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On the zeta function of a hypersurface

On the zeta function of a hypersurface ON THE ZETA FUNCTION OF A HYPERSURFACE II) By B~.RNARD DWORK This article is concerned with the further development of the methods of p-adic analysis used in an earlier article [i] to study the zeta function of an algebraic variety defined over a finite field. These methods are applied to the zeta function of a non- singular hypersurface .~ of degree d in projective n-space of characteristic p defined over the field of q elements. According to the conjectures of Weil [3] the zeta fimction of .~ is of the form (I) ~(.~, t) = P(t)(--1)n/'l-It (I'~ --fit) li=O where P is a polynomial of degree d-~{ (d--i)~+t + (--i)~+~(d--i) }, (here n2o, d2 I, for a discussion of the trivial cases n=o,I see w 4 b below). It is well known that this is the case for plane curves and for special hypersurfaces, [3]- We verify (Theorem 4.4 and Corollary) this part of the Well conjecture provided I = (2, p, d), that is provided either p or d is odd. In our theory the natural object is not the hypersurface alone, but rather the hypersurface together with a given choice of coordinate axes X1, X2, ..., X, +1. If for each (non-empty) subset, A, of the set S = { I, 2, ..., n + I } we let.~A be the hypersurface (in lower dimension if A4=S) obtained by intersecting ~ with the hyperplanes {Xi=o}ieA , then writing equation (I) for -~A, we define a rational function PA by setting ira(A) (2) ~(~a, t) = PA(t)(--1)m(A)(I --q'"(alt)/H__ ~ (i--qg), where i + re(A) is the number of elements in A. If.~A is non-singular for each subset A of S and if the Weil conjectures were known to be true then we could conclude that PA is a polynomial for each subset A. Our investigation rests upon the fact that without any hypothesis of non-singularity we have (4.33))/~n+l(/) = (I -- t) l~PA(qt), (a) This work was partially supported by National Science Foundation Grant Number G7o3o and U.S. Army, Office of Ordnance Research Grant Number DA-ORD-BI-I24-6I-G95. BERNARD DWORK the product on the right being over all subsets A of S and ZF is the characteristic series of the infinite matrix [2] associated with the transformation ~ --- ~boF introduced in our previous article [I] and studied in some detail in w 2 below. We recall that z~(t) =y.F(t)/zF(qt) and the fundamental fact in our proof of the rationality of the zeta function is that ZF is an entire function on ~, the completion of the algebraic closure of Q', the field of rational p-adic numbers. In w 2 we develop the spectral theory of the transformation x and show that the zeros of ZF can be explained in terms of primary subspaces precisely as in the theory of endomorphisms of finite dimensional vector spaces. In this theory it is natural to restrict our attention to a certain class of subspaces L(b) (indexed by real numbers, b) of the ring of power series in several variables with coefficients in ft. The definition of L(b) is given in w 2, for the present we need only mention that if b'>b, then L(b') eL(b). An examination of (4.33) shows that if the right side is a polynomial and if0-1 is a zero of that polynomial of multiplicity m then (Oqi)-1 must be a zero of ZF of multiplicity m(n+i). This is << explained ~> by the existence of differential operators Dr, ..., Dn+ 1 satisfying (4-35) ~oD~ = qD~o~ ln+t The space L(b)/~=1D,L(b )_ is studied in w 3 d (in a slightly broader setting than required for the geometric application), for i/(p-- I)<b<p/(p-- I), the main results being Lemmas 3- 6, 3. io, 3. i i. This is applied in w 4 to show that if ~A is non-singular for each subset A of S then the right side of (4.33) is a polynomial of predicted degree and is the characteristic polynomial of E, the endomorphism of L(b)/ED~L(b) obtained from ~ by passage to quotients. (Theorems 4. i, 4.2, 4.3) We emphasize that this result is valid for all p (including p----2). The main complication in our theory lies in the demonstration (Theorem 4-4 and corollary) that if I = (2, p, d) then Ps(tq) is the characteristic polynomial of Es, the restriction of E to the subspace of L(b)/Y.DiL(b ) consisting of the image of LS(b) under the natural map, LS(b) being the set of all power series in L(b) which are divisible by X1X~... X,+ 1. This result is of course based on the study (w 3 e) of the action of the differential operators on LS(b). This study is straightforward for p t d but for p ld the main results are shown to be valid only for special differential operators. We must now explain that for a particular hypersurface we have many choices for the operator ~ (see w 4 a below) but once ~ is chosen the differential operators satisfying (4.35) are fixed. With a simple choice of ~ the eigenvector spaces lies in L(~--~) while for a more complicated choice ofc~ the eigenvector space is known to lie in L(#-~). The special differential operators referred to above in connection with the case p]d are those which correspond to the simple choice of ~. for which the ON THE ZETA FUNCTION OF A HYPERSURFAGE eigenvector space lies in L(P__I~. Unfortunatelyp--I< I ifp=2 (and fortunately, \P/ p p--I only in that case). Thus for p = 2, if ~ ] d we cannot apply the results of w 3 e to determine the action of the special differential operators on L(I/2). Finally (w 4c) using an argument suggested by J. Igusa, we show that our conclusions concerning P=Ps remain valid without the hypothesis that -~A is non- singular for each choice of A. This completes the sketch of our theory. We believe that our methods can be extended to give similar results for complete intersections. We note that the Well conjectures for non-singular hypersurfaces also assert that the polynomial P in equation (i) has the factorization P(t)= l] (I--O~t) such that i0~1 = q(,~-l)12 for each i (Riemann Hypothesis) Oi--~qn-1/Oi is a permutation of the Oi (functional equation). We make no comment concerning these further conjectures. In fulfillment of an earlier promise we have included (w I) a treatment of some basic function theoretic properties of power series in one variable with coefficients in f2. It does not appear convenient to give a complete table of symbols. We note only that throughout this paper, Z is the ring of integers, Z+ is the set of non-negative integers and R is the field of real numbers. w L P-adic Holomorphic Functions. Let f2 be an algebraically closed field complete under a rank one valuation x~ord x. This valuation is a homomorphism of the multiplicative group, f~*, of f2 into the additive group of real numbers and is extended to the zero element of f2 by setting ord o = -b o0. Furthermore ord(x +y) < Min(ord x, ordy) for each pair of elements x, y in f~ and the value group, (5, of fl (i.e., the image of s under the mapping x-+ord x) contains the rational numbers. For each real number b, let I'b={xE lordx=b } U b ----- {x~) [ ord x>b} Cb ={x~ Iordx>b}. As is well known, f~ is totally disconnected, and each of these sets are both open and closed. However by analogy with the classical theory it may be useful to refer to the set C b (resp: Ub) as the closed (resp: open) disk of additive radius b. U_ oo will be understood to denote f2 and clearly F b is empty if b does not lie in the value group of f~. We further note that U b, (resp: Cb, ) is a proper subset of U~ (resp: Cb) ifb'>b. If be(5 then U b=C b. 8 BERNARD DWORK The power series in one variable with coefficients in ~, (x.x) F(t) = ~ A S S~0 will be viewed as an f~ valued function on the maximal subset of f~ in which the series converges. (This is to be interpreted as a remark concerning notation, the power series and the associated function cannot be identified unless (cf. Lemma 1.2 below) the series converges on some disk, Ub, b>oo). It is well known that F converges at xEf2 if and only if lirnA~x~=o. An obvious consequence may be stated : Lemma x.i. -- F converges in C b if and only if (x .2) lim (oral Aj§ = 0% provided b e(5. The series converges in Ub if and only if (I. 3) lim inf (ord Aj)/]> -- b. j--~ 0o We may now prove the analogue of Cauchy's inequality as well as the analogue of the maximum principle for closed disks. Lemma x.2. -- IfF converges on C b and be~ then (i .4) Min ord F(x) = Min (ord Aj+jb) xEP b O~i<~ Furthermore Min ord F (x) --= Min ord F (x). xEr b zEC b Proof. -- Since Fb is not compact it is not immediately obvious that ord F(x) assumes a minimum value at some point of Pb" However the existence of the right side of (I.4) is an immediate consequence of Lemma I.I. Let M = Min (ord Aj +jb), then ord (Aixi)~M for all xEP~ and hence ordF(t)__M on F b. Let S be the set of all jEZ+ such that ordAj+jb = M. By definition S is not empty and Lemma i shows that S is finite. Let g(t) = ~, Ajt j, f(t) --=F(t)--g(t). Lemma I also shows that yes there exists ~>o such that ordAj-]-jb>M-t-~ for each j~S. Hence ord f(t)>M+r everywhere on P b. Let nEFb, n'eF M and let gl(t) =g(nt)/n'. Let Bj be the coefficient of t ~ in gl. For jES, ordBj=ord Aj§ Thus the coefficients of gl are integral and the image of gl in the residue class field of ~2 is non-trivial. Since the residue class field is infinite there exists a unit x in f~ such that ordgi(x ) =o. This shows that ord g(r~x) = M. However r~xEP b and hence ord F(t) assumes the value M on Pb. This shows that the left side of (I . 4) exists and is equal to the right side. The assertion concerning C b follows from the obvious fact that for b'>b, we have ordAj+jb'>ord Ai+jb for each jEZ+ and hence Min ord F(x)>Min ord F(x), which implies the assertion of the lemma. ~Erb' ~Erb ON THE ZETA FUNCTION OF A HYPERSURFACE As in [I], the ring of power series in one variable, t, with coefficients in f~, ~{t}, is given the structure of a complete topological ring by letting the subgroups {Cb{t}+ tmf~{t}}belt,,,eZ§ constitute a basis of the neighborhoods of zero. This topology will be referred to as the weak topology of ~{t}. It may also be described as the topology of coefficientwise convergence. We now obtain an elementary, but useful relation between convergence in the weak topology and uniform convergence in the function theoretic sense. Lemma z. 3. -- Let ft,f2, ..., be a sequence of elements of f2{t}, each converging in Cb, be(5. (i) If the sequence converges uniformly on C b to a function F then a) The sequence is uniformly bounded on C b. b) The sequence converges in the weak topology to fell{t} which itself converges on C b and f(x)=F(x) for all xeC b. (ii) Conversely, if a) the sequence is uniformly bounded on Cb, b) the sequence converges in the weak topology to fe~{t} then f converges in U b and for each ~>o the sequence converges uniformly to f on Cb+ ~. Pro@ -- Let .f(t) = ~ A~,itJ for i---- I, 2, ... i=0 (i) Since the sequence converges uniformly on C b and since, by Lemma i .2, fl is bounded on Cb, we may conclude that the sequence is uniformly bounded on C b. By hypothesis, given N>o there exists neZ+ such that ord (f~(t)---~,(t))>N for all t~C b and all i, i'>n. Hence by Lemma i .2, for i, i'>n and for alljeZ+ (x. 5) ord(A~, i--A~., i) > N--jb. For fixed j, (5) shows that {A,,i}~=l, ~ .... is a Cauchy sequence and hence converges to an element A i of fL It now follows from (I .5), letting i'--~oo that for i>n and all jeZ+ (x. 6) ord (A~,j-- Ai) > N --jb. Let f(t) = ~ A/i. Iff does not converge on C b then we may suppose N chosen such that ord A~.+jb<N for all j in some infinite subset, T, of Z+. Let i be a fixed integer, i>n. Since f~ converges in Cb, we know that ordAi,~.§ for all jeZ+--T' where T' is a finite (possibly empty) subset of Z+. For jeT~T', ord Ai, j>ord Aj, which together with (i.6) shows that ordAj-t-jb>N. Hence T--T' must be empty, a contradiction, which shows that f converges on C b. Lemma x .2, together with equation (x.6), shows that for i>n, ord(f(t)--f(t));>N everywhere on C b. In parti- cular for fixed teCb, letting N-+oo we conclude that f(t)=limf(t)=F(t). This completes the proof of (i). 2 lo BERNARD DWORK (ii) By hypothesis the sequence is uniformly bounded on C b and hence by Lemma 1.2 there exists a real number, M, such that ord Ai, i +jb ~ M (x.7) for all i, jeZ+. Furthermore, writing f= ~ AitJ , we know that for each j~Z+, limi_, oo Ai, j = A i. For each jsZ+, therefore, there exists i (depending on j) such that ord (A~,j--A~) >M--jb. Hence by comparison with equation (i.7) we may conclude that (I. 8) ord A i-k-jb> M for all jeZ+. This shows that f converges in U b. Now let ~ be a real number, ~>o. Given a real number N, let jo~Z+ be chosen such that jo e-[-M>N. Then by (I.7) and (i.8) we have ord Aj+j(b +a)>N, ord A~,j +j(b +r for all ieZ+ and all J>Jo. Hence ord (A,,j--A~)+j(b+r for all J>Jo, i~Z+, while since limA~.j=A~., we may conclude that there exists n~Z+ such i-+oo that ord(Ai, i--Aj)+j(b+r for all j<jo, i>n. Hence for i>n, jeZ+, ord (A~..i--Ai) +j(b + r and hence by Lemma i. 2, ord (fi(t) --f(t))>N everywhere on Cb, which shows that the sequence converges uniformly tofon Cb+ ~. This completes the proof of the lemma. With F(t) as in equation (I. I) we define thef h derivative ofF (for jeZ+) to be the power series F/J)(t)-= - ~ s(s--I)... (s--j+ I)A~t s-j andlet F[fl(t)= ~=0(~)Ast~-J where (~) 8~j denotes the binomial coefficient of t ~ in the polynomial (i + t)*. Clearly F C~/=j ! F [fl, the notation F [fl being convenient if the characteristic of ~2 is not zero. We now prove an analogue of Taylor's theorem. Lemma I. 4. -- If Fsf~{t} converges in Cb, (be(5) then (i) F is a continuous function on C b and is the uniform limit of its partial sums. (ii) F It converges in C b for each jeZ+. ,, (iii) For fixed xeCb, the polynomials Pn(t) ----- ~] F[fl(x)(t--x) i (n = I, 2, ...) converge Qo j=0 weakly in f~{t} to F(t). The element L(Y)= Y, FEfl(x)YJEf~{Y} converges for all Y~C b and F(t)=L(t--x) for each teC b. ~=0 Proof. ~ (i) In the notation of equation (I. I), we conclude from (1.2) that given N>o, there exists neZ+ such that ord Aj+jb>N for all j>n. Hence byLemma I .2, ord (F(t)--~ Ait~)>N everywhere on C b. Hence F is the uniform limit on C b of its partial sums and thus continuity of F follows from the continuity of polynomials. Assertion (ii) is a direct consequence of Lemma x.i. 10 ON THE ZETA FUNCTION OF A HYPERSURFACE [I (iii) For jeZ+, let M i= Min (ord A s +sb). Since F converges on Cb, Lemma I . 1 *>i shows that M~.--->oo as j~oo. Lemma i .2 shows that for xeCb, ord Fti](x) > Min {ord (~) +ord A~+ (s--j)b}. -- 8>i Hence Min ord FtJ](x) > Mj--jb, Mr+t> Mj. xCC b Hence by Lemma i. i, the series L(Y) converges for all yEC b and hence by part (i), P~(t) converges uniformly to L(t--x) on C b (as n-+oo). Thus in view of part (i) of Lemma 1.3, the proof is completed if we can show that P~(t) converges weakly to F(t) as n-+~. Let Pn(t)-----~', An, st ~. We must show for fixed s that lim A n ~=A s. From the definitions ~=0 (I. IO) A., s = Z i=0 We now write F=F,~-}-G,~, where F~(t)= Y~Aiti. Clearly A~,s=A'n,~-k-A',,',, , where i=O A~,~ (resp. A','s) is given by the right side of(I. io) upon replacing F by F,~ (resp. Gn). Since Taylor's theorem is formally true for polynomials, A',,~=A~ for s~n, A~,,,= o for s>n. On the other hand for all j~Z+, ord (Gt, i](x))~M~--jb and hence ord A','s~M,,--sb. Hence for n>s, ord(A~--A~,s)=ordA~,~>M~--sb oo as n-+oo. This completes the proof of the lemma. We can now give some equivalent definitions of the multiplicity of a zero of a power series. Lemma x .5. -- If F converges in Cb, meZ+ and xeC b then the following statements are equivalent :r lim F(t)/(t--x)" exists. l "-~ x ~) Ft~l(x) = o for i = o, I, ..., m-- i. "() The formal power series, F(t)(i--t/x) -m converges in C b if x:~o while if x=0, t" divides F(t) in ~{t}. Proof. -- By Lemma i .4 for t~Cb, t~ex, we have F(t)/(t--x)~=~tFtiJ(x)/(t--x)m-' + ~ r[q(x)(t--x) '-'~. i=0 i=m Hence, by the continuity of power series, the limit exists if and only if (~) is true. Thus (0c) and (~) are equivalent. If x=o then (~) and (y) are clearly equivalen t. Hence we may suppose that x4:o. Let f~{t}, f(t)(i--t/x)m=F(t). Since the rules of multiplication of formal power series and of convergent power series (in the function theoretic sense) are the same, it follows that iffconverges in C b then as a function, f(t) = F(t)/(i--t/x) ~ for all t~Cb--{x }. The continuity of convergent power series now 11 i~ BERNARD DWORK shows that (u implies (0c). To complete the proof we show that (~) implies (y). It follows from (~) and Lemma I. 4 that in the weak topology F(t) = lira ~2 F~l(x)(t--x) i and Y~---~ OO ~ ~ hence in that topology, F(t)(I --t/x) -"= (--x)" lim ~ FtJl(x)(t--x) j-'. The coefficient B~ of t s is clearly B,= ~ Ftfl(x)(J-;"~)(~x) i-" so that by (i.9), f--m ord B~ > MingM.--sb/. -- i>s t ~ .i Thus ordB,+sb>M, and since Ms-+m with s, this shows that F(t)(i--t/x)-" converges in C b. IfF converges in Cb, XeCb, we say that x is a zero of multiplicity m>o if FEd(x) =o for i=o, i, ..., mui, while Ft"l(x)~eo. In particular if H converges in Cb, x+o, H(x) +o and F(t)= (t--t/x)mH(t) then x is a zero of F of multiplicity m. Let F be an element of f2{t} which converges in U b for some b<oo (i.e., the domain of convergence of F is not the origin). We assume with no loss in generality that Fei +tfl{t}. In the notation of equation (I.i), the Newton polygon of F is the convex closure in R � R (=two dimensional Euclidean space with general point (X, Y)) of the positive half of the Y axis and the points (j, ord A~.), j = o, i, ..., it being recalled that ordAi= q-oo if Aj= o. The Newton polygon will have a second vertical side of infinite extent if F is a polynomial of degree m>o. In this case the boundary of the Newton polygon (excluding the vertical sides) is the graph of a real valued function, h, on the closed interval [o, m]. Likewise if F is not a polynomial then the boundary (excluding the vertical side) is the graph of a real valued function, h, on the positive real line. In either case, h is continuous, piecewise linear with monotonically increasing derivative. Furthermore equation (i. 3) shows that the graph of h is asymptotic (if F is not a polynomial) to a line of slope -- b, where b is the minimal element of the extended real line such that F converges in Ub. If x is not an end point of the interval on which h is defined then h'(x--o)<h'(x+o). The points at which the strict inequality holds are called the vertices of the polygon. The abscissa, j, of a vertex is an integer and the vertex is then (j, ord AS). Finally, if l is the line obtained by extending in both directions a non-vertical side of the Newton polygon of F then for each jeZ+, the point (j, ord As) lies on or above the line l. Lemma x.6. -- Let F(t)= ~ AjtJ: I] (i--t/~i) be a polynomial of degree n>o, #=o j=t with constant term i. Let Xx<X2<...<X s be the distinct values assumed by ord0~ -1 as i runs from I to n and for j= i, 2, ..., s, let rj be the number of zeros, ~, off (counting multiplicities) such that ~ ord 0c = Xj. The vertices of the Newton polygon of F are the origin P0, and the s points (x.xI) P.= ri, r~;~ i i = a-~- I~ 2, . . ., s. 12 ON THE ZETA FUNCTION OF A HYPERSURFACE Proof. -- Let the zeros of F be so ordered that ord ~-l<ord e~-l<... <ord ~-l. The proof may be simplified by letting r 0 = o, X 0 be any real number, say 7~-- I. Then P, = r~, r~X~ for a = o, i, ..., S. Let j, be the abcissa of P,, then Ai, is the sum i ~ Ja of all products of the ~-i taken j, at a time. This sum is dominated by I-I ~-l. Hence Ja Ja i=1 ordAi=ordIle~-t= ZriN. If a>o,L_t<j< L then i=l /,=0 ] a--I ord Ai>ord 1I ~i-1= Z r(h4-X,(j--j,_t ) /:1 i=0 and hence the point (j, ord Ai) lies on or above the line since the equation Pa -1 Pa of that line is a--i (x. x2) Y-- X r~X~ = X.(X--L_I). i,-O Thus the Newton polygon is the convex closure of the s + I points P0, P1, 9 9 P, and the point (o, 4-oo). Equation (I. 12) shows that the slope does change at the points Pa, P2, ..., P~-i and this completes the proof. Corollary. -- The numbers {ord ~-1}7= ~ are precisely the slopes of the non-vertical sides of the Newton polygon ofF. If X is such a slope then the number of zeros ~ ofF such that ord ~ = -- X is the length of the projection on the X-axis of the side of slope Z. We now prove a refined form of a well-known theorem [4, Theorem IO, p. 4 I] which states roughly that two polynomials of equal degree have approximately the same zeros if the coefficients of the polynomial are approximately equal. Lemma x. 7. -- Let f and g be elements of ~[t] and let X be an element of the value group of ~) such that a) f(o) =g(o) = I b) The number (counting multiplicities) of zeros off on Px is a strictly positive integer, n. If N is a strictly positive real number such that (*. x3) Min ord (f(x) --g(x)) >nN, xEP x then each (multiplieative) eoset of I + C N contains the same number of zeros off in P x as of g. Proof. -- Let at, ..., ~, be the zeros of fin Px, let ~(t, --., g,~be the (possibly empty) set of" zeros off in U z and let S be the set of zeros off outside Cx. Clearly for ~S, ord~<X and hence if ~zPx, ord(I--~/~)=o. Since ord~i>Z, we have ord (i--~/yi) =ord (~/gi) =X--ord u for i= I, 2, ..., m if ~Px. Since f(t) = fl fI (,-t/v,). II i=l i=l zttE8 18 14 BERNARD DWORK we may conclude that for ~eFz, ordf(~) = ~ ord (i--~/e~) + ~ (Z--ord Yl). Letting m i.=l i =I c----- 2] (--?,+ord y~), we note that c is independent of ~er x. Letting 0~, ~, ..., ~,~, 4=1 be the (possibly empty) set of zeros of g in F x we conclude by the same argument as above that there exists a constant c'>o such that for ~EF x ft ord f(~)=--c-}- Z ord(:--~/~) (x. '4) ord g(~) = -- c'+ Y~ ord (I -- ~/0d), 4=1 it being understood that ordg(~)=--c' if n'-----o. It is easy to see that n'+o for otherwise ord g(%) = --c'< o<nN<ord(f(~l)--g(o~l) ) = ord g(0cl) , a contradiction. Let ~,, ..., ~, be chosen in Px such that ~l(I -r ..., ~e(I +CN) are disjoint and such that their union contains all zeros of f and g in F z. If e>i, ord (I--~/~l)<N for j=2, 3, --- e and hence there exists e>o such that (I.IS) o<ord(I--[~i/~l)<N--r for 2<j<_e. If e= x, we interpret this condition to mean simply o<~<N. With r so chosen we shall for the remainder of the proof let ~ be a variable element of Fx satisfying the condition (I. I6) N-- a< ord(1 -- ~1/~) < N. We now show that if aa~i(1 -}-CN) then ord(1--~/~l) if i= I (I.IT) N>ord (I--~/~)=ord(i__~j~i) if i4:1. For i=I this follows from ~/}----(,/~I)(~X/})E(~J~)(I+CN), while by (1.I6) (~l/}) 6(I -~-CN). For i>2, we have 0t/~G(~,/~)(i +CN) ~ (~J~t)(~l/~)(i +CN) while by (i. 15) and (:. 16) ord (I--3J~l)<N--r (I--~l/~). This completes the proof of (I. 17). In particular if ~ is a zero of fin F x then, by (I. :7), ord (: --~/~)<N and hence by (i. i4) since c>o, ordf(~)<nN. From (I. 13) we now see that ordf(~) =ordg(~) and thus equation (i. I4) shows that ?t t (x.x8) --c+ ~ ord (I--~/0t~):--c'+ Y~ ord (I +~/0~) i =l 4=1 For j---- x, 2, ..., e, let n~ (resp. n~) be the number of zeros off (resp. g) in ~i(I -~- CN). Equations (I. 17) and (i. 18) now give (I. I9) (n 1 -- n~) ord ( I -- ~/~l) = C -- C" -~- i~2 (n; -- ni) ord( I -- ~i/~l) the right side being simply c--c' if e = 1. As ~ varies under the constraints of (1.16), ord (x- ~i/~) varies at least over the rational points in the open interval (N--C, N) 14 ON THE ZETA FUNCTION OF A HYPERSURFACE I5 while the right side of (I. I9) is independent of ~. This shows that nl----n~ and by the same argument n~ = n~' for i = 2, 3, 9 9 e. This completes the proof of the lemma. As an immediate consequence we state the following corollary. Corollary. q Let f and g be elements of~[t] such that f(o) =g(o) = I. Let b be an element of if) and let m be the number (counting multiplicities) of zeros off in C b. ~. If Minord (f(x)--g(x))>o then the sides of the Newton polygon off of slope not xCC b greater than --b coincide with the corresponding sides of the Newton polygon of g. 2. If N is a strictly positive real number and Min ord(f(x) --g(x))>mN xEC b then each coset of I -+-C N in C b contains the same number of zeros off as of g. We can now demonstrate the main properties of the Newton polygons of power series. Theorem I.I. -- Let b'<b<~, bEff) and let F be an element of ~{t} converging in Ub, , F(o) = i. Let m be the total length of the projection on the X axis of all sides of the Newton polygon of F of slope not greater than --b. There exists a polynomial G of degree m, (G(o) = I) and an element H of ~{t} such that the zeros of G lie entirely in C b and (i) H converges in U~,, ord H(t)=o everywhere in C b. (ii) F = GH. These conditions uniquely determine G and H. Furthermore : (iii) The Newton polygon of G coincides with that of F for o < X < m while the polygon of H is obtained from the set: (Polygon of F) -- (Polygon of G) by the translation which maps the point (m, ord Am) into the origin. (iv) If K is a complete subfield of s which contains all the coe ficients of F, then GeK[t]. (v) If for each partial sum, Fn, of F we write F n =G,H,,, where G, is the normalized polynomial whose zeros are precisely those of F n (counting multiplicities) in Cb, then G,, converges to G in the weak topologv of ~{t}. (vi) If neZ+ and N is a strictly positive real number such that ord (F(t)--F,(t))>mN everywhere on Cb, then each coset of I + C~ in C b contains as many zeros of F as of F,. Proof. -- We follow the procedure of part (v). For n>_m the Newton polygon of F, coincides with that of F in the range o < X < m and furthermore all sides of the polygon of F, of slope not greater than --b occur in that range. This shows that for n~m, F,, has m zeros in C b. Since the sequence {Fn} converges uniformly on C b to F, we conclude that given N>o, there exists nleZ , nl>m , such that ifn and n' are integers not less than n 1 then ord (Fn--F,,)>mN everywhere on C b. We may conclude from the corollary to the previous lemma that each coset of I + Cs in C b contains as many zeros ofF,, as of F,, and hence the same holds for G, and Gn,. This shows that for n > m we may write Gn(t ) = 1-[ (i--t/a,,~) where the zeros 0%1 , ..., ~,,,,, of G, are so ordered i=l 15 I6 BERNARD DWORK that lima,,~=a~ exists for i= I, 2,...,m. This shows that G, converges to G, a n --~ OO polynomial of degree m whose Newton polygon coincides with that of F,1 and hence with that of F for o<X<m. For each n~Z+, H,(t) is a product of factors of type (i--t/a) where ord 0c<b. Hence (x. 2o) ord Hn(t ) = o everywhere on C b. G, is a product of factors of type (i --tin), where ~eC b and hence if ord t<b then ord Gn(t)<o (equality holds if G,(t)= x). If then b"effj, b>b">b', then ord G,(t) < o everywhere on Fb,, and hence ord H,~(t) = ord F,~(t) -- ord G.