 <~ i ~ t, 1 <~ j <<. e} is a minimal set of generators of aM~ -~ i~. e ~' N. It 88 C.M. SKINNER, AJ. WILES follows that et <~ en, whence t ~< n. However as R (~ | K has dimension zero it follows that t = n and that R ~~ | K is a complete intersection. [] Remark 5.10. -- The circuitous route to this proposition via a counting argument is forced on us by the lack of a natural K~sl, ..., s,]]-algebra structure on R~. Only the elements {tl, ..., t~} are naturally defined in Ror The structure assumed in (5.4v) is an artifice which is not assumed to be related to the action of Ao | K on M(s \') | K, except for the compatibility with the subring Ba | K. 6. Estimates of cohomology groups In this section we consider a representation (6.1) p : Gz = GaI(Fz/F) , GLe(A ) where A -~ k[Dv]]. Here we are using the notation and assumptions of w so that, in particular, F is a totally real field. We let K be the field of fractions of A, and we recall that if p is ramified at w { p then we distinguish the tbllowing possibilities for plI. : peA 0| *) _ l, typeB pOK[Iw ~(t _ ~q ) , gtq non-trivial of finite order. Throughout this section we make the following assumptions on p: (6.2) (i) p | K is irreducible and of type A, type B or unramified at each prime w{p, (ii) ~ := p mod ~. = P, for some c as in (2.1), (iii) Z contains the primes dividing p and all primes at which ~ is ramified, (iv) p is of type A or type B precisely where j~ is, (v) det p = )C, with )~ as in w (vi) Pll)~ -~ (xl x2* ) with )~1/;(2 of infinite order for each rip. Lemma 6.1. --The Gal(Fz/F)-module W = ad~ /s irreducible. In particular P | K is not "dihedral" (i.e., is not inducedfiom a character over a quadratic extension). Proof -- By condition (vi) we see that there is an element c E D,, C Gz such that (X1/)~2)(a) has infinite order. Here we may choose v dividing p such that )~]D, is non-trivial by condition (ii). It follows from the existence of| and the self-duality W --~ Homg(W, K) that any invariant subspace of W has a complement. So if W is reducible then either W -~ Yl O Y9 with Y_9 of dimension 2 and irreducible, or W "-' Yi q)Y,2 q)Y3. The self-duality also shows that in the former case YI is acted on by Gz via a quadratic character, possibly trivial, and in the second case that there is also a unique subspace, YI say, on which Gz acts via a quadratic character. RFSII)UAI,LY REDUCIBLF REPRESENTATIONS AND MODUI2kR FORMS Now let z be a complex conjugation and pick a basis for p such that -I ) . Suppose first that z acts trivially on Yi. Then we may identify Y, with {(-a a)} C_M2(K) and we see that imp C {(* ,), (, *)} C GL2(K ). In particular, either im 9 is abelian, which contradicts assumption (i), or imp has a subgroup H of index 2 for which the action is abelian. In the latter case, H acts via two characters gt and ~g~(~z(~)= gt(~-I~x)) for any xjEH. Thus p = Ind~Z~. Now suppose that z acts non-trivially on YI. This time YI C {(b a)} _C M2(~). An easy calculation shows that if ~ = 1 on Y l then p(a)= (O~ca ~) with a~ = d~. Using that HI = { (r : o = 1 on YI } is a group we check that p(Hi) is abelian. Thus just as above, 9 -- Ind ~ for some character ~. Now consider 9 restricted to D~.. Then the quadratic field associated to P (i.e. the fixed field of H or HI in the two cases) is not split at v as otherwise PlI~,. = ~g | ~ll~. and this contradicts assumption (vi). So letting H~ be H N D,, or Hi n D,. in the two D., cases, we see that PIp,, = IndH',~, where ~, = ~ID~. Again this contradicts assumption (vi) since if the ratio of the two characters on H~, had infinite order then Pit), would be irreducible. [] Now let F' be the splitting field of det p adjoin all p-power roots of unity, and let F + be the subfield of F' fixed by the complex conjugation Zl. Lemma 6.2. --- The restriction of 9 | K to GaI(F~/F +) /s neither reducible nor dihedral. Proof. --- Let V be the representation space for p | K. Suppose first that p Q K restricted to Gal(F~/F*) has an invariant subspace V0. Then since Gal(F~:/F +) is normal in Gr we see that for any o E G~-, oV0 is also invariant. As Zl acts by + 1 on V0 and by the opposite sign on V/V0 there are at most two invariant subspaces. So either V0 is invariant by Gz or p | K is dihedral, but in each case this contradicts Lemma 6.1. Suppose next that 9 | K restricted to Gal(F~/F +) is dihedral. Then there is a subgroup H C Gal(Fz/F ~) of index 2 which has two fixed spaces. From the form of Pc and the definition of F' we see that the splitting field of p | K generates an extension of F' which is pro-p, whence H = Gal(F~:/F'). So H acts on the two fixed spaces via a character ~ and its inverse ~g-l (and the two spaces are unique if ~g ~: 1 as ~ cannot be of order 2). So H acts on W via the characters { 1, ~2, ~-'2 }. Either gt is trivial, in which case Gzc acts on W via the abelian group Gal(F'/F), or the subspace of W corresponding to the character 1 is invariant under Gy. as Gal(F~-/F') ,~ Gr.. In either case we get a contradiction to Lemma 6.1. [] 90 C.M. SKINNER, AJ. WILES Lemma 6.3. (i) There exists o E Gal(Fz/F') such that the eigenvalues of p(o) have infinite order and are in A. (ii) There exists o E Gal(F~/F+)\Gal(Fz/F ') such that the eigenvalues of p(o) have infinite order and are in A. Proof. -- First we prove parts (i) and (ii) without requiring that the eigenvalues are in A. (i) If o E Gal(Fz/F') has eigenvalues of finite order then the eigenvalues must be 1 as the image of Gal(Fz/F') is a pro-p group and K has characteristic p. Assume no o as in the Lemma exists. Pick a ~ E Gal(Fz/F') such that p(~) ~: 1, which can be done as p is not abelian. Pick a basis for P@K such that p(~)= (~ ~) with a:t= 0. any o E Gal(Fz/F') we have trace p(oz)= 2, so if p(o) = ( a~ s ) then Then for CO aa + aca + d~ = 2 = ao + d~. It follows that ca : 0 for all o E GaI(Fz/F'), contradicting Lemma 6.2. Thus there exists a o E GaI(Fz/F') such that P(O) has eigenvalues of infinite order. (ii) Assume otherwise. Then as in part (i), we see that there are only finitely many possibilities for the trace of p(o) with o E S = Gal(F~:/F+)\Gal(F~./F'). Fix a "c E GaI(Fz/F') such that p(~) has eigenvalues of infinite order. Choose a basis for p | K such that (p@K)(~)= ([~ ~-1). I~br any oES, ifp(o)= (~ ~), then we have trace p(~"o) = 13"a~ + 13-"d~. Since there are supposed to be only finitely many choices for the trace, a~ = da = 0 for all o E S. It follows easily that p | KIG.~(Fz/V+ ) is dihedral, contradicting Lemma 6.2. To complete the proof of the lemma, note that in (ii) the eigenvalues will necessarily be in A. This follows from Hensel's lemma using that the two eigenvalues are distinct modulo ~.. Then part (i) follows also by taking the square of any o obtained in part (ii). 5 Lemma 6.4. - /fG is a narmal subgroup OfGaI(Fz/F +) offinite index then p @ Kl~. is irreducible. Proof. -- Suppose that V is the representation space for p @ KIGaI(I,z/F, ) and V0 is a subspace invariant by G. By Lemma 6.3(ii) there exists an element of G whose eigenvalues are 13, 13-1 with 13 of infinite order. Arguing as in Lemma 6.2, we deduce that either V0 is invariant under Gal(Fz/F) or the representation is dihedral, contradicting Lemma 6.2. [] RESIDUALIN REDUCIBLE REPRESENTATIONS AND MOI)UI~kR FORMS 91 Let ;~ = ad~ = {f E ad P "trace f = 0 } where as usual we identify ad 0 with Homa(~gg , '?~, ), '~Z being the representation space for P (more precisely qZ is a free A-module of rank 2). Let o~-, = ~'/~.". Lemma 6.5. -- There exists an integer N 2 with the following propero~. If M C 3", is a submodule for some n and )v~m ~- 0 for some a > N2 and m E M, then )v"- ~-N2),~'- C M. The same holds if M is a G-submodule of 57", for G a normal subgroup of Gal(Fx/F +) of finite index, N2 depending only on G. Proof. -- Suppose x E ad~ - ~.ad~ Then by Lemma 6.4, A[G]x D ~/ad~ for some minimal r = r(x). Define a function f : ad~ - Xad~ , Z by f(x) = r(x). Then f is continuous and hence imf is finite. Let N2 be the greatest value of imf Now 5F', = a d~ and we pick s maximal such that E~y = m for some y E .~7",. So a + s < n. By the definition of N 2 we see that ~,Ne.~7", C p(G)y, whence which completes the proof. ~, Remark 6.6. -- When combined with Lemma 6.3 this shows in particular that #(JT",) 6 is bounded independent of n. As above, let qZ be the representation space for P. This is a free A-module of rank 2 having for each v[p a filtration 0 C ~'~l,v C ~/, such that '~2Zl,,, is a free A-module on which Dv acts via a character reducing to ~ modulo ~.. The quotient "~Z2, v = ~/qg" 1,~ is a free A-module on which D, acts via a character reducing to 1 modulo ~,. If P | K is type A at w, then there is a filtration 0 C ~l w C q~' such /tJ that both ~Z~ ~ and the quotient ~Z~ L'= ~Z/~ZI ~ are frec A-modules on which I,, acts trivially. If p | K is type B at w, then qZ decomposes as '~?[ = ~ ~):g/,~ 2' with I~ acting on the first factor via ~ and acting trivially on the second factor. Also as above, let ;7"= {f E adp: trace(f) = 0}. Let ,~7 "~ = {f E .'7: ./'('~) C 'ggl,~}- Similarly, ifp| is type A or type B at w, then let .~u,= {f E .~': f(~ C qgl'}. We write ~F.~-ord w ord n ,~, /~, .~u,/~.., respectively Let ,~,,, ,~.., ~,, and "~,i for 3"/V, and H,,(.~,,) = Hi(Iv, ~. / ~176 r/[ v ~1, 0 ] ' and let {tI (Iw, ifp| typeAor type B atw, Hw('qT") = o otherwise. 92 C.M. SKINNER, AJ. WII,ES For each w E Y-, put L.,(~7"~) = ker{ Hl(Dw, 57~) , Hw(5~'.) }. We define a Selmer group for ."2". by H~:(g'.) = {or E HI(Fz/F, 57".) : res 0t E Lw(~57,,,) for each w E Z}. For each place w E s denote by L~(J'.) the orthogonal complement of Lw(~.) under local duality (so L~(~7.) C_ HI(Dw, .7,,(1))), and put H~(J'.) = {or E H'(Fx/F, .~.(1)): reSwOt E L*,(:~.) for each w E Z}. By the argument for [TvVI, Proposition 1.6], which generalizes easily to the case of an arbitrary totally real field F, (6.3) #H~:('~3") - ho~(57.) H h~(.~,,), #Hx(3".) weX where h~(.~"n) = #H~ "~")" (#H~ "~'~"(I)))[F~ #H~ .~'~.(1)) h~,(g'.) = #--H~ :7"(1))'#Lw('~") #HI(Dw, .'~,,) We now estimate these factors. For two positive quantities B and C (possibly depending on n and Z), we write B << C to mean that the ratio B/C is bounded independently of n and the places in Z (it may, however, depend on p and #Z). Similarly, we write B ~ C to mean that max(B/C, C/B) << 1. A simple computation using our hypotheses on p shows that (6.4) Almost by definition, hw(27~) = #H~ ,~.(1)) if p NK is unramified at w. (6.5) Suppose that P | K is type A or type B at w. From the definition of L~(S'n), it is I w clear that H~(Fw, .~7"~' )~--~ I.w(.~). The order of the quotient Lw(.~)/H'~w, .~F':") is I)zt bounded by the order of K. , where K,, = ker{ H'(Iw, ~,,,) ---+ H'(Iw, z~,,,/o~)}. 93 RESIDUALLY REI)UCIBI,E REPRESENTATIONS AND MOI)ULAR FORMS The exact sequence o--, 37' , ,o ,gives rise to an exact sequence H'(I~;, ~.~) --- J'~' , (.~,/.Y'7) ~' ~- A/~." ~ A/~. = ~ Kn ---+ 0 ifp| typeAat w, and --Iw ,-,., zt: I 0 , )" , Ht(I~, .~.'") , K. ,0 I w ~, = #(~~ /~7s ) ~ 1, if P | K is type B at w. In the former case it follows that #Kn and in the latter case it follows that #K,I~ 'L= #H1(I,:,.57"~) I)~ = #H~ ,~,,w(1))x 1. It now follows from local duality that (6.6) h~(~-~,) ~ 1 if P | K is type A or type B at w. It remains to estimate h~(.'W,,) for riP. To do so, consider the diagram H'(D~, .~,,) n~ v 7l/ n, t) ]" Our hypothesis (vi) of (6.2) implies that #cok(q)) ~ 1. It follows that #HI(D,., ~7".) (6.7) #L,(o~',) = # ker(V ) = #im(~) ( ~ / ,~ord\l,, #H'(D~, ~57,,,) 9 #Ht(F,,, ,. ,/.. ,,,, j ) (7-ord ~./~7 "~ From the long exact cohomology sequence for 0 ~ ,_ . ,, ~ ~7. ), n /,]~l n, v ) 0 and the fact that D+ has cohomolog-ical dimension two, one finds #H~(D,,, .SK./.5~~ #H2(D,,,.'~,,) . " hi*" n, V ] " #im(q0) - #H2(D,,, , o~.a ord 9 ) ) #H2(D,, Substituting this into (6.7) yields (6.8) h~,(:~.) #H2(D~,, ~;~ord] - ,,,,, ~ .#(A/~,")-'elv,,:~ #(A/V)2rF,,:%I Here, we have again used hypothesis (vi) of (6.2) (really its implication that )CIX2 -1 ~ e). 94 C.M. SKINNER, AJ. WILES Now writing Z = EoUZ' with { v[p} C_ ~,, p ramified at each prime in Z0\{ v[p}, and unramified at each prime in Z' and combining (6.3)-(6.6) and (6.8) yields (6.9) #Hz(~7.) x #ns 9 II #H~ D,~, ~.(1)). wCZ' The exact sequence 0 ,0 gives rise to the commutative diagram 0 0 0 l 1 l kn l 1 1 0 , HI(Fx/F,.Tn) H1 (Fx/F, j~m) X" , ltl(Fz/F,.,Tm_n) l l l X ~ GwEN~Mw, n "--~ , Hw,,--n,"~" ' @weXoHw(.L~'m) ' (~wEXtlltw(.~P"m-n) whose last two rows are exact. Each M~,, ts a finite group such that #Mw, n ~ 1. It is apparent from the diagram that (6.10) Itz(~7"~) ~ Ux(,~,,,) and #Hx(,~) x #Hz(.~m)[)~"]. Similar considerations show that (6.11) #H~,(.~,,) � #H~,(.'~"m)[~.~]. We will combine the above computations with the following lemma to deduce some results about "divisible ranks" of various Selmer groups. Lemma 6.7. - The groups Hx(,7,,) and H~(,~) are finite A-modules whose minimal number of generators is bounded in terms of #Y, but independent[r of n. Proof. -This follows from (6.10) and (6.11). Note that as U~:(.Tn)is a submodule * t" of" Hx0(o~'~ ) it suffices to prove that the number of generators of H~:,,(,Tn) is bounded independently of n. [] A refinement of this lemma using also (6.9) is the following result. Lemma 6.8. -- Forming limits with respect to the obvious maps (7~) -- (K/A) T | X and lim (.7,,,) = (K/A)' | X* lim Hx~ ~ H~ "~ tl n with r < oc and X and X" finite groups. wweX~m RESIDUAI~IN REI)UCIBLE REPRESENTATIONS AND MODULAR FORMS 95 The following lemma is an analogue of [W1, Proposition 1.1 l] and it occupies a similar place in the proof of the main result of this section. Lemma 6.9. -- Let E be the splitting field of p, and let E~ be the extension of E obtained by adjoining all p-th power roots of unit)'. There exists an integer Nl > 0 such that for each n, H~(E~/F, .~(1)) /s annihilated by )(~'1. Proof. --- Let F + and F' be as defined prior to Lemma 6.2. There is an exact sequence 0 , H~(F+/F, .~7",(1) G~0':~/v+)) , Hl(Eoo/F, ,r (6.12) , H~(E~/F + , .~,,(1)). The first term in this exact sequence is bounded independent of n by Remark 6.6. Now consider the last term of the sequence (6.12). Let A = Gal(F'/F +) ~ Z/2. There are isomorphisms (6.13) H~(E~/F + , ,~(1)) ~ H'(E~:/F', ~(1)) A=' -~ H~(E~/F ', ,~Tn) A=-I , the first by restriction and the second by the fact that 27,(1) and .~ are iso- morphic as Gal(E~/F')-modules. Note that Gal(E~/F +) and GaI(E~/F') project iso- morphicaUy onto subgroups H + and H' of Gal(E/F), respectively. In particular, HI(E~/F ,, .~,)A=- l = HI(H ,, :7-)A=-l, and an element of the latter corresponds to an equivalence class of representations into GL2(A | aA/V) having trivial determinant and reducing to p modulo e. Here A| is given a ring structure by setting ~2 = 0. This correspondence is given by a E Ht(H ', .~P",) ~ , Pa : H' , GL2(A | e_A/~"), Pa(o) = p(o)(1 + ea(o)). Put H = H+\H ', and define a map q0" H , H', q0(o) = lim (ol~P"). n~oc Consider the open set HM = { a E H: trace(p(~)) ~ 0mod~ M }. By Lemma 6.3(ii), HM is non-empty if M is large. Fix such an M. We will show that the closure of q0(HM) has positive measure (with respect to the Haar measure of H'). For each n, write Pn for p mod)~n, and write H2, H',, and HM,, for the respective images of H + , H', and HM under p,. Being open, Hut contains a translate of an open subgroup of H', so there exists a constant C > 0 such that HM (6.14) - - >/C for all n. #H'. 96 C.M. SKINNER AJ. WILES Suppose h, h' E HM are such that pdh)~: p,(h') but q0(h) = q~(h')mod~. ". It is not hard to deduce that trace p,,(h) = trace pn(h' ). With respect to a basis for P,, such that p,(h) = (1~ _~-~), p,(h') = g-l([~ _~-1 )g for some g E GL2(A/~. '~) commuting with 9,,(q~(h)) = (1~ [~_l ) but not with (1 -~)" Therefore, given x E H:, we find that #{ h E HM, ,~" q0(h) = x } ~< #(A/K'~M). Let SM = q0(HM), and let aM, n be the image of SM under p,. Combining (6.14) and (6.15) shows that J~SM, n >/C 9 #(A/K TM) -~ > 0, #H'. from which it follows that g = lim SM M . ,n, the closure of q~(HM), has positive measure. Fix c~ E Ht(H ', g-)a--1. For (~ E HM, trace 9a(q0((~)) = trace p(q~((~)). As trace(.) is continuous, this equality holds for all s E SM" Fix a ~0 E SM having infinite order (this is possible provided M is large enough). Choose a basis for 9 such that P((~o) = (~ ~-1 ). Then Pa((~0) = .([~ [~-1 ) (l+Exew 1 -ezex) with ~.Mx = 0. Put O~ 1 = ~,Mo~. It follows that Pa I ((~0) is diagonalizable with eigenvalues [3 and [~-i. Pick a basis tbr pa I such that Pat(~ = ( ~ ~-~ ). As SM has positive measure, there must be some r > 0 such that X, = S M 7/(~0 SM also has positive measure. For any (~ E X, Y r trace Oh, ((~) = trace p((~) and trace pa, ((~0r~) = trace p((~0(~). Write p(o)= (a~,:o dob'~) and pa,(o)= "~ It Follows that both (~2r -1)a(o~x)o and ([3 ~r - 1)d(oq)o are in A. Thus if 0~2 = K'-'M~oq, then a~ = a(o~2)o and do = d(o~2)o for all ~ E X,. There exists ~ E Xr such that b, cT ~-O. If this were not so, then the image of pill+ would be triangular on a set of positive measure and also on the group generated by the set. This is ruled out by Lemma 6.4. Let nL = ordx(b~). If o~3 = ~n, c~.;, then b(a3)~ = b~(1 + et) for some t. Rescaling the basis for Pa~, we can assume that b((x3)z = bz. Now put a4 = ~. nl o~g. AN bzc(o~3) , : 1 - a(Ix3)~d(a3) x = 1 - ardz E A, it tbllows that c(cxt)z = c:. In particular, p(x) = Pa,(x). Now pick an integer s such that Y, = Xr M ~-'Xr has positive measure. As the eigenvalues of p(x) have infinite order and p(z) is not triangular, p(x') is not triangular. d(a,)o/'b(al)~ (a(al)o,~a,)o RESIDUALLY REI_)UCIBLE REPRESENTATIONS AND MOI)UI.AR FORMS In particular, b,,, c v ~= 0. Moreover, if ~ E Y, then by considering 9a,(#~) we see that (6.16) a.va ~ + bz.~c(lX4) ~ = av, c~ E A and cz, b(a4)~ + dx, d~ = ~' E A. Let n2 = max(ordx(bT,), ordx(q,)). Put a5 = k"'~cz4. It follows from (6 9 that for a E Y~, b(a5)~ = b~ and c(a5)a = c~. In other words, 9(6) = 9a~(o) for all ~ E Ys. The same holds for all elements ~ in the subgroup G generated by Y, This subgroup has positive measure and hence has finite index 9 Choose a subgroup G' C_ G of finite index that is normal in H +. Consider the exact sequence 0 , Ht(H'/G', (.~)G') , HI(H ', 3",) res H'(G', .5;r~). We have shown that if a E HL(H ', .:~7"~) n=-l, then res(o~) E HI(G ', 2W,) is annihilated by ~3.',t,+2,~+~.2. By Remark 6.6 there is an integer N2 (depending on G') such that k x' annihilates (.~n)G'. Therefore, 3 3M~+N~ +2~,+~0 H t(H , ' .~,,)~=-, = 0. Combining this with (6.12) and (6.13) yields the lemma. [] The next result, the principal result of this section, will enable us to control the ranks of various "tangent spaces" in the auxiliary deformation rings and Hecke rings that appear in the proof of the fundamental isomorphism (see section w Proposition 6.10. -- Let 6 E Gal(F~-/F') be an element such that the eigenvalues of 9(~) are in A and have infinite, order, as in Lemma 6.3(i). Then there exists an integer r = r(9 ) such that for each m > 0 there are infinit@ many sets Q = { Wl,..., w, } such that (i) Nm(wi) = 1 modp '~ for each i. (ii) 9p(Frob~) = 9p(6) mod 3. m for each i. (iii) lira Hx~t(,~,) _~ (K/A)' q~ Xx~t with I2o_ = 120 U Q, #X~:o_ < C(6, r) < e~ for some constant C(6, r) depending only on ~ and r. Moreover r is given by I2~nma 6 9 Proof. -- If the eigenvalues of c~ are cx and 13 then let )d be the highest power of ~. dividing (0~/13- 1). We fix a fi"ee, rank one quotient ,Ag of .57(1) on which ~ acts trivially. We denote by xn the projection of ,~,(1) onto ,/Zg, = ,~#g/~n. Write 9 * ~ Y O Xr~ hmH (Jn) (K/A) * ~0 i as in Lemma 6.8 9 Let f be the smallest integer such that kf annihilates X~. Let N~ be as in Lemma 6.9 and let N2 be as in Lemma 6.5 (for the group Gal(Fz/F+)). 98 C.M. SKINNER, AJ. WILES As the natural map H~0(~,,, ) , H~0(.~,,, ) for n' > n has kernel of order bounded independent of n we can choose n > m sufficiently large that H~ (..~,,) '~ t~(A/),5) @ X~ i= l with ri > N~ + N9 + e +f Let [el] E H~)(.Cz',) be a cohomology class of exact order ~, rl (where the ri are indexed arbitrarily) where Cl is a cocycle representing [cx]. Let E, denote the field generated by the splitting field of p mod ~,n together with a primitive p"-th root of unity. Suppose the annihilator of res([cd) E HI(Fz/E,, .~(1)) is k q. Then by Lemma 6.9 st > rl - N1. The restriction res([cl]) determines a homomorphism f E t [om(Gal(Fx/E,,), .~7,(1))C;ai(Vz/}). Since f has order s imf _D ~,a~,7,,(1) with n - al = sl - N2 by Lemma 6.5. Let Ml be the fixed field of kerr and pick "q E Gal(MI/E,) such that =,(f (~L)) has order at least ~,'1-N~. Let h = sl - N2. Put ir 0 gl = t~IM 1 "Zl otherwise. Then r~,,(cl(gl)) has order > ~,q. We choose a prime wl of F such that p is unramified at wl and Frob=,~ = gl in Gal(M1/F). By the choice ofg~, Nm(w0 = lmodp ~. Moreover the image of [el] in HI(Fw~, .//g,(1)) has order /> ~). r t Now suppose n' > n. Write H~0(.r ) _ "~ (~(A/~, r') | X* ~. We may assume that i= 1 there is a cohomology class [C'l] E H~,(.r of exact order ~.'~ such that ~m~ [C'~] is the image of [cl] in H~(~',,) for some ml ) Yl --rl. It follows that the order of res(~, m' [all] ) E HI(Dwj, ,-57n'(1)) is at least )v q-'. As tl-e = r~-NI-N2- e > 0 the order of res( [C'l] ) is at least ~,~-NL-X~-~. Let Yq = ~, U { w~ }. It then follows that l'_lm tI~t (..~'~,) ~ (K/A) r-I 9 X* -- 32 1 n t with ~:~X~l x( #X~.,0. #(A/~, Nl +'~'2+t'+'J). Suppose that inductively we have picked primes wl,...,wj such that for Ej= zo u lim tt~2(~ ) --~ (K/A) r-j Q X~ n RESIDUAIJN REDUCIBLE REPRESENqATIONS AND MOI)UIAR I,'ORMS 99 with #x;, .< We repeat the construction given for w~ and obtain a new prime wj+l. When we reach j = r we have lim H~,(.'U,) ~ X* #X~r <~ #X~0. #(A/k N' +x"+~+f)/. From this it follows that #H~;(~) ~ #X~, and by (6.9) #Hz~(3~) x #(A/k")r 9 #X~;. The proposition now follows from this together with (6.10). [] 7. Nice primes at minimum level In this section we assume that F is a totally real field of even degree. Associated to a cohomology class c as in (2.1) is a deformation datum ~ = (6s, Z, c, ../~g). We suppose that we are given a prime p C_ T~ which is nice for ~ in the sense of w rain Note that since p E p, p must come from a prime of T~ , and we also use p to denote this prime. min Now T~ ~ acts on the module M~ = M~J~ )m defined at the end of w Furthermore, T~ ~ is a finite, torsion-free A6:,-algebra. On the other hand, associated to ~ we have a deformation ring R~ n defined in w which is also a Ar~y-algebra. There is no nalural map R~ n ~ T~" since we have no natural representation with coefficients in T~)" ~. However there is a pseudo-representation with coefficients in T~" inducing the horizontal map in mi,l R~ ,,, ~ ) Tc(/min (7.1) rain N, N r~ Rff n min (cf. (3.4)). The map r~ is the one which induces the pseudo-representation associated to p~ n (see w mm Since Is is nice, T~ /Is is of dimension one and p C p. So the integral closure min A of T~/p is isomorphic to k'[[kll for some finite extension k' of k and some k. Furthermore the assumption that Is is nice ensures that under the composite map mill Ad , T~ /p ~-~ A the ring A is finite over A~ . (This is because the associated representation PplDi has at least one character of infinite order on the diagonal.) Writing A6 = C[[Zl, ..., Zm'l] 100 C.M. SKINNER, AJ. WILES let us suppose that zi t ~ s E A with ui a unit or zero for each i. Then we may take ri > 0 tbr each i (as follows from the definition of the A(# -action on Ts in w and we may assume, after possibly renumbering, that ul is a unit. Set (7.2) a~ = C'[IW~, ..., W,,]l, where C' = ~ | W(k'). There is a map ~-e~. -+ A defined by Wl , ~ ~. and Wi ~ ~ 0 for 2 ~< i<~ m. Define a homomorphism Ae~, ' Ae~ by z~ w--~W Iul, zi~ >--Wi+Wlui for 2~<i~<m. is finite and flee over Here ui denotes any fixed choice of lift of ui to Aes.'. Then ~.c A and we have a commutative diagram of rings ,< A. -< rain T~ From this diagram we deduce the existence of a prime p of ~n | Ar extending p. Similarly we deduce the existence of a prime p~ of R~ ~ @Ac, A~ extending p~, where p~ is the prime of R~ n associated to p as defined at the end of w For simplicity of notation, we may write p to denote p~ if the context in min makes this usage clear. It will also be convenient to write ~ for T~ QA~.~ AS' and /~_ minM. -- min R(,gr;~mi .... for R~in | A.# from now on. Let v ~ Jp and (T~ ~ denote the localization ~_min ~ nfin and completions of ~ and T~, respectively at p~ and p. Definc R.~ p~, ~, and (P.,~ p~)~ similarly; where pP~ is the inverse image of p under r~n: R~ p~ --+ R~ ~. Lemma 7.1. - There is a natural local surjective map rain N min (7.3) t~(~, p) 9 (R~ ~ --, (T~)3 rain rain , under which trace p.@ (Frobe)| 1 ~ T(g)| 1 and detp~ (Frobe)| 1 ~ S(g)Nm(g)| l Jbr a//g ~X. RI;SIDI,ALLY REDUCIBLE RI~PRESI'NTATI()NS AND MODUI..XR FORMS 101 Proof. -- Let rc be a uniformizer of ~", and let P = (Tt, W2, ..., Wm) C ~-C, 9 Write A~,, e for the localization and completion of Ac~ at P. Note that P = ~ f-/A,<,,, l,et C-J0 be the set of primes q of ~,~, o[ dimension two contained in P and such that each quotient *r /q is again a regular ring. It is easy to see that if q E G0, then qA~, p is also a prime ideal and that the set ~ 0 is Zariski-dense in spec (~-r~,P). min Let Q be a minimal prime of T~ contained in ~'. Let B be the integral closure min of T~/Q in its field of fractions. Observe that B is a finite, torsion-free ~.c -algebra (cf. [G, 7.8.3]). Let Pl,...,P, be the (finitely many) primes of B extending ~. For each i = 1,...,s let g~, be the set of primes of B of dimension two contained in Pi and extending some prime in ~C,~0. Note that given any q E ~)'0 there exists a prime D. E ~i extending q (cf. [Mat, Theorem 9.4(ii)]). For each i = 1,...,s we have a commutative diagram 1 1 The arrows are the obvious maps. That the top arrow is an injection Follows from the observation in the preceding paragraph. That the rightmost vertical arrow is also an injection follows t~om the fact that Bpi is a finite A~ p-algebra and that each A~. ~/q is a one-dimensional domain. If the bottom arrow were not an in~ection, then neither would be its composition with the leftmost vertical arrow, since Bpi is a domain and an integral extension of Xr,v (cf. [G, 7.8.3]). Choose t.~ E ~i. Let T = B/D and let R be the integral closure of T in its field of fractions. Let q = D KI At. and let A = A~/q. Note that A is regular. Since T is a finite, torsion-free A-algebra so is R. Let r ...,r be the (finitely many) primes of R extending pi. By Proposition 2.15, for each j = 1,...,ti there exists an extension R+ of R, Rf a domain, an extension K':j of C, and a prime q3j of Rf extending glj such that there exists a detbrmation Pi into GL2(R;) that is (~'.i, Z, c, ,//g )-minimal and whose associated pseudo-deformation is just that obtained from the one into T~ '~ described in w via the obvious map T~ n -* T. We have natural injections ti Tp, ~ II R% j=l and t t. t t. j=l .= , q3~ 9 102 C.M. SKINNER, AJ. WII,ES We claim that both maps remain injective upon passing to completions of the rings in question. For the first map this is an easy consequence of the t~ct that both T and R are finite A-algebras. For the second map we note that each Rq3j is an integrally closed one-dimensional domain and hence so is each R~. On the other hand, each R* is reduced, so the kernel of the induced map R%---+ . + must be either (0) or J, ~ J, 9i ~3j. It cannot be the latter, as we have R%. ~ R j, + V*.~ ~ ~+ i, ~+" This proves the desired injectivity. We have an induced injection t i (7.5a) j= l ' . We also have a map RT" | Cj ~ Rj inducing 9f. It is easy to see that the inverse image of ~3j+ in R~ n is just p(/. We thus have a map ti /~nlin~ n ~.+ (7.50) n . +- j:I'LJ" j, g3j mill Composing (7.5a) with the map (R~ p~,~ --~ Tp, coming from =~ and composing /~nlin,~ (7.5b) with the map (R~ p.,)~,~ ~ t ~ #~ (see Proposition 2.11 for surjectivity) yields the same map. Therefore we haw ~. a map (~min,~ ~p. through which (R~p~)~,~ ~ ~'p factors. Combining this with the injectivity of the bottom row of (7.4) yields a map 3" mill I "m l through which the map (R~ p.,~,, --+ 1-[~=1B,, coming from x~n factors. The image of lllin this last map is just that of the natural map fi'om (T~ ~/Q. As the latter map is injective, we obtain a surjection _ min\ ~ min - -- min mm through which the map (R~ ~)~,., ~ (T~)~/Q coming from n~ factors. From this we obtain a map (7.6) - rain N rain 1-I(T RESII)12ALLY RE1)U(,~IBLI~ REPRESENTATIONS AND MOI)ULAR FORMS lnin where the product is over the minimal primes of T~ contained in ~'. Since the maps rain ~ min (R,~ p,)~p~ ~ (T~)~/Q factor through the surjection (R~ r,.~)~v, -, (T~)~ coming from min -- min ~ , the image in (7.6) is the same as that of the natural map from (T.~,)~. This latter map is injective, and we therefore have a surjection min ~ min min through which the map (R~ p~)~,~ ~ ('r~)~ factors. [] Remark 7.2. -- The same result holds for ~ replaced by any of the auxiliary deformation data ~ 0v with the same proof. Since p is nice for ('_~, there is some choice of basis for pp such that pp has image in GL2(A ) and the corresponding representation into GI,2(A ) is a deformation of type-(~:', Z, c, ,~N). From here on we write pp for this deformation. We will assume for the rest of this section that ~0~ = (~J,. (i.e., that E = Z~ is the set of primes at which p~ ramifies together with ~)o = {v~" vilp}, and that = = Next we define the rings and modules which will be used for patching. Let R~ p~ denote the ring R(_/p~ @ A,~,; and let M~ = M~ @ A(~,. Then we define A( A(~" (7.7) N (~ = im{M~ , (M~)~/P} where (M~)~ is the localization and completion of M~ with respect to p C T~ and P C_ ~,~ is the prime defined in the proof of Lemma 7.1. There is a commutative triangle (7.8) 111[11 ~ nlil| R(_/ r " (R~)~/~-F0, P) where F0 = Fitt ( (M~ ~ ~ /Dmi"~ -- (R~-'i~ 25J C ~l,~ j~ is the Fitting ideal of (M~)~ as an The maps r and q)2 are the obvious ones. We define rings (7.9) R (~ = im(q)2), R tr(~ = im(qh). Now we introduce auxiliary levels. First we fix a c~ E Gal(F~/F') as in Lemma 6.30). Then there exists an integer r = r(pp) as in Proposition 6.10 with the following C.M. SKINNER, AJ. WILES I04 property. For each odd integer N there is a set of primes Qx = {w(~/, ..., w;(~} of F satisfying Nm(w:/~) = l(modp x) as well as property (ii) and (iii) of Proposition 6.10. We can and do choose the sets Qx to be disjoint from each other as well as from Z. For such a set Q = Q.~, we earlier introduced a deformation problem 6_~o as well as associated Hecke and deformation rings T~ and R~o ~. In particular at the end of w we associated a T~Q-module M~-Q to ~Q. We now set M~Q = M~ct ~A6 min as before and note that this is a T~Q-module. For each wi = wl ~ E Q = QN there is an associated element &o, E End T~,~, -,r (M~Q) as in Lemma 3.21. We let si = 8w,- 1. We can then define, for each odd integer 1 ~ a ~< N, (7.10) M~ = im{ , (M~c~/(P , s I , ...,s, 1)} rnin where the completion is as a T~osmodule (with respect to p) and Q = QN. Then a+l) M~ ) is a module over the ring a~ = A[[Sl, ...,s~]]/(s~ +l,..., sr by construction. Let 9~ denote the universal pseudo-deformation associated to 6~Q. Let 6~,~ E Gal(F/F) be such that ~w, E Iw~, the inertia group of wi, and ~w, is a generator of the p-part of tame inertia as in Lemma 3.21. Then there is a map of rings (7.11) X~ [Ill,..., tr]] ' I-(~s -- Rc~"(~ ~ A,.' @ X(~/ given by ti, , trace (9~,(~w;) - 2) @ 1, where here Q = Qx = { w,,...,/~)r }- We have a commutative diagram (7.12) a+[ a+l ~min "7" =2- _i~ " (R~ct)r tl ,..., tr , "p'Fx) 9 2, a RESIDUALLY REI)UC1BI,E REPRESENTATIONS AN1) MODUI~AR FORMS 105 where Q = Q~ and Fx = Fitt((M~Q)~) is the Fitting ideal of (M~o)~ with respect to /~_min ,~ --a --a the ring ~ ~Q2~. The maps are the obvious ones. We detine rings R (~) and R tr(N) by R~IN~ 9 , (N),, Rtar(N) 9 (N~ (7.13) = xm(~p.), ~), = tm(%, ~). a+l a+l These rings are both algebras over B~ = A[[h, ..., t~/(t~ ~r-, ..., t, T) by construction. 9 - GL / t'D min, " Let P~o. GaI(F/F) --+ 2~,.~o)~) be the representation obtained from p~Q. %-1 for some character ~/i. It is easyto see that for each wiE O_)~'~cah% ~- ( qi ) Define a map Al[sl, ..., s,]] -+ R~ ~ ~A K by si ~ ~i(cw,)- I. This makes R] "< | K into an A~ | K-algebra compatibly with its structure as a B~-algebra. Moreover, it is obvious that each si maps to 0 under the canonical map R~ "~? | K---+ R (~174 K. ,-~min ~ ~j... Q. The action of R~ IN) on M~ is obtained as follows. First ~(~ acts on nin ~ min whence it acts on M~ \3 through the image of T'~, z in (T~o) ~. Now we have homomorphisms a+l a+l /'7 11~ k"x~/ ps ~min ~min R~c ~ * T~o " " crwQ)o/(p, tTe-,...,tT,~FN) a+l a+ I -" min p -7- c_ , tT , ..., t, , and the diagram commutes by Lemma 7.1 and the remark following it. So by Lemma 3.21 we get an induced action of R~ (''~ on M~ v' which is compatible with the A~-action of the subring B~. Now put 2 t M(0', = | N (~ i--- 1 where N (~ is as in (7.7). The same reasoning shows that R 'r(~ acts on M (~ and a diagram as in (7.14) holds for N = 0. We define R~ \; and ~'0 by R~\9= R (~ and R~ rr = R '~(~ , -, ~ ~2 r rib define M't~' we first define a natural map M~o , M~ (Q = Qx). Let = = = ...,w)'}. l'br each 0 ~< i < r define a ~.r c-~ i c.c~ i where ~i {w~ ~, '?" l-l+ fl r rain "~2 + rnin map 1,~,o~i, -- --+ H~(U~+,) by (f(g),f2(g))' ' f(g) +J2 (g(l X,,1)) where 106 C.M. SKINNER, AJ. WILES ~,i§ E U/v | Z is the element chosen in the definition of the Hecke operator T(g~,,+~). Repeated composition of these maps gives a map H+(U~ ) 2r ~ H+(U~Q) and taking .... 2r Pontryagin duals and tensoring with A(~ gives the desired map M~ ~ M~. We define ~,,N) ~ ~ 2 r 9 "~0 to be the image of M~(~ , (M~)~/P. Clearly M:~: C_ M :~ We now verify that these constructions satisfy the properties in w needed for :x) formal patching. A bound for the number of generators of R (x! = R x is given by dimk(Hl(F~-~Q/F, ad~ which is easily bounded independent of N. A bound for the number of generators of R m~ = R tr(N) x is more subtle. When N = 0 this follows from Lemma 2.10, which bounds the number of generators of R~ ~,~. In general a similar argument applies using R~ ~,~, and a uniform bound independent of N can be given for example by applying Lemma 2.10 with E replaced by Z U Q. We'. can choose the generators in each case so that (5.2ii) and (5.6ii) hold by subtracting suitable elemenu of ~: [[WI]]. The other properties in (5.2) and (5.4) follow from the definitions. Next we consider the properties (5. I0) of M2 \!. Properties (iii)-(v) are straightfor- ward, as are the first two assertions of (ii), but we need to check (i), (vi), and the last assertion of (ii). Property (vi) follows immediately from Lemma 3.19 provided the hy- pothesis that U'/F � N U' acts without fixed points on D� holds. Here we need to take U successively as U~0, U~, ..., U~, = U~z where U~ = U~, .~-2~i = f~,, and rci = {w('~!,...,~ N} and check the conditions of Lemma 3.19 with v= w(,~ for U = U~ ,. However these conditions need not hold, and instead we consider modules with an auxiliary level structure at primes gl, ...,g~ )~Z U Q~ chosen so that M~(U~,), with U"= U~ A U1(gl "" "g~), is related to M~(U~) in a simple way. To achieve this, pick primes g 1, .-., g.~ satisfying the hypotheses of Corollary 3.6 as well as satisfying the conditions (7.15) (i) Nm(s 0 ~ l(modp) (ii) pp(Frobef) = 9p(tJ0)mod~. ~ for e sufficiently large, where tJ0 is chosen as in Lemma 6.3(ii). (In order that condition (ii) make sense, we identify 9p with 9~ modp.~, which in turn we view as taking values in GL~(A).) Condition (ii), for a sufficiently large, ensures that (7.16) trace pp(Frobt: S ~ detpp(Frobe~)(1 + Nm(gi))'2Nm(g,.) -~. ]'his, together with (i), ensures that M~(U~)p ~- M~,(U~,)~ ~ by Lemma 3.29, where the isomorphism is of A~;p0[Aw, J-modules..Now Lernma 3.19 and Corollary 3.6 can be used to check that Moo(U~i)~ is free over A(.~,)[A~,J for RESIDUALLY REDUCIBI,E REPRESENTATIONS ,k\'I) MOI)UI.&R FORMS 107 each wi = w i ' E QN. We then deduce the same result for M~(U~,)p and so also tbr (M~)~ and M~N;| K over A:, p[A+,] and Aa | K respectively. The second property (:,' of (5.10vi) follows similarly from Lemma 3.19(ii). Property (5.10i) follows from property (5.10vi) which was just established. It remains to show that the last assumption of (5.10ii) holds. For this we pick some I~ 1 E Gal(F/P b) such that pp(ch) has infinite order. That such a ch exists is an easy consequence of Lemma 6.3. I~t P~" be the pseudo-deformation associated to T~". It nlill follows from Lemma 3.26 that z = trace p~ (~Ji) - 2 annihilates the cokernel of" the map M~o ~ > M~ in the definition of M~ "~'. Therefore z also annihilates the cokernel of M~,2. ' M(~ and hence also that of M~ rq/ ---~ M (~ By our choice of ~, z fiSp. Note that z is independent of N. The properties in (5.9) are consequences of those in (5.10) as well as of the definitions of the M'X/'s. Next we verit) properties (5.5i) and (5.3i). Let dl(a) = dlmK(M 2 ' | K). This is independent of N by (5.10vi). Again using (5.10vi) dl(1)/> I-IRj@AK(J.VIa @A K) where as before ps(X) denotes the minimal number of generators of the S-module X. Now ~dl~a', annihilates M] ~' | K and hence 17" ...(N) ~(/l(a)Ul(1; C 1 lttR:.~ . (Mo | K). -- a @AK From the definition of R] ~ | K it follows that I = 0 in this ring so we may take d(a) = dl(a)dl(1) + 1. Now we check (5.5ii) and (5.3ii). Recall that we are given a set Q,~ = {w/~,...,wl N)} of primes satisfying the hypotheses of Proposition 6.10, and that by the same proposition (7.17) lim Hr.Q(.7,) ~ (K/Ay (9 X~z with s = Z U QN and #Xz~z bounded independent of N. Now let Q = QN and "-'min ~ ~ rain suppose that #~ ~z' #2 ~z are the primes corresponding to p in R~Q and R~r z. Then we also have the usual isomorphism in the style of [WI, Proposition 1.2] /': umin /: ~'min '~2 k-"A/A) -% Hz,z(.'~,, ). (7.18) X( 4 = Itom A k~v.~,2_/kv(,/.ct: , P), The isomorphism is obtained as follows. To an element ~ E ~o~ we associate the representation (7.19) pe Gal(F/F) +m~,, "--m~,, 2 9 , GL,,(R+ct/((I~o ) , P, ker ~o). ~dl~+)dl+t)+ 108 C.M. SKINNER, AJ. WII,ES ]'his is a deformation of pp with values in A[e]/0J'e , e 2) and its associated coho- mology class lies in Hz~2(;'7,). Conversely to such a cohomology class, we obtain a deformation with values in A[e]/(~,"e, e"), and hence by universality a homomorphism R~Q| ~' ~ A[E]/(~."e, E2). Extending scalars we get an A-algebra homomorphism ~xmin /p ~nin P,.~/P , A[e]/(~."e, c 2) which factors through ~/ . Restricting to p~ we recover an element of XQ. The restrictions on classes in H~V.~.7,,,) correspond to the restrictions required for p to be of type -~0_ e~ mi- " In particular, the "min" condition, together with reduction mod P, ensure that P~0 does not deform the determinant. This is why the cohomolog-y class we obtain is associated to "~,,, = ad~ ' rather than adp/~.". We omit the details and refer to [Wl] for a more detailed argument in a similar situation. Of course (7.18) also holds with ~ o__ replaced by ~ and s by s To apply this we use the sequence of homomorphisms -rnin/, ~min,2 .(N)/,.(N),2 p(0)/(p(0))2 P'2gQ/~P~.~Q. ) P) . p(N)/(p(N))2 ~min/, ~min,2 P~ ///P~ ) ,P) Here Q = QN. Tile horizontal maps are surjections arising from the definitions in (7.9) and (7.13). Tile maps ~O_. and 130 are surjective and 130 is an isomorphism after tensoring with K. That [~e is also an isomorphism upon tensoring with K follows from Proposition 6.10 and (7.18). Properties (5.5ii) and (5.3ii) then follow from (7.17) and (7.18). Next we verify property (5.7). Using (7.18) the condition in (5.7) translates into the requirement that #GN, where G~ = {[P~] :trace P~ = trace Pp, q0 E fro_), is bounded independent of N and n. (Here as before Q = Qx). Fix a basis for Pp such that Pp(zl) = (1 -1-) and pp(O') = -(. .u)- for some unit uEA � where zt is a complex conjugation and c' is some element of Gal (F/F). With respect to this basis write p,(o) = (a~co ~)" Fix a ~ such that c, flu 0. Now suppose that [Pt0] is a class in GN. The class [P~] has a representative P~0 such that and b'ccz E A for all o E Gal(F/F). Hence c~ annihilates [P~0]. Since the number of generators of GN over A is bounded independent of N and n (as ibllows, for example, from (7.17) and (7.18)) we obtain the desired bound on #GN. RESIDUALIN REI)U(:IBLE REPRESENTATIONS AN'D MODULAR FORMS 109 It remains to prove the existence of an element x tN) as in (5.11). Let us write M ~ min for M~, which is a T~ -module. Let J = ker{ (M/P) 2~ ~ M (~ Since M (~ ~ (M~)~/P by definition, J is just the kernel of M2'/P , (M/P)~. Let m~,...,m, be generators of J as a 1'~-module. ~min For each i, choose xi ~_ T~ , xi~'p, such that x,'mi = 0. Put x = xl".x~. Clearly, x annihilates J. Also, x )Ep since each xi ~'p. Now set MN = M~ where Q = Qx. Then suppose that we have an element ~c, in y('~ C ~<t with the property that ~ z , (7.20) y :N, ker{Mx/(Sl ..., ST) > (M/P) 2~} = 0. It would follow that xy (N) would annihilate ker{MN/(S~, ..., sT) , M (~ where x is a lift --rain ~ ~N' 9 9 (N) of x to T_~:( Q, and so xy' ' would also annihilate ker{M~ /(Sl, ...,s~) > M(~ Thus it would satisfy condition (i) of (5.11) except of course that it is not in the desired ring. Our construction of such ay (N' is an involved procedure. We begin by introducing auxiliary level structures much as we did in the proof of property (5.10v i). Let g.