(t) > ord Fn(t ) everywhere on Pb,,. Lemma I-3 shows that F,~(t) is uniformly bounded on Fb,, and hence the same holds for H.(t). Hence by Lemma 1.2 the sequence Ha, H2, ... is uniformly bounded on Cb,,. We show that the sequence H1, H2, ... converges in the weak topology of f2{t}. This follows from the fact that x +trY{t) is a complete multi- plicative group under the weak topology. Certainly F,~F and G,--->G in that topology and hence H,,=F,/G,, converges weakly to the power series H=F/GEI +trY{t}. It now follows from Lemma I-3 (part ii) that H converges in Ub,, (and hence letting b"-+b', in Ub, ) and that for each r H, converges uniformly on Cb,+~ to H. Using equation (I .2o), it is now clear that H(t) is a unit everywhere on Cb. This completes the proof of parts (i), (ii), (v). Assertion (iii) has been verified for G, its verification for H follows from Lemma t .6 and the fact that H,-+H. Assertion (vi) follows from the construction of G, the corollary to Lemma I. 7 and from the fact that the zeros of F in Cb are precisely those of G. To verify (iv) it is enough to show that G, rK[t] for each neZ+ since then G = lim G, eK[t]. Since the valuation in a finite field extension of K is invariant under automorphisms which leave K pointwise fixed, we may conclude that the coefficients of G. are purely inseparable over K. Thus we may suppose K is of characteristic p 4: o. If g is a root of G, then it is a root of F n of the same multiplicity and hence the multiplicity must be a multiple, mp r, of a power ofp such that o~P" is separable over K. This shows that the coefficients of G n are separable over K which now shows that G,~K[t]. This completes the proof of the theorem. Part (v) of the above theorem has an important generalization which is the analogue of a theorem of Hurwitz. Theorem x.2. -- Let b'<b<o% ben and let fl,J~,.., be a sequence of elements of f~{t}, each converging in C b, such that fi(o)=x for each jeZ+ and such that the sequence converges uniformly on C b, to Fsf2{t}. By the preceding theorem, F=GH, fi=gjh i where G (resp. g~) is a polynomial whose zeros are precisely those ofF (resp. fi) in Cb, and G(o) =g~(o) = I. The conclusion is that G =~irng~ and that for i large enough, & and G are polynomials of equal degree. 16 ON THE ZETA FUNCTION OF A HYPERSURFACE t7 Proof. -- Let degree G = m and for each jEZ+, let F i be the jth partial sum of F and let .~,i be the jth partial sum of f~. Let N be a strictly positive real number. Pick j~Z+ such that ( x. 2x ) ord (F (t) -- Fj(t) ) > mN everywhere on C b. Part (vi) of Theorem I. i shows that F i has m zeros in C b. Pick i 0 such that for each i>i o (I. 22) ord (F --~)>mN everywhere on C b. Pick ueZ+ such that for given i>i o (x .23) ord(f --fl, u)>mN everywhere on C b. We may conclude from these three relations that (x. 24) ord (F~.--f~, u) > mN everywhere on Cb, and the Corollary to Lemma 1.7 now shows that each coset of i § C N in C 6 has as many zeros of F~ as offi, u and in particularf~,u has m zeros in C 0. Equation (I .33) together with part (vi) of Theorem I.I now shows that~ has m zeros in C 0. Furthermore equations (~.2I) and (I .23) and part (vi) shows that each coset of I +-C N contains as many zeros of F in C 0 as of Fj and as many zeros of.~ as offi,~. We may now conclude that each coset of I -t- CN contains as many zeros of F in C 0 as off~ for each i>i o. It is now clear that g~--~G and that deg g~ = m tbr i large enough. Corollary. -- Under the hypothesis of the theorem,for i large enough, the zeros ~, t, ~i, 2, 9 9 9 ~,,, of .~ in C o may be so ordered that lime i j=~i,j= I, 2, ... m and ~1, ..., %, are the zeros of F in C b. We conclude by recalling that in our previous article we left two propositions unverified. Proposition 2 of [I] is contained by Theorem I.I above. We now demonstrate Proposition ~. Proposition. -- If b'< b < oe and F converges in U b, but is never zero in Ub, then the series I/F converges in U b. Proof. -- As before we may assume A0= I. The Newton polygon of F has no side of slope less than --b and hence ord Aj>--jb. The conditions A0= i, ord Aj>--jb define a subgroup of I +trY{t} and hence are satisfied by the formal power series I/F. This shows by Lemma I. I that I/F converges in U b. w 2. Spectral Theory. Let Q' be the field of rational p-adic numbers, f~ the completion of the algebraic closure of Q', the valuation of f~ being given by the ordinal function x--~ord x which is normalized by the condition ordp-= + I. Let q, n, d be integers q> I, d> I, n>o which will remain fixed throughout this 3 ~8 BERNARD DWORK section. Let 3; be the set of all u = (u0, ul, ..., u,)EZ~_ +1 such that du o >u 1 + ... + u,. The set, Z~_ +1 may be viewed as imbedded in n+ I dimensional Euclidean space, I! "+1 and let a be the projection (Yo,Yl, ...,Y,)---~Yo of R "+1 onto R. We formalize and reformulate in a manner convenient for our present application the methods appearing in the second half of the proof of Theorem i [I]. Lemma 2. I. -- Let c,~ be the minimal value of as (u (t), ..., u (")) runs through all sets o fro distinct elements of %. Then c"[m~oo as m-~ oo. Let 9J~ be an infinite matrix with coefficients 9)lu, v (in f~) indexed by ~ � 3; which have the property ord 9J~,,~>� where � is a strictly positive real number. When convenient we write 9J~(u, v) instead of 9Jlu,~. Lemma 2.2. -- (i) IfgJ~' is any.finite submatrix of 9J~ obtained by restricting the indices (u, v) to ~' x ~' where ~.' is a finite subset of ~, then the coefficient ~'m, oft" in det(I --t?O~') satisfies the condition: ordy,,>� m. Hence for tEf~, ord det(I--tg)~')> Min(mordt+� an estimate depending only on ord t and the constants� q, d, n, but independent orgY'. In particular for each bounded disk of f~, det(I -- t?Ol') is uniformly bounded as ~.' varies over all finite subsets of ~.. (ii) If (u, v) ~' � ~', then the minor of (u, v) in the matrix (I--tgJ~') is a polynomial ~]u v)t" and ord V (u, v) > + � , Hence for tef~, ord (minor of (u, v) in (I--tgJ~') ) > q� § where c is a constant inde- pendent of ~i~' and ~; (if ord t is fixed). Proof. -- (ii) The coefficient, ,("(u, v) is a sum of products P =-t-1-I ~lJl(u (~), v(~)), where {u, u (t), ..., u (")} is a set of m + i distinct elements of 3;' and {v, v (t), ..., v (")} is a permutation of that set. Hence � ~ a(qu(')--v (')) =a{q u (~) -- v+ v (') --(qu--v)}= i=i '= i a{(q--I)(V+ ~ v (')] --(qu--v)}~qa(v--u) ~- (q-- I)C". i=1 / Lemma ~'.3. -- For N~Z+, let ?Ot~ be the submatrix of ?Ol obtained as in the previou~ lemma by letting ~;'={ue~l~(u)gN }. Let ~J~N be the matrix obtained from 9J~ by replacing ~l~u, , by zero whenever a(qu--v)>(q--i)N. Then lira det(I--t~s)= lim det(I--t~), N-~oo N.-~ oo the limit being in the sense of uniform convergence on each bounded disk of ~. The limit is an entire function, ~ y,~t m, and ord y,,~ (q-- I)� "=0 The remaining proofs may be omitted since they are consequences of the methods of [i]. Lemma 2.3 follows from Lemma 2.~ and Lemma i. 3 (part (ii)) once it is verified that the two sequences converge weakly to the same limit. However the details concerning weak convergence are very similar to the proof of Lemma ~. ~. (We note 18 ON THE ZETA FUNCTION OF A HYPERSURFACE I9 that the method used in [I, equ. (2o.2)] to show weak convergence cannot be used here as that proof made use of the geometrical application.) Let f~ {X} be the ring of power series and ~)[X] the ring of polynomials in n+ i variables X0, X1, ..., X~ with coefficients in f~. If u= (u0, ul, ..., u,)eZ~. +l, let X ~ denote the monomial l-I XUq Let + be the endomorphism of fl {X} or f~[X] ~=0 Io if q~u as linear space over ~) defined by ~b(X ") = t X"/a if q lu" For each ordered pair of real numbers (b, c), let L(b, c) be the additive group of all elements Y,A,X"sf~{X} such that (i) A~=o if ur (ii) ord A,> bu o + c. Let L(b)=,U L(b, c), E be the subspace of f2[X] spanned by {X"},ez. For each ell integer N>o, let 2~ (s) be the subspace of E consisting of elements of degree not greater than N as polynomials in X 0. Let ~(b, c) =~nL(b, c), ~(Sl(b, c) =E(mnL(b, c). If Heft{X}, let +oH denote the linear transformation ~--~+(H~) of ~{X} into itself. Lemma 2.4. -- Let ~ be any mapping of ~[X] into the real numbers such that.for ~1, ~2, c4:o, + =--oo (2. x) = + < Max If s is an integer, s> t, x is a non-zero element oJ f~ and ~ is a polynomial such that (2.2) (I--z-l+oH)"~ = o, (H 4: o) then ~(~)< v.(H) l(q-- ~). The proof may be omitted as it follows trivially from the fact that for ~Efl{X}, V.(,.,b(H~)) < (o.(H) + ~(~))/q. In particular if h is a linear homogeneous function on R~. +1 and if for each ~f~{X}, ~(~q) is the maximum value assumed by h(u) as X" runs through all monomials occurring in ~, then ~z satisfies the conditions of Lemma 2.4. In particular if He~ (N(q-1)l, then letting h(u)=ut+... +un--duo, we may conclude that if ~ satisfies (2.2) then ~ lies in E and letting h(u) = u 0 we may conclude that ~ lies in E(N). Thus the definition of det (I--t+oH) appearing in our earlier work is unchanged if (+oH) is restricted to ~(') for any integer m>N. Now let � be a strictly positive rational number. Let F-----Y~A~X" be an element of L(x, o) which will remain unchanged in the remainder of this section. We associate 19 ~o BERNARD DWORK with F a power series Zv, the characteristic series of qJoF which generalizes the characteristic polynomial appearing in the case in which F is a polynomial. For each integer N>o, t X" if u0<N let T n be the linear mapping of L( oo) into E(n) defined by TN(X")= (o otherwise " Let eN be the mapping ~--~qJ(~(Tn(q_t)F)) , and let ~ be the mapping ~---~T~(+(~F)) of (say) ~(N) into itself. If in the terminology of Lemma 2.3, we set 9J~,,, = Aq~_, for all (u, v) e 3; � 3;, then relative to a monomial basis of ~(N) the matrix form of a n is gJ~s while that of ~ is gJl~. Hence lira det (I--ten) and lim det (I--ta~) both exist and are equal by Lemma 2.3- The characteristic series, Xr, is defined to be this common limit. Lemma 2.3 shows that XF is entire and lies in ~{t}, ~ being the ring of integers of f~. The mapping ~ :~---~+(F~) of f2{X} into itself will now be examined. We first show by a general example that a satisfactory theory cannot be obtained if we allow ~ to operate on the entire space f2{X}. If F has constant term I then let Qt) G(X)=I:IF(Xqi). Clearly, F(X)=G(X)/G(X q) and hence if X~o, X~f~ then i=0 = ~ XiX0r is a non-zero element of f~ {X}, while e~ = X~. Thus as an operator i=0 on f~{X} each non-zero element of f2 is an eigenvalue of e. We shall show that Xr can be explained by restricting e to L(qz). However to obtain a complete theory it will be necessary to assume that the coefficients of F lie in a finite extension of Q'. Let Q, be the field of rational numbers. The value group of f2 is the additive groupofQ. For x= (x0, x~, ..., x~)ef2 ~+~, let ordx= (ordx0, ordxt, ..., ordx~)eQ "+~ if none of the x~ are zero. If a and a' are elements of Q,+l, we define the usual inner product (2.3) o(a, a') = Z a,a;. i=0 If ~ef~{X}, let S t be the set of all aeQ, n+l such that ~ converges at x if ord x =a. Writing 4= ~ BuX", (2.4) u~Z~b +t we have a generalization of Lemma I.I :If aeQ n+t then aeSg if and only if ord B u q- p (u, a) ~ -k oo as u-+ oo in Z~_ + 1. It is convenient to introduce a partial ordering of On+i. If a and a' are elements ofQ n+l, we write a'>a if a;>ai for i=o, I, ..., n. It is clear that if a'>a and aeS~ then a'~S~. We easily check that for 4, ~efl{X}, Sr162 ~qSr S~ D S~r~ S., (2.5) 20 ON THE ZETA FUNCTION OF A HYPERSURFACE Let g be a mapping of Z~_ +* into the set of two elements, {o, I } in f~. Let ~. be the f~ linear mapping of f~{X} into itself defined by y(X") =g(u)X u. (2.6) For such a mapping we have (2.7) Sv(~)~ St. For each aeS~, let M(~, a) = Min ord ~(x). ordx~ The generalization of Lemma I. 2 may be stated without proof. Lemma 2. 5. -- For aeS~, ~ as in (2.4), M(~, a)= Min (ordB,+p(u, a)). uEZ~_ +l If a'> a then M(~, a') ~ M(~, a). We easily verify for ~, ~q~f~{X}, u as in (2.6) that (2.8) M(~q, a)> M(~, a)+ M(~q, a) if aeS~nS~ (2.9) M(y~, a)~M(~, a) if a~S~ (2. xo) M(+~, a)> M(~, a/q) if a/qeS~ (=.xx) M(~ +~q, a)>Min{M(~, a), M(~, a)} if aeSenS~ and equality holds in (2. ii) if M(~, a) 4=M(~q, a). Let S={aeQ"+tlao>--q� da i+ao>-qx , i= i, 2, ..., n} Elementary computations show that if c is a real number, ~;cL(qx, c) then S~z S (2. x2) ) M(~q, a) >c for aeS, and I SFD q-iS (2"x3) M(F, a/f)2o if aeS. It follows from (2.8), (2. IO) and (2.13) that (~,. x4) M(~, a) > M(~, a/q) if aeS n qS ~. This relation remains valid if, is replaced by coy or yo~, the composition of ~ with g on either right or left side. Let 9{X} be the ring of power series in X0, ..., X,, with coefficients in 9, the ring of integers in ft. Let L' be the space of all elements of f~ {X} which converge in a polycylinder of radii greater than unity (i.e. an element ~Ef~ {X} lies in L' if and only if there exists a rational number b>o such that (--b,--b, ...,--b)eS~). We 21 ~ BERNARD DWORK note that L'~L(b) for all b>o but L' is not the union of such subspaces since the monomials, X ~, in L' need not satisfy the condition uz~. Lemma 2.6. ~ Let ~q~L(q~, --q(q--I) -1 ord X), where X is a non-zero element of ~), and let ~ be an element of L'nO {X} such that (2. xs) We may then conclude that ~r --q(q--I) -a ord X). Note. ~ The same conclusion would hold if ~ in (2.15) were replaced by ~oy or by ~,o~, with y as in (2.6). In particular, ~ may be replaced by ~{~. Proof. ~ Writing (2.r5) in the form ~---=--~+X-l~, we see from (2.5) that S~3S~c~S~3S~nqSF~3S~nqSFnqS 0 and hence by (2. i2) and (2.I3) we have (2. I6) S~3 S r~ qS~. By hypothesis, ~eL' and hence there exists b>o such that a(~ (~b, --b, ..., --b)eS t. If aeS then there exists an integer, r>o, so large that q-ra>a (~ and hence q-'a~S~. Let r be the minimal element of Z+ such that the displayed relation holds. If aeS then q-~a, q-2a, etc., lie in S and hence if r>I then q-('-~)a lies in S as well as in qS o so that by (2.16) we have q-('-l)aeS~, contrary to the minimality of r. This shows that r=o and hence ScS~. Since q-aScS, we may also conclude that ScqS~. Equations (2. r4) and (2. I5) show that (2.t7) ordX+M(~+~,a)>M(~,a/q) if a~S. We write ~ as in (2.4) and we assert that for a~S, vaZ]_ +~, (~. x8) ord Bo + O (v, a) > -- q(q-- I ) --1 ord X. To prove this we think of a as fixed and consider two cases. Case 1. -- M(~, a)>M(~q, a) In this case Lemma 2. 5 and equation (2. r2) give a direct verification of (2.18). Case 9. -- M(~, a)<M(~q, a). Here we may use (~. Ii) and deduce from (~. 17) that (m. x9) ord X + i(~, a)> M(~, a/q). Lemma ~-5 shows that there exists a particular element, ueZ~_ +t (depending upon a) such that M(~, a/q) ----ord B,+ p(u, a/q). On the other hand M(~, a)<ordB~+p(v, a), for each vsZ~_ +~. Thus we have (~. ~,o) ord B~ + p (v, a) + ord X> ord B~, + p (u, a/q), 22 ON THE ZETA FUNCTION OF A HYPERSURFACE ~3 for a particular u and for all veZ~_ +1. In particular (2.~o) holds for v=u and this gives ord X_>(q -1- I)p(u, a). (,.21) We recall that by hypothesis ~e~{X} and hence ordB.>o. Equation (2.I8) now follows from (2.2o) and (2,2I). This completes our verification of (2.18) for all aeS. Now let c be a rational number, e>o, let %, ax, ..., a, be rational numbers, ao>--qx, a,=c--d-l(qx+ao) for i=i,2,...,n. Then a=(ao, at,...,a,)eS and p(v, a)=ao(vo--d -1 ~ ,)+ (c--d-tq� ~ ,, i=l i=l which shows that if v0<d-t]~ v i then p(v, a)--~--oo as a0---~+oo if c is kept fixed. i=1 n Applying this to (2. I8) we see that ord B~----- -t-oo if v0<d -t Y, vl, i.e. i=1 (2.22) B~=o if v~ 3;. With c>o as before, let ao=--qx+c,a~=o for i-=I,2, ...,n. Once again a----- (a0, al, ..., an)ES and thus (2.18) shows that ord Bo> v0(qx--c ) --q(q--I) -1 ord X for each c>o. Taking limits as c-+o, (2.23) ord B~> qxv0-- q( q- I) -t ord ~. Relations (2.22) and (2.23) show that ~eL(qx,--q(q--I)-tordZ), as asserted. Note. -- If ~=o in the statement of the lemma, then equation (~. 19) is valid for all a~S. Since (o, o, ..., o)~S, it follows that ord X> o. Theorem 2.x. -- Let 11,...,;% be a set of non-zero elements of ~ and let e= ~ ord Xi+ (q--I) -1 Max ordk e Let ~ be an element of L'o~){X} such that ,/.=1 l <~ i<~ s (2.24) = then ~eL(qx,--e). Proof. ~ The theorem is a direct consequence of the previous lemma if s= I. Hence we may suppose s>I and apply induction on s. Let ordXl>...>ordX~ 8=1 and let ~q=(0~--X,I)~. Since ~qeL'~g){X} and II (I--x~-x~)~=o, we may i=1 conclude that ~eL(qx,--e'), where e'=e--ordX s. We may choose yeO such that ord-r=e'--(q--I) -1 ord X,. Clearly T;~;-l~qeL(qx,--q(q -I) -1 ord X,), while ~(7~)-----Xs(~'r 9 Since y;~T~e~, we may conclude from the previous lemma that -(~eL(qx,--q(q--I)-lordX,). The proof is completed by checking that --ord 7--q(q-- I) -1 ord ),~ = --e. 28 BERNARD DWORK "4 Note. -- Although not needed for our applications, we note that we had shown with the aid of Lemma 2.4 that if F~ (N(q-1)l and ~ is a polynomial satisfying (2.~4), then ~ lies in ~(N). We can now show that if ~ is known to satisfy (2.24) and is known to lie in L' then it must be a polynomial (and hence lie in ~CNI). If Fe~ (NIq-l~) then there exists y~K) such that yF~9[X] and hence if r~Z+,p'N(q-llyF~L(r, o). If ~,---- ~bop~N(q-1)yF, then fi (I--k~r)~---- o, where ~, __~ypm(q-1) and hence the theorem ~=1 shows that ~ lies in L(qr,--e--(s+ i)(ordy+rN(q--I))). Hence ~=ZB~X ",u~X and ord B,,~qruo--e--(s--I ) ordy--rN(q--I) (sq- I) for each r~Z+. Letting r-+o% it is clear that B,~-o if u0>N(q--I ) (s+ I)/q, which shows that ~e~. Theorem 2.2. -- If the coefficients of F lie in a field, K0, of finite degree over Q' and if k -1 is a zero qf order ~ Of XF, then the dimension of the kernel in L(qx) of (I--~-1~) ~ is not less than ~, indeed the kernel contains g. linearly independent elements which lie in L(qx) n K0(k ) {X}. Proof. -- We may suppose that ~ I. Since XF~){t}, ZF(O) = I, we may conclude from Theorem I.I that kes Let z~(t)----det(I--t~). We recall that Lemma 2.3 shows that Z~-+ZF uniformly on each bounded disk. There exists a real number, p>o so large that X~ has no zero distinct from ),-~ in ?,-a(~ +C~). The proof of Theorem ~. 2 shows that for N large enough (as will be supposed in the remainder of 9 ~-~ . k-~ of the proof) there exist (counting multiplicities) precisely ~ zeros, ~,~,. -, ,,N Z~ in )~-~(I -4- C~). Since XF, Z~ and the set ~-~(~ +Co) are all invariant under auto- morphisms of ~ which leave K0(k ) pointwise fixed, we conclude that the polynomial f~(t)= 1-I (I--kr is also invariant under such automorphisms and hence lies in K0(k ) It). Let K be the composition of all field extensions in f~ of K(k) of degree not greater than [z. Theorem ~.~ shows that k is algebraic over Ko, hence deg(K0(~)/Q')~oo. This shows that deg(K/K0)(oo and hence deg(K/O')(oo. The conclusion is that ),~,NeK, limk~N~X for i~ I, 2, ..., [z and that K is locally N .._~ o o , compact. Furthermore fN is relatively prime to zN/f~. We now restrict ~N to K[X] n2 (N), This does not change the characteristic equation of a N and letting W~ be the kernel in that space of ~N ~ I-[ (I--)~N) , we conclude that the dimension of W N (as K-space) is B- An element, 4, of W~ will be said to be normalized if it lies in ~ {X} and at least one coefficient is a unit. If ~ is such a normalized element of W~ then by Theorem 2. I, ~L(ffz, --e), where e= (~+ (q--I) -~) ord k. If we write ~ ~BuX ~ then ord B~Uo--e and hence Bu must be a unit for at least one element ua~.,={wSg[Vo~e/(q� Conversely if B, is a unit then ueS~. It is clear that a subspace W of K[X] of dimension b~ has a basis ~, ..., ~ in s for which there exist distinct elements ut, ..., u~ of Z~_ +~ such that the coefficient of X u~ in ~i is the Kronecker 8i.i(i,j= ~, 2, ..., b~)- Hence for each N there exists a set of t* linearly independent elements {~,s}i=,,~ ..... ~ in Ws corresponding to 24 ON THE ZETA FUNCTION OF A HYPERSURFACE ~5 which there exist ~ distinct elements, {u~,N}i=~, 2 ..... ~ in Z~ such that ~.,~es and the coefficient of X ui,~ in ~i,N is 3i,~. for i, j = i, 2, ..., tz. Since 3:, (and hence 3;~) is a finite set, there exists by the pigeon hole principle, an infinite subset, 9.I, of Z+ such that u~=u~, N is independent of N for each N in the subset and i= I, 2, ..., ~. In the following N will be restricted to this infinite subset. Now let fl3=K{X}r~L(qx,--e). Generalizing the definition of w I, we may define the weak topology of K{X} and by the local compactness of K and the theorem of Tychonoff, ~3 is compact under the induced topology. Thus ~3 ~, the ~ fold cartesian product of ~3 is also compact under the product space topology. Clearly the ordered set ~(~/= (~,s, ~,~, ..., ~,~) ef~ and hence an infinite subsequence of the sequence {~(~/}Ne~ must converge. Hence there exists an infinite subset, 92[' of 2[ such that {~(N)}N~9 l, converges to an element (~1, ..., ~)e~3 ~. For j= I, 2,..., ~ we have {~i,N}~e~,-+~ and since the coefficient of X u~ in ~',s is 3i.~., the same holds for ~.. This shows that ~,..., ~ are elements of ~3 which are linearly independent over f2. Furthermore ~N~,N=O for each N~' and hence taking limits as N-+oo in 9.[', we conclude that ~t, ..., ~ lie in the kernel of ~ = (I--X-~) ~ in L(qx). Now let to~, ..., to,, be a minimal basis of K over K0(X ). If then there exist ~1, ..., ~%~K0(X){X } such that ~q = ~ ~to~ and since the basis is minimal, ~L(qx,--e--i) for i=I,~, ...,m. Ifo=~then o= ~to~ andsince ~,~K0(X){X } for i= i, ~,..., m, we can conclude that ~, lies in the kernel of ~. Applying this argument to ~t, ..., ~ we conclude that the D.-space spanned by them is spanned by elements of the kernel of ~ in L(qx)nK0(X){X }. This completes the proof of the theorem. To complete our description of ZF in terms of a spectral theory for ~, we must prove a converse of the previous theorem. Theorem 2.3. -- Let ~. be an integer, ~. > i and X a non-zero element in fL The dimension of the kernel in L' of (I--Z-l~) ~ is not greater than the multiplicity of X -1 as zero of z F. We defer the proof except to note that we may assume that the kernel of (I--X-l~)~ in L' may be assumed to be of non-zero dimension and to show that X~s If the kernel of (I--?,-'~) ~ is not {o} then by an obvious argument, the same holds for the kernel of (I--X-t0Q. Hence there exists ~eL' such that ~=-X~, ~4=o. Since tEL' there exists u such that y~E~)(X}. Hence it may be assumed that ~e~3{X}. Thus Xr~=~r~ for each feZ+ and since e maps~{X} into itself, we conclude that ~4:o, Z~f){X} for all rcZ+. This shows that XeD. Theorem 2. I now shows that we can replace L' in the statement of the theorem by L(qx). Before resuming the proof we must recall some formal properties of matrices. Let A be an m � matrix with coefficients in some field of characteristic zero. For each subset H of {i, 2, ..., m}, let (A, H) be the square matrix obtained by deleting the jth row and column of A for each j~H. Let [H] denote the number of elements 4 26 BERNARD DWORK in H and let t be trancendental over the field, K, generated by the coefficients of A. If [H]=m, we define det(A,H)=i and for o<[H]<m, I--t(A,H) denotes ((I-- tA), H). Lemma 2.7. -- For I < r < m 1-I t---- det(I--ta) )'r! Y, det(I--t(A, H)), (2.25) /r-1/ddt (,=o( (m--i) )) =(--, [H]=~ the sum on the right being over all subsets, H, of{l, 2, ..., m} such that [H] =r. Proof. -- We recall the classical result that ifB is an m � m matrix whose coefficients are differentiable functions of t then (2.26) det B = Y~ det Bj where B i is the m X rn matrix obtained from B by" differentiating each coefficient in thej t~ row and leaving the other rows unchanged. Thus ~det(It--A) ---- Y, det(It--(A, H)). as [~ =t However t -m det(I--tA) =det(t-lI--A) and therefore det(I--tA)+t-mddet(I--tA)=--t -2 2~ det(t-lI--(A,H))= a$ [H] = t --t-~t -('-11 2 det(I--t(A, H)). [iq = i The assertion for r= I follows immediately. We may therefore suppose r>i and use induction on r. Hence ,rs, )) (2.~'7) r!-ll II [t----(m--i) det (I--tA) = \~=0\ dt (--~)r-lr-'(td--(m-(r--i))) Z det(I--t(A, H)). [H] = r - t The lemma is known to be true for r= I and hence for given H such that [H]=r--1, since (A, H) is an (m--r+ I) x (m--r+i) matrix, (td--(m--r + I)] det(I--t(A, H))=--Y~ det(I--t((A, H), H")), H ,I the sum being over all H"c{I,2, ...,m}--H such that [H"]----I. However ((A, H), H") = (A, H') where H' = H" u H and hence the sum over H" may be replaced by ~] det (I--t(A, H')), the sum now being over all H' such that H'D H, [H'] = r. Thus H' the right side of (2.27) is (--I)rr-lW,~]det (I--t(A, H')), the sum being over all H H H' such that [H]----r--I and over all H'D H such that [H'] = r. But given H' such that [H'] = r there exists exactly r distinct subsets H of H' such that [H] = r-- I. Thus the right side of (2.27) is (--I) r ~] det(I--t(A, H')), which completes the proof of the lemma. [H'] = r 26 ON THE ZETA FUNCTION OF A HYPERSURFACE With the previous conventions, let Si~l(A),j=o, ~,..., m denote the elements of the field K generated by the coefficients of A which satisfy the formal identity (2.~8) det(I + tA) = ~. S(~')(A)t i i=0 We observe that SIm-J)(A) -- ~] det(A, H),j = o, i, ..., m-- I [H] = j the sum being over all subsets H of {I, 2, ..., m} such that [H] =j. Let ~> I be a rational integer, let ~ be a primitive /h root of unity in some extension field of K. For (io, it, ..., i~-l) eZ~-, let g(io, il, ..., i~_1) =~ where tt--I ~--I r= Y~ si~. Since det(I--PA ~) = l-[ det(I--tco-~A), we have 8=1 8=0 m I~,--1 m (2.3o) Z tJ~SI~)(--A~) = H Z tJSI~/(--o~-'A). ~=o s=oj=o For o < i <m, by comparing coefficients of t ~!"-i) on both sides of (~. 3o), we conclude that g--1 (~.3 I) S(m-'t(A ~) = (S("-~)A)~ + (-- i)~/~-l/X'g(i0, ..., i~,_1) I-[ s(m--i,)A, s=0 I.