~, ..., g, )~g tO Q, where Q = QN, be primes satisfying the hypotheses of Corollary 3.6. as well as (7.15) and (7.16). We can and do choose the g; to be independent of N. Now let U' rain ' = M~(U~ )m | A~'- Let = U~Q V/UI(gl "" "g,) and put M N c-C/(Z <Q. t ~ min also ~_q4' = (C', E', c, J'd) with Z' = Z tO {g~, ...,gs}. Let T N denote T~(~. It is clear that M~ is a T~-module. t 2s There is a natural map My > M x defined analogously to the map MN > M 2r used in the definition of "'0~a(N)" Composing these maps we obtain a similar map , ) M e~s My . Arguing just as in the verification of (5.10ii) (see the first full paragraph ,~min following (7.16)) we find that there is some y(~') E ~a such that (7.21a) y?/. coker{M~ > M~} = 0. and 0 :~ ordx(y] N3 modp) is independent of N. (7.21l)) .(N) I We next construct Y2 E T N such that ~N~ } = 0 (7.22) Y2 " ker {M~/(s,, ...,st) > (M/P) ~'+' for then fiN) =y/~} .y~) will satisfy (7.20). (We view si as an element of End(M~) just as we did for MN. These actions are compatible under the map M'.x > MN2'.) 110 C.M. SKINNER, AJ. WILES Let U~Q = U~"N Uo(w,,...,Ws)71 Ul(el,...,er) and write M~ for the module M~(U~)m @A,< A~ . By our choice of gt, ..., g,, U'~a satisfies the hypotheses of Lemma 3.19, from which we deduce that M~/(s~,..., Sr) --- M~. Let Ix be the set of minimal primes q ofT' such that M" :~ 0. Let IC Ix be the N N,q -- -- min subset consisting of the inverse images of the minimal primes of T~. It is relatively straightforward to see that for q E I~\I the representation Pq is type A at each of the (w, primes gi and Jw: . If .-<.~: is the field of fractions of A~ then we have M x~ M~ | ~ = II M~,q qG1N M ~ M | <~ = H Mq. qEl The kernel of the map M~ | ~' , (M | J~):' is just qei.~\iH M"g, q. Therefore, ify~ '~' E ["l q, then qEIN\I (7.23) y;'. :.-,~ ker(M~ ~ M~, +, ) = 0. We choose y~X> as follows. Let P.',' be the pseudo representation associated to T-". Let "ci E Gal(F/I:) be a lift of Frob~..N, ' and let 5i be a lift of Frob~: Put Y~") = II (.l.~,, _ a,,,,qi" :(N) + 1) 2) " 1-I (T2e~- de,(Nm(gi)+ 1) 2) i= 1 i= I where T,, i = trace p~.('r,i) . , qi(N) = Nm(wi iN; ), dwi = det p',4(~i), qi(N)-' and similarly tbr Tti and dtr Then y~\/ E q for every q E Ix\I as can be seen by examining the actions (N; of Dw and D~, (decomposition groups at wi = w i and gi, respectively) on the Galois representations associated to such primes q. From our choice of wi (and our choice of o as in Proposition 6.10) 9 O O 1 [,i - d"i (q(]'~' + 1)2 = trace pp((5)- - 4 ~ 0 mod (p, ~:) for some sufficiently large e independent of N. This, together with (7.16) shows that ,iN ~ (7.24) 0 :~ ordx~y~ Jmodp) is independent of N. RESIDUAI,LY REDUCIBI,E RI,iPRESENTATIONS ANI) MOI)UI,AR I"ORMS Finally ar,,~ing as we did to establish (5.10ii) as well as (7.21a, b) shows that there is some y~')E T~ n such that (7.25a) (x~ 9 coker{M ,, x ~ M 2'+' } = 0 and (7.25b) 0 :~ orally 4 modp) is independent of N. Let y.~"q and ,4~"' be lifts ofy~ w' and y4 \~' to T x. Combining (7.23) and (7.25a) shows that y~' =y:; -'g" .y~Y) satisfies (7.22) 9 Moreover by (7.24) and (7.25b) ,N), 0 ~: ord)~(y,) modp) is independent of N. We may then take ~6 ~? to be any lift of x .y:'~: to R(. p,. Its image in W ~u'~' satisfies to, Q (5.1 li, ii). We have now verified all the hypotheses in w and are thus in a position to prove the main result of this section. Proposition 7.3. -Suppose that F is a totally real field of even degree. Suppose that .~ is a deformation datum and that ~ = ~. Suppose finally that p C T~ is a prime which is nice for ~ in the sense of w Then rain ~ rain ~ rain (i) gt(~, p) 9 (R~)~ , (T~)~ is an isomorphism and (R(/)~ is a complete intersection over ~,~, l, and reduced; (ii) M~, ~ /s a flee Cr -H,i,, ~ )~-module. Proof. -- Our constructions give the following identifications: rnin R,',0; | K = (R~)~/(pF0, P) rain RW0: | K )/P 2 r N~~ K = (I~)F/P, M(~ K = iO(N ~~ Q A K). By Lemma 5.2 the natural map (7.26) R '''~ @a K ' R (~ | K is an isomorphism. By Proposition 5.8, M (~ | K is free over R (~ | K. As the action of R (~ | K on M (~ | K tZactors through the composite map R ;~ | K ~ R ''(~ @,t K c~min~/p /~,min, ~,) , 9 $/ we conclude that M (~ | K is a free ix(,, )~/l-moame and that gt(~, p) 112 C.M. SKINNER, AJ. WII,ES min ~ min induces an isomorphism (R~)~/(pF0, P)--~ ('I'~)~/P. Picking generators of M(~174 as an R ''(~ | K-module and lifting them to (M~)3 we get a map (for some minimal s) A 9 (7.27) ('I~)~. ~ (M~)3 which is an isomorphism modulo P. Since (M~)~ is free over A~,p it follows that min (7.27) is an isomorphism. In particular (T~)3 is free over .g,~; p. As observed, the reduction mod P of the map (7.28) - min ~ min (R~)J(pF0) , (T~)~ mlil induced by ~(c.ffr p) is an isomorphism. Using that ('I'~)~ is free over A.~ we now deduce that (7.28) is an isomorphism. Under (7.28) F0 maps to zero as M~,o is a free min ~ min ('l'~)~-module. So F0/pF0 = 0 whence F0 = 0. Finally (R~)~ is a complete intersection rain ~ min rain since (R~)~/P is by Proposition 5.9. (Note that ('r~)3 is reduced as Ty is reduced.) This completes the proof of the proposition. [] 8. Raising the level for nice primes 8.1. Preliminaries In this section we complete the proof that property (P1) holds for a defi)rmation datum c~.~. However, before doing so we need some auxiliary results. We begin by imposing a partial ordering on the deformation data. If 6.~1 = (], Z1, Cl, .AN l) and ~-c2~2 = ((92, E2, c2, d/~2) are data, then we write 6.~1 /> ~-~2 to mean ~l = 82, cl = c2, Zl _D Z2, and ~t/~l C_ Jfg2. If #(ZI\E2) + #(,/f~2\.///~1) = 1, then we say that the inequality ~l /> .ff(~. is strict. Let ~ = (~-', Z, c, Jtg) be a deformation datum and suppose that p C T~ is nice for ~. As p E p, p is the inverse image of a prime of T~" which we also denote by p. We adopt the notation and conventions from the beginning of w The primary goal of this section is to prove the following proposition. rain Proposition 8.1. -- /f p C_ T~ is nice for c~j, then the map ~(c~, p). (R~)3 min (T~)p in Proposition 7.1 is an isomorphism. For ~ = ~c.@'c this was proven in w - w (see Proposition 7.3). We will deduce the general result from this case by a generalization of the arguments in [W1, Chapter 2]. RESIDUALLY RH)UCIBLE REPRESENTATIONS AND MOI)ULAR FORMS 113 8.2. Congruence maps A key ingredient in our proof that V(c"J, p) is an isomorphism will be a lower bound for the length of a certain "congruence module". In this subsection we construct min maps between various T~-modules that will be instrumental in obtaining this lower bound. We first fix a sequence of deformation data ("~.-~ = 6~ ,< .~ ,< ... ,< G.q4~. = such that .~r i ) c.(~ i_ j is a strict inequality for 1 ~< i <, n. Put min ~ m Ri = (R~) ~, Ti = ('~,@i) ~' Mi = (M~,)~, and M i = (M~.)~, where M~/, is defined as is M~ but using M ~ + instead. Let P = ~N~.4 ~. Lemma 8.2. -- Each Mi and M,~ is a free At,, e-module. Also, there exists an integer s ,7,~ t, 2 ~ such that M~' -~ Horn x (M[, Ar p)- and M i ~ Hom x (Mi, AC. p)2' as Ti-modules. (',, P (;, Proof. -- Choose a set of places {rl,..., rs}, distinct from those in Z, satisfying the hypotheses of Corollary 3.6 and such that 0J(Frob ) z~ 1 and pp(Frob) has eigenvalues of infinite order for each i. Put U, = U~iN rl(yl,...:,?~r ) and U7 i"= U'~' n U~(r~,...,r,). '9 By Lemma 3.2., To~(Ui, ~')p -~ T~,p and nlin 2 s rain § ,.~ 1~1-+. 2 2 Mo~(U, )p "~ M~,.p and Ms(U,. )p -- lvL~i,p. [] The lemma now follows from Proposition 3.3 and (3.17). Fix an s as in Lemma 8.2. Now let 2 if wi C_ ,~.f~ i- 1 {w3 = z,\z,_, u 3 if Wi E ~,i\~,i_l, and (qi - 1) (T(gi) 2 - S(g/)(q, + 1) 2) if wi E Zi\Y'. i_ 1 if wi E J/Ni-1 and Z[r,,, = 1 rli = (qi- 1) (qi + 1) (q;- 1) if w, ~ '~i-I and )~]I% ~ 1. Here gi = ~,, and qi = Nm(gi). Next we define maps of Ti-modules ' ~ t q,~ q.v 2 .~ rt 0," Mi2_'l ) M~, Oi" M~ > Mi_ 1 iS v,., o,-MfZ 114 C.M. SKINNER, AJ. WILES such that (8.1) a) ~i is injective with Xm, i,-free cokernel and ~ is surjective. 9 M~,,,(Ti_ !) with the image of det(Ai) b) ~)i O $30 (I) i O I'P i -- not a zero-divisor in Tv. 6") iITl(~I j, O ((I~i_ 1, "", Oi-1)) = im(Oi_ l, ..., Oi_t). Let ~,, = ~(~'i/ be as in the definition of T(gi). For any f: GD(A/) , R (R an ~- Xi))" We define *i to be module) let a3f: GD(M) R be given by (cq~(x) =f(x( 1 r i tile direct sum of 2' copies of the localization of the map (lim q~) | 1 : Mc.,~,_l ~)Ac ~'(~ , M~, | AO = M~, where *7" eI-I~ - I, a), c"~) r~ , eH~ a), ~') is given by (f ,f2) ~--*f + ~i~ if r, = 2 and (f ,.~ ,f~) ~----*f + ~if2 + (x~fa if ri = 3. We define 4- + + q~i " Mi+-'i ' * M i similarly and take for ~i the dual map obtained from ~i by applying Homx~r e(. , A~, ~,). Similarly, let ~, be the dual of q~i. We now verify (8.1a). Choose n to be an ideal such that U0(n) _D U,, D U(n). If u)i E ]gi\~i-I (SO ri = 3) then by Lemma 3.28 for a sufficiently large both the kernel of a ~a q~i and the cokernel of ~, are annihilated by T(g) - 1 - Nm(g) for any prime e that splits completely in the ray class field of conductor p~-n-c~. Here ~i is the adjoint of ~ with respect to the pairings defined in w Let F~ be the ray class field of conductor p~ 9 n- cxD and let F~ = UF~. Choose c~ E Gal (Fx/F~). Such a ~ is the limit of a sequence of Frobenii {Frobeo } of primes g~ splitting in Fa. In fact such a sequence rood. . can be chosen so that Frobeo = Frob~b (b/> a) in Gal (F~/F). As trace p~ ~) is the limit mod ~l~" ~ a of {T(eo)}, it follows that trace 9~,/ ) - 1 - ~(a) annihilates ker (*~) and coker(q~i). rood. . As pp is neither reducible nor dihedral it must be that trace 9~, t~)- I -e(~))~t~ for some ~ 9 Gal (F/F~). It follows that both ker(lim*~) and coker (lim~) vanish when localized at p. Thus ~i is injective and ~, is surjective. A similar argmment shows that ~i is injective and ~i is surjective. This proves (8. l a). Now if wi 9 .-.4r then it tbllows from Lemma 3.27 that ker (~i) and coker(~i) are isomorphic to submodules of (I~(U'))~' where m'm {(: bd) e, ei',)-l} U' = U~ 3- l " 9 GL?( ~v, ~i) : a -- 1, c 9 g~� with r(Si) as in the definition of lJ'~. By considering 9~!4,,~ one sees that p is not a prime in To,)(U', (~-~') (so T~(U', ~')~ = 0) whence M~(U')p = 0. This proves that RFSIDUALIN RFI)UCIBI.E REPRESENT.\TIONS AND MOI)ULAR FORMS 115 ~i and ~i are, respectively, injective and surjective in this case. The same argument applies to ~i and ~i, thereby establishing (8. l a) in this case as well. 2 s ~l/t Next we define q~i and Oi. If wi E Y.i\Ys_I, then we put ~, = | and | = ~--'l| where -s(e,) o o o o s ,) T(e,) -S(e,)-' , 01= 0 s(e,)-' . ( 01) ( !) --qi 0 -- 1 0 If wi E ././g ,_ l, then we take , (T(g:) 0 ) , (0 1) q2i = -qi T(gi) -1 ' Oi = 1 0 " We note that while T(gi) is not included in the definition of Ti-t if wi E ~/:g:-l, it is in fact in Ti_I and is a unit, so the definition of q~i makes sense in this case. To see this, let Q be any minimal prime of T,..~ ~_~. Then T(gi) is the eigenvalue of the action of Frob:, on the maximal unramified quotient of 0oJDe. (This can be checked on the representations associated to algebraic primes containing Q.) As wi E ,//~,._ i, the min representation 9~, .~ [I)t. has a non-trivial maximal unramified quotient, and it follows that the image in Ti_I under gt(~, p) of the eigenvalue of Frobe, on this quotient must be T(gi). As (8. l c) is obvious from the definition of ~Fi, it remains to verify (8. l b). Suppose first that wi E Zi\s A straightforward calculation shows that ~i o ~i is a direct sum of 2' copies of T(gi) 2 - S:g, ,/,~,v" + 1) '~ T:g qi(qi + 1) oqi S(~ i)- 1 T(g i)qi q:(qi + 1) T(g,)q, ) / '~2 -: -2 T~gi/ Skgi) -- S(gi)-l( 1 + qi) T(gi) S( g i)- l qi q,(qi + 1) Thus nil,) Oi o ~i o ~i o q~i = ____~_ , det(Ai) = (qiS(gi)-2)2'.q2'-1. IrXi That the determinant of Ai is not a zero-divisor in To is easily checked. As To is reduced, we need only verify that det(Ai)~Q for all minimal primes Q of T%. Suppose that det(Ai) is contained in such a Q. Let P C_ T~o be an algebraic prime containing Q (and hence det(Ai)). Let t = T~ei:modP : ~ and s = S(gi)modP. We will show that t e -s(1 + qi) 2 :~ 0. Let a and [3 be the eigenvalues of 9p(Frobf ). Recall that 116 C.M. SKINNER, AJ. WII,ES t = o~ + [3 and sq, = ~. If t 2 ~1 + qi) 2 = 0, then either et -- -- = qi or -- = qi. But both possibilities violate (3.3). It follows that det (A,)~P and hence det (A~)~Q. The verification of (8.1 b) in the case where wi E ,.//~ i-~ is done similarly. We are now in a position to define our "congruence maps". Put I ~ = ker{T, , To} and P" = AnnT,,I ~ Put also 1 <~j<~i 5., 2s 2 s Define @('~' 9 ~2 ~,, 9 "~0 > M i and ~('~' " *'~0~z'e"; M i recursively by ~'J) = ~1, ~(a) = @i o ~l, and (I) (z~ -- (D i O ((I) (i-1) X 9 9 9 X (I) (i-I)) ,i,('~ = (r o '-I.'~) o ($(~-'~ x ... x ,i,('-'/). Define ~i) . M~' ' ,'~0 and "M[ ~ ~'-0 in the same way but using ~i and Oi and reversing the order of composition as appropriate. Put ~,,~ -~,,,, $ = $~,,~ $~',. @=@ , @~ =@ , , and~ = Put also "l"lr r = r(n) and rl = l[ ni- l <~i<~n Lemma 8.3. - (i) im(q)~ ) = M, [I ~ 2: and coker(~ ) is a flee ~,~, p-module. (ii) @~ is surjective. / \ (iii) ~ o @~ = / (unit)xTI *] E M2,r(T0 ) with det(A) not a zero-divisor. (iv) ker($~) = M,[In'w]2'. Proof. -- Part (ii) follows from (8.1a). Part (iii) follows from (8.1b). We leave the details to the reader noting only that det(A) is a product of powers of the det(Ai)'s and the qi's. The freeness over A~, p of the cokernel of ~ also follows from (8.1). It remains to prove the first assertion of part (i) and part (iv). RESIDUALIN REDUCIBI,E REPRESENTATIONS AND MODULAR FORMS 117 Note that by (8. l c), im(@~) = im(~), so it suffices to prove that im(~) = Mn[Iold]. Next note that 9 is the localization of (lim ~) Q 1 where ~ is defined as is 9 but with ~ replacing ~i. Now let I~ ~ = ker {T2(U~. ~, C') > T2(U~ .... ~9~)}. It follows from the theory of "new vectors" that im(@ a) | K = eH~ ~), K) [I~ (For a more detailed proof in a similar situation see the proof of Lemma 3.29). Now consider the commutative diagram [imO ~ lim eH~ ~), ~?)~ ~> lim eH~ ~), (Y') [Ia ~ --o C a g l 1 0 , lim im(~") | K , lim eH~ ~), K) [I ~ ,0 a a having exact rows and with the vertical arrows being the natural maps. Applying the snake lemma we find that C embeds into a quotient ofN = lim ker(@~QK/-(~5). Arguing as in the proof of Lemma 3.29 shows that Np = 0 and hence Cp = 0. Now let I~ = ker{Too(U~, ~') , Too(U~, ~)}. It follows from the preceding remarks that the quotient M~ [I~a]/im(lim @~) vanishes upon localizing at p. As I ~ -- -ooI~ part (i) follows. To prove (iv) we first note that Mo[I"~w ] = 0 by Lemma 8.2, for M0[I"CW]is a T0/InCWT0-module and hence a torsion Acc,;p-module. Therefore M,[I n~w ] C ker(~ ). On the other hand, it follows from (i) and (iii) that @~ maps M,,[I ~ ] | .5~ isomorphically onto M(~)| where ~ is the field of fractions of/1r r. As M,| = (M, [ I ~ ] @ 2~ ) 9 (M, [ I ~ ] | cf~ ) it follows that the quotient ker (~)/M, [ I n~'~ ] is a Aes, v-torsion module, from which we easily conclude that ker(~_c,,~) = M,[In~']. (The tensor products are as ~:~,,-modules.) [] 8.3. An auxiliary result We now state (and prove) a simple restflt in commutative algebra. This result will be important in the proof of the main result of this section (the proof of Proposition 8.1). Let A = B~Xl,...,x~]] be a power series ring over a complete DVR B of characteristic 0. Suppose that (Al, A2, [3, NL, N,2, r, qo, ~) is an 8-tuple consisting of 118 C.M. SKINNER, A.J. WILES 9 complete local finite A-algebras At and A2 with AI reduced, 9 a surjection [~ : A,) ~ Al of A-algebras, 9 for each i = 1,2 an Ai-module N, with each Ni finite and free over A and with NI free over A1, 9 an integer r >/ 1 and maps of A.,-modules q) : N' I ~ N,) and ~ : N2 ~ N] such that ~ o q~ E Mr(AI) C_ EndA(N~) and det(~ o q~) is not a zero-divisor in At. We further require that 9 im(q~) = N2 [ I ] and ker (~) = N2 [J ] where I = ker ([3) and J = AnnA. , (I), 9 coker(q~) is A-free. Lemma 8.4. -- l~br each 0 <~ t <~ s there exist yl, ...,yt E A such that (i) ~Yl, ...,Yt) is a prime ideal of A, (ii) yl, ...,Yt generate a t-dimensional subspace of mA/(m~A, rn~) (iii) A,/(Yl, ...,Yt) is reduced, (iv) A N (I +J,yl, ...,yt) ~: (y,, ...,yt), (v) det(~ o q0)mod(yt, ...,yt) is not a zero-divisor in At/(yl, ...,y,), (vi) ker (} mod (yt, -..,yt) ) = N2 [J ]/(yb...,yt), (vii) im(q~ mod (yt, ...,yt)) = (Nz/(yl,-.-,Y3)[I]. Proof. -- Our proof will be by induction on t. Note that if t = 0 then all the conclusions are satisfied by the hypotheses on Ai and Ni. Suppose then that we have foundyl,...,yt, t < s, satisfying the lemma. We will show how to findyt.l. Let ~) C_ A' = A/~I, ...,Yt) be a prime ideal such that 4 (Y!, ...,Y~,Y) satisfies (i) and (ii), b) (y) does not contain AnnA,(aA,/!,., ...,y,!/A')' c) (I +J,y~, ...,yt,y) N A =~ (Yl,---,Yt,Y), d) (det(~ o q)), y,, ...,yt,Y) ~- ~Y,, ...,yt, y). Clearly all but finitely many (.iv) satisfy a) -- d), and since there are infinitely many possibilities for (y) some (y) has the desired properties. Note that Al/(yl, ...,yt) is a finite and free A'-module because N1 is finite and free over A and also free over A~. Hence the hypothesis that At/(y~, ...,yt) is reduced is equivalent to ~'~al/(Yl, ...,_,'t)/A' being a torsion A'-module (here we use the fact that char(B) = 0). We now show that one may take for yt+l any y such that (y) satisfies a) - d). Properties (i) and (ii) follow trivially. Property (iii) is a simple consequence of b). Property (iv) follows from c) and property (v) from d) once we know that A1/(yl,...,yt,y) is reduced. Property (vi) is immediate. It remains to prove property (vii). It follows from (v) that ~ maps im(q0mod(y~,...,yt,y))| FA,, isomorphically onto N]/(y~, ...,Yt,Y) | FA,,, where A" = A'/(y) and FA,, is its field of fractions. If we RESIDUAI,LY REI)UCIBI.E REPRESENTATIONS AND MODULAR FORMS 119 can show that (8.2) (N,)/(yl, ...,yt,y))[I] 71 ker (~mod (y~, ...,y,,y)) = 0 then it will also follow that ~ maps (N2/(yl,...,yt,y))[I] | FA,, isomorphically onto N]/(yl, ...,y,,y) | FA,, whence im(q0 mod (y~, ...,y,,y))| F^,, = (N2/(y~, ...,y,, ...,y))[I] | FA,,. The desired equality will follow from this one since im(q)rood (yl,-..,Y,,Y)) is contained in (N2/(y~,...,yt,y)[I] with A"-torsion-free cokernel. (Here we are using that coker (q~) is A-free.) To prove (8.2) we need merely note that the intersection in question is contained in (Nz/(yl, ...,Yt,Y))[I +J] which must be zero as it would be simultaneously a torsion-free A"-module if non-zero and annihilated by 0 :~ (I +J)A A". (The latter is non-zero by c)). [] 8.4. q/(~_~, p) is an isomorphism We now complete the proof that V(_~, p) is an isomorphism. To do so we return to the notation of w By Proposition 7.2 M0 is a free T0-module and V(cJc, p) : R0 ,.o To. Moreover, To is a reduced complete intersection over ~,e~, P. Let A = ~,~,p. Note that A = B[[W2, ...,Wm]] where B is the localization and com- pletion of C'[[WI]] at the prime ideal (~). Let ~ : T,, ~ To be the natural surjection. It 2 s 2.~ follows from the results of w that the 8-tuple (Tn, To, ~, M, , M 0 , r, O~ , ~ ) satis- tics the hypotheses of w Therefore by Lemma 8.4 there are elementsyt, ...,ym-1 C A such that (8.3) (i) A/(yl, ...,Ym-l) ~ B (ii) T0/(yl, ...,ym-l) is reduced. (iii) im(O~ mod (Yl, ...,ym-l)) = ~vl,/(yt, ...,ym-t)) [I ~ ]. O~,~ o ON ((unit) x 11 * ] (iv) = E Mr(T0). (v) det(O~ o O~) is not a zero-divisor in T0/(yl, ...,Ym-~). Put Ri = Ri/(yl,...,ym-~), Ti = Ti/(yj,...,ym-,), and M i = Mi/(Yl,..., ym-l) for each 0 ~< i ~< n. Let Q be a minimal prime of T o . As T O is rcduced and "q is not a zero-divisor in T O (11 being a divisor of det(O~ o ON )) q ~Q. Let C be the integral closure of T0/Q in its field of fractions. As To is a free A-module, T o is a free B-module. Thus C is a complete DVR and a finite flat extension of B. Put R'i=Ri| Ti=Ti| andM i=M i| 120 C.M. SKINNER, AJ. WILES Let ~' = (M~) ~' ~ (M,'y ~ and ~" (M'~ 2s ~ ~'-0J ,--,,, rx~'~2'r be the maps induced from q~ and ~, respectively. We have maps R,X' W'~ ~ T'ffAnnT,(M',) ~ a T 0' _~ R 0' and 5" T~ ~ C. Put [3' = c~o% Here ~' is the map induced by ~(.~,p), D' is the map induced from [3 " T, ~ To, and 8 is induced from the reduction of To modulo Q. That [3' factors through T,',/AnnT,(M',) is a consequence of the surjectivity of ~' and of M0 being a free T0-module. Put T" = T,',/AnnT;(M'~). This is a free C-module. Now put H0 = ker(8), Go = AnnT~ ' ker(8), H, = ker(8 o ~' o tg'), G~ = AnnT,, ker(8 o a). As T(~) is a reduced complete intersection ove.r C, it follows from [DRS, Criterion 1] that (8.4) gc(H0/t-I~) = gc(C/8(G0)), where for any C-module X, go(X) denotes the length of X as C-module. Our goal is to prove a similar equality for gc(H,/H~) and go(C~(8 o o0(G,,)). First we prove that (8.5) go(C/(8 o ~)(G,))/> gc(C/8(G0)) + ec(c/~(n)). We prove this as follows. Let I = ker(a) and J = AnnT, ~,(I). It follows from the definition of T,," that M,[I' o~d] = M,,[I].' Therefore, by Lemma 8.3(i), ( 2~ C_ im(~'). In particular, ifj E.J and m E M~, then there exist ml,..., nor E M~ such that (m) A m is r As det (A)mi = 0 for i = 2, ..., 2Sr we have that mi = 0 for i = 2,..., 2'r since det (A) is not a zero-divisor in T; and M~ is a free T;-module. We conclude that.lMi, c_ -qM; and hence (8.6) 2r,', c (n). Now suppose that g E G,. Then ct(g) E (rl) by (8.6) since g annihilates I C kcr(8 o a). Write a(g) = fix. Since ~x annihilates a(ker(8 o tx)) = ker(5) = H0 and since 11 is a 3,.,,,jTx~'~ RESIDUALLY REDUCIBLE REPRESENTATIONS AND MOI)UI,AR FORMS 121 non-zero divisor in T~ it must be that x E Go. We have thus shown that a(G,) C_ "qG0. It follows that (8 o ot)(G,) C_ 8(riG0). The inequality (8.5) is an immediate consequence of this. Next we show that (8.7) <. gc(Ho/H ) + We will prove this by comparing the lengths in question to those of various cohomo- logy groups. First we note that p~ : Gal (F~:/F) , GL2(R~ ) determines a represen- tation p : Gal (Fr/F) , GL~(C) obtained from the composition map ~mhl -- -- ' 5~176 C. R~ ,(R~)p=R.~R,, ~R | -+ Fix a basis for p~ such that P~(zl) = /{t -1 \) and P~(g0) = /'{, "'"\~), u0 E (~� for some go E Gal(F~:/F) fixed. Let k be a uniformizer of C. A C- algebra homomorphism f: R', , C | eC/k m (e'-' = 0) determines a represen- tation 9f:Gal(F~/F) , GLg(C | eC/k m) such that p = 9fmode. Write pf(~) = p(~)(1 + e~/f(~)), ~,j(~) E M,~(C/k'). It is readily checked that a, , ~'f(~) is a 1-cocycle of Gal (Fr]F) with coefficients in Mv(C/V') -~ ad p/k '~. We first claim that f~---~(cocycle (:lass of yf) determines an embedding Homc_~g(R,',, C | eC/k m) '--+ H' (F~/F, ad p/kin). (8.8) Here, and in what follows, all cohomology groups are the usual group cohomology; we do not require the cocycles to be continuous. To see that (8.8) is an embedding first note that if YJi and gf2 are cohomologous, then Pfl. and p& are equivalent. Thus there must be some A = (ac ~) E GL2(C @ eC/k":) such that AptiA -I = P.t~. Since pfl(zl)= (l -1), it follows that A = (a d)" We also have that (* (a/g,),0) = APfl(g0)A-I = p/2(g0) = (:: ~0"/). Thus a = d and A is a scalar, whence PJi = 9/.3. This implies that f = J) since any map R~ '~ , C | eC/k" is completely determined by the images of the elements in the set (~o bo) {a~, b6, c6, d~ 9 a E Gal(Fr/F)} (p~(6) = ,,, d, ) by Lemma 2.5. This proves injectivity of (8.8). Recall that there is a decomposition adp = ad~ @ C where ad~ are those elements in M2(C) with trace zero. It follows from the definition of R~ ~ that if 1 A ,- rain w E Z\?]' then ?~- ue~ ~,~ is unramified at w. The same is then true of Z- ~ "det pf ibr every fE Homc_~lg(R'~, C (D C/~.'e). Therefore (8.9) resw('/j ) e H'(D~,,, ad~ m) gw E E\~. 122 C.M. SKINNER, A.J. WILES Let V be the representation space for p. This is a free C-module of rank . For each w 9 .///~\,~'g there is a filtration 0 g Vw g V such that Vw and V~w = V/Vw are free C-modules of rank 1 such that I~ acts via ~11~ on VI,,. For each w E Z\Z~ U ,~ c\,//~ define Uw C_ ad~ by = ~ Homc(Vlw, Vw) if w 9 ,//gc\,//~ and ZlI~ = 1 Uw ( 0 otherwise. the definition of R~" and R~ one easily checks that if resw(7/) = 0 in Using Hl(Iw, ad~ D~ tbr all w 9 2\2,. U ,/Pgc\,//~, then ffactors through R~. It follows that gc(Homc_~g(R' ., C 9 EC/;~m)) - gc(Homc_~g(R~, C | ~C/~.')) (8.10) ~< ~ ec(H' (I~,, (ad~ D~) wEZ\Zcu'/lg c\' Ig .< gc(c/8(n), z.=). The last inequality follows from an explicit calculation of gc(Ht(I=,, (ad~ D"') for each w. As 8(rl) 3 k 0 we see that gc(c/8(n), ~.") = gc(C/8(n)) for large m. Next we note that there are canonical isomorphisms Homc(H0/(H~, ~m), Clam) ~, Homc_.~g(l~, C | r ~) and Homc(H,/(H~, ~Y), C/~, m) ~-- Homc_alg(R~n, C 9 s It follows from this and from (8.10) that (8.11) gc(HJ(H~))- gc(H0/H~)~< g,:(e/8(n)). Combining (8.11) with (8.4) and (8.5) shows that ec(H./H~) ~< to(C/(8 o ~)(G.)). It now follows from [DRS, Criterion I] that 7 o # :R', -+ T~' is an isomorphism of complete intersections of C-algebras. Therefore ~/ must also be an isomorphism of complete intersections. Since C is faithfully fiat over B we conclude that the map R ~ T induced from ~t(~,, p) is also an isomorphism of complete intersections, and hence YL,...,Y~-L ks a regular sequence in T~. It then follows easily that ~(~, p) is itself an isomorphism. This completes the proof of Proposition 8.1. The following proposition is a simple consequence of that one. RESIDUALLY REDUCIBLE REPRESENTATIONS AND MOI)ULAR FORMS Proposition 8.4. --- If ~J.(~ is a deformation datum for F then property (P1) holds for 6~ . Proof. -- Suppose that p C_ T~ is a prime that is nice for 6.~. Let pN C_ R~ be the prime associated to p as in w Let Q c_ lJ~ be any minimal prime and let p = p~ modQ. Put R = R~/Q. As in w let Ly/F be the maximal abelian p-extension of F unramified away from Y. and let Nx be the torsion subgroup of Gal(Lx/F). Fix a finite character V: Gal(Fz/~] , R � of p-power order such that ~-1. det(p | gt) is trivial on Nx. Corresponding to the deformation 13 | V rain is a homomorphism R~ ) R that factors through R~. The kernel of this homomorphism, say Ql, is contained in p~. It then follows easily from Proposition 8.1 that QI is pro-modular. By Lemma 3.17 there is some map T~(U~ NUt(cond~?(V)2),6 ":) , R inducing the pseudo-detbrmation associated to p. To show that P t and hence Q is pro- modular it is enough to show that U~ C_ Ul(cond~)(V)2). To establish this inclusion we first note that since gt has p-power order conde)(gt) is square-free. Moreover, if .~.~ = (~, E, c, ,//~) and if glcond~i(~), then g E Z\~/~. It then follows from the definition of U~ (see w that U~ C_ UI(g2). Thus U~ C_ Ul(cond~)(gt)2). We have thus shown that any minimal prime of R~ contained in p.~. is pro- modular. The same is then true of any prime of R~ contained in p~. [] A. A useful fact from commutative algebra The following result, in the guise of its corollary stated below, is the linchpin in our proof of the Main Theorem. Proposition A.1. [Ray, Corollaire 4.2] -- If A is a local Cohen-Macaulay ring of dimension d, and zf I = (f, ...,f) is an ideal of A with r <~ d- 2, then spec(A/I) \ {mA} is connected. We are indebted to M. Raynaud for providing us with the reference to a proof of this proposition. Suppose now that A and I are as in the proposition. Let ~' be the set of irreducible components of spec (A/I). Corollary A.2. -- If ~r = ~fl II ~ is a partition of ~ with ~i and W,2 non-empty, then there exist irreducible components CI E ~i and C2 E ~,) such that CI O C2 contains a prime of dimension d-r- 1. Proof of Corollary. -- Our proof is by induction on d- r. If d-r = 2, then the assertion of the corollary is an immediate consequence of the proposition. Now 124 C.M. SKINNER, AJ. WILES suppose d- r > 2. The conclusion of the proposition implies that there exist C' 1 E ~i and C,5 E ~2 such that C'~ (1 C,5 contains a prime p of dimension 1, which we may view as a prime of A. Now consider spec(Ap/I). Let W' be the irreducible components of spec(Ap/I). The embedding spec(Ap/I)~-+ spec(A/I) of topological spaces induces a decomposition of g~': g)' = W'l tl g:~;, ~,! = {C' = C C'l spec (AJI) 9 C E <}. By the choice of p, C~ E ~"i, so this is a non-trivial decomposition. As dim Ap = dim A- 1, the conclusion of the corollary now follows from the induction hypothesis together with the fact that Ap is also Cohen-Macaulay and that the dimension of a prime of Ap/I is one less than the dimension of the corresponding prime of A/I. [] Index of selected terminology admissible cocycle .......................... w 6c~ ................................ 2.3 ................................ 2.3 ~Q-minimal ............................ 2.3 good pair .............................. 4.2 good prime ............................. 4.2 nice deformations .......................... 2.3 nice for e_~ ............................. 4.2 nice prime ............................. 4.2 permissible extension ......................... 2.1 permissible maximal ideal ....................... 3.3 min x~, n~ .............................. 3.6 min ~r r~ .............................. 2.4 REFERENCES H. CARA~X)L, Sur les repr6sentations g-adiques associ~es aux formes modulaires de Hilbert, Ann. Sci. Ec. [Cl Norm. Sup. IX" Ser. 19 (1986), 409-468. [Ca] P. CARTIER, I,a conjecture locale de Langlands pour GL(2) et la ddmonstration de Ph. Kutzko, in Bourbdd Semina~ [bl. 1979/1980, Lecture Notes in Math., 842, Springer, (1981), 112-138. 125 RESII)UAIJN REI)U(/IBLE REPRESENTATIONS ~M\'L) MOI)UI2kR FORMS C.-L. CH.~, Arithmetic minimal compactification of the Hilbert-Blumemhal moduli spaces, Ann. oJ" Math. [Ch] (2) 131 (1990), no. 3, 541-554. J. (:o,~:s, p-adic L-functions and Iwasawa's theory, in Algebraic number fieldr: [,~mctions and Galois properties [co] (Proc. @topos., l]nh~. Durham, l'~trham, 1975), Academic Press (1977), 269-353. P I)ELU;.XE, Formes modulaires et reprfisentations de GL(2). in ,!,lodular Jimctions of one variable, II, lecture JOel Notes in Math., 349, Springer; (1973), 55-105. [D-R] P D~:I.IGYE, K. Rl~E'r, Values of abel[an L-fimctiorls at negative integers over totally real fields, hwent. Math. 59 (1980), no. 3, 227-286. B. Dr: S~nT, K. RuniN, R. S(:Hoor, Criteria for complete intersections, in Modular forms and Fermat's Ix2st [DRS] 7heorem, Springer (1997), 343-356. E Di,~xlOXD, On defbrmation rings and Hecke rings, Ann. of Math. (2), 144 (1996), no. I, 137-166. [DII E DLa.xto.','u, The Taylor-Wiles construcdon and multiplicity one, Invent. Math. 128, (1997), no. 2, [D21 379-391. E l)L,~to.xv, R. T,wl.OR, Non optimal levels of mod g modular representations, Invent. Math. 115 (1994), [DT] no. 3, 435-462. J.-M. FO.';T,XINE, B. M~xZt'R, Ge()metric Galois representations, in Ell[pile Curves, modular forms, and Fermat~ I~'t 7heorem (Hong Abng, 1993), Internat. Press (1995), 4[-78. S. G~:I,~T, Automorphic Forms on Adele Groups, Ammls of Math. Studies, Vol. 83, Princeton University [Ge] Press (1975). P. J.-P. L-xBESSE, The solution of a base change problem for GL(2) (following Langlands, [GL] Saito, Shintani), in Automorphic forms, representations and I.functions, Proc. Symp. Pure Math., XXXII1, part 2, 115-133. [(q A. El6ments de la gdomOtrie algebrique IV, /;;tude locale des scb6mas et des morpbismes de schc~mas (deuxi~me partie), lh~bL Math. & I'IHES 24 (1965). H. t-In)a, On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. (2) 128, (1988), [I411 no. 2, 295-384. H. Hm% On nearly ordinary Hecke algebras for GL(2) over totally real fields, in A~ebraic number theorr, [He] Adv. Stud. Pure Math., 17, Academic Press (1989) 139-169. H. HID,t, Nearly ordinary Hecke algebras and Galois representations of several variables, in A~ebraic [H31 analysis, geometry, and number theory (Baltimore, MD 1988), John tlopkins Univ. Press (1989) 115-134. K. IWASAWA, On Zf extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. [tl H. J..WQLET, R. Ia.~(;L~'~r)s, Automorphic forms on GL(2), Lecture ~,~btes in Math., 114, Springer (1970). [J-LI P. Ku'rzKo, The Langlands conjecture for GI,2 of a local tield, Ann. of3,Iath. (2) 112, (1980), no. 2, [Ku] 381-412. H. MAI'St;MURA, Commutative Ring 7beVy, Cambridge Studies in Advanced Mathematics, 8, Cambridge [M~t] Univ. Press (1989). B. M^zuR, Deforming Galois representations, in Galois Groups over Q, vol. 16, MSRI Publications, [Ml Springer (I989). M. N^c~'r~, L~,cal Rings, lnterseience Tracts in Pure and Applied Mathematics, no. 13, Interscience Publishers (1962). R. On a variation of Mazur's deformation functor, Comp. Math. 87 (1993), 269-286. M. IL~vxAt:I}, 'I'h~ortmes de Lefschetz en cohomolo~e cohtrent et en cohomologie 6tale, Bull. Soc. ~ayl Math. France, Mgm. no. 41. Suppldment au BuU. Soc. Math. France, qbme 103, Socitt~ Mathtmatique de France (1975). K. Rindr, Congruence relations between modular forms, Proc. Int. Cong. of~Iath. 17 (1983), 503-514. [Ri] M. NznLrss~y(,z~, Functors on Artin rings, 7runs. A~IS 130 (1968), 208-222. [Sch] .l-R Sr:~.~, Sur le r6sidu de la function z~ta p-adique d'un corps de nombres, CR. Acad. Sc. Pari~ 287, [Sel Serie A (1978), 183-188. R,x.x~..~rd~sn.~a, (~.ROTHENDIEf',K, GErb~.II.DIN, 126 C.M. SKINNER, AJ. WILES [Shi] H. SHIMIZU, Theta series and modular forms on GL2, J. Math. Soc. Japan 24 (1973), 638-683. [Sh] G. SmMue, a, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. j. 45 (1978), 637-679. [sw] C. SKINNER, A. WILES, Ordinary representations and modular forms, Proc. Nat. Acad. Sci. USA 94 (1997), no. 20, 10520-10527. [rw] R. TAYLOR, A. WILES, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), 141 (1995), no. 3, 553-572. [Wal] M. A lower bound for the p-adic rank of the units of an algebraic number field, in Topics in classical number theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. J/mos Bolyai, 34, North-Holland (1984), 1617-1650. L. WASHTNCTON, The non-p-part of the class number in a cyclotomic Zp-extension, Invent. Math. 49 [wa] (1978), no. 1, 87-97. [We] A. WEIL, Basic Number Theory, Springer (t967). A. WILES, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. (2), 142 (1995), 443-551. [W1] A. WinEs, On ordinary )~-adic representations associated to modular forms, Invent. Math. 94 (1988), [W2] 529-573. A. WILES, On p-adic representations for totally real fields, Ann. of Math. (2), 123 (1986), 407-456. [W3] C.M.S. School of Mathematics Institute for Advanced Study Princeton, NJ 08540 USA AJ.W. Department of Mathematics Princeton University Princeton, NJ 08544 USA Manuscrit refu le 23 ddcernbre 1998. WALDSCUMIDT, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Residually reducible representations and modular forms

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Springer Journals
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Copyright © 1999 by Publications Mathematiques de L’I.H.E.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/BF02698855
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Abstract

6 C.M. SKINNER, A d. WIIJ'S 1. Introduction In this paper we give criteria for the modularity of certain two-dimensional Galois representations. Originally conjectural criteria were tbrmulated for compatible systems of ~.-adic representations, but a more suitable formulation for our work was given by Fontaine and Mazur. Throughout this paper p will denote an odd prime. Conjecture (Fontaine-Mazur [FM]). -- Suppose that p 9 Gal(Q,/Q) , GL2(E ) is a continuous representation, irreducible and unrami~ed outside a ~nite set of primes, where E is a/~Tite e~Tension of Q#. Suppose also that (i) PlI# ~' ( ; t )' where Ip is an inertia group at p (ii) det p = ~s ~- 1 for some k >1 2 and is odd, where ~ is the cyclotomic character and ~ is of ~nite order. Then p comes from a modular form. To say that p comes from a modular form is to mean that there exists a modular form /with the property that T(g)/= tracep(Frob~)/for all g at which p is unramified. Here T(g) is the g,h tIecke operator, and an arbitrary embedding of E into C is chosen so that tracep(Frobe) can be viewed in C. Fontaine and Mazur actually state a more general conjecture where condition (i) is replaced by a more general, but more technical, hypothesis. The condition which we use, which we refer to as the condition that p be ordinary, is essential to the methods of this paper. If we pick a stable lattice in E 2, and reduce p modulo a uniformizer )~ of ~:E, the ring of integers of E, we get a representation ~ of Gal(Q,/Q) into GL2(~-' E/~.). If is irreducible, then it is uniquely determined by p. In general we write Oss tbr the semisimplification of ~, and this is uniquely determined by p in all cases. Previous work on this conjecture has mostly focused on the case where ~ is irreducible (cf. [Mill, [D1]). In that case the main theorems prove weakened versions of the conjecture under the important additional hypothesis that ~ has some lifting which is modular. This hypothesis, which is in fact a conjecture of Serre, is as yet unproved. In this paper we consider the case where ~ is reducible, and we prove the following theorem. Theorem. -- Suppose that p 9 Gal(O~/Q) , GL2(E ) is a continuous representation, irreducible and unramilied outside a tinite set of primes, where E is a tinite extension of Qp. Suppose also that ~' ~_ 1 | x and that (i) XlD~ ~: 1, where Dp is a decomposition group at p, (ii) PIt, -~ (0 1)' (iii) det p = ~s k- t for some k >1 2 and is odd, where e is the cyclotomic character and ~r is of linite order. Then p comes from a modular form. RESIDUALLY REI)U(I[BLE REPRESEN'IATI()NS AND MOI)ULAR FORMS We also prove similar but weaker statements when Q is replaced by a general totally real number field: see Theorems A and B of w Ill the irreducible case the proof consists of identifying certain universal deforma- tion rings associated to ~ with certain Hecke rings. However in the reducible case even for a fixed ~s _~ 1 (9 Z we have to consider all the deformation rings corresponding to the possible extensions of Z by 1. These deformation rings are not nearly as well- behaved as in the irreducible case. They are not in general equidimensional. Indeed there is a part corresponding to the reducible representations whose dimension grows with Z, the finite set of primes at which we permit ramification in the deibrmation problem. Just as in the irreducible case, we do not know whether there is an irreducible lifting for each extension of Z by 1, but happily we do not need to assume this. In a previous paper [SW] we examined some special cases where we could identify the deformation rings with Hecke rings. These cases roughly corresponded to the condition that there is a unique extension of 1 by Z- In this paper we proceed quite differently. In particular we do not identify the deformation rings with Hecke rings. As we mentioned earlier, we consider the problem over a general totally real number field. This is not just to extend the theorem but is, in thct, an essential part of the proof. For it allows us by base change to restrict ourselves to situations where the part of the deibrmation ring corresponding to reducible representations has large codimension inside the full deformation ring. It should be noted that the base change we choose depends on Z. We now give an outline of the paper. In w we introduce and give a detailed analysis of certain deformation rings R~. These are associated to an extension c of Z by 1. They are given as the universal deformation ring of the representation where the implied extension is given by c. Here Qz is the maximal extension of Q unramified outside Z and oc although in the main body of the paper Q is replaced by a totally real field F. More precise definitions are given in w In w we give a corresponding detailed analysis of certain nearly ordinary Hecke rings introduced by Hida. We say that a prime of R,/ is pro-modular if the trace of the corresponding representation occurs in a Hecke ring in a sense that is made precise in w If all the primes on an irreducible component of Re/ are pro-modular then we say that the component is pro-modular. If all the irreducible components of R~ are pro-modular then we say that R2/ is pro-modular. The above theorem is deduced from our main result which establishes the pro- modularity of R@ for suitable c,,~. There are three main steps in the proof of this latter result: (I.M. SKINNF, R, AJ. WII,ES (I) We show that if p is a "nice" prime of R~ then every component containing p is pro-modular. (The definition of a nice prime is given in w it includes the requirement that p itself is pro-modular). (II) We show that Rj has a nice prime p. (III) We show that Rj is pro-modular. The proof of step (I) is modelled on that for the residually irreducible case and is given in w The point is that the representation associated to R~/p is irreducible of dimension one and pro-modular. However tile techniques of the irreducible case have to be modified as this representation, which we now view as our residual representation, takes values in an infinite field of characteristic ,0. We should note also that the analog of the patching argument of [TW] is here performed on the detbrmation rings rather than on the Hecke rings. The proof of step (III) is given in Proposition 4.1. Steps (I) and ([I) show that some irreducible component at the minimum level is modular. Then we use a connectivity result of M. Raynaud (see w to show that there is a nice prime in every component at the minimum level. By step (I) again we deduce pro-modularity at the minimum level. A more straightforward argument then shows that there is a nice prime in every component of R~, so that we can again apply step (I) to deduce pro-modularity. For step (II) we proceed as fi)llows. First we show, using the main result of w (which in turn uses techniques for proving the existence of congruences between cusp forms and Eisenstein series), that R(/ has a nice prime tbr some extension Co of Z by 1. Using commutative algebra we show that there are primes in the subring of traces of R~ which correspond to representations with other reduction types, i.e. corresponding to a different extension c (the pair 1, )~ are fixed though). We make a construction to show that we can achieve all extensions in this way, and hence find nice primes for all extensions c. These primes are necessarily primes of the ring of traces which do not extend to R(/ itself. The proof of step (II) is given in Proposition 4.2. At the start of the proof of this proposition is a more detailed outline of how we carry out step (II). We now briefly indicate the extra restriction in the case of a general totally real field F. We need to be able to make large solvable extensions of F0~), the splitting field of )r with prescribed local behavior at a finite number of primes and such that the relative class number is controlled. When F(Z ) is abelian over Q we can do this using a theorem of Washinoon about the behavior of the p-part of the class number of Zt-extensions. In the general case such a result is not known. Finally we note that the ordinary hypothesis which is essential to our method is frequently satisfied in applications. I~br example, suppose that 9 (with ~ reducible) arises as the ~,-adic representation associated to an abelian variety A over Q with a field of endomorphisms K~--~ EndQ(A)| Q such that dim A = [K : Q.]