~--1 the sum, E', being over all (io, ...,i~,_l)eZ~_,i~<m such that ]~ i,-----~.i, but i 0 = i I ..... i~_ 1 --= i is explicitly excluded. , =0 Proof (Theorem 2-3). -- We first outline the proof. Let W be the kernel of (I--X-la) ~ in L' (and hence by Theorem 2. I) in L(qx). Suppose dim W>r>o for some reZ+. We must show that Z~-I)(X -1) =o for s----I, 2,...,r. Let gJl~ (for each NeZ+) denote the matrix relative to a monomial basis corresponding to the linear transformation ~k----=Tso~ of 2/~/. Explicitly, for each ve3;s, a~(X ~) =EgJtk(u, v)X", the sum being over all u~3;s. Let xN(t)=det (I--tg)~). We know that for all seZ+, lim ?(~l(X -~) =Z~/(X -1) and thus we must show that lim )~-~/(X -1) = o for s = I, 2, ..., r. Letting N' be the Ig --->- 0o dimension of ~(s), equations (2.25) and (2.~9) show that it is enough to prove that (2.32) lim S/N'-~/(I--Z-1932~) =o for i=o, I, ..., r--I. We shall prove the existance of a constant c independent of N, such that for i~o, I, ...,r--I ord c +� I)N (2.33) and prove (2.32) by using (2.33) and (2.31) to deduce the existence of a constant c' independent of N such that for i--o, I, ..., r--i (2-34) ord S(Z~'-0(I--X-t~lJ~)2c' +� I)N[~ '+'. 27 ~8 BERNARD DWORK Let 9J/~' = (I--X-~gTt~) ~. We may view 9J/ff as a matrix whose rows and columns are indexed by the set ~s of all u e3; such that u 0< N. If H is any subset of ~, H 4= 35r~, we may, following our previous convention, denote by (922~', H) that square matrix obtained from 9Jl{~ by deleting all rows and columns indexed by elements of H. We shall show that if H is any set of not more than r--~ elements of 3; then c may be chosen independent of H and N such that ord det (!IR~', H) _>c +x(q-- ~)N (2.35) whenever H (if not empty) is contained properly by 35 N. Equation (2.29) shows that equation (2.35) implies (2.33)- Our first object is the proof of equation (~ .35). Let H be a set of no more than r--I elements of ~E. We know that there exist ~1, ..., ~,, a set of r linearly independent elements in W. Let ~i=ZB,,iXu, j=I, 2, ..., r, the sum being over all ue3;. The (possibly empty) set of [H] equations ]~ aiBu, i=o i=1 for each ueH, in r unknowns a 1, a~., ..., a, certainly has a non-trivial solution in f~ (since r> [H]). Since ~t, -.., 4, are linearly independent, we conclude that ~---- ~ ai~ ~ j=i is a non-trivial element of W. Since o + ~eL(q~), ~ may be normalized so that ~es and at least one coefficient of ~ is a unit. Thus there exists a normalized element ~=ZB~X" in W such that B,=o foreach ueH. Theorem ~. ~ shows that, forall ue~;, (~. 3 6) ord B~> q� , where e=~tordX+(q--I)-aordX. Hence if N>No=e/qk, we may conclude that T~ is also normalized and the coefficients of ~N=TN~ satisfy (2.36). For typographical reasons we shall when convenient denote the coefficient of X u in ~ (resp. F) by B(u) (resp. A(u)) instead of B, (resp. A,). For given integer j>i, (0@ ~T N ~ = (T~ o~)JT N ~ ---- ZXW(/)B (w (~ A (qw I1)- w I~ A(qw C~)- w I1)) ... A(qw lil- w I j- 1)) the sum on the right being over all (w (~ w (1), ..., w (i)) ~7~ +~. We may write TN(s as a similar sum except in this case the sum is over all ((w (~ w (1), ..., w!i-~)), w Ij)) e3J x 3;N- Since ord A~>� 0 for all ue3;, we have by (2.36), i-1 X--1 ~176176 " " A(qw(i)--w(i-1)) } >-- --� qw(~ +,~=o (qw{'+ l)--w(i)) ) = Thus we If w/~ w(1), ..., w Ci-t) do not all lie in 3;N then certainly a w/~) ) >N. i= can conclude (using only the fact that ~eL(qx,--e)) that TN(~i~) -- (TNOe)~TN~ mod Y~ X~'C(� (q--~)� (2.37) u~ N 28 ON THE ZETA FUNCTION OF A HYPERSURFACE where for each real number, b, C(b) is used in the sense of C~ in w I. Since o= (I--X-te)~= Y. (__?-t)i(~)~i~, i=0 we have tx :=o i=o (I--X-%t~)~TN~ mod N X"C(xquo--~e+(q--i)xN ). For each element (u, v)s~l~ x ~N, let ~'(u, v) denote the coefficient of the matrix ~l~' in u t~ row and v tb column. We have for each vs~N, (I--X-10t~)~X'=]~Jt~'(u, v)X ", t* the sum being over all ue~ N. Thus (I--X-~a~)~TN~=Y~B~ v)X ", the sums being over all ue~ N and all w~s. We conclude that for each ue~I~ , ]~lJ~(u, v)B~-=o mod C(xqu0--~ e+ (q~)xN), the sum being over all vs~ N. We recall that B,-----o for wH and hence if N" is the number of elements in ~--H, the system of N" congruences indexed by ue~N--H , (2.38) ~p-~'~ v)B,---o mod C(--2 e+ (q--I)xN), (the sum being over all VEZN--H), has a non-trivial solution if N>N 0 since B, is a unit for at least one W~N--H. The ring of integers, s of Y~ is not a principal ideal ring, but finite sums of principal ideals are principal. Hence the theory of elementary divisors may be applied to the matrix E N indexed by (~--H) � (%N--H) whose ,,general" coefficient is EN(U , v)-=p-"q~"gJl~z'(u, v). If r are the elementary divisors of E N then (2.38) shows that (~'.39) CN"- O mod C(--2 e+ (q-- I)� Since our object is to prove (2.35), we may assume det(gX~', H) 4: o. Hence o 4 = det EI~ , O4:sN,,. If U and v lie in ~N--H, let (EN, (u, v)) denote the matrix obtained from E N by deleting row u and column v. Let ((gX~', H), (u, v)) denote the corresponding matrix associated with (gYt~', H). It follows from the definitions that det(Es, (u, v))/det Et~ =p~"det((gJl~', H), (u, v))/det(gJ~, H). (2.40) Ideal theoretically, (detEN)=(SN,,)E(det(EN, (u,v))), the sum being over all (u, v)e(~N--H) 2. Thus -- ord r ord det(Es, (u, v))- ord det EN, the mini- mum being over all (u, v)~(%N--H) 2. This together with (2.4 o) shows that (2.4I) --ord ,z~,,----Min {� + ord det((g3~', H), (u, v)) }--ord det(~It~;, H), the minimum being over the same set as before. This together with (2.39) would Z~''''~-~tN(u, BERNARD DWORK 3 o give the proof of (2.35), if it were known that c may be chosen independent of N and H, u and v such that (2.42) xqu o -}- ord det ((~Q~', H), (u, v) ) ~ c + 2 e. Thus the proof of (2.35) has been reduced to that of (2.42). We observe that 93~(u, v) =Aqu_ . and hence ord 9JUs(u, v)~x~(qu--v). It is easily verified that if two square matrices (each indexed by 25s) satisfy this estimate then so does their product since xa(qu--w)-+-xa(qw--v)~xa(qu--v). Thus tt tlt --~ ~N = (I--x-~)" = I + 9J~N x , where 9J~' is a square matrix indexed by X~--H satisfying the condition (2.43) ord ?/J~"(u, v) ~x~(qu ~v) for all (u, v)e(B:s--HH) ~. Equation (2.42) now follows directly from Lemma 2.2 (ii). This completes the proof of (2.42) and hence of (2.35). As we have noted previously, this implies the validity of (2.33). We must now show that (2.33) implies (2.34). This is clearly the case for r= I. Hence we may assume that r> I and that (2.34) has been verified for i = o, i, ..., r~2. Replacing A by I--X-l~lJ~ in (2.3 I), we have (SIS'-r --),-~gJ~)) ~ = S/S'-I'-t))((I--X-a931~) ") --Z'g(i0, ..., i~_t) I] S(N'-i,)(I--X-t~) the sum, Z', on the right being over all i0, iD...,i~_~ in {I, ~,..., N'} such that IX--1 5~ i~=~t(r--I), but io=i t .... =i~_x is excluded. In each term in the sum, Z', at $~0 least one factor S(N'-is)(I--Z-x~J~) occurs such that is<r--i, while the remaining factors are ~--x in number and each of type SIS'-i/(I--X-x~ff~), j~(r--i). The assertion follows from the induction hypothesis provided we verify the existence of a finite lower bound for ord S Is'-i)(I-x-tgJ~) independent of N and valid for j~.(r-- ~). The existence of such a lower bound is an obvious consequence of equation (~. ~9) and Lemma ~.2 (i). This completes the proof of the theorem. Note. ~ No use has been made in Theorem ~. 3 of compactness and no hypothesis concerning the field generated by the coefficients of F is needed. On the other hand we do not know if Theorem ~. ~ is valid without that hypothesis. We now summarize some of our information. Theorem 2.4. -- For each non-zero element, ~, of ~, let sx be the multiplicity of ~-1 as zero of XF. If the coeffcients of F lie in a finite extension, K0, of Q', then for s~s x the space Wx=kernel in L' of (I _)-l~)s isindependentof s, lies in L(xq) andis of dimension s x. Further- more W x has a basis consisting of elements of K0(~,){X }. Proof. -- For given )~2", let W cs) be the kernel of (I--),-1~) ". Theorem 2.2 shows that for s~sx, dimWIS)~sz, while Theorem 2.3 shows that dimWC~l~sx for all 30 ON THE ZETA FUNCTION OF A HYPERSURFACE s>~. Since W(~)cW (~+1) for all s>~ it is clear that W (") is independent of s and has dimension s x for s >s x. The remainder of the theorem follows directly from Theorem ~. ~. Corollary. -- If G is an element of K0(X ) such that for some real number b>o both G and ~/G lie in L(b) and if H(X) =F(X)G(X)/G(X~) then Z~=7.a, it being understood that F and K 0 satisfy the conditions of the theorem. Proof. -- Let e=Min(� b). It is clear that ~G.~ is a mapping of L(c) onto itself. The corollary now follows from the theorem and the fact that each ~eL(c), d?(H~) = G(X) -'. ~(~). G(X). We have shown that the zeros of Z~ can be explained in terms of spectral theory if F satisfies the condition of Theorem 2.4. If it were known (as is the case in the geometrical application) that the coefficients of F and the zeros of Zr all lie in a finite extension, f20, of Q,' then the zeros of ;(~ can be explained entirely on the basis of the spectral theory of ~ as operator on L"=f~0{X}nL(q~z ). We make no assertion of the type: L" is a sum of primary subspaces corresponding to :r Our next result serves as a substitute for a statement of this type. Theorem ~,. 5. -- If X is a non-zero element of ~2 which is algebraic over O', /f X -1 is of multiplicity ~ as a zero of )F, if the eoe~cients of F lie in a finite extension, K0, of Q' and if K is anr finite extension of K0(X ) then (2.t4) (I--X-'~)~+* (K{X}n L(qx)) = (I--X-*~)~(K{X}n L(qx)) 9 In particular if ~=o (i.e., Z~(X -~) 4:o) then K{X}na(qx) =(I--X-*:c)(K{X}na(qx)). Proof. -- Let K' = K0(X ). By hypothesis K is a finite extension of K'. For given K we show that (2.44) holds if and only if it is valid when K is replaced by K'. Let col, ..., r be a minimal basis of K over K'. Suppose (2.44) is valid with K replaced by K'. If ~eK{X}nL(qx) then ~= ~ co~., where ~eK'{X}nL(qx), i= i, 2, ..., m. i=t each in K'{X}nL(q� such that Hence by hypothesis there exist ~t,''',~m (I--X-~0c)~'+l~qi = (I--X-la)~., i= I, 2, ..., m, and hence ~= Y~ r ) i=1 and furthermore (I--Z-~)~+l~= (I--X-~e)~. This shows that (I--X-'0c)~+t (K{X}nL(q� ~ (I--X-*~)~(K{X}n L(qx)) and since inclusion in the opposite direction is clear, we may conclude that (2-44) is valid for K if it is valid for K'. Conversely if (2-44) is valid for K, then given ~eK'{X}nL(qx) there exists ~eK{X}nL(q~z) such that = (I--x-t~)~. The relative trace, S, which maps K onto K' may be extended to a mapping of K{X} onto K'{X} in an obvious way. The trace, S, commutes with ~ and hence (I--X-le)~'~=(I--X-la)t'+tS(~/m) since S(~)=m~. Since S(~q)eK'{X}nL(qx)we may conclude that (2.44), if valid for a given K, is also valid for K'. 31 3~ BERNARD DWORK We have shown that it is enough to prove the theorem for one finite extension, K, of K'. If k-1 is not a zero of ZF, let K = K'. If X-1 is a zero of ZF then following the procedure of the proof of Theorem 2.2, let ZN(t)=det(I--taN) , let p be real, p>o such that Z~ has no zeros in k-l(~ + Cp) distinct from k-*. For all N large enough, k-1 . k-1 in ~.--l(I Ji-Cp), these are zeros ofa polynomialfN ZN has precisely ~ zeros 1,N, 9 9 , ~,~ of degree ~ which divides ZN and is relatively prime to zN/fN" Let K be the composition of all extensions of K' of degree not greater than ~. We know that k-~l,N, 9 9 -, k~,N-1 lie in K, approach k -1 as N-+oo and are distinct from all other zeros of ZN. In the following a n will be restricted to K[X] ni~ CN). ~t Let ~n be the endomorphism II (I--k~aN) of K[X]n~ CN) (~N=I if a=o). 4=1 Since ~N annihilates the primary components of K[X]n~ (N) relative to a N corresponding to the cigenvalues )'I,N, ---, k,,N, it is clear that ~NIK[X]nEIN)) is the direct sum of the primary components of K[X]r~(N) corresponding to the remaining eigenvalues of a n. Hence if a~' denotes the restriction of a n to ~N(K[XJo~tN)), we can conclude that (2-45) det(I--ta~') = det (I--ta N (I --th.N). Let ~ be a given element of K{X}nL(qx). We must find ~q in the same space such that (I--Z-~a)~+:~= (I--Z-xa)~. We may suppose that ~eL(qx, o). Let ~N----TN~. Since k is not an eigenvalue of a~, there exists BNeK[X]ns such that (2.46) (I --Z -~ aN) ~N~n = ~z~ ~N" Eventually we shall complete the proof by taking the limit of this relation as N-+~. The main problem is the demonstration that ~s may be chosen such that its limit lies in L(q~). We note that ~N~N is uniquely determined by (2.46) and hence ~N is uniquely determined modulo the kernel, WN, of ~N in K[X]n~ <N), a subspace of dimension ~. We shall show that there exists a real number c' independent of N such that ~N can be chosen so as to satisfy the further requirement (2.47) ~NeK[X]~L(qx, c') for an infinite set of integers, N. We first construct a basis of ~N(K[X]~IN)). For each Ue~:N, let Y,,N----~N X~. The set {Y~,N} indexed by ue~n, spans ~N(K[X]o~IN/) but does not (unless ~t =o) constitute a basis of that space. In the proof of Theorem 2.2, it was shown that there exists an infinite subset, 91, of Z+ and a set S of ~z elements of ~5 such that for each Ne91, the kernel, WN, of ~N in (K[X]c~ Is/) has a basis {g,,N}~es consisting of elements of K[X]n~(N)(qx, --e)n~[X] indexed by S such that for each veS the coefficient of X ~ in g~,z~ is the Kronecker 3,,,. (In the previous remark, e= (~ -]- (q-- I) -~) ordk, precisely as in the proof of Theorem 2.2.) Thus for each u e S, we have (N being assumed in the remainder of the proof to lie in 91), (~' .48) g.,~ = X"+ ZE~(w, u)X w, 32 ON THE ZETA FUNCTION OF A HYPERSURFACE We may the sum being over all we3;~--S, and furthermore ord EN(W , u)>qXwo--e. now conclude that for each ueS, since o= ~s(g,,N), that (e-49) --Y.,N = ZEN(w, u)Yw,~, the sum being over all WE~N--S. Thus the set {Yw,~} indexed by ~N--S spans ~N(K[X]ns (N/) and hence must be a basis of that space, since it contains the correct number of elements. We have noted that if ~z~ is a solution of (2.46), then the sum of ~N and any K-linear combination of the g~,s is also solution of (2.46). Equation (2.48) shows that ~qN may be chosen such that the coefficient of X" in ~ is zero for each uES. (In fact these additional conditions uniquely determine ~qN). Thus we may write aqN=ZB~,NX v, the sum being all vE3;N--S. By hypothesis ~EL(qx, o) and we write ~---ZG,X ~, the sum being over all vz%. Thus ~N=TN~=ZG,X *, the sum now being over ZN, and ord G~> qxv 0. Thus ~N~N = y~ GvY~,N = Y~ G~Y~,N + ~] G~Y~,N" Applying (2.49) v~ZN v~ZN--S u~S we now obtain ~N~N----=ZY,,N{G~-- 2 G~EN(V , u)}, the sum being over all ve~s--S. u@8 Thus ~N~=ZGv, NY~,N, the sum being again over all ve3;N--S. Here G~,~ = G,-- Z g~(v, u)G. u@S and hence ord G~,N>q� , c being a real number independent of N. We now determine the matrix of ~' relative to the basis {Yo,N},eZ,,-s of ~N(K[X]ns Since ~N commutes with [~, we have 0~N'Yv, N = 0~N~N xv ~--- ~N~N xv = ~N Z Aqw_vX w. w~N With the aid of equation (2.49), it is easily seen that for W~s--S (2.5 o) a~'Yv, N ---- EASt(w, v)Yw, rr, the sum being over all WeZN--S, where for (w,v)a(3;N--S) ~, A~(w, v) = Aqw_ v- ?~ Ez~(w, u)Aq,,_~. uES It is easily verified that ord A~r(w , v)>� Let A~ be the square matrix indexed by 35N--S whose w, v coefficient is AN(w ,' v). Equation (2.5 o) shows that det(I--X-~A~) = det(I--^'-laN)."' Since ~F(t) = lim det(I--taN) , we conclude from (2.45) that lim det(I--X-~A~) is the value N-o,- oo N~oo assumed at t=X -~ by zr(t)/(I--Xt) ~. This value is not zero since ~z is the multiplicity of X -1 as zero of Zr and hence for N large enough, det(I--X-lA~) is bounded away from zero. Explicitly there exists a rational number c" such that for N large enough, (2.5 I) ord det(I--X-lA~) <c". 33 34 BERNARD DWORK Equations (2.46) and (2.5 o) show that the set {B,,s} indexed by veX~--S is a solution of the system of equations indexed by w~3;~--S Z(8~,~--x-*a~(w, v))B,,N = Gw, z~, (~. 52) the sum being over w3;s--S, it being understood that ~,~ is the Kronecker ~ symbol. To verify equation (2.47), we apply Cramer's rule to equations (2.52) and estimate ordB,,s for each w3;s--S. For each element (w, v) of (3~--S) ~, let((I--X-tA~), (w, v)) be the square matrix obtained from I--x-~A~ by deleting row w and column v. Clearly --1 , B,,~.det(I--X A~)=Z+det((I--X-tA~), (w, v))G~,~ the sum being over all w~3;~--S. In view of (2.5i) it is enough to show that there exists c'" independent of N such that (e.53) ord det((I--X-~A~), (w, v))+ord G~.~>_q� for all (w,v)~(%N--S) 2. Equation (2.53) is however a direct consequence of Lemma 2.2 (ii) and our estimates for ordG~, N and ordA~(w, v). This completes the proof of (2.47). Since K{X}nL(qx, c') is compact, we conclude that the infinite sequence {~N} has a limit point v? in L(qx, c'). Taking the limit of equation (2.46) as N~oo over a suitable infinite subset of Z+, we obtain (I--X-l~)~+tzq =(I--X-t~)~. Thus we have Shown that (I--X-I~)~+~(K{X}nL(q�215 This completes the proof of the theorem. Corollary. -- In the notation of the theorem, let R = K{X}nL(q� and let W be the kernel of (I--X-t~) ~ in R. For each integer j, j>I we have (R=W+ (I--Z-l~)J~ (e 54) tWn(i__Z-~e)jR= (I__X-~)jW. If (I--X-'~)'W={o} then (2.55) (I--X-t~) ~+~R = (I--X-re) ~R- Proof. -- For simplicity let us use the symbol 0 for (I--X-le). The theorem shows that given ~ there exists ~ such that 0~=0~+t~, which shows that 0~(~--0~) =o and therefore ~q~W-t-0~. This shows that ~cW-+-0!~ and hence using the fact that 0WoW we easily see that RcW+0~'R if j>I. This proves the first half of equation (2.54). Writing this with j= I and applying 0" to both sides we obtain 0v!R----0~W-I-0~+t~R, which proves (2.55), since 0~W=o. If ~ER and 0i~EW then 0i+~0~W={o} and using Theorem 2. 4 we see that ~W, which shows that 0~~e0iW. This completes the proof of (2.54). 34 ON THE ZETA FUNCTION OF A t-IYPERSURFACE w 3. Differential Operators. a) Introduction In this section we modify the notation of the previous section so as to facilitate the application of our results to projective varieties. Let Q' and ~ be as before. Let ~,) be a finite extension of Q' in f2, whose absolute ramification is divisible byp -I. Let n>o,d>I be fixed integers as before. Let Z now be the set of all U -==- (U(), Ul, ..., Un+I) EZ~ +2 such that duo=Ux+... +u,+ x. The definitions ofL(b, c), L(b), ~, ~2 (N), PJl(b, c) are now precisely as in w 2 except that the set Z is given a new meaning and furthermore these additive groups now lie in DO{X 0, X1,..., X,+~} instead of ~{X0, X1, ..., X,}. Let S be the set {I,2,...,n+I}. For each subset A of S (including the empty subset), let M A be the monomial l-Ix~, (M o = i) and let iffA LA(b, c), LA(b), ~A, ~A,(NI, ~A,(NI(b ' C) be the subsets of the previously defined sets which satisfy the further condition of divisibility by M t in f~o{X0, Xa, ..., X,}. Let S'~-{o,i,...,n+I}, S"={o,I,...,n}. Let ~0 be the ring of integers in DO and let K be the residue class field of D 0. Let E i be the derivation ~ X~X] ~ of DO{X0, ..., X,+a}, i=o, ,, ..., n+ i~ A homo- geneous form fin D0{X~,..., X,+~} will be said to be regular (with respect to the variables Xa, ..., X,+~) if the images in K[Xt, ..., X,+~] of the polynomials f, E~f, ..., E,+a/ have no common zero in n-dimensional projective space of characteristic p. Let ~ be an element of1~ 0 such that ord ~= ~/(p--I), f a form of degree d in ~)0[X~, ..., X,+I] which is regular with respect to the variables X1, ..., X,+ 1 and let H be an element of L(i/(p--x)) such that HeDO{X} H - ~zX0fmod Xo 2 H~=E,H~L(p/(p--~),--~), i=o, ~, ..., n+,. For i=o, I,..., n+I, let D i be the differential operator ~--~Ei~+~.H~, mapping L (--oo) into itself. It is easily verified that dD 0 = D 1 + ... + D,+~, that D~oD~ = DioD~ and that D~ maps L(b) into itself for b~p/p-- I. The object of this section is the study n+l of the factor space L(b)/Y, D~L(b). To make use of the regularity of f we must recall some well-known facts about polynomial rings. b) Polynomial Ideals. If A is any set and G is an additive group then a set of elements ~. i in G indexed by A� will be said to be a skew symmetric set in G indexed by A if ~.,i=--~i.~, ~,i=o for all i,j~A. Let JR be a field of arbitrary characteristic, and let a be a homogeneous ideal in R[X]~-R[Xa, ..., X,+J. The ideal a has an irredundant decomposition into 35 3 6 BERNARD DWORK homogeneous primary ideals, a= n q/. The dimension of a is defined to be i=1 Max dim q~ and dimension here is in the projective sense. We recall [5], I. If geR[X] then (a:g)=a if and only if gisqi ,i-=I,2,... r. II. If g is a non-constant homogeneous form then dim (a + (g)) = dim a-~ if g lies in no primary component q~ of maximal dimension, while otherwise dim(a + (g)) = dim a. III. (Unmixedness Theorem): If a= (gl, g2, 9 .., gt), t<n+ i and dim a-~n--t then each primary component of a is of dimension n--t. Lemma 3. x. -- If gl, 9 9 g,+l are non-constant homogeneous forms in R[X1, . .., X,+t] with no common zero in n-dimensional projective space of characteristic equal to that of R and if {P~}~EA is a set of polynomials indexed by a subset a of S={I, ~,..., n+ I} such that Pig/= o then there exists a skew symmetric set ~/,i in ~R[X] indexed by A such that P/= $'. ~%gi for each i~A. Furthermore/f{Pi}iea consists of homogeneous elements such that deg(P/gl) = m is independent of i, then each ~i,j may be chosen homogeneous of degree m--deg(gigi). Proof. -- Let a,. = (g,, 9 gr), I < r < n + I. By hypothesis dim a,+ 1 = -- i, while by II, dimar--I~dimar+t~dimar for r=I,e, ...,n. Also by II, dima,~n--x. These inequalities show that dim ar=n--r for r= i, 2, ..., n-I- i and that dim a,+ t = dim r i for r~n. Hence by III, the primary components of a, are all of dimension n--r and by II, gr+, does not lie in any primary component of ar for r=I, 2, ...,n. Hence by I, (ar:gr+l)=a~" Furthermore since dim%=n--I, we know gl:~~ If a--{I} then Pl=~ and hence we can assume a={I,a, ...,r+I}, r:>x. Since (a, :gr+~)=a~, P,+lea~ and hence there exist polynomials h,, h2, ..., h~ such that P,+~= ~ h/gi. Thus ~ (Pi+higr+,)g~=o. Using the obvious induction i=1 i=l hypothesis on r, there exists a skew symmetric set ~/,i in R[X] indexed by {x, 2, ..., r} such that P~+h~g~+~= ~ ~%g~. for i= I, 2, . .., r. Let ~+~,/=h~, ~i,r+t-~--h~, for i= i= i, 2, ... r and let ~+~,~+~, r=o. The assertion follows directly. The valuation of ~0 can be extended to a valuation of the polynomial ring ~0[X] in the usual way, if g(x) =Za,,x", let ordg=Min ord a,. Lemma 3. ~. -- Let g~ , ..., gn + 1 be non-constant homogeneous forms in s ..., X, + t] whose images in K[X] have only the trivial common zero. Let A be a non-empty subset of S and let g be an element of the ideal Z (g~) of x20[X ]. Then there exist elements {hi},e ~ of ~0[X] such that g = Y~ g/hi and such that ord hi~ ord g for each /cA. i@A Proof. -- We may suppose that g 4= o and hence that ord g = o. By hypothesis g = Y, g/hi, hie~0[X ]. Let e be the absolute ramification of ~0 and let -- b = e. Min ord h;. Clearly b is an integer and we complete the proof by showing that if b>o then there 36 ON THE ZETA FUNCTION OF A HYPERSURFACE exists a set of elements {h~}i~ A indexed by A in f20[X ] such that g-= Y~ g~h~ and such i@A that e. Minordh;~--b+ I. Let II be a prime element of f20. By definition, iEA l-lbhi~s for each j~A and if b>o then ZgiHbh~=Ilbg-omod(II). Let G~ be the image ofg i and let ~ be the image of IPh i in K[X] for each leA. Thus in K[X], o= Y~ Gi~ i and so by Lemma 3. i, there exists a skew symmetric set, {~,i}, in K[X] iEA indexed by A such that ~= Y~ ~,jG i foreach ieA. We now choose a skew symmetric set i@A {~, j} in s indexed by A such that ~, i is the image in K[X] of ~'~ for each (i, j) cA � A. Hence Hbh~ = E ~igimod(II) for each i~A. We now define a set of elements {hl}~E A in f~0[X] by the equations Ilbh~ = 1-Ibh~ + Y, ~',igJ for each icA. Clearly iEA IIbh~=omod(II) and hence e. Minordh~--b+I. On the other hand the skew i EA symmetry of the set ~q~'j shows that g= Y~ gih~ which completes the proof of the lemma. ~A Corollary.- If ga, ...,g,+~ satisfy the conditions of the above lemma and {P~},cA is a set of elements of I20[X ] such that ~ Pig~=o, then the skew symmetric set ~i,i of Lemma 3. may be chosen such that Min ord ~q~,~> Min ord P~. i,i Letfbe the form of degree d in 230{X~, ..., X,,+~} which is regular with respect to the variables Xa, Xz, ...,X,+~. Let f0=f,f=EJ for i=I,2, ...,n+I. Since df0=fl+f~+..--q-f,+a, it is clear (lettingf~ be the image off~ in K[X~, ..., X,+~]) that the regularity off is equivalent to (i) f0,f~, .-.,~ have only the trivial common zero (ii)fl,f~, ...,f,+~ have only the trivial common zero if ptd. Condition (ii) is simpler for most of our applications but will not be used since it would limit our results to the case in which d is prime to p. However we do note that in any case the regularity off implies the triviality of the common zeros in f~ off~,fz, .- 9 ,f,+ ~. Thus Lemma 3. x shows that f~,f~, ...,f,+~ are linearly independent over D~ 0 (and f~). The following lemma refers to ideals in either f~0[X] or in K[X]. To simplify the statement we use the same symbol for f and f~. Lemma 3.3. -- Let B be a non-empty subset of S = { i, ~, ..., n + I }., (i) (M,)n Y~ (f) = Y, (M~f) + X (Msf/X~) i~ A iuA --B /~Ac~B if A is any non-empty subset of S, provided the characteristic does not divide d (i.e. the assertion holds in any case in ~2o[X ] and if ptd in K[X]). (ii) In either characteristic, if A + S then (M,) o ((fo) + X (f)) = (MBJo) + X (M,f) + X (M~f/X~.). i~A i~A--B i~A ('IB unless both AoB=S and A contains n elements. 