. Then the nearly ordinary hypothesis will hold provided A is semistable at p, or even if A acquires RESIDUALLY REI)UCIBLE REPRESENTATIONS AND MOI)ULAR FORMS 9 semistability over an extension of Qp with ramification degree < p- 1. This can be verified by considering the Zariski closure of ker(~.) in the Neron model of A. 2. Deformation data and deformation rings 2.1. Generators and relations Let F be a totally real number field of degree d. For any finite set of finite places Z, let F~ be the maximal extension of F unramified outside of X and all vle~. For each place v, fix once and for all an embedding of P into ]~v. Doing so fixes a choice of decomposition group Dv and inertia group Iv for each finite place v and a choice of complex conjugation for each infinite place. Let zx, ..., ze be the d complex conjugations so chosen, and let vl,..., vt be the places dividing p. Write Di and Ii for the decomposition group and inertia group chosen for the place vi. Let di be the degree of Fv~ over Qp. Normalize the reciprocity maps of Class Field Theory so that uniformizers correspond to arithmetic Frobenii and write Frob fi)r a Frobenius at a place v. Suppose that k0 is a finite field of characteristic p and that )~ : GaI(F/F) ~ k~ is a character such that 9 )~l~i ~ 1 for i= 1,...,t 9 )~(zi) = - 1 for i = I, ..., d. A deformation datum for F is a 4-tuple ..~ = (~(5 ~ , E, c, ../A~) consisting of the ring of integers C of a local field with finite residue field k containing k0, a finite set of finite places E containing all those at which 3~ is ramified together with ,~ = {vl, ..., vt}, a non-zero cohomology class res t'TN TTI/T'~ (2.1) 0 ~ cE ker {H~(Fx/F, k(~-~))----+ t~l;~n/~i, k(X-~))}, i= 1 and a set of places ~ C X\?) '~ at each of which either c is ramified or ZII,, is non- trivial. For future reference write Hx(F, k) for the kernel of the map in (2.1). A cocycle class c E Hx(F, k) is called admissible. Let F(g) be the splitting field of ?~. There is a canonical isomorphism (via the restriction map) HI(Fx/F, k(~-L))~, HI(Fx/F~), k(~-l)) Gal(F0~)/I'~. Using this identification, one sees that for any cocycle c there is a unique representation 9,:" Gal(Fx/P; ' GL2(k), 9~ = X 10 C.M. SKINNER, A.J. WILES such that 9~(z~) = ( 1 -1) p~(a)= (1 c(1)) for ~E Gal(Fx/F(; 0). If c is admissible, then Pc also satisfies (1) i= 1,...,t. P, IDi -~ Z ' A deformation of p~ is a local complete Noetherian ring A with residue field k and maxima] ideal mA together with a strict equivalence class of continuous representations p 9 Gal(F/F) ~ GL2(A ) satisfying p~ = pmodmA. Such a deformation is oftype-~ if 9 A is an C-algebra, 9 P is unramified outside of Z and the places above 0% v~i) with )~ = ~q mod ma for each i, and 9 ( ") 2 for each w E .//g. Here ~ denotes the Teichmtiller lift of X to A. We usually denote a deformation by a single member of its equivalence class. For any deformation datum ~, there is a universal deformation of type-~ p.g 9 GaI(Fz/F) , GL2(R~,~ ). We omit the precise formulation of the universal property as well as the proof of existence as these are now standard (see [M], [R], [W1]). A totally real finite extension F' of F is permissible for ~ provided 9 o, loal(~/v", is non-split; 9 if v E dg and X]I,, =~ 1, then ZII. ~: 1 for each place w of F' dividing v; 9 if v E./E~ and P,[I~, q= 1 but zlI~ = 1, then P~lI,, =~ 1 for each place w ofF' dividing v ; 9 if w is a place of F' dividing p, then ?~lD~, :~ 1. Remark 2.1. - - If F' is permissible for ~, then ~ determines a deformation datum ~' = (L "t2j , Z', c, ./f/~') for F' with Z' and ~fd' being the sets of places of F' dividing those in X and ,/dd~, respectively. Clearly, if p " Gal(F/F) , GL2(A ) is a deformation of type-~, then Pl(;al~F/v'i is a deformation of type-.~'. In this subsection we give a preliminary analysis of the structure of Ry as an abstract ring. ~Ib start, we analyze the versal deformation rings associated to RESII)UALLY REDUCIBI,E REPRESENTATIONS AND MODULAR FOR-MS representations p : D: ~ GL~(A) satisfying p modmA = )C | 1 and det9 = ~. Such a deformation is a local ~:-deformation if A is an ~'-algebra, and it is nearly ordinary if ( ") in addition p _~ g~' ~r with 1 = tg mod mA. Applying the criteria of Schlessinger as in [M], one sees that there is a versal local ~:-deformation and a versal nearly ordinary deformation 9:': D~ , GL2(R':; ) and 9oi{.a "D,. , GLg(R~'~a) respectively. The representative [ord(~) can be chosen so that Pord = ~u- 1 9 The following lemma gives a ring-theoretic description of Rot d. Lemma 2.2. -- Let co be the character giving the action of D: on the pth roots of unity. There is an isomorphism " ..., = = R(,) , ,~, [[Xl, x24+2]]/(f) if )~ II)i 03 or if m 1, {~ --or~ -- ~i, [[xl,..., xzu/+l]] otherwise. Proof. -- Our proof follows along the lines of that of [M, Proposition 2]. Let V be the representation space of p01Di where P0 = 9c with c = 0. Clearly, V ~ k | k(x ). Let ado() = Hom~(V, V) be the adjoint representation, and let ad~ be the submodule consisting of homomorphisms whose trace is zero. The reduced tangent space of ~(') a~-or d has dimension equal to r = dim k ker { U 1 (Di, ad~ , H [ (Di, Homk(k(~), k)) }. A simple calculation using local class field theory and local Galois duality shows that 2di + 2 if X IE~, = co or if co = 1, r = 2di + 1 otherwise. It follows that R"o';~d is a quotient of the power series ring P = ~" [[x~, ..., x(l] by some ideal I. Consider the exact sequence 1~(') 0 ~ I/mI , P/mI ~ ""ord ~ 0 ~('? for deformations where m is the maximal ideal of P. The universal deformation ring ~-1 of the trivial character satisfies gy,, ...,A]/(h) 12 C.M. SKINNER, A d. WILES where (2.2) s= dimkHl(Di, k) and if co =~ 1 then h= 0. This follows immediately from local class field theory. There is a natural map o/,~ corresponding to the character ~g. Choose a compatible homomorphism R('l ~ ~ ~"ord ~' ~Y~,...,Ys] ' P/mI. This induces a continuous character * : Di ' (P//mI, h))� projecting to q~. Choose a (continuous) set-theoretic map 0:D~ , GL 2 (P/(m], h)) _ (,) projecting to Pond such that * ) Define a 2-cocycle ~: Di - ' I/(mI, h)(z) by 0(~j, a2)0(~2)_, 0(6L)-1 = (1 ~(~1~ r ) and consider its class ['/] in H2(Di, (I/(mI, h))~))-~ H2(Di, k(z))| h). The map (2.3) (I/(mI, h))* , H2(Di, k(Z)) , f~----. (1 | is injective. Here the superscript '*' denotes the k-dual. For if f E (I/(mI, h))* maps to zero, then 7mod(mI, h, ker f) equals d~ fbr some map ~ : Di , (I/(mI, h, ker f))(~), and 0' = (1 ~)0 is a representation into GL~(P/(mI, h, ker f)) that is clearly a nearly ordinary deformation. By the versality of R~ d there is then a homomorphism R(o0rd ~ P/(mI, h, ker f) inducing 0', and its composition with the ~(z? is an isomorphism. Comparing maps on reduced projection P/(mI, h, ker f) --, ~'ord tangent spaces shows that ""ord D(z) -- ~ P/(mI, h, ker f), which is possible only if f= 0. Let g = dim k I/mI. This is the minimal number of generators of the ideal I. By (2.2) and the injectivity of (2.3), 1 if ~=1 g ~< dim k H2(Di, k~)) + 0 otherwise 1 if co=l dim~I[~ k(z-lc~ + 0 otherwise. I if co=l orif )~=o~ 0 otherwise. This proves the lemma. [] RESIDUAI,LY REDUCIBI,E REPRESENTATIONS AND MOI)ULAR FORMS 13 r~('; of R ('~ ideal generated by Corollary 2.3. -- The ring L,or d i,Y a quotient by an 2 if ~ll), = o) = ~-1 ]I), d, + 1 if ZID~ = O) :~ Z-IID~, or o~= 1 zl,,i ,o = Z- ID , 0 otherwise e/ements. Proof. ---The ring R (i) is a quotient of C'~ [[YL,-..,Ye]] with r' = dim kHI(Di, ad~ 3 if ~=0~=Z -1 =3di+ 2 if 0~=1 or ~=0~fik~-I or ~:~o~=)U 1 1 otherwise. Combining this with the previous lemma and the fact that "o~d ~('~ is a quotient of R/'l yields the corollas. [] The above lemma and its corollary, together with minor variations of the methods used to prove them, yield the following ring-theoretic description of Re. Let ~- be the Zp-rank of the Galois group of the maximal abelian pro-p-extension of F unramified away from primes above p. Proposition 2.4. Suppose that ~ = (8, Y., c, J~) is a deformation datum. There exist integers g and r, depending on 5~, such that Re Ix,, ..., ...,frr) and g- r/> d+ fly - 2t- 3 9 #rig. Recall that t is the number of places of F dividing p. Proof. -- First we introduce an auxiliary deformation problem. A deformation P : Gal(ff/F) , GL2(A ) of 9c is of auxiliary O~pe-~ if 9 A is an ~-algebra 9 det p = 9 p is unramified outside of Z and the places above oo. There is a universal deformation of auxiliary type-~.~ RI~X aLIX 9e " Gal (Fz/F) ~ GL2(R e ). 14 C.M. SKINNER, AJ. WILES &tax aux Clearly, there are natural maps q0i " R('; ~ R.~ corresponding to P.~ ID, for i = I, ..., t. Let J2 be the kernel of the projection R(') ' and let J be the ideal generated by ug)i(Ji). It follows from Corollary 2.3 that (2.4) J is generated by ~_~(di + 2) = d + 2t elements. t'= l Now p~modJ is clearly a deformation of type-~.~ ', where c~, = (~:, y., c, ~). Using the versality of the various rings one finds that R~, ~ R~X/J | ~2 [[Gal(L(Y.)/F)]] (2.5) where L(Z) is the maximal abelian pro-p-extension of F unramified away from Z. (One difference between deformations of type-6_~ r' and auxiliary deformations of type-~ is that the former include deformations of the determinant whereas the latter do not.) It is easy to see that there is an isomorphism (2.6) [[Gal (L(X)/F)]] ~_ ~' [[x~ , ..., x~ r , y, , ..., y~]/(g, , ..., g,). Let I be the kernel of R~, ~ R~. For each v G ~ let % E I~ be a generator of the p-part of tame inertia. Choose for each v E ,/tg a basis for 9~' such that P~'(%) m~ = (' *)1 . Write P~'(%) = (a~c~ d~b~) with respect to the basis. Clearly, I is generated by the set { a~ - 1, d~ - 1, q, 9 v E ,.rig }. It follows that (2.7) I is generated by 3. #~/fg elements. Arguing as in [M, Proposition 2] shows that 1Ix,, ..., ...,&) where e' = dimk H' (Fz/F, ad~ r' ~< dim~ H2(Fz/F, ad ~ Combining this with (2.4), (2.5), (2.6), and (2.7) shows that ..., x ll/(j5, ...,f) where g=g'+S+~F and r=r'+s+d+2t+3.#Jg. The desired bound for g- r is a consequence of the global Euler characteristic formula for ad~ [] --o~aP('?, RESIDUALIN REI)UCIBI,E REPRESENTATIONS AND MODULAR FORMS 15 We conclude this subsection with two simple facts about deformation rings. Suppose that ~ is a deformation datum. Fix a basis for p~. With respect to the basis write p~ (~)= (aoco ~) for each (~ E Gal(F/F). Let R' C_ R~ be the ~;-subalgebra generated by { do, bo, co, d~](~ ~ Gal~/F)}. Let m' = m~ 71 R', where m~ is the maximal ideal of R~. Let R~ = R~, and denote by R~ the completion of Rl at its maximal ideal. The inclusion Rt C_ Rg, induces a map i" P--I ~ R~. Lemma 2.5. -- The map i" P,-I , R~ /s surjective. Proof. -- Let m, be the maximal ideal of P,,~. Let P, " Gal(F/F) , GL2(R.~) by PI((~) = (c~ bzz)" Clearly, composing Pl with the homomorphism be defined GL2(13~t) , GL~(R~) induced by i yields p~. It follows from the definitions of RT and Pl that plmodml = p~. Let a = rn~R~. The deformation p~moda is the same as the deformation obtained by composing ptmodmt with the homomorphism GL,(k) = GL.2(P,I/m~ ) , GI,.~(R~/tt) obtained from i. As plmodm~ = Pc, it follows from the universality of R~ that there is a unique map R~ ~ R~/ml whose kernel is necessarily m~. The composition R~ ~ Rt/ml ~ R~/a must be the same as the canonical map R~ ~ R~/a. Therefore a = m~. This proves that dim k (R~/mlR~ ) = 1, from which it follows that R~ is generated as an Rwmodule by one element (cf. [Mat, Theorem 8.4]). [] I~br future retkrence we also record the following fact. i.emma 2.6. -- /fp C_ Ry ~ not the maximal ideal, and zfp~ = Rt 71p, then dimRi/pl /> 1. Proof. -- If" Pl is maximal, then Pl, and hence also p, contains a uniformizer of r ~. In the deformation p~ modplR~ the matrix entries are in k. Therefore the deformation p~2 ~modp is obtained by composing p~ modm~ with the natural inclusion k ~ R~/p. From the universality of R~ it then follows that p = m~. [] 2.2. Reducible deformations A reducible deformation of p~ is a deformation p such that p -~ (zl *) X2 ' In this subsection we analyze the universal reducible deformation of type- ~c.~r where = (~, Z, c, ,~). Write red red p~ 9 6al(Fz/F) , GL:(R~) for the universal reducible deformation. A consequence of our analysis of p~ will Dred be an upper bound for the dimension of l,,~. This bound will be important in our subsequent analysis of R~. 16 C.M. SKINNER, AJ. WILES Choose a basis for p~ such that p~ (zt) = /~1 --1 \~" For a E Gal(Fx/F) write 9~ (a)= (o~ ~ ), and let I be the ideal generated by the c~'s. Clearly, Rred red ~ =R~/I and 9~ =9~ modI. 1Dred Unfortunatel~ this description of ~,~ does not easily yield a non-trivial bound for the we take a more pedestrian approach. dimension. Therefore Let L(Z) be the maximal abelian pro-p-extension of F unramified away from Z. Write g~ G = Gal (L(E)(Z)/N ~-- A x r x Z~ r' where A _~ Gal(F(x)/I"), r is a finite p-group, and L(Z)(X) = L(E). F(X). Let M be the maximal abelian pro-p-extension of L(Z)~) unramified away from Z\.5~ and such that A acts on Gal (MILd)Of,)) via the unique representation over Zp associated to )U 1. Any reducible deformation of type-6.~ factors through Gal(M/F). Put A = Zp]]-G]] ~ Zp ][A x 1-]] [[I',, ..., T~v] ]. group H = Gal(M/L(s is a finitely generated A-module generated by The m elements where m = dim e H/mA , } = dimtker{ HI(Fx/F, k(X-1)) i= l = dim~ Hx(F, k). Note that by our hypothesis that g[Di 36 1, tll(Di, k(x -i) --~ H~(Ii, k(X-~)) Di. Here mA is the maximal ideal of A corresponding to X-I and k' is the residue field of Zp[A] associated to )C -1. Fix a presentation rtl a ~ ~ Aei ~ H i= 1 such that em projects to an element ~ of H tbr which pc(bin) = (1 u)t with u :~ 0 and such that if i:~ m then ei projects to an element hi of H for which pc(hi) = (l ) 1 " Choose an element u0 E r reducing to u. Put Al = ~'[[1-]] ~-Wl, ... , TaF]], A2 = ~' ~ [[Sl, ..., S~F]] and fix embeddings q0i 9 A ~ A | ~" ~ Ai RESIDUALIN REDUCIBLE REPRESEN~IATIONS AND MODULAR FORMS 17 where for (DI the map Zp[A] , C ~ is that induced by ~-i and for ~p9 it is that induced by ~. Let J be the ideal of AI ~_x~, ..., Xm-~]] generated by { qIl(al)Xl + q~l(a2)x2 +''" + q)l(am-1)Xm-I + (pl(am)U0 :Zaiei ~ a }. Fix a homomorphism of A-modules : H , B = AI [[Xl, ..., Xm-l~/J; ei I ~ xi, i= l,...,m- 1 era1 ~ UO. Put R = B~ A 2. Observe that Gal(M/F) _~ G ~ H. We may therefore define a reducible deformation of type-_q~', ~' = (65~, Z, c, ~) p 9 Oal(M/F) , GL2(R ) by p(g) = (~P1(g) | q~2(g) ) q~z(g) ' g E G, 9(h) = (1 z(h)) 1 ' hEH. The deformation P is readily seen to be the universal reducible deformation of type- ored 6_~. As easy consequences of this explicit description of ~,~, we obtain the following estimates. Dred Iz, mma 2.7. --- I4~ have dim ,,~ ~< 1 + 23F + dim k H~-(F, k). Lemma 2.8. -- If q C_ R~ is a prime containing p such that p~ mod q is reducible and its determinant has finite order, then dim R~/q ~< 8~, + dim k H~(F, k). A diagonal deformation of the representation 1 G X is a representation P : Gal(F/F) ~ GL2(A), A a complete local Noetherian ring with residue field k, such that P = (,1 ~2) with 1 = 001 modmA and Z = 002modmA" Such a representation P: Gal(F~/F) ~ GL2(A ) is a diagonal deformation of type-(G, 2) if A is a local O ~- algebra (C~ -~' and Y~ as in the definition of a deformation datum). There is a universal diag diagonal deformation of type-(C, s 9(~:~ ~ag I " Gal(Fz/F) , GL~(R/e~,~/). Later we shall need to know an upper bound for the dimension of various primes of R~ g, ~1" The proof of the following lemma is similar to, but much simpler than, the proofs of Lemmas 2.7 and 2.8 and hence is omitted. D diag _ D diag Lemma 2.9. -- (i) dim 1,~6~, ~, ~< 1 + 28r. (ii) If q C ~,~, ~ is a prime containing p such -- diag 1 .. ndiag j that detp(~:,y.)moaq hasfinite ordo;, then aamt~(~ ,z)/q ~< ~v- 18 C.M. SKINNER. AJ. WILES 2.3. Some special deformations In the course of our analysis of the rings R(/ we shall sometimes have to consider some augmented deformation problems as well as deformations of various restricted type. Here we introduce these deformations and, when applicable, their universal deformation rings 9 Let .r = (6~.", Z, c, .//~) be a deformation datum and Q a finite set of finite places, Q = { w~, ..., w~ }, disjoint from X. A deformation p of type-(~ ~' , Z U Q, c, ..~//~) is of type-~ o if 9 detp is unramified at each wicQ. There exists a universal deformation of type-r1(2: p~Q : Gal (FxuQfF) ' GL2(R~,)). For a deformation datum ~ = (~:, E, c, ,///), Zc C Z is the subset of places at which p, is ramified together with the set ,O ~. Similarly, ~/fgc = Z,\ ?)~. Also, we write Z0 for the set of finite places at which )~ is ramified together with ,?~. Given ~ we write .~c for the deformation datum .~c~,. = (~, Z,., c, ,J/gc)- A deformation p " Gal(Fz/F) ----0 GL,)(A) of type-~ is nice if 9 A is a one-dimensional domain of characteristic p, 9 p is a deformation of type-C'~_~, (*) 9 PlDi ~- ~li) ~i) with V( ~/~l/i ~; having infinite order for i= 1, ..., t. For a deformation datum ~ = (~, E, c, ./f~) let Lx/F be the maximal abelian pro-p-extension of F unramified away from 2, and let Nz be the torsion subgroup of Gal(Lx/F). A deformation p 9 6aI(Fx/F) , GL~(A) of type-CJ~o., is .~Q-minimal (6~.'-rninimal if Q= 0) if det P is trivial on Nz. Let min P~a 9 Gal (FxuofF) --, GL2(R~o mi~ ) be the universal ,f~o-minimal deformation. If Q = 0, then we just write p~n and mm rain R~. There is a simple relation between R~z and R~Q. We fix for each s a free Zz-summand Hy, C_ Gal(Lr./F) such that Gal(Lr/F) -~ Hx (9 Nr,. We choose the Hz's to be compatible with varying Z. Let Wz " Gal(Lr/F) ~ Ny denote the character min obtained by projecting modulo Hx. The representation P~)_ | ~z " Gal@'z/F) , GL2(Rff~ ~ | ~:'[Nz]) is easily seen to be a deformation of type-~oc It follows from min the universal properties of R~Q and R~Q that ~-~ min ~ min R~ a _ R~Q | ~' [Nz] and p~Q _ p~za | q%. RESI1)UALIN REDUCIBIA~; REPRESENTATIONS AND MO1)UI.AR FORMS 19 Suppose that ~ is a field and that p : Gal(F/F) , GL)(.~) is a represen- tation. For each place w { p at which p is ramified we distinguish for future reference four possibilities for 0Ix,,: TypeA pllw"( 1 *) _ 1 , ,q=O. TypeB pli,,~"( - ~ I ) , ~ a tinite character. TypeB' PII,-~(* 0-1), ~afinitecharacter. |,'.~ Type C PI,~ = Indv'e~ where F=._, is the unique unramified quadratic extension of F=, and V is a character of Gal(Fw/Fd, ) such that Vl~,,. 3 k ~:r'"'"'lX,,., V[l,. has p-power order, and det P[Dw has order prime to p. Note that Type C can only occur if the characteristic of .-Srg / is zero. 2.4. Pseudo-deformatlons For our purposes, following [W2] a (2-dimensional)pseudo-representation of Gal(F/F) into a topological ring A is a set 9 = {a, d, x} of continuous thnctions a, d : Gal(F/F) , A and x : Gal(F/F) 2 , A such that 9 a(~) = a(<a(~) + x(~, ~), 9 d(o~) : d(o)d(~) + x(~, ~), 9 x(o, ~>(~, ~) = ~(~, [3>(n, ~), 9 x(ox, ~) = a(o)a(13>(x, ~) + a(13)d(x>(o, a) + a(o)d(~>(x, 13) + d(x)d(~)x(o, [3), 9 a(1)= 1 = d(1), 9 a(z.) = 1 - -d(zl), and 9 x(~,g)=0=x(g,o) if g= 1 or zl. The trace and determinant of p are trace 9(0) = a(o) + d(o) and det p(o) = a(o)d(o) - x(o, o). Suppose that P " Gal(F/F) ~ GL2(A ) is a continuous representation such that 9(z,) = (1 _,). Write P(O) = ('~ by). The functions a(o)= a~, d(o)= do, and x(o, z) = boc~ form a pseudo-representation, and the trace and determinant of this pseudo-representation are merely the trace and determinant of the representation p. Let P0 be the pseudo-representation associated to the representation 1 9 X (i.e., a = 1, d = Z, and x = 0). A pseudo-deformation of P0 is a pair (A, 9) consisting of a local complete Noetherian ring A with residue field k (which we assumed finite) and maximal ideal mA and a pseudo-representation p of Gal(F/F) into A such that p mod mA = P0. We often just write p to mean such a pair (A, p). A pseudo-datum for F is a pair ~_~.m = (C, E) consisting of the ring of integers 6 ~ of some local field having 20 C.M. SKINNER, AJ. WILES residue field k and a finite set of finite places X of F containing ~ and those places at which )~ is ramilied. A pseudo-deformation 9 of P0 is of type-~.q~ v~ if 9 A is an c"~:-algebra and 9 p is unramitied outside of Y. (i.e., a, d, and x factor through Gal(Fx/F)). It is relatively straightforward to verify that the functor F~ o~ from the category of local complete Noetherian ~'-algebras with residue field k to the category of sets given by F~p~(A) = {pseudo-deformations into A of type-e_qr vp~} satisfies the criteria of Schlessinger [Sch]. The only non-trivial point is the finiteness of the tangent space, and this is provided by the following lemma. Lemma 2.10. - - Let k[~] be the "dual numbers". Then #F~ps(k[e]) = (#k) T, where r ,< 4(#GaI(F(x)/F)) + 2(dim k HI(Fz/F(x), k))2 + 4. Proof. -- If P = {a, d, x} E F~ps(k[e]), then a=l+~al, d=x+adl, and x=exl. If p' = {a', at, x'} is another such pseudo-deformation, and if cx E k, then {1 + eo~(al + a/l), X + ~cx(d, + ~), ecx(xl + ~l)} is in F~ p~(k[e]). In particular I"~p~(k[e]) is a k-space. Let G x = Gal(Fz/F(x) ). From the relations defining pseudo-representations it follows that Xl[Gn� determines an element of Hom(G;c, Hom(Gx,k)) via xl , , {g, ~ xl(',g)}. If P is in the kernel of the k-linear map FNp~(k[E]) , Hom(Gx, Hom(G~, k))given by p, , XI[Gxx(Ix , then aliGn, dl[Gx E Hom(G x, k). Thus .< #{ p" = = x, = 0 }, where s = dim~ H 1 (Fz/F(~), k). Now let G = Gal(F~:/F) and suppose that p = {a,d,x} E F~p,(k[~]) satisfies allcx = dl[G x = XIIG? x -" 0. Then XI[GxxG determines a 1-cocycle G , Hom(G:~, k(%)) via g~ ~ xl(', g). Moreover, this cocycle vanishes upon restricting to G x. Thus the number of possibilities for Xl [G~� is at most #Hom(Gx, k). A similar argument shows that the number of possibilities for xl IG� is also bounded by the same quantity. Thus #F~ ~(k[E]) ,< (#k) '~+4'. #{ p: a~ IGn = d~ L~ = x, LG~ � = x~ IG� = 0 } <~ (#k)S2 +4s+4"#Gal(F(X)/F). RESIDUALLY REDUCIBLE RI'PRESENTATIONS AND MODULAR FORMS 21 Here we have used that for any pseudo-deformation O = { a, d, x} satisfying aLlGx = dl]c, x = Xl]Gx� = Xl ]G� x = 0 the functions al, dz and xl are constant on cosets of G x in G. [] There is therefore a universal pseudo-deformation (R~ p.,, 9~ p,,) of type-("-/yps. Clearly, any deformation 9 9 Gal(F~/F) , GL2(A ) of some 9c with A an &"-algebra gives rise to a pseudo-deformation of type-(~ ~ , E). Choose a basis for 9 such that p(z,) = (1 -1)-Write p(a)= (~ ~). As we have previously noted, { a~, d~, x~,z = b~cz } is a pseudo-representation, and its reduction modulo ma is P0, so it is a pseudo-deformation. One easily checks that it is also of type-(~", s The entries of p(~) with respect to any other basis for which p(z~) = {1 -l ) are obtained by \ / conjugating the chosen basis by a diagonal matrix. Such a conjugation does not change a~, d~ or b~c,. We call { ~, d~, xa., = b~c, } the pseudo-deformation associated to 9 and sometimes denote it by 9 as well. There is a unique map R~p~ , A ((_~P~ = (r I2)) inducing the pseudo-deformation associated to 9. This argument shows that to any deformation p of p,, where c is some cocycle in H~(F~/F, k(z-~)), one can associate a well-defined pseudo-deformation. In particular, if ~r = (c~, Z, c, ,/~) is a deformation datum, and if ~_9~ "p~ = (r 12), then we obtain a unique map r~ 9 RNp~ ~ R~ min rnin inducing the pseudo-deformation associated to p~. We write r~ : R~ ps ~ R~ for min the composition of r~ with the canonical map R~ ; R~. Let ~P~ = (~, 12) be a pseudo-datum and let Q be a finite set of finite places disjoint from s A pseudo-deformation O of type-(~, Z tO Q) is of type-~-c-~ if 9 det P is unramified at each w E Q. There exists a universal pseudo-deformation of type-!~Q" (R~, p~ ps ). If 6_~ = 6.~Q (~, Z, c, JPg) is a deformation datum, then as in the preceding paragraph there is a unique map r~Q : R~ ps ~ R~ Q inducing the pseudo-deformation associated to P~ ~z" Of course, if Q = 0, then R~ ~_ = R~ p~, p~ ~ = 9~ v~, and r~ Q = r~. Suppose !~ = (~, Z, c, J~g) is a deformation datum and 6~p~ = (G, E). The following proposition reflects the relation of R~ v~ to R~ Q Q" Proposition 2.11. -- If p C_ R~ o - is a one-dimensional prime such that pg Qmodp is __1 A irreducible, and if pps -- rc.c.~ Q(p) C R6c~ ~, then the canonical map R~ ~, pp~ ~ R~ Q, p is surjective. Proof. -- As p~ modl0 is not reducible, pps is not the maximal ideal mps. Therefore mpSR~Q ~ p. Let A = R~Q/p and A p~ = R~ ~/pP~. Let K ps be the field of fractions of A ps. The map r~ : R~ ps ~ R~ o_ induces an inclusion A ps ~ A. Q 22 C.M. SKINNER, AJ. WILES As we have observed, mP~A ~= 0, whence A/mPSA is a zero-dimensional Noetherian local ring with residue field k. It follows that #(A/toP'A) < oc and hence that A is a finite APe-module (cf. [Mat, Theorem 8.4]). Thus A is an integral extension of A I'~ and dim A ps = dim A = 1. Fix a basis of p~Q such that P~(zl) = ,(1 ,) and 9~(o0) = (* u) -I * * u E 6: � for some c0 E Gal(F/F). With respect to this basis write 9~ o(~) = ao tiff It is easily checked that { ao, do, co, bob I o, 1: E GaI(F/F)} is contained in im(r~ ). Let R' C_ R~Q be the subring generated by im(r~) and the set { bo I r E Gal(F/~ }. Let m' = R' M m~ o_, where rn~ ,~ is the maximal ideal of R~ oj Put R~ = R~,. Let p' = R ~ f3 p and P0 = R4~ C'l p. It is a standard tact about localizations that P0 = p'R0. Let A' = R'/p' and ?to = Rq)/p0. Our first claim is that A0 = A and p0R~ o_ = P" "Ib see this, first note that there are inclusions A ps C_ A0 C A. Since A is a finite AP~-module and A vS is Noetherian, A0 is also a finite AP~-module as is any ideal of Ao. It fi)llows that Ao is a Noetherian ring and that it is complete as an AP~-module. Since A0 is local and the radical of rnPSA0 is the maximal ideal of Ao, it follows that Ao is a complete local Noetherian domain. It now follows from I~mma 2.5 that the map from A0 to R~ ~Jp0R~ ~ is surjective, so we have A0 -~ R~ ~t/p0R~ Q -~ R~ ~/pR~ c~ = A. The claim follows. We next claim that the canonical map R~p., p~,~ > R'p, is surjective. As Q, p~ modp is irreducible there exists some % such that % ~ r~ tv :'P'~x j. It follows easily that { bo [ o E Gal(F/F) } C R* = im(R~ p, , R;,). Therefore the image of the canonical map R' ~ R~, is contained in R*. The inverse image in R' of the prime pP~R* is just p', whence localization induces a map Rp, ~ R* whose composition with the inclusion R* ~ R~, is the identity map. It follows that R* = Rp,. As a consequence we have ppsR; = p'R~,. Combining the results of the preceding two paragraphs yields - ' = =pR~ ppSR~ o_, p - P R~ o~, v p0R~ o, v <), p" We also find that A p~, A', A0 and A all have the same field of fractions, namely K p~. It follows that dimKp~(R~Q,p/pP~R~,p ) = I. Therefore the canonical map R~ ~,~ ~ > R;/~ Q,~ is surjective (see [Mat, Theorem 8.4]). Q, PP As a corollary of this we have the following important result. Corollary 2.12. --/fQC R~ is any prime such that p~ modQ is irreducible, and if QpS= r)l (Q) c_ R~ ps, then dim R~p~/QP~ > dimR~/Q with equality holding if Q is a dimension one prime. RESII)UALI,Y Rt-I)UCIBLE REPRESI'NTATIONS AND MOI)UI.AR FORMS 23 Proof. - Equality of dimR~,,,/02 '~ and dim R~/Q when Q is a dimension one prime was shown in the first paragraph of the proof of Proposition 2.1 1. We may therefore assume that Q is a prirrie of dimension at least two. Choose a prime p _D Q of R~ having dimension one and such that P~ modp is also irreducible. Let pps _- r~ 1 (p). We then have dimR~/Q= l+dimR~,p/Q ~< l+dimR~p~p)>~/Q v~ ~< dim R~ p~/Qp~ with the first inequality following from Proposition 2.11. [] We now collect a few results connecting deformations and pseudo-deformations. Suppose that A is a complete DVR with residue field k. Let K be the field of fractions of A, and let ~ be a uniformizer of A. Suppose that 9 : Gal(Fz/F) ---~ GL2(K ) is a continuous representation. As Gal(F~/F) is compact, there exists a Gal(F~/F)- stable A lattice L in the representation space of p. Such a lattice, being a free A-module of rank 2, gives rise to a representation PL" Gal(Fz/F) ~ GL2(A ) such that Pu | K ~-- 9- It is well-known that whereas the reduction 9L = PL rood ~. is not --SS --S~ necessarily independent of L, its semisimplification 9L is. We call 9L the reduction of 9. Ix'mma 2.13. -- Suppose that the reduction of p is 1 | )~ and that 9 is irreducible. (i) There exists a Gal(F~/F)-stable lattice L m the representation space of p such that = ( 1 Z(~jba 1"~ for all ~, and -Or, has scalar centralizer. (ii) For two lattices L, and L2 as in (i), the classes in II~(Fz/F, k(z-~)) of the coqycles (yl , )~(~)-lbi(o), i= 1, 2, are non-zero scalar multiples of one anoth~ Proof - - Choose any Gal(Fz/F)-stable lattice L and pick a basis for PL such that ( ) (~o ~) and let n= min ord~,(bo). As p is irreducible, PI.(Zl)-" 1 --1 " Write P1,(~J) = co ' o n < oo. Let L' be the lattice obtained by scaling L by (~'--'~ 1)" The representation PL' is just = ( )~-" 0, ,) ,) The representation clearly has the properties desired for part (i). To prove part (ii) it suffices to show that the representations 9I.~ and 9L2 are equivalent. Choose bases for the 9Li'S such that PL,(Zl) = (1 -1 )" As 9L, | K "~-- 9, there exists g G GL2(K ) such that -1 (l) (1) g 9L~g = 962. Since g must commute with - l , one may assume that g = a 9 Write 9L1(~)= ~.(a~ ~/b'~] By hypothesis, there exists some 60 such that b,0 is a unit. ClJ As ab~ o E A, it must be that ord~.Ia )/> 0. As the reduction of ab~ is not always zero, it must be that ord~(a) ~< 0. Thus a is a unit and 9L~ and Pb~ are equivalent. [] 24 C.M. SKINNER, AJ. WILES Corollary 2.14. -- IrA is a complete DVR with residue field k as above, and zf(A, q~) is a pseudo-deformation of Po unram~fied away from 2 for which x(a, "c) is not identical~ zero, then there exists a non-zero coqycle c E H'(Fr./F, k0u')) and a deformation Pq~ : Gal(Fy./F) , GL2(A ) of Pc whose associated pseudo-deformation is q). Moreover, c is unique up to multiplication by a non-zero scalar. Proof. --We need only observe that there is some irreducible representation p : Gal(F~/F) ~ GL2(A ) whose associated pseudo-representation is q0, for then the claim follows from the lemma. Fix ~0, ~0 C Gal(Fr./F) such that ord~.x(~0, ~0) is minimal. Define P by = a) " [] Our next result also associates deformations to pseudo-deformations. Suppose that R is a local complete Noetherian domain with residue field k and maximal ideal m. Suppose that P = { a, d, x} is a pseudo-representation of Gal(F~-/F) into R such that P0 = P modm (i.e., (R, P) is a pseudo-deformation). Let p be a prime of R such that the dimension of R/p is one. Let A be the integral closure of" R/p in its field of fractions K. This is a complete DVR with residue field a finite extension of k, say k'. Suppose that xmodp is not identically zero. By Corollary 2.14 there exists a cocycle 0 :~ c in HI(F~:/F, /fix-l)) and a deformation Pp : Gal(Fz/F) , GL2(A ) of Pc such that the pseudo-deformation associated to pp is 9 rood p. We will construct a local complete Noetherian domain R + having the same dimension as R, an injective local homomorphism R ~ R +, and a deformation 9 + : Gal(FyJF) , GL2(R +) of 9c whose associated pseudo-deformation is P. Moreover, R + will have a prime p+ of dimension one such that p = R n p+ and Pp = p+modp +. Let L be the field of fractions of R. Pick c~, ]3 E m, ]3 ~ p, such that or/[3 is a uniformizer of A. Put R' = R[ot/[3] C L. This is a Noetherian domain with maximal ideal m' = (m, c~/]]). To see that m' is in fact a maximal ideal, let q~' : R' -* A be given by ~'(f(ot/~))=f(o~/]3) for any polynomial fwith coefficients in R. Here the "bar" denotes reduction modulo p. This is well-defined, for iff(o~/[3) = 0, then f(o~/~) = 0 as can be seen by first clearing denominators and then reducing. Let p' be the kernel of q0', and let I be the ideal of R' generated by the set { x(r~, ~) }. Let { il, ..., i~ } be a set of generators of I taken from among the x((~, x)'s. As Pp is irreducible, the image of I under q)' is non-trivial. Pick an i E { il, ..., i, } whose image has minimal valuation in A. Define R* by R* = R'[iL/i, ..., ir/t] C_ L. This is a Noetherian integral domain with maximal ideal va* defined as the inverse image of the maximal ideal of A under the homomorphism q0* : R* ~ A given by RESIDUALLY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS f(i,/i, ..., ir/Z)~---~f(-it /-i, ..., i/i) for any polynomial f with coefficients in R'. (Now the "bar" denotes reduction modulo p'.) Let p* be the kernel of the map q)*. Let R + = ~., and let rn + be its maximal ideal and p* the kernel of the induced map R + -~ A. The ring R ~ is clearly a local complete Noetherian ring with residue field k. Moreover, the inclusion R ~-~ R + is a local homomorphism. To see that the dimension of R + is the same as that of R, observe that it follows fi'om the construction of R* that R~. = Rp, whence dimR* =dimR~. =dimR~.+l =dimRp+ 1 =direR. Unfortunately, R § need not be a domain. However, since the "going-down" property holds for the pair R~. and R* (see [Mat, Theorem 9.5]), there is a minimal prime q+ of R + contained in p+ and such that q+ 7/R~. = (0) and dimR+/q + = direR +. We replace R + by the quotient R+/q +. This ring has all of the desired properties. In the ring R 4 the ideal IR + = {i} is principal. As i = x(t~0,"c0) for some t~0, "Co E Gal(Fz/F), one can define a representation 9 + : Gal(Fz/F) ~ GL2(R +) by : o/ " The reduction of p*modm + is non-semisimple as P*(Z,) = (1 -1) and P+(~0) = (* I ). Thus 9+ is a deformation of 9e for some 0 :~ d 9 HI(Fy./F, k'(z -l)). Reducing p+ modulo p+ gives a deformation of Pe into GL2(A ) whose associated pseudo- deformation is p modp. It follows from Corollary 2.14 that c' is a non-zero scalar multiple of c. Thus, after possibly replacing 9+ by a conjugate, we may assume that c = c' and p+modp § = pp. Finally, suppose that A is an U/-algebra with ,~;' the ring of integers of some local field having residue field k' and that ~///~, Q c E are sets of finite places (possibly empty) such that (2.8) (i) Jig C Y~\~ consists of places w such that 9c is ramified at w ; (ii) Q c_ z\?~ u .Ill consists of places w at which 9c is unramified. Let ~9r be the field of fractions of R +. It is easily checked that if (2.9) (i) p+ | [D,"~ ( "~xgl'i *) ~/%i ' %"ira~ = 1, i= 1,...,t, p+ -~llw -~ (1 *'~ for all wEJtg and (ii) \ / (iii) det 9 + II,, = 1 for all w C Q, then c is admissible and 9 + is of type-,~Q where ~ = (r Z, c, ~fg). l%r ease of reference we summarize these results in a proposition. 26 C.M. SKINNER, AJ. WILES Proposition 2.15. -- Suppose (R, 9) is a pseudo-deformation of type-(~;, Z). Suppose also that p C_ R is a prime of dimension one such that x(~, ~)modp /s not identical!y zero (9 = { a, d, x } ). Let k ~ be the residue field of the integral closure of R/p in its field of fractions. There exists a cofycle 0 ~- c E HI(Fx/F, /((~-l)) unique up to multiplication by a scalar, a local complete Noetherian (~-domain R + with residue field k ~ and haviN the same dimension as R, a local homomorphism R~-~ R ~ of ~;-algebras, a dimension one prime p~ C_ R ~ extending p, and a deformation 9 + 9 Gal(Fx/F) , GL2(R +) of p~ whose associated pseudo-deformation is that induced from 9. Moreover, if Q, ,/2,g C_ Z are sets of places satisfying (2.8), and if 9 + @ ~, .~ thefield of fractions of R +, satisfies (2.9), then 9 + is a deformation oftype-~o_ with ~_~ = (C, E\Q., c, JAg), with 6~' = (~;' | W(k'). For a finite field F, W(F) denotes the ring of Witt vectors of F. 2.5. The Iwasawa algebra In this subsection we describe how each of the deformation rings R~ and R~ ~z is an algebra over a certain multivariate "Iwasawa algebra". Let LI be the maximal abelian pro-p-extension of F unramified away from ?~. Let I C Gal(L0/F) be the subgroup generated by the images of the inertia groups Ii, i = 1, ..., t. We fix once and for all a maximal free Zp-summand I0 of I (necessarily of rank 5v). Fix also a free Zp-summand Go of Gal(L0/F) containing I0 (this also has rank 5/). Finally, fix elements 'YI, ..., "/5 F E Gal(F/F) whose images in Gal(L0/F) generate Go and for which there exist integers rl ' ..., rgv such that 3'( ~ '""Ygv /~v generate I0. For each 0~<i~< t fix once and for all y(~ ' , "",Ydi E Ui (the units of F~i) generating a free Zp-summand of rank di. Put A~ = ~'~T,, ..., %F, Y~t', .-., Y~I]I. The rings R~ (and hence the R~(z) are algebras over A~ via 9 Ti . , det p~ (Ti) - 1, i = 1, ..., ~SF; 9 Yf' ' ~g21Y))-- 1, where 9~ I.), - v]~/~ v~/ and U, is identified with the inertia subgroup of D ab via local reciprocity. Suppose that ~ = (6;, E, c, rig) and cj, = (~:,Z', c,.~/g') are deformation data with E C_ E' and .//g~ C_ d,g. The natural map R~, ~ R~ is a map of A~: -algebras. Each universal pseudo-deformation ring R~ p,, and R~ ~ is a A~-algebra in a manner compatible with the canonical maps R~ p., ~ R~ and R~ (p~ ~ R~ ~. To see this, for each i = 1,...,t fix gi E Di such that x(gi):~ 1 and for each ,,g j = 1,..., di let ~j!'? C Di be a lift of yj~. By the choice of gi the polynomial RESII)UALLY REI)UCIBI,E REPRFSFNTATIONS AND MODUI2kR FORMS 27 X 2- tracep~p,(gi)X + detp~p~(g/) has distinct roots in Rcj:p~, say ~i and ~i with 0~i reducing to Z(gi) modulo the maximal ideal of R~ p~. (The images of c~i and ~i in R~ are just the eigenvalues of p~ (gi)). We define a map A~ ~ R~ p~ by * Ti~ ~ det Pcj p~('fi)- 1, i= 1, ..., ~iF, ' ' 0) * Yf, , (trace p~ ,,~(g.~}i/) _ a,. trace p~ p,(~) ))/(~i- o~i)- 1. The compatibility with the A~-algebra structure of R~ is clear. Also, if d;~ ps .~I~, = (~.,, Y'I) and -~2 = (~', Z2) are two pseudo-data with Y.,~ D Z1, then the natural map R~!~ ~ R~ ~,~ is a map of A~ -algebras 9 ~ I 3. Nearly ordinary Hecke algebras and Galois representations 3.1. Modular forms and Hecke operators We keep our previous conventions for the field F. We write A and Af for the adeles and the finite adeles of F, respectively. If G is any algebraic group over F, then we identify G(A) with the restricted direct product of the groups G(F~.) with respect to the subgroups G(~F, ,1,) (for finite w), writing .'cw for the w-component of x E G(A), and similarly for G(Aj). For a finite place w, we sometimes write xp for xw with p the prime ideal of F corresponding to w. Let I be the set of infinite places of F (equivalently; the set of embeddings "c 9 F ~ R). This description of G(A) identifies G(F | R) with G(R) I. We also fix an algebraic closure Qr of Qp and an embedding of Q = F into Qp. l~br an ideal n of ~';v we define various standard open compact subgroups of GL2(Aj) as tbllows: Uo(n) ={(: b) EGL2(~;F| moan}, 9 a-- 1 modn}, and {(++ 9 d- 1 modn}. U(n) = c ,1 E U t(n) For k= Zk~ E Z[-l] and x E (31 write x ~ for the product II~. Let t= E~. To each k = Zk~, with each k~/> 2 and having the same parity as the others, we associate quantities m, v E Z[I] and la E Z as follows: m=k-2t and v=Zv~'~, v~)0, somev~=0; m+2v=g.t. 28 C.M. SKINNER, A.J. V~qI,ES Let H denote the complex upper-half plane. Define j" GL2(F | R) x H l ~ C I by , = = (o, C (F | R). j(u~ z) (c~z~ + d~), u~ ~. Definc also an action of GL~(F | R) = GL~(R) I on H I by uoo(z)= + b~'~ Denote by z0 the point (i, ..., z) G H I. We now recall the notion of a (holomorphic) modular form on GL 2. First, for any congruence subgroup F C_ GL2(F), denote by Mr(F) and St(F) the spaces of (classical, Hilbert) modular forms and cusp forms on H I, respectivel)4 of weight k (cf. [Sh]). For a functionf : 6L2(A ) , C and u = uf- uo,z G GL2(,A ) - GL2(Af). GL2(F| ) we define fl . by (/Itu)(g) = j(u~ , Zo) -t det(u~)V+t-t f (gu-'). Write Co~ for the subgroup (R � -SO2(R)) I C_ GL~(F| A functionf 9 GL2(A ) , C satisfying f Itu = f fbr all u E Coo gives rise to a function ~: I-I I > C for each x E GL2(A/): f~(z) =j(Uoo, Zo) t det(uoo)t-t-V f(xuoo), uo~(Zo) = z. Let U C_ GL2(Af) be an open compact subgroup. A function f: GL2(A ) , C is a modular form of weight k and level U if 9 f(ax) =f(x) Va E GL2(F ), 9 f]tu=f VuEU'C~, 9 f~(z) E Mk(F,), Fx = GL2(F) nxU-GL~(F| Vx E GL2(At). Such a function is a cusp form if f~(z) E Sk(Fx) for all x E GL2(Aj). Denote by Mk(U) and Sk(U) the spaces of modular forms and cusp forms of weight k and level U, respectively. For more on such forms see [Sh] and [H1]. If U = Ut(n), then Mk(U) and Sk(U) are just the spaces Mk(n) and Sk(n) defined in [Sh]. For each n choose once and for all representatives t (') E A x of the ideal classes of F (i = 1, ..., h) such that t~ ) = 1 for each place w lNm(np).oo. Put xi = ( t(O 1) and write Fi for the subgroup Fx i and for each f E Mr(n) write f for f,. There are h h isomorphisms Mk(n) --~ I-[ Mk(F;) and Sk(n) -~ I-[ Sk(F;) given by f~--~ (f). Each f(z) has i= 1 i= I a Fourier expansion of the form f(z) = ai(O)+ ~+ai(g)e(g" z) where (P)) is the ideal go(#)) of F associated to the idele t/'l, the sum is over totally positive elements of (t(')), and (a~z.