37 3 8 BERNARD DWORK Proof. -- In both statements the ideal on the right side clearly lies in the ideal on the left side. To prove (i) it is clearly enough to show that if MB.he ~ (f) then IEA (3.,) 2 (f,)+ 2 (f, lX,) IGA--B i~Af~B Let B n A = C, B' = B-- C. Let h' = Mch. We first show that h' E Y~ (f). This is clear iEA if B' is empty, hence we may use induction on the number of elements in B'. If j~B', then letting B"= B'--{j}, h" = Mw,h' then Xjh" = Mwh'e ~ (f) and if we can show that h"e Y, (f~) then by the induction hypothesis we may conclude that the same holds for h'. Thus we consider jCA, Xjh"e Y~ (f) and recall that Xj, {f}~.j,~s is a set ~A of n § I non-constant polynomials with no non-trivial common zero (since the charac- teristic does not divide d) and hence Lemma 3.i shows that h"eZ (f). Hence iEA Mche Y~ (f) as asserted. If d---- I then if C is empty, (3- i) is trivial, while if jeC, i@g then f//Xj is a non-zero constant which again shows that (3. I) is trivial. Hence it may be supposed that d> i, in which case f~ =f/X i is a non-constant form for eachieS. We may assume that C={I,2,...,r},A={i, 2,...,t},r<t<n+I Thus t t X,X2...Xrhe 2 (f) and hence for some polynomial hi, XI(X2...Xfi--f(ht)e 2 (f). i=l /=2 We now apply Lemma 3- I to the n+ i polynomials, Xl,f2 , ...,f,f+l, 9 9 .,f,+t and conclude that X2...Xfl~(f() + ~ (f) (the left side is h if r= I). Now suppose for i=2 some s, I<s< r, X~+lX,+2...XrhE ~ (fi') @ Z (f/). Then there exists a polynomial, i=1 ]=s+l h~+,, such that X~.t(X~+e...X fl h~+tf~'+l)~ (f() + ?~ (.~). The n+I poly- i:= 1 i=s+2 nomials fl' ,f2', 9 9 9 ,f,', X~. t,f, + 2, 9 .. ,J~ + 1 are non constant forms satisfying the conditions s+l t of Lemma 3.~ and hence X~2...Xfle xZ (f') + Y, (f). This completes the proof i=l i=s+2 of (3. I), and hence of the first part of the lemma. (ii) Here it is enough to show that if MBhe (f0) + 2 (f~) then (3.2) 2 z (Z). i@A--B iCA~B Let C and B' be defined as before and let h'= Mch. We first show that (3.3) h' (f0) + z (f,). To show this, it is enough (as before) to show that if ICA and Xth"e Z (f) + (f0) then /@A the same holds for h". By hypothesis B' is empty if A contains n elements and hence for the proof of (3-3) it may be assumed that A does not contain n elements. Thus 38 ON THE ZETA FUNCTION OF A HYPERSURFACE Ata{I} contains at most n elements. Let C' be a subset of S disjoint from {x} which contains A and consists of exactly n-- I elements. The n + I polynomials, f0, X~, {f~}~c' satisfy the conditions of Lemma 3. I and hence h"e 52 (f) + (f0). This proves (3.3)- 4cA If C is empty then (3-2) is trivially true, hence we may assume C not empty. If d= I the f/'=I for each ieAc~BoeD and hence it may again be assumed that d>i. We may now let C={I, 2, ...,r},A={I, 2, ...,t},r<t<n. Since (3.3) now shows that X1.--X,h~(f0)+ }2 (fi), we have for some polynomial, h~, i=t Xl(X2...Xrh--hlft')c(L ) ~- ~a (fi)" i--2 The set of n + ~ polynomials, (f0, X~,f~, ... ,f,) satisfy the conditions of Lemma 3- I and hence X2...X~he(f0)+ (f~')+ Z (f). We now suppose that for some s, I<_s<r, i=1 we have X~+t...Xrh~(fo) § ~] (f()§ Y~ (f). Then for some polynomial i~l i=s+l h~+~,X~+~(X,+2...X~.h--fj+th~)e(fo)+~(f()+ Z (L). The n-k-~ polyno- i=l i=s+2 mials f0,f~',...,fj, X~+l,fs+-~,-..,f~ satisfy the conditions of Lemma 3.i and s+l t hence X~+2...X~he(f0)§ ~] (f')+ Y~ (f) which completes the proof of (3.2) i=1 i=s+2 and hence of the lemma. C) P-adic Directness. Let W be a vector space of dimension N over f~0 which has a <( naturally >~ preassigned basis. For the purpose of the immediate exposition, we may let W be the space all N-tuples, Do s, with coefficients in ~0- However in the applications in the following parts of this section, W will be a subspace of f~0[X] whose << natural ~ basis is a finite set of monomials. Let ~ be the ~)0-module, ~2, in W and let q~ be the natural map of ~ onto the K-space, W*= K N. For each subspace W t of W there exists a subspace, W~ =9 (Wa r~ ~B), of W*. The correspondence WI~W ~ maps the set of all subspaces of W onto the set of all subspaces of W* and preserves dimension. If W t and W~ are subspaces of W then (W 1 nW2)*c W~ n W.~, but equality need not hold If however W~ nW~ = {o }, then equality must hold and hence Wtr~W 2 ={o}. We shall say that W 1 + W 2 is a p-adically direct sum, written W~[+]W2, if W~c~W~={o}. In particular if Wt[+]W2=W then we shall say that Wz is p-adically complementary to W 1 in W. It follows from the above remarks that given a subspace W a of Q, there exists a subspace of W which is p-adically comple- mentary to W 1 in W. The notion of p-adic directness is introduced because of the metric naturally associated with W. If w=(wt, ..., wN) is an element of W then let ord w=Min ord w i. 39 40 BERNARD DWORK If weW'+W", (W' and W" being subspaces of W), then w=w'§ where w'eW', w"EW". Certainly ord w> Min(ord w', ord w"), but if the sum W'+ W" isp-adically direct then ord w=Min (ord w', ord w") and hence ord w'>_ord w. d) General Theory'. Let 9/be the ideal (f0,fx, -..,f,) in ~0[X1, ..., X,+,]. For each integer m~o, let W (m) be the space of forms of degree dm in f~0[X1, ..., X,+I] , let 9/~ =W(m~c~9/ and let V <'~; be a subspace of W (m) p-adically complementary to ~Im in W ('), with respect to the monomial basis of W Ira). Since (f0, fl, ... ,f,) have no common nontrivial zero in ~, 9/ must contain all homogeneous forms of high enough degree and hence there exists an integer, No, such that V Ira) ={o} for m>N 0. We shall show eventually that we may take N o to be n. We note that V~ We now let V ~ ~ X~'V/"), a subspace ofE (N~ and for each pair of real numbers b, c, m=0 let V(b, c)=Vt~L(b, c). It follows from Lemma 3.2, the construction of V and the regularity of the polynomial f that if Q is a homogeneous form in ~0[X1, ..., X,+I] of degree dm, then Q=P+ ~ Pif, where P~V (m), P0, P1, ..., P, each lie in W (m-l) i=0 and ord P>ord O, ord P~ord Q for i=o, i, ..., n. We now proceed with the analysis of the differential operators introduced in part a) of this section. We recall that H~EL(p/(p-- ~), -- I), and that H~ has no constant term. It follows easily that if b<p/(p-- x) then H~L(b, --e), where e= b-- (p-- ~)-~. Lemma 3-4. -- L(b, c) = V(b, c) + ~ H4L(b, c + e) if b<p/(p-- i), e = b-- I/(p-- i). i=0 Proof. -- It is clear that the left side contains the right side. If ~ is an element of L(b, c), we show that for each N~Z+ there exists ~qNeV(b, c)n~ Is/ and a set ~,N-a of elements in L(b, c+e) indexed by iES"={o, I,..., n} such that (3-4) ~ - ~N + ~ H,~,,N-1 mod(X0 N+~) (3.5) i~-1 - ~z~ mod X0 N I~,N-~- ~,N-z mod X0 N-1 for each ieS". Let pl0) be the constant term of ~, then (3.4) holds for N = o if we set ~0 = P;~ b, c) and ~i,_l=o for each ieS". We now suppose N>o and use induction on N. Then ~(N)=~--(~N_ 1+ ~ Hi~.,N_21 lies in L(b, c)and is divisible by X0 s. Let p(N)be the \ / {=0 coefficient of X N in ~(N). Clearly ord P(N)~bN+c and as noted above there exists O~N)~V(N),p~0~-I) ' ...,p~N-1) each in W (N-~) such that P(N)=O~)+E~p~N-')f, where ord O~)~ bN ~- c, ord p~N--1)> bN + c-- (p-- ~)-1 = (b -- I )N ~- c -}- e for each i e S". We now let ~=~N_~+X0nO~N)eV(b, c), and for each ieS" let ~"N-- ID(N--I) ~ I /I~ 40 ON THE ZETA FUNCTION OF A HYPERSURFACE 41 and compute ~-- ~-q- H~ ~,~ t = i 0 = o mod Xo s +' 2 Xo ( ) This completes the proof of (3-4) and (3-5)- The proof is completed by taking weak limits, ~,s-+~eL(b, c+e) for each ieS", ~N-+~eV(b, c) and hence ~=~+ ~ H~. i=0 Lemma 3-5.- Vn~ H,L(b)={o} /f b<p/(p--i). i=O Proof. -- Let ~ lie in the intersection, then ~ = ~ Hi,i, ~eL(b) for each i~S". ~=0 Let m be the minimal integer such that the coefficient P~) of X~ in ~i is not zero for at least one ieS". For given ~ it may be assumed that t0' 9 9 ~. have been chosen in L(b) such that m is maximal. For m'<m -t- I it is clear the coefficient of X~' in ~ is zero. Let (m) (m + i) O~ ''+l) be the coefficient ~ in 4- Clearly O~"+~)=r~ Y~f~P~ eV ng.Im+l={o}. i=0 It follows from Lemma 3-x that there exists a skew symmetric set {Bi, i} indexed by S" in W tm-~) such that P~m/=r~Y,~B~,j for each ieS". Let ~I,j-=B~,jX~ -x, ~' =~i-- ~ Hj.~.f~L(b), then ~= ~ Hi~=~ ~ ~ H,~ and for each i~S" the coefficient /=0 i=0 i=0 n of X~' in ~ is zero for m'< m and the coefficient of X~ is P!~)-- 7: ]~Bi, i = o, contra- dicting the maximality of m. j=0 Lemma 3.6. -- L(b,c)=V(b,c)+ ~ D,L(b,c+e) i=0 if (p--I)-~<b<p/(p--I), e---b--I/(p--I). Proof. -- Certainly L(b, c) contains the space on the right side. We first prove inclusion in the reverse direction if e>o (i.e. b>i/(p--1)). Given ~eL(b,c) we construct a sequence of elements indexed by reZ+, (~(r), ~(~), ~(0r), ..-, ~))~L(b, c+re)� c+re)� (L(b, c+ (r+ l)e)) "+1 by letting ~(0)= ~ and the following recursive method. Given ~(r)~L(b, c+ re) we choose by Lemma 3.4, ~(r)ev( b, c+re) and ~)~L(b, c+ (r+ i)e) for i=o, I, . .., 7"/ such that ~(')=~(~)+ ~ Hi~ r). We now define ~(r+l) by i=O (3.6) ~(r+t) = ~(r)__@r)___ X Di~Ir). i=o 41 4~ BERNARD DWORK We must show that ~(r+t)eL(b,c+(r+i)e). We note that t(~+')= - ~ E,~)eL(b, c + (r + I)e) i=0 and this establishes our recursion process. Writing equation (3.6) for r=o, I~ . . .~ h and adding, we obtain h h (3"7) ~(h+l)=~(0) X ~(r) ~ D, Y~ ~!~). r=O i=0 ~ =0 For e>o, ~ ~(r/ converges in V(b,c) and ~ ~,1 converges in L(b,c+e) for each r=0 r=0 ieS". Furthermore t(~'+~l-->o as h~oo and thus taking limits as h~oo, equation (3.7) shows that ~ lies in the right side of the equation in the statement of the lemma. We now consider teL(b, c), b = I/(p-- I). If NeZ+, ~>o, s<N then s(s/N+b) +c--e<sb+c and therefore TNtSL/N/(b+~/N, c--e), which shows since b + s/N> i/(p-- i) that there exists ~q(N)cV(e/N + b, c--s), ~INlcL(b + z/N, c--~ + ~/N) for each i~S" such that (3.8) TN~=~ IN/-4- ~ D,~ Ni. i=0 The space V(b,c--e)� c--e)) ~+~ is compact in the weak topology, which shows that the sequence (~(N/, ~N/ ~(N/~ has an adherent point " " "~ ~n /N=0,1,...~ (vl ("), t~), ..., ~)) in that space. Hence taking limits we obtain from equation (3.8), (3.9) t = ~(~/+ ~ D~t~). i=O We now let ~ run through a monotonically decreasing sequence of positive real numbers with limit zero. The use of compactness shows that the sequence (~(~/, ~1, ..., ~/) indexed by e has an adherent point. Restricting our attention to a converging subsequence we conclude that the adherent point (~, t0, ~1, ..-, ~n) lies in V(b, c--e) � (L(b, c--e)) "+1 for each e in an infinite sequence of positive real numbers with limit o. Thus taking limits in equation (3.9) we obtain t =~q + ~ D~t~, "~eV(b, c), ~.eL(b, c) for each ieS". This completes the proof of the lemma, i=0 We defer for the moment the discussion of Vn ~ D~L(b). i=0 Lemma 3-7. -- Let p,c,b be real numbers, b<p/(p--i), N an element of Z+, e=b--I/(p--i), p+e>_c and let A be a proper subset ofS', A#S. Let {~,}~sA be a set of elements in XoN~20{X}nL(b, c) indexed by A such that ~ Hi~eL(b , p). Then there exists iEA a set of elements {~,},SA in (XoN~20[Xa, ...,X,+t])nL(b , p+e) indexed by A, and a 42 ON THE ZETA FUNCTION OF A HYPERSURFACE skew symmetric set ~ in (XN-~Y~o[X~, ..., X,,+~])nL(b, c +e) indexed by A such that if we set (3" IO) iGA ] for each leA then Z H~;eL(b, p) and ~ lies in L(b, c) and is divisible by X~o+:for each /cA. Proof. -- It is quite clear that if the elements ~i are chosen in L(b, p +e) and the ~qi, j are chosen in L(b, c+e) then ~ as given by (3.IO) certainly lies in L(b, c) and 2~ Hi~ ~ =EH~--NH~i~L(b, p). Thus the only important condition to be satisfied by ~ is that of divisibility by X0 ~+t For each leA, let p~N/be the coefficient of X0 ~ in ~ and let Q~S+~/be the coefficient of X(0 N+~I in ~ H~,. Hence ord p!N/> Nb +c, ord O~N+al> (N + I)b + p, O~ s+~/=~ y,f~p~N~. i@A i~A Lemma 3.2 now shows that there exists a set of homogeneous forms of degree dN, {C~}~A such that o~N+~I=~z ~fC~, ord C~>Nb+9+e. Thus o: ~f(C~-P~)) and hence i~A leA by the corollary of Lemma 3-2, there exists a skew symmetric set of forms of degree d(N--I), {B~,i} indexed by A such that for each leA. (3. ") p+N/= C,-t- ~ N B, if and ord B~i> (N-- i) b + c + e (since by hypothesis, p + e >c). We now let ~i,~ = Bi,~X0 ~-' for each (i,j)~A x A and ~,----C,X0 ~ for each leA. It is clear that X0 ~ divides ~ (as given by equation 3. IO), while the coefficient of X0 s in ~ is p~i~)__ C,--~ 2~ B,,ifi = o. This completes the proof of the lemma, iCA Lemma 3.8. -- Let b, c, p be real numbers, b< p/(p-- i), e=b-- I/(p--1), e +p> c. Let A be a proper subset of S', A 4= S and let {~},eA be a set of elements of L(b, c) indexed by A such that Y~ Hi~.eL(b, p). Then there exists a set of elements {:qi} in L(b, p + e) indexed by A iEA and a skew symmetric set ~i~i in L(b, c +e) indexed by A such that ~ = ~ + 2~ Hj~ i iEA for each i~A. Proof. -- Let _.i~(~ for each icA. It is clear that Lemma 3-7 gives a recursive process by which for each NcZ+ we may construct a set {~sl} in 9 ~(N-1)~ (X~g~0[X1, . X,+l])nL(b , p+e) indexed by A and a skew symmetric set t'J~,j in (X0~-lf~0[Xl, ..., X~+t])nL(b, c+e) indexed by A such that for each icA, (3.x,) = X ~* jCA ~-*ij ) ~SlcL(b, c), X0 s divides ~sl, ~2 H,~N/ is divisible by X~ and lies in L(b, p). Let iGA ~ = ~ ~I ~), ~,j = ~ for each i, jcA, convergence being obvious in the weak topology. N =0 N =0 ,~i,~(~lj 44 BERNARD DWORK Clearly ~q~eL(b, p+e), ~r c+e). For reZ+, we write equation (3.i2) for N---=o, i,..., r and add. This gives N =1 i~A N =1 The lemma now follows by taking limits as r--~oo since lim ~!r+l) = o in the weak topology. I. ,,,~ oo Lemma 3.9. -- Let b, c, p be real numbers such that p > c, I/(p-- I) < b <p/(p-- ~) and let e = b-- I/(p-- I). Let A be a proper subset of S', A + S and let ~i be a set in L(b, c) indexed by A such that Y-, Di~ieL(b , p), then there exists a set {:qi} in L(b, p+e) indexed by A and a i~A skew symmetric set {~} in L(b, c+e) indexed by A such that ~=~-t- y~ D~:%~. Proof. -- There exists a unique element N of Z+ such that (N--I)e + c<p<Ne + c. For each integer r, o<r<N we construct a set {~)} in L(b, c+re) indexed by a and, for o~r<N a set {~)} in L(b, c+ (r+ I)e) indexed by A and a skew symmetric set {~!"}} in L(b, c-t- (r-? ~)e) indexed by A such that (letting ~=- ~ D~.) ~= Z D~ 0 for o<r<N, (3. x3) for r<N, (3- I4) ~(r-}- 1) __-- ~r)__ Z ~(r) for r<N, i Djqij iEA and such that ~0) ~i for each leA. Suppose the set f~(r)~ in L(b, c-t-re) I.~i JiCA satisfying (3. I3) is given for some integer r, o~r<N. We then have Z H,~") = ~-- Z E~!')eL(b, p) +L(b, c-t-re) =L(b, c+re). i~h ich Hence by Lemma 3.8, elements ~r)in L(b, c~-e(r+I)) and ~:} in L(b, c-t-(r+I)e) may be chosen such that equation (3.i4) is valid for each i~A. If ~r+l) is now defined by equation (3.I5) then certainly ~= Z D~ r+l) and furthermore, ~/r+l) (r) Y~ Ei~I~EL(b , c+ (r+ i)e). This completes the construction of ~i r) for i ~ ~i -- r----o, I, N, since ~0) is specified, and also of ~r) and ~!r) for r-= o, I, N--I. In particular, ~!meL(b, c+Ne) eL(b, 9) and therefore Z H~)=f-- E E~S)eL(b, p). i@A i@A Since 9 +e_> c + Ne, we may conclude from Lemma 3.8 that there exists a set {W~m} in L(b, p+e) indexed by A and a skew symmetric set f~(N)~ in L(b,c+(N+I)e) tqi, j J" indexed by A such that equation (3. I4) is valid for r= N. If now we define for each i~A, ~i~(~+1) by setting r=N in equation (3.i5) we have ~I~+1)-=~ s)- Z E~)~L(b, p+e) +L(b, c+ (N+ I)e)=L(b, p+e). 44 ON THE ZETA FUNCTION OF A HYPERSURFACE If now we write equation (3.I5) for r=o, ~,..., N and add, we obtain after (!) obvious cancellation, ~lY+~)=~--i~ D~ ~ 0~ ) . The lemma follows directlyby setting p +e) and :C c +e). r=0 Lemma 3.~o. -- If A is a proper subset of S',A4:S; b, c are real numbers, ~/(p--I)<b~p/(p--I) and if {~} is a set in L(b, c) indexed by A such that ~, D~=o then iGA there exists a skew symmetric set {~,i} in L(b, c-+-e) indexed by A such that ~= ~, Dd% i for each i~A. ie~ Proof. -- Let p be any real number, then ~ D~i~L(b, ~) and hence if O>c there iEA exists a set {~,/} in L(b, p+e) indexed by A and a skew symmetric set {._(0)X in ql, ~j L(b, c+e) indexed by A such that (3-x6) ~ ~)+ E D~(~) " i Ai, j jGA for each ieA. Let p run through an infinite sequence of real numbers towards +m, then by the compactness of the cartesian product of copies of L(b, o) indexed by A and of copies of L(b, c + e) indexed by A � A there exists an infinite subsequence such that if p is restricted to the subsequence, then, as p-+ov, ~q~) converges (necessarily to o) and ~ql~ converges to ~iEL(b, c+e). Clearly the set {~i} is skew symmetric and taking limits in equation (3-16) as p-+ov, the assertion is proved. Lemma 3.ii.- For b)l/(p--I), Vn ~ D,L(b) ={o}. Proof. -- Let ~ be an element of the intersection. It may be assumed that i/@-- i)<b~p/(p--i). With b fixed in this range, let p be chosen such that ~L(b, p), ~r If ~:~o then p certainly exists. Since ~e~D~L(b), Lemma 3.9 i=0 shows that there exist ~0, ~1, ---, ~n in L(b, pWe) such that ~= ~ D~ i. Thus i=0 ~-- ~ H~,---- ~ Edq~L(b , p+e). Lemma 3.4 shows that there exists ~'~V(b, p+e), i=0 i--0 ~o, .-., ~',~ in L(b, 9+2e) such that ~-- ~ H~=~'~- ~ H~ o. This shows that ~--~,' i=0 i=0 lies in Vn ~ HiL(b), and hence by Lemma 3.5, ~--~'=o. Thus ~=~'eL(b, p+e), i--0 which contradicts the choice of 9. Hence ~----o. This completes the (( general )) theory of the differential operators. We note that if b<p/(p--I) then for each subset A of S, the subspace LA(b) of L(b) is invariant under each D i. The action of the differential operators on these subspaces must now be discussed in greater detail. 45 4 6 BERNARD DWORK e) Special Theory. In this section we cannot avoid distinctions (1) depending upon whether or not p divides d. Furthermore some of our results will be valid only if H and the H~ = E~H are subject to further restrictions. To avoid confusion, for each ieS, let H~ =r~'X0fi, and let ~3~ be the mapping ~ Ei~ + ~H~, where r:'e f~0, ord r:' = x/(p-- I). For each subset, A, of S = { I, 2, ..., n -t-I }, let X A be the set of variables {X,}~ A. The ring, f20[XA] , of polynomials in the variables X A with coefficients in f~0, is viewed as a subring of ~0[Xs] = f20[X1, ..., X,+I] and in particular if A is empty then f~0[XA] is the field f~0. Let 3A be the homomorphism of f~0[Xs] onto f~0[XA] defined by ~A(X,)=iXi if icA 1 o if ieA As before W iml denotes, for each meZ+, the space of forms of degree dm in f20[Xa]. For each subset A of S let Wk m/=.~A(W/~/) and for each subset B of A, let W~'I"/=W~"/c~(MB), where (MB) denotes the principal ideal in f~0[Xs] generated by the monomial M B = I-I X~. (Unfortunately, our notation permits the same space to be designated by several symbols. Thus if o is the empty subset of S, then W] "/= W~ '/''l and W(m) _ w(m) _ ~O,(m)~ -- ''S -- ''A 1" For each subset A of S let 23~ '('') be a subspace of W~ '(m) which is p-adically comple- mentary (with respect to the monomial basis of W~ '(m)) in W~ '(') to W~'(m)t~3A(9~). Thus we have (a.,7) = For each subset, A, of S, let (3. xs) = the sum being over all subsets, B, of S which contain A. Lemma 3. x2. -- Let A be a subset of S. (i) Ws A'(m)= Y~ W~ '(m), the sum being over subsets, B, of S which contain A. ADB (ii) WsA'(")r (Kernel of UA) = ~] W~ '(~), the sum being over all subsets, B, of S which ~zx contain but are not equal to A. . (iii) W]'Im!n (~ag.t) = ~a(9~caW~'Iml). Proof. -- The first assertion is trivial. For (ii) we observe that a polynomial, ~, lies in the kernel of g~ if and only if each monomial, X ~, appearing in ~ is divisible by at (x) The theory in the case pld is hampered by the fact that Lemma 3-3 fails to give an explicit basis for the n+l ideal (Ms) N (f~,~ .... ,f~) in K[X]. This ideal contains but is not necessarily equal to (Msa~) + ~] (Msa~/Xi) , i=1 a counter-example being given (for n = 3) by ~ =J~Sx(~4--I)fa +fa~(I--~,)f2--A~,~4A, where for i = t, 2, 3, 4, 81 is the specialization of K[X1, X2, Xa, X4] defined by 8iXi = o, ~iXj = X i for j* i. 46 ON THE ZETA FUNCTION OF A HYPERSURFACE least one variable X/such that iES--A. If X ~' is also divisible by M A then certainly there exists a subset, B, of S containing A properly such that M B divides X ~. For the proof of (iii), we use Lemma 3.3 (i) which shows that as an ideal in ~0[XA], (MA) n~n(9~ ) ----  (MA(~Af/)/X~). Intersecting both sides of this last relation with W(A m), i@A we see W]'(m)n 3A9~ is the set of all homogeneous polynomials of degree dm of the form ~] g~MA(~Af~)/X~, the gi being elements of f~0[XA]. By homogeneity it may be assumed i@A that giMAfJXi is a form of degree dm in f20[Xs] and hence lies in 91nW~ '(~). Thisshows that the left side of (iii) lies in the right side, which completes the proof since inclusion in the reverse direction is trivial. Lemma 3.13. -- Let A be a subset of S (i) ~'(~)=Z[+]~ '(m), the sum being over all subsets B of S which contain A. (ii) W A' (m~ __ ~3A,(m)| S -- S (g[c~W~ '(~)) and the sum isp-adically direct ifp[d. Proof. -- (i) The definition of 2~ '(m) shows that it is enough to prove the p-adic directness. For each set B containing A, let ~B be an element of V~ '(~t such that ord (Z~B)> o. Let C be a minimal subset of S which contains A such that ord ~c <o. Clearly ord ~c ---- ord(,~cZ~) > ord(Z~B) > o, which shows that ord ~ > o for each B. (ii) We first prove this assertion without any claim concerning directness. The assertion is equivalent to equation (3.I7) if A=S. Thus we may assume that A:t:S and use induction on the number of elements in A. By Lemma 3-I2 (i), W~'~m)=W~'/")+ Y, WB s'(~). Equation (3.17) and Lemma 3.i2 (iii) show that BDA W2 '("!=~AmA'Im)+~a(2Ic~W~'(m) ) and since -~a acts like the identity on W2 '!m), we may conclude that W2'(m) c2~2' (r~) -t- 2I n Ws A'(m) + (Kernel ~A) n Ws A'(m). Lemma 3- 12 (ii) now shows that W2'(")c 2~]'('~)+ ?I n Ws a' (~) + ~] Ws B'('~) and it is clear from the previous relations BDA that W2 '(m) also lies in this space. The induction hypothesis now shows that ,-~ *Art (m) W~'(")cf~]'I")+'anvv+ ' + Y~ (~B'(m)-t-!~InWff'(m)). Equation (3.I8)now shows that BDA W~'(m~ cm~'~-,s ('*! ~ ~ 2 r WA's ('*) and equality is clear. To show directness (in the ordinary sense) of the sum, let ~ be an element in ~32'(m)n(2oW~'("/). Equation (3.18) shows that for each set B containing A, there exists ~,e~3 B'('/ such that ~=Z~n. Let C be a minimal set containing A such that ~c+o. Clearly ~c=~c~c2[ and hence ~ce(WcC'(m)n~cg/)c~3 c'(~)c , which shows by equation (3.~7) that ~c=O. This contradiction shows that ~ =o for all B and hence ~ =o. Let ~ be an element of~s ~,(m/ and B an element of W~t'/m)ng/, both in s such that ord (~--~q)>o. To complete the proof of the lemma, we must show (ifpld) that ord ~>o. By definition, for each set B containing A there exists ~e2~ '(") such that ~=Z~u. We show that ord ~B>O for each B. Suppose otherwise, then there exists a minimal set C containing A such that ord ~c=O. Then ord(~c--~c~)>o , 47 4 8 BERNARD DWORK while ord(~c~--~c~)>ord(~--~)>o. Thus ~ce~0[Xc] and ord(~c--~c~q)>o. Let ~c be the image of ~c in K[Xc] under the residue class map. Clearly ~c is divisible by M c and lies in the image in K[Xc] of ~0[Xc] n~cg.I. Using the asterisk to denote images in K[Xc] under the residue class map, we may conclude from Lemma 3.3 (i) (since p t d) that there exist a set of forms of degree din, {g~} in K[Xc] indexed by C such that ~c = y' giMc(~cf~)*/Xi '. Choosing forms G i of degree dm in ~0[Xc] which represent the g~ and setting ~ = ~ GiM c (~cf)/Xi eWc' (")n ~cg.I, we have ord (~c-- 4;) > o. iEC Since ~ce2~ c'("), this contradicts equation (3.17) and so the proof of the lemma is completed. For each subset A of S, let Vs A'(m) = ~sA,(~) if p t d, while otherwise let Vs A, (") be chosen in W~ 'Ira) p-adically complementary to (9.InW~'(m)). (Clearly we may let VsS'(")=~s s'(m) in any case.) It follows from the definitions and Lemmas 3.2, 3.3 and 3-13 that if AcS"nS and PeWs A'(m) then there exists O~")eVs A'(m/ and a set of homogeneous elements {Pi} indexed by S" in f~0[Xs] such that (3- I9) P = (~'~)+ Z P4f~M~/X4 + Z pJ~MA, i@A i~S'" --A ord O~m)>ord P, ord P~>ord P. If A is any subset of S, there exists O~m)e~3s ~'(m) and a set of homogeneous elements {P~} indexed by S in f20[Xs] such that (3-20) P = (~) + Z P~fM~/X~ + Y~ P~fM~, i~A iG S --A but in this situation the previous estimates for ord O~ ~) and ord P~ do not hold unless p does not divide d. Finally let Vs~= ~ Vsa'(m)X~, ~s a = ~ Vs~( b, c) = Vs~nL(b, c). In parti- ng=0 m=O cular the space, V, defined previously, may in our present notation be written V~. We shall write V A (resp. ~) instead of V~ (resp. ~Bs ~) and likewise V (resp. ~3) instead of V ~ (resp: ~B") whenever there is no danger of confusion. In particular ~3 = Z~], the sum being over all subsets A, of S. We note that for each subset A of S, V A and ~3 A lie in ~(s~ and have equal dimension. Lemma 3- 14. -- If b <p/(p-- I), and A is any subset of S n S" then LA(b, c) =VA(b, c) + ~ H~LA-(~)(b, c+e) + Y~ H~LA(b, c+e). i~A i@S" --A If p t d then L s (b, c) = V s (b, c) + Z H~L s- ~ (b, c + e). i@S The proof is a step by step repetition of that of Lemma 3.4 and therefore may be omitted. We note that the statement of Lemma 3.4 is obtained from this lemma by setting A = o. Lemma 3.x5. -- If (p--I)-l<b<p/(p--i) and if A is any subset of SnS" then LA(b, r =VA(b, c) + Y, D~LA-~i~(b, c+e) + 2 D~La(b, c+e) IEA i~S" --A ~smA'('~)Y~"-0, ON THE ZETA FUNCTION OF A HYPERSURFACE If p I d then LS(b, c) =VS(b, c) + Z D, LS-{'/(b, c+e). ~s This generalization of Lemma 3.6 follows from Lemma 3. I4 in precisely the same way that Lemma 3-6 follows from Lemma 3.4. We must now overcome some of the difficulties caused by the incompleteness of Lemma 3. I5. Lemma 3.I6. -- For each subset A of S and each NeZ+, N>N0, s =~3A + X ~2 A-{~)'(N-t~ + y. ~A,I~--ll ~6A iES --A and the equality remains valid if 2~ A is replaced by V A. Proof. -- Since the left side of our assertion clearly contains the right side it is enough to shows that the right side contains the left side. We show this inclusion for each NeZ+. This is trivial for N----o since ~A't0)--~A'(0)-----vA'(0)-----{O} (resp. ~0) if A+o (resp. A=o). We now suppose that N>o and use induction on N. Let ~e~ A't~I and let P be the coefficient of X0 ~ in ~. Let homogeneous forms Qsl, {Pi}i~s be chosen as indicated by equation (3.20) (with m replaced by N). Let ~=X~-~MAP~/X~ for i~A and ~=X~-IMAP~ for i~S--A. Let ~=O~X0~e~ A and n+l then ~-- (~ + Y, ~3~.) e~ A' IN-l/. This shows that i=l ~A, (N) C~B(A) + ~ ~)~A-- {~}, (:~ --I) + y, ~A, :N --I) + ~A, :~ --I~ ~EA iES --A and the assertion now follows from the induction hypothesis. The above argument can be used for ~A replaced by VIA), since O~ N) may be choscn in V A' IN) instead of 2~ A' (S). The following ]emma is a special case of Lemma 3-: : unless p divides d. n+l Lemma 3.x7. -- For b~I/(p--I), ~n ]~ ~31L(b)=o. n+l n+t Proof. -- The previous lemma shows that V C~/l~~ ---- ~ + 2~ ~3~ IN~ c~ + Y~ ~3~L (b). We may assume that b<p/(p--I) and use Lemma 3.6 which shows that n+l n+l L(b) =V + 2~ ~3~L(b) and thus conclude that L(b) =~ + 2~ ~3~L(b). Lemma 3. ~ i shows i=1 i=l n+l n+l that V ~ L(b)/]~ ~3~L(b) ~ 23/(~n Z ~3~L(b)). Since V and ~ are vector spaces of the i=I i=i same (finite) dimension, this completes the proof of the lemma. Our next lemma is a weak form of Lemma 3. I5 of interest only if p divides d. Lemma 3.I8. -- IrA is any subset of S and if ~[(p--x)<b<p/(p--x) then n+l L~(b)c2~A+ Y~ ~),L(b). 7 5 ~ BERNARD DWORK Proof. -- By Lemmas 3. I5 and 3.16 we have if A# S, n+l n+l n+l LA(b)cVA+ Z ~L(b)ct~A'(N')- b Z ~)~L(b)c~A + Z ~L(b). i=1 i=i i=I To prove the lemma for A=S, let B={I, 2, . .., n} and let ~+l denote the mapping of f~0{Xs} onto f~0{XB} obtained by replacing X,+~ by o. For each ieB let ~ be the mapping ~E~+~,,+~H~ of ~)0{Xn} into itself. For ~en0{Xs} , ieB we have ~,,, l~)i~==~)~nq_l ~, while ~,,+l~)n+l~=O. If ~eLS(b) then from the part of the lemma n+l already proven, there exists ~e~s B such that ~e~ Jr- ~] ~iLs(b). Applying U,+t to this i=1 relation we have o=~,+t~-k ~LB(b ). However equation (3.I8) shows that {=1 ~3~ =~3~-b~ ss andhence ~,+~:qe~3~ and henceliesin ~,e~ ~ ~L~(b), which according i=1 to Lemma 3. ~ 7 (with S replaced by B) is {o} since b> ~ [(p-- ~). Thus ~, +t~ ---- o, which shows that ~e~3s s . This completes the proof of the lemma. f) Exact Sequences. The object of this section is the computation of the dimension of the space V s defined in the previous section. For this purpose we shall need a theorem concerning exact sequences which will be used again in the geometric application of our theory. Let R be a field of arbitrary characteristic and let W be a vector space over R with an infinite family of subspaces indexed by both Z and by the subsets of S={I,2, ...,n+i}. That is, for each t~Z and each subset, A, of S, let W(A,t) be a subspace of W. Let 9a, .-., %+1 be a commutative set of endomorphisms of W with the property (3.2x) 9,W(A, t) cW(Au{i}, t-b i) for each itS, teZ, and each (not necessarily proper) subset, A, of S. For each reZ+ and each pair of subsets A, B of S such that o+A_CB, let ~(t,r;A,B) be the space of all antisymmetric functions g on A r such that g(al, ..., ar)eW(--t--r , B--{ai, a2, ..., at}), it being understood that ~?(t, o; A, B) is to be identified with W(--t,B). For r>i, let 3(t,r;A, B) be the mapping of ~?(t, r; A, B) into ~(t, r--~;A, B) defined by (3.22) (3(t, r; A, B)g)(al, ..., at_l) = Y~ 9ig(a,, ..., ar_l,j) /CA for each ge~(t, r; A, B). This mapping shall be denoted by ~ when no confusion can arise. Theorem 3. x. -- If the sequence 8 8 9 .__). ~(t,r@2,A,B) ~(t,r+~ A,B) ~(t,r;A,B) is exact when r = o for all pairs of subsets A, B of S such that o + AcB then the sequence is exact for all reZ+. 50 ON THE ZETA FUNCTION OF A HYPERSURFACE 5~ Proof. -- We must show that Kernel ~(t, r+ ~; A, B) =Image ~(t, r+2; A, B). We show that the right side is contained by the left side by showing that ~(t,r+i;A,B)3(t,r-b~;A,B)=o. Let g~(t,r+~,A,B), then (~(t, r+ i; A, B)~(t, r+2; A, B)g)(a~, a2, ..., a~) = Y~ %.(8(r+~; A, B)g)(al, ..., a,, j) = ~, ?,?ig(al, ..., a,,j, i)=o iGA ~,iGA by the commutativity of the endomorphism q~ and the skew symmetry of g. To complete the proof we must show: Kernel 3(t, r-t- I; A, B) cImage ~(t, r+2; A, B). This is true by hypothesis for r = o and hence we may assume that r > I. Antisymmetry shows that if A contains just one element then ~(t, r+ I;A, B) =~(t, r+2; A, B) =o for r>_I. The assertion is thus trivial if A contains only one element. We now may assume that A contains at least two elements, that r> I and we use induction on r for all t. Let ge Kernel ~(t, r + i ; A, B). Renumbering the elements of S if neces- sary we may suppose that A={l, 2, ..., s}, s>2 and hence o= ~ q~g(al, ..., at,j) for all (al, ...,a~)~N. With a 1 fixed, say al=I , we consider the mapping (a~, ..., a~+l)-+g(I, a2, ..., at+l) as a function on (A--{I}) ~, indeed as an element of ~(t + I, r; A--{i }, B--{I }) since it is skew symmetric in the <( variables,, a2, ..., a,+ 1 and g(I, a2,...,ar+l)eW(--t--r--I,B--{I}--{a2,...,a~+l}). In this sense the mapping lies in Kernel S(tq- I, r; 1--{i}, B--{I}) and hence by induction on r there exists h'e~(t+ I, r-t- I;A--{I}, B--{I}) such that (~(t-t- I, r-}-I ; A--{I}, B--{I })h')(a2, ..., at+l) =g(I, a2, ..., dr+l) for all (a2, ...,ar+I)E(A--{x}) ~. Let h be the function on {I}� "+1 defined by h(I, a~, ..., a~+z) =h'(a2, ..., a,+2) for all (az, ..., a~+l)e(A--{i}) ~+t. Let a 1 be the set of all (bl, bz, ..., b,+2)eA r+2 such that at least one (( coordinate ,, is i. By anticommutativity, h may be extended uniquely to a mapping (again denoted by h) ofA 1 into W. Furthermore it is easily verified that if (b~, ..., b~+ 1) � A cA I (Le. at least one b,=i) then g(b~, bz, ..., b,+~)= Y~ ~pih(bl, b2, ..., b~,~,j). If (61, ..., br+2) Ea iEA then h(b~, bz, ..., b~ + 2) eW(-- t-- (r + 2), B--{bl, ..., b~+z}) as follows directly from the corresponding property of h'. For each integer m, i<rn<s, let Am={(a~, ...,a~+2)eN+2la~e{~,2, ...,m} for at least one ie{I, 2, ...,r+2}}. Let A~={(aa, ...,ar+l)eA~+llaie{i,2 , ...,m} for at least one iE{I, 2, ..., r + I}}. Suppose (second induction hypothesis) that h has been extended to a skew symmetric function on A,, such that for all (al, ..., a,+2)eA,, and (bl, ..., b,+l)eA ~ wehave h(ax, ..., a,+2)eW(--t--(r+2), B--{al, a.,, ..., ar+~}) and g(b~, ..., b~+~) = Y, ~ih(b~, ..., b~+l,j). If m=s, we are done, i.e. h~(t, r+2; A, B) iGA and 3(t,r+2;A,B)h=g. Hence we may assume that i<m<s. If rn=s--I then 51 BERNARD DWORK since r+ 22 2, g(m+ I, a,~, ..., ar+l) ~---O unless (m+ I, a2, .... ar+l)eA~. Likewise h is defined on A,, and can be extended to an anticommutative mapping of A '+~- into W by letting h map elements of A '+2 not in A,, into o. Thus for (m+ 1, a2, ..., a,+~)eA "+~, g(m+ 2, a2, ..., ar+~) = Z ?~h(m+ I, a2, ..., at+t, i) since this is certainly true if iEA (m + I, a2, 9 9 a,+t) EA~, while otherwise m + i = a2 = 9 9 9 = a,+ t and hence both sides are zero. Thus our second induction hypothesis naay be applied to the case in which i<m<s--I. We know that Y. 9ig(m+ I, a2, ..., a~,j) =o for all (a2, ..., a~)~A ~-1. ~'GA We restrict (a.,,..., at) to (A--{I, 2,..., m}) '-t. For j<m, the second induction hypothesis gives g(m+ i, a,,, ..., a~,j)= ~ 9ih(m+ i, a2, ..., at,j, i) and hence i=1 o= ~ ~ 9ig, h(m + I, a2, . . ., a~,j, i) + X 9ig(m + I, a2, . . ., a,,j). j=li=l /=m+l The anticommutativity of h on A,, shows that o= ~ ~ e~j?ih(m + I, az, ..., a,,j, i) j-ii=l and hence O~ Y~ 9i g(m+l, a2, . . ., a~,j) + 9ih(m + I, a2, . . ., a,, i,j) . i=m+l i=l Since m+ I<S, this last relation may be written o= Y, g'(m+I, a2, ..., a~,j) whereg' i=m+2 is the mapping (a2, ..., ar+l) --> g(m+ I, a.2, ..., a,+l)-- ~ 91h(m+ I, a2, ..., a~+t, i), i=1 of (A--{I, 2, ..., m-t- I}) r into W. It is easily verified that g'ea(t+ 2, r; A--{~, 2, ..., m+ 2}, B--{m+ ~}) and we have just shown that g' lies in the kernel of a(t+I,r;A--{I, 2, . . ., m+ I}, B--{m@ I}) and hence by induction on r, there exists h"e~(t+ r,r + I; A--(I, 2, ...,m+ I),B--{mq- I}) such that a(t+I,r+i;A--{i,2, ...,m+I},B--{m+I})h"=g'. Thus g(m+ 2, a2, ..., a,+l) = k ~h(m+ i, a~, ..., at+, i) + Z ~Y'(a2, ..., a~+,, k) i=1 k=m+2 for all (a2,...,ar+l)e(A--{2,2, ...,I+m}) r. We now define for all (a2, ..., a~+l)e(A--{1, 2, ..., m+ 1}) r+l, h(m+ I, a2, ..., a,+i, a~+2)=h"(a2, ..., at+z) and extend h by antisymmetry to F={(al,..., ar+2)~(A--{I, 2,..., rn}) ~+2 such that at least one a~=m+ i}. (We note that Pt~Am=o while FoAm=Am+t). Thus h is 52 ON THE ZETA FUNCTION OF A HYPERSURFACE now well defined and anfisymmetric on A,,+l. If now (oa,...,ar+l)eA~+ 1 then g(al,..., a,+l)= ]~ ?~h(al,..., ar+l,i ) since this is known by the inttuction hypo- thesis to be true if (aa, ..., a,+x)sA ~ and hence we may assume that (al, ..., ar+l)e(A--{I , 2, ..., m}) ~+t and that at least one of the a i is m + i in which case we may use our relation involving h", our extension of h and the antisymmetry of both h and g. Finally we note that for (aa, ..., a,+ 2)eAm+,, h(a,, ..., a,+~)eW(--t--r--2, B--{a,, ..., at+2} ) since this holds by the induction hypothesis if (al, ..., a,+2)eA =, while otherwise we may suppose a,=m+ I, (a2,..., a,+~)e(A--{I, 2,...,m+ I})" so that h(a , ..., = ..., which lies in the asserted space since h" ~ (t + i, r +i ; A-- { i, 2,..., m +i }, B-- {m + I }). This completes the proof of the theorem. For subsequent applications it is convenient to make available a weaker form of the theorem. Let W now be a vector space over K with an infinite family of subspaces, W(t), indexed by teZ. Let q~l, 9 .-, q0,+t be a commutative set ofendomorphisms of V with the property 9~W(t) cW(t+ I) for each i~S, tEZ. For each r~Z+ and each non-empty subset, A, of S, let ~(t, r; A) be the space of all antisymmetric functions, g, on A ~ such that g(a~, ..., ar)~W(--t--r), it being again understood that ~(t, o; A) is to be identified with W(--t). For r>I, let 3(t, r; A) be the mapping of ~(t, r; A) into ~(t, r-- i ; A) defined as in equation (3. ~). The second corollary follows directly from the theorem. Corollary. -- If the sequence ~(t, r-i- 2; A) ~(t, r+I;A) ~(t, r; A) r~Z+. is exact when r=o for each non-empty subset A of S then it is exact for all For our final result of this section we use the notation of w 3 e. Lemma 3-I9. -- Consider the polynomial, Y"+t(I--Y~-I)"+I/(I __y),+l ____ y~yiyi in one variable, Y. Then for each meZ, dim ~s s' (m) _-- y,,a, and hence dim 23s s ----d-l{ (d-- I) n+l -I-(-- I) n+l . (d--I) }. Proof. -- In the notation of Theorem 3. I, let W----E and let W(t, A)=Ws A'(t) for each tEZ and for each subset A of S. For each i~S, let q~ be the mapping ~-+fi~ of W into itself. It is clear that condition (3-2 I) is satisfied. To apply the theorem we must verify that if o + AcB cS then Kernel ~(t, I ; A, B) = Image ~(t, 2 ; A, B). This is equivalent to the assertion that if h i is a set of elements of W(s -t-t) indexed by A such that h~(MB/XI) and such that Y~ h~f~----o then there exists a skew symmetric set {~ii} 88 54 BERNARD DWORK in W(s -t-2) indexed by A such that h~ = Z ~,js for each ieA and such that ~,.ie(MB/X~X~). iEA This assertion may be proven without difficulty by means of Lemma 3.3 (i), using the fact that the proof of Lemma 3.i shows that (fl, ...,f,):f~+l=(f,,-.-,f~) for r= I, 2, ..., n. Thus, Theorem 3- i may be applied and denoting 8(--m, r; S, S) by 8~ for fixed m, and ~(--m, r; S, S) by ~, we may conclude that (3.33) Kernel 8 r = Image 8r+ 1 for r= i, 2,... Furthermore ~, being the space of all skewsymmetric functions, g, on S ~ taking values in W (m-') such that g(al, ..., a,) aW s - {" ...... ,,}. (m-,), we easily compute (3.34) dim ~ = ("+~) (~(m--~)+ ~--1) Since Image 8r~-~{~,/Kernel 8, and since ~, is of finite dimension, we have (3.35) dim 3, = dim Kernel 8~ + dim Image 8,. Writing [Im 8~] (resp: [Ker 8~]) for dim Image 8~ (resp: dim Kernel 8~), we now have as power series in Y, ~ Y" dim ~'= ~ Y'[Im 8~]-k- ~ Y'[Ker 8,]. Equation (3.33) now gives , = 1 , = 1 , = t (3.3 6) ~ Y~ dim ~ =Y[Im 8~] + (I --~ y-t) ~ Y'[Im 8~]. Since dim ~, = o for r>n + I, this equation is a relation between polynomials and hence setting Y = -- i in (3.36), we have -- [Im 81] = ~ (-- I)r dim ~r. By definition r=l sS.(m) is isomorphic to the factor space /~ s J and Lemma 3.3 (i) shows that 91nW~d(m)=Image81. Furthermore 30=W~s '(") and hence dim ~3 s'(=) = dim ~o-- [Im at]. We may conclude that dim ~s,(~> = ~ (--I)~ dim ~. (a.37) r=O It is easy to verify with the aid of (3.34) that the right side of (3.37) is the coefficient y~ of Y~ in the polynomial h(Y) =Y"+~(I --Ya-~)"+~/(i _y),,+t _y,+a (i +Y + ... + yd-2),+~. OD Clearly dimes s = 52 y~=d-i~h(o), the sum beingover the d th roots of unity. Clearly I--r d-t =--r and hence h(o~) = (-- 1) "+1 if ~. I, while h(i) = (d-- I) n+l. This completes the proof of the lemma. We now observe that dim V = dim ~ = Y~ dim ~], the sum being over all subsets A of S. In particular for A = o, dim ~o ~ = I and this coincides with the formula of the ~4 WS'(m)/tg-InWS'(m)~s ON THE ZETA FUNCTION OF A HYPERSURFACE previous lemma if we replace n -k- I by o. It is easily verified that dim V ---- d", a result that could have been obtained directly by an argument similar to that of the lemma in which the corollary of Theorem 3. I is used instead of Theorem 3. I. Since the polynomial h in the proof of the previous lemma has the property h (Y) = Y+(" + 1)h(Y-1) it is clear that Ye(n+tt-~=Yi for all jEZ. In particular Y,,~ = Y(n+l-m)a for all m, a result which may be related to the conjectured functional equation of the zeta function. We also note that Ya --= o if and only if d<n -k- I, a fact related to the results of Warning. w 4. Geometrical Theory. The notation of w 3 shall be used whenever possible. In this section q =pa, aeZ+, a> i. The first subsection involves power series in one variable, t, with coefficients in ~2. Such a power series, Xy,+t ~, will be said to lie in L(b, c) if ord ym>mb +c for all m~Z+. a) Splitting functions. In [i] we gave two examples of a power series, 0, in one variable satisfying the conditions (i) OzL(� o),� (ii) O(I) is a primitive pm root of unity. (iii) If yP~-----,( for some integer s, s>o then s-1 8~21~'PJ H 0(v pi) = 0(I) =0 j=0 (iv) The coefficients of 0 lie in a finite extension of Q'. A power series in one variable satisfying these four conditions will be called a splitting function. We shall construct an infinite family of such functions indexed by Z*={+oe}u{s~Zls>i}. Indeed the theory of Newton polygons shows that for each seZ*, the polynomial (or power series), yp /p has a zero, Ys, such that ord y+ = I/(p-- I). While there are p-- I such zeros, we shall suppose one has been chosen for each seZ*. For each s~Z* we now set (4" I ) 0 s (t) = exp ty~) P'/p' . 8g 5 6 BERNARD DWORK Lemma 4. x. -- For each seZ*, 0, is a splitting function. Proof. -- In the following the symboly shall denote a parameter to be chosen in f~ subject to the condition ordy = I/(p-- I). For each seZ+, let gs(t,y) = exp{-- (ty)~/p'~}. It is easily verified that g, eL(as, o), where (4.2) a~ = (p-- I)-l--p-"(s + (p-- I) -t) for seZ+, while a~ is taken to be (p--I) -1 for later use. For seZ+ let G,(t,y)= fi gj(t,y). Since ai+t>a i for eachj~Z+, we conclude /=s+t that G,(t,y)eL(as+t, o). Let E(t) denote the Artin-Hasse exponential series E(t) =exp ! ~ot@P~ I. (4.3) It is well known that E(t) eL(o, o) (4-4) E(t)-=-I +tmodt~Q'{t}. Let h~(t,y)=E(ty) and for seZ+, s> I let h~(t,y) =h| (4.5) and so for seZ* (4.6) hs ( t, y) = exp l i~= o ( tY)Pi [P~ t. Clearly h~(t,y)eL(a| o) and for seZ, s>I, equation (4.5) shows that (4.7) hs(t,y) eL(as+t, o). Since a 2-- (p-- I)/p~>o, we may conclude that hs(t,y ) converges for ord t>o. Further- more equation (4.5) shows that (4.8) hs(t,y ) + (ty)P~+l/p "+t - h~(t,y) mod tl+P~+~ f2{t}. Combining this relation with (4.7) we conclude that for s> I ord(h,(i,y) +yPS+l/ps+x--h~o(I,y) ) > a~+1(i +p~+ ~). (4.9) Since h~(I,y) =E(y), we conclude with the aid of (4-4), and (4.2) that for seZ+, s>I ord(h,(I,y)- I) =I/p-- I (4-io) and (4-4) shows that (4-Io) is valid for all seZ*. Furthermore equation (4.6) shows that for seZ ~ (4. xx) log hs(I,y ) = ZyPJ/p j i=0 and hence loghs(i , 7~) =o. 56 ON THE ZETA FUNCTION OF A HYPERSURFACE Since 08(t)=hs(t,'~8), we conclude from (4.7) that 08eL(a,+~,o), and from (4. II) that 0~ (I) is a p~-- th root of unity for some r while (4. I o) shows that 08 ( i ) is a primitive pt~, r 1 . root of unity, r-~ v ypJ Ify p~ =y where reZ, r~1 then as a power series iny, II hs(yPl, y) = h~(i,y) i =o /=0 as may be seen from equation (4-6). Replacing y by ~'8 we conclude that 08 satisfies condition (iii) in the definition of a splitting function. We have already verified conditions (i) and (i_i). Finally we note that Q'(y~) is a purely ramified extension of Q' of degree p--I, while condition (ii) shows that Q,'(Ys) contains a primitive pth root of unity. We conclude that for each ssZ* the coefficients of 08 lie in the field ofp th roots of unity. This completes the proof of the lemma. If geI +trY{t}, let ~(t)= IIg(tPi), an infinite product which converges in the i=0 formal topology of f~{t}. Clearly g(t)=fi,(t)/~(t p) and if q=p~, a>x then a--1 (4- I2) H g(t pi) = ~ (t)/~ (tq). i=l It follows from the definitions that for each seZ* 0~(t) =exp ,i=078,i , (4. x3) 8-1 tP I where vs, j = Z v '/p (4. x4) i=0 It is worth observing that =(p--~)-lpJ+l--(j-+1) ord u C4. I5) In particular 01 = exp (glt). In the application use will be made only of 0oo and 01. b) Let f(X) be a homogeneous polynomial of degree d in n + I, (n~o) variables, X1, X~, ..., X,+ 1 whose coefficients are either zero or (q-- i)--th roots of unity in f~. We may write (4-i6) f(X) = Z A,M~, i=1 where A~=AI and Mi is a monomial in Xl, ...,Xn+ 1 for i=I, 2, ...,p. Let 6, denote n dimensional projective space of characteristic p and let .~ be the variety in ~ defined over the field k of q elements by the equation f(X)- o mod p. For n=o, extending in the obvious way the usual identifications associated with projective coordinates, 60 consists of just one point which is of course rational over the prime field. In any case .~=6, iffis trivially congruent to zero modp. Iffis not trivial modp then ~ is a hypersurface in 6,, to which we attatch the conventional meaning if n~2, while if n = o then .~ is empty and if n = i then .~ is a set of at most d points on the 57 5 8 BERNARD DWORK projective line which are algebraic over k and closed under field automorphisms which leave :he elements of k fixed. For n>o we say that ~ is a non-singular hypersurface of degree d in ~, if the U gf (mod p) have no common zero in ~,,. For n > 2 this polynonfials f, ~-Xl , ..., ~Xn+ 1 coincides with the usual definition, while for n = I it means that .~ is a set of d distinct points and for n-----o, it means that .~ is empty (i.e. f is not trivial mod p). Let ~(53, t) be ~be zeta function of ~ as variety defined over k and let P(.~, t) be the rational function defined by (4.I7) P(5, t)(-1)'=~(5, t)(I --q"t) ~ (I (~--q't) i=O According to the Well hypothesis, if n> 2 and ~ is a nonsingular hypersurface of degree d in ~, then P(.~, t) is a poly~..,mial of degree d-'{(d--I) "+~ + (--I)"+~(d--I)}. Using the above conventions this hypoJ-,c.sis is easily verified fi r n = o, I as for n = o, -- I I ": ~ is e,~,pty I~) l) (4 9 (l--t) ' if 5=6 while if n = i and .~ consists of d distinct points, then ~ ~:- ;~ union oi e disioint sets of points, the i th subset consisting of b/points conjugate ovm ,, ,,,,d each point generating an extension of k of degree bi. In this case d = ~ b i and i=1 (4" I9) ~(~, t)= fi (I--tb/) -1 /=1 Thus if ~ is a non-singular hypersurface of degree d in 6, then ) i/~[I1 ()) if n=o (4.20) P(5, t) = i--tbi /(r--t) if n= I, which is precisely the Well hypothesis in these trivial cases. We know from [I] that the zeta function of~ is related to the linear transformation +oF, where p a--I (4.2I) F(X) = H II 0((XoA, MjPi), i-lj -o O being any splitting function. If 0 is defined as before then since A~ = Ai, (4.22) F(X) = P (X)/P (Xq) where fI 6(A, XoM;) i=1 58 ON THE ZETA FUNCTION OF A HYPERSURFACE ~9 then F takes the If we take the splitting function to be 0s, s= i, 2, ..., +o% form oxply0 I where -~ is the Frobenius automorphism over Q' of a sufficiently large, unramified extension field. Since 0~eL(as+l, o), equation (4-12) shows that Os(t)/O,(tq)eL(pa~+l/q, o). It follows without difficulty that F/X) = F~(X)/~'8(Xq ) eL(pa~+:/q, o) in the sense of w 3, a~+ 1 being given by (4.2). We now recall and clarify the geometrical significance of the characteristic series, ZF, where F is given by (4.21) If gea{t}, let g~' be the power series g(qt) and if g~: +tn{t}, let g~ be the power series gl-~=g(t)/g(qt). n+l If ~' is the << hypersurface )), l-I Xi=o in ~, then by [i, equation (2I)] i=l (recalling that although F now involves a total of n + 2 variables, we are now counting points in projective rather than affine space) (4"2~) ~(5--5', q~):)~F-(--~)n+l(I-t) -( ~)n For each non-empty subset A of S={I, 2, ...,.n+I}, let I+m(A) be the number of elements in A and let ~a be the variety in ~,~(AI defined by the equation in X A, ~Af --= O mod p and let ~[ be the hypersurface II Xl = o in ~,,(AI" Let A A be the power series in one variable defined by ~A , ~ - (- ~)t + re(A) t. __ t) - (- ~),~(A) (4" 25) ~(-~A---~A, qt) = a A ' ~, The precise formulation of A A as a characteristic series in the sense of w 2 does not concern us here, except that we observe that 5s=5, As =ZF. To simplify notation let PA(t) denote P(SA, t) as defined by (4. I7), so that m(A) (4.26) PA(t) (- 1)re(A) ~--- ~(SA, t)(I --q"(A)t)--: l-[ (I --fit). /=0 If B is a non-empty subset of S then (4-27) "~" =A~, (~A--Si), a disjoint union indexed by all non-empty subsets, A, of B. We may conclude with the aid of (4.25) that ~(5~, qt)= I] ~(SA--~A, qt)= 1-] {A~-(-8)l +'n(A)(I--t)--(--8)m(A)}. AcB ACB But an elementary computation gives }2 --(--S)"(A)=8-:(qo:+m(~)--i) and hence BCA ('1-28) ~(~I~, qt) = (I --t) ~ --1 ($1+m(B)_ 1) l-I A2(--~)l+m:h) , AcB 59 60 BERNARD DWORK while equation (4.26) shows that (4" 29) ~ (SB, qt) = { PB (t) (- 1)re(B) ( I --t) -- ~ -1(1 -- ?re(B))}~ comparing (4.28) and (4.29) we obtain (4.3 o) (I--t)PB(t)~(--1)m(B)~--- II A~-(-8)t+m(A) ACB Relations such as (4.3 ~ can easily be inverted by an analogue of the MSbius inversion formula. Explictly if A-+(5 a is a mapping of subsets of S into a mulfiplicative abelian group and if for each subset B of S (4.3 I) G B = I] (5 a, A CB the product being over all subsets, A, of B, then (4.3 2) (~B= 1-[ G(-1) ~(B)-m(AI ACB The inductive proof of (4.3 2) may be omitted since it depends entirely on the well known fact that ~ (~)(--i)~=--~ for each integer r~ I. Applying this to (4.3 o) i=1 and letting B=S, we obtain As(-~)l+"=I-[{Pa(t)~(-1)"Ia)(z--t)}(-1)"-"IAI. Since ZF = As we obtain A 9 s~+" -- (I--t) II PA(qt), (4- 33) LF -- the product being over the non-empty subsets, A of S. (A similar formula appeared in an earlier work [6, equation 2I].) We believe this equation is quite significant since XF is entire even if ~ is singular. Since ~(~A, t) is rational, PA is also rational and hence (4-33) shows that the zeros of ZF and the (q--I)p roots of unity generate a finite extension, ~0, of O.'. With this choice of ~0, the results of w 2 show that the zeros of ZF are explained by the action of ~oF as linear transformation of L (q� if F ~ L (� o). We now fix s~Z*, let F=F~ so that � F=exp H, where s--1 H = Y~ "(s, iX0P~J(XPJ). We shall assume unless otherwise indicated that f is a regular i=0 polynomial (1). Equation (4. I5) shows that H satisfies the conditions of w 3. It follows from (4.22) that a=~oF, may be written (4.34) = while with this choice of H, the mappings D i of w 3 are simply ~-+F-1Ei(~F ). Since qEio~b=~oEi, we conclude for i=o, i, ..., n+I that (4.35) 0coD~ ----- qD, ooc. (1) Thi~ condition on f is equivalent to the condition that ~A is non-singular for each non-empty subset, A, of S. It will be shown that this condition involves no essential loss in generality. 60 ON THE ZETA FUNCTION OF A HYPERSURFACE 6i If X is any non-zero element of Do, let W z be defined as in Theorem 2.4, i.e. Wx={o } if X -~ is not a zero of X~, while Wx=Kernel of (I--X-~) ~ in L(qx) if z- ~ is a zero of multiplicity ~. We note that � ~, F, the D~, H, and the spaces W x depend upon our choice of s. The maximum value of qx is p/(p--i) and corresponds to s=~. The minimum value of qx is (p--I)[p and this exceeds I/(p--~) unless p ~ 2. This minimum value of qx corresponds to s = I. It is assumed in the following that qx> I/(p-- I). Lemma 4.2. -- If A is any subset of S and o<b<qx then Wxta Z DiL(b ) = Z DiWz/q icA leA Proof. -- Let {~} be a set of elements in L(b)indexed by A such that Y~ D,~. = ~cW x. lea Let ~=max{dimW z, dim Wx/q}. It follows from the corollary to Theorem ~.5 that for each icA there exists ~cWz/~ and ~cL(b) such that ~i = ~, + (I-- (X/q) -~ 0~)~;. Thus (I--X -t ~)~' Y, D~ = ~] D~-- Y, D~ = ~-- Y, Di~, which lies in Wx by hypothesis, icA icA icA icA choice of the ~ and equation (4.35). We may now conclude from equation (2.54) that (I--X-I~) ~ ~] Di~q~cWxta (I--?~-~)~L(b) = (I--k-la)~Wx={o}. This shows that leA ~-- ~ BiB i ---- o and hence ~ ~] D~Wx/q. Thus Wxn ~ D~L(b) c Y, DiWz/q and equality iGA ~cA i@A i~A follows without difficulty. Lemma 4.3. -- If A is a non-empty subset of S and {~}icA is a set of elements in W x such that ]~ D~. = o then there exists a skew symmetric set {~%} in Wx/q indexed by A such that icA ~= ]~ D~q~ i for each icA. iCA Proof. -- Let A={I, 2, ..., r}, I <r<n-~- I. Ifr=I then D1~1=o , ~lcL(qx) and hence Lemma 3.IO shows that ~1=o. We may therefore assume r>I and use induction on r. Lemma 3. io shows that there exist ~i, .-., ~:-1 in L(qx) such that r--1 ~, = Z D~. i=1 Since ~rcWx, the previous lemma shows that the ~ may be chosen in Wx/q. Hence r--1 o = Z Di(~+D~.' ) and since ~i+D~'cW x for i= i, 2, ..., r--l, the inductionhypo- i=l thesis shows the existence of a skew symmetric set {~,i} in Wx/q indexed by { i, 2, ..., r- I } such that for i= I, 2, . . ., r--I r--1 ~ + D,~.' = ~] Di~ j i=1 61 6~ BERNARD DWORK We now extend the skew symmetric set by defining ~ql, ~ = --~ = --~%.~ for i = I, 2, . . ., r-- I and ~r.~ = o. It is readily seen that the ~i.j satisfy the conditions of the lemma. Let X be an eigenvalue of ~. We now compute the dimension (as vector space In+ 1 over ~0) of the factor space Wz/i~lDiWz/q. [ In+l \ n+l Lemma 4.4. -- DimtWx/,EtDiWx'q)'= ' = r=oX ('+l) (-- l)r dim Wk/qr. Proof. -- In the statement of the Corollary of Theorem 3. i, let W = L(qx) and for each t~Z, let W(t)=Wxqt , q01=D ~ for i= I, 2, ..., n+ I. The previous lemma shows that the sequence of the Corollary is exact when r = o and hence the Corollary may be applied. In this application ~(o, r; S) is the space of all skew symmetric maps of S' into W(--r)=Wx/qr and hence dim~(o, r; S)= (n+l) dimWx/qr. The corollary may be used to obtain an identity similar to equation (3.36), where ~,=~?