___5~ RESIDUALLY REI)UCIBI,E REPRESENTATIONS AND MOI)UI2kR FORMS 29 g'z = Xz(g)-z,. For a ring A _C C, let Me(n, A) be the space of modular formsfE M~(n) such that each f has Fourier coefficients in A. Define Sk(n, A) similarly. Shimura has shown that Mk(n, A) = Mk(n, Z) | A and Sk(n, A) = Sa(n, Z) | A. For a ring R C Qr define Mk(n, R) and Sk(n, R) by Mk(n, R) = Ma(n, Z)| R and Sk(n, R) = Sk(n, Z)| R. If R C_ Q as well, then this agrees with the earlier definitions, as Shimura's result shows. From now on we require each U to satisfy U = II Uw, U~ C_ GL2(~F,w). ?/,' f OC We also require that U(n) C U C U0(n) for some n. Next we recall the connection between modular forms on GL~ and automorphic representations of GL 2. For details and definitions the reader should consult [De], [Ge], and [J-L]. Let ,_~0 be the space of all cusp forms on GL 2 (over F, of course) of weight k. The group GL2(Af) acts on ,.d~ via (~)(x) =f(xg). Under this action ,~ is an admissible representation of GL2(Af). Moreover, ~4~ decomposes into a direct sum ~ = (~VTr where, for each g, V~ is an irreducible admissible representation ]I of GL2(Af) (which we often denote just by re), and the Vn are all non-isomorphic. For an open subgroup U C_ GL2(O,; | 7.)let Ilk(U)= {~ I V~ ~( 0 }. Clearly the space (~ V~ is just Sk(U). We recall that each rc E FIt(U) can be written as a restricted tensor product ~ = I~)lt~ where v runs over the finite places of F and each ~ is an irreducible admissible representation of GL2(F~ ). Let V~ - | ~ be the corresponding tensor product decomposition of V,~. Clearly V~ ~ = @ V~U,~ It follows from the theory /; of newforms that dim V L'~ -- 1 for each place v for which U~ = GL2(~F ' ,). l%r each g E GL2(Aj) define a Hecke operator [UgU'] 9 Mk(U) , Mk(U') by X -1 (3.1) [UgU']f(x) = ~_.,f( "gi ), UgU' = I lUg,. gi Of course, [UgU'] maps St(U) to Sk(U'). For each prime ideal g of F choose an element ~.(g)E r | such that )~{)is a uniformizer of (~v,t and ~t)= 1 for p :~ g. If pL0 then we require that ;~P) also be an element of ~F such that X(~P)) ~: 1 and that )~) E (J~.~,,, for each p'~ but p' =~ p. (For y E F~ we define )~(y) to be the value obtained from composing )~ with the local reciprocity map F~ , Gal(F~pb/Fp).) Denote by T(g)and S(g)the operators [U( 1 ~(~))U] and [U(~'(t) ~(t))U], respectively. These operators commute one with another. Moreover, it is easy to see that T(g) and S(g) are independent of the choice of ~(t/ if Ut = GL2(~"v,t). Also, if V C_ U and if 30 C.M. SKINNER, AJ. WILES GL.~(~ F, C) = Ve = Ue, then the inclusion M~(U)C_ Mk(V) respects the actions of T(g) and S(g). If U = U1 (n), then T(g) and S(g) (for g r n) are just the Hecke operators defined and so denoted in [Sh]. These operators stabilize each M~(n, R) and S~(n, R). As S~(U)= (~ V~ we find that the Hecke operator [UgU] stabilizes each V~ ~, ~EFIk(U) the action being given by (3,1 t) [UgU]x -- E -1 U ~(~i )x, x 9 UgU=UUg,. gz l:br each place v let g,. be the v-component ofg. Under the tensor product decomposition gl7 u,. u. = | lUgU] decomposes as [-UgU] = @ [U,.g,.UJ with [ld, g,U,] 9 End(V~,~) being given by = x E V~'~, U~g~ U~. = [IU~hi. hi 3.2. Nearly ordinary Hecke algebras Keeping the conventions for U introduced in the preceding subsection, for each positive integer a define U~, U~, and Uo by "e'~ = U r] Uo(pa), U~ = U ["] UlOOa), and U~ = U N U(p~'), respectively. Suppose that U~ = GL2(~::v,~ ) for each v[p. There is an action of the group G~U~) = U a" ~ C ~ r/Ua" � ~V � on M~(Ua) with x.y acting via the operator (UaxUa) = (~r -~) [IS~xUa] where x = (i b) E U~ andy E d~-. Here co" Gal(F"b/F) ~ F� '12" is the Teichmaller character, and c0(a,.) is defined by composing co with the global ,>: X reciprocity map taking d; v.l. to the inertia group of v in Gal(F~U/F). Recall that we -- -- X have fixed an embedding of F into Q~, so co, which a priori takes values in Qp, can be considered as taking values in F � e t el Let (p) = p,, ...,p, be the prime factorization of (p) in F with Pi the prime corresponding to the place vi. For each i = 1, ..., t define an operator T(~(pi) on Mk(U,) by = (~P'~)-VT(pi). Define an operator T0(p) similarly by T.(p)= p-v[U~(~ ~)U~] where p, = p if v[p and p,, = 1 otherwise. Note that T0(p) differs from FI T.(pi)" by i= 1 multiplication by some ~,. [U~ (e) 1 Ua] where ordp,(~,) = 0 and ep, E (?'� F, p, for each pi. RESIDUALLY REDUCIBI,E REPRESENTATIONS AND MOI)UI.AR FORMS 31 As discussed in the previous subsection, these operators act on each V~, rc E Ht(U,). If V _C U and if V~, = GL2(dv,,, ) for each vlp, then the inclusion Mt(U~) C Mt(V~)(b /> a) is compatible with the natural homomorphism G(Vb) , G(U,) and with the actions of T~ (p) and the T0(Pi)'s. Let k = Y~kr with each k~ >/ 2. Let n E II~(U~). Suppose that vlP and that p~ is the corresponding prime ideal. It is an easy consequence of the classification of local automorphic representations that if there exists a line in V~('I/ on which T0(p,) acts via an element of F that is a unit in the ring of integers of/~r via the fixed embedding F~--~ O-~0, then the line is unique. Call such a line v-good. A v-good line exists if and I I only if n~ is either a principal series representation ~(n~l 9 , 9 or a special 1 I representation rc({~ I 9 I[ ~ , {~l" I~)such that in either case ~.~-v{~()~-I)is a unit in the ring of integers of O~ (cf. [H3, Corollary 2.2]). Here ~.~. is the uniformizer of ~' r',~ chosen in the definition of T0(p~). The representation n is said to be nearly ordinary if a v-good line exists in V~,~ for each v dividing p. Similarly, a newformfE Sx(U~) is called nearly ordinary if the corresponding automorphic representation nj is nearly ordinary. Let H""a/ll ~ C rlk(U~) be the subset of nearly ordinary representations. A representation k k ''~a] -- ord n E 11 k (U~) is said to be ordinary if ~ is unramified at v for each v~. Similarly, a newform f E Sk(U~) is ordinary if the corresponding automorphic representation is ordinary. Fix an identification ofC with Q~ extending the fixed embedding of 0 into O-~0" For each rc E II~rd(ua)let w(rc)= | v) E @V;)'I ~ be a vector such that l' w(x, v) spans a v-good line for each v~b. Each w(n) is an eigenw:ctor for the Hecke operators T0(Pi) for i = 1, ..., t, T0(p), T(g) and S(g) for each prime ideal g r tbr which Ue = GL2(tg?v, g), and for each element of G(U~), and the corresponding eigenvalues are integers in Qr Let S~ (Uo) c S~(U,,) be the subspace spanned by the w(n)'s (recall that Sk(ga) = (~ v~Ta). Let Tk(ga) C Endc(S~.(U,) ) be the subalgebra generated over Zp /IE I-lk(l_'a; by the aforementioned Hecke operators. The ring Tk(U~) is a finite, flat, commutative, reduced Zp-algebra. In fact, we have an injection ~C-II k (l a/ Note that the definition of Tk(U~) is independent of the choice of the w(n)'s. Also, if V _D U is another open compact subgroup, then there is a canonical homomorphism Tk(Vb) ' Tk(Ua) (b >/a). 32 C.M. SKINNER, AJ. WII3~;S For ~' the ring of integers of some finite extension K of Q0, put T~(U,, 8) = T~(Ua) | ~:. Put also T~(U, ~') = lim T2(U~, 8:). Suppose k is parallel (i.e., v = 0). We now give an alternate (but equivalent) definition of Tk(Uo, 8'). As we shall see, both definitions will have their uses. Let Tk(Ua) be the subring of Endc(M~(Ua)) generated over Z by T0(Pi) tbr i= 1,..., t, T0(p), the action of G(Ua), and by the operators T(g) and S(g) for each prime ideal g ~ p for which Ue = GL2(~r, e ). Let T~.(Ua) be the quotient ring obtained by restricting the action of the Hecke operators to the space Sk(U,) of cusp forms. These rings are finite, flat, commutative Z-algebras. Put Tk(Ua, ~,~) = 'rk(Ua) @ ~' and T~(Ua, ~') = T~(U~) | C~ with 8' as in the preceding paragraph. For all sufficiently divisible integers rn, the operator e lim F0(p) e v, -, exists in Tk(U,,, ~;) and is independent of m. Moreover, n~cx3 * T e is an idempotent. Put T~(U~, .~,') = eTk(Ua , 8). That this definition of T~(Uo, ~) yields the same ring as did the previous one can be seen as follows. The ring T~(U~, C ~) is the subring of Endc(Sk(Ua)) generated over ~' by T0(p/) for i = 1, ..., t, T0(p) , the action of G(Ua), and by T(g) and S(g) for each g Cp such that Ue = GL~(~'F.e). In particular, e is identified with an idempotent in Endc(Sk(Ua)) and eT~(Ua, ~)is just the image of T~(Uo, ~)in Endc(eSk(Wa)). Write Ua ord eSk(Ua) = ~ eVn . From the definition of e we have that eV~" = 0 if r~ r I1, (Ua) r~El Ik(Ua) Fl~ T ~ Ua and that if g E "'k ~,,J then eV= = { x = | 9 x~ spans a v-good line Vv~0 }. It is now immediate that eT~(Ua, -~;) agrees with the first definition of Tk(U~, ~/). Let G(U) = lira G(Ua), where the transition maps are the maps induced from the inclusions U~ C U~, a >/b. There is a homomorphism ~" [[G(U)~ , T~(U, r Put t/= u a (&)� = uo n (A2� , Z(Ua) = Ur. and Z(U) = lim Z(Ua). The map (~ b), , (~(a_ld)~, a)induces isomorphisms G(Ua) --~ (~'F/pa) � X Z(Ua) and G(U) ~ (.~f,4' F ~) Zp) � x Z(W). RESIDUALIN REDUCIBLE REPRESENTATIONS ANI) MODULAR FORMS For (y, 1) E G(U) we write T~. for the corresponding Hecke operator. Similarly, we write Sx for the operator corresponding to (1, x) E G(U). Let ez = n c_ | zp) � = n v i v i where ~ ~ C ~� is the subgroup of units congruent to one modulo vi. Let y} ~ E '~.~ -- J F, v i ' ' "'"YgF ' respectively, via the be as in w Let x~ ..., x~ v E Z(U) be the images of ~'(~ /~v global reciprocity map (for the definition of Yi and ri see w The xi's generate a maximal Zp-free direct summand of Z(U). The ring T~(U, (9') is an algebra over the ring A' = ()? [[-Xl, ..., X~F:, Y(I 1), Y~ via Xi, , S,,- 1 and ~'3, , T (,: - 1. 9 .., )) The principal goal of this subsection is to show that T~(U, ~) is a finite, torsion- ti'ee A~.-module. We only prove this for F having even degree, although the result is true in general. Our proof involves analyzing modular forms on a twisted-form of GL 2. Suppose that F has even degree. Let D be the unique quaternion algebra over F ramified at every infinite place and unramified at all finite places, and let R be a maximal order of D. Let G D be the unique algebraic group over F such that GD(F) = D x. Let VD " G D ~ Gm be the reduced norm. For each finite place v fix an isomorphism R | ~'F. 0 --~ M2((~F.,). This induces an isomorphism GD(Af) _~ GL2(Af) which we use to identify these two groups. For each open compact subgroup U _C GL~(Af) put c~'~ = {f: D� , 13 }. (Note that D� is a finite set.) We distinguish a subspace ID(u) = {f E c~'D(u): ffactors through GD(Af)/U-~(Af)� }. For any g E GD(Af) -~ GL2(Af) there is a Hecke operator [UgU'] : J3~D(u) , Jfi'~)(U') defined as in (3.1). It is easy to see that [UgU'] maps ID(U) to ID(u'). A theorem of Jacquet, Langlands, and Shimizu [J-L], [Shi] states that there is a system of isomorphisms sD(u) = s2(u) compatible with the action of the tIecke operators [UgU']. Thus T_~(Ua) can be identified with the subring of Endc(SD(Ua)) generated over Z by T0(p), T0(P/) for i= 1,...,t, G(U~), and T(g) and S(Q for all prime ideals ~ for which Ue = GL2(~v,e). Put X(U) = D� and define H~ Z) = {f E .r taking values in Z }. 34 C.M. SKINNER, AJ. WII,ES This is a free Z-module of rank equal to #X(U). I~br any Z-module R, put H~ R) = H~ Z)| R. Note that H~ C) = ,~)(U). The action of [UgU] on H~ (3) stabilizes H~ Z) and hence induces an action of [UgU l on H~ R) for any Z-module R. If R is an ~-module then the operator e= l'ma[U,( 1 ~)U~] f~-l/ exists in Endr176 R)) for sufficiently divisible m. tl Moreover, e annihilates ID(U,, Z)| R, where ID(Ua, Z) = { f E ID(Ua) taking values in Z}. Let T(U~, (~") be the C-subalgebra of End,~ (H~ ~:')) generated over C by T0(p), T0(Pi) for i= 1,...,t, G(U~), and T(g) and S(g) for all prime ideals g {p such that Ue = GL,2(~Yr, e). It follows that T2(U~, ~') can be identified with eT(U~,, ~) (equivalently, with the image of T(U,, ~'~) in End~,, (eH~ ~))). Put H~(U) = lim eH~ K/~). (K is the field of fractions of ~'.) This is a T~(U, ~)-module. For any open subgroup U put U = U/UnF � For each x E GD(Ap) put c (x) = #{u E U = x}. Let R be any ~'-algebra. If each cu(x) is invertible in R, then define a pairing ( , )~:" H~ R) � H~ R) , R by (f, g)u = ~_~ Cu(X)-l f(x)g(x) - xGX(U) This is a non-degenerate pairing, and the map f. , (f, .)t: determines an isomorphism H~ R) -~ HomR(H~ R), R) that is functorial in R. The pairing ( , )u is not Hecke-equivariant, but a straight-forward calculation shows that ([UgU]f, h)t: = (f, [Ug-~U]h}u for any g E GD(Af). It follows that for each t E T(Ua, ~') there exists t § E Endr (H~ ~')) such that (t .f, h)u~ = (f, t +" h)u~ for all f, h E H~ C). Let T+(Ua, ~) C_ End(H~ ~)) be the ~;'-subalgebra generated by {t + 9 t E T(Ua, ~:)}. Clearly the map t ~ ~ t* determines an isomorphism of ~- algebras T(U~, ~') ~-T+(U~, ~). For any C'-module R write H~ R) ~ for the T(U~, ~')-module whose underlying ~-module is just .~s~ ~ 9 ~'~J, R) but the action of t E T(Ua, ~') is via t +. It follows that the pairing ( , )t:, induces a perfect RESIDUAI,LY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 35 Put pairing (,)G: eH~ R)x eH~ R)+ , R of T2(U~, (~)-modtfles. H+ (U~) = limeH~ K/U) +. If V C_ U is any subgroup, then we define a trace map tr(V, U) 9 H~ R) H~ R) by tr(V, U)f(x) = ~_,f(xx~-l), U = kJVxi. If V = Vb and U = U~ (b /> a) then it is easy to see that this is independent of the chosen coset representatives and that it is compatible with the actions of t and t + for t E T2(V, ~'). The pairings ( , )v and ( , )v satisfy the following compatibility whenever they are both defined: the diagram ( , )v " H~ R) x H~ R) , R T tr(V, II ( , )v" H~ R) x H~ R) , R commutes. Since Cuo(X) = 1 for all x if a is sufficiently large, it follows that by putting Mo~(U) = lim eH~ ~),";:' ~ where the transition maps are just the trace maps tr(Ub, U~)(b ) a), we have an identification of T~(U, ~')-modules Ms(U) -~ lim Hom(:~ (eH~ C), r162 _~ lim Hom~. (eH~ K/~::'), K/(Y ') _~ Hom~ (lim eH~ K/~"), K/~ ~) --~ Hom~: (H~(U), K/C'), the Pontryacn dual of H~(U). Putting ML(U) = lim eH~ ~) we obtain a similar identification of M~(U) + with the Pontryagin dual of Ha(U). + The following proposition is due to Hida [H2, Theorem 3.8]. Proposition 3.3. --If the action of every element 0fU/UVIF � on D� f) isfixed-point M~(U) are ' -modules of rank equal to flee, then Mo~(U) and + flee A~, { order of the torsion "~ rank6:~ eH~ ~) x \ subgroup of G(U) 1" 36 C.M. SKINNER, AJ. WII,ES Proof. -- We first claim that the Pontryagin dual of eH~ K/~ ~:) is a free E[[G(U~)~-module of rank equal to the ~'-rank of the Pontryagin dual of eH~ K/~). Clearly; it suffices to prove the claim without having ap- plied the operator e, in which case it is a simple consequence of the fact that H~ K/6 ~') = H~ K/r ~tJ~ and that #X(U~) = #X(U~ #G(Ua), the latter equality a consequence of the assumption that G(U~) acts freely on X(U,). The assertion of the proposition for Mo(U) will follow if we can establish that eH~ K/~") = eH~ K/~), for then we will have that the dual of eH~ K/~ ~) is a free r of rank equal to the ~-rank of the dual of eH~176 K/~ ~') which in turn equals rank6~ H~176 E). Now, if a /> 2, (1) 0 0(1 ) 0 then U~ ~ U a = U a ~ Ua_l, so T0(p). H~ K/#) C_ H~ K/6~), whence eH~ K/eft) = eH~ K/(~ ~) as desired. The same argument applies to the Pontryagin dual of eH~ K/~) + yielding the assertion of the proposition for M~(U). + [] Corollary 3.4. -- For any U, To(U, ~') is a finite, torsion-flee A~-module. In particular To(U, ~) is a semilocal ring complete with respect to its radical. Proof. - Choose a prime g of F for which V = U nU(g) is such that V/V N F � acts freely on D� The existence of such an g is proven in Lemma 3.5 below. The induced map Mo(U)~ Mo(V) is compatible with the action of To(V, ~'~) and hence is a map of A~-modules. As M~(V) is a free A'F-module by Proposition 3.3, Mo(U) is a finite, torsion-free A~-module. As there is an injection To(U, ~)"-~ EndA~.~ (Mo(U)), the same is therefore true of To(U, (Y). [] Lemma 3.5. --/fg ~6 is an unramifiedprime ideal ofF, then U(e)/U(e)NF � o~ts3~ee~ on D � \GD(Az). Proof. -- If~x= xu for some ~ E D � x E GD(Af), and u E U(g), then fi E Fx where Fx = D � n xU(g)x -l. We claim that Fx/Fx n F � has finite order. To see this, note that the canonical injection i : D� � ~ GD(A)/A � identifies D� � with a discrete subgroup of GD(A)/A � . Now let V = Gr~(R | F)/(R | F) � � (xU(e)x-~/U(s n (A/)� This is a compact open subgroup of GD(A)/A � so W = V N im(i) is a finite group, and it is clear from the definitions that Fx/Fx n F � ~ W. This proves the claim. Thus some power of 5 lies in F � and the same is true ofu. By the choice of g, u must itself be in F � . [] Corollary 3.6. -- Let M be the exponent of the torsion subgroup OfD� � . If{g,, ..., e, } is a set of unramfied primes of F such that (i) gi { 6 and (Nm(gi) - 1, M) = 2~i for each i, RESIDUALIN REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 37 (ii) for each y E ~ ~, that is totally positive some g~ does not split in F(v~) , then U 1 (e I ... es) / g I (e l ... e~) ['-'l F x acts freely on D � \GD(A~). Proof. -- If 8x = xu for some 8 E D � , x E GD(Az), and u E [Jl(gl ,-,gs), then 8*x = xu ~ for e = II(Nm(g~) - 1). Since u ~ E U(g~ ...g,) it follows from I~emma 3.5 that i= I u ~ E F � and therefore ~ E F � . It then follows from our hypotheses on gi that 6 2' E F � for some r. If r= 0, then ~, u E F � We may therefore suppose that r/> 1 but that 2 r-I Fx, o2 Fx" 2r-I r F � Put 7 = 62'' and to = u . Then 7, o) ~ but y2, E Let ot and II be the eigenvalues of 7. As ((x/~) ~ = 1 it must be that either a = II or ot = -13. If a = [3 then 7 E F � since 2ct = 0t + 1~ E F. Therefore a = -ll. Note that det(7) = oc13 must be totally positive. Since o~ and [~ are also the eigenvalues of co and since co E U~(gl ... gs) we find that a and II are in Fe~ for each i and that all E ~; F" � Therefore each g, splits in F(x/-2~) = F(v/~), contradicting our hypotheses. [] Suppose k : T~(U, G) , Qr is a homomorphism of" ~'-algebras such that = klz(u 1 and q0 = kl(~,~.|215 are finite characters. (Recall that we have identified G(U) with Z(U)x (-~v | Z~) � It is not difficult to deduce from the definition of T~(U, C?) that ~ factors through some T~(U~, (~) and hence corresponds to some rl~ 1 rr~ ~ That is to say; there exists a unique n E ..~ ~,~ and an eigenvector ~ *'2 \"~a]" Ua v E V~ for T~(U, 62P.) such that the eigenvalue of each t E T~:(U, C) acting on v, viewed as an element of Qr is just ~(t). The existence of such a n follows from the ord definition of T~(U~, ~(2v). It is also easy to see that any ~ E I-I~. (Ua) determines such a homomorphism ~.. Therefore there is a correspondence between homomorphisms as at the start of this paragraph and nearly ordinary autornorphic representations ord n E U I-I~ (U,). This correspondence generalizes to other weights k as summarized in the following remarkable result of Hida [H2, Corollary 2.5]. Proposition 3.7. -- /f X " T~(U, ('~) , Qr an ~'-a~ebra homomorphism such that ~.[z~/ = 9 e~, g >/ 0, with ~ and g~ = ~.[(~v~zp~� finite characters, then there exists a nearly ordinary automorphic representation n of weight k = (g + 2). t for which ~.~F(g) ) and k(S(g)) equal, respectively, the eigenvalues of T(g) and S(g) acting on the newform associated to n for all prime ideals g r p for which Ue = GL2(~'F.t). Here, as in the preceding section, e denotes the cyclotomic character giving the action of Gal(F~b/F) on the Zp-module lim~p., ~p, being the group of p"th roots of unity. The character ~ factors through Gal(F~)/F) where F~) is the maximal abelian extension of F unramified outside of those places dividing p and oc. Global reciprocity determines a homomorphism Z(U) ~ GaI(F~//F) via which we view ~ as a character on Z(U). 38 C.M. SKINNER, AJ. WILES We continue to assume that the degree of F is even. A prime P of Toe(U, C) that is the kernel of a homomorphism as in Proposition 3.7 is called an algebraic prime. The associated element k = m + 2t E Z[I] is called the wei~t of P. For an algebraic prime P of T~(U, ~') there are finitely many homomorphisms ~. : T..,o(U, C') > whose kernel is P since T~(U, 65')/P is a finite extension of ~5 ~:. Let .~(P) denote the set of such homomorphisms. The set of algebraic primes of T~(U, ~') is Zariski dense, as the following lemma shows. Lemma 3.8. -- Let Q be a minimal prime of T~(U, ~) and let ,2g'(Q) be the set of algebraic primes of weight 2 containing Q. The set 32/(Q) is Zariski dense in spec(T~(U, 6':)/0,). Proof. -By Corollary 3.4, T~(U, ~?)/Q is an integral extension of A~:. Call a prime IJ C_ A} algebraic (of weight 2) if it is of the form p = A} NP for some algebraic prime P of T~(U, U'.')/Q of weight 2. The algebraic primes of A~.: are just those corresponding to kernels of homomorphisms A' , ~ sending 1 +YJf, , q)~vj") and 1 + Xi, , ~(xi) for finite characters q0 and V of (~?v | Zp) � and Z(U), respectively. That such primes are Zariski-dense in spec(A~. ) is immediate. [] Corollary 3.9. -- Let Q be a minimal prime 0fToo(U, ~). /fV D U is such that some )~ E ,~ (P)factors through the map T~(U, ~>') , T~(V, ~')for all P in a subset of .3g'(O~ that is Zariski-dense in spec(Too(U, ~')/Q), then e ~ the inverse image of a minimal prime of T~(V, r 3.3. Hecke algebras, representations, and pseudo-representatlons In this subsection we assume that F has even degree. Let U C_ GL2.(6~: v | Z) be as in the preceding subsection. Write n for the product of those prime ideals g . Vl~ fl " T for which Ut ~= GLo(Cv.e). Suppose that k = ~:- t with  /> 2. Let r~ E --h ~,oa) and let )v 9 Tk(Ua, .~) > ~ be the corresponding homomorphism. Suppose that 7t is ordinary. In [W2] it was shown that there exists a continuous, irreducible representation p,~" Gal(F/F) , GLg(O_.# ) such that (3.{~) 9 Dn(Zl) = (1 --1) 9 p~ is unramified at all primes g {np 9 trace o~(Frobe) = ~.(T(g)) for all g ~np 9 det p~(Frobt) = X~S(e))Nm(e) for all g ~np 9 det On(X) = ~.(Sx)e(x) tbr all x E Z(U) 9 Pro[l), ~ t~{i) V~) with ~'?(y)= )v(Ty) for all y E ~"~., ~,~ and v9 v"~, ) = ~.(F0(Pi) ) for all i = 1, ..., t. RESIDUALLY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS ord Now suppose that n is any element of rI~ (U~). Given any finite set S of finite places of F distinct from those dividing p, there exists a finite character ~ unramified at S and such that n | ~ is ordinary. The representation p~ = (p~)| ~ is independent of ~, and by varying S one finds that (3.2) also holds for this p~. Now, for any representation p 9 Gal(F/F) ~ GL2(()~0 ) and tbr any finite place v ~p let n~(p) be the automorphic representation of GL2(F~, ) corresponding to Pit), via the local Langlands' correspondence (see [Ca] and [Ku]) normalized as in [C]. So, I 1 in particular, if OID~ ~ (gl g2* ) then n,.(p)= n(gl' ] 9 If ~ , ~j~211 9 Iv2). For any ordinary ord representation n E FI t (U) it was shown in [W2, Theorem 2.1.3] that n~(prc) ~ n~. Now 1-;otd/! -~ suppose that n is any element in H k ~uj and that qt is a finite character unramified at v such that n | ~ is ordinary. It follows from the preceding observations that n~, | qt~. = (n | qt)~ _~ n,(Pn| = n,(pn | qr -1) = n,.(pn) | qty. We therelbre have that vI~ ,'~ (3.3) n~.--~ n,,(pTt), n E "'k t'-~J, vCpoo. ]'he representation pn can be generalized as follows. Suppose that Q is a prime of T~(U, ~'). Let R = T~(U, ~')/Q and let L be the field of fractions of R. Note that R is a complete local domain. Hida has shown that there is a continuous, semi-simple representation pQ" Gal(F/F) , GL2(L ) such that (3.4) (i) PQ(Zl) = (1 -1) (ii) Po is unramified at all primes g ~ np (iii) trace pQ(Frobg) = T(g) mod Q for all g ~ np (iv) det po(I"robe) = S(e) Nm(g) mod Q for all e ~ np (v) det pQ(x) = S~ ~(x) mod Q for all x E Z(U) V~) with ~r = ~Iy mod Q for all y E C,"v ~, ,, (vi) p~,D%( v~'' " ) ' and qt 2 (~i ') = T0(p/) mod Q for all i = 1,..., t. By pQ being continuous we mean that there is a finitely generated Gal(F/F)-stable R-module ,////~ in the underlying representation space of PQ. such that Gal(F/F) acts continuously on J/-g. We give a proof of the existence of Pe. in the next few paragraphs. If Q is an algebraic prime of weight 2, then the desired representation follows immediately from the existence and properties of the representations pn. For let ~. E .5gr (Q) and let n be the automorphic representation corresponding to ~.. The homomorphism ~." T~(U, ~:~) , O p determines an embedding R~--* Qp which extends to an identifcation of L with Qp. Under this identification we may take 94 = P,t. Properties (3.4i-vi) follow from (3.2). Suppose now that Q is a minimal prime. Using Lemma 3.8 one then deduces the existence of PQ as in the proof of [W2, Theorem 2.2.1]. Of the properties of PQ 40 C.M. SKINNER, AJ. WILES listed in (3.4) the only one that is not immediate from the construction is the final one concerning the restrictions 9e[Di. This can be deduced from the corresponding properties of the 9p's, P an algebraic prime of weight 2, as follows. First, arguing as in the proof of [W2, Lemma 2.2.4] shows that the semisimplification of 9QID, is the sum of two characters gtl '~ and " (') ~ � V2 with ~)(y) = Ty for each y E ~F.v,- Moreover, (,?, we may assume that gt 1 II,,i ~ for otherwise there would be nothing to prove. Choose ~0 E Iv, such that gtl')(~0):~ gt~)(~0). Let ~" C f(Q) be the Zariski-dense subset of primes P for which ~i~('c0) ~: q~)('t0)mod P. Choose a basis of 9Q such that pQ('c0) = (~ [3 ) and Po[,~, = (x~ ~2) with either Z~ = ~]~? or Z, = ~?. If 9QIDi is split, then property (vi) is immediate, so assume otherwise. Let ~0 E Di be such that 9Q(~j0 ) = (* bo ) with b0 * 0, and let go E Gal(F/F) be such that PQ(g0) = (co *) with co :~ 0. Let R C L be a finite integral extension of R containing a and 13. Let ~- be the set of primes in R consisting of those primes P such that P M R is in ~'. Let C ~- be the subset consisting of primes P such that boco E R{. The set ~ is non-empty. For each P E o~; it is not difficult to see that pQ = b0 pQ bo t ) takes values in GL2(I~). As 9}l]DmodP is nonsplit for each prime P E ~; it follows that for P E o~;, )~lmodP = vl0modP. Since V~ ) ~ ~')modP and either ;(l = ~i '~ or gt~ ~ we conclude that ;~ = gt/~ '). Note that arguing again as in [W2, Lemma 2.2.4] (z)/,~ (Pi),~ _. = shows that either V2,'~,, T0(p/)modQ or gt?()~"/)) T0(Pi)modQ. Arguing as before shows that the former must hold. This proves property (vi). Note that the representation pQ gives rise to a pseudo-representation into R = T.~(U, ~)/Q. This is just the pseudo-representation associated to pQ (cf. w For a non-minimal O~ C__ Q the representation pe, can be constructed in the usual way (cf. end of the proof of [W2, Lemma 2.2.3]) from the pseudo-representation into T~(U, (~)/Q' obtained by reducing modulo O~ the pseudo-representation associated to pQ. The only property that is not immediate is (3.4vi). For this we note that if pQ, is reducible then there is nothing more to prove (as one of the characters has the desired property), so assume that pa is irreducible. Let R be a finite integral extension of R containing the values of ~t~"' and gt~ ), and let (~ be an extension of O~ to R. It is easy to see that the semisimplication of Ooj I~i is the sum of the characters ~t:i ') and ~ " (') modulo (~. If ~:2 = gt~)m~ then there is nothing more to prove. If gi'~ ~,)modQ~, then for a suitable choice of basis oo takes values in R~, and satisfies OoJi~ = (~r * ) ~r - Reducing modulo Q~ yields the representation 9Q'. Property (3.4vi) is now immediate. Suppose now that m is a maximal ideal of T~(U, ~). Patching together the pseudo-representations for the various minimal primes Q contained in rn yields a ~l/~?[I~.~ RESIDUALIN REDUCIBLF REPRFSENTATIONS AND MODUI,AR I,'ORMS 41 pseudo-representation p'~'"~ into T~(U, ~"),, (here we have used the fact that T~(U, ~) is reduced) satisfying (3.5) (i) pmod is unramified at primes g ~ np, m~ ~ = T(g) for all g ~ np, (ii) trace p~ ~ .... tJ mod r (iii) detpm (F ob~) = s(e)Nm(e) for all g r np. Let ~ and k be as in w Henceforth #' is the ring of integers of a finite extension of O~ having residue field k. A maximal ideal of T~(U, #:~) is permissible if m A #:: JIG(U)]] is the maximal Zo ~ - i ideal corresponding to the character G(U) ~ Z(U) , k, if m contains T0(p~)- 1 tbr each i = 1, ..., t, and if p~ -~ X | 1. Such a maximal ideal, if it exists, is unique. For this reason we will drop the subscript m from the notation for p,,od whenever m is permissible 9 Suppose that m is a permissible maximal ideal of T~(U, 6~'). The ring T~(U, ~')m is an algebra over A~ "~4a 1 + Y!'~ ~ ~ T (i: and 1 + T i D , det prnod(~). The - .I Yj 9 r. r. homomorphism A' C ~ A,< determined by 1 + Xi~ . (1 + Tj)P~e(~ 7#) is compatible with the A~.-algebra structure of To~(U, ~)m and makes A~ a free A~-module of rank r = 2r i. Consequently, we obtain the Ibllowing lemma. Lemma 3.10. -- If m is a permissible maximal ideal OfT~(U, ~'), then (i) T~(U, G)m is a torsion-flee, finite A~ -algebra, (ii) Jbr U satisfying the hypotheses of Proposition 3.3, M~(U)m and M~(U)m are .free Ar -modules of equal rank 9 Let Z be the places of F for which U,: 3(GLT(~F.,.) together with vl,..., v~. If m is a permissible maximal ideal of T~(U, ~'), then it is easy to see that pmoa is a pseudo-deformation of type-6.~ p~ = ( ;G., Z). Consequently, there is a map (3.6) R~ p~ , T~(U, ~:~)m inducing p m~ Lemma 3.11. --- Suppose that m is a permissible maximal ideal of T~(U, c<~'). /f S is any finite set of primes of F containing all those for which Ut ~ GL.,(~'v,e), then the ring T~(U, -#~)m is generated over he': by the operators { T(e), S(e)" e r S }. -- Let T s C T~(U, ~)m be the subring generated over A~ by Proof. 9 g)ES}. Note that T s is a local, complete A~-algebra 9 Let p m~ = { T(e), s(e) { x(o, t) } and let Z = S U { v~, ..., v, }. The pseudo-representation p m~ factors GaI(Fx/F). Since Gal(Fx/I:) is topologically generated by through Z} and since trace p m~ and detp rand are continuous maps, it follows { Frobt : 42 C.M. SKINNER, AJ. WILES that T s contains trace prnod(ty) and detpm~ for every ~ E Gal(F/F). It remains to show that T s contains T~' tbr eachy E (~'~,QZp) � as well as T0(p/) for each i = 1, ..., t. Let gi E Di be such that )~(gi) :~ i. Let 0~i and 13i E T s be the roots of the polynomial X 2- trace p'n~ + detpm~ with a~ reducing to 1 modulo m. One then has Ty = ([~i- oti) ([~i trace pm~ -- trace pm~ E T S ~ x ab fory E .~ v, ,~,, where % E Di is anv lift of the element of Gal(Fo/F~) corresponding to y via local reciprocity. Similarly, one also has T0(Pi) = ([3i- 0~i) (~i trace ~3nl~ -- trace 9m~ E T s where ~i E Di is any lift of an element of Gal(~ b/F,:) corresponding to (X~P'~). These expressions for 'F~, and T0(pi) can be checked for each Po., Q a minimal prime of T~(U, ~)m, using (3.4) 9 [] , T~,(U, ~)~ /s Corollary 3.12. - /f V C_ U, then the natural map T~], ~')., surjective. Corollary 3.13. The map (3.6) is surjective. We conclude this subsection with a t~w results about the "level" of a prime of T~(U, ~'). The first of these is a generalization of Carayol's n~, _~ n(~,:) result (see [C]). Indeed, its proof boils down to Carayol's result as generalized in [W2]. For w a finite place of F write gw for the prime ideal of ~v associated to w and write A~,, for the Sylow p-subgroup of (.~:' F/gw) � . Proposition 3.14. -Let w ~ p be a finite place of F. Suppose that U C_ GL2(~ v Q Z) /s such that U~, _D {(a b) E GL2(~v ~)" c, a- 1 E g~. } for some s. Given a minimal prime Q c_ T~(U, U) there exists a subgroup V D_ U such that Q is the inverse image of a prime of To~(V, C) and V satisfies (i) /foe ir unramfied at w, then V _D GLe(6"F,w); 9 c E ~,~ ] (ii) f pe /s type a at w, then V D_ I ( ci b) E GL2(6-:' F 9 "') a rood g ~,, E Aw f; is type B at w, then V D I (~i b) (iii) E GL2(r - 1,c E wh e e2 I,', w) 9 a if Pe is the conductor 0f~ = detpoli,,; -t 5- . ~. 9 (iv) if 9(2 is type C at w, then V D_ ,. ,1 EGL2(C~I,',w) a- 1, c E g~ S RESII)UALLY REDU(JIBI,E RI'PRESFNTATIONS AND MOI)UI,AR FORMS 43 Proof. -- Recall that types A, B, and C were defined in w We first claim that for P E .:~'(Q) the representation pp is of the same type at w as 9o. and that if , with condw(,) = gw then PPlI~ -~ ,, with condw(q~') = g~, as well. In light of Corollary 3.9 it then suffices to show that some ~, E ,~f (P) factors through T~(V, ~') for some V as in the statement of the proposition. To prove the claim, first assume that OQ is unramified at w. In this case it is obvious that each 9P, P E .~'(Q), is also unramified at w. This can be seen, for instance, by observing that the pseudo-representation associated to 9Q. is trivial on Iw and hence the same is true of the pseudo-representation associated to Pc- As pp is irreducible, this tbrces pp to be trivial on Iw. Next assume that pe, is type A at w. If P E J~)'(Q), then it is easily deduced that the semisimplification of 9vi~,. is just 9 egt for some character ~ unramified at w. If PPID, were unramified then this would contradict (3.3). Therefore, it must be that OI~ID,,. is ramified, and it follows from the description of its semisimplification that it must be of type A. Now suppose that Po is type B at w. Write 9oJl~ = ~lg?| 1 with ~l of order prime to p and gt2 of p-power order. Note that cond~,,(gtl~?)= max(cond (~), cond (~2)) and that both It./1 and g2 take values in T~(U, C)/Q. It follows that PPI~= -~ (~L~r mod P) | 1. As p ~ P one sees that cond,,(gtl~t? modP) = max(cond (gtl), cond (~)), proving the claim in this case. The remaining case (i.e., po being of type C at w) is proved similarly. Now let P E ~,q~/(Q) and choose ~, E .~,7r Let n be the automorphic representation corresponding to ~.. rib prove that ~, factors through T~(V, ~") for Fl~ some V as in the statement of the proposition we need only show that rc E ..,) t,:. In other words, we need to show that if W;~ is the underlying representation space for =, then W v ~: 0. Let x = | E W~; and let x' w E W~,= be the new vector at w. It Uzr follows from our hypotheses on U~ and the theory of newforms that x' E W=, w. Put y = @ x~ | ~,,. We claim that y is fixed by a subgroup of the desired type. For this we v:t: t0 note that it follows easily from (3.3) that x' w is fixed by a subgroup of GL~(U"v,w) of the desired type necessarily containing Uw. [] As a variant of the above we have the following result. For a place w of F, let (~)',,/gw) � = Aw x A'. (Recall that A w is the p-Sylow subgroup of (~v/gw) � .) Proposition 3.15. -- Let w ~ p be a finite place of F and let U ~,, = { ~c E GL2(~ F,w) " c E g~, ad-~modg~, E A'w}. Suppose that Q c__ To~(U, ~) is a minimal prime such that oczl,,~ (* ,-1 ) with, of p-power ord . Put U' = rI x U;. There exists a minimal z~ w prime Q~ c_ T=(U', ~:) such that 9c2 ~- 9Q' and such that Q and O~ have the same inverse image in T~(U M U', ~). Proof. -- We prove the existence of a minimal prime Q~ c T~(U', ~)') such that Q~ and Q have the same inverse image in T~(U M U', ~,~). Clearly the assertion that 44 C.M. SKINNFR, AJ. WILES 9o~ -~ 9Q will follow from this. Upon replacing U by U N U' and Q by its inverse image in T~(U M U', ~:') we need only show that Q is the inverse image of some minimal prime in T.~(U', ~). By Corollary 3.9 it then suffices to show for each P C .~'(Q) and ll'~ T '~ ~. C ,~_~gS (P) that )~ factors through T~(U', ~'). Fix such a P and ~. Let u C ..,~ ~,~j be the automorphic representation associated to )~. To know that ~ factors through ord ,I l-'a T~(U', ~) it suffices to know that rc C 11 2 (U~). Now suppose that x = | E V~ . U' ;t We will establish the existence of a non-zero vector x' w C V m ~,.. The non-zero vector U' U' y = @ x~ | ~,. will then lie in V~ ~ showing that V~ ~ :~ 0, from which it follows that l: z~ [U ]lOrd/| It '~ g ~ ""2 \"~a]" ,,_ ~ with We now establish the existence of ~,. First we note that ppll ~ (*' ) 00' a character of p-power order. To see this observe that by hypothesis PqlI,, factors through a quotient of Iw of p-power order and detp@i~ = 1. Hence the same is true of the pseudo-representation associated to 9% As the pseudo-representation associated to 9P is obtained by reducing modulo P the one associated to pQ, it follows easily that PPlI,~ factors through a finite quotient of p-power order and that detpvll,, = 1. That P~'I~, has the form asserted is now immediate. It follows from (3.3) that ~,,, is a S: X principal series representation n(gl, g'2) with ILlg2 trivial on ~' v, ~, and each gi trivial on a subgroup of ~'~ x ~,, (~ v, of index a power of p. Let v0 E V~, ~, be the vector corresponding to a new vector of ~w | 00 where 00 is a finite character such that 00]~ v. ~ w _ ~ g~-l I~- � 9 I:, It' h\ It follows that {: ~t)vo = g,(a)g'2(d) for all [c )) E U~,. Thus x' w = v0 is the desired .e / \ 1_,'. vector in V~, w. [] Next we record for later reference the following relations between PQII~, and the subgroups U~. Lemma 3.16. -- Suppose that w ~ p is a finite place of F. Suppose also that QC_ T~(U, ~:) is a minimalprime and that UwD_ {(~ ~):cE g~, amodg'w E Aw} J'br some r >1 1. (i)/f Pal',, ~- ( *~ **2 ) with 001 and 002 of p-power order and 00 non-trivial of order prime to p, and if cond (00) = g~,, then 001 and 002 are trivial. (ii)/f OQII),,, -~ ( E0  ) and if (Nm(w) - 1, p) = 1, then 001ia. is a finite character of order prime to p, and if r = 1, then 001Iz,, = 1. Proof. -- Let R = T~(U, ~')/Q. We first prove (i). The characters 00~, 002 and 00 take values in R, so they may be reduced modulo P for any P E ,~'(Q). We denote these reductions by 001,P,002, p, and 00P, respectively. The reduced characters have the same orders as the corresponding non-reduced characters. Now fax a choice RESII)L'ALIN REDUCIBI,E REPRESENTATIONS AND MOI)UI2kR FORMS 45 of P E ,2Z'(Q) and ~. E .~ (P). Let ~ be the nearly ordinary automorphic representation c~176 t~ )~" It is easy t~ see that PI'[', "~ %'p- ( *P*2, v) from. It follows this description of PPIL, and from (3.3) that cond(n,,,) = condw(qbt, Cp,) p)cond,.(Ol,p ) = (b) gw" cond(Ol,p). However, since by hypothesis x~,. has a vector fixed by { ~ ,1 " c E g,~,, amodgi~ E A~,} it follows that cond(xw)l e2 and that the restriction to Iw of the central character of x~: has order prime to p. From this we deduce that qbl, p and r are both trivial. As 0h,~ and ~2, P have the same order as ~1 and ~2, respectively, the latter are trivial as well. We now prove (ii). Our hypothesis on Nm(w) ensures that ~ takes values in R. As in the proof of (i) we write r for the reduction of ~ modulo P. The character 0Op[i~. has the same order as does {~[I~,. Again, fix a P E ,542'(Q) and a )~ E ,~ (P). Let n be the automorphic representation corresponding to ~.. From the hypotheses on U we find that cond(nw)]g~,, and that the restriction to I~. of the central character of n~ has order prime to p. It is easy to see that PPID,--~ (eOr, *t,* )" It follows from (3.3)that cond(n,,.) = max(g~,,, cond(qb~,) ~) and that the restriction to I~. of the central character of xw is just @ L,." From this, one deduces that Cvilw, and hence OO]L, , has order prime to p. And moreover, if r = 1, then cond(r [g .... hence 0Ov]]~ (and so q~[l~) is trivial. [] We conclude this subsection with a simple observation about twists of the representations pQ. /emma 3.17. - - Suppose that Q c_ p c_ T~(U, ~:) are primes with Q minimal. Let L be the field of fractions OfT~(U, ~)/Q and suppose that R C L is a finite integral extension of T~(U, &~')/Q./f ho " Gal(F/F) , R � is a character of finite order, then there exists primes O~ C p' _C T~(U 71 U1 (cond'~/(qJ)2), (5 :') with O~ minimal and such that PQ' ~- PQ | ho and pp, ~ pp | u?. Here cond~~ denotes the prime-to-p part of the conductor of q~. Proof. -- Let V = U 71Ul(cond(q~) 2) and let R0 = To~(U, u~,')/Q. Let .3~ '(Q) be the set of primes of R extending those in ,~'(Q). For each P E .3~"(Q) we write ,Sref (P) for ,~ (P 71 Ro). Now, for each P E ,~"(Q), let hop = ho mod P. Let n be the product of those primes g such that either glcond*)('e) or Ue :1: GL2(6"'F,g). We claim that tbr each P E ,3~"(Q) there exists a homomorphism zp : T~(V, ~') , R/P such that (3.7) 9 zp(T(g)) = ~I'(g) rood P). ~Fp(Frobt) for all g r np; ,p(S(g)) = (S(g) mod P) 9 Udp(l~robt) 2 . for all g { np ; r = (Sx mod P). ho~,(x) for all x E Z(U); t. (Pi)\- 1 zi,(T0(Pi)) = (T0(Pi)modP) 9 ~FpCLpi ) for i = 1,..., t; "~P('I~r) = (T~' mod P)- hop[y) for all y E r and each i = 1,..., t. 46 C.M. SKINNER, AJ. WILES ord We construct zp as follows. Let )L E ~ (P) and let zc E I12 (U~) be the corresponding automorphic representation. We fix an embedding R/P ~ O~ extending the embedding C L coming from We thus view as taking values in CL � hence in ~x (via the fixed embedding F~--~ ~). Clearly 7t | q~p E II~"d(vb) for some b /> a. Therefore there exists an algebraic prime P~ of Too(V, O) whose corresponding representation is just PP,v -~ Pp | ~Pp- Let zp E o~(P,v) be the homomorphism corresponding to rc | ~Fp. Viewing R/P as an O-subalgebra of O~ as above, we see from the fact that 9Pv ~- PP | ~Fp that zp takes values in R/P and satisfies (3.7). Now consider the map "c 9 Too(V, G) > I-I R/P given by z(t) = Ilzp(t). PE.5~"(O0 It is easily deduced from (3.7) that the image of z is contained in the image of the diagonal embedding R~-+ II R/P. In particular, x determines a homomorphism pc.