(o, r; S), 8r =8(0, r; S) and the assertion follows without difficulty since [ / '~+1 \ dim |Wz/,~t \ - -'= DiWx~q ' ] / = dim ~~ [Im 3'] = ~=0 (--I)r dim ~r" 31+n We can now show that ZF is a polynomial. n+l Theorem 4. x. -- For each X~f~o, let b x = dim Wz/.= * Z t DiWx,q, then z~rl+" = II(i_xt)b~ the product being over all eigenvalues X of ~. Proof. -- Let X be an eigenvalue of ~ with the property that X/q" is not an eigenvalue for any r>I. For each eigenvalue, X', of~, there exists an eigenvalue X with this property such that X'=qiX for some i~Z+. Let aj=dimWzq~ for each j~Z+. The factors of ZF corresponding to terms of type (I--Xqrt), r~Z+ may be written ai cp i Hx(t) = fi (I--t~.qi)ai=(I--tX) i=~ i=O The previous lemma shows that n+J. bxqi = Z (-- I)i("+J)ai_r j=0 and hence (I--~) n+l ~a ai(p i= ~ bxqie~'. i=0 i=O It follows that b~qi (~ i Hx(t) ~n+l _-- (i--~kt)/=0 This completes the proof of the theorem. 62 ON THE ZETA FUNCTION OF A HYPERSURFACE 63 Equations (4.26) and (4-33), together with the known rationality of zeta functions, show that ZSF t+" is a rational function. The theorem shows that the function is entire in the p-adic sense and hence it must be a polynomial. n+, Let ~ be the factor space L(q� DiL(qn). For qx> I/(p--I), we have shown n+, i=' in w 3 that dim ~B =d ". Since 5", D~L(qx) is a subspace of L(qx) which is invariant i--1 under e, there exists an endomorphism Y of g8 deduced from e by passage to quotients. Theorem 4.2. ~,-I-n ZF = det (I--tl), provided qx> I/(p-- I). Proof. -- It is quite clear that the characteristic equation of ~ is independent of D 0 and hence it may be assumed that f20 contains the zeros of det (I--t~). For each non-zero element X of f~0, let ~03 x be the primary component of X in ~ with respect to e. To prove the theorem it is enough in view of Theorem 4. I to show that / n+l \ (4.3 6) dim ~Bx = dim (Wx/~ D,Wx,,,q) Under the natural mapping, J, of L(q� onto 2B, W x is mapped into ~3 x with kernel n+l n+l Wxn ~] D~L(qx), which by Lemma 4.2 is ~] DiWzi q. This shows that dim~3x is at i--1 i--1 least as large as the right side of (4.36) 9 To complete the proof it is enough to show that ~x is the image ofW x under J. To prove this let ~'~, hence there exists r~I such that (I--x-l~)r~'=O. Let ~ be a representative of ~' in L(qx), then n+l (I--X-~)'~ ~ D~L(q� Hence there exists elements ~qt, .-., ~+~ in L(qx) such that i=1 n+, (I--X-~)~ = 2] Di~ ~. i--1 Let ix be the multiplicity of (X/q)-' as zero of )~, then n+I (I--X-'~)r+~ = ~] D~(I--qX-~)~;. i=i i ! Theorem 2.5 shows that there exist ~1, ..., ~,+1 in L(q� such that for i = I, 2, . .., n ~- I (i_qX-,~)~+ r~q~ = The last two displayed formulas show that / n+l ,\ q =o. n-l-I This shows that ~Wx+ ]~ DiL(qx) and hence ~' =J(~)aJ(Wx), which completes the i=1 proof of (4.36 ) and hence of the theorem. 63 6 4 BERNARD DWORK Theorem 't-3. -- The mapping, ~, is a non-singular endomorphism of !lB (and hence X~ ~+" is a polynomial of degree d'~). Proof. -- It is enough to show that K(~B) = 2B, which by Lemma 3.6 is equivalent to the assertion that n+l (4.37) ~Vq- Z D,L(qx)~V. i=l We recall that 0~ depends upon the choice of s~Z* in our construction of F=F~, but .$1+, is clearly independent of s and Theorem 4.2 therefore shows that the degree of/~r dim K(~) is independent of s provided q� I/(p--I). Since dim ~ is also independent of s (subject to the same condition) we conclude that if equation (4.37) holds when s = oo then it holds for all s such that q~> I/(p--I). We may suppose in the remainder of the proof that s ~ oo. Let -: be an extension, which leaves fixed a primitive pth root of unity, to f20 of the Frobenius automorphism over Q' of the maximal unramified subfield of gl 0. Our proof is based on the fact that while F(X)/F(X q) lies in L(p/q(p-- I), o), F(X)/F'(X p) lies in L(I/(p-- i), o). Let +p denote the mapping + with q replaced by p, (i.e. d? = d?~). Let ~p be the mapping X~-+X p" of no{X } onto itself. Let Co, ~o be the Q'-linear mappings of Do{X } into itself defined by We note that ~0 and [~0 are endomorphisms of DO{X} as Q'-space, not (necessarily) as f20-space. In view of our previous remarks we easily verify since F (X)/F'(X v) 9 L( I [(p--I) ) that i ~oL(p/(P-- I)) cL(I/(p-- i)) (4.38) ~0L(iI(p - i)) r i)) and since +poOr= I, we conclude trivially that (4.39) ~oO~o = I. Since .:a leaves F invariant, the definitions show that (4.40) = = Equations (4.38) and (4-39) give L(p/ (p-- I ) ) =~o~oL(p/ (p-- i ) ) c%L(i/(p--i)) r (p-- i ) ) which shows that %L(I/(p-- i)) = L(p[(p-- I)). (4-4 I) Furthermore, the definitions show that for i = o, i, , , ,, n + I (4" 42) ~0oD~ =pD, o%. 64 ON THE ZETA FUNCTION OF A HYPERSURFACE 65 n+l Lemma 3.6 shows that L(I/(p-- i)) =V + ~ DiL(i/(p--I)) ; applying % to both sides of i=l this relation and applying (4.41 ) and (4.42) we find n+l (4.43) L(p/(p-- i)) c~0V + Y, D,L(p/(p-- I)). i=1 [n+l ) n+l (4.44) 0 ,XlD, L(p/<p--i)) c X D,L(p/Ip--II). i=1 Since VcL(p/(p--I)), we may conclude that for j=o, i, ..., a--i n+l a~Vc0@tV + Z D,L(p/(p-- i)) i=l an elementary consequence of which is n+l Vc~V+ Y~ D~L(p/(p--I)). i=1 Since L(p/(p--I)) is stable under 9 and V may be assumed to have been constructed so as to be stable under v a, equation (4.4 o) and this last relation give n+l Vc~V+ ~] D~L(p/(p--i)), i=1 which is the form taken by (4- 37) when s = oo. This completes the proof of the theorem. We have thus shown that iff is a regular polynomial then (I--t)IIPA(qt) (the product being over all non-empty subsets, A, of S) is a polynomial of degree d"; and if s is chosen such that q� (p--I) -t then this polynomial is simply the characteristic equation of ~. Since � =pa~+t/q, equation (4- 2) shows that qx certainly exceeds (p-- I) t if s>i (resp. s>3) when p>2 (resp. p=2). We now propose to investigate the factor Ps(qt) under the restriction that the hypersurface is of odd degree if the characteristic is 2. To do this we now specialize s. Ifpdividesdlets= I. Ifpdoesnotdividedletsbesolargethat qx>I/(p--i) (says=m). For each subset A of S, a ring homomorphism, ~A of f~0[Xs] onto D~0[Xa] was defined in w 3. We now use the same symbol to denote the extension of this homo- morphism to one of f~0{X0, Xs} onto a0{X0, Xa} which is defined by ~A(X0)-----X0. For each subset, A (including the empty subset) of S and for each subset B of A and each real number b, let LA(b ) = ~AL(b) L](b) ={~ELA(b ) such that M B divides ~}. For ieau{o}, let D,, A be the mapping ~--->~AD,~ of f~0{X0, XA} into itself. Let "A be the mapping ~--->~A(,~) of LA(qx ) into itself. Using an obvious analogue of equation 4.35, the subgroup ~ Di, ALA(qx) of LA(qx ) is mapped into itself by % and hence by passage to quotients we define an endomorphism 0c- A of the factor space ~BA = LA(q~)/i~A Di'A LA (q~)" (Thus in the notation of Theorem 4.2, ~B = ~gs, 0~ = 5s)- 9 65 BERNARD DWORK Now let ~B] be the image in 213 A of L](qx). We note that L](qx) is mapped into itself by % and hence YA maps ~] into itself. Let Y] be the restriction of ~A to ~]. For the empty subset, ~, of S, we have ~AF=I, Lo(qx ) =S0, D0,oLo(q~ ) ={o}, O,.~, ~o=~3o~f~0, % is the mapping ~--->+~ of f~0 into itself. Clearly % operates as the identity mapping on f20 and hence det(I--t~) = i--t. Theorem 4.4. det(I--t~) = II det(I--t~), the product being over all subsets, A, of S. Proof. -- Lemmas 3. i i, 3. :5, 3. :7, 3. :8 show that under the natural mapping of LA(qx ) onto ~3A, ~ is mapped isomorphically onto ~I~]. The proof oflemma 3.17 shows that ~BmN =Z~] and here the isomorphism is given by the natural map of L s (~) = L(qx) onto ~3. For each subset A of S, let ~3A be a basis of N] and let ~ = u ~3:. Lemma 3-I3 shows that ~3 is a basis of ~. We use this basis to construct a matrix corresponding to Y. For each coe~3 we may write (by virtue of Lemmas 3.15 and 3. i8) (4.'t5) s(co)e X 9X(co, co')co'+ 2 D,L(qx), r /GS where 9~(~, r 0. It follows from Lemmas 3.1I and 3-~7 that this relation uniquely determines 9~(r r If M A divides ~ then ~(~o)eL~(q� and hence by Lemmas 3- :5, 3- 18, ~: ~(r r A, which shows that 9~(r r =o unless M~ divides ~'. We now order the elements of ~ so that the elements of ~ preceed those of ~B if the number of elements in B exceed the number in A and such that for A 4= B no element of ~B lies between two elements of ~. Let ~ be the matrix indexed by � ~ with general coefficient 9~(r co') and with the elements of ~ ordered as indicated. Let 9~ be the submatrix obtained from 9~ by restricting (co, ~') to ~ � ~. It is clear that 9~A is a square matrix, its diagonal lies along the diagonal of ~ and the coefficients of 9~ lying below 9~ are zero since these coefficients are of type 9~(r ~o'), where ~o'e~ and r is divisible by M B for some B not contained by A. It now follows that (4.`16) det (I--tg"J~) = II det(I--tgX~), the product being over all subsets, A, of S. It follows from (4.45) that det (I--tE)= det (I--t93l). For r e ~A if we apply ~A to both sides of equation (4.45) we obtain s](co) =%(o~) =..~(~z~)~ Y, 9cJl(% o~')~'+ 2~ D,,ALA(q� ). co' G ~3A i GA Since ~A is a set of representatives of a basis of ~B], this shows that for each subset A det(I-- tgJlA) = det(I--t~]). The theorem now follows from (4.46) 9 66 ON THE ZETA FUNCTION OF A HYPERSURFACE Corollary. Ps(qt) = det (I--t~s s) deg Ps = d-l{( d-I)n+~ + (d--i)(--i )n+ 1} Proof. -- Theorem 4-2, equation 4.33 and Theorem 4.4 show that for each non-empty subset B, of S, H det (I--t~) = H PA(qt) A A the products being over all non-empty subsets A of B. This system of relations can be solved for P~(qt)/det (I--t~) by means of equation (4.32). This gives the first assertion of the corollary. The assertion concerning the degree follows from the compu- tation of dim ~3s s (Lemma 3.19) and the proof (Theorem 4-4) that Y (and hence ys) is non-singular. c) Let k (as previously) be the field of q elements and let us extend the notion of regularity (in the obvious way) to polynomials in k[X1, ..., X,+I]. We have verified a part of the Weil hypothesis for a non-singular hypersurface, 9, in ~ defined over k provided d is odd if p = 2 and provided the defining polynomial f ek[X] of ~ is regular. (f = image forf under the residue class map). We now consider the situation in which f is not necessarily regular. Let A = (%) be an (n + i) � (n + i) matrix whose coefficients are algebraically independent over k[X1, ..., X,+L]. We consider the coordinate transformation n+l X~= Z %Yj, j= I, 2, ..., n + I i = 1 and consider f as a polynomial in Yt, ..., Y,+ 1 with coefficients in k(%, ..., a, + 1,, + 1). We easily compute n+l Of rT_vOf -- ~ Xt~,%A1Jdet A i,t=l i where Aij is the cofactor of % in A. Our problem is to specialize the matrix A subject to the conditions (i) det A 4= o (2) f, (det A)f~', ..., (det A)f,'+l have no common zero in ~n. Let U be the set of all A with coefficients in the algebraic closure of k which fail to satisfy these conditions, i.e. U is the set of all A such that either detA=o or J~ (det a)j-~, ..., (det a)f-~+ 1 have a common zero in ~,. It follows from elimination theory that U is an algebraic variety in ~m, where m = (n + i)2 i. On the other hand it is known ([7], Chap. VIII, prop. 13) that the generic hyperplane section of a non- singular variety is non-singular and therefore U 4= ~m. Hence the dimension of U is at most m--I (and hence must in fact be m--i). Thus if k~ is the field of qr elements, the number of points of U rational over k r is no greater than b(qr(m-li--I)/(q~--I) for 67 68 BERNARD DWORK some fixed real number b. On the other hand there are (q"--I)/(q~--I) points in ~,, rational over k,. Thus there exists an integer r 0 such that for each r>ro, there exists a point of~ m rational over k~ but not in U. This means that for each r>r o there exists a coordinate transformation rational over k, such that ~ is defined by a regular polynomial with respect to the new coordinates. For each integer r, let ~, be the zeta function of as hypersurface over k~ and let P, be the rational function defined by n--1 P~(t)(-1)"----~r(t) 1-[ (I--q'it). i=O It follows that for each r~Z, r_) L(t) = II P~(vt~/'), vr=l the product being over all r th roots of unity, ,~. Furthermore if r>r o then P~ is a polynomial of a certain predicted degree m'. If 1)1 is a polynomial then clearly it must also be of degree m', and hence to complete our treatment of Pt it is enough to show that P~ is a polynomial. Since t)1 is a power series with constant term I, we may write V,(t) = fi (I--b,t) (I--blt) i=1 where the b; are distinct from the b~. Consider b~. If Pr is a polynomial then there must be an r th root of unity, v, such that b~v----b~ for some integer i, i < i< c. Let r run through c q- i distinct primes each greater than r 0. By the pigeon hole principle there exists one integer i such that b~v'= b~-----b~v", where v' (resp. ~") is a p'-th (resp. p"-th) root of unity, p', p" being distinct prime numbers. It is clear that v'=v"= t and b~ =bl, contrary to hypothesis. It is now clear that for the treatment of a non-singular hypersurface, the hypothesis that the defining polynomial is regular is no essential restriction. REFERENCES [I] B. DWORK, On the rationality of the zeta function of an algebraic variety, Amer. 07. Math., vol. 82 (t96o), pp. 631-648. [2] J.-P. SERRE, Rationalit3 des fonctions z~ta des varidtds algdbriques, S6minaire Bourbaki, x9592x96o, n ~ i98. [3] A. Wsm, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., vol. 55 (1949), PP. 497-5 o8, [4] E. ARTm, Algebraic numbers and algebraic functions, Princeton University, New York University, i95o-i951 (Mimeographed notes). [5] W. GR6BNER, Moderne Algebraische Geometric, Wien, Springer, i949. [6] B. DWORK, On the congruence properties of the zeta function of algebraic varieties, .7. Reine angew. Alath., vol. 23 096o), PP. I3O-I42. [7] S. LANO, Introduction to algebraic geometry, Interscience Tracts, n ~ 5, New York, I958. Refu le 15 aodt 1961. The Johns Hopkins University. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

On the zeta function of a hypersurface

Publications mathématiques de l'IHÉS , Volume 12 (1) – Aug 4, 2007

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Springer Journals
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Copyright © 1962 by Publications mathématiques de l’I.H.É.S
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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/BF02684275
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Abstract

ON THE ZETA FUNCTION OF A HYPERSURFACE II) By B~.RNARD DWORK This article is concerned with the further development of the methods of p-adic analysis used in an earlier article [i] to study the zeta function of an algebraic variety defined over a finite field. These methods are applied to the zeta function of a non- singular hypersurface .~ of degree d in projective n-space of characteristic p defined over the field of q elements. According to the conjectures of Weil [3] the zeta fimction of .~ is of the form (I) ~(.~, t) = P(t)(--1)n/'l-It (I'~ --fit) li=O where P is a polynomial of degree d-~{ (d--i)~+t + (--i)~+~(d--i) }, (here n2o, d2 I, for a discussion of the trivial cases n=o,I see w 4 b below). It is well known that this is the case for plane curves and for special hypersurfaces, [3]- We verify (Theorem 4.4 and Corollary) this part of the Well conjecture provided I = (2, p, d), that is provided either p or d is odd. In our theory the natural object is not the hypersurface alone, but rather the hypersurface together with a given choice of coordinate axes X1, X2, ..., X, +1. If for each (non-empty) subset, A, of the set S = { I, 2, ..., n + I } we let.~A be the hypersurface (in lower dimension if A4=S) obtained by intersecting ~ with the hyperplanes {Xi=o}ieA , then writing equation (I) for -~A, we define a rational function PA by setting ira(A) (2) ~(~a, t) = PA(t)(--1)m(A)(I --q'"(alt)/H__ ~ (i--qg), where i + re(A) is the number of elements in A. If.~A is non-singular for each subset A of S and if the Weil conjectures were known to be true then we could conclude that PA is a polynomial for each subset A. Our investigation rests upon the fact that without any hypothesis of non-singularity we have (4.33))/~n+l(/) = (I -- t) l~PA(qt), (a) This work was partially supported by National Science Foundation Grant Number G7o3o and U.S. Army, Office of Ordnance Research Grant Number DA-ORD-BI-I24-6I-G95. BERNARD DWORK the product on the right being over all subsets A of S and ZF is the characteristic series of the infinite matrix [2] associated with the transformation ~ --- ~boF introduced in our previous article [I] and studied in some detail in w 2 below. We recall that z~(t) =y.F(t)/zF(qt) and the fundamental fact in our proof of the rationality of the zeta function is that ZF is an entire function on ~, the completion of the algebraic closure of Q', the field of rational p-adic numbers. In w 2 we develop the spectral theory of the transformation x and show that the zeros of ZF can be explained in terms of primary subspaces precisely as in the theory of endomorphisms of finite dimensional vector spaces. In this theory it is natural to restrict our attention to a certain class of subspaces L(b) (indexed by real numbers, b) of the ring of power series in several variables with coefficients in ft. The definition of L(b) is given in w 2, for the present we need only mention that if b'>b, then L(b') eL(b). An examination of (4.33) shows that if the right side is a polynomial and if0-1 is a zero of that polynomial of multiplicity m then (Oqi)-1 must be a zero of ZF of multiplicity m(n+i). This is << explained ~> by the existence of differential operators Dr, ..., Dn+ 1 satisfying (4-35) ~oD~ = qD~o~ ln+t The space L(b)/~=1D,L(b )_ is studied in w 3 d (in a slightly broader setting than required for the geometric application), for i/(p-- I)<b<p/(p-- I), the main results being Lemmas 3- 6, 3. io, 3. i i. This is applied in w 4 to show that if ~A is non-singular for each subset A of S then the right side of (4.33) is a polynomial of predicted degree and is the characteristic polynomial of E, the endomorphism of L(b)/ED~L(b) obtained from ~ by passage to quotients. (Theorems 4. i, 4.2, 4.3) We emphasize that this result is valid for all p (including p----2). The main complication in our theory lies in the demonstration (Theorem 4-4 and corollary) that if I = (2, p, d) then Ps(tq) is the characteristic polynomial of Es, the restriction of E to the subspace of L(b)/Y.DiL(b ) consisting of the image of LS(b) under the natural map, LS(b) being the set of all power series in L(b) which are divisible by X1X~... X,+ 1. This result is of course based on the study (w 3 e) of the action of the differential operators on LS(b). This study is straightforward for p t d but for p ld the main results are shown to be valid only for special differential operators. We must now explain that for a particular hypersurface we have many choices for the operator ~ (see w 4 a below) but once ~ is chosen the differential operators satisfying (4.35) are fixed. With a simple choice of ~ the eigenvector spaces lies in L(~--~) while for a more complicated choice ofc~ the eigenvector space is known to lie in L(#-~). The special differential operators referred to above in connection with the case p]d are those which correspond to the simple choice of ~. for which the ON THE ZETA FUNCTION OF A HYPERSURFAGE eigenvector space lies in L(P__I~. Unfortunatelyp--I< I ifp=2 (and fortunately, \P/ p p--I only in that case). Thus for p = 2, if ~ ] d we cannot apply the results of w 3 e to determine the action of the special differential operators on L(I/2). Finally (w 4c) using an argument suggested by J. Igusa, we show that our conclusions concerning P=Ps remain valid without the hypothesis that -~A is non- singular for each choice of A. This completes the sketch of our theory. We believe that our methods can be extended to give similar results for complete intersections. We note that the Well conjectures for non-singular hypersurfaces also assert that the polynomial P in equation (i) has the factorization P(t)= l] (I--O~t) such that i0~1 = q(,~-l)12 for each i (Riemann Hypothesis) Oi--~qn-1/Oi is a permutation of the Oi (functional equation). We make no comment concerning these further conjectures. In fulfillment of an earlier promise we have included (w I) a treatment of some basic function theoretic properties of power series in one variable with coefficients in f2. It does not appear convenient to give a complete table of symbols. We note only that throughout this paper, Z is the ring of integers, Z+ is the set of non-negative integers and R is the field of real numbers. w L P-adic Holomorphic Functions. Let f2 be an algebraically closed field complete under a rank one valuation x~ord x. This valuation is a homomorphism of the multiplicative group, f~*, of f2 into the additive group of real numbers and is extended to the zero element of f2 by setting ord o = -b o0. Furthermore ord(x +y) < Min(ord x, ordy) for each pair of elements x, y in f~ and the value group, (5, of fl (i.e., the image of s under the mapping x-+ord x) contains the rational numbers. For each real number b, let I'b={xE lordx=b } U b ----- {x~) [ ord x>b} Cb ={x~ Iordx>b}. As is well known, f~ is totally disconnected, and each of these sets are both open and closed. However by analogy with the classical theory it may be useful to refer to the set C b (resp: Ub) as the closed (resp: open) disk of additive radius b. U_ oo will be understood to denote f2 and clearly F b is empty if b does not lie in the value group of f~. We further note that U b, (resp: Cb, ) is a proper subset of U~ (resp: Cb) ifb'>b. If be(5 then U b=C b. 8 BERNARD DWORK The power series in one variable with coefficients in ~, (x.x) F(t) = ~ A S S~0 will be viewed as an f~ valued function on the maximal subset of f~ in which the series converges. (This is to be interpreted as a remark concerning notation, the power series and the associated function cannot be identified unless (cf. Lemma 1.2 below) the series converges on some disk, Ub, b>oo). It is well known that F converges at xEf2 if and only if lirnA~x~=o. An obvious consequence may be stated : Lemma x.i. -- F converges in C b if and only if (x .2) lim (oral Aj§ = 0% provided b e(5. The series converges in Ub if and only if (I. 3) lim inf (ord Aj)/]> -- b. j--~ 0o We may now prove the analogue of Cauchy's inequality as well as the analogue of the maximum principle for closed disks. Lemma x.2. -- IfF converges on C b and be~ then (i .4) Min ord F(x) = Min (ord Aj+jb) xEP b O~i<~ Furthermore Min ord F (x) --= Min ord F (x). xEr b zEC b Proof. -- Since Fb is not compact it is not immediately obvious that ord F(x) assumes a minimum value at some point of Pb" However the existence of the right side of (I.4) is an immediate consequence of Lemma I.I. Let M = Min (ord Aj +jb), then ord (Aixi)~M for all xEP~ and hence ordF(t)__M on F b. Let S be the set of all jEZ+ such that ordAj+jb = M. By definition S is not empty and Lemma i shows that S is finite. Let g(t) = ~, Ajt j, f(t) --=F(t)--g(t). Lemma I also shows that yes there exists ~>o such that ordAj-]-jb>M-t-~ for each j~S. Hence ord f(t)>M+r everywhere on P b. Let nEFb, n'eF M and let gl(t) =g(nt)/n'. Let Bj be the coefficient of t ~ in gl. For jES, ordBj=ord Aj§ Thus the coefficients of gl are integral and the image of gl in the residue class field of ~2 is non-trivial. Since the residue class field is infinite there exists a unit x in f~ such that ordgi(x ) =o. This shows that ord g(r~x) = M. However r~xEP b and hence ord F(t) assumes the value M on Pb. This shows that the left side of (I . 4) exists and is equal to the right side. The assertion concerning C b follows from the obvious fact that for b'>b, we have ordAj+jb'>ord Ai+jb for each jEZ+ and hence Min ord F(x)>Min ord F(x), which implies the assertion of the lemma. ~Erb' ~Erb ON THE ZETA FUNCTION OF A HYPERSURFACE As in [I], the ring of power series in one variable, t, with coefficients in f~, ~{t}, is given the structure of a complete topological ring by letting the subgroups {Cb{t}+ tmf~{t}}belt,,,eZ§ constitute a basis of the neighborhoods of zero. This topology will be referred to as the weak topology of ~{t}. It may also be described as the topology of coefficientwise convergence. We now obtain an elementary, but useful relation between convergence in the weak topology and uniform convergence in the function theoretic sense. Lemma z. 3. -- Let ft,f2, ..., be a sequence of elements of f2{t}, each converging in Cb, be(5. (i) If the sequence converges uniformly on C b to a function F then a) The sequence is uniformly bounded on C b. b) The sequence converges in the weak topology to fell{t} which itself converges on C b and f(x)=F(x) for all xeC b. (ii) Conversely, if a) the sequence is uniformly bounded on Cb, b) the sequence converges in the weak topology to fe~{t} then f converges in U b and for each ~>o the sequence converges uniformly to f on Cb+ ~. Pro@ -- Let .f(t) = ~ A~,itJ for i---- I, 2, ... i=0 (i) Since the sequence converges uniformly on C b and since, by Lemma i .2, fl is bounded on Cb, we may conclude that the sequence is uniformly bounded on C b. By hypothesis, given N>o there exists neZ+ such that ord (f~(t)---~,(t))>N for all t~C b and all i, i'>n. Hence by Lemma i .2, for i, i'>n and for alljeZ+ (x. 5) ord(A~, i--A~., i) > N--jb. For fixed j, (5) shows that {A,,i}~=l, ~ .... is a Cauchy sequence and hence converges to an element A i of fL It now follows from (I .5), letting i'--~oo that for i>n and all jeZ+ (x. 6) ord (A~,j-- Ai) > N --jb. Let f(t) = ~ A/i. Iff does not converge on C b then we may suppose N chosen such that ord A~.+jb<N for all j in some infinite subset, T, of Z+. Let i be a fixed integer, i>n. Since f~ converges in Cb, we know that ordAi,~.§ for all jeZ+--T' where T' is a finite (possibly empty) subset of Z+. For jeT~T', ord Ai, j>ord Aj, which together with (i.6) shows that ordAj-t-jb>N. Hence T--T' must be empty, a contradiction, which shows that f converges on C b. Lemma x .2, together with equation (x.6), shows that for i>n, ord(f(t)--f(t));>N everywhere on C b. In parti- cular for fixed teCb, letting N-+oo we conclude that f(t)=limf(t)=F(t). This completes the proof of (i). 2 lo BERNARD DWORK (ii) By hypothesis the sequence is uniformly bounded on C b and hence by Lemma 1.2 there exists a real number, M, such that ord Ai, i +jb ~ M (x.7) for all i, jeZ+. Furthermore, writing f= ~ AitJ , we know that for each j~Z+, limi_, oo Ai, j = A i. For each jsZ+, therefore, there exists i (depending on j) such that ord (A~,j--A~) >M--jb. Hence by comparison with equation (i.7) we may conclude that (I. 8) ord A i-k-jb> M for all jeZ+. This shows that f converges in U b. Now let ~ be a real number, ~>o. Given a real number N, let jo~Z+ be chosen such that jo e-[-M>N. Then by (I.7) and (i.8) we have ord Aj+j(b +a)>N, ord A~,j +j(b +r for all ieZ+ and all J>Jo. Hence ord (A,,j--A~)+j(b+r for all J>Jo, i~Z+, while since limA~.j=A~., we may conclude that there exists n~Z+ such i-+oo that ord(Ai, i--Aj)+j(b+r for all j<jo, i>n. Hence for i>n, jeZ+, ord (A~..i--Ai) +j(b + r and hence by Lemma i. 2, ord (fi(t) --f(t))>N everywhere on Cb, which shows that the sequence converges uniformly tofon Cb+ ~. This completes the proof of the lemma. With F(t) as in equation (I. I) we define thef h derivative ofF (for jeZ+) to be the power series F/J)(t)-= - ~ s(s--I)... (s--j+ I)A~t s-j andlet F[fl(t)= ~=0(~)Ast~-J where (~) 8~j denotes the binomial coefficient of t ~ in the polynomial (i + t)*. Clearly F C~/=j ! F [fl, the notation F [fl being convenient if the characteristic of ~2 is not zero. We now prove an analogue of Taylor's theorem. Lemma I. 4. -- If Fsf~{t} converges in Cb, (be(5) then (i) F is a continuous function on C b and is the uniform limit of its partial sums. (ii) F It converges in C b for each jeZ+. ,, (iii) For fixed xeCb, the polynomials Pn(t) ----- ~] F[fl(x)(t--x) i (n = I, 2, ...) converge Qo j=0 weakly in f~{t} to F(t). The element L(Y)= Y, FEfl(x)YJEf~{Y} converges for all Y~C b and F(t)=L(t--x) for each teC b. ~=0 Proof. ~ (i) In the notation of equation (I. I), we conclude from (1.2) that given N>o, there exists neZ+ such that ord Aj+jb>N for all j>n. Hence byLemma I .2, ord (F(t)--~ Ait~)>N everywhere on C b. Hence F is the uniform limit on C b of its partial sums and thus continuity of F follows from the continuity of polynomials. Assertion (ii) is a direct consequence of Lemma x.i. 10 ON THE ZETA FUNCTION OF A HYPERSURFACE [I (iii) For jeZ+, let M i= Min (ord A s +sb). Since F converges on Cb, Lemma I . 1 *>i shows that M~.--->oo as j~oo. Lemma i .2 shows that for xeCb, ord Fti](x) > Min {ord (~) +ord A~+ (s--j)b}. -- 8>i Hence Min ord FtJ](x) > Mj--jb, Mr+t> Mj. xCC b Hence by Lemma i. i, the series L(Y) converges for all yEC b and hence by part (i), P~(t) converges uniformly to L(t--x) on C b (as n-+oo). Thus in view of part (i) of Lemma 1.3, the proof is completed if we can show that P~(t) converges weakly to F(t) as n-+~. Let Pn(t)-----~', An, st ~. We must show for fixed s that lim A n ~=A s. From the definitions ~=0 (I. IO) A., s = Z i=0 We now write F=F,~-}-G,~, where F~(t)= Y~Aiti. Clearly A~,s=A'n,~-k-A',,',, , where i=O A~,~ (resp. A','s) is given by the right side of(I. io) upon replacing F by F,~ (resp. Gn). Since Taylor's theorem is formally true for polynomials, A',,~=A~ for s~n, A~,,,= o for s>n. On the other hand for all j~Z+, ord (Gt, i](x))~M~--jb and hence ord A','s~M,,--sb. Hence for n>s, ord(A~--A~,s)=ordA~,~>M~--sb oo as n-+oo. This completes the proof of the lemma. We can now give some equivalent definitions of the multiplicity of a zero of a power series. Lemma x .5. -- If F converges in Cb, meZ+ and xeC b then the following statements are equivalent :r lim F(t)/(t--x)" exists. l "-~ x ~) Ft~l(x) = o for i = o, I, ..., m-- i. "() The formal power series, F(t)(i--t/x) -m converges in C b if x:~o while if x=0, t" divides F(t) in ~{t}. Proof. -- By Lemma i .4 for t~Cb, t~ex, we have F(t)/(t--x)~=~tFtiJ(x)/(t--x)m-' + ~ r[q(x)(t--x) '-'~. i=0 i=m Hence, by the continuity of power series, the limit exists if and only if (~) is true. Thus (0c) and (~) are equivalent. If x=o then (~) and (y) are clearly equivalen t. Hence we may suppose that x4:o. Let f~{t}, f(t)(i--t/x)m=F(t). Since the rules of multiplication of formal power series and of convergent power series (in the function theoretic sense) are the same, it follows that iffconverges in C b then as a function, f(t) = F(t)/(i--t/x) ~ for all t~Cb--{x }. The continuity of convergent power series now 11 i~ BERNARD DWORK shows that (u implies (0c). To complete the proof we show that (~) implies (y). It follows from (~) and Lemma I. 4 that in the weak topology F(t) = lira ~2 F~l(x)(t--x) i and Y~---~ OO ~ ~ hence in that topology, F(t)(I --t/x) -"= (--x)" lim ~ FtJl(x)(t--x) j-'. The coefficient B~ of t s is clearly B,= ~ Ftfl(x)(J-;"~)(~x) i-" so that by (i.9), f--m ord B~ > MingM.--sb/. -- i>s t ~ .i Thus ordB,+sb>M, and since Ms-+m with s, this shows that F(t)(i--t/x)-" converges in C b. IfF converges in Cb, XeCb, we say that x is a zero of multiplicity m>o if FEd(x) =o for i=o, i, ..., mui, while Ft"l(x)~eo. In particular if H converges in Cb, x+o, H(x) +o and F(t)= (t--t/x)mH(t) then x is a zero of F of multiplicity m. Let F be an element of f2{t} which converges in U b for some b<oo (i.e., the domain of convergence of F is not the origin). We assume with no loss in generality that Fei +tfl{t}. In the notation of equation (I.i), the Newton polygon of F is the convex closure in R � R (=two dimensional Euclidean space with general point (X, Y)) of the positive half of the Y axis and the points (j, ord A~.), j = o, i, ..., it being recalled that ordAi= q-oo if Aj= o. The Newton polygon will have a second vertical side of infinite extent if F is a polynomial of degree m>o. In this case the boundary of the Newton polygon (excluding the vertical sides) is the graph of a real valued function, h, on the closed interval [o, m]. Likewise if F is not a polynomial then the boundary (excluding the vertical side) is the graph of a real valued function, h, on the positive real line. In either case, h is continuous, piecewise linear with monotonically increasing derivative. Furthermore equation (i. 3) shows that the graph of h is asymptotic (if F is not a polynomial) to a line of slope -- b, where b is the minimal element of the extended real line such that F converges in Ub. If x is not an end point of the interval on which h is defined then h'(x--o)<h'(x+o). The points at which the strict inequality holds are called the vertices of the polygon. The abscissa, j, of a vertex is an integer and the vertex is then (j, ord AS). Finally, if l is the line obtained by extending in both directions a non-vertical side of the Newton polygon of F then for each jeZ+, the point (j, ord As) lies on or above the line l. Lemma x.6. -- Let F(t)= ~ AjtJ: I] (i--t/~i) be a polynomial of degree n>o, #=o j=t with constant term i. Let Xx<X2<...<X s be the distinct values assumed by ord0~ -1 as i runs from I to n and for j= i, 2, ..., s, let rj be the number of zeros, ~, off (counting multiplicities) such that ~ ord 0c = Xj. The vertices of the Newton polygon of F are the origin P0, and the s points (x.xI) P.= ri, r~;~ i i = a-~- I~ 2, . . ., s. 12 ON THE ZETA FUNCTION OF A HYPERSURFACE Proof. -- Let the zeros of F be so ordered that ord ~-l<ord e~-l<... <ord ~-l. The proof may be simplified by letting r 0 = o, X 0 be any real number, say 7~-- I. Then P, = r~, r~X~ for a = o, i, ..., S. Let j, be the abcissa of P,, then Ai, is the sum i ~ Ja of all products of the ~-i taken j, at a time. This sum is dominated by I-I ~-l. Hence Ja Ja i=1 ordAi=ordIle~-t= ZriN. If a>o,L_t<j< L then i=l /,=0 ] a--I ord Ai>ord 1I ~i-1= Z r(h4-X,(j--j,_t ) /:1 i=0 and hence the point (j, ord Ai) lies on or above the line since the equation Pa -1 Pa of that line is a--i (x. x2) Y-- X r~X~ = X.(X--L_I). i,-O Thus the Newton polygon is the convex closure of the s + I points P0, P1, 9 9 P, and the point (o, 4-oo). Equation (I. 12) shows that the slope does change at the points Pa, P2, ..., P~-i and this completes the proof. Corollary. -- The numbers {ord ~-1}7= ~ are precisely the slopes of the non-vertical sides of the Newton polygon ofF. If X is such a slope then the number of zeros ~ ofF such that ord ~ = -- X is the length of the projection on the X-axis of the side of slope Z. We now prove a refined form of a well-known theorem [4, Theorem IO, p. 4 I] which states roughly that two polynomials of equal degree have approximately the same zeros if the coefficients of the polynomial are approximately equal. Lemma x. 7. -- Let f and g be elements of ~[t] and let X be an element of the value group of ~) such that a) f(o) =g(o) = I b) The number (counting multiplicities) of zeros off on Px is a strictly positive integer, n. If N is a strictly positive real number such that (*. x3) Min ord (f(x) --g(x)) >nN, xEP x then each (multiplieative) eoset of I + C N contains the same number of zeros off in P x as of g. Proof. -- Let at, ..., ~, be the zeros of fin Px, let ~(t, --., g,~be the (possibly empty) set of" zeros off in U z and let S be the set of zeros off outside Cx. Clearly for ~S, ord~<X and hence if ~zPx, ord(I--~/~)=o. Since ord~i>Z, we have ord (i--~/yi) =ord (~/gi) =X--ord u for i= I, 2, ..., m if ~Px. Since f(t) = fl fI (,-t/v,). II i=l i=l zttE8 18 14 BERNARD DWORK we may conclude that for ~eFz, ordf(~) = ~ ord (i--~/e~) + ~ (Z--ord Yl). Letting m i.=l i =I c----- 2] (--?,+ord y~), we note that c is independent of ~er x. Letting 0~, ~, ..., ~,~, 4=1 be the (possibly empty) set of zeros of g in F x we conclude by the same argument as above that there exists a constant c'>o such that for ~EF x ft ord f(~)=--c-}- Z ord(:--~/~) (x. '4) ord g(~) = -- c'+ Y~ ord (I -- ~/0d), 4=1 it being understood that ordg(~)=--c' if n'-----o. It is easy to see that n'+o for otherwise ord g(%) = --c'< o<nN<ord(f(~l)--g(o~l) ) = ord g(0cl) , a contradiction. Let ~,, ..., ~, be chosen in Px such that ~l(I -r ..., ~e(I +CN) are disjoint and such that their union contains all zeros of f and g in F z. If e>i, ord (I--~/~l)<N for j=2, 3, --- e and hence there exists e>o such that (I.IS) o<ord(I--[~i/~l)<N--r for 2<j<_e. If e= x, we interpret this condition to mean simply o<~<N. With r so chosen we shall for the remainder of the proof let ~ be a variable element of Fx satisfying the condition (I. I6) N-- a< ord(1 -- ~1/~) < N. We now show that if aa~i(1 -}-CN) then ord(1--~/~l) if i= I (I.IT) N>ord (I--~/~)=ord(i__~j~i) if i4:1. For i=I this follows from ~/}----(,/~I)(~X/})E(~J~)(I+CN), while by (1.I6) (~l/}) 6(I -~-CN). For i>2, we have 0t/~G(~,/~)(i +CN) ~ (~J~t)(~l/~)(i +CN) while by (i. 15) and (:. 16) ord (I--3J~l)<N--r (I--~l/~). This completes the proof of (I. 17). In particular if ~ is a zero of fin F x then, by (I. :7), ord (: --~/~)<N and hence by (i. i4) since c>o, ordf(~)<nN. From (I. 13) we now see that ordf(~) =ordg(~) and thus equation (i. I4) shows that ?t t (x.x8) --c+ ~ ord (I--~/0t~):--c'+ Y~ ord (I +~/0~) i =l 4=1 For j---- x, 2, ..., e, let n~ (resp. n~) be the number of zeros off (resp. g) in ~i(I -~- CN). Equations (I. 17) and (i. 18) now give (I. I9) (n 1 -- n~) ord ( I -- ~/~l) = C -- C" -~- i~2 (n; -- ni) ord( I -- ~i/~l) the right side being simply c--c' if e = 1. As ~ varies under the constraints of (1.16), ord (x- ~i/~) varies at least over the rational points in the open interval (N--C, N) 14 ON THE ZETA FUNCTION OF A HYPERSURFACE I5 while the right side of (I. I9) is independent of ~. This shows that nl----n~ and by the same argument n~ = n~' for i = 2, 3, 9 9 e. This completes the proof of the lemma. As an immediate consequence we state the following corollary. Corollary. q Let f and g be elements of~[t] such that f(o) =g(o) = I. Let b be an element of if) and let m be the number (counting multiplicities) of zeros off in C b. ~. If Minord (f(x)--g(x))>o then the sides of the Newton polygon off of slope not xCC b greater than --b coincide with the corresponding sides of the Newton polygon of g. 2. If N is a strictly positive real number and Min ord(f(x) --g(x))>mN xEC b then each coset of I -+-C N in C b contains the same number of zeros off as of g. We can now demonstrate the main properties of the Newton polygons of power series. Theorem I.I. -- Let b'<b<~, bEff) and let F be an element of ~{t} converging in Ub, , F(o) = i. Let m be the total length of the projection on the X axis of all sides of the Newton polygon of F of slope not greater than --b. There exists a polynomial G of degree m, (G(o) = I) and an element H of ~{t} such that the zeros of G lie entirely in C b and (i) H converges in U~,, ord H(t)=o everywhere in C b. (ii) F = GH. These conditions uniquely determine G and H. Furthermore : (iii) The Newton polygon of G coincides with that of F for o < X < m while the polygon of H is obtained from the set: (Polygon of F) -- (Polygon of G) by the translation which maps the point (m, ord Am) into the origin. (iv) If K is a complete subfield of s which contains all the coe ficients of F, then GeK[t]. (v) If for each partial sum, Fn, of F we write F n =G,H,,, where G, is the normalized polynomial whose zeros are precisely those of F n (counting multiplicities) in Cb, then G,, converges to G in the weak topologv of ~{t}. (vi) If neZ+ and N is a strictly positive real number such that ord (F(t)--F,(t))>mN everywhere on Cb, then each coset of I + C~ in C b contains as many zeros of F as of F,. Proof. -- We follow the procedure of part (v). For n>_m the Newton polygon of F, coincides with that of F in the range o < X < m and furthermore all sides of the polygon of F, of slope not greater than --b occur in that range. This shows that for n~m, F,, has m zeros in C b. Since the sequence {Fn} converges uniformly on C b to F, we conclude that given N>o, there exists nleZ , nl>m , such that ifn and n' are integers not less than n 1 then ord (Fn--F,,)>mN everywhere on C b. We may conclude from the corollary to the previous lemma that each coset of I + Cs in C b contains as many zeros ofF,, as of F,, and hence the same holds for G, and Gn,. This shows that for n > m we may write Gn(t ) = 1-[ (i--t/a,,~) where the zeros 0%1 , ..., ~,,,,, of G, are so ordered i=l 15 I6 BERNARD DWORK that lima,,~=a~ exists for i= I, 2,...,m. This shows that G, converges to G, a n --~ OO polynomial of degree m whose Newton polygon coincides with that of F,1 and hence with that of F for o<X<m. For each n~Z+, H,(t) is a product of factors of type (i--t/a) where ord 0c<b. Hence (x. 2o) ord Hn(t ) = o everywhere on C b. G, is a product of factors of type (i --tin), where ~eC b and hence if ord t<b then ord Gn(t)<o (equality holds if G,(t)= x). If then b"effj, b>b">b', then ord G,(t) < o everywhere on Fb,, and hence ord H,~(t) = ord F,~(t) -- ord G.(t) > ord Fn(t ) everywhere on Pb,,. Lemma I-3 shows that F,~(t) is uniformly bounded on Fb,, and hence the same holds for H.(t). Hence by Lemma 1.2 the sequence Ha, H2, ... is uniformly bounded on Cb,,. We show that the sequence H1, H2, ... converges in the weak topology of f2{t}. This follows from the fact that x +trY{t) is a complete multi- plicative group under the weak topology. Certainly F,~F and G,--->G in that topology and hence H,,=F,/G,, converges weakly to the power series H=F/GEI +trY{t}. It now follows from Lemma I-3 (part ii) that H converges in Ub,, (and hence letting b"-+b', in Ub, ) and that for each r H, converges uniformly on Cb,+~ to H. Using equation (I .2o), it is now clear that H(t) is a unit everywhere on Cb. This completes the proof of parts (i), (ii), (v). Assertion (iii) has been verified for G, its verification for H follows from Lemma t .6 and the fact that H,-+H. Assertion (vi) follows from the construction of G, the corollary to Lemma I. 7 and from the fact that the zeros of F in Cb are precisely those of G. To verify (iv) it is enough to show that G, rK[t] for each neZ+ since then G = lim G, eK[t]. Since the valuation in a finite field extension of K is invariant under automorphisms which leave K pointwise fixed, we may conclude that the coefficients of G. are purely inseparable over K. Thus we may suppose K is of characteristic p 4: o. If g is a root of G, then it is a root of F n of the same multiplicity and hence the multiplicity must be a multiple, mp r, of a power ofp such that o~P" is separable over K. This shows that the coefficients of G n are separable over K which now shows that G,~K[t]. This completes the proof of the theorem. Part (v) of the above theorem has an important generalization which is the analogue of a theorem of Hurwitz. Theorem x.2. -- Let b'<b<o% ben and let fl,J~,.., be a sequence of elements of f~{t}, each converging in C b, such that fi(o)=x for each jeZ+ and such that the sequence converges uniformly on C b, to Fsf2{t}. By the preceding theorem, F=GH, fi=gjh i where G (resp. g~) is a polynomial whose zeros are precisely those ofF (resp. fi) in Cb, and G(o) =g~(o) = I. The conclusion is that G =~irng~ and that for i large enough, & and G are polynomials of equal degree. 16 ON THE ZETA FUNCTION OF A HYPERSURFACE t7 Proof. -- Let degree G = m and for each jEZ+, let F i be the jth partial sum of F and let .~,i be the jth partial sum of f~. Let N be a strictly positive real number. Pick j~Z+ such that ( x. 2x ) ord (F (t) -- Fj(t) ) > mN everywhere on C b. Part (vi) of Theorem I. i shows that F i has m zeros in C b. Pick i 0 such that for each i>i o (I. 22) ord (F --~)>mN everywhere on C b. Pick ueZ+ such that for given i>i o (x .23) ord(f --fl, u)>mN everywhere on C b. We may conclude from these three relations that (x. 24) ord (F~.--f~, u) > mN everywhere on Cb, and the Corollary to Lemma 1.7 now shows that each coset of i § C N in C 6 has as many zeros of F~ as offi, u and in particularf~,u has m zeros in C 0. Equation (I .33) together with part (vi) of Theorem I.I now shows that~ has m zeros in C 0. Furthermore equations (~.2I) and (I .23) and part (vi) shows that each coset of I +-C N contains as many zeros of F in C 0 as of Fj and as many zeros of.~ as offi,~. We may now conclude that each coset of I -t- CN contains as many zeros of F in C 0 as off~ for each i>i o. It is now clear that g~--~G and that deg g~ = m tbr i large enough. Corollary. -- Under the hypothesis of the theorem,for i large enough, the zeros ~, t, ~i, 2, 9 9 9 ~,,, of .~ in C o may be so ordered that lime i j=~i,j= I, 2, ... m and ~1, ..., %, are the zeros of F in C b. We conclude by recalling that in our previous article we left two propositions unverified. Proposition 2 of [I] is contained by Theorem I.I above. We now demonstrate Proposition ~. Proposition. -- If b'< b < oe and F converges in U b, but is never zero in Ub, then the series I/F converges in U b. Proof. -- As before we may assume A0= I. The Newton polygon of F has no side of slope less than --b and hence ord Aj>--jb. The conditions A0= i, ord Aj>--jb define a subgroup of I +trY{t} and hence are satisfied by the formal power series I/F. This shows by Lemma I. I that I/F converges in U b. w 2. Spectral Theory. Let Q' be the field of rational p-adic numbers, f~ the completion of the algebraic closure of Q', the valuation of f~ being given by the ordinal function x--~ord x which is normalized by the condition ordp-= + I. Let q, n, d be integers q> I, d> I, n>o which will remain fixed throughout this 3 ~8 BERNARD DWORK section. Let 3; be the set of all u = (u0, ul, ..., u,)EZ~_ +1 such that du o >u 1 + ... + u,. The set, Z~_ +1 may be viewed as imbedded in n+ I dimensional Euclidean space, I! "+1 and let a be the projection (Yo,Yl, ...,Y,)---~Yo of R "+1 onto R. We formalize and reformulate in a manner convenient for our present application the methods appearing in the second half of the proof of Theorem i [I]. Lemma 2. I. -- Let c,~ be the minimal value of as (u (t), ..., u (")) runs through all sets o fro distinct elements of %. Then c"[m~oo as m-~ oo. Let 9J~ be an infinite matrix with coefficients 9)lu, v (in f~) indexed by ~ � 3; which have the property ord 9J~,,~>� where � is a strictly positive real number. When convenient we write 9J~(u, v) instead of 9Jlu,~. Lemma 2.2. -- (i) IfgJ~' is any.finite submatrix of 9J~ obtained by restricting the indices (u, v) to ~' x ~' where ~.' is a finite subset of ~, then the coefficient ~'m, oft" in det(I --t?O~') satisfies the condition: ordy,,>� m. Hence for tEf~, ord det(I--tg)~')> Min(mordt+� an estimate depending only on ord t and the constants� q, d, n, but independent orgY'. In particular for each bounded disk of f~, det(I -- t?Ol') is uniformly bounded as ~.' varies over all finite subsets of ~.. (ii) If (u, v) ~' � ~', then the minor of (u, v) in the matrix (I--tgJ~') is a polynomial ~]u v)t" and ord V (u, v) > + � , Hence for tef~, ord (minor of (u, v) in (I--tgJ~') ) > q� § where c is a constant inde- pendent of ~i~' and ~; (if ord t is fixed). Proof. -- (ii) The coefficient, ,("(u, v) is a sum of products P =-t-1-I ~lJl(u (~), v(~)), where {u, u (t), ..., u (")} is a set of m + i distinct elements of 3;' and {v, v (t), ..., v (")} is a permutation of that set. Hence � ~ a(qu(')--v (')) =a{q u (~) -- v+ v (') --(qu--v)}= i=i '= i a{(q--I)(V+ ~ v (')] --(qu--v)}~qa(v--u) ~- (q-- I)C". i=1 / Lemma ~'.3. -- For N~Z+, let ?Ot~ be the submatrix of ?Ol obtained as in the previou~ lemma by letting ~;'={ue~l~(u)gN }. Let ~J~N be the matrix obtained from 9J~ by replacing ~l~u, , by zero whenever a(qu--v)>(q--i)N. Then lira det(I--t~s)= lim det(I--t~), N-~oo N.-~ oo the limit being in the sense of uniform convergence on each bounded disk of ~. The limit is an entire function, ~ y,~t m, and ord y,,~ (q-- I)� "=0 The remaining proofs may be omitted since they are consequences of the methods of [i]. Lemma 2.3 follows from Lemma 2.~ and Lemma i. 3 (part (ii)) once it is verified that the two sequences converge weakly to the same limit. However the details concerning weak convergence are very similar to the proof of Lemma ~. ~. (We note 18 ON THE ZETA FUNCTION OF A HYPERSURFACE I9 that the method used in [I, equ. (2o.2)] to show weak convergence cannot be used here as that proof made use of the geometrical application.) Let f~ {X} be the ring of power series and ~)[X] the ring of polynomials in n+ i variables X0, X1, ..., X~ with coefficients in f~. If u= (u0, ul, ..., u,)eZ~. +l, let X ~ denote the monomial l-I XUq Let + be the endomorphism of fl {X} or f~[X] ~=0 Io if q~u as linear space over ~) defined by ~b(X ") = t X"/a if q lu" For each ordered pair of real numbers (b, c), let L(b, c) be the additive group of all elements Y,A,X"sf~{X} such that (i) A~=o if ur (ii) ord A,> bu o + c. Let L(b)=,U L(b, c), E be the subspace of f2[X] spanned by {X"},ez. For each ell integer N>o, let 2~ (s) be the subspace of E consisting of elements of degree not greater than N as polynomials in X 0. Let ~(b, c) =~nL(b, c), ~(Sl(b, c) =E(mnL(b, c). If Heft{X}, let +oH denote the linear transformation ~--~+(H~) of ~{X} into itself. Lemma 2.4. -- Let ~ be any mapping of ~[X] into the real numbers such that.for ~1, ~2, c4:o, + =--oo (2. x) = + < Max If s is an integer, s> t, x is a non-zero element oJ f~ and ~ is a polynomial such that (2.2) (I--z-l+oH)"~ = o, (H 4: o) then ~(~)< v.(H) l(q-- ~). The proof may be omitted as it follows trivially from the fact that for ~Efl{X}, V.(,.,b(H~)) < (o.(H) + ~(~))/q. In particular if h is a linear homogeneous function on R~. +1 and if for each ~f~{X}, ~(~q) is the maximum value assumed by h(u) as X" runs through all monomials occurring in ~, then ~z satisfies the conditions of Lemma 2.4. In particular if He~ (N(q-1)l, then letting h(u)=ut+... +un--duo, we may conclude that if ~ satisfies (2.2) then ~ lies in E and letting h(u) = u 0 we may conclude that ~ lies in E(N). Thus the definition of det (I--t+oH) appearing in our earlier work is unchanged if (+oH) is restricted to ~(') for any integer m>N. Now let � be a strictly positive rational number. Let F-----Y~A~X" be an element of L(x, o) which will remain unchanged in the remainder of this section. We associate 19 ~o BERNARD DWORK with F a power series Zv, the characteristic series of qJoF which generalizes the characteristic polynomial appearing in the case in which F is a polynomial. For each integer N>o, t X" if u0<N let T n be the linear mapping of L( oo) into E(n) defined by TN(X")= (o otherwise " Let eN be the mapping ~--~qJ(~(Tn(q_t)F)) , and let ~ be the mapping ~---~T~(+(~F)) of (say) ~(N) into itself. If in the terminology of Lemma 2.3, we set 9J~,,, = Aq~_, for all (u, v) e 3; � 3;, then relative to a monomial basis of ~(N) the matrix form of a n is gJ~s while that of ~ is gJl~. Hence lira det (I--ten) and lim det (I--ta~) both exist and are equal by Lemma 2.3- The characteristic series, Xr, is defined to be this common limit. Lemma 2.3 shows that XF is entire and lies in ~{t}, ~ being the ring of integers of f~. The mapping ~ :~---~+(F~) of f2{X} into itself will now be examined. We first show by a general example that a satisfactory theory cannot be obtained if we allow ~ to operate on the entire space f2{X}. If F has constant term I then let Qt) G(X)=I:IF(Xqi). Clearly, F(X)=G(X)/G(X q) and hence if X~o, X~f~ then i=0 = ~ XiX0r is a non-zero element of f~ {X}, while e~ = X~. Thus as an operator i=0 on f~{X} each non-zero element of f2 is an eigenvalue of e. We shall show that Xr can be explained by restricting e to L(qz). However to obtain a complete theory it will be necessary to assume that the coefficients of F lie in a finite extension of Q'. Let Q, be the field of rational numbers. The value group of f2 is the additive groupofQ. For x= (x0, x~, ..., x~)ef2 ~+~, let ordx= (ordx0, ordxt, ..., ordx~)eQ "+~ if none of the x~ are zero. If a and a' are elements of Q,+l, we define the usual inner product (2.3) o(a, a') = Z a,a;. i=0 If ~ef~{X}, let S t be the set of all aeQ, n+l such that ~ converges at x if ord x =a. Writing 4= ~ BuX", (2.4) u~Z~b +t we have a generalization of Lemma I.I :If aeQ n+t then aeSg if and only if ord B u q- p (u, a) ~ -k oo as u-+ oo in Z~_ + 1. It is convenient to introduce a partial ordering of On+i. If a and a' are elements ofQ n+l, we write a'>a if a;>ai for i=o, I, ..., n. It is clear that if a'>a and aeS~ then a'~S~. We easily check that for 4, ~efl{X}, Sr162 ~qSr S~ D S~r~ S., (2.5) 20 ON THE ZETA FUNCTION OF A HYPERSURFACE Let g be a mapping of Z~_ +* into the set of two elements, {o, I } in f~. Let ~. be the f~ linear mapping of f~{X} into itself defined by y(X") =g(u)X u. (2.6) For such a mapping we have (2.7) Sv(~)~ St. For each aeS~, let M(~, a) = Min ord ~(x). ordx~ The generalization of Lemma I. 2 may be stated without proof. Lemma 2. 