~2,(Q) z 9 Too(V, ~) , R such that "c(T(g)) = (T(g) mod Q). ~(Frobe) (3.8) for all g ( np "c(S(g)) = (S(g) mod Q). ~2(Frobe). Let O~ = ker('c). By (3.8) we have 9o2 "~ Pc)_ | u?. Moreover, by comparing dimensions one sees that O~ is minimal. Let P l be any prime of R extending p. Let p' be the kernel of the composition Too(V, ~) ~> R , R/pl. Obviously p' D Q~. Also, it follows from (3.8) that Pp' ~- Pp | tF as well. [] 3.4. Eisenstein maximal ideals (existence) In this subsection we establish sufficient conditions for the existence of a permissible ideal of Too(U, ~). We continue to assume that the degree of F is even. An equivalent definition of permissible maximal ideal is a maximal ideal m of Too(U, G) such that (3.9) 9 m V/& [[G(U)]] is the maximal ideal corresponding to the character ~r l a(u) ,z(u) ,k, 9 m contains T0(p/)- 1 for i = 1, ..., t, and 9 m contains T(g) - 1 - ~(Frobe) for each g {p for which Ut = GL2(~F,e). We will also call a maximal ideal m of any T2(Ua, ~) satisfying (3.9) a permissible maximal ideal since any such ideal determines a permissible maximal ideal of Too(U, ~). Clearly, to conclude that Too(U, 8) has a permissible maximal ideal it suffices to show that T2(U~, ~) does for some a (and hence for all sufficiently large a). This we do, provided a certain p-adic L-function is not a unit. Let rt be the prime-to-p part of the conductor of ;(o) -1. For each prime gin let t4t)lrrt and write Ae for the Sylow p-subgroup of (~v/g) X, which we think of as a RESII)UALLY REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 47 subgroup of (~s:v/g~e") � . Define an open compact subgroup U x = 1-I U~ C GL2(~)v@Z) as follows: if gin, U~= {{(:GL,)(~'v,t) :) E GL2(~;v't) c E g~e',, amodg~g)GAg} otherwise. Let Lp(F, s, ~to) be the p-adic L-function associated to ~to (cf. [Co], [D-R]). Let n denote a uniformizer of ~. Proposition 3.I8. -- /f ordn(Lp(F, - 1, Xm)) > 0, then some T2(U~, ~') (and hence T~(U x, ~*)) has a permissible maximal ideal. Proof. -- Let ~r = )~t0 -1. For each integer n /> 2 let Mn(pn, gt) C Mn(pn) be the subspace of modular forms having nebentypus character gt. Define S,(pn, ~) similarly. For a pair of characters ~l and r for which ~)1r = ~ and cond(~l)cond(r let E,(q~l, ~2) E M,(pn, gt) be the Eisenstein series whose associated Dirichlet series is L(F, s, q),)L(F, s-n+ 1, *2) (cf. [Sh]). A complement for the space S,(n, ~) in M,(n, ~) is spanned by the set { E,(r r It is well-known that (3.10) ai(E,(1, V), 0) = 2-dL(F, 1 - n, ~) Nm(ti) '~ where the ai(E,(1, ~/), 0) are the constant terms of the Fourier expansions of E,(1, ~) described in w Let "/be a generator of the Galois group of the cyclotomic Zp-extension of F, and let f= ordp(e(~/) - 1). Now let n = 2 +pf(p - 1)m E Z be so big that there exists a modular form g E M,(np, gt) M M,(np, C) such that (3.11) ai(g, 0)= 2-dNm(ti) ~ , i= 1, ...,h. The existence of such a g for large n is proven in [Ch, w Let E0 = E,(1, ~). It follows from our choice of n and standard facts about p-adic L-functions that ord~(L(F, 1 - n, V)) > 0 as well. Thus E0 E M,(np, ~'). Consider the form f= L(F, 1 -n, gt)g-Eo. By (3.10) and (3.11) this form satisfies ai(f , O) = O, i= 1,...,h. We also have f E M,(np, ~). Let e E ~',(U, (n)71 U(p), ~) be the operator defined in w It is easily checked that E~ = eE0 is the modular form whose associated Dirichlet series is just ~v(s)L:P)(s -- n + 1, ~), where for an ideal a we write L"( 9 ) to mean that the Euler factors at places dividing ct have been removed. In particular, El is an eigenform for the ring T,(U~(n) M U(p), ~':). Let e0 E T,,(U~(n) N U(p), ~:) be the 48 C.M. SKINNER, AJ. WII,ES idempotent associated to the corresponding maximal ideal. The tbrmfcan be expressed as f= F + G with F E Sn(np, ~) and G a linear combination of Eisenstein series, say G = Zc(,)E,,(qb, ,-Iv) , where the sum is over those ~b such that cond(~b), cond(qb--Iv)lnp. It follows that e0G = Zd(~.)E,,(~., ~.-1V) where the sum is now over the unramified characters of p-power order. Since the constant terms off are zero, the same is true of those of e~ Thus e0G must have all constant terms zero (i.e. a~(e0G, 0) = 0 for i = 1, ..., h). Arguing as in the proof of [W3, Proposition 1.6] shows that e0G = 0. It follows that FI = e~)ef is a cusp form. Moreover, since ~IL(F, 1 - n, V) by hypothesis, we have Fl = El modrc. As El ~ 0mode, FI ~= 0. As El is an ordinary eigenform (in the sense of [W2]) and Fl = El ~ 0, it is easily seen that there must be an ordinary newform f such that ~0 f :~ 0. The tbrm ef is a p-stabilized newform in the sense of [W2]. Let 99I be the maximal ideal of the ring of integers of Qr The non-vanishing of e0 f means that the coefficients of the Dirichlet series Ln(ef, s) associated to ef are congruent modulo 9)I to those of n np ~v(s)L (F, s- n + 1 V). By the theory of "A-adic forms" developed in [W2] (see especially [W2, Theorem 1.4.1]) there is some p-stabilized newform~ E S2(np ", V), for some large a, such that the coefficients of L"(~, s) are congruent to those of L"(ef, s) modulo 9N, and hence are congruent to the coefficients of ~v(s)L n ,p (F,s-n+ , 1,V). Being a p-stabilized newform, ~ spans a v-good line in the associated local automorphic representation for each place viP. Thus ~ is an eigenform for the ring of operators T2(Ua, ~) (U = Ut(n)). Let m be the corresponding maximal ideal. We claim that m satisfies (3.9). The second and third properties listed in (3.9) are consequences of the connection between the eigenvalues of the Hecke operators and the coefficients of the Dirichlet series L"(J~, s). The first property listed in (3.9) follows from the t~ct that f, is ordinary (so the operators T~, Y E (~ F @ Zp)x, act trivially) and that J~ E S,,(np a, V) (so S~, x E Z(U), acts via )~0~-t(x)). It remains to show that this maximal ideal occurs in T2(U~, ~). This follows from the fact that n/,, the automorphic representation associated to the p-stabilized vl~ I)C\ newformJ~, is in "-2 ~. This last fact can be seen by considering the possibilities for pp,,lD e at primes g I n (P,) being the algebraic prime of T~(UI(n), ~') corresponding toJ~) and invoking [W2, Theorem 2.1.31 or (3.3). [] 3.5. Some miscellaneous results We keep the conventions of the previous sections. Suppose that U = IIU~, C_ -1 5)~ GL2(C~ v | Z) is a compact open subgroup as usual. In this subsection we consider the effect of altering U at one selected place w. RESIDUALIN REDUCIBLE REPRESEN'FATIONS AND MODULAR FORMS 49 Let w r p be a place of F such that U~ = GL2(~F,w). Let A~ be the Sylow p-subgroup of (~'v/g~,) x and let A 2 be a complementary subgroup (so (~F/G) � ~_ Aw x A'~,,). Put 9 c E f~,} U' w = {(~ ~) E GL2(~r,,) and : ad-lmodg~ E A'}. Put also U'=U'-IIu~ and U"=U'~',.I-IU~;. There is a natural isomorphism (3.12) U'o/U~ ~ , a~ given by (~ ~) ~-* (image of (ad-l)~, in a~). Recall that U'~ = U' N U(p ~) and similarly for U'a. The group U'~ acts on S~(U~') with g E U'~ acting via thc Hccke operator '" "" - 1, Till I It [tGg tJ~l. This action clearly factors through the quotient U~/U~ and hence determines via the isomorphism (3.12) an action of A~ on Sk(U;'). Under the Jacquet- Langlands correspondence (see w this action is compatible with usual action of Uta/Uta , on {f: Dx\GD(Ad)/U'~ ' , (3 } = H~ (3) given by (gf)(x) =f(xg) for , H 0 'q g E Ua. This action clearly stabilizes (X(U~,, Z) and hence we obtain an action of" H~ r"~ R) for any Z-module R. It is straight-forward to check that the action Aw on ~- -\ ~ a/, of Aw commutes with that of G(U~') and the Hecke operators T0(p/), i= 1, 9 t, and T(g) and S(g) for primes g Cpw tbr which U, = GL2(CF, e ). Moreover, the action of A~ is compatible with varying a. We also have that (3.13) 0 -, 0 ,, H (X(U~), R) A~ = U (X(U~), R). If every element of U'/F x ffl U' acts without fixed points on D� then much more is true, as the following lemma shows 9 Lemma 3.19. -- If each element of U'/F � n U' acts without fixed points on D � \G D (At) then (i) ModU") and M*~(U") are fiee A~ ~&,,]l-modules, (ii) M~(U )a,,; = M~(U') and + " ,, Moo( U )a,~ = Moo(U + ' ). Proof. --By (3.13) we have 0 it #H (X(Ua) , 4"~/rQG = #H0(X(UI{), (~)/gn)G = #H~ ,,, 4"47 /r('). 50 C.M. SKINNER, AJ. WILES On the other hand, it follows from the hypothesis on U'/F � N U' that #X(U~') = #Aw #XCU'~). Combining these observations we find that H~ .~.'/n") is a free C/g"[[Aw]]-module. The lemma follows from this, the definitions of M,,~(U") and + II M~(U ), and Proposition 3.3. [] It is a consequence of the lemma that many characteristic p primes of T~(U", (2:) (i.e., primes containing p) come from primes of T~(U', (2:'). We state this more precisely in the next proposition. Note in particular that we are not assuming anything about U'/U' N F � . Suppose that p _C T~(U", ~') is a prime such that 9 pep, 9 det 9p = Z, 9 Pplr~,.--~ (~z ~2" ) with (~l/~t2)l,, having infinite order for some vlp, 9 9p is irreducible but not dihedral (i.e., not induced from a one-dimensional representation over a quadratic extension). Proposition 3.20. -- The prime p is the inverse image of a prime 0fT~(U', ~"). Proof. - Choose o E Iw such that co(o) = 1, detpp(~ ) = 1, and pp((y) has infinite order. Such a ~ exists by the hypothesis on 9p]l~.. We claim that there exists r E Gal(F/F) for which c0(r) :~ 1 and an n such that 9p(gnr) has infinite order. To see ,r this, choose a basis of 9p for which 9p(o) = (a u-1 ). If pp(r) E ~(* *)} for each "c such that c0(r) ~: 1, then it would follow that 9p is dihedral. Therefore there exists some (o('c0) 1, for which pp(r0) = ( ~ b) with either a ~: 0 or d ~: 0. Suppose now that "go, 9p(o"~0) always has finite order. In this case, the roots of X 2- (oCa + a-"d)X + )C('c0) are roots of unity lying in some quadratic extension of the field of fractions of To.(U", ~')/p. As there are only finitely many such roots of unity, there are only finitely many possible values for o~"a + a-"d, which is easily seen to be absurd. Let s be the set of finite places v such that U'/ :~ GL,)((2~F, ,:). Fix now an no for which pp(c"~'c0) has infinite order. Let Frob e E Gal(F~-/F) be a Frobenius element such that g { 6 is unramified in F, det pp(Frobe) = )~(r0), m(Frobg) = (o(r0) :~ 1, and pp(Frobg) has infinite order. Such a prime g can be found by choosing a Frob t sufficiently close to o'er0 in GaI(Fz/F). Put V' = U' 71 U(g) and V" = U" N U(g). The prime p of To~(U", ~:') determines a prime of T~(V", ~:') (the inverse image of p) which we also denote by p. We now claim that p comes from T~(V', 6:::). "lb see this, note that by Lemma 3.5 and Lemma 3.19, M~(V") is a free A~,. [lAw]l-module. Let T C EndA) ~ (M~(V")) be the ring generated by As. and T~(V", 6'). This is a finite RESIDLTALLY REI)UCIBI,E REPRESEN'IATIONS ANI) MOI)UI,AR FORMS 51 integral extension of T~(V", ~J'). Let Pl be an extension of p to T. As p 9 p, it is clear that { 8 - 1 : 8 E A~: } C Pl. As Mec(V", ~9) is a thithlhl T-module, we have that Fit~/p, (M~(V", ~)/p,) = 0. Since T/pl is a domain, it follows that it acts faithfully on M,,c(V", C)/pl. Put B = T.~(V", C)/p C T/p~. It follows that B acts faithfully on Moo0g", ~:')/P~. On the other hand, M~(V", ~r is a quotient of M~(V", ~;)a,, = M~(V', ~') by Lemma 3.19(ii). As the action of T~(V", eg) on M~(V', 6"') is via the natural map T~(V", ~") -+ TooOg', ~:) we have T~(V", ~)/p = T~(V', ~')/im(p) which proves the claim. Write P2 for the corresponding prime of T~0g' , ~'~) (so P2 = im(p)). Our final claim, which proves the proposition, is that P2 is the inverse image of a prime of To~(U', ~'). Let Q c p~ be a minimal prime of T~(V', ~). [t suffices to prove that Q. comes from a prime of T~o(U', ~'~). Consider po_[D e . As f does not divide p, p { (Nm(g) - 1), and pp., --~ pp is unramified at g, there are three possibilities for POJDe : (i) pQID e is unramified at f, (ii) polD e is of type A, (iii) polD e is of type C. If the first possibility holds, then the desired claim is a consequence of Proposition 3.14. We will now show that the second and third possibilities cannot occur. If P@Dg were of type A, then the eigenvalues of pQ(cg) (~e a lift of Frob~), say ~ and 13, would 0~ satisfy ~ = e(g) or -- = e(g) -~ The same would then be true of Pp~(~t). However, since P9 contains p and detpp., =)~ it would follow that the eigenvalues of ppe(~) would have finite order, contradicting our choice of g. Similarly, if Po were of type C at s then trace Po(~e) = 0, but we have chosen f so that trace Pp(~e) ~: 0. ]'his final contradiction completes the proof of the proposition. [] We now assume that T~(U, e'~) has a permissible maximal ideal. The same is then true of T~(U', eg:) and T~(U", ~5: ). We also assume that the place w satisfies z(Frob ) = 1 as well as (~(Frobw) = 1. Let ~,,. ~ I,,. be a generator of the p-part of tame inertia. We identify (~w with ~,~.� (m,,) an element of ~ F, ~ via local reciproci~. The element 8w = , generates &,, via (3.12). Recall that both &,. and T~(U", ~'~)m act on the module M~(U")m. Lemma 3.21. - The element trace pm~ E T~(U", ~')~, acts on M~.(U")~ via 8w+8~ ~. Remark 3.22. -- Since both Toc(U", ~'~)m and A~ are contained in Ende 7 (Mo~(U")m), the lemma identifies &, + 8~,i I with an element of T~(U", ~s')m. Moreover, this identi- fication behaves well with respect to varying U. 52 C.M. SKINNER, AJ. WILES Proof. -- Let V _C eH~ ~i:)m | ~ be an eigenspaee for the action of T~nt", U). The space V is stable under zXw. Under the Jacquet-Langlands correspondence V is identified with a subspace of S2(U~'). Let ~." T~(U", ~;) , O.~ be the homomorphism gi~fing the action of T~(U", C) on V. Clearly ~ factors through l-lOrd ~ ttr Tg(U~', ~i) and hence is an algebraic homomorphism of weight 2. Let 7t E ,) (U~) be the corresponding automorphic representation. The space V is identified with a subspace of V~ 9 We now determine the action of 8~ on V, which is via ~ ( ~' ] ) 9 First we note that ~, cannot be supercuspidal. To see this, let P = ker0v ). The prime P is clearly contained in m. It then follows from (3.3) that if ~. is supercuspidal then Ppll),,, is type C, but clearly this can only occur if x(Frobw) = -1 contradicting our assumptions on w. Now suppose that ~. = 9(its, It~l J~ -1) is a special representation. It follows from the definition of U~. that gl is unramified. From this we find that n~,(~w 1 ) = 1. If ~,. "~_ ~(gl, It,.,)is a principal series representation then it must be \ / that It! and It7 are tamely ramified and ItlIt~ is unramified. Moreover, the action of (oo) 1 on V~.~, is by either gt(~,) or It2(cw) = gTl(gw). Now it follows from (3.3) that if n is either a principle series representation or a special representation then pl,(~,) = (gl(~,) ~]-~(~w) * ) . Thus we find that trace pp(Gu, ) - Itl (cu,) + ItTl(G,r) -= 8,,~ + 821 [] 3.6. The rings T~ and T~ ~ In this subsection we associate Hecke rings to various deformation data. Essentially this is done by first defining a suitable open compact subgroup of GI,,(6~v | Z) and then localizing the corresponding Hecke ring at a permissible maximal ideal. To ensure the existence of such a maximal ideal we henceforth assume that Lp(F, - 1, Xm) is integral but not a unit (see Proposition 3.18). We are, of course, also assuming that the degree of F is even. Suppose that ~ Q = (~)~, Z, c, .//g)e is an (augmented) deformation datum 9 As in the previous subsections, for each finite place w we write gw for the prime ideal of F corresponding to w and we write A~, for the Sylow p-subgroup of (~;'v/gw) � which we identify with a subgroup of ((J ~" v/g,,) r � for any r/> I. We also write A t for ~ x __ t a complementary subgroup of (8 v/g~,) (so (~'F/g~) � ~ &L, X ZX~,). We define r(w) by RESIDUALIN RE1)UCIBI,E REPRESENqTkTIONS AND MODULAR FORMS 53 r(w) g~. ][cond(z~,'). 'Ore define a subgroup U~,~ = 1-I U~ C GL.~(~v | Z') by putting if w ~(Z\~') U Q if w ~ Z\(.~, 7~ U ,/gg) ]ii'" i ,> > U~ q,. = 'l[ ~ ~ m~ ew". max(l , <w) ~ )/~w~ Aw'} , if w ~ .~g. ifw6 Q. {(c a :)~GL~(~',:,w):C~g~., ad-'modgw~A:} Let m be a permissible maximal ideal of To,(U~ ,.t' ~)" Put T~ o_ = T~(U~ a, r mod We define T~ to be T~ 0. We write 9~Q for the pseudo-representation into T~Q described in (3.5). This is in fact a pseudo-deformation of type-C_Z~, 6.~Z~. = (~, Z)Q, and we write n~ : R~ p., ~ T~ and n~ct " R~ ~ ~ T~o_ for the corresponding maps Suppose now that 6_~Q. = (~, Z, c, .-.tg)O - is an augmented deformation datum. rain of T~a. This quotient At times it will be necessary to work with a quotient T~Q is defined as follows. As in w let L~/F be the maximal abelian p-extension of F unramified away from the places in Z\M. Let Gal(L~/F) _~ H~ (9 NN be the decomposition fixed in w (Ny is the torsion subgroup). It is a consequence of our definition of U~ t that if q C_ T~ z is a minimal prime, then ~-~ detgq factors through Gal(L~/F). Let ,.//g(~-~e) be the set of minimal primes q of T~o and let ,_/~min(~..@yQ) min be the subset of those q for which (%- ~ detpq)lNz is trivial. Define T~o " by rnim __ T~Q T~Q/ A q" For this defmition to make sense we must show that dgmi~(6.~f@ :~ 0. TO this end, fix another decomposition Gal(L~/F) ~ M~ x N~ with M~ the free Zp-summand 1 mod __ generated by Tl,-.., 78F" (For the definition of the Ti's see w Write aet 9~ Q O-W. with W trivial on N~ and O trivial on M~. Let ~ be a square root of O (i.e., qi-' = O). It follows from Lemma 3.17 and from the definition of U~q that given any q C ,//~ (6_~e) there exists some qo E ~r (6~e) such that 9q. ~- On | ~-~ Clearly ~-~-detpq . is trivial on Nx, and so q,~ C ./~min(~@. This proves that/~min((~Q) is non-empty. GL2(~TF'w):Ca 54 C.M. SKINNER, AJ. WILES rain We now relate T~o - to T~ a more directly. Let I~/F be the splitting field of O. A priori, Gal(Le/F) is a quotient of N~. We claim that Gal(Le/F) ~ N~. To see this, let { 9 N~ , ~� be any character. Extend { to a character of Gal(F/F) by first setting it to be trivial on M~ and then composing with the projection of Gal(F/F) onto Gal(L~/F). Choose q E .~,gmm(~_~g, @ and P E '2g~'(q). By Lemma 3.17 there is an algebraic prime P{ of T~ Q such that Pp~ "~ PP | {. By the choice of q, ~-l det Pe is trivial on N~Q, whence ~-l -detpe~ -~ ~2. It follows that {2 = OmodPg. As ~2 can be any character of N~, | has trivial kernel. This proves the claim. Now let X(~'.~) be the group of O_,r215 characters of Ng, which we view as characters of Gal(F/F) that factor through Gal(Le/~. For each q E ./Ig(~J)let R(q) = T~o/q and let L(q) be the field of fractions of R(q). We identify Qp with an ~'-subalgebra of L(q). In this way we may view each ~ E as taking values in L(q). }'or each q E ~//~(66ge) and ~ E X(~) let R(q, ~) be the subring of L(q) generated by R(q) and the values of {. This is again a complete local Noetherian domain. By Lemma 3.17 there is a prime q~ E ./;/g)(~e) such that pq~ --~ pq | ~. We next claim that the set ,//~g' = { qg'q E ,//gmin(6-~./Q), ~ E X(~ r) } is just ,/l,g(~Q). For let q E ,/r (~@, and let { E X(~) be the unique character such that 9 = {mod q. Let q' = q~ ~. Clearly q' E ./~g ram(D@. Also, q~ = q since pq~ ~ pq, (x) ~ '~ pq @ ~-1 @ ~ = pq. This proves the claim. Given a prime q E J~gmi~((-g2ge) and a character { E X(~) we have used that Lemma 3.17 ensures that there is a prime q~ E ./r such that 9% -~ Pq | {. However, more is true. It was shown in the proof of Lemma 3.17 that there is a homomorphism x(q, ~) : T~ a ~ R(q, {) whose kernel is qg and such that (T(e)) = (T(g) mod q). ~(Frobe) (3.14) z(q, (S(e)) = (S(g)mod q). {2(Frobt) for an primes g ~ ZuQ. There is also a homomorphism ,(q, ~) 9 T~e~m~ | ~::[N~] , R(q, ~) such that 0(q, ~) (T(g) | Frobr) = (T(e) mod q)- ~(Frobr) (3.15) O(q, ~) (S(g) @ Frobr) = (S(g)rood q). ~9(Frob,). Now define "c "T~/Q ~ I-I R(q, ~) qE,IL mi.(~ @ X(C_2~7,@ RESIDUAl,IN REI)UCIBLE REPRESENTATIONS AND MODULAR F()RMS 55 and 9 T rnin a(q, , II by "c : 1-I "c(q, ~) and * = II *(q, {), q,~ q,~ respectively. It follows from (3.14) and (3.15) that im(,) = im(r from which one deduces the following proposition. (Note that O(Frob) and ~(Frob~) are mapped by each "~(q, ~) to ~9(Frob~) and ~(Frobr) , respectively.) Proposition 3.23. There is an isomorphism 0fA~ min -- -algebras T~ Q| ~; [N~ ] ~ T~ Q such that T(g) | n, , T(g). r 1 9 n) and S(g) | n, , S(g)- 19(Frob~ -l 9 n). nin Corollary 3.24. -- T~Q is a finite, torsion-flee A~;-algebra. rain Lernma 3.25. -- Under the isomorphism in Proposition 3.23 the element 5~, + 5-w I E Tg Q maps to 8~ + 8~-~ E T ~ Q. We now define a T~ Q-module M~Q for each deformation datum c.5~Q. The obvious choice for M~Q is M.~(U~ Q)m, where m is the permissible maximal ideal of T~(U~ ~). Itowever, for technical reasons we find it better to define M~ Q to be min Mg (~ = Mo~(U~ o)m, min ~ umin where U~ Q I-[ ~ o_. w C_ GL2(~' v | 7,) is such that wtec or if w ~ X a max(l r(w)) if w E mi. U~q,w. 1 "amodgw '' GA~ otherwise. c g E GL2(~F,~) " c = 0modg2~, min The module M~o_ is a T~ o-module (and hence a T~ Q-module by Proposition 3.23) min via the natural map T~ Q -~ T~(U~ o)m. We write ~" R~ ps ~ T~" for the composition of ~ with the canonical surjection T~ -~ ~c9 n. 56 C.M. SKINNER, AJ. WILES 3.7. Duality again We now nmke some important observations concerning the modules introduced in w Fix an open subgroup U C_ GL2(.r F | Z) as in the preceding subsections. We assume that U/U OF � acts freely on DX\GI~(Af). Thus by Proposition 3.3 Mo~(U) and M~(U) are free ~:[[G(U)]]-modules of (the same) finite rank. Let tr(a): C [[G(U~)]] , (9 ~ be the "trace map" given by Zxgg, , X~d (where "id" is the identity element in G(U~)). We have an identification of T~(U, ~')-modules Hom~ ~c,(u,n(ML(U), ~' [[G(Ua)]] ) = Homm ~G(Ua)](eH~ ("~), 6? [[G(Ua)]]) tr(a)o(.) Hom~ (eH~ r 6) ( , )U a = eH~ C) +. Denote by Ka this identification of Home Cw,~.)~(M+(U), ~' [[G(U.)]]) with eH~ 6) +. For b/> a we have a commutative diagram of T~(U, ~]-modules Itom,~ ~c,(bsB(M+~(U), #~ ~G(Ub)~) ~ eH~ ~;)+ 1 ~tr(U,, Ua) ltomr~ IG(v:H(M~(U), ~'llG(Ua)]]) ' eH~ ~?)§ where the left vertical arrow is induced from the natural projection CI[G(Ub) ll ' 6"~ I[G(Ua)]]. We obtain therefore an identification )~o~ 9 Homr~ IG(U)u(M+~(U), ~:[[G(U)~) --~ M~(U) satisfying L~(tm) = t~(m) for all t 9 To~(U, ~"). Recall that there is an isomorphism G(U) ___ (Cv | Zp) � x Z(U) inducing an identification C [[G(U)]] = A~ [[Z0]] with Z0 a finite group 9 Composing K~ with the isomorphism (NI~o(U), A~) ~_ Hom~;, Uc(u)~(M~(U), (~[[G(U) 1 ) I IomA~ + ' + coming from the trace from ~ [[G(U)]] to A~ induces an isomorphism Boo(U) : Homn~ " (M~(U), + A n ) _~ Moo(b). 9 Mo~(V ) is any map compatible with the canonical map Moreover, if q0 M+(U) , + C_flU) ~ CffV) then q~ can be written as q) = lim q)a with % " eH~ 6) RESI1)UALIN REDUCIBI.E REPRESENTATIONS AND MOI)UI,AR FOICMS eH~ 6~), and there is a commutative diagram 13~(U) HomA~: (M+~(U), A~ ) T~ = lim ~a ~ , M~o(V) HOmA/~ (NIL[v), A'~. ) where ~ is the adjoint of % with respect to the pairings ( , )t:~ and ( , )v~- Now suppose that p C_ Too(U, ~) is a prime. Let P = A~: 71 p. It is easily deduced from the above that ~loo(U) induces an identification (3.16) Moo(U)p ~- Horn&, ~ ^' (Moo(U), , A~,p) e<~, 1' of q'~(U, G)p-modules. Recall that we defined in w an injection A~:. ~ AC which identifies A~: with A~ [Z1] for some finite group Z1. Suppose that m is a permissible maximal ideal of T~(U, C). By Lemma 3.10 both Moo(U),, and M+~(IJ)"` are free Ar so composing with the trace map from A~ to A~ induces an isomorphism HomA~: (Moo(U)"`,+ A(;, ) _~ HOmA~ (M~(U)"` ,+ A~, ) of T~(U, ~;~)"`-modules. Combining this with (3.16) yields an isomorphism (M~(U)"`, A#::) (3.17) Mo~(U)"` 2 Homhr + of To~(U, ~;)"`-modules. This will be important in our later computation of various congruences. 3.8. Congruence maps In this subsection we prove a number of results that will be helpful in our analysis of "congruences" between Hecke rings in w As always, U C_ GL~(6~F | Z.) is a compact open subgroup such that U = FIUw and Uo(n) D U D U(n) for some n. Let w~,p be such that U~, = GL2(C~?~v,~), let g = g~, and let X= ~(t) be as in the definition of T(f). For any f: GD(Af) ,R (R an .r (Ixf)(g)=f (g(l ~.)). Let V = U n U0(e). Consider the map ~1 " H~ R) 2 ) H~ R) given by ~l(f,g) =f+ txg. The following is the analog of Ihara's Lemma (cf [Ri]) in our setting. ~)q) - 1 - Nm(q) for an), prime ideal q of F that splits complet@ in the ray class field of conductor n . oo. C.M. SKINNER, AJ. WILES Proof. Our proof of this lemma is a straight-forward generalization of [DT, Lemma 2, p. 445]. Put 8 = (1 x-l). Suppose that (f,J?) E ker(~l). We first claim that f(gu) = f(g) for all u E &~GLg(d;v.~.)& This is an easy calculation: if /2 ---- 8-1Z/l" 8 E 8-1GL2(6"F,w)8, then f(g.) = -k(gu8 = -s *.') = -s =A(g). As SL2((~'~ F, w) and &ISL2(~rF, w)8 generate SL2(Fw) it follows that (3.18) S(gu) =f(g) for ail u E U. SL2(F~,,). Now let q be a prime that splits in the ray class field of conductor n. (x~. It follows from class field theory that such a prime has a uniformizer n E F that is totally positive and satisfies x = 1 mod n. Suppose now that 7 E GD(Aj) is any element such that Vl)(y) = x-l. For any g E GD(Af), &)g,/g-1 E GIl)(Af), where GI] C_ G D is the kernel of the reduced norm vi) and 80 E D � is such that VD(80) = X. Such a 8o exists as is totally positive (cf. [We, XI, w Proposition 3]). As G D is a twisted form of SL2 for D .- 8tglAg- 1 which G l (F~) = SL2(Fw), it follows from strong approximation that 80gyg -1 for some 8' E D � and u C U. SL2(F,,,). We have then by (3.18) that (3.19) f (g),) =f(8o'8'~) =f(gu) =f(g). Nm(q)+ 1 VD(gi) = ~, it follows from (3.19) and the definition X(q) i__[Jl Ug~ of [U(' k(q))Ul that [U(1 k(q) )Ulf =(1 + Nm(q))f. The lemma follows. [] ,A Now put U @ = U 71UI(U), r)0. Lemma 3.27. -- For r >>. 1 the sequence , H~ R) 2 ~, H~ R), H~ ('- ')), R) = o~f +f2, is exact. with 8(f) = (f, - co f) and 7(f ,J)) Proof. - To establish exactness, it suffices to prove that if (f ,~) is in the kernel of 7 thenf E H~ R). For any functionf : GD(Aj)~ R put c~-lf(g)=f (g(1 ~-1 )). Suppose that , = = -0~ ~. Now observe (f j~) is in the kernel of 7. As czf -J~ we also have f -1 RFSII)UALIN RI"DUCIBLE REPRESENTATIONS AND MODUI2kR FOILMS 59 t'_ that (x-IJ~(gu) = (z-lf)(g) for all uE U'= ((: :) E U:a-1 E g',cEs beg, It follows that f E H~ R) where W is the subgroup of U generated by U (~) and U'. This subgroup is just U (~-1~. [] Now consider the map {: H~ R) 3 , H~ R) given by {(f, J~, A) =f + (zJ~ + a2j~. As a consequence of Lemmas 3.26 and 3.27 we obtain the following. )~(,.) U - 1 - Nrn(r)for any Lemma 3.28.- The kernel of { is annihilated by IU( 1 ) ] prime ideal r of F that splits completely in the ray class field of conductor n . oo. Proof. - We can write ~ as the composite H~ R) ~ ~) H~ R)4~'~ ~ H~ R) 2 ~)H~ R) where ~(f, ~, .~) = (0, .)~, f, if2). It follows from Lemma 3.27 that {f, O, O, -f} C H~ R) 4 surjects onto the kernel of 7. If ~(f, J~, J~) E ker(7 o (~, if) ~t), then there exists somef E H~ R) such that (-f, J~, f, ff_9+f) E ker(~l O~i). Therefore by Lemma 3.26, f, J~, f, J~ +f are annihilated by the operators in question. This proves the lemma. [] We conclude our discussion of }~ongruence maps with an important application of Lemma 3.26. Let U C GL,)(~r | Z) be as at the start of this subsection, only now we assume that (n, p) = 1. Suppose that p C T~(U, ~:') is a prime such that *pEp 9 9p is irreducible and not dihedral. For simplicity we shall also assume that 9 p is contained in a permissible maximal ideal. Suppose that g is a prime ideal of F such that * gr 9 p ~ (Nm(g) -- 1) 9 the ratio of the eigenvalues of pp(Frobe) does not equal Nm(g) or Nm(g) -l. Put U (~ = U N U0(g) and U (t) = U N Lrj (~). Lemma 3.29. (i) Too(U (l), ~")p --~ Too(U (~ (9~)p --~ To~(U, ~)p. To~(U 0), (r Proof. -- I,et m be the permissible maximal ideal of T~(U, (~) containing p. Write m for the inverse image of this maximal ideal in _~T n~l(0,, ~*) and Too(U (1) , C). 60 C.M. SKINNI:R, AJ. WILES By Corollary 3.12 we have surjections T~(U ''), ~)m -" To~(U '~ 6~)m --~ To~(U, ~').,. To prove part (i) it suffices to show that every minimal prime of T~(U (l' , ~)m contained in p is the inverse image of a prime of T~(U, ~")~. Let Q c_ T~(U (1), ~i~),, be such a prime. An analysis of the possibilities for pQ.[D~ shows that po must be unramified at g. It then follows from Proposition 3.14 that Q is the inverse image of a prime of 0 ~1~ We now prove part (ii). For each a > 0, let ~ 9 eH~ C) 2 ,eH CX(U'~ '), &?) be given by ~a(f,g) =f+ O~g where cz is as in Lemma 3.26. Let ~ = lim{~ 9 + 2 + (V M~(U) ~ M~(U '). Also for each a > 0, let I~ = ker{ --2/~'1" r,.(o), ~) > T2(Ua, ~) }. Then I = lim I, C T~(U ~~ , ~) is just ker{ T~(U '~ , ~') > T~(U, ~'~) }. We claim that (3.20) + 2 Mo~(U)p = ML(U(~ For this we note that M+~(U(~ = limeH~ C)[I~]. Recall that we have fixed an identification of O~ with C (see w The map ~ extends to a map ~@e~J 13 : eH~ 13)2 ~ eH~ 13). By the Jacquet-Langlands correspondence (see w we have T2(Ua, ~')-equivariant isomorphisms U '0) Ua eH~ 13) _~ (~ V~ ~ and eH~ 13) ~_ (~ V~ . ~ord _ (0) ~ordf... ~CII 2 (tJ a ) ~tell 2 ,to a) u(O) a 1-l~ T It is easy to see that V~ [Ia] ~: 0 if and only ifnE..2 ~'~). On the other hand, if l-I~ I " U(0) [Ja lJa nE 2 ~'~), thenV~ ~ =V~ +a(V~),whence eH~ C) [I~] = im({~ | C). Let K be the field of fractions of ~. It follows that (3.21) eH~ K) [I~] = ilTl(~ a @ K). Now consider the commutative diagram lim ~a lirneHO(X(Ua), ~),, a lim eHO(X(U],)) ' C)[Ia] , C , 0 1 1 0 , lim im({a @ K) ~, lim eH~ K)[Ia] , 0 , 0 RESIDUALLY REDUCIBI.E REPRESEN'Ea.TIONS ANI) MOI)UIAR FORMS 61 where the vertical arrows are the natural ones. Applying the snake lemma we find that C embeds into a quotient of limker(~a Q K/~?). Now, for each a > 0 let F~ be the ray class field of conductor gnp a 9 oo. Let F~ = UFa. Let F' be the maximal extension of F unramified away from places dividing gnp. ec. Any element ~ E Gal(F'/F~) is the limit of a sequence of Frobenii { Frob,, } with r~ splitting completely in F~. It follows that trace p,,(o) is the limit of the sequence { T(r~)} and ~(~) is the limit of { Nm(r~)}. It then follows from Lemma 3.26 that trace pm(o )- 1-~(o) annihilates Cm for all ~ C Gal(F'/F~). Thus if Cp ~: 0, then it must be that trace Pm(~)- 1 - e(o) is in p for all o E Gal(F/F~). It is easily deduced from this that PlGaI(-r is reducible, and hence 9p is either reducible or dihedral, contradicting our assumptions on pp. Therefore Cp = 0. The same argument shows that ker({)p = 0. This proves (3.20). It follows from part (i) that M+oo~U%Jp = M+ (U0)p[i] = Moo( U+ 0)[i]p, whence (3.22) + (0) + 2 Mo~(U )p ~- M~(U)p. We next prove that (3.23) + (l) Mo~(W )0 "~ M~(W(~ 9 For this we note that M+(U (~ = M+(U('~)[S(g) - 1]. By part (i), 1~/I + tl T(1)~ + (1) + -(1) ...~,._. ,p = M~(U )p[S(g)- 1] = M~o(U )[S(g)- 1]p, [] from which (3.23) follows. A similar argument shows that M~(U(~)p -~ M~(U)p. 4. The Theorems 4.1. Pro-modularity and primes of R~ We assume throughout w that F, ?~, and k are as in w and that the degree of F is even unless indicated otherwise. In this subsection we also assume that Lp(F, - 1, ~o) is not a unit (so T.~ exists for any 5~r). Suppose that ~ = (~, Z, c, J/g) is a deformation datum for F. Let 6_c_~m = (G, Y.). Let q be a prime of R~. There is a map % : R.~ p~ ~ R~/q corresponding to the pseudo-deformation associated to P~ mod q. The prime q is pro-modular if % factors through r~ : R~ps ~ T~. That is, q is pro-modular if there is a homomorphism 0q :T~ ~ R~/q such that (4.1) (pq = Oq 0 /1~_~ . 62 C.M. SKINNER, AJ. WII,ES (Throughout this section, if a deformation datum ~ = (eS', Z, c, ,l/g) is ,given, then ~vs will denote the pseudo-datum c~_ZJv~ = (~,Z).) Note that in (4.1) 0q(T(g)) = trace P~ (Frobt) mod q for all g 9~ Y.. Similarly, a detbrmation p : Gal (Fx/F) , GL9(A ) of type-~ is a pro-modular deformation if the kernel of the corresponding map R~ , A is a pro-modular prime. It is immediate from the above definition that if q is a pro-modular prime of R~ and if p D q is another prime ideal, then p is also pro-modular. In particular, if a minimal prime of R~ is pro-modular, then so is every prime ideal on the corresponding irreducible component of spec(R~ ). In this case we say that the component is pro-modular. 4.2. Good data and properties (P1) and (P2) Our primary goal is to show that for certain "good" deformation data ~ the components of spec(R~ ) are all pro-modular provided the data have certain properties (labeled (P1) and (P2) below). In this subsection we describe these "good" data and the relevant properties. Let ~ = (~;, Z, c, ,~g) be a deformation datum for F. The pair (F, ~) is good if 9 the degree d of F is even 9 L~(F, - 1, )~t0) E ~' but I,p(F, - 1, )~co) ~ C � 9 d > 2 + 6v + 8. (#E + dim k H~)(F, k) ) 9 for each vilp the degree d~. i of F~. i over Qp satisfies d~ > 2 + 2t + 7 9 (#Z + dimx HL~(F, k) ) 9 if PclI., ~ 1 and w ~p, then either )~[i w z~ 1 or )~[t),~ = 1. As before, t = #?J), where ;~ = {vi} is the set of places of F over p, X~ is the set of finite places at which Z is ramified together with .~/~, and I'(~S I H~,(F, k)= ker{H~(F~,/F, k(X-t))---+ t~H (D,,,, k(z-')) }. i=1 Note that if ~ is good and if ~_~' = (~", E', c, ,//g') is another datum with s C_ Y~, then (F, ~') is also good. However, being good does not behave well with respect to change of fields, meaning that if (F, c~) is good and if L/F is permissible for ~ (as defined before Remark 2.1), then it can happen that (L, c-,~l,) is not good. On the other hand, it can also happen that (F, .~) is not good but (L, ~.~SL) is. This will be a key ingredient in our reduction in w of Theorems A and B to the Main Theorem. Let 6_~ = (t~,~', y~, c, .~d/g) be a deformation datum for F. Let p be a dimension one prime of T~. Let Pp be the representation described in w Let A be the integral closure of T~/p in its field of fractions K. If 9p is irreducible, then Lemma 2.13 RESIDUAI,LY REDU('IBLE RFPRESENTATIONS ANI) MODULAR FORMS 63 associates to pp a representation p : Gal (Fz/F) , GLg(A) such that 9 | K --~ 9p and 9 is a deformation of some 9c' for some cocycle 0 ~: c'C HI(Fz/F, k'(z-l)), k ~ some finite extension of k. We claim that c' is admissible and that 9 is a deformation of type- (~', E, c', 0), where ~' has residue field k ~. To see this, let vi be one of the places over p. Choose c~i E Di such that Z(ai) ~: 1 and choose a basis for 9 such that 9(~i) = ( c~ ) with ~ modmA = )~(Ci). /ks p | K "~ 9p it follows from (3.4) that with respect to this basis either 9[Di is split, 9[Di is non-split and 9[Di = ('1 ,2 * ) , or 9[Di is non-split and ,~) If 91I~, is split then clearly 0 resz.(c' ) E H~(Di, k'(~-l)) and 9[D~ satisfies the desired criteria. If PlDi = ( ,1 02" )' then ~,~()~i)modm,_ . = ?~(~'~i) ~: 1 . (Here ~'~i is the uniformizer of ~i,, ~i chosen for the definition of T0(p/)-- see w However, as 9 | K _~ pp, it follows from (3.4) that T0(pi)modp = ~20~:~) and by the permissibility of the maximal ideal of T~, T0(Pi)modmA = 1. This contradiction shows that if PIp, is non-split, then 91,)~ = (**l ,2) with *~modmA = Z. One sees immediately that res (c') = 0 and that PlDs satisfies the desired hypotheses. Therefore c' is admissible and p is a deformation of type-((9 ~', Y,, c', l~). We say that the prime p is nice for ~:~ if 9 p is a dimension one prime of T~, 9 9p is irreducible, 9 p is the inverse image of a prime of T~i (where c~ is the deformation datum defined in w 9 d is a scalar multiple of c, 9 some conjugate of 9 is a nice deformation of type-(~', 2, c, ,~fg) in the sense of w A prime p of R~ is good if 9~ modp is nice in the sense of w Such a prime is nice if it is also the inverse image of a pro-modular prime of R~. If p is nice for c,~, then the universality of R~ yields a unique map R~ -----* A inducing a conjugate of p. We denote by p~ the kernel of this map. This is a nice prime. The first of the aforementioned properties of ~ is that if p C_ T~ is any prime that is nice for r (P l) then any prime Q c p~ c R~ is pro-modular. The second important property of ~ is that there exists a pro-modular prime of R~ whose I c (P2) corresponding detbrmation is nice in the sense of w 4.3. The key proposition The following proposition is the key ingredient in our proof of the Main Theorem. 64 (3.M. SKINNER, AJ. WILES Proposition 4.1. -- Let ~ be a deformation datum Jbr F. /f (F, (~) is good, and if (P l) and (P2) hold for ~.~ and ~c, then every prime of R~ is pro-modular. Proof. Let L'~ be the set of irreducible components of spec (R~) and let ~mod C_ W'~ be the subset consisting of pro-modular components. The assertion of ff.e~l rood the proposition is equivalent to U~ = t~-~ . We be~n by proving the proposition for the case ~ = ~. (Note that since (F, c~) is good, so is (F, c~,).) The proof consists of two steps. In the first, we show that any component of spec(R~,) containing a nice prime is itself pro-modular. As a consequence of this and of rp9x ~c.,mod ~...; we have that ,:, ~, ~: (~. In the second step we combine step one with our analysis of the structure of the ring R~ to conclude that -lnod Suppose that p is a nice prime of R~. By the definition of pro-modularity of p there is a unique map 0p : T~ c ~ R~c/10 inducing the pseudo-deformation associated to p~,modp. Call the kernel of this map P l. Clearly, P l is nice for e.G'Yc. It follows from (P1) that if Q c_ p is any prime of Rs[, then Q is pro-modular. In particular, any minimal prime of R~, contained in p is pro-modular. This completes step one. Combining this with (P2), which asserts the existence of a nice prime of R~c, yields .~mod ~mod .~t c.~'-~, \ ~m~� The next step is to prove that ~'~ = ~_,~, . Put ~ ~, = r~ ~,\~ . If ~ = 0, then there is nothing to prove, so assume otherwise. It follows from Proposition 2.4 and <~moa U'~, such that C 1 NC 2 Corollary A.2 that there are components Ct G ,~ (/, and C,_, E contains a prime Q of dimension d-2t+ 8v- 3-#,~#g,.. Let I~ be the ideal generated by the set {p; det 9~0(Y~)- 1 I i = 1, ..., 8v}. Let Q~ be a minimal prime of R~/.J(Q, Il). The dimension of Q1 is at least d- 2t - 3. #J/g, - 1 > 1 + ~v + (#Z + dim k H~;c~(F, k)), the inequality by (G). It follows from Lemma 2.6 that pgmod Q1 is irreducible. Since Q~ E C~, O~ is pro-modular. The prime Ql determines a prime Q~Od of T~. The prime Q~Od is the kernel of 0Ol :T~, ~ R~/QI. Moreover, since P~ modQi is irreducible as remarked in the preceding paragraph, it follows from rl-, /g-~rnod Proposition 2.12 that dim ~,/td. 1 >/ dim R~/QI. Recall that T~, is an integral ~n-v!l~ y/t~ Tt, ...,T~F]I (cf. Corollary 3.4). By construction extension of Ae~: = ~ tt ~l ,..., dr, Qmoa ~_ . '~ Y('~ it would follow I I,~,~e contains T~,...,Ta v. IfOm~ also contained Y'~',..., ai that the dimension of O~ ~ would be at most d- di. Comparing this with the lower bound for the dimension of Oi obtained earlier and recalling that the dimension of QI is at most that of O~ TM, one finds that d, ~< 21 + 3 9 #~,,Pg~, + 1 which contradicts (G). Thus, after possibly reordering the Yf's we may assume that Y'( ~ Q~ for each i= 1,...,t. RESIDUALIN RI'~I)UCIBI~E REPRESI';NTATIONS AND MO1)UI_AR I"ORMS 65 ( ) t'o~ bo~ Fix now a basis for p~ for which P~(zl) = 1 -1 9 Write pM(o) = \ca do/. As p~ mod Q~ is irreducible, there is some o0 for which co,, ~ Q~. Let p _D Q~ be a prime of dimension one not containing coo, Y:i 1), ---, Y;f. Such a p always exists. As p E C1 it is pro-modular. We claim that it is also good. By construction p contains p, and it is, of course, a prime of R~, so it remains to check the conditions at each Di. I,et A = R~/p and let p : Gal(F/F) , GL2(A ) be the deformation p~ modp. Consider pjl)i~ (Vli) ~) * ) . By definition V~'~/[ .... ') equals 1 + Y:r which has infinite order in A. Thus gt~ ) is a character of infinite order. On the other hand, det p(~.) = 1 for j = 1, ..., 5F, SO, as char A = p, det P = )~- It follows that ~il'? = )~ 9 gt~ ~-1, whence (~) !i: g4/~:2 has infinite order. Therefore p is a nice prime of R~. As p E C2 it follows ~,mod from step one that C2 E ~r contradicting the assumption that C2 E ~~. This ~r rood proves that 5W~z~ = ,J ~ . "1 We now prove the proposition in its full generality. We first show that any component of spec (R~) containing a good prime is pro-modular. For this we use the proposition in the case ~ = CJc. We then combine this with our previous analysis of rood Rj to conclude that ~'~ = ,~ . Suppose that p is a good prime of R~. It follows that p is the inverse image of a prime ~1 of R~ under the canonical map R~ -~ R~,. By the proposition in the case 6~ = ~,., P l is a pro-modular prime. Thus there is a map 0p~ : T~ ~ R~/pl = R~/p inducing the pseudo-deformation associated to p~,mOdpl = p~modp. Composing Op~ with the canonical map T~ ---T~. yields a map 0p : T~ ~ R~/p inducing the pseudo-deformation associated to P~ modp. Let P2 be the kernel of O~. It follows from the definition of P2 that it is nice for ~, whence by (P1) any prime Q c p~, .~ c R~ is pro-modular. As p = P2, ~, it follows that any component of spec (R~) containing p is also pro-modular. In our final step we complete the proof of the proposition in its full generality. Let Q be a minimal prime of R~. Let I~ C R~ be the ideal defined as follows. Choose a basis for P~ such that P~(z~) = (l _~). Write P~(o)= (~ ~) and p~(o) = (~ ~(o)"~ ) . For each place v E Z\,+:~ fix a generator ~, E I,, of the pro-p-part of tame inertia at v. Let I~ be the ideal generated by the set {p; c% - 1, bz,: - u.r,~, %, d~,,; det p~ (~) - 1 I v E E\,~, j = 1, ..., gF}. Let Q~ be a minimal prime of R~/(Q, I.~). By Proposition 2.4 the dimension of Q~ is at least d- 7. #Z - 1. It follows ti-om this and from (G) that the dimension of Q~ is at least ~iv + #Z+ dimk H:~(F ) + 1 from which it follows by Lemma 2.6 that p~ mod Q.2 is not reducible. Moreover, it is clear from the fact that Q2 _D I~ that P~ mod Q., is a deformation of type-~:. It follows from the proposition in the case ~ = ~_~ that Q.2 is 66 C.M. SKINNER, AJ. WILES pro-modular. Arguing as in step two of the proof in the case ~ = ~ shows that Q2 is contained in a good prime. As Q c_ Q2, the same is true of Q. The conclusion of the preceding paragraph now implies that Q is pro-modular. Therefore, every minimal prime of R~ is pro-modular. This completes the proof of the proposition. [] 4.4. Conditions under which (P2) holds In this subsection we establish the following criteria for (P2) to hold for a .given deformation datum G~. Proposition 4.2. --Let ~ = ((r Z, c, .//~) be a deformation datum. If (F, ~) /s a good pair, and if (P 1) holds for each datum (C', s c', ,/W~ ') with E' C_ E and C' D_ ~', then (p2) holds for Proof. -- The proof of this proposition consists roughly of three steps. In the first we prove that (P2) holds for some deformation datum ~J0 -- (8", Z0, Co, Jg0) with ~'P _D (r From this, together with the hypotheses of the proposition and Proposition 4.1, we obtain that if .~2;' = (~-~r Z', c0,,//g') with Z' C_ E then every prime of R~, is pro-modular. In the second step we combine step one with the existence of suitable reducible deformations to show that there exists a prime P l of Tcj~ 1 (where 6-~1 = ((2~:', Ec, co, ~'~gl) for a suitable JW~ 1) such that the pseudo-deformation associated to P l comes from the pseudo-deformation associated to a deformation 131 of type- (J(~, Ec, c, 0). In the third step we prove that 91 is actually of type-~'_~c and that Pl is essentially the inverse image of a prime of T~, thereby proving that (P2) holds for c~. We now prove that (P2) holds for some deformation datum c_Z0 = (~', Z0, co, ,/IN 0). (Recall that Z0 is the set of finite places at which )~ is ramified together with the places Vl,...,o t over p and that d/~-0 = Z0\{Vl, ..., ut}. ) Let g x C_ GLy(~/'v | 7..) be as in w Since the pair (F,~) is good, L#(F, - 1,Zr is not a unit in ~', so it follows from Proposition 3.14 that T~(U x , 8') has a permissible maximal ideal m. Recall that by Corollary 3.4, T z = T~(U ~, r is an integral extension of A~,. = ~ [[Y(I I), y',0 T1,..., TSF~. Let Q c Tx/(p , "1"~, Ta~.) be a minimal prime. By 9 ", d t ' -- ..., its choice, the dimension of Q is at least d. Let R = Tx/Q. The pseudo-representation associated to pQ. determines a pseudo-deformation into R of type-(~ ;~ , Z0). We denote this pseudo-deformation by 9-~ = {a(~), d(~), x(c, z)}. We claim that x(~, z) is not iden- tically zero. If it were then p" Gal (F~,/F) , GL,)(R) defined by 9(~) = (~(~) ~1(~)) would be a diagonal deformation of type-(~:', Z0) (see w Therefore, there would Ddiag be a map ~/: ~,(~, x) ~ R inducing p. Since it follows from Lemma 3.11 that R is generated (pro-finitely) by the set {trace p(~)} = {trace pQ(o)}, T must be surjective. Ddiag Thus the kernel of 7 would be a prime q of ,,(~-~, z/ of dimension at least d. However , diag 9 , by the choice of Q, det9 (and hence fletp(~iz/moclq) has finite order. Lemma 2.9 would now imply that d ~< 1 + gF, but this contradicts (G). This contradiction implies RESII)UALIN REDUCIBLE REPRFSENTATIONS AND MODUI2kR FORMS 67 that there exists some c0 and z0 such that x((s0, z0) ~: 0. Now let p _D Q be a dimen- sion one prime of T)~ not containing x(60, "Co), y,l),..., y(~. Let A be the normalization of Tx/p (this is a complete DVR with residue field k' a finite extension of k). Let ~7' = ~' | W(k~) 9 I,et q0 = p-'Qmodp be the induced pseudo-deformation into A of type-(~ ~:', E0). This is nothing more titan the pseudo-representation associated to pp. By Corollary 2.14 there exists a cocycle 0 ~= Co E HI(Fr~,/I ', k'(3(-1)) and a deformation p`0 9 Gal(Fx0/F) ~ GL2(A ) of P~0 whose associated pseudo-deformation is q0. Let K be the field of fractions of A (equivalently, the field of fractions of Tz/p). Comparing traces we find that P`0 | K -~ pp. Arguing as in the second full paragraph of w (the paragraph describing primes of T~ that are nice for r shows that co is admissible and that P`0 is a deformation of type-(~r v', Y--o, Co, 0). We claim that it is in fact a nice deformation of type-C:'.~0, where 6_~0 = (~", Y--0, Co, ~/g~o). Recall that dg,~ is nothing more than the set of finite places other than vl, ..., v, at which ~ is ramified. Therefore if w E J/~0, then one sees easily that p`0[,,, _~ (r ~002" ) with r and r finite characters of p-power order. However, since the characteristic of A is p it must be that r = 1 = r This shows that P`0 is of type-C*~0. Moreover, since 9`0@K_~pp it fbllows from (3.4) that p~0,t) _~ (01 i) * ) (,~, ~1 9 ~i) with r " a character of finite order and ,~('~l,.('~x = 1 + Y?, which is an element of infinite order in A � To "f2~l s conclude that 9`0 is a nice defbrmation it remains to check that the corresponding prime of R~0 is of dimension one. This follows from Lemma 2.12. By construction p`0 is a pro-modular deformation of type-6_~0 (since T~, = T:t | ~'~'). It follows that the prime of R~ 0 corresponding to 9~0 is a nice prime. This completes step one. If the cocycle c is a scalar multiple of co, then (P2) holding for 6.~0 easily implies that (P2) holds for 6Ad~ and hence also for ~_Z), as was to be proved. For let P,0 be the deformation of type-CJ0 described in the preceding paragraph. There exists a conjugate 9`0 of 9`0 that takes values in GL2(B) with B C A an r with residue field k and that is a deformation of type-(~ v , ,To, c, JeSt0). Since (~~, 120, c, '//g~0) = 6-~c, this shows that P~ is a nice, pro-modular deformation of type-.~ (since T~ | C' = T~0). Suppose from now on that c is not a scalar multiple of Co. Let ~-~'1 = (~', I?~, Co, d-g ~) with .//{~ the set of finite places w G Z~ other than vl, ..., vt such that )~l[w ~: 1. Since (F, is good so is (F, ~r). Having shown that (P2) holds for c2c_~0 we see that (P2) also holds for .~ZI. Combining this with the hypothesis that (P1) holds for ~0 and 6.~1, and with Proposition 4.1, yields that every prime of R~ is pro-modular. As both c and co are classes in Hz~(F, k') they can both be viewed as Gal (F(,z)/F)- equivariant homomorphisms GaI(F00~)/F~)) ~ k'(g-l), where F0~) is the minimal field over which every cocycle c' G Hz:(F) becomes trivial. FLx Gal(F~)/F)-generators (~l,...,(~, of Gal(F0(z)/F(z)). Then any cocycle in Hz(F, k') is determined completely by its values on the 6i's. Let {(oh,j,...,ot.,.j) ~ k 's,1 ~<j ~< s-2} be s-2 linearly ~.~z;'Y) 68 CM. SKINNER, AJ. WILES s k independent vectors such that ~ o~i, jCo(~,) = 0 and o~i, jc(~i) = 0. Note that i= I i: 1 (4.2) s - 2 = dim e Hz,.(F, k') - 2 ~< #Z~ + dim~ Hz~,(F, k). Fix a lift ~i,j of each oti, a to (S 3''. Fix now a basis of 13~, such that 13~l(Zl) = (t ) Write 13~,(~)= (('~ b~) -- 1 " ca d~ " Let I C_ R~t be the ideal generated by {p; detp ,(' e) lJ = i= 1 g = 1, ..., 8~: }. It follows from (G), (4.2), and Proposition 2.4 that any minimal prime of R~/I has dimension at least dim R~, -(dim k Hz(F, k) - 2) - 8v - 1 > d+ 7 - 3 9 #,//g~ - 4-dimk HL(F , k) - 2t (4.3) > d + 7 - 7 9 (#E~ + dim k H~,(F, k)) > 8v + dimk Hz,(F, k). Comparing this estimate with that in Lemma 9.6 shows that any minimal prime of R~I/I corresponds to an irreducible (pro-modular) deformation. Now, there exists a reducible deformation 13 : Gal(FzJF) , GLg(k'[[x~) of 13q~ given as follows. Let c and Co be cocycle representatives of c and co such that c(za) = 0 = c0(zl). Define 13 by p(o)=(1 X(o).(~0(o)+'~o)X)).X(o) Clearly, 9 is a deformation of type-~l, so 13 corresponds to a dimension one prime p of R~/I. Let Q be a minimal prime of R~/I contained in p. As we observed in the preceding paragraph, 13~lmod Q is irreducible. Let Qr be the inverse image of Q under r~ 9 Ro-zp., ) R~. Let Atr = R~,~/O~ r and let A be the integral closure of --1 N r in its field of fractions L. The ring A is a KruU domain [N, (33.10)]. Let K be the field of fractions of R~/Q. Choose 131,...,13~ E k' such that ~[3ic(~i) = 0 but ~ico(~i)~: 0. Fix a lift i= 1 i= 1 ~i of each ~i to (~-@t. Choose a basis for 13~, such that 13y,(zl) = (1 -1) and 2 ~ibrj i = z/~ E ~("~,,x,~ where 13~I (~): (aaco ~ )" Put x((~, "c)= bocz. Suppose that P is a height one prime of A for which ~ "~ix(,~i, (~p) (/ P for some gl, E Gal(FzJF). Then, since ~'~ix((~i, (~r) = u0%,, %, ~/ P. It follows that b~ E Ap for all ~ E Gal(Fz/F). In particular, the matrix entries of each 13~t(~) modQ are in Ap. Thus, if such a oF RESIDUAI~LY REDUCIBLE RI,;PRESI{NTATIONS r MODUI.*~R FORMS existed for each height one prime P of A, then p~ ]mod Q would have matrix entries in MAp = A. It would then follow from Lemma 2.6 that any dimension one prime of R~I/Q pulls back to a prime of A, and hence to one of Atr, of dimension at least one. However, this is impossible as the non-maximal prime p of R~,/Q pulls back to the maximal ideal of Atr. Therefore, there must exist a height one prime P0 of A" for which { ~ ~Jix((yi, T.)" T, E Gal (Fx,/~ } C_ P0. i = 1 Suppose that x(o, z) E P0 for all o and z. It would follow that the representation Pro defined by Pv0(o) = ( a~ do) EGL9(N~/Po) would be a diagonal deformation of P0 = (1 x) of type-(~;', E,) having determinant equal to X. As Nr/p0 is (pro-finitely) generated by the traces of PP0 it follows that the l~diag ) NVP0 would be a surjection, whence by Lemma 2.9 (ii) the natural map -'-re, , z,) dimension of N"/P0 would be at most 1 + 3v. However dim A'~/P0 = dim Atr -- 1 ) dim R~r l/Q- 1 > 8r" + dime Hz~(F, k') >By+ 1, the first inequality coming from Proposition 2.12, the second from (4.3), and the last from the fact that c and co span a two-dimensional k'-subspace of Hzc(F,/4) by hypothesis. This contradiction implies that there is some o' and z' for which x(o', r P0. We next claim that after possibly renumbering ';, 4 we can assume that Y~I ') ~ P0. l~br this we recall that since Q is a prime of R~ it is pro-modular, so m~ m A ('~mod Atr = T~/Qmod for some prime Q~,od C_ T~ such that p~ ou ,.~ is the pseudo- representation associated to p~]modQ. Recall also that T~, is an integral extension of A~. By the choice of Q, Qmod contains (T1,...,T~v,p). Hence so does P0- If P0 also contained (Y(~'), ..., Y~) then the dimension of P0 would be at most d-d~- 1. Hence the dimension of Qmod (and hence of N r) would be at most d- di. However, as the dimension of Q is at least d + 7 - 3 9 #,M/N 1 -- 4" dim k Hz~(F) - 2t, it would then follow from Proposition 2.12 that di ~< 7. (#Z~ + dimkHz0(F))+ 2t, contradicting (G). This proves the claim. Now let P l be a dimension one prime of N' containing P0 but not containing Yl ,...,Y';, or x(o', "(). Let B be the integral closure of A'~/p] in its field of fractions L. Let k" be the residue field of B. By Corollary 2.14 there is a representation P l " 70 C.M. SKINNER, AJ. WII,ES Gal (Fz~/F) , GL2(B ) whose associated pseudo-deformation comes from P~ ii~ mod P l and for which plmodmg = 9,1 for some cocycle 0 3( cl E HI(Fz:/F, k"(x-I)). Recall that since Q is a pro-modular prime of R~I there is a map T~ ~ A t'. inducing the pseudo-deformation p~p~modO~ r. Thus Pl corresponds to a prime of T~, which we also denote by Pl. It is clear that 91 | L _~ Ppj. Arguing as in the paragraph describing primes of T~ that are nice for ~ shows that Cl is admissible and that Pl is a deformation of type-(.(.; ?'', E:, cl, 0), where C" = ~" | W(k"). We next claim that Cl and c differ by a scalar, or, in other words, Pc "" Pq. Recall that { 2~ix(oi, ~), E ai, jx(oi, z) lz E Gal(F/F); j = 1, ...,s- 2 } is contained in p,. Suppose that 2 oti,./Cl(Oi) 3(_ 0 for some j. Fix a basis for Pl for which Pl(Zl) = (I ) --1 " Write Pl(o) = ('~co ~) . From our supposition it follows that bj = ~ oti, jbo, is a unit in B. But we also have (~ ai, jbo,)co = F, c~i, jx(o~, o) = 0 in B. It follows that co = 0 tbr all o and hence that x(o, ~) = 0 in A for all o and ~, contradicting the assumption that x(o-', ~') ~ Pl. A similar argxlment shows that ~ ~icl(oi) = 0. It follows that cL restricted to Gal(Fz,/F(x) ) is a scalar multiple of c. This proves the claim since restriction determines an isomorphism H' (Fz,/F, k~- 1) ) ,.~ Hom (Gal (Fz,/F0~) ), /doZ -1))c~1 (vz:/v~. Therefore, after possibly replacing Pl by a conjugate, we may assume that Cl = c and that Pl takes values in GL,)(B') with B' a ~::-subalgebra with residue field k and hence that Pl is a deformation of type-(~", E~, c, 0). This completes step two. We now prove that Pl is a nice deformation of type-6.~Y:. The only thing needing proof is that Pl is actually of type-~c, for the desired properties of p~[D~ follow from the isomorphism Pl @ L ~ PpI' and that the corresponding prime of R~: will have dimension one will follow from Lemma 2.12. As .//gc consists only of those finite places other than vl, ..., vt at which Pc is ramified one finds that each w C ,//g~ satisfies exactly one of the following possibilities: (4.4) (ii))~lI~ = 1, XID~, = c0-1= 1. If w E Jgc satisfies (i) then it is easily seen that P l 1I w e~ (| ~) (here we have used that char (B') = p). If w E J/g,. satisfies (ii) then pc[D. is type A. We want to show that the same is true for Pl. Since det Pl = X there are two possibilities for Pl | LID,: it is either type A or type B'. If it were type B' then Pl | I]lI u, ~ (1~ ,-1 ) with a finite character of p-power order. However, since char (L) =p any such ~ must in fact be trivial, from which it would follow that P l is unramified at w, contradicting the assumption that Pc (and hence DI) is ramified at w. Thus it must be that 131 @ L[I),~ is type A. It is now straightforward to show that since [clI)~ = (1 .] is non-split, I,,] RESII)UALLY REDUCIBLI'~ REPRESENTATIONS AND MOI)UI~R FORMS 71 as well. For let u ~ I~: be a generator of the pro-p-part of tame inertia. It follows that p~(u) = (1 b{)) for some 0 :~ b0 ~ k. Let V be the underlying representation space tbr p~ (so V is a free B'-module of rank 2). Let U C_ V be the B'-submodule annihilated by u- 1. That U =~ 0 follows easily from Pl | L being type A at w. That U =~ V follows from the fact that p,.(u) 5 k 1. It follows that U -~ B' and V/U --~ Bq Let et, e2 E V be such that el generates U as a B-module and e2 generates V/U as a B'-module. Thus el and e2 form a basis of V as a B'-module. With respect this basis u acts via a matrix of the form (1 ~). Since the reduction el and to of el and e2 modmB form a basis for V = VmodmB,, the k-space underlying Pc, and since u does not act trivially on V, b is a unit in B ~. After possibily scaling el and e2 by units in B ~ we may assume that b reduces to b0. In this case el and e2 form a basis for follows from the deformation Pl with respect to which pl(u) = (1 ~) (b ~: 0). It now the well-known action of Dw on tame inertia that p,lDw = -(1 *).- (Here we have used that char(B') =p and that det f)l = Z)- This completes the proof that P l is of type-,~c. Let P2 _C R~, be the prime corresponding to Pl. We have shown that p~ modp,2 is nice. We now show that P2 is pro-modular. Put ~ = (r X,, c, .~r Clearly T~, 2 Qnj ~" = T~. From the choice of p~ and the definitions of pe and p~ we have a commutative diagram R~ p~ ~ T~ ,2 ~ T~ ~A~"~A~/p l (4.5) J. r~ ,, B' ~ B R~ 2 ~ R~, where the map T~ ---A tr induces the pseudo-detbrmation associated to p~ l mOdQ and R~ ~ B' corresponds to Pl. Denote by p.~ the kernel of the map T~ ~ ~ At"/pl in (4.5). We need to show that P3 is the inverse image of a prime of T~ c under the canonical surjection Tc~ ~ ~ T~. Let Ql C_ p:~ be a minimal prime of T~ 2" We describe the possibilities for p% ID,, when w C ~/]g,.. First, note that PQI is ramified at every place in .liNe since pp:~ --~ pp~ is. Second, recall that by (G) every place w E J/g~ satisfies one of the two possibilities listed in (4.4). If w satisfies (4.4i) then w E ~//Ni (by the definition of ,~tg 1), and it is easy to see thai 9% [I~ ~- (001 ~02 ) with q~t and ~2 finite characters of p-power order. As w E ,M~Y~Yl, Ur,[c,2, w = c C E g~, ' , amodgw E AL,. especially for notation.) It follows from Lemma 3.16 that q~l and 0~ are in fact trivial. This shows that (4.6a) if ZII~ ~: 1 then pQ~li~ ~ ( 1 ) ' ~ " 72 C.M. SKINNER, AJ. WILES Finally, suppose that w C ,//g~ satisfies (4.4ii). In particular m(Frob) = 1. A straightforward analysis of the possibilities for pe.~ II),,. using that p% I~, factors through the pro-p-part of tame inertia at w shows that there exists a finite character ~ of Dw of p-power order such that Po_a ID,, | is either type A, type B, or type C. As type C can only occur if m(Frob ) = -1, this case is impossible. It follows that if ZIP,, = m--i = 1, then either (1) PQII~, ~- (r ,.~), 01 and r finite characters of p-power order, or (4.6b) ,) O , ~ a finite character of p-power order. Now write det9% = ~" qb.~ where q~ is finite of p-power order and ~ has infinite order and factors through a free Zp-extension ofF (hence gt is ramified only at places in ?/~). It follows from (4.6 a, b) that 0 is ramified only at places in .~/g,.\-tg i and in .~/~. Fix a character 0~ : Gal(Fx,./F) , (T~.,/Qt) � ramified only at places in E~\.~g~ and such that ~b-~1 = ~-1. By Lemma 3.17 there are primes Q:, _c p~ c_ Tj., with Q,~ minimal such that Pe.., ~- 9% ~ qbl and 9p~ ~- Pp:~ | 01. As 9p:~ | ~1 = Dp3, it follows that Pt = P3. Thus Q,2 is contained in P3- It follows from (4.6 a, b) and the definition of ~t that (i) if X!i, + 1, then pQ.,lI,~.~ ( 1 ~), (4.7) (ii) if ZID= = m-1 = 1, then either _ (' "" , OF b) 9Q21Dw ~ (* ,-1 ) with ~)a finite, character of p-power order. Next we introduce some new subgroups of GL2(~v | Z). Write U~ = II U~. ~.. t,' Let ~f" be the set of places w C ./lg,.\,~gl tbr which (4.7iib) holds, l?br w E ~/" define c E g~,, ad-lmodgw E A~w} (see w for the definition of A'w). Put (4.8) U' = II u~,,~, x II u~,, and u" = U~, flU'. z,,4 7/" ~eT/" Let m' and m" be the permissible maximal ideals of T~(U', ~) and T~(U", t~), respectively, obtained by pulling back the permissible maximal ideal of T~(U~c, ~') RESIDUALIN REI)UCIBI,E REPRESENTATIONS AND MOI)UI,AR FORMS 73 via the canonical projections. There is a commutative diagram (4.9) T / S T~(U', 6n~).,, T" = T~(U", d~j.,,, where all the maps are the canonical ones and are surjective. Let ~ C_ p.~' C_ T" be the inverse images of Q2 and P3. It follows from (4.7), (4.8), and Lemma 3.15 that there are primes Q~2 c_ p~ c_ T' whose inverse images in T" are just O~'~ and p~', respectively. It now follows from Proposition 3.20 that p.~ is the inverse image of a prime Pc of T~ c By the commutativity of (4.9) and the fact that P"3 is the inverse image of both P3 and p.~ it follows that P3 is the inverse image of p~.. This proves that the map T~., ---, Atr/p I in (4.5) factors through the canonical surjection T~ 2 ~ T~,, completing the proof that the deformation 91 is nice and pro-modular of type-r This completes the third and final step in the proof of the proposition. [] 4.5. The Main Theorem We now state and prove our Main Theorem. In this subsection and the next, we forego the convention that F has even degree. We will, however, assume property (P1) for certain fields. That this property holds is proven in w (see Proposition 8.4) which are independent of w Main Theorem. -- Suppose that F is a totally real field and that ~.~ = (~ , Z, c, ./Z/g) is a deformation datum for F. Suppose also that 9 9 GaI(Fr./F) , GL2(~' ) is a deformation of type-~ such that 9 p is irreducible 9 det p = gt~ ~t with g >, 1 an integer and V a finite character @) with V'2 II~ of finite order for each i= 1, ..., t. If there exists an extension L/F of totally real fields such that (i) the Galois closure of L over F is solvable (ii) L has even degree over Q (iii) L is permissible for (iv) (L, ~ r,) it a good pair in the sense of w then p | Qp is a representation associated to a Hilbert modular newform. 74 C.M. SKINNER, AJ. WILES Proof. -- Let P~ = [~lGal(L/L)" As L is permissible for F'~, P~ is a deformation of type-~t. Since L has even degree over O~ it follows from Proposition 8.4 that (P1) holds for any deformation datum for L. As the pair (L, ~-~1.) is good it then follows from Propositions 4.1 and 4.2 that 9~ is a pro-modular detbrmation. In particular, there is a map ~. : TCz L > ~> inducing the pseudo-representation associated to P l. By the conditions imposed on p in the statement of the theorem, the map ~. satisfies the hypotheses of Proposition 3.7. Thus there is a Hilbert modular newformf (over L) such that (4.10) trace pt(Vrobt) = (eigenvalue ofT(g) acting on f) for g f~EL. Let PJi " Gal(L/I~) ) GL2(Qp ) be the representation associated to f. (If is the automorphic representation associated to f, then 9Ji is just Pn, the latter being the representation described in (3.2)). This representation satisfies (4.11) trace 9j~(Frobt)= (eigenvalue ofT(g) acting on f) for g f~Ei.. As P is irreducible by assumption and odd, PI (and hence 91 Q Qp) is also irreducible. It therefore follows from (4.10) and (4.11) that Pl Q Q~ ~- 9fL- Now, as the Galois closure of L/F is solvable, it follows from the known cases of base change for 0aolomorphic) Hilbert modular forms (cti [GI,]) that there is a newform f over F such that PJ~GaI',~/L', #.o Pfl (here pj" Gal(F/F) > GL.>(Op) is the representation associated to f). Since 9f]~.~l~/i.l -~ P | Qp ](;~J',~/g) and these are irreducible, it is easy to see that -- --X ~ -- there is some finite character ~ : Gal (F/F) , Qp such that P/| ~) = P | Q/,- As 9f| is the representation associated to the newform corresponding to the twist off by ~), this proves the theorem. [] In the next subsection we will deduce the following theorems from this one. Theorem A. -- Let F be a total~ real abelian extension of Q. Suppose that p is an odd prime and that P " Gal (F/F) ~ GL,,(/Dp) is a continuous, irreducible representation unramified away from a finite number of places ofF. Suppose also that the reduction of P satisfies ~ ~- )~, | If 9 the splittingfield F(~l/~2 ) of)~l/)~ 2 ~ abelian over Q, 9 ~1/)@ (z) = --1 .)Co, each complex conjugation z, 9 ()~1/%'2)Ir),, ~- If or each v[p, ( ") 9 o li) with factoring through a pro-p-group of and of 2 2 finite order for each v~, 9 det p = g~k-1 with k >>. 2 an integer and ~r a character of finite order, then p is a representation associated to a Hilbert modular newform. RI"SIDUALI,Y REI)UCIBI,E REPRESENTATIONS AND M()DUI~'~,R FORMS 75 A critical ingredient in the proof of Theorem A is a result of Washin~on on the boundedness of the p-part of the class group of a cyclotomic Z:-extension of an abelian number field (cs [Wa]). A similar result for any totally real field would yield the same theorem but with the omission of the hypotheses that F and F()~I/Z2 ) be abelian. l~br our next theorem, we make the following hypothesis, which plays a role similar to that of Washington's theorem in the proof of Theorem A. We believe that this hypothesis will be easier to establish than the analog of Washing-ton's theorem, though the latter would yield a stronger result. Hypothesis H. --- There exists 0 < ~. < ~ and a constant c(e) > 0 such that given a totally real field K and a finite set S of finite places of K there is an imaginary quadratic extension L of K having prescribed behavior at each place in S and such that the relative class group of L/K has p-rank at most c(e)deg(K/Q) l-e Theorem B. -- Let F be a total{y real extension of Q. Assume Hypothesis H for all solvable, total~ real extensions ofF. Suppose that p is an odd prime and that 9 : Gal (F/F) , GL2(Qp ) is a continuous, irreducible representation unramified away from a finite number of places of F. Suppose alto that the reduction of 9 satisfies ~ ~- )tl | )t'2. If " (~i/)t~,)(z) = -l for each complex conjugation z, 9 O~,/Z2)[D. has even order for each v~).. 9 911~, ~- ( ~g~)Z~ ~g~v)~ *) 2 with W~ ') factoring through a pro-p-extension ofF, and W.~:[I, of finite order for each 9 det 9 = V ek- ~ with k >1 2 an integer and ~ a character of finite order, then 9 is a representation associated to a Hilbert modular newform. 4.6. Proofs of Theorems A and B We now prove Theorems A and B. In both cases this is done by reducing to a situation to which the Main Theorem applies. Proof of Theorem A. -- Put Pl = 9@Z2 and )C = )~l/X2. I,et Z be the set of finite places at which 91 is ramified together with the places over p. There exists a finite extension K of Q~ such that for some choice of basis 9~ takes values in GL2(~" ) with ~"' the ring of integers of K. Such a basis can be chosen so that the reduction ~ P' m~176 the maximal ideal (~')~ ~, Pl = P'm~ satisfies 91 = ( 1 ~) and is non-split. Let k be the residue field of ~'. It follows that 91 ~- 9c for some cocycle 0 :~ c E H~(Fx/F, LOUt)). The hypotheses on Pll), ensure that c is an admissible cocycle. Thus after possibly replacing Pl by a conjugate we may assume that Pl is a 76 C.M. SKINNER, AJ. WILES deformation of type-C~ ~ , where ~ = (~', Z, c, ~). Clearly 91 satisfies the hypotheses of the Main Theorem. The conclusion of Theorem A will thus follow from that of the Main Theorem if we exhibit an extension L/F of totally real fields that (i) has solvable Galois closure over F, (ii) has even degree over Q, (iii) is permissible for ~, and (iv) is such that (L, ~I~) is a good pair. We will construct such an L. Let E/F be any even extension that is permissible for .(/' and is such that E/Q is abelian, each place v[p of F splits in E, and if w is a place of F at which 9c is ramified and )CID~ is unramified then )dDw, = 1 for each place w'lw of E. It is easy to find such fields: take for example, E = F. E' where E' is a real cyclic extension of Q of sufficiently divisible degree in which p splits completely and all primes q :~ p divisible by a place in Z are inert. Choose an odd rational prime g such that g ~ #k � and g is not divisible by any of the places in E. For each positive integer n let E, be the cyclotomic Z/g"-extension of E. It is easily checked that E, is permissible for ~. Let Z, be the set of places of E,, dividing those in Z, and let .~n be the set of places of E,, dividing p. Let rn denote the p-rank of the i~- l-isotypical piece of the p-part of the class group of E,,(X) and let p'" denote the order of the p-part of the class group of E,. From the theory of cyclotomic extensions we know that there exist integers s and t such that (4.12) #Z,=s and #?/,=t for n>>0. Similarly, it follows from [Wa] that there exist r and c such that (4.13) cn =c and r,=r for n>>0. As E,,/F is a Galois extension, we also have that (4.14) deg En, ~./Qp/> g"/t V v[p. Let p~" be the number of p-th power roots of unity in E,,(~p), ~p a primitive p-th root of unity. As the degree of E,/E is a power of g, there is an integer e such that (4.15) e. = e Vn. For each n/> 1 choose a set S,, of en + cn + 1 finite places of E, disjoint from Z, and such that (4.16) * pe,+c,+ t [ (Nm(w) - 1) V w E S,, 9 )c(Frob,~,) = 1 V w E S,, 9 there exists an abelian p-extension L,/En of degree at most pe.+2c.+,, unramified away from Sn, and such that the subgroup of Gal(Ln/E,~) generated by {Iw:w C S~} is isomorphic to (Z/p) e'+c'+l. Note that I~ is necessarily ramified at each place in Sn. The existence of such a set S,, follows easily from Class Field Theory. RESIDUALLY REI)UCIBLE REPRESENTATIONS AND MOI)UI~kR FORMS 77 Let E,, C_ Hn C_ L, be the maximal unramified subextension of L,. It follows that L(L~, - 1, Zro-') = II L(H., - 1, Zeo-~qb) where ~ runs over the characters of GaI(L,JH,,)--~ (Z/p) enfcn+l . Let pC, be the number of p-th power roots of unit)' in H,,(~p). Note that g, ~< e, + c,. It follows from well- known congruences for p-adic L-functions that if Xo-I ~= 1 then L(H,,, - 1, X0~-l~) E Z0 [~c0-1q~] for all , and if Xc0- 1 ~_ 1 then L(H,, - 1, Xm --l*) E Zp [~] for, non-trivial and /"L(H,, - 1, Zc0 -l) E Zp (cf. [Co], [D-R], or [Se]). Here, for any character 0, Zp [0] denotes the ring obtained by adjoining the values of 0 to Zp. We also have by our choice of S~ that if ~ is non-trivial and if n, is a uniformizer of ZpL'~oJ-l~] then L(H., - 1, )~(0--11~)) ~ L(H., - 1, zo-l) I-I (1 - xto- 1,(Frobw)Nm(w)) = 0 mod n,, where S,(~) is the set of places of H, at which ~ is ramified. Combining this with the earlier expression tbr L(L,,, - 1, ;(co -l) we obtain that L(L,, - 1, Xo) -1) E Zp[zr -1] and L(L,, - 1, Xr -1) = 0mode., ~. a uniformizer of Zp[zco-l]. We have thus shown that (4.17) Lp(L~, - 1, Xr E ~\& � Since E, is permissible for 6~, the field L,z is as well. Moreover, it is a simple exercise in p-groups to show that dim~ Hy.o(Ln, k) ~< #Gal (L,/E,) 9 dim k Hr.o(L,, k) Gal(I'n/Fn) (4.18) ~< #Gal (L,/E,). dim k H~,us,(E,, k) <<, pe"+2c"~l(r, + e, + c, + 1). Now choose no so large that g"~ > 2 + 8(f+~c+l(s + r+ e+ c+ 1)) (4.19) t,o > t(2 +pe~2c~I(t+ 7(S+ r+ e+ c+ 1))) and such that the equalities in (4.12) and (4.13) hold. Let L = L, o. By construction the Galois closure of L/F is solvable and the degree of L is even. As noted above, L is permissible for !~. It remains to verify that (L, ~-/Yl.) is a good pair. For this let dl. be the degree of L. By construction dE >/ g,0. Also, by 78 C.M. SKINNER, AJ. WII,ES di It follows that [Wal], ~L ~ --"" dl, ~> 6L + g"/2 > 2 + 8(pe+2'+'(s + e+ lc+ 1 "b?l)'t-~) L (4.20) >/2 + ~ir, + 8(#ZL + dim k HE,(L, k)), the last inequality following ti'om (4.12), (4.13), (4.15), and (4.18). Suppose that v is a place of L dividing p. Let d,, be the degree of L~. The number of such places v is at most pe+')c+lt, so it follows from (4.14) and (4.19) that d~ >1 g'~/t > 2 +p~ 7. ooe+2c+l(s+ r+ e+ lc+ 1)) >1 2 + pe+2~+~t + 7(#EL + dim k Hr~(L, k) ). That (L, C_YYL) is good now follows from this, from (4.17) and (4.20), and from the choice of E. [] Proof of Theorem B. - Let X = Zt/Z2. It follows from base change that it suffices to prove the theorem with F replaced by the maximal totally real subfield F + of F(X ). By the hypotheses in the theorem ZII),, ~: 1 for each place vb0 of F +, and therefore we may assume that X is quadratic. Put Pl = P | ~-1. Let Z be the set of finite places of F at which Pl is ramified together with those dividing p. As in the proof of Theorem A, there exists a finite extension K of Qp with integer ring C~' such that, for a suitable choice of basis, 91 takes values in GL.,(C) and is a deformation of type-C2~ for some ~ = (~, Z, c, 0). The conclusion of the theorem will follow from the Main Theorem if we can find an extension L/F that (i) has solvable Galois closure over F, (ii) has even degree over Q, (iii) is permissible for 6~r and (vi) is such that (L, ~-~I,) is a good pair. Arguing as in the proof of Theorem A shows that we can find a solvable permissible extension E/F that has even degree over Q and is such that (4.21) 9 Lp(E, -1, Zoo) E ~-~' and is not a unit, and wCp, then either X[t,, :~ 1 or X[I),, = 1, for w a place of E. 9 if Pc[I~ :~ 1 Let Z' be the primes of E above those in Z. We now construct a solvable permissible extension L of E such that (L, ~e) is good. By Hypothesis H there is a totally imaginary quadratic character ~g over E such that 9 if w E s and w~p then V is unramified at w and X~g[D~, ~- 1, 9 if vb0 and X[l, :~ 1 then ~ is unramified at v and v(Frob) :~ 1, 9 if v[p and XlI~. = 1 then gt is ramified at v, and 9 the p-rank of the relative class group of" E(~g)/E is at most @)21-~deg(E/Q). Let L, be the splitting field of the character X~g over E. This is a totally real quadratic extension of E and clearly permissible for c~. Let ZI be the set of places RESII)UALIN REDUCIBLE REPRESENTATIONS AND MODULAR FORMS 79 of L1 above those in Y/. It follows from the choice of V that #E~ = #E'. It is also relatively easy to see that dimk HxI(L1, k) ~< dimkHx~(E , k) + c(e)2 I-E deg(E/Q.)+ #Z'. Proceeding inductively, one constructs in the same manner for each n > 1 a totally real quadratic extension L,, of L,_l such that (4.22) 9 L, is permissible, 9 #Zn = #Z I, where s is the set of places of L,, over those in Z', 9 dim k Hx,,(L,,) < dim k Hz,(E, k) + c(e)~,~l 2i(l-e" deg(E/Q)+ #E'. Now choose no so large that (4.23) 2 "~'-~ > 2 + 17-#E' + 8. dim k H~(E, k) + c(g)2 ('~'+1!(1-e)+6 deg(E/Q). dL Put L = Ln0 and ZL = E, 0. Let dE be the degree of L over Q. By [Wal] 5L <~ ~-, so by (4.22) and (4.23) we have that d1. >/~i. + 2 "~ deg(E/Q) (4.24) > gL + 2 + 9. #ZL + 8. dim k HxL(L, k). If v is any place of L dividing p, then d~ = deg (Lv/Qp) is at least 2 '~'. It follows from (4.22) and (4.23) that (4.25) d,, > 2 + 9 9 #ZL + 8. dim~ HzI.(L , k). Since L has even degree by construction, combining (4.25) with (4.24) and (4.21) shows that (L, c~L) is good. [] 5. A formal patching argument In the next four sections we give the proof of property (P1) (see Proposition 8.4). These sections do not make use of any results from w In this section we will describe a formal patching argument which is a variant on the patching argument in [TW] and its refinement in [D2]. The extra complexity in our case is caused by the fact that we are considering the localizations of deformation rings and Hecke rings and not the original rings themselves. In particular the residue fields are not finite. We will, in section w apply our patching argument to localizations of deformation rings (in contrast to [TW] where it is applied to Hecke rings), but in this section we will just axiomatize what is assumed (and later proved) about these rings and consider only the formal aspects of the argument. Let k be a finite field of characteristic p and let A = k[[T]J. Let K be the field of fractions of A. Let .5/' = {N} be a sequence of strictly increasing odd integers together with zero. Let n be a fixed positive integer. 80 C.M. SKINNER, AJ. WILES We introduce rings Ax, BN (for each N E ,5gJ) given by Ax A[Is,, s,]] / (S1 N+I N+I.~ = ..., ,...,S, j, Ao = A Bx = A[[6, ..., t,]] / (~+li/2 ~.X+l~/2. . ,...,t; ' ), B0=A. There is a homomorphism BN ' At,- given by ti, , (1 + si) + (1 + $i)-1 _ 2 which we use to identify BN as a subring of AN. We assume that we are given a ring R ~ for each NE cf~ of the form (5.1) R/'3 = A[x~,..., Xm]l/d a~ with rn independent of N. Furthermore we assume that R Cx) has the following properties: (5.2) (i) R IN) is finite and free as an A-module, (ii) a (x~ G (x,,..., x,,), (iii) 3 a surjective map R (x) -+ R/~ of A-algebras, (iv) R/\3 is a BN-algebra for N > 0. Now letting p(-',3 be the prime of R ~ corresponding to (Xl, ...,Xm) (which we usually abbreviate to p if the N is clear from the context) we assume two further (and less formal) properties of RC'W': (5.3) (i) 3 d(0) > 0 such that pd(0) = 0 in R '~ (ii) p(.X~/(p(X))2 ~ A n | Tor,,~, where the free summand A" is spanned by xl, ..., x, and "For(y) is a finite group whose order is bounded independent of N. For each odd 0 ~< a ~< N together with zero we assume given a ring R~ "4/ which has the following properties: (5.4) (i) R] ~'3 is finite and free as an A-module, (ii) ~(~ R~ -x) :0' "-x =Ri~, =R , (iii) there are surjective maps of Bx-algebras Ri m .., (iv) R(ff 3 is a Ba-algebra (compatible with BN + Ba) such that if" a > 1, then a-I a-I | KI/(tY, ..., tT ) RaT2 | K, (v) R~ ~ @A K is an Aa @A K-algebra satisfying (via the map in (iii)) el N) @A g/(s,, ...,Sn)~ Rff3 ~ @A K, RESII)UALIX REDUCIBLE REPRESEN'IATIONS AND MODULAR FORMS 81 Letting p(~\) denote the prime corresponding to (xl,..., xm) (which we again write as p if a and N are clear from the context) we assume two further properties: (5.5) (i) 3 d(a) > 0 independent of N such that I aJ(~i = 0 in R~ :<, / / -x (ii) '~ ~x;2 A" Pa /(Pa ) ~- {t) TOr(N,a), where the free summand A" is spanned by Xl, ..., x, and Tor:x, ~3 is a finite group whose order is bounded independent of N and a. Associated to the rings R __~ ;x~' are certain subrings R*] ~-'~ (of "traces" in the application) which are assumed to satisfy the following conditions. First we assume given an A-subalgebra R '~(a') C R ix) satisfying (5.6) (i) R '~''~ : A ~Yl, ".,Ym~/[~''N~ (same m as in (.5.1)) where b ~? C Iv,, ...,Y,,,), (ii) R tr(N) is a Bx-algebra for N > 0 compatible with the algebra structure on R i'xi. Setting q~.x, = p(X', 71R t'~'",' (thus the prime corresponding to (Yl, ...,Y,,,) ) we assume in addition the property (5.7) coker" q~r~)/(q(I,0)2 ~ p(N)/(p(N))P, has order bounded independent of N. For 0 ~< a ~< N, a odd or zero, we set R~ 'Ix' =im {R ~''') , R;\'~}. Observe that __~R t'rN~' inherits a B~-algebra structure from R~ "x?. Then we deduce from (5.7) that (N'., (Ng 2 'N'; (N~ 2 coker{% /(q~ ) , p~ /(p~ ) } has order bounded (5.8) independent of N and a, where q]<' = p::.x~ A R~,t~'x'. Associated to the rings we have described we will assume given a set of modules as follows. First we assume given an integer 7, independent of N. Then we assume we are given M i'x~, a finite R<'~)-module, satisfying the hypotheses that (5.9) (i) M $'~ is a free A-module of rank equ~ to the rank of A~, (ii) M (r'~: is an Ax-module compatible with the. Bx-structure via R ''c'; , (iii) there is a map M '''? ~ M '~ of R<a;-modules. For 0 ~< a ~< N (a odd or a = 0) we assume that we are given a Rt'~'NLmodule quotient ',N: of M ix) denoted M a satisfying (5.10) (i) M] v) is a ti'ee A-module of rank equal to the rank of A~, (ii) M::'; (x~ M(O~ x = M(X), M0 C and there exists some z E W '(~ independent of N, ordl.(Z mod q(0)) ~ 0, such that z M (~ C M 'x) 41--- ~ ~ ..., (iii) there are surjective maps of RU','~-modules M~ \) M:i \: M (x) 82 C.M. SKINNER, AJ. WILES (iv) M~ "~) is an R~"N"-module (compatible with the R'r"N"-structure), (Nh (v) M~ is an Aa-module (compatible with the Ba-structure induced in (iv)) in such a way that the maps in (iii) are compatible with A0 ~- Al --- A3..., and the actions of Rtd ('x) and An commute on M~ v', (vi) M2 "~' @A K is a free A~ | K-module and M~ s; | K/(sl, ...,s,) --~ M~ ~ @A K. Furthermore, we assume there exists x ~''; E R trtr~) such that (5.11) (i) ~ annihilates ker{M a /(Sl, s,,) "", *'~0 J, (ii) OrdT(X~'~mod q(r~) = t < oc with t independent of N. We now derive some simple properties of the above rings and modules. Lemma 5.1. - rankAR~ "47 ~< g(a) where e(a) depends only on a. Proof. -- This follows immediately from (5.5i). [] Lemma 5.2. -- There exists an E(a) independent of N such that TE(a)R~ ? C R t'(~3 ---a 9 In particular R tr(~ @A K = R(s "3 | K. Proof. -- Since pd(~) = 0 in R2 ~ by (5.5) it follows that it is enough to check that coker~r, where ~r is the natural map z (N),r// (N)2~r+l (N) r (N) r+l f~r" ~qa ) /~qa ) ' (Pa ) /(Pa ) , is finite and bounded independent of N for r < d(a). For r = 1 this is given by (5.8). A similar bound follows for r = 2 by picking generators for im(~l), lifting them to (N) (r,0, 2 elements, say Zl,...,Zs, in (q~)/(% ) and considering the map (N) ,~) 2 s (qa /(qa )) ' (q~"))2/((q~X))2n(p~'~))3), (al,...,a,), , Zaizi which is surjective. The property for r = 2 can now be deduced from the property for r = 1, and we proceed by induction up to r = d(a) - 1. [] From this lemma we deduce immediately that for any c the kernel and cokernel of (5.12) R~ r('~/T c , R~ "9/T c is annihilated by T E'~). Now let t, x ;~3 be as in (5.11) and let z be as in (5.10ii). Let dl = OrdT(zmodq(~ ). RESIDUALIN REDUCIBLE REPRESEN'IATIONS ANI) MOI)ULAR FORMS Iz, rnma 5.3. --- T annihilates both the kernel and the cokernel of the map ~N~ M,0!/TC M~ / (T c, SI,... , St) --""+ for any c. Proof. --- Let ~ be a lift of z to R t~N~. By (5.11) and the definition of dl, x ~x~- ~- uT t+J~ E q(X) for some unit u E A x . So T t+d' = u Ix(N)~ + v, v E q(N) Hence T d''~)(t+6? = w. x~N)~ + v a~) for some w E R my). By (5.5) ~',~) = 0, so the result follows from the defining properties of x ~'x) and z. [] Next we introduce level structures which we will use to make a patching argument similar to the one in [TW]. A level-(a, c) structure J(N, a, c) is a collection of data comprising (i) Ba-algebras, R~(. ~, = R~(N)/q'c, R~ =, R~<N)/T,' M(N) M(N)/T c ,, mN~ , , (ii) an A~-module _._~ , c = --a /-- that is also an r~o~ -moclme, (iii) a map of B~-algebras R~"~ , R'~"~, tr(~ ~) (iv) a map of Ra' c -modules :.x/ M~O~/TC M~, c/(Sl,..., s,) ' compatible with the actions of A~ and A via A~ ~ A, ..., , R~, c/(xl,..., x,,) ~ A, (v) elements {Xl, Xm} of R]') such that ~.x) n tr(N]// (vi) elements {y,, ...,Ym} of Rt;~) ) such that r%,~ /l.h, ...,Y~)----- A. Let ~o = c~, = {N} as at the be~nning of the section. Let .~'~(0)= .5;f0 and define ~'l(C) C ,~(c- l) inductively as tbllows. We require that 5~l(C ) should be an infinite strictly increasing subsequence of integers from Sr 1) with the property that J(N', 1, c) = J(N", 1, c) for N', N" E .~'l(c). The equality here sig-nifies that the J(N, a, c)-structures for N = N', N" can be identified (non-canonically). Since the total number of such non identifiable structures for fixed a and c is finite a choice of Yg~l (c) can be made. Then define .~', = {N,: N~ E ,Z~L(i)} again with the Ni strictly increasing. Finally we can define (~a for an odd a > 13 inductively by .~(0) = c~'o_~ and defining ~,(c) inductively in the same manner as ~.gf~l(c). We set R~ ~ = lim R~"(c ~ R~ = lim R (N~) , Mo = lim M (m3 p fl, c a,c " 4~)<t+d') 84 C.M. SKINNER, AJ. WILFS Lemma 5.4. a) R~ r | K = R, | K, Ro | K = R '~ | K. b) R, @A K is a quotient ofK[[xl, ...,x,,]]. c) M. | K is a flee An | K-module of rank r and OM. @a K) / (SI, ..., St/) = M a) | K. Proof a) By Lemma 5.2 for any c we have natural maps of A-modules , R;,] , R~, c whose composite is multiplication by T L~a) with E(a) independent of c. Taking projective limits and tensoring with K gives the isomorphism. b) By construction there are elements {xl,..., Xm} of R~ such that R~/(Xl,..., Xm) --~ a. Let pa = (x,, ...,Xm) C Ro. Letting p(~7~ } denote the ideal generated by {Xl,...,Xm} in , we see easily that (No) , ~ , ., 2 a p"N-/(p'X~ F lira P~,c = Pa, lira ~,~ .... c, --- Pa/P 9 +--- Then by (5.5) we deduce that p~/p2 ~_ A n | T, where T is a finite group and Xl, ..., xn span the free summand. Hence (po | K)/(p~ | K) '~ ~ K" by (5.5) which ensures that Ra, p~, p] are all finite A-modules. Part b) follows by Nakayama's lemma. c) By Lemma 5.3, (5.13) (M, @A U)/(Sl, ..., Sn) ~ M (~ | K. By construction, dimK(Ma | K) ) ranKA~lVl ~ ) for large enough c and Nc E oS;~Ja. By (5.10i) the right-hand side has rank equal to rankA(A~). But r is the K-rank of the right-hand side of (5.13). The result follows. [] Lemma 5.5. - l~br odd a > 1, there are surjections a) Ra ~ Ra-2 of A[[xl, ...,Xm]]-algebnz~ and Ba-algebras. b) R~ -+ R~_9_ of A[[yl, ...,ym~-atgebras and Ba-algebras. RESIDUAIAN REDUCIBLE REPRESENTATIONS AND MOI)ULAR FORMS 85 c) M~ ~ M~_z of R~-algebras compatible with b) and of A~-algebras. Here the B~ action via R~ and Aa are the same, and the R~ ~ action commutes with the Aa action. The same holds for a = 1 with a - 2 replaced by O. We have that ,~-2 = {Ni} and ~f~ = {Nj} C_ ~o 2. Note that ji >1 i. Proof -- By our choice of Y~a and ~-2, l(a, - #)~) i --.* R~-2, i = Ra_), (x~ i [] whence taking limits yields R~ -+ R~_2. The same works also for R7 and M~. NOW we set, taking limits over odd integers a, R'. = Ra | K = R~ | K, M'~ = Ma | K, Roo = lim R'., M~ = lim M'~. a a Thus M'~ is an R'~-module and M~ is an R~-module. Lemma 5.6. (i) Roo = K [Ix,,..., x, ifl. (ii) Mor is a free R~-module. Proof. -- By Lemma 5.4 c) there is a map K[[sl, ..., s,]] r --* M~ which is seen to be an isomorphism. Consequently M~ is also a free K[[tt,...,t,]]-module. Thus K[[tl, ..., t,]] ~ R~/AnnR~0VI~ ). On the other hand there is a surjection K[[xl, ..., x,]] --- R.~. By Krull's dimension theorem we deduce part (i). (Note that R~/AnnR~(M~ ) is a finite K[[h, ..., t,]]-module.) Then part (ii) follows from the Auslander-Buchsbaum Theorem ([Mat, Theorem 19.1])since depthR~(M~)>/ n ({h,...,&} is a regular R.-,o- sequence for M~). [] Proposition 5.7. - -/f N >> 1 then dimK(R~ x) | K) >/ 2" dirnK(R~ x) | K). -- ~ N @A K. Choose maps Proof. Fix a projection K[[xl, ..., x,]] R (~ , , such that ( ~---+ (1 + ~)+ (1 + 4) -1- 2 and such that the images of K[[~,...,s',]] and K[I~,..., ~]] in R~ | K are just those of AN @a K and BN | K respectively in the 86 C.M. SKINNER, AJ. WILES sense that the images of ~ and ti are the same, and so are the images of ~ and si. It follows from (5.10 vi) that Bg | K > R~" | K is injective, for by (5.10) the action of Bx | K on M~ ) | K factors through its image in R~ ? | K. So if N+I n--I 9 (N) '' >/n (dXmK(R ~ @A K/(t'~,...,t',)) n = n "-t dimK(R;/'' | K)" then the hypotheses of Lemma 4.1 of [DRS] hold and using (5.4iv) we get that (5.14) K[[Xl, ..., x,]] / (t'l, ... , t',) "" R(~ ~ | K. Applying the Auslander-Buchsbaum Theorem again we find that K[[xl, ..., x,,]] is a free K[[fl,... , ?,]]-module of some rank d. Applying the same theorem yet again we find that K[[x~, ..., x,]] is a free KI[s'~, ..., s',]]-module of rank d/Z". It follows that a = 2 n dimK(K~-Xl, ... , xn]] / (s'l, ..., s',)) 2" dim, (R | ...,s~ ) z mnaK~v, v | K), the last inequality by (5.4v). Combined with (5.14) this proves the proposition. [] Proposition 5.8. -- M ~~ | K ~ (R '~ | K) ~ where e = rkR~(M~ ). Proof. -- We set R := R~/(tl, ..., t,) and := Mo~/(h,...,t~) = M~/(s~,...,s~). Thus M is a free R-module of rank e. Now R --~ R' t since the t~'s arc zero in R't, so (5.15) dimK P-/> dimK R'I ) dimK(R([ "? | K), for any sufficiently large N E 5JJL. (More precisely if T i annihilates the A-torsion submodule of R~l we can take N ) Ni+ l E ,~ (i + 1).) Now let gs(X) for any ring S and S-module X denote the minimum number of generators of X as an S-module. Let ~'~ Ar Then we have the inequality e := rkRoo(Moo ) = g~(M)/> g~(M/(sl, ...,sn)) >>, el. Also we have the inequality (5.16) 2"el dimK~.." /1~(~ | K) /> 2" dimK(M ;0' | K). RESIDUAl,IN REDUCIBI,E REPRESENTATIONS AND MODULAR FORMS 87 Now as remarked at the beginning of the proof of l~emma 5.6, M~ is a free K[[sl,..., s,]]-module of rank r, where r = dimK(M~~ | K), whence (5.17) 2" dimK(M(~ @A K) = dimK(l~ ) = e dimK(P, ). Combining the inequalities (5.15), (5.16), (5.17) with Proposition 5.7 gives el )e. Since also e/> el we have equality and all the inequalities just cited are equalities. In particular el = e, edimK(R(~174 K) = dimK(M (~174 K). It follows that M (~ @A K ~ (R '~ | K) ' as claimed. [] Proposition 5.9. -- R (~ | K is a complete intersection at a K-a~ebra. Proof. -- We recall what we have proved so far. By Lemma 5.6, (5.18) M~ --~ R e and Roo is a power series ring over K of dimension n. By construction we have elements {Sl, ..., s,} acting on M,,~, and (5.19) sl,..., s~ E EndRo~(M~). By Lemma 5.4 c), we have (5.20) M~/(s~,..., s,)M~ --~ M (~ | K. The action of R~ on M (~ | K is via R (~ | K and (5.21) M (~ | K ~ (R~ ,) | K)" by Proposition 5.8. Now let a = ker 9 Roo > R (~ | K. Let N = Z siM~ C Mo~. Then M~/N _"~ M '~ | K _"~ Re/ a by (5.20) and (5.21). Since Mo~ ~ R~ it follows that N _~ a e as R~-modules. (Consider the map q~ : Mo~ --~ R~. of (5.18). Then the above isomorphism easily implies that a e Let wi, ..., we be an R~-basis of Mo~. ]'hen N is generated as an R~-module by the set {siwj : 1 <<. i <~ n, 1 <~ j <<. e}. In particular a set of minimal generators has cardinality ~< en. Let {xl,...,xt} be a minimal set of generators of a. Then {xiwj : 1 <~ i ~ t, 1 <~ j <<. e} is a minimal set of generators of aM~ -~ i~. e ~' N. It 88 C.M. SKINNER, AJ. WILES follows that et <~ en, whence t ~< n. However as R (~ | K has dimension zero it follows that t = n and that R ~~ | K is a complete intersection. [] Remark 5.10. -- The circuitous route to this proposition via a counting argument is forced on us by the lack of a natural K~sl, ..., s,]]-algebra structure on R~. Only the elements {tl, ..., t~} are naturally defined in Ror The structure assumed in (5.4v) is an artifice which is not assumed to be related to the action of Ao | K on M(s \') | K, except for the compatibility with the subring Ba | K. 6. Estimates of cohomology groups In this section we consider a representation (6.1) p : Gz = GaI(Fz/F) , GLe(A ) where A -~ k[Dv]]. Here we are using the notation and assumptions of w so that, in particular, F is a totally real field. We let K be the field of fractions of A, and we recall that if p is ramified at w { p then we distinguish the tbllowing possibilities for plI. : peA 0| *) _ l, typeB pOK[Iw ~(t _ ~q ) , gtq non-trivial of finite order. Throughout this section we make the following assumptions on p: (6.2) (i) p | K is irreducible and of type A, type B or unramified at each prime w{p, (ii) ~ := p mod ~. = P, for some c as in (2.1), (iii) Z contains the primes dividing p and all primes at which ~ is ramified, (iv) p is of type A or type B precisely where j~ is, (v) det p = )C, with )~ as in w (vi) Pll)~ -~ (xl x2* ) with )~1/;(2 of infinite order for each rip. Lemma 6.1. --The Gal(Fz/F)-module W = ad~ /s irreducible. In particular P | K is not "dihedral" (i.e., is not inducedfiom a character over a quadratic extension). Proof -- By condition (vi) we see that there is an element c E D,, C Gz such that (X1/)~2)(a) has infinite order. Here we may choose v dividing p such that )~]D, is non-trivial by condition (ii). It follows from the existence of| and the self-duality W --~ Homg(W, K) that any invariant subspace of W has a complement. So if W is reducible then either W -~ Yl O Y9 with Y_9 of dimension 2 and irreducible, or W "-' Yi q)Y,2 q)Y3. The self-duality also shows that in the former case YI is acted on by Gz via a quadratic character, possibly trivial, and in the second case that there is also a unique subspace, YI say, on which Gz acts via a quadratic character. RFSII)UAI,LY REDUCIBLF REPRESENTATIONS AND MODUI2kR FORMS Now let z be a complex conjugation and pick a basis for p such that -I ) . Suppose first that z acts trivially on Yi. Then we may identify Y, with {(-a a)} C_M2(K) and we see that imp C {(* ,), (, *)} C GL2(K ). In particular, either im 9 is abelian, which contradicts assumption (i), or imp has a subgroup H of index 2 for which the action is abelian. In the latter case, H acts via two characters gt and ~g~(~z(~)= gt(~-I~x)) for any xjEH. Thus p = Ind~Z~. Now suppose that z acts non-trivially on YI. This time YI C {(b a)} _C M2(~). An easy calculation shows that if ~ = 1 on Y l then p(a)= (O~ca ~) with a~ = d~. Using that HI = { (r : o = 1 on YI } is a group we check that p(Hi) is abelian. Thus just as above, 9 -- Ind ~ for some character ~. Now consider 9 restricted to D~.. Then the quadratic field associated to P (i.e. the fixed field of H or HI in the two cases) is not split at v as otherwise PlI~,. = ~g | ~ll~. and this contradicts assumption (vi). So letting H~ be H N D,, or Hi n D,. in the two D., cases, we see that PIp,, = IndH',~, where ~, = ~ID~. Again this contradicts assumption (vi) since if the ratio of the two characters on H~, had infinite order then Pit), would be irreducible. [] Now let F' be the splitting field of det p adjoin all p-power roots of unity, and let F + be the subfield of F' fixed by the complex conjugation Zl. Lemma 6.2. --- The restriction of 9 | K to GaI(F~/F +) /s neither reducible nor dihedral. Proof. --- Let V be the representation space for p | K. Suppose first that p Q K restricted to Gal(F~/F*) has an invariant subspace V0. Then since Gal(F~:/F +) is normal in Gr we see that for any o E G~-, oV0 is also invariant. As Zl acts by + 1 on V0 and by the opposite sign on V/V0 there are at most two invariant subspaces. So either V0 is invariant by Gz or p | K is dihedral, but in each case this contradicts Lemma 6.1. Suppose next that 9 | K restricted to Gal(F~/F +) is dihedral. Then there is a subgroup H C Gal(Fz/F ~) of index 2 which has two fixed spaces. From the form of Pc and the definition of F' we see that the splitting field of p | K generates an extension of F' which is pro-p, whence H = Gal(F~:/F'). So H acts on the two fixed spaces via a character ~ and its inverse ~g-l (and the two spaces are unique if ~g ~: 1 as ~ cannot be of order 2). So H acts on W via the characters { 1, ~2, ~-'2 }. Either gt is trivial, in which case Gzc acts on W via the abelian group Gal(F'/F), or the subspace of W corresponding to the character 1 is invariant under Gy. as Gal(F~-/F') ,~ Gr.. In either case we get a contradiction to Lemma 6.1. [] 90 C.M. SKINNER, AJ. WILES Lemma 6.3. (i) There exists o E Gal(Fz/F') such that the eigenvalues of p(o) have infinite order and are in A. (ii) There exists o E Gal(F~/F+)\Gal(Fz/F ') such that the eigenvalues of p(o) have infinite order and are in A. Proof. -- First we prove parts (i) and (ii) without requiring that the eigenvalues are in A. (i) If o E Gal(Fz/F') has eigenvalues of finite order then the eigenvalues must be 1 as the image of Gal(Fz/F') is a pro-p group and K has characteristic p. Assume no o as in the Lemma exists. Pick a ~ E Gal(Fz/F') such that p(~) ~: 1, which can be done as p is not abelian. Pick a basis for P@K such that p(~)= (~ ~) with a:t= 0. any o E Gal(Fz/F') we have trace p(oz)= 2, so if p(o) = ( a~ s ) then Then for CO aa + aca + d~ = 2 = ao + d~. It follows that ca : 0 for all o E GaI(Fz/F'), contradicting Lemma 6.2. Thus there exists a o E GaI(Fz/F') such that P(O) has eigenvalues of infinite order. (ii) Assume otherwise. Then as in part (i), we see that there are only finitely many possibilities for the trace of p(o) with o E S = Gal(F~:/F+)\Gal(F~./F'). Fix a "c E GaI(Fz/F') such that p(~) has eigenvalues of infinite order. Choose a basis for p | K such that (p@K)(~)= ([~ ~-1). I~br any oES, ifp(o)= (~ ~), then we have trace p(~"o) = 13"a~ + 13-"d~. Since there are supposed to be only finitely many choices for the trace, a~ = da = 0 for all o E S. It follows easily that p | KIG.~(Fz/V+ ) is dihedral, contradicting Lemma 6.2. To complete the proof of the lemma, note that in (ii) the eigenvalues will necessarily be in A. This follows from Hensel's lemma using that the two eigenvalues are distinct modulo ~.. Then part (i) follows also by taking the square of any o obtained in part (ii). 5 Lemma 6.4. - /fG is a narmal subgroup OfGaI(Fz/F +) offinite index then p @ Kl~. is irreducible. Proof. -- Suppose that V is the representation space for p @ KIGaI(I,z/F, ) and V0 is a subspace invariant by G. By Lemma 6.3(ii) there exists an element of G whose eigenvalues are 13, 13-1 with 13 of infinite order. Arguing as in Lemma 6.2, we deduce that either V0 is invariant under Gal(Fz/F) or the representation is dihedral, contradicting Lemma 6.2. [] RESIDUALIN REDUCIBLE REPRESENTATIONS AND MOI)UI~kR FORMS 91 Let ;~ = ad~ = {f E ad P "trace f = 0 } where as usual we identify ad 0 with Homa(~gg , '?~, ), '~Z being the representation space for P (more precisely qZ is a free A-module of rank 2). Let o~-, = ~'/~.". Lemma 6.5. -- There exists an integer N 2 with the following propero~. If M C 3", is a submodule for some n and )v~m ~- 0 for some a > N2 and m E M, then )v"- ~-N2),~'- C M. The same holds if M is a G-submodule of 57", for G a normal subgroup of Gal(Fx/F +) of finite index, N2 depending only on G. Proof. -- Suppose x E ad~ - ~.ad~ Then by Lemma 6.4, A[G]x D ~/ad~ for some minimal r = r(x). Define a function f : ad~ - Xad~ , Z by f(x) = r(x). Then f is continuous and hence imf is finite. Let N2 be the greatest value of imf Now 5F', = a d~ and we pick s maximal such that E~y = m for some y E .~7",. So a + s < n. By the definition of N 2 we see that ~,Ne.~7", C p(G)y, whence which completes the proof. ~, Remark 6.6. -- When combined with Lemma 6.3 this shows in particular that #(JT",) 6 is bounded independent of n. As above, let qZ be the representation space for P. This is a free A-module of rank 2 having for each v[p a filtration 0 C ~'~l,v C ~/, such that '~2Zl,,, is a free A-module on which Dv acts via a character reducing to ~ modulo ~.. The quotient "~Z2, v = ~/qg" 1,~ is a free A-module on which D, acts via a character reducing to 1 modulo ~,. If P | K is type A at w, then there is a filtration 0 C ~l w C q~' such /tJ that both ~Z~ ~ and the quotient ~Z~ L'= ~Z/~ZI ~ are frec A-modules on which I,, acts trivially. If p | K is type B at w, then qZ decomposes as '~?[ = ~ ~):g/,~ 2' with I~ acting on the first factor via ~ and acting trivially on the second factor. Also as above, let ;7"= {f E adp: trace(f) = 0}. Let ,~7 "~ = {f E .'7: ./'('~) C 'ggl,~}- Similarly, ifp| is type A or type B at w, then let .~u,= {f E .~': f(~ C qgl'}. We write ~F.~-ord w ord n ,~, /~, .~u,/~.., respectively Let ,~,,, ,~.., ~,, and "~,i for 3"/V, and H,,(.~,,) = Hi(Iv, ~. / ~176 r/[ v ~1, 0 ] ' and let {tI (Iw, ifp| typeAor type B atw, Hw('qT") = o otherwise. 92 C.M. SKINNER, AJ. WII,ES For each w E Y-, put L.,(~7"~) = ker{ Hl(Dw, 57~) , Hw(5~'.) }. We define a Selmer group for ."2". by H~:(g'.) = {or E HI(Fz/F, 57".) : res 0t E Lw(~57,,,) for each w E Z}. For each place w E s denote by L~(J'.) the orthogonal complement of Lw(~.) under local duality (so L~(~7.) C_ HI(Dw, .7,,(1))), and put H~(J'.) = {or E H'(Fx/F, .~.(1)): reSwOt E L*,(:~.) for each w E Z}. By the argument for [TvVI, Proposition 1.6], which generalizes easily to the case of an arbitrary totally real field F, (6.3) #H~:('~3") - ho~(57.) H h~(.~,,), #Hx(3".) weX where h~(.~"n) = #H~ "~")" (#H~ "~'~"(I)))[F~ #H~ .~'~.(1)) h~,(g'.) = #--H~ :7"(1))'#Lw('~") #HI(Dw, .'~,,) We now estimate these factors. For two positive quantities B and C (possibly depending on n and Z), we write B << C to mean that the ratio B/C is bounded independently of n and the places in Z (it may, however, depend on p and #Z). Similarly, we write B ~ C to mean that max(B/C, C/B) << 1. A simple computation using our hypotheses on p shows that (6.4) Almost by definition, hw(27~) = #H~ ,~.(1)) if p NK is unramified at w. (6.5) Suppose that P | K is type A or type B at w. From the definition of L~(S'n), it is I w clear that H~(Fw, .~7"~' )~--~ I.w(.~). The order of the quotient Lw(.~)/H'~w, .~F':") is I)zt bounded by the order of K. , where K,, = ker{ H'(Iw, ~,,,) ---+ H'(Iw, z~,,,/o~)}. 93 RESIDUALLY REI)UCIBI,E REPRESENTATIONS AND MOI)ULAR FORMS The exact sequence o--, 37' , ,o ,gives rise to an exact sequence H'(I~;, ~.~) --- J'~' , (.~,/.Y'7) ~' ~- A/~." ~ A/~. = ~ Kn ---+ 0 ifp| typeAat w, and --Iw ,-,., zt: I 0 , )" , Ht(I~, .~.'") , K. ,0 I w ~, = #(~~ /~7s ) ~ 1, if P | K is type B at w. In the former case it follows that #Kn and in the latter case it follows that #K,I~ 'L= #H1(I,:,.57"~) I)~ = #H~ ,~,,w(1))x 1. It now follows from local duality that (6.6) h~(~-~,) ~ 1 if P | K is type A or type B at w. It remains to estimate h~(.'W,,) for riP. To do so, consider the diagram H'(D~, .~,,) n~ v 7l/ n, t) ]" Our hypothesis (vi) of (6.2) implies that #cok(q)) ~ 1. It follows that #HI(D,., ~7".) (6.7) #L,(o~',) = # ker(V ) = #im(~) ( ~ / ,~ord\l,, #H'(D~, ~57,,,) 9 #Ht(F,,, ,. ,/.. ,,,, j ) (7-ord ~./~7 "~ From the long exact cohomology sequence for 0 ~ ,_ . ,, ~ ~7. ), n /,]~l n, v ) 0 and the fact that D+ has cohomolog-ical dimension two, one finds #H~(D,,, .SK./.5~~ #H2(D,,,.'~,,) . " hi*" n, V ] " #im(q0) - #H2(D,,, , o~.a ord 9 ) ) #H2(D,, Substituting this into (6.7) yields (6.8) h~,(:~.) #H2(D~,, ~;~ord] - ,,,,, ~ .#(A/~,")-'elv,,:~ #(A/V)2rF,,:%I Here, we have again used hypothesis (vi) of (6.2) (really its implication that )CIX2 -1 ~ e). 94 C.M. SKINNER, AJ. WILES Now writing Z = EoUZ' with { v[p} C_ ~,, p ramified at each prime in Z0\{ v[p}, and unramified at each prime in Z' and combining (6.3)-(6.6) and (6.8) yields (6.9) #Hz(~7.) x #ns 9 II #H~ D,~, ~.(1)). wCZ' The exact sequence 0 ,0 gives rise to the commutative diagram 0 0 0 l 1 l kn l 1 1 0 , HI(Fx/F,.Tn) H1 (Fx/F, j~m) X" , ltl(Fz/F,.,Tm_n) l l l X ~ GwEN~Mw, n "--~ , Hw,,--n,"~" ' @weXoHw(.L~'m) ' (~wEXtlltw(.~P"m-n) whose last two rows are exact. Each M~,, ts a finite group such that #Mw, n ~ 1. It is apparent from the diagram that (6.10) Itz(~7"~) ~ Ux(,~,,,) and #Hx(,~) x #Hz(.~m)[)~"]. Similar considerations show that (6.11) #H~,(.~,,) � #H~,(.'~"m)[~.~]. We will combine the above computations with the following lemma to deduce some results about "divisible ranks" of various Selmer groups. Lemma 6.7. - The groups Hx(,7,,) and H~(,~) are finite A-modules whose minimal number of generators is bounded in terms of #Y, but independent[r of n. Proof. -This follows from (6.10) and (6.11). Note that as U~:(.Tn)is a submodule * t" of" Hx0(o~'~ ) it suffices to prove that the number of generators of H~:,,(,Tn) is bounded independently of n. [] A refinement of this lemma using also (6.9) is the following result. Lemma 6.8. -- Forming limits with respect to the obvious maps (7~) -- (K/A) T | X and lim (.7,,,) = (K/A)' | X* lim Hx~ ~ H~ "~ tl n with r < oc and X and X" finite groups. wweX~m RESIDUAI~IN REI)UCIBLE REPRESENTATIONS AND MODULAR FORMS 95 The following lemma is an analogue of [W1, Proposition 1.1 l] and it occupies a similar place in the proof of the main result of this section. Lemma 6.9. -- Let E be the splitting field of p, and let E~ be the extension of E obtained by adjoining all p-th power roots of unit)'. There exists an integer Nl > 0 such that for each n, H~(E~/F, .~(1)) /s annihilated by )(~'1. Proof. --- Let F + and F' be as defined prior to Lemma 6.2. There is an exact sequence 0 , H~(F+/F, .~7",(1) G~0':~/v+)) , Hl(Eoo/F, ,r (6.12) , H~(E~/F + , .~,,(1)). The first term in this exact sequence is bounded independent of n by Remark 6.6. Now consider the last term of the sequence (6.12). Let A = Gal(F'/F +) ~ Z/2. There are isomorphisms (6.13) H~(E~/F + , ,~(1)) ~ H'(E~:/F', ~(1)) A=' -~ H~(E~/F ', ,~Tn) A=-I , the first by restriction and the second by the fact that 27,(1) and .~ are iso- morphic as Gal(E~/F')-modules. Note that Gal(E~/F +) and GaI(E~/F') project iso- morphicaUy onto subgroups H + and H' of Gal(E/F), respectively. In particular, HI(E~/F ,, .~,)A=- l = HI(H ,, :7-)A=-l, and an element of the latter corresponds to an equivalence class of representations into GL2(A | aA/V) having trivial determinant and reducing to p modulo e. Here A| is given a ring structure by setting ~2 = 0. This correspondence is given by a E Ht(H ', .~P",) ~ , Pa : H' , GL2(A | e_A/~"), Pa(o) = p(o)(1 + ea(o)). Put H = H+\H ', and define a map q0" H , H', q0(o) = lim (ol~P"). n~oc Consider the open set HM = { a E H: trace(p(~)) ~ 0mod~ M }. By Lemma 6.3(ii), HM is non-empty if M is large. Fix such an M. We will show that the closure of q0(HM) has positive measure (with respect to the Haar measure of H'). For each n, write Pn for p mod)~n, and write H2, H',, and HM,, for the respective images of H + , H', and HM under p,. Being open, Hut contains a translate of an open subgroup of H', so there exists a constant C > 0 such that HM (6.14) - - >/C for all n. #H'. 96 C.M. SKINNER AJ. WILES Suppose h, h' E HM are such that pdh)~: p,(h') but q0(h) = q~(h')mod~. ". It is not hard to deduce that trace p,,(h) = trace pn(h' ). With respect to a basis for P,, such that p,(h) = (1~ _~-~), p,(h') = g-l([~ _~-1 )g for some g E GL2(A/~. '~) commuting with 9,,(q~(h)) = (1~ [~_l ) but not with (1 -~)" Therefore, given x E H:, we find that #{ h E HM, ,~" q0(h) = x } ~< #(A/K'~M). Let SM = q0(HM), and let aM, n be the image of SM under p,. Combining (6.14) and (6.15) shows that J~SM, n >/C 9 #(A/K TM) -~ > 0, #H'. from which it follows that g = lim SM M . ,n, the closure of q~(HM), has positive measure. Fix c~ E Ht(H ', g-)a--1. For (~ E HM, trace 9a(q0((~)) = trace p(q~((~)). As trace(.) is continuous, this equality holds for all s E SM" Fix a ~0 E SM having infinite order (this is possible provided M is large enough). Choose a basis for 9 such that P((~o) = (~ ~-1 ). Then Pa((~0) = .([~ [~-1 ) (l+Exew 1 -ezex) with ~.Mx = 0. Put O~ 1 = ~,Mo~. It follows that Pa I ((~0) is diagonalizable with eigenvalues [3 and [~-i. Pick a basis tbr pa I such that Pat(~ = ( ~ ~-~ ). As SM has positive measure, there must be some r > 0 such that X, = S M 7/(~0 SM also has positive measure. For any (~ E X, Y r trace Oh, ((~) = trace p((~) and trace pa, ((~0r~) = trace p((~0(~). Write p(o)= (a~,:o dob'~) and pa,(o)= "~ It Follows that both (~2r -1)a(o~x)o and ([3 ~r - 1)d(oq)o are in A. Thus if 0~2 = K'-'M~oq, then a~ = a(o~2)o and do = d(o~2)o for all ~ E X,. There exists ~ E Xr such that b, cT ~-O. If this were not so, then the image of pill+ would be triangular on a set of positive measure and also on the group generated by the set. This is ruled out by Lemma 6.4. Let nL = ordx(b~). If o~3 = ~n, c~.;, then b(a3)~ = b~(1 + et) for some t. Rescaling the basis for Pa~, we can assume that b((x3)z = bz. Now put a4 = ~. nl o~g. AN bzc(o~3) , : 1 - a(Ix3)~d(a3) x = 1 - ardz E A, it tbllows that c(cxt)z = c:. In particular, p(x) = Pa,(x). Now pick an integer s such that Y, = Xr M ~-'Xr has positive measure. As the eigenvalues of p(x) have infinite order and p(z) is not triangular, p(x') is not triangular. d(a,)o/'b(al)~ (a(al)o,~a,)o RESIDUALLY REI_)UCIBLE REPRESENTATIONS AND MOI)UI.AR FORMS In particular, b,,, c v ~= 0. Moreover, if ~ E Y, then by considering 9a,(#~) we see that (6.16) a.va ~ + bz.~c(lX4) ~ = av, c~ E A and cz, b(a4)~ + dx, d~ = ~' E A. Let n2 = max(ordx(bT,), ordx(q,)). Put a5 = k"'~cz4. It follows from (6 9 that for a E Y~, b(a5)~ = b~ and c(a5)a = c~. In other words, 9(6) = 9a~(o) for all ~ E Ys. The same holds for all elements ~ in the subgroup G generated by Y, This subgroup has positive measure and hence has finite index 9 Choose a subgroup G' C_ G of finite index that is normal in H +. Consider the exact sequence 0 , Ht(H'/G', (.~)G') , HI(H ', 3",) res H'(G', .5;r~). We have shown that if a E HL(H ', .:~7"~) n=-l, then res(o~) E HI(G ', 2W,) is annihilated by ~3.',t,+2,~+~.2. By Remark 6.6 there is an integer N2 (depending on G') such that k x' annihilates (.~n)G'. Therefore, 3 3M~+N~ +2~,+~0 H t(H , ' .~,,)~=-, = 0. Combining this with (6.12) and (6.13) yields the lemma. [] The next result, the principal result of this section, will enable us to control the ranks of various "tangent spaces" in the auxiliary deformation rings and Hecke rings that appear in the proof of the fundamental isomorphism (see section w Proposition 6.10. -- Let 6 E Gal(F~-/F') be an element such that the eigenvalues of 9(~) are in A and have infinite, order, as in Lemma 6.3(i). Then there exists an integer r = r(9 ) such that for each m > 0 there are infinit@ many sets Q = { Wl,..., w, } such that (i) Nm(wi) = 1 modp '~ for each i. (ii) 9p(Frob~) = 9p(6) mod 3. m for each i. (iii) lira Hx~t(,~,) _~ (K/A)' q~ Xx~t with I2o_ = 120 U Q, #X~:o_ < C(6, r) < e~ for some constant C(6, r) depending only on ~ and r. Moreover r is given by I2~nma 6 9 Proof. -- If the eigenvalues of c~ are cx and 13 then let )d be the highest power of ~. dividing (0~/13- 1). We fix a fi"ee, rank one quotient ,Ag of .57(1) on which ~ acts trivially. We denote by xn the projection of ,~,(1) onto ,/Zg, = ,~#g/~n. Write 9 * ~ Y O Xr~ hmH (Jn) (K/A) * ~0 i as in Lemma 6.8 9 Let f be the smallest integer such that kf annihilates X~. Let N~ be as in Lemma 6.9 and let N2 be as in Lemma 6.5 (for the group Gal(Fz/F+)). 98 C.M. SKINNER, AJ. WILES As the natural map H~0(~,,, ) , H~0(.~,,, ) for n' > n has kernel of order bounded independent of n we can choose n > m sufficiently large that H~ (..~,,) '~ t~(A/),5) @ X~ i= l with ri > N~ + N9 + e +f Let [el] E H~)(.Cz',) be a cohomology class of exact order ~, rl (where the ri are indexed arbitrarily) where Cl is a cocycle representing [cx]. Let E, denote the field generated by the splitting field of p mod ~,n together with a primitive p"-th root of unity. Suppose the annihilator of res([cd) E HI(Fz/E,, .~(1)) is k q. Then by Lemma 6.9 st > rl - N1. The restriction res([cl]) determines a homomorphism f E t [om(Gal(Fx/E,,), .~7,(1))C;ai(Vz/}). Since f has order s imf _D ~,a~,7,,(1) with n - al = sl - N2 by Lemma 6.5. Let Ml be the fixed field of kerr and pick "q E Gal(MI/E,) such that =,(f (~L)) has order at least ~,'1-N~. Let h = sl - N2. Put ir 0 gl = t~IM 1 "Zl otherwise. Then r~,,(cl(gl)) has order > ~,q. We choose a prime wl of F such that p is unramified at wl and Frob=,~ = gl in Gal(M1/F). By the choice ofg~, Nm(w0 = lmodp ~. Moreover the image of [el] in HI(Fw~, .//g,(1)) has order /> ~). r t Now suppose n' > n. Write H~0(.r ) _ "~ (~(A/~, r') | X* ~. We may assume that i= 1 there is a cohomology class [C'l] E H~,(.r of exact order ~.'~ such that ~m~ [C'~] is the image of [cl] in H~(~',,) for some ml ) Yl --rl. It follows that the order of res(~, m' [all] ) E HI(Dwj, ,-57n'(1)) is at least )v q-'. As tl-e = r~-NI-N2- e > 0 the order of res( [C'l] ) is at least ~,~-NL-X~-~. Let Yq = ~, U { w~ }. It then follows that l'_lm tI~t (..~'~,) ~ (K/A) r-I 9 X* -- 32 1 n t with ~:~X~l x( #X~.,0. #(A/~, Nl +'~'2+t'+'J). Suppose that inductively we have picked primes wl,...,wj such that for Ej= zo u lim tt~2(~ ) --~ (K/A) r-j Q X~ n RESIDUAIJN REDUCIBLE REPRESENqATIONS AND MOI)UIAR I,'ORMS 99 with #x;, .< We repeat the construction given for w~ and obtain a new prime wj+l. When we reach j = r we have lim H~,(.'U,) ~ X* #X~r <~ #X~0. #(A/k N' +x"+~+f)/. From this it follows that #H~;(~) ~ #X~, and by (6.9) #Hz~(3~) x #(A/k")r 9 #X~;. The proposition now follows from this together with (6.10). [] 7. Nice primes at minimum level In this section we assume that F is a totally real field of even degree. Associated to a cohomology class c as in (2.1) is a deformation datum ~ = (6s, Z, c, ../~g). We suppose that we are given a prime p C_ T~ which is nice for ~ in the sense of w rain Note that since p E p, p must come from a prime of T~ , and we also use p to denote this prime. min Now T~ ~ acts on the module M~ = M~J~ )m defined at the end of w Furthermore, T~ ~ is a finite, torsion-free A6:,-algebra. On the other hand, associated to ~ we have a deformation ring R~ n defined in w which is also a Ar~y-algebra. There is no nalural map R~ n ~ T~" since we have no natural representation with coefficients in T~)" ~. However there is a pseudo-representation with coefficients in T~" inducing the horizontal map in mi,l R~ ,,, ~ ) Tc(/min (7.1) rain N, N r~ Rff n min (cf. (3.4)). The map r~ is the one which induces the pseudo-representation associated to p~ n (see w mm Since Is is nice, T~ /Is is of dimension one and p C p. So the integral closure min A of T~/p is isomorphic to k'[[kll for some finite extension k' of k and some k. Furthermore the assumption that Is is nice ensures that under the composite map mill Ad , T~ /p ~-~ A the ring A is finite over A~ . (This is because the associated representation PplDi has at least one character of infinite order on the diagonal.) Writing A6 = C[[Zl, ..., Zm'l] 100 C.M. SKINNER, AJ. WILES let us suppose that zi t ~ s E A with ui a unit or zero for each i. Then we may take ri > 0 tbr each i (as follows from the definition of the A(# -action on Ts in w and we may assume, after possibly renumbering, that ul is a unit. Set (7.2) a~ = C'[IW~, ..., W,,]l, where C' = ~ | W(k'). There is a map ~-e~. -+ A defined by Wl , ~ ~. and Wi ~ ~ 0 for 2 ~< i<~ m. Define a homomorphism Ae~, ' Ae~ by z~ w--~W Iul, zi~ >--Wi+Wlui for 2~<i~<m. is finite and flee over Here ui denotes any fixed choice of lift of ui to Aes.'. Then ~.c A and we have a commutative diagram of rings ,< A. -< rain T~ From this diagram we deduce the existence of a prime p of ~n | Ar extending p. Similarly we deduce the existence of a prime p~ of R~ ~ @Ac, A~ extending p~, where p~ is the prime of R~ n associated to p as defined at the end of w For simplicity of notation, we may write p to denote p~ if the context in min makes this usage clear. It will also be convenient to write ~ for T~ QA~.~ AS' and /~_ minM. -- min R(,gr;~mi .... for R~in | A.# from now on. Let v ~ Jp and (T~ ~ denote the localization ~_min ~ nfin and completions of ~ and T~, respectively at p~ and p. Definc R.~ p~, ~, and (P.,~ p~)~ similarly; where pP~ is the inverse image of p under r~n: R~ p~ --+ R~ ~. Lemma 7.1. - There is a natural local surjective map rain N min (7.3) t~(~, p) 9 (R~ ~ --, (T~)3 rain rain , under which trace p.@ (Frobe)| 1 ~ T(g)| 1 and detp~ (Frobe)| 1 ~ S(g)Nm(g)| l Jbr a//g ~X. RI;SIDI,ALLY REDUCIBLE RI~PRESI'NTATI()NS AND MODUI..XR FORMS 101 Proof. -- Let rc be a uniformizer of ~", and let P = (Tt, W2, ..., Wm) C ~-C, 9 Write A~,, e for the localization and completion of Ac~ at P. Note that P = ~ f-/A,<,,, l,et C-J0 be the set of primes q of ~,~, o[ dimension two contained in P and such that each quotient *r /q is again a regular ring. It is easy to see that if q E G0, then qA~, p is also a prime ideal and that the set ~ 0 is Zariski-dense in spec (~-r~,P). min Let Q be a minimal prime of T~ contained in ~'. Let B be the integral closure min of T~/Q in its field of fractions. Observe that B is a finite, torsion-free ~.c -algebra (cf. [G, 7.8.3]). Let Pl,...,P, be the (finitely many) primes of B extending ~. For each i = 1,...,s let g~, be the set of primes of B of dimension two contained in Pi and extending some prime in ~C,~0. Note that given any q E ~)'0 there exists a prime D. E ~i extending q (cf. [Mat, Theorem 9.4(ii)]). For each i = 1,...,s we have a commutative diagram 1 1 The arrows are the obvious maps. That the top arrow is an injection Follows from the observation in the preceding paragraph. That the rightmost vertical arrow is also an injection follows t~om the fact that Bpi is a finite A~ p-algebra and that each A~. ~/q is a one-dimensional domain. If the bottom arrow were not an in~ection, then neither would be its composition with the leftmost vertical arrow, since Bpi is a domain and an integral extension of Xr,v (cf. [G, 7.8.3]). Choose t.~ E ~i. Let T = B/D and let R be the integral closure of T in its field of fractions. Let q = D KI At. and let A = A~/q. Note that A is regular. Since T is a finite, torsion-free A-algebra so is R. Let r ...,r be the (finitely many) primes of R extending pi. By Proposition 2.15, for each j = 1,...,ti there exists an extension R+ of R, Rf a domain, an extension K':j of C, and a prime q3j of Rf extending glj such that there exists a detbrmation Pi into GL2(R;) that is (~'.i, Z, c, ,//g )-minimal and whose associated pseudo-deformation is just that obtained from the one into T~ '~ described in w via the obvious map T~ n -* T. We have natural injections ti Tp, ~ II R% j=l and t t. t t. j=l .= , q3~ 9 102 C.M. SKINNER, AJ. WII,ES We claim that both maps remain injective upon passing to completions of the rings in question. For the first map this is an easy consequence of the t~ct that both T and R are finite A-algebras. For the second map we note that each Rq3j is an integrally closed one-dimensional domain and hence so is each R~. On the other hand, each R* is reduced, so the kernel of the induced map R%---+ . + must be either (0) or J, ~ J, 9i ~3j. It cannot be the latter, as we have R%. ~ R j, + V*.~ ~ ~+ i, ~+" This proves the desired injectivity. We have an induced injection t i (7.5a) j= l ' . We also have a map RT" | Cj ~ Rj inducing 9f. It is easy to see that the inverse image of ~3j+ in R~ n is just p(/. We thus have a map ti /~nlin~ n ~.+ (7.50) n . +- j:I'LJ" j, g3j mill Composing (7.5a) with the map (R~ p~,~ --~ Tp, coming from =~ and composing /~nlin,~ (7.5b) with the map (R~ p.,)~,~ ~ t ~ #~ (see Proposition 2.11 for surjectivity) yields the same map. Therefore we haw ~. a map (~min,~ ~p. through which (R~p~)~,~ ~ ~'p factors. Combining this with the injectivity of the bottom row of (7.4) yields a map 3" mill I "m l through which the map (R~ p.,~,, --+ 1-[~=1B,, coming from x~n factors. The image of lllin this last map is just that of the natural map fi'om (T~ ~/Q. As the latter map is injective, we obtain a surjection _ min\ ~ min - -- min mm through which the map (R~ ~)~,., ~ (T~)~/Q coming from n~ factors. From this we obtain a map (7.6) - rain N rain 1-I(T RESII)12ALLY RE1)U(,~IBLI~ REPRESENTATIONS AND MOI)ULAR FORMS lnin where the product is over the minimal primes of T~ contained in ~'. Since the maps rain ~ min (R,~ p,)~p~ ~ (T~)~/Q factor through the surjection (R~ r,.~)~v, -, (T~)~ coming from min -- min ~ , the image in (7.6) is the same as that of the natural map from (T.~,)~. This latter map is injective, and we therefore have a surjection min ~ min min through which the map (R~ p~)~,~ ~ ('r~)~ factors. [] Remark 7.2. -- The same result holds for ~ replaced by any of the auxiliary deformation data ~ 0v with the same proof. Since p is nice for ('_~, there is some choice of basis for pp such that pp has image in GL2(A ) and the corresponding representation into GI,2(A ) is a deformation of type-(~:', Z, c, ,~N). From here on we write pp for this deformation. We will assume for the rest of this section that ~0~ = (~J,. (i.e., that E = Z~ is the set of primes at which p~ ramifies together with ~)o = {v~" vilp}, and that = = Next we define the rings and modules which will be used for patching. Let R~ p~ denote the ring R(_/p~ @ A,~,; and let M~ = M~ @ A(~,. Then we define A( A(~" (7.7) N (~ = im{M~ , (M~)~/P} where (M~)~ is the localization and completion of M~ with respect to p C T~ and P C_ ~,~ is the prime defined in the proof of Lemma 7.1. There is a commutative triangle (7.8) 111[11 ~ nlil| R(_/ r " (R~)~/~-F0, P) where F0 = Fitt ( (M~ ~ ~ /Dmi"~ -- (R~-'i~ 25J C ~l,~ j~ is the Fitting ideal of (M~)~ as an The maps r and q)2 are the obvious ones. We define rings (7.9) R (~ = im(q)2), R tr(~ = im(qh). Now we introduce auxiliary levels. First we fix a c~ E Gal(F~/F') as in Lemma 6.30). Then there exists an integer r = r(pp) as in Proposition 6.10 with the following C.M. SKINNER, AJ. WILES I04 property. For each odd integer N there is a set of primes Qx = {w(~/, ..., w;(~} of F satisfying Nm(w:/~) = l(modp x) as well as property (ii) and (iii) of Proposition 6.10. We can and do choose the sets Qx to be disjoint from each other as well as from Z. For such a set Q = Q.~, we earlier introduced a deformation problem 6_~o as well as associated Hecke and deformation rings T~ and R~o ~. In particular at the end of w we associated a T~Q-module M~-Q to ~Q. We now set M~Q = M~ct ~A6 min as before and note that this is a T~Q-module. For each wi = wl ~ E Q = QN there is an associated element &o, E End T~,~, -,r (M~Q) as in Lemma 3.21. We let si = 8w,- 1. We can then define, for each odd integer 1 ~ a ~< N, (7.10) M~ = im{ , (M~c~/(P , s I , ...,s, 1)} rnin where the completion is as a T~osmodule (with respect to p) and Q = QN. Then a+l) M~ ) is a module over the ring a~ = A[[Sl, ...,s~]]/(s~ +l,..., sr by construction. Let 9~ denote the universal pseudo-deformation associated to 6~Q. Let 6~,~ E Gal(F/F) be such that ~w, E Iw~, the inertia group of wi, and ~w, is a generator of the p-part of tame inertia as in Lemma 3.21. Then there is a map of rings (7.11) X~ [Ill,..., tr]] ' I-(~s -- Rc~"(~ ~ A,.' @ X(~/ given by ti, , trace (9~,(~w;) - 2) @ 1, where here Q = Qx = { w,,...,/~)r }- We have a commutative diagram (7.12) a+[ a+l ~min "7" =2- _i~ " (R~ct)r tl ,..., tr , "p'Fx) 9 2, a RESIDUALLY REI)UC1BI,E REPRESENTATIONS AN1) MODUI~AR FORMS 105 where Q = Q~ and Fx = Fitt((M~Q)~) is the Fitting ideal of (M~o)~ with respect to /~_min ,~ --a --a the ring ~ ~Q2~. The maps are the obvious ones. We detine rings R (~) and R tr(N) by R~IN~ 9 , (N),, Rtar(N) 9 (N~ (7.13) = xm(~p.), ~), = tm(%, ~). a+l a+l These rings are both algebras over B~ = A[[h, ..., t~/(t~ ~r-, ..., t, T) by construction. 9 - GL / t'D min, " Let P~o. GaI(F/F) --+ 2~,.~o)~) be the representation obtained from p~Q. %-1 for some character ~/i. It is easyto see that for each wiE O_)~'~cah% ~- ( qi ) Define a map Al[sl, ..., s,]] -+ R~ ~ ~A K by si ~ ~i(cw,)- I. This makes R] "< | K into an A~ | K-algebra compatibly with its structure as a B~-algebra. Moreover, it is obvious that each si maps to 0 under the canonical map R~ "~? | K---+ R (~174 K. ,-~min ~ ~j... Q. The action of R~ IN) on M~ is obtained as follows. First ~(~ acts on nin ~ min whence it acts on M~ \3 through the image of T'~, z in (T~o) ~. Now we have homomorphisms a+l a+l /'7 11~ k"x~/ ps ~min ~min R~c ~ * T~o " " crwQ)o/(p, tTe-,...,tT,~FN) a+l a+ I -" min p -7- c_ , tT , ..., t, , and the diagram commutes by Lemma 7.1 and the remark following it. So by Lemma 3.21 we get an induced action of R~ (''~ on M~ v' which is compatible with the A~-action of the subring B~. Now put 2 t M(0', = | N (~ i--- 1 where N (~ is as in (7.7). The same reasoning shows that R 'r(~ acts on M (~ and a diagram as in (7.14) holds for N = 0. We define R~ \; and ~'0 by R~\9= R (~ and R~ rr = R '~(~ , -, ~ ~2 r rib define M't~' we first define a natural map M~o , M~ (Q = Qx). Let = = = ...,w)'}. l'br each 0 ~< i < r define a ~.r c-~ i c.c~ i where ~i {w~ ~, '?" l-l+ fl r rain "~2 + rnin map 1,~,o~i, -- --+ H~(U~+,) by (f(g),f2(g))' ' f(g) +J2 (g(l X,,1)) where 106 C.M. SKINNER, AJ. WILES ~,i§ E U/v | Z is the element chosen in the definition of the Hecke operator T(g~,,+~). Repeated composition of these maps gives a map H+(U~ ) 2r ~ H+(U~Q) and taking .... 2r Pontryagin duals and tensoring with A(~ gives the desired map M~ ~ M~. We define ~,,N) ~ ~ 2 r 9 "~0 to be the image of M~(~ , (M~)~/P. Clearly M:~: C_ M :~ We now verify that these constructions satisfy the properties in w needed for :x) formal patching. A bound for the number of generators of R (x! = R x is given by dimk(Hl(F~-~Q/F, ad~ which is easily bounded independent of N. A bound for the number of generators of R m~ = R tr(N) x is more subtle. When N = 0 this follows from Lemma 2.10, which bounds the number of generators of R~ ~,~. In general a similar argument applies using R~ ~,~, and a uniform bound independent of N can be given for example by applying Lemma 2.10 with E replaced by Z U Q. We'. can choose the generators in each case so that (5.2ii) and (5.6ii) hold by subtracting suitable elemenu of ~: [[WI]]. The other properties in (5.2) and (5.4) follow from the definitions. Next we consider the properties (5. I0) of M2 \!. Properties (iii)-(v) are straightfor- ward, as are the first two assertions of (ii), but we need to check (i), (vi), and the last assertion of (ii). Property (vi) follows immediately from Lemma 3.19 provided the hy- pothesis that U'/F � N U' acts without fixed points on D� holds. Here we need to take U successively as U~0, U~, ..., U~, = U~z where U~ = U~, .~-2~i = f~,, and rci = {w('~!,...,~ N} and check the conditions of Lemma 3.19 with v= w(,~ for U = U~ ,. However these conditions need not hold, and instead we consider modules with an auxiliary level structure at primes gl, ...,g~ )~Z U Q~ chosen so that M~(U~,), with U"= U~ A U1(gl "" "g~), is related to M~(U~) in a simple way. To achieve this, pick primes g 1, .-., g.~ satisfying the hypotheses of Corollary 3.6 as well as satisfying the conditions (7.15) (i) Nm(s 0 ~ l(modp) (ii) pp(Frobef) = 9p(tJ0)mod~. ~ for e sufficiently large, where tJ0 is chosen as in Lemma 6.3(ii). (In order that condition (ii) make sense, we identify 9p with 9~ modp.~, which in turn we view as taking values in GL~(A).) Condition (ii), for a sufficiently large, ensures that (7.16) trace pp(Frobt: S ~ detpp(Frobe~)(1 + Nm(gi))'2Nm(g,.) -~. ]'his, together with (i), ensures that M~(U~)p ~- M~,(U~,)~ ~ by Lemma 3.29, where the isomorphism is of A~;p0[Aw, J-modules..Now Lernma 3.19 and Corollary 3.6 can be used to check that Moo(U~i)~ is free over A(.~,)[A~,J for RESIDUALLY REDUCIBI,E REPRESENTATIONS ,k\'I) MOI)UI.&R FORMS 107 each wi = w i ' E QN. We then deduce the same result for M~(U~,)p and so also tbr (M~)~ and M~N;| K over A:, p[A+,] and Aa | K respectively. The second property (:,' of (5.10vi) follows similarly from Lemma 3.19(ii). Property (5.10i) follows from property (5.10vi) which was just established. It remains to show that the last assumption of (5.10ii) holds. For this we pick some I~ 1 E Gal(F/P b) such that pp(ch) has infinite order. That such a ch exists is an easy consequence of Lemma 6.3. I~t P~" be the pseudo-deformation associated to T~". It nlill follows from Lemma 3.26 that z = trace p~ (~Ji) - 2 annihilates the cokernel of" the map M~o ~ > M~ in the definition of M~ "~'. Therefore z also annihilates the cokernel of M~,2. ' M(~ and hence also that of M~ rq/ ---~ M (~ By our choice of ~, z fiSp. Note that z is independent of N. The properties in (5.9) are consequences of those in (5.10) as well as of the definitions of the M'X/'s. Next we verit) properties (5.5i) and (5.3i). Let dl(a) = dlmK(M 2 ' | K). This is independent of N by (5.10vi). Again using (5.10vi) dl(1)/> I-IRj@AK(J.VIa @A K) where as before ps(X) denotes the minimal number of generators of the S-module X. Now ~dl~a', annihilates M] ~' | K and hence 17" ...(N) ~(/l(a)Ul(1; C 1 lttR:.~ . (Mo | K). -- a @AK From the definition of R] ~ | K it follows that I = 0 in this ring so we may take d(a) = dl(a)dl(1) + 1. Now we check (5.5ii) and (5.3ii). Recall that we are given a set Q,~ = {w/~,...,wl N)} of primes satisfying the hypotheses of Proposition 6.10, and that by the same proposition (7.17) lim Hr.Q(.7,) ~ (K/Ay (9 X~z with s = Z U QN and #Xz~z bounded independent of N. Now let Q = QN and "-'min ~ ~ rain suppose that #~ ~z' #2 ~z are the primes corresponding to p in R~Q and R~r z. Then we also have the usual isomorphism in the style of [WI, Proposition 1.2] /': umin /: ~'min '~2 k-"A/A) -% Hz,z(.'~,, ). (7.18) X( 4 = Itom A k~v.~,2_/kv(,/.ct: , P), The isomorphism is obtained as follows. To an element ~ E ~o~ we associate the representation (7.19) pe Gal(F/F) +m~,, "--m~,, 2 9 , GL,,(R+ct/((I~o ) , P, ker ~o). ~dl~+)dl+t)+ 108 C.M. SKINNER, AJ. WII,ES ]'his is a deformation of pp with values in A[e]/0J'e , e 2) and its associated coho- mology class lies in Hz~2(;'7,). Conversely to such a cohomology class, we obtain a deformation with values in A[e]/(~,"e, e"), and hence by universality a homomorphism R~Q| ~' ~ A[E]/(~."e, E2). Extending scalars we get an A-algebra homomorphism ~xmin /p ~nin P,.~/P , A[e]/(~."e, c 2) which factors through ~/ . Restricting to p~ we recover an element of XQ. The restrictions on classes in H~V.~.7,,,) correspond to the restrictions required for p to be of type -~0_ e~ mi- " In particular, the "min" condition, together with reduction mod P, ensure that P~0 does not deform the determinant. This is why the cohomolog-y class we obtain is associated to "~,,, = ad~ ' rather than adp/~.". We omit the details and refer to [Wl] for a more detailed argument in a similar situation. Of course (7.18) also holds with ~ o__ replaced by ~ and s by s To apply this we use the sequence of homomorphisms -rnin/, ~min,2 .(N)/,.(N),2 p(0)/(p(0))2 P'2gQ/~P~.~Q. ) P) . p(N)/(p(N))2 ~min/, ~min,2 P~ ///P~ ) ,P) Here Q = QN. Tile horizontal maps are surjections arising from the definitions in (7.9) and (7.13). Tile maps ~O_. and 130 are surjective and 130 is an isomorphism after tensoring with K. That [~e is also an isomorphism upon tensoring with K follows from Proposition 6.10 and (7.18). Properties (5.5ii) and (5.3ii) then follow from (7.17) and (7.18). Next we verify property (5.7). Using (7.18) the condition in (5.7) translates into the requirement that #GN, where G~ = {[P~] :trace P~ = trace Pp, q0 E fro_), is bounded independent of N and n. (Here as before Q = Qx). Fix a basis for Pp such that Pp(zl) = (1 -1-) and pp(O') = -(. .u)- for some unit uEA � where zt is a complex conjugation and c' is some element of Gal (F/F). With respect to this basis write p,(o) = (a~co ~)" Fix a ~ such that c, flu 0. Now suppose that [Pt0] is a class in GN. The class [P~] has a representative P~0 such that and b'ccz E A for all o E Gal(F/F). Hence c~ annihilates [P~0]. Since the number of generators of GN over A is bounded independent of N and n (as ibllows, for example, from (7.17) and (7.18)) we obtain the desired bound on #GN. RESIDUALIN REI)U(:IBLE REPRESENTATIONS AN'D MODULAR FORMS 109 It remains to prove the existence of an element x tN) as in (5.11). Let us write M ~ min for M~, which is a T~ -module. Let J = ker{ (M/P) 2~ ~ M (~ Since M (~ ~ (M~)~/P by definition, J is just the kernel of M2'/P , (M/P)~. Let m~,...,m, be generators of J as a 1'~-module. ~min For each i, choose xi ~_ T~ , xi~'p, such that x,'mi = 0. Put x = xl".x~. Clearly, x annihilates J. Also, x )Ep since each xi ~'p. Now set MN = M~ where Q = Qx. Then suppose that we have an element ~c, in y('~ C ~<t with the property that ~ z , (7.20) y :N, ker{Mx/(Sl ..., ST) > (M/P) 2~} = 0. It would follow that xy (N) would annihilate ker{MN/(S~, ..., sT) , M (~ where x is a lift --rain ~ ~N' 9 9 (N) of x to T_~:( Q, and so xy' ' would also annihilate ker{M~ /(Sl, ...,s~) > M(~ Thus it would satisfy condition (i) of (5.11) except of course that it is not in the desired ring. Our construction of such ay (N' is an involved procedure. We begin by introducing auxiliary level structures much as we did in the proof of property (5.10v i). Let g.~, ..., g, )~g tO Q, where Q = QN, be primes satisfying the hypotheses of Corollary 3.6. as well as (7.15) and (7.16). We can and do choose the g; to be independent of N. Now let U' rain ' = M~(U~ )m | A~'- Let = U~Q V/UI(gl "" "g,) and put M N c-C/(Z <Q. t ~ min also ~_q4' = (C', E', c, J'd) with Z' = Z tO {g~, ...,gs}. Let T N denote T~(~. It is clear that M~ is a T~-module. t 2s There is a natural map My > M x defined analogously to the map MN > M 2r used in the definition of "'0~a(N)" Composing these maps we obtain a similar map , ) M e~s My . Arguing just as in the verification of (5.10ii) (see the first full paragraph ,~min following (7.16)) we find that there is some y(~') E ~a such that (7.21a) y?/. coker{M~ > M~} = 0. and 0 :~ ordx(y] N3 modp) is independent of N. (7.21l)) .(N) I We next construct Y2 E T N such that ~N~ } = 0 (7.22) Y2 " ker {M~/(s,, ...,st) > (M/P) ~'+' for then fiN) =y/~} .y~) will satisfy (7.20). (We view si as an element of End(M~) just as we did for MN. These actions are compatible under the map M'.x > MN2'.) 110 C.M. SKINNER, AJ. WILES Let U~Q = U~"N Uo(w,,...,Ws)71 Ul(el,...,er) and write M~ for the module M~(U~)m @A,< A~ . By our choice of gt, ..., g,, U'~a satisfies the hypotheses of Lemma 3.19, from which we deduce that M~/(s~,..., Sr) --- M~. Let Ix be the set of minimal primes q ofT' such that M" :~ 0. Let IC Ix be the N N,q -- -- min subset consisting of the inverse images of the minimal primes of T~. It is relatively straightforward to see that for q E I~\I the representation Pq is type A at each of the (w, primes gi and Jw: . If .-<.~: is the field of fractions of A~ then we have M x~ M~ | ~ = II M~,q qG1N M ~ M | <~ = H Mq. qEl The kernel of the map M~ | ~' , (M | J~):' is just qei.~\iH M"g, q. Therefore, ify~ '~' E ["l q, then qEIN\I (7.23) y;'. :.-,~ ker(M~ ~ M~, +, ) = 0. We choose y~X> as follows. Let P.',' be the pseudo representation associated to T-". Let "ci E Gal(F/I:) be a lift of Frob~..N, ' and let 5i be a lift of Frob~: Put Y~") = II (.l.~,, _ a,,,,qi" :(N) + 1) 2) " 1-I (T2e~- de,(Nm(gi)+ 1) 2) i= 1 i= I where T,, i = trace p~.('r,i) . , qi(N) = Nm(wi iN; ), dwi = det p',4(~i), qi(N)-' and similarly tbr Tti and dtr Then y~\/ E q for every q E Ix\I as can be seen by examining the actions (N; of Dw and D~, (decomposition groups at wi = w i and gi, respectively) on the Galois representations associated to such primes q. From our choice of wi (and our choice of o as in Proposition 6.10) 9 O O 1 [,i - d"i (q(]'~' + 1)2 = trace pp((5)- - 4 ~ 0 mod (p, ~:) for some sufficiently large e independent of N. This, together with (7.16) shows that ,iN ~ (7.24) 0 :~ ordx~y~ Jmodp) is independent of N. RESIDUAI,LY REDUCIBI,E RI,iPRESENTATIONS ANI) MOI)UI,AR I"ORMS Finally ar,,~ing as we did to establish (5.10ii) as well as (7.21a, b) shows that there is some y~')E T~ n such that (7.25a) (x~ 9 coker{M ,, x ~ M 2'+' } = 0 and (7.25b) 0 :~ orally 4 modp) is independent of N. Let y.~"q and ,4~"' be lifts ofy~ w' and y4 \~' to T x. Combining (7.23) and (7.25a) shows that y~' =y:; -'g" .y~Y) satisfies (7.22) 9 Moreover by (7.24) and (7.25b) ,N), 0 ~: ord)~(y,) modp) is independent of N. We may then take ~6 ~? to be any lift of x .y:'~: to R(. p,. Its image in W ~u'~' satisfies to, Q (5.1 li, ii). We have now verified all the hypotheses in w and are thus in a position to prove the main result of this section. Proposition 7.3. -Suppose that F is a totally real field of even degree. Suppose that .~ is a deformation datum and that ~ = ~. Suppose finally that p C T~ is a prime which is nice for ~ in the sense of w Then rain ~ rain ~ rain (i) gt(~, p) 9 (R~)~ , (T~)~ is an isomorphism and (R(/)~ is a complete intersection over ~,~, l, and reduced; (ii) M~, ~ /s a flee Cr -H,i,, ~ )~-module. Proof. -- Our constructions give the following identifications: rnin R,',0; | K = (R~)~/(pF0, P) rain RW0: | K )/P 2 r N~~ K = (I~)F/P, M(~ K = iO(N ~~ Q A K). By Lemma 5.2 the natural map (7.26) R '''~ @a K ' R (~ | K is an isomorphism. By Proposition 5.8, M (~ | K is free over R (~ | K. As the action of R (~ | K on M (~ | K tZactors through the composite map R ;~ | K ~ R ''(~ @,t K c~min~/p /~,min, ~,) , 9 $/ we conclude that M (~ | K is a free ix(,, )~/l-moame and that gt(~, p) 112 C.M. SKINNER, AJ. WII,ES min ~ min induces an isomorphism (R~)~/(pF0, P)--~ ('I'~)~/P. Picking generators of M(~174 as an R ''(~ | K-module and lifting them to (M~)3 we get a map (for some minimal s) A 9 (7.27) ('I~)~. ~ (M~)3 which is an isomorphism modulo P. Since (M~)~ is free over A~,p it follows that min (7.27) is an isomorphism. In particular (T~)3 is free over .g,~; p. As observed, the reduction mod P of the map (7.28) - min ~ min (R~)J(pF0) , (T~)~ mlil induced by ~(c.ffr p) is an isomorphism. Using that ('I'~)~ is free over A.~ we now deduce that (7.28) is an isomorphism. Under (7.28) F0 maps to zero as M~,o is a free min ~ min ('l'~)~-module. So F0/pF0 = 0 whence F0 = 0. Finally (R~)~ is a complete intersection rain ~ min rain since (R~)~/P is by Proposition 5.9. (Note that ('r~)3 is reduced as Ty is reduced.) This completes the proof of the proposition. [] 8. Raising the level for nice primes 8.1. Preliminaries In this section we complete the proof that property (P1) holds for a defi)rmation datum c~.~. However, before doing so we need some auxiliary results. We begin by imposing a partial ordering on the deformation data. If 6.~1 = (], Z1, Cl, .AN l) and ~-c2~2 = ((92, E2, c2, d/~2) are data, then we write 6.~1 /> ~-~2 to mean ~l = 82, cl = c2, Zl _D Z2, and ~t/~l C_ Jfg2. If #(ZI\E2) + #(,/f~2\.///~1) = 1, then we say that the inequality ~l /> .ff(~. is strict. Let ~ = (~-', Z, c, Jtg) be a deformation datum and suppose that p C T~ is nice for ~. As p E p, p is the inverse image of a prime of T~" which we also denote by p. We adopt the notation and conventions from the beginning of w The primary goal of this section is to prove the following proposition. rain Proposition 8.1. -- /f p C_ T~ is nice for c~j, then the map ~(c~, p). (R~)3 min (T~)p in Proposition 7.1 is an isomorphism. For ~ = ~c.@'c this was proven in w - w (see Proposition 7.3). We will deduce the general result from this case by a generalization of the arguments in [W1, Chapter 2]. RESIDUALLY RH)UCIBLE REPRESENTATIONS AND MOI)ULAR FORMS 113 8.2. Congruence maps A key ingredient in our proof that V(c"J, p) is an isomorphism will be a lower bound for the length of a certain "congruence module". In this subsection we construct min maps between various T~-modules that will be instrumental in obtaining this lower bound. We first fix a sequence of deformation data ("~.-~ = 6~ ,< .~ ,< ... ,< G.q4~. = such that .~r i ) c.(~ i_ j is a strict inequality for 1 ~< i <, n. Put min ~ m Ri = (R~) ~, Ti = ('~,@i) ~' Mi = (M~,)~, and M i = (M~.)~, where M~/, is defined as is M~ but using M ~ + instead. Let P = ~N~.4 ~. Lemma 8.2. -- Each Mi and M,~ is a free At,, e-module. Also, there exists an integer s ,7,~ t, 2 ~ such that M~' -~ Horn x (M[, Ar p)- and M i ~ Hom x (Mi, AC. p)2' as Ti-modules. (',, P (;, Proof. -- Choose a set of places {rl,..., rs}, distinct from those in Z, satisfying the hypotheses of Corollary 3.6 and such that 0J(Frob ) z~ 1 and pp(Frob) has eigenvalues of infinite order for each i. Put U, = U~iN rl(yl,...:,?~r ) and U7 i"= U'~' n U~(r~,...,r,). '9 By Lemma 3.2., To~(Ui, ~')p -~ T~,p and nlin 2 s rain § ,.~ 1~1-+. 2 2 Mo~(U, )p "~ M~,.p and Ms(U,. )p -- lvL~i,p. [] The lemma now follows from Proposition 3.3 and (3.17). Fix an s as in Lemma 8.2. Now let 2 if wi C_ ,~.f~ i- 1 {w3 = z,\z,_, u 3 if Wi E ~,i\~,i_l, and (qi - 1) (T(gi) 2 - S(g/)(q, + 1) 2) if wi E Zi\Y'. i_ 1 if wi E J/Ni-1 and Z[r,,, = 1 rli = (qi- 1) (qi + 1) (q;- 1) if w, ~ '~i-I and )~]I% ~ 1. Here gi = ~,, and qi = Nm(gi). Next we define maps of Ti-modules ' ~ t q,~ q.v 2 .~ rt 0," Mi2_'l ) M~, Oi" M~ > Mi_ 1 iS v,., o,-MfZ 114 C.M. SKINNER, AJ. WILES such that (8.1) a) ~i is injective with Xm, i,-free cokernel and ~ is surjective. 9 M~,,,(Ti_ !) with the image of det(Ai) b) ~)i O $30 (I) i O I'P i -- not a zero-divisor in Tv. 6") iITl(~I j, O ((I~i_ 1, "", Oi-1)) = im(Oi_ l, ..., Oi_t). Let ~,, = ~(~'i/ be as in the definition of T(gi). For any f: GD(A/) , R (R an ~- Xi))" We define *i to be module) let a3f: GD(M) R be given by (cq~(x) =f(x( 1 r i tile direct sum of 2' copies of the localization of the map (lim q~) | 1 : Mc.,~,_l ~)Ac ~'(~ , M~, | AO = M~, where *7" eI-I~ - I, a), c"~) r~ , eH~ a), ~') is given by (f ,f2) ~--*f + ~i~ if r, = 2 and (f ,.~ ,f~) ~----*f + ~if2 + (x~fa if ri = 3. We define 4- + + q~i " Mi+-'i ' * M i similarly and take for ~i the dual map obtained from ~i by applying Homx~r e(. , A~, ~,). Similarly, let ~, be the dual of q~i. We now verify (8.1a). Choose n to be an ideal such that U0(n) _D U,, D U(n). If u)i E ]gi\~i-I (SO ri = 3) then by Lemma 3.28 for a sufficiently large both the kernel of a ~a q~i and the cokernel of ~, are annihilated by T(g) - 1 - Nm(g) for any prime e that splits completely in the ray class field of conductor p~-n-c~. Here ~i is the adjoint of ~ with respect to the pairings defined in w Let F~ be the ray class field of conductor p~ 9 n- cxD and let F~ = UF~. Choose c~ E Gal (Fx/F~). Such a ~ is the limit of a sequence of Frobenii {Frobeo } of primes g~ splitting in Fa. In fact such a sequence rood. . can be chosen so that Frobeo = Frob~b (b/> a) in Gal (F~/F). As trace p~ ~) is the limit mod ~l~" ~ a of {T(eo)}, it follows that trace 9~,/ ) - 1 - ~(a) annihilates ker (*~) and coker(q~i). rood. . As pp is neither reducible nor dihedral it must be that trace 9~, t~)- I -e(~))~t~ for some ~ 9 Gal (F/F~). It follows that both ker(lim*~) and coker (lim~) vanish when localized at p. Thus ~i is injective and ~, is surjective. A similar argmment shows that ~i is injective and ~i is surjective. This proves (8. l a). Now if wi 9 .-.4r then it tbllows from Lemma 3.27 that ker (~i) and coker(~i) are isomorphic to submodules of (I~(U'))~' where m'm {(: bd) e, ei',)-l} U' = U~ 3- l " 9 GL?( ~v, ~i) : a -- 1, c 9 g~� with r(Si) as in the definition of lJ'~. By considering 9~!4,,~ one sees that p is not a prime in To,)(U', (~-~') (so T~(U', ~')~ = 0) whence M~(U')p = 0. This proves that RFSIDUALIN RFI)UCIBI.E REPRESENT.\TIONS AND MOI)ULAR FORMS 115 ~i and ~i are, respectively, injective and surjective in this case. The same argument applies to ~i and ~i, thereby establishing (8. l a) in this case as well. 2 s ~l/t Next we define q~i and Oi. If wi E Y.i\Ys_I, then we put ~, = | and | = ~--'l| where -s(e,) o o o o s ,) T(e,) -S(e,)-' , 01= 0 s(e,)-' . ( 01) ( !) --qi 0 -- 1 0 If wi E ././g ,_ l, then we take , (T(g:) 0 ) , (0 1) q2i = -qi T(gi) -1 ' Oi = 1 0 " We note that while T(gi) is not included in the definition of Ti-t if wi E ~/:g:-l, it is in fact in Ti_I and is a unit, so the definition of q~i makes sense in this case. To see this, let Q be any minimal prime of T,..~ ~_~. Then T(gi) is the eigenvalue of the action of Frob:, on the maximal unramified quotient of 0oJDe. (This can be checked on the representations associated to algebraic primes containing Q.) As wi E ,//~,._ i, the min representation 9~, .~ [I)t. has a non-trivial maximal unramified quotient, and it follows that the image in Ti_I under gt(~, p) of the eigenvalue of Frobe, on this quotient must be T(gi). As (8. l c) is obvious from the definition of ~Fi, it remains to verify (8. l b). Suppose first that wi E Zi\s A straightforward calculation shows that ~i o ~i is a direct sum of 2' copies of T(gi) 2 - S:g, ,/,~,v" + 1) '~ T:g qi(qi + 1) oqi S(~ i)- 1 T(g i)qi q:(qi + 1) T(g,)q, ) / '~2 -: -2 T~gi/ Skgi) -- S(gi)-l( 1 + qi) T(gi) S( g i)- l qi q,(qi + 1) Thus nil,) Oi o ~i o ~i o q~i = ____~_ , det(Ai) = (qiS(gi)-2)2'.q2'-1. IrXi That the determinant of Ai is not a zero-divisor in To is easily checked. As To is reduced, we need only verify that det(Ai)~Q for all minimal primes Q of T%. Suppose that det(Ai) is contained in such a Q. Let P C_ T~o be an algebraic prime containing Q (and hence det(Ai)). Let t = T~ei:modP : ~ and s = S(gi)modP. We will show that t e -s(1 + qi) 2 :~ 0. Let a and [3 be the eigenvalues of 9p(Frobf ). Recall that 116 C.M. SKINNER, AJ. WII,ES t = o~ + [3 and sq, = ~. If t 2 ~1 + qi) 2 = 0, then either et -- -- = qi or -- = qi. But both possibilities violate (3.3). It follows that det (A,)~P and hence det (A~)~Q. The verification of (8.1 b) in the case where wi E ,.//~ i-~ is done similarly. We are now in a position to define our "congruence maps". Put I ~ = ker{T, , To} and P" = AnnT,,I ~ Put also 1 <~j<~i 5., 2s 2 s Define @('~' 9 ~2 ~,, 9 "~0 > M i and ~('~' " *'~0~z'e"; M i recursively by ~'J) = ~1, ~(a) = @i o ~l, and (I) (z~ -- (D i O ((I) (i-1) X 9 9 9 X (I) (i-I)) ,i,('~ = (r o '-I.'~) o ($(~-'~ x ... x ,i,('-'/). Define ~i) . M~' ' ,'~0 and "M[ ~ ~'-0 in the same way but using ~i and Oi and reversing the order of composition as appropriate. Put ~,,~ -~,,,, $ = $~,,~ $~',. @=@ , @~ =@ , , and~ = Put also "l"lr r = r(n) and rl = l[ ni- l <~i<~n Lemma 8.3. - (i) im(q)~ ) = M, [I ~ 2: and coker(~ ) is a flee ~,~, p-module. (ii) @~ is surjective. / \ (iii) ~ o @~ = / (unit)xTI *] E M2,r(T0 ) with det(A) not a zero-divisor. (iv) ker($~) = M,[In'w]2'. Proof. -- Part (ii) follows from (8.1a). Part (iii) follows from (8.1b). We leave the details to the reader noting only that det(A) is a product of powers of the det(Ai)'s and the qi's. The freeness over A~, p of the cokernel of ~ also follows from (8.1). It remains to prove the first assertion of part (i) and part (iv). RESIDUALIN REDUCIBI,E REPRESENTATIONS AND MODULAR FORMS 117 Note that by (8. l c), im(@~) = im(~), so it suffices to prove that im(~) = Mn[Iold]. Next note that 9 is the localization of (lim ~) Q 1 where ~ is defined as is 9 but with ~ replacing ~i. Now let I~ ~ = ker {T2(U~. ~, C') > T2(U~ .... ~9~)}. It follows from the theory of "new vectors" that im(@ a) | K = eH~ ~), K) [I~ (For a more detailed proof in a similar situation see the proof of Lemma 3.29). Now consider the commutative diagram [imO ~ lim eH~ ~), ~?)~ ~> lim eH~ ~), (Y') [Ia ~ --o C a g l 1 0 , lim im(~") | K , lim eH~ ~), K) [I ~ ,0 a a having exact rows and with the vertical arrows being the natural maps. Applying the snake lemma we find that C embeds into a quotient ofN = lim ker(@~QK/-(~5). Arguing as in the proof of Lemma 3.29 shows that Np = 0 and hence Cp = 0. Now let I~ = ker{Too(U~, ~') , Too(U~, ~)}. It follows from the preceding remarks that the quotient M~ [I~a]/im(lim @~) vanishes upon localizing at p. As I ~ -- -ooI~ part (i) follows. To prove (iv) we first note that Mo[I"~w ] = 0 by Lemma 8.2, for M0[I"CW]is a T0/InCWT0-module and hence a torsion Acc,;p-module. Therefore M,[I n~w ] C ker(~ ). On the other hand, it follows from (i) and (iii) that @~ maps M,,[I ~ ] | .5~ isomorphically onto M(~)| where ~ is the field of fractions of/1r r. As M,| = (M, [ I ~ ] @ 2~ ) 9 (M, [ I ~ ] | cf~ ) it follows that the quotient ker (~)/M, [ I n~'~ ] is a Aes, v-torsion module, from which we easily conclude that ker(~_c,,~) = M,[In~']. (The tensor products are as ~:~,,-modules.) [] 8.3. An auxiliary result We now state (and prove) a simple restflt in commutative algebra. This result will be important in the proof of the main result of this section (the proof of Proposition 8.1). Let A = B~Xl,...,x~]] be a power series ring over a complete DVR B of characteristic 0. Suppose that (Al, A2, [3, NL, N,2, r, qo, ~) is an 8-tuple consisting of 118 C.M. SKINNER, A.J. WILES 9 complete local finite A-algebras At and A2 with AI reduced, 9 a surjection [~ : A,) ~ Al of A-algebras, 9 for each i = 1,2 an Ai-module N, with each Ni finite and free over A and with NI free over A1, 9 an integer r >/ 1 and maps of A.,-modules q) : N' I ~ N,) and ~ : N2 ~ N] such that ~ o q~ E Mr(AI) C_ EndA(N~) and det(~ o q~) is not a zero-divisor in At. We further require that 9 im(q~) = N2 [ I ] and ker (~) = N2 [J ] where I = ker ([3) and J = AnnA. , (I), 9 coker(q~) is A-free. Lemma 8.4. -- l~br each 0 <~ t <~ s there exist yl, ...,yt E A such that (i) ~Yl, ...,Yt) is a prime ideal of A, (ii) yl, ...,Yt generate a t-dimensional subspace of mA/(m~A, rn~) (iii) A,/(Yl, ...,Yt) is reduced, (iv) A N (I +J,yl, ...,yt) ~: (y,, ...,yt), (v) det(~ o q0)mod(yt, ...,yt) is not a zero-divisor in At/(yl, ...,y,), (vi) ker (} mod (yt, -..,yt) ) = N2 [J ]/(yb...,yt), (vii) im(q~ mod (yt, ...,yt)) = (Nz/(yl,-.-,Y3)[I]. Proof. -- Our proof will be by induction on t. Note that if t = 0 then all the conclusions are satisfied by the hypotheses on Ai and Ni. Suppose then that we have foundyl,...,yt, t < s, satisfying the lemma. We will show how to findyt.l. Let ~) C_ A' = A/~I, ...,Yt) be a prime ideal such that 4 (Y!, ...,Y~,Y) satisfies (i) and (ii), b) (y) does not contain AnnA,(aA,/!,., ...,y,!/A')' c) (I +J,y~, ...,yt,y) N A =~ (Yl,---,Yt,Y), d) (det(~ o q)), y,, ...,yt,Y) ~- ~Y,, ...,yt, y). Clearly all but finitely many (.iv) satisfy a) -- d), and since there are infinitely many possibilities for (y) some (y) has the desired properties. Note that Al/(yl, ...,yt) is a finite and free A'-module because N1 is finite and free over A and also free over A~. Hence the hypothesis that At/(y~, ...,yt) is reduced is equivalent to ~'~al/(Yl, ...,_,'t)/A' being a torsion A'-module (here we use the fact that char(B) = 0). We now show that one may take for yt+l any y such that (y) satisfies a) - d). Properties (i) and (ii) follow trivially. Property (iii) is a simple consequence of b). Property (iv) follows from c) and property (v) from d) once we know that A1/(yl,...,yt,y) is reduced. Property (vi) is immediate. It remains to prove property (vii). It follows from (v) that ~ maps im(q0mod(y~,...,yt,y))| FA,, isomorphically onto N]/(y~, ...,Yt,Y) | FA,,, where A" = A'/(y) and FA,, is its field of fractions. If we RESIDUAI,LY REI)UCIBI.E REPRESENTATIONS AND MODULAR FORMS 119 can show that (8.2) (N,)/(yl, ...,yt,y))[I] 71 ker (~mod (y~, ...,y,,y)) = 0 then it will also follow that ~ maps (N2/(yl,...,yt,y))[I] | FA,, isomorphically onto N]/(yl, ...,y,,y) | FA,, whence im(q0 mod (y~, ...,y,,y))| F^,, = (N2/(y~, ...,y,, ...,y))[I] | FA,,. The desired equality will follow from this one since im(q)rood (yl,-..,Y,,Y)) is contained in (N2/(y~,...,yt,y)[I] with A"-torsion-free cokernel. (Here we are using that coker (q~) is A-free.) To prove (8.2) we need merely note that the intersection in question is contained in (Nz/(yl, ...,Yt,Y))[I +J] which must be zero as it would be simultaneously a torsion-free A"-module if non-zero and annihilated by 0 :~ (I +J)A A". (The latter is non-zero by c)). [] 8.4. q/(~_~, p) is an isomorphism We now complete the proof that V(_~, p) is an isomorphism. To do so we return to the notation of w By Proposition 7.2 M0 is a free T0-module and V(cJc, p) : R0 ,.o To. Moreover, To is a reduced complete intersection over ~,e~, P. Let A = ~,~,p. Note that A = B[[W2, ...,Wm]] where B is the localization and com- pletion of C'[[WI]] at the prime ideal (~). Let ~ : T,, ~ To be the natural surjection. It 2 s 2.~ follows from the results of w that the 8-tuple (Tn, To, ~, M, , M 0 , r, O~ , ~ ) satis- tics the hypotheses of w Therefore by Lemma 8.4 there are elementsyt, ...,ym-1 C A such that (8.3) (i) A/(yl, ...,Ym-l) ~ B (ii) T0/(yl, ...,ym-l) is reduced. (iii) im(O~ mod (Yl, ...,ym-l)) = ~vl,/(yt, ...,ym-t)) [I ~ ]. O~,~ o ON ((unit) x 11 * ] (iv) = E Mr(T0). (v) det(O~ o O~) is not a zero-divisor in T0/(yl, ...,Ym-~). Put Ri = Ri/(yl,...,ym-~), Ti = Ti/(yj,...,ym-,), and M i = Mi/(Yl,..., ym-l) for each 0 ~< i ~< n. Let Q be a minimal prime of T o . As T O is rcduced and "q is not a zero-divisor in T O (11 being a divisor of det(O~ o ON )) q ~Q. Let C be the integral closure of T0/Q in its field of fractions. As To is a free A-module, T o is a free B-module. Thus C is a complete DVR and a finite flat extension of B. Put R'i=Ri| Ti=Ti| andM i=M i| 120 C.M. SKINNER, AJ. WILES Let ~' = (M~) ~' ~ (M,'y ~ and ~" (M'~ 2s ~ ~'-0J ,--,,, rx~'~2'r be the maps induced from q~ and ~, respectively. We have maps R,X' W'~ ~ T'ffAnnT,(M',) ~ a T 0' _~ R 0' and 5" T~ ~ C. Put [3' = c~o% Here ~' is the map induced by ~(.~,p), D' is the map induced from [3 " T, ~ To, and 8 is induced from the reduction of To modulo Q. That [3' factors through T,',/AnnT,(M',) is a consequence of the surjectivity of ~' and of M0 being a free T0-module. Put T" = T,',/AnnT;(M'~). This is a free C-module. Now put H0 = ker(8), Go = AnnT~ ' ker(8), H, = ker(8 o ~' o tg'), G~ = AnnT,, ker(8 o a). As T(~) is a reduced complete intersection ove.r C, it follows from [DRS, Criterion 1] that (8.4) gc(H0/t-I~) = gc(C/8(G0)), where for any C-module X, go(X) denotes the length of X as C-module. Our goal is to prove a similar equality for gc(H,/H~) and go(C~(8 o o0(G,,)). First we prove that (8.5) go(C/(8 o ~)(G,))/> gc(C/8(G0)) + ec(c/~(n)). We prove this as follows. Let I = ker(a) and J = AnnT, ~,(I). It follows from the definition of T,," that M,[I' o~d] = M,,[I].' Therefore, by Lemma 8.3(i), ( 2~ C_ im(~'). In particular, ifj E.J and m E M~, then there exist ml,..., nor E M~ such that (m) A m is r As det (A)mi = 0 for i = 2, ..., 2Sr we have that mi = 0 for i = 2,..., 2'r since det (A) is not a zero-divisor in T; and M~ is a free T;-module. We conclude that.lMi, c_ -qM; and hence (8.6) 2r,', c (n). Now suppose that g E G,. Then ct(g) E (rl) by (8.6) since g annihilates I C kcr(8 o a). Write a(g) = fix. Since ~x annihilates a(ker(8 o tx)) = ker(5) = H0 and since 11 is a 3,.,,,jTx~'~ RESIDUALLY REDUCIBLE REPRESENTATIONS AND MOI)UI,AR FORMS 121 non-zero divisor in T~ it must be that x E Go. We have thus shown that a(G,) C_ "qG0. It follows that (8 o ot)(G,) C_ 8(riG0). The inequality (8.5) is an immediate consequence of this. Next we show that (8.7) <. gc(Ho/H ) + We will prove this by comparing the lengths in question to those of various cohomo- logy groups. First we note that p~ : Gal (F~:/F) , GL2(R~ ) determines a represen- tation p : Gal (Fr/F) , GL~(C) obtained from the composition map ~mhl -- -- ' 5~176 C. R~ ,(R~)p=R.~R,, ~R | -+ Fix a basis for p~ such that P~(zl) = /{t -1 \) and P~(g0) = /'{, "'"\~), u0 E (~� for some go E Gal(F~:/F) fixed. Let k be a uniformizer of C. A C- algebra homomorphism f: R', , C | eC/k m (e'-' = 0) determines a represen- tation 9f:Gal(F~/F) , GLg(C | eC/k m) such that p = 9fmode. Write pf(~) = p(~)(1 + e~/f(~)), ~,j(~) E M,~(C/k'). It is readily checked that a, , ~'f(~) is a 1-cocycle of Gal (Fr]F) with coefficients in Mv(C/V') -~ ad p/k '~. We first claim that f~---~(cocycle (:lass of yf) determines an embedding Homc_~g(R,',, C | eC/k m) '--+ H' (F~/F, ad p/kin). (8.8) Here, and in what follows, all cohomology groups are the usual group cohomology; we do not require the cocycles to be continuous. To see that (8.8) is an embedding first note that if YJi and gf2 are cohomologous, then Pfl. and p& are equivalent. Thus there must be some A = (ac ~) E GL2(C @ eC/k":) such that AptiA -I = P.t~. Since pfl(zl)= (l -1), it follows that A = (a d)" We also have that (* (a/g,),0) = APfl(g0)A-I = p/2(g0) = (:: ~0"/). Thus a = d and A is a scalar, whence PJi = 9/.3. This implies that f = J) since any map R~ '~ , C | eC/k" is completely determined by the images of the elements in the set (~o bo) {a~, b6, c6, d~ 9 a E Gal(Fr/F)} (p~(6) = ,,, d, ) by Lemma 2.5. This proves injectivity of (8.8). Recall that there is a decomposition adp = ad~ @ C where ad~ are those elements in M2(C) with trace zero. It follows from the definition of R~ ~ that if 1 A ,- rain w E Z\?]' then ?~- ue~ ~,~ is unramified at w. The same is then true of Z- ~ "det pf ibr every fE Homc_~lg(R'~, C (D C/~.'e). Therefore (8.9) resw('/j ) e H'(D~,,, ad~ m) gw E E\~. 122 C.M. SKINNER, A.J. WILES Let V be the representation space for p. This is a free C-module of rank . For each w 9 .///~\,~'g there is a filtration 0 g Vw g V such that Vw and V~w = V/Vw are free C-modules of rank 1 such that I~ acts via ~11~ on VI,,. For each w E Z\Z~ U ,~ c\,//~ define Uw C_ ad~ by = ~ Homc(Vlw, Vw) if w 9 ,//gc\,//~ and ZlI~ = 1 Uw ( 0 otherwise. the definition of R~" and R~ one easily checks that if resw(7/) = 0 in Using Hl(Iw, ad~ D~ tbr all w 9 2\2,. U ,/Pgc\,//~, then ffactors through R~. It follows that gc(Homc_~g(R' ., C 9 EC/;~m)) - gc(Homc_~g(R~, C | ~C/~.')) (8.10) ~< ~ ec(H' (I~,, (ad~ D~) wEZ\Zcu'/lg c\' Ig .< gc(c/8(n), z.=). The last inequality follows from an explicit calculation of gc(Ht(I=,, (ad~ D"') for each w. As 8(rl) 3 k 0 we see that gc(c/8(n), ~.") = gc(C/8(n)) for large m. Next we note that there are canonical isomorphisms Homc(H0/(H~, ~m), Clam) ~, Homc_.~g(l~, C | r ~) and Homc(H,/(H~, ~Y), C/~, m) ~-- Homc_alg(R~n, C 9 s It follows from this and from (8.10) that (8.11) gc(HJ(H~))- gc(H0/H~)~< g,:(e/8(n)). Combining (8.11) with (8.4) and (8.5) shows that ec(H./H~) ~< to(C/(8 o ~)(G.)). It now follows from [DRS, Criterion I] that 7 o # :R', -+ T~' is an isomorphism of complete intersections of C-algebras. Therefore ~/ must also be an isomorphism of complete intersections. Since C is faithfully fiat over B we conclude that the map R ~ T induced from ~t(~,, p) is also an isomorphism of complete intersections, and hence YL,...,Y~-L ks a regular sequence in T~. It then follows easily that ~(~, p) is itself an isomorphism. This completes the proof of Proposition 8.1. The following proposition is a simple consequence of that one. RESIDUALLY REDUCIBLE REPRESENTATIONS AND MOI)ULAR FORMS Proposition 8.4. --- If ~J.(~ is a deformation datum for F then property (P1) holds for 6~ . Proof. -- Suppose that p C_ T~ is a prime that is nice for 6.~. Let pN C_ R~ be the prime associated to p as in w Let Q c_ lJ~ be any minimal prime and let p = p~ modQ. Put R = R~/Q. As in w let Ly/F be the maximal abelian p-extension of F unramified away from Y. and let Nx be the torsion subgroup of Gal(Lx/F). Fix a finite character V: Gal(Fz/~] , R � of p-power order such that ~-1. det(p | gt) is trivial on Nx. Corresponding to the deformation 13 | V rain is a homomorphism R~ ) R that factors through R~. The kernel of this homomorphism, say Ql, is contained in p~. It then follows easily from Proposition 8.1 that QI is pro-modular. By Lemma 3.17 there is some map T~(U~ NUt(cond~?(V)2),6 ":) , R inducing the pseudo-detbrmation associated to p. To show that P t and hence Q is pro- modular it is enough to show that U~ C_ Ul(cond~)(V)2). To establish this inclusion we first note that since gt has p-power order conde)(gt) is square-free. Moreover, if .~.~ = (~, E, c, ,//~) and if glcond~i(~), then g E Z\~/~. It then follows from the definition of U~ (see w that U~ C_ UI(g2). Thus U~ C_ Ul(cond~)(gt)2). We have thus shown that any minimal prime of R~ contained in p.~. is pro- modular. The same is then true of any prime of R~ contained in p~. [] A. A useful fact from commutative algebra The following result, in the guise of its corollary stated below, is the linchpin in our proof of the Main Theorem. Proposition A.1. [Ray, Corollaire 4.2] -- If A is a local Cohen-Macaulay ring of dimension d, and zf I = (f, ...,f) is an ideal of A with r <~ d- 2, then spec(A/I) \ {mA} is connected. We are indebted to M. Raynaud for providing us with the reference to a proof of this proposition. Suppose now that A and I are as in the proposition. Let ~' be the set of irreducible components of spec (A/I). Corollary A.2. -- If ~r = ~fl II ~ is a partition of ~ with ~i and W,2 non-empty, then there exist irreducible components CI E ~i and C2 E ~,) such that CI O C2 contains a prime of dimension d-r- 1. Proof of Corollary. -- Our proof is by induction on d- r. If d-r = 2, then the assertion of the corollary is an immediate consequence of the proposition. Now 124 C.M. SKINNER, AJ. WILES suppose d- r > 2. The conclusion of the proposition implies that there exist C' 1 E ~i and C,5 E ~2 such that C'~ (1 C,5 contains a prime p of dimension 1, which we may view as a prime of A. Now consider spec(Ap/I). Let W' be the irreducible components of spec(Ap/I). The embedding spec(Ap/I)~-+ spec(A/I) of topological spaces induces a decomposition of g~': g)' = W'l tl g:~;, ~,! = {C' = C C'l spec (AJI) 9 C E <}. By the choice of p, C~ E ~"i, so this is a non-trivial decomposition. As dim Ap = dim A- 1, the conclusion of the corollary now follows from the induction hypothesis together with the fact that Ap is also Cohen-Macaulay and that the dimension of a prime of Ap/I is one less than the dimension of the corresponding prime of A/I. [] Index of selected terminology admissible cocycle .......................... w 6c~ ................................ 2.3 ................................ 2.3 ~Q-minimal ............................ 2.3 good pair .............................. 4.2 good prime ............................. 4.2 nice deformations .......................... 2.3 nice for e_~ ............................. 4.2 nice prime ............................. 4.2 permissible extension ......................... 2.1 permissible maximal ideal ....................... 3.3 min x~, n~ .............................. 3.6 min ~r r~ .............................. 2.4 REFERENCES H. CARA~X)L, Sur les repr6sentations g-adiques associ~es aux formes modulaires de Hilbert, Ann. Sci. Ec. [Cl Norm. Sup. IX" Ser. 19 (1986), 409-468. [Ca] P. CARTIER, I,a conjecture locale de Langlands pour GL(2) et la ddmonstration de Ph. Kutzko, in Bourbdd Semina~ [bl. 1979/1980, Lecture Notes in Math., 842, Springer, (1981), 112-138. 125 RESII)UAIJN REI)U(/IBLE REPRESENTATIONS ~M\'L) MOI)UI2kR FORMS C.-L. CH.~, Arithmetic minimal compactification of the Hilbert-Blumemhal moduli spaces, Ann. oJ" Math. [Ch] (2) 131 (1990), no. 3, 541-554. J. (:o,~:s, p-adic L-functions and Iwasawa's theory, in Algebraic number fieldr: [,~mctions and Galois properties [co] (Proc. @topos., l]nh~. Durham, l'~trham, 1975), Academic Press (1977), 269-353. P I)ELU;.XE, Formes modulaires et reprfisentations de GL(2). in ,!,lodular Jimctions of one variable, II, lecture JOel Notes in Math., 349, Springer; (1973), 55-105. [D-R] P D~:I.IGYE, K. Rl~E'r, Values of abel[an L-fimctiorls at negative integers over totally real fields, hwent. Math. 59 (1980), no. 3, 227-286. B. Dr: S~nT, K. RuniN, R. S(:Hoor, Criteria for complete intersections, in Modular forms and Fermat's Ix2st [DRS] 7heorem, Springer (1997), 343-356. E Di,~xlOXD, On defbrmation rings and Hecke rings, Ann. of Math. (2), 144 (1996), no. I, 137-166. [DII E DLa.xto.','u, The Taylor-Wiles construcdon and multiplicity one, Invent. Math. 128, (1997), no. 2, [D21 379-391. E l)L,~to.xv, R. T,wl.OR, Non optimal levels of mod g modular representations, Invent. Math. 115 (1994), [DT] no. 3, 435-462. J.-M. FO.';T,XINE, B. M~xZt'R, Ge()metric Galois representations, in Ell[pile Curves, modular forms, and Fermat~ I~'t 7heorem (Hong Abng, 1993), Internat. Press (1995), 4[-78. S. G~:I,~T, Automorphic Forms on Adele Groups, Ammls of Math. Studies, Vol. 83, Princeton University [Ge] Press (1975). P. J.-P. L-xBESSE, The solution of a base change problem for GL(2) (following Langlands, [GL] Saito, Shintani), in Automorphic forms, representations and I.functions, Proc. Symp. Pure Math., XXXII1, part 2, 115-133. [(q A. El6ments de la gdomOtrie algebrique IV, /;;tude locale des scb6mas et des morpbismes de schc~mas (deuxi~me partie), lh~bL Math. & I'IHES 24 (1965). H. t-In)a, On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. (2) 128, (1988), [I411 no. 2, 295-384. H. Hm% On nearly ordinary Hecke algebras for GL(2) over totally real fields, in A~ebraic number theorr, [He] Adv. Stud. Pure Math., 17, Academic Press (1989) 139-169. H. HID,t, Nearly ordinary Hecke algebras and Galois representations of several variables, in A~ebraic [H31 analysis, geometry, and number theory (Baltimore, MD 1988), John tlopkins Univ. Press (1989) 115-134. K. IWASAWA, On Zf extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. [tl H. J..WQLET, R. Ia.~(;L~'~r)s, Automorphic forms on GL(2), Lecture ~,~btes in Math., 114, Springer (1970). [J-LI P. Ku'rzKo, The Langlands conjecture for GI,2 of a local tield, Ann. of3,Iath. (2) 112, (1980), no. 2, [Ku] 381-412. H. MAI'St;MURA, Commutative Ring 7beVy, Cambridge Studies in Advanced Mathematics, 8, Cambridge [M~t] Univ. Press (1989). B. M^zuR, Deforming Galois representations, in Galois Groups over Q, vol. 16, MSRI Publications, [Ml Springer (I989). M. N^c~'r~, L~,cal Rings, lnterseience Tracts in Pure and Applied Mathematics, no. 13, Interscience Publishers (1962). R. On a variation of Mazur's deformation functor, Comp. Math. 87 (1993), 269-286. M. IL~vxAt:I}, 'I'h~ortmes de Lefschetz en cohomolo~e cohtrent et en cohomologie 6tale, Bull. Soc. ~ayl Math. France, Mgm. no. 41. Suppldment au BuU. Soc. Math. France, qbme 103, Socitt~ Mathtmatique de France (1975). K. Rindr, Congruence relations between modular forms, Proc. Int. Cong. of~Iath. 17 (1983), 503-514. [Ri] M. NznLrss~y(,z~, Functors on Artin rings, 7runs. A~IS 130 (1968), 208-222. [Sch] .l-R Sr:~.~, Sur le r6sidu de la function z~ta p-adique d'un corps de nombres, CR. Acad. Sc. Pari~ 287, [Sel Serie A (1978), 183-188. R,x.x~..~rd~sn.~a, (~.ROTHENDIEf',K, GErb~.II.DIN, 126 C.M. SKINNER, AJ. WILES [Shi] H. SHIMIZU, Theta series and modular forms on GL2, J. Math. Soc. Japan 24 (1973), 638-683. [Sh] G. SmMue, a, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. j. 45 (1978), 637-679. [sw] C. SKINNER, A. WILES, Ordinary representations and modular forms, Proc. Nat. Acad. Sci. USA 94 (1997), no. 20, 10520-10527. [rw] R. TAYLOR, A. WILES, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), 141 (1995), no. 3, 553-572. [Wal] M. A lower bound for the p-adic rank of the units of an algebraic number field, in Topics in classical number theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. J/mos Bolyai, 34, North-Holland (1984), 1617-1650. L. WASHTNCTON, The non-p-part of the class number in a cyclotomic Zp-extension, Invent. Math. 49 [wa] (1978), no. 1, 87-97. [We] A. WEIL, Basic Number Theory, Springer (t967). A. WILES, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. (2), 142 (1995), 443-551. [W1] A. WinEs, On ordinary )~-adic representations associated to modular forms, Invent. Math. 94 (1988), [W2] 529-573. A. WILES, On p-adic representations for totally real fields, Ann. of Math. (2), 123 (1986), 407-456. [W3] C.M.S. School of Mathematics Institute for Advanced Study Princeton, NJ 08540 USA AJ.W. Department of Mathematics Princeton University Princeton, NJ 08544 USA Manuscrit refu le 23 ddcernbre 1998. WALDSCUMIDT,

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