5. -- For aeS~, ~ as in (2.4), M(~, a)= Min (ordB,+p(u, a)). uEZ~_ +l If a'> a then M(~, a') ~ M(~, a). We easily verify for ~, ~q~f~{X}, u as in (2.6) that (2.8) M(~q, a)> M(~, a)+ M(~q, a) if aeS~nS~ (2.9) M(y~, a)~M(~, a) if a~S~ (2. xo) M(+~, a)> M(~, a/q) if a/qeS~ (=.xx) M(~ +~q, a)>Min{M(~, a), M(~, a)} if aeSenS~ and equality holds in (2. ii) if M(~, a) 4=M(~q, a). Let S={aeQ"+tlao>--q� da i+ao>-qx , i= i, 2, ..., n} Elementary computations show that if c is a real number, ~;cL(qx, c) then S~z S (2. x2) ) M(~q, a) >c for aeS, and I SFD q-iS (2"x3) M(F, a/f)2o if aeS. It follows from (2.8), (2. IO) and (2.13) that (~,. x4) M(~, a) > M(~, a/q) if aeS n qS ~. This relation remains valid if, is replaced by coy or yo~, the composition of ~ with g on either right or left side. Let 9{X} be the ring of power series in X0, ..., X,, with coefficients in 9, the ring of integers in ft. Let L' be the space of all elements of f~ {X} which converge in a polycylinder of radii greater than unity (i.e. an element ~Ef~ {X} lies in L' if and only if there exists a rational number b>o such that (--b,--b, ...,--b)eS~). We 21 ~ BERNARD DWORK note that L'~L(b) for all b>o but L' is not the union of such subspaces since the monomials, X ~, in L' need not satisfy the condition uz~. Lemma 2.6. ~ Let ~q~L(q~, --q(q--I) -1 ord X), where X is a non-zero element of ~), and let ~ be an element of L'nO {X} such that (2. xs) We may then conclude that ~r --q(q--I) -a ord X). Note. ~ The same conclusion would hold if ~ in (2.15) were replaced by ~oy or by ~,o~, with y as in (2.6). In particular, ~ may be replaced by ~{~. Proof. ~ Writing (2.r5) in the form ~---=--~+X-l~, we see from (2.5) that S~3S~c~S~3S~nqSF~3S~nqSFnqS 0 and hence by (2. i2) and (2.I3) we have (2. I6) S~3 S r~ qS~. By hypothesis, ~eL' and hence there exists b>o such that a(~ (~b, --b, ..., --b)eS t. If aeS then there exists an integer, r>o, so large that q-ra>a (~ and hence q-'a~S~. Let r be the minimal element of Z+ such that the displayed relation holds. If aeS then q-~a, q-2a, etc., lie in S and hence if r>I then q-('-~)a lies in S as well as in qS o so that by (2.16) we have q-('-l)aeS~, contrary to the minimality of r. This shows that r=o and hence ScS~. Since q-aScS, we may also conclude that ScqS~. Equations (2. r4) and (2. I5) show that (2.t7) ordX+M(~+~,a)>M(~,a/q) if a~S. We write ~ as in (2.4) and we assert that for a~S, vaZ]_ +~, (~. x8) ord Bo + O (v, a) > -- q(q-- I ) --1 ord X. To prove this we think of a as fixed and consider two cases. Case 1. -- M(~, a)>M(~q, a) In this case Lemma 2. 5 and equation (2. r2) give a direct verification of (2.18). Case 9. -- M(~, a)<M(~q, a). Here we may use (~. Ii) and deduce from (~. 17) that (m. x9) ord X + i(~, a)> M(~, a/q). Lemma ~-5 shows that there exists a particular element, ueZ~_ +t (depending upon a) such that M(~, a/q) ----ord B,+ p(u, a/q). On the other hand M(~, a)<ordB~+p(v, a), for each vsZ~_ +~. Thus we have (~. ~,o) ord B~ + p (v, a) + ord X> ord B~, + p (u, a/q), 22 ON THE ZETA FUNCTION OF A HYPERSURFACE ~3 for a particular u and for all veZ~_ +1. In particular (2.~o) holds for v=u and this gives ord X_>(q -1- I)p(u, a). (,.21) We recall that by hypothesis ~e~{X} and hence ordB.>o. Equation (2.I8) now follows from (2.2o) and (2,2I). This completes our verification of (2.18) for all aeS. Now let c be a rational number, e>o, let %, ax, ..., a, be rational numbers, ao>--qx, a,=c--d-l(qx+ao) for i=i,2,...,n. Then a=(ao, at,...,a,)eS and p(v, a)=ao(vo--d -1 ~ ,)+ (c--d-tq� ~ ,, i=l i=l which shows that if v0<d-t]~ v i then p(v, a)--~--oo as a0---~+oo if c is kept fixed. i=1 n Applying this to (2. I8) we see that ord B~----- -t-oo if v0<d -t Y, vl, i.e. i=1 (2.22) B~=o if v~ 3;. With c>o as before, let ao=--qx+c,a~=o for i-=I,2, ...,n. Once again a----- (a0, al, ..., an)ES and thus (2.18) shows that ord Bo> v0(qx--c ) --q(q--I) -1 ord X for each c>o. Taking limits as c-+o, (2.23) ord B~> qxv0-- q( q- I) -t ord ~. Relations (2.22) and (2.23) show that ~eL(qx,--q(q--I)-tordZ), as asserted. Note. -- If ~=o in the statement of the lemma, then equation (~. 19) is valid for all a~S. Since (o, o, ..., o)~S, it follows that ord X> o. Theorem 2.x. -- Let 11,...,;% be a set of non-zero elements of ~ and let e= ~ ord Xi+ (q--I) -1 Max ordk e Let ~ be an element of L'o~){X} such that ,/.=1 l <~ i<~ s (2.24) = then ~eL(qx,--e). Proof. ~ The theorem is a direct consequence of the previous lemma if s= I. Hence we may suppose s>I and apply induction on s. Let ordXl>...>ordX~ 8=1 and let ~q=(0~--X,I)~. Since ~qeL'~g){X} and II (I--x~-x~)~=o, we may i=1 conclude that ~eL(qx,--e'), where e'=e--ordX s. We may choose yeO such that ord-r=e'--(q--I) -1 ord X,. Clearly T;~;-l~qeL(qx,--q(q -I) -1 ord X,), while ~(7~)-----Xs(~'r 9 Since y;~T~e~, we may conclude from the previous lemma that -(~eL(qx,--q(q--I)-lordX,). The proof is completed by checking that --ord 7--q(q-- I) -1 ord ),~ = --e. 28 BERNARD DWORK "4 Note. -- Although not needed for our applications, we note that we had shown with the aid of Lemma 2.4 that if F~ (N(q-1)l and ~ is a polynomial satisfying (2.~4), then ~ lies in ~(N). We can now show that if ~ is known to satisfy (2.24) and is known to lie in L' then it must be a polynomial (and hence lie in ~CNI). If Fe~ (NIq-l~) then there exists y~K) such that yF~9[X] and hence if r~Z+,p'N(q-llyF~L(r, o). If ~,---- ~bop~N(q-1)yF, then fi (I--k~r)~---- o, where ~, __~ypm(q-1) and hence the theorem ~=1 shows that ~ lies in L(qr,--e--(s+ i)(ordy+rN(q--I))). Hence ~=ZB~X ",u~X and ord B,,~qruo--e--(s--I ) ordy--rN(q--I) (sq- I) for each r~Z+. Letting r-+o% it is clear that B,~-o if u0>N(q--I ) (s+ I)/q, which shows that ~e~. Theorem 2.2. -- If the coefficients of F lie in a field, K0, of finite degree over Q' and if k -1 is a zero qf order ~ Of XF, then the dimension of the kernel in L(qx) of (I--~-1~) ~ is not less than ~, indeed the kernel contains g. linearly independent elements which lie in L(qx) n K0(k ) {X}. Proof. -- We may suppose that ~ I. Since XF~){t}, ZF(O) = I, we may conclude from Theorem I.I that kes Let z~(t)----det(I--t~). We recall that Lemma 2.3 shows that Z~-+ZF uniformly on each bounded disk. There exists a real number, p>o so large that X~ has no zero distinct from ),-~ in ?,-a(~ +C~). The proof of Theorem ~. 2 shows that for N large enough (as will be supposed in the remainder of 9 ~-~ . k-~ of the proof) there exist (counting multiplicities) precisely ~ zeros, ~,~,. -, ,,N Z~ in )~-~(I -4- C~). Since XF, Z~ and the set ~-~(~ +Co) are all invariant under auto- morphisms of ~ which leave K0(k ) pointwise fixed, we conclude that the polynomial f~(t)= 1-I (I--kr is also invariant under such automorphisms and hence lies in K0(k ) It). Let K be the composition of all field extensions in f~ of K(k) of degree not greater than [z. Theorem ~.~ shows that k is algebraic over Ko, hence deg(K0(~)/Q')~oo. This shows that deg(K/K0)(oo and hence deg(K/O')(oo. The conclusion is that ),~,NeK, limk~N~X for i~ I, 2, ..., [z and that K is locally N .._~ o o , compact. Furthermore fN is relatively prime to zN/f~. We now restrict ~N to K[X] n2 (N), This does not change the characteristic equation of a N and letting W~ be the kernel in that space of ~N ~ I-[ (I--)~N) , we conclude that the dimension of W N (as K-space) is B- An element, 4, of W~ will be said to be normalized if it lies in ~ {X} and at least one coefficient is a unit. If ~ is such a normalized element of W~ then by Theorem 2. I, ~L(ffz, --e), where e= (~+ (q--I) -~) ord k. If we write ~ ~BuX ~ then ord B~Uo--e and hence Bu must be a unit for at least one element ua~.,={wSg[Vo~e/(q� Conversely if B, is a unit then ueS~. It is clear that a subspace W of K[X] of dimension b~ has a basis ~, ..., ~ in s for which there exist distinct elements ut, ..., u~ of Z~_ +~ such that the coefficient of X u~ in ~i is the Kronecker 8i.i(i,j= ~, 2, ..., b~)- Hence for each N there exists a set of t* linearly independent elements {~,s}i=,,~ ..... ~ in Ws corresponding to 24 ON THE ZETA FUNCTION OF A HYPERSURFACE ~5 which there exist ~ distinct elements, {u~,N}i=~, 2 ..... ~ in Z~ such that ~.,~es and the coefficient of X ui,~ in ~i,N is 3i,~. for i, j = i, 2, ..., tz. Since 3:, (and hence 3;~) is a finite set, there exists by the pigeon hole principle, an infinite subset, 9.I, of Z+ such that u~=u~, N is independent of N for each N in the subset and i= I, 2, ..., ~. In the following N will be restricted to this infinite subset. Now let fl3=K{X}r~L(qx,--e). Generalizing the definition of w I, we may define the weak topology of K{X} and by the local compactness of K and the theorem of Tychonoff, ~3 is compact under the induced topology. Thus ~3 ~, the ~ fold cartesian product of ~3 is also compact under the product space topology. Clearly the ordered set ~(~/= (~,s, ~,~, ..., ~,~) ef~ and hence an infinite subsequence of the sequence {~(~/}Ne~ must converge. Hence there exists an infinite subset, 92[' of 2[ such that {~(N)}N~9 l, converges to an element (~1, ..., ~)e~3 ~. For j= I, 2,..., ~ we have {~i,N}~e~,-+~ and since the coefficient of X u~ in ~',s is 3i.~., the same holds for ~.. This shows that ~,..., ~ are elements of ~3 which are linearly independent over f2. Furthermore ~N~,N=O for each N~' and hence taking limits as N-+oo in 9.[', we conclude that ~t, ..., ~ lie in the kernel of ~ = (I--X-~) ~ in L(qx). Now let to~, ..., to,, be a minimal basis of K over K0(X ). If then there exist ~1, ..., ~%~K0(X){X } such that ~q = ~ ~to~ and since the basis is minimal, ~L(qx,--e--i) for i=I,~, ...,m. Ifo=~then o= ~to~ andsince ~,~K0(X){X } for i= i, ~,..., m, we can conclude that ~, lies in the kernel of ~. Applying this argument to ~t, ..., ~ we conclude that the D.-space spanned by them is spanned by elements of the kernel of ~ in L(qx)nK0(X){X }. This completes the proof of the theorem. To complete our description of ZF in terms of a spectral theory for ~, we must prove a converse of the previous theorem. Theorem 2.3. -- Let ~. be an integer, ~. > i and X a non-zero element in fL The dimension of the kernel in L' of (I--Z-l~) ~ is not greater than the multiplicity of X -1 as zero of z F. We defer the proof except to note that we may assume that the kernel of (I--X-l~)~ in L' may be assumed to be of non-zero dimension and to show that X~s If the kernel of (I--?,-'~) ~ is not {o} then by an obvious argument, the same holds for the kernel of (I--X-t0Q. Hence there exists ~eL' such that ~=-X~, ~4=o. Since tEL' there exists u such that y~E~)(X}. Hence it may be assumed that ~e~3{X}. Thus Xr~=~r~ for each feZ+ and since e maps~{X} into itself, we conclude that ~4:o, Z~f){X} for all rcZ+. This shows that XeD. Theorem 2. I now shows that we can replace L' in the statement of the theorem by L(qx). Before resuming the proof we must recall some formal properties of matrices. Let A be an m � matrix with coefficients in some field of characteristic zero. For each subset H of {i, 2, ..., m}, let (A, H) be the square matrix obtained by deleting the jth row and column of A for each j~H. Let [H] denote the number of elements 4 26 BERNARD DWORK in H and let t be trancendental over the field, K, generated by the coefficients of A. If [H]=m, we define det(A,H)=i and for o<[H]<m, I--t(A,H) denotes ((I-- tA), H). Lemma 2.7. -- For I < r < m 1-I t---- det(I--ta) )'r! Y, det(I--t(A, H)), (2.25) /r-1/ddt (,=o( (m--i) )) =(--, [H]=~ the sum on the right being over all subsets, H, of{l, 2, ..., m} such that [H] =r. Proof. -- We recall the classical result that ifB is an m � m matrix whose coefficients are differentiable functions of t then (2.26) det B = Y~ det Bj where B i is the m X rn matrix obtained from B by" differentiating each coefficient in thej t~ row and leaving the other rows unchanged. Thus ~det(It--A) ---- Y, det(It--(A, H)). as [~ =t However t -m det(I--tA) =det(t-lI--A) and therefore det(I--tA)+t-mddet(I--tA)=--t -2 2~ det(t-lI--(A,H))= a$ [H] = t --t-~t -('-11 2 det(I--t(A, H)). [iq = i The assertion for r= I follows immediately. We may therefore suppose r>i and use induction on r. Hence ,rs, )) (2.~'7) r!-ll II [t----(m--i) det (I--tA) = \~=0\ dt (--~)r-lr-'(td--(m-(r--i))) Z det(I--t(A, H)). [H] = r - t The lemma is known to be true for r= I and hence for given H such that [H]=r--1, since (A, H) is an (m--r+ I) x (m--r+i) matrix, (td--(m--r + I)] det(I--t(A, H))=--Y~ det(I--t((A, H), H")), H ,I the sum being over all H"c{I,2, ...,m}--H such that [H"]----I. However ((A, H), H") = (A, H') where H' = H" u H and hence the sum over H" may be replaced by ~] det (I--t(A, H')), the sum now being over all H' such that H'D H, [H'] = r. Thus H' the right side of (2.27) is (--I)rr-lW,~]det (I--t(A, H')), the sum being over all H H H' such that [H]----r--I and over all H'D H such that [H'] = r. But given H' such that [H'] = r there exists exactly r distinct subsets H of H' such that [H] = r-- I. Thus the right side of (2.27) is (--I) r ~] det(I--t(A, H')), which completes the proof of the lemma. [H'] = r 26 ON THE ZETA FUNCTION OF A HYPERSURFACE With the previous conventions, let Si~l(A),j=o, ~,..., m denote the elements of the field K generated by the coefficients of A which satisfy the formal identity (2.~8) det(I + tA) = ~. S(~')(A)t i i=0 We observe that SIm-J)(A) -- ~] det(A, H),j = o, i, ..., m-- I [H] = j the sum being over all subsets H of {I, 2, ..., m} such that [H] =j. Let ~> I be a rational integer, let ~ be a primitive /h root of unity in some extension field of K. For (io, it, ..., i~-l) eZ~-, let g(io, il, ..., i~_1) =~ where tt--I ~--I r= Y~ si~. Since det(I--PA ~) = l-[ det(I--tco-~A), we have 8=1 8=0 m I~,--1 m (2.3o) Z tJ~SI~)(--A~) = H Z tJSI~/(--o~-'A). ~=o s=oj=o For o < i <m, by comparing coefficients of t ~!"-i) on both sides of (~. 3o), we conclude that g--1 (~.3 I) S(m-'t(A ~) = (S("-~)A)~ + (-- i)~/~-l/X'g(i0, ..., i~,_1) I-[ s(m--i,)A, s=0 I.~--1 the sum, E', being over all (io, ...,i~,_l)eZ~_,i~<m such that ]~ i,-----~.i, but i 0 = i I ..... i~_ 1 --= i is explicitly excluded. , =0 Proof (Theorem 2-3). -- We first outline the proof. Let W be the kernel of (I--X-la) ~ in L' (and hence by Theorem 2. I) in L(qx). Suppose dim W>r>o for some reZ+. We must show that Z~-I)(X -1) =o for s----I, 2,...,r. Let gJl~ (for each NeZ+) denote the matrix relative to a monomial basis corresponding to the linear transformation ~k----=Tso~ of 2/~/. Explicitly, for each ve3;s, a~(X ~) =EgJtk(u, v)X", the sum being over all u~3;s. Let xN(t)=det (I--tg)~). We know that for all seZ+, lim ?(~l(X -~) =Z~/(X -1) and thus we must show that lim )~-~/(X -1) = o for s = I, 2, ..., r. Letting N' be the Ig --->- 0o dimension of ~(s), equations (2.25) and (2.~9) show that it is enough to prove that (2.32) lim S/N'-~/(I--Z-1932~) =o for i=o, I, ..., r--I. We shall prove the existance of a constant c independent of N, such that for i~o, I, ...,r--I ord c +� I)N (2.33) and prove (2.32) by using (2.33) and (2.31) to deduce the existence of a constant c' independent of N such that for i--o, I, ..., r--i (2-34) ord S(Z~'-0(I--X-t~lJ~)2c' +� I)N[~ '+'. 27 ~8 BERNARD DWORK Let 9J/~' = (I--X-~gTt~) ~. We may view 9J/ff as a matrix whose rows and columns are indexed by the set ~s of all u e3; such that u 0< N. If H is any subset of ~, H 4= 35r~, we may, following our previous convention, denote by (922~', H) that square matrix obtained from 9Jl{~ by deleting all rows and columns indexed by elements of H. We shall show that if H is any set of not more than r--~ elements of 3; then c may be chosen independent of H and N such that ord det (!IR~', H) _>c +x(q-- ~)N (2.35) whenever H (if not empty) is contained properly by 35 N. Equation (2.29) shows that equation (2.35) implies (2.33)- Our first object is the proof of equation (~ .35). Let H be a set of no more than r--I elements of ~E. We know that there exist ~1, ..., ~,, a set of r linearly independent elements in W. Let ~i=ZB,,iXu, j=I, 2, ..., r, the sum being over all ue3;. The (possibly empty) set of [H] equations ]~ aiBu, i=o i=1 for each ueH, in r unknowns a 1, a~., ..., a, certainly has a non-trivial solution in f~ (since r> [H]). Since ~t, -.., 4, are linearly independent, we conclude that ~---- ~ ai~ ~ j=i is a non-trivial element of W. Since o + ~eL(q~), ~ may be normalized so that ~es and at least one coefficient of ~ is a unit. Thus there exists a normalized element ~=ZB~X" in W such that B,=o foreach ueH. Theorem ~. ~ shows that, forall ue~;, (~. 3 6) ord B~> q� , where e=~tordX+(q--I)-aordX. Hence if N>No=e/qk, we may conclude that T~ is also normalized and the coefficients of ~N=TN~ satisfy (2.36). For typographical reasons we shall when convenient denote the coefficient of X u in ~ (resp. F) by B(u) (resp. A(u)) instead of B, (resp. A,). For given integer j>i, (0@ ~T N ~ = (T~ o~)JT N ~ ---- ZXW(/)B (w (~ A (qw I1)- w I~ A(qw C~)- w I1)) ... A(qw lil- w I j- 1)) the sum on the right being over all (w (~ w (1), ..., w (i)) ~7~ +~. We may write TN(s as a similar sum except in this case the sum is over all ((w (~ w (1), ..., w!i-~)), w Ij)) e3J x 3;N- Since ord A~>� 0 for all ue3;, we have by (2.36), i-1 X--1 ~176176 " " A(qw(i)--w(i-1)) } >-- --� qw(~ +,~=o (qw{'+ l)--w(i)) ) = Thus we If w/~ w(1), ..., w Ci-t) do not all lie in 3;N then certainly a w/~) ) >N. i= can conclude (using only the fact that ~eL(qx,--e)) that TN(~i~) -- (TNOe)~TN~ mod Y~ X~'C(� (q--~)� (2.37) u~ N 28 ON THE ZETA FUNCTION OF A HYPERSURFACE where for each real number, b, C(b) is used in the sense of C~ in w I. Since o= (I--X-te)~= Y. (__?-t)i(~)~i~, i=0 we have tx :=o i=o (I--X-%t~)~TN~ mod N X"C(xquo--~e+(q--i)xN ). For each element (u, v)s~l~ x ~N, let ~'(u, v) denote the coefficient of the matrix ~l~' in u t~ row and v tb column. We have for each vs~N, (I--X-10t~)~X'=]~Jt~'(u, v)X ", t* the sum being over all ue~ N. Thus (I--X-~a~)~TN~=Y~B~ v)X ", the sums being over all ue~ N and all w~s. We conclude that for each ue~I~ , ]~lJ~(u, v)B~-=o mod C(xqu0--~ e+ (q~)xN), the sum being over all vs~ N. We recall that B,-----o for wH and hence if N" is the number of elements in ~--H, the system of N" congruences indexed by ue~N--H , (2.38) ~p-~'~ v)B,---o mod C(--2 e+ (q--I)xN), (the sum being over all VEZN--H), has a non-trivial solution if N>N 0 since B, is a unit for at least one W~N--H. The ring of integers, s of Y~ is not a principal ideal ring, but finite sums of principal ideals are principal. Hence the theory of elementary divisors may be applied to the matrix E N indexed by (~--H) � (%N--H) whose ,,general" coefficient is EN(U , v)-=p-"q~"gJl~z'(u, v). If r are the elementary divisors of E N then (2.38) shows that (~'.39) CN"- O mod C(--2 e+ (q-- I)� Since our object is to prove (2.35), we may assume det(gX~', H) 4: o. Hence o 4 = det EI~ , O4:sN,,. If U and v lie in ~N--H, let (EN, (u, v)) denote the matrix obtained from E N by deleting row u and column v. Let ((gX~', H), (u, v)) denote the corresponding matrix associated with (gYt~', H). It follows from the definitions that det(Es, (u, v))/det Et~ =p~"det((gJl~', H), (u, v))/det(gJ~, H). (2.40) Ideal theoretically, (detEN)=(SN,,)E(det(EN, (u,v))), the sum being over all (u, v)e(~N--H) 2. Thus -- ord r ord det(Es, (u, v))- ord det EN, the mini- mum being over all (u, v)~(%N--H) 2. This together with (2.4 o) shows that (2.4I) --ord ,z~,,----Min {� + ord det((g3~', H), (u, v)) }--ord det(~It~;, H), the minimum being over the same set as before. This together with (2.39) would Z~''''~-~tN(u, BERNARD DWORK 3 o give the proof of (2.35), if it were known that c may be chosen independent of N and H, u and v such that (2.42) xqu o -}- ord det ((~Q~', H), (u, v) ) ~ c + 2 e. Thus the proof of (2.35) has been reduced to that of (2.42). We observe that 93~(u, v) =Aqu_ . and hence ord 9JUs(u, v)~x~(qu--v). It is easily verified that if two square matrices (each indexed by 25s) satisfy this estimate then so does their product since xa(qu--w)-+-xa(qw--v)~xa(qu--v). Thus tt tlt --~ ~N = (I--x-~)" = I + 9J~N x , where 9J~' is a square matrix indexed by X~--H satisfying the condition (2.43) ord ?/J~"(u, v) ~x~(qu ~v) for all (u, v)e(B:s--HH) ~. Equation (2.42) now follows directly from Lemma 2.2 (ii). This completes the proof of (2.42) and hence of (2.35). As we have noted previously, this implies the validity of (2.33). We must now show that (2.33) implies (2.34). This is clearly the case for r= I. Hence we may assume that r> I and that (2.34) has been verified for i = o, i, ..., r~2. Replacing A by I--X-l~lJ~ in (2.3 I), we have (SIS'-r --),-~gJ~)) ~ = S/S'-I'-t))((I--X-a931~) ") --Z'g(i0, ..., i~_t) I] S(N'-i,)(I--X-t~) the sum, Z', on the right being over all i0, iD...,i~_~ in {I, ~,..., N'} such that IX--1 5~ i~=~t(r--I), but io=i t .... =i~_x is excluded. In each term in the sum, Z', at $~0 least one factor S(N'-is)(I--Z-x~J~) occurs such that is<r--i, while the remaining factors are ~--x in number and each of type SIS'-i/(I--X-x~ff~), j~(r--i). The assertion follows from the induction hypothesis provided we verify the existence of a finite lower bound for ord S Is'-i)(I-x-tgJ~) independent of N and valid for j~.(r-- ~). The existence of such a lower bound is an obvious consequence of equation (~. ~9) and Lemma ~.2 (i). This completes the proof of the theorem. Note. ~ No use has been made in Theorem ~. 3 of compactness and no hypothesis concerning the field generated by the coefficients of F is needed. On the other hand we do not know if Theorem ~. ~ is valid without that hypothesis. We now summarize some of our information. Theorem 2.4. -- For each non-zero element, ~, of ~, let sx be the multiplicity of ~-1 as zero of XF. If the coeffcients of F lie in a finite extension, K0, of Q', then for s~s x the space Wx=kernel in L' of (I _)-l~)s isindependentof s, lies in L(xq) andis of dimension s x. Further- more W x has a basis consisting of elements of K0(~,){X }. Proof. -- For given )~2", let W cs) be the kernel of (I--),-1~) ". Theorem 2.2 shows that for s~sx, dimWIS)~sz, while Theorem 2.3 shows that dimWC~l~sx for all 30 ON THE ZETA FUNCTION OF A HYPERSURFACE s>~. Since W(~)cW (~+1) for all s>~ it is clear that W (") is independent of s and has dimension s x for s >s x. The remainder of the theorem follows directly from Theorem ~. ~. Corollary. -- If G is an element of K0(X ) such that for some real number b>o both G and ~/G lie in L(b) and if H(X) =F(X)G(X)/G(X~) then Z~=7.a, it being understood that F and K 0 satisfy the conditions of the theorem. Proof. -- Let e=Min(� b). It is clear that ~G.~ is a mapping of L(c) onto itself. The corollary now follows from the theorem and the fact that each ~eL(c), d?(H~) = G(X) -'. ~(~). G(X). We have shown that the zeros of Z~ can be explained in terms of spectral theory if F satisfies the condition of Theorem 2.4. If it were known (as is the case in the geometrical application) that the coefficients of F and the zeros of Zr all lie in a finite extension, f20, of Q,' then the zeros of ;(~ can be explained entirely on the basis of the spectral theory of ~ as operator on L"=f~0{X}nL(q~z ). We make no assertion of the type: L" is a sum of primary subspaces corresponding to :r Our next result serves as a substitute for a statement of this type. Theorem ~,. 5. -- If X is a non-zero element of ~2 which is algebraic over O', /f X -1 is of multiplicity ~ as a zero of )F, if the eoe~cients of F lie in a finite extension, K0, of Q' and if K is anr finite extension of K0(X ) then (2.t4) (I--X-'~)~+* (K{X}n L(qx)) = (I--X-*~)~(K{X}n L(qx)) 9 In particular if ~=o (i.e., Z~(X -~) 4:o) then K{X}na(qx) =(I--X-*:c)(K{X}na(qx)). Proof. -- Let K' = K0(X ). By hypothesis K is a finite extension of K'. For given K we show that (2.44) holds if and only if it is valid when K is replaced by K'. Let col, ..., r be a minimal basis of K over K'. Suppose (2.44) is valid with K replaced by K'. If ~eK{X}nL(qx) then ~= ~ co~., where ~eK'{X}nL(qx), i= i, 2, ..., m. i=t each in K'{X}nL(q� such that Hence by hypothesis there exist ~t,''',~m (I--X-~0c)~'+l~qi = (I--X-la)~., i= I, 2, ..., m, and hence ~= Y~ r ) i=1 and furthermore (I--Z-~)~+l~= (I--X-~e)~. This shows that (I--X-'0c)~+t (K{X}nL(q� ~ (I--X-*~)~(K{X}n L(qx)) and since inclusion in the opposite direction is clear, we may conclude that (2-44) is valid for K if it is valid for K'. Conversely if (2-44) is valid for K, then given ~eK'{X}nL(qx) there exists ~eK{X}nL(q~z) such that = (I--x-t~)~. The relative trace, S, which maps K onto K' may be extended to a mapping of K{X} onto K'{X} in an obvious way. The trace, S, commutes with ~ and hence (I--X-le)~'~=(I--X-la)t'+tS(~/m) since S(~)=m~. Since S(~q)eK'{X}nL(qx)we may conclude that (2.44), if valid for a given K, is also valid for K'. 31 3~ BERNARD DWORK We have shown that it is enough to prove the theorem for one finite extension, K, of K'. If k-1 is not a zero of ZF, let K = K'. If X-1 is a zero of ZF then following the procedure of the proof of Theorem 2.2, let ZN(t)=det(I--taN) , let p be real, p>o such that Z~ has no zeros in k-l(~ + Cp) distinct from k-*. For all N large enough, k-1 . k-1 in ~.--l(I Ji-Cp), these are zeros ofa polynomialfN ZN has precisely ~ zeros 1,N, 9 9 , ~,~ of degree ~ which divides ZN and is relatively prime to zN/fN" Let K be the composition of all extensions of K' of degree not greater than ~. We know that k-~l,N, 9 9 -, k~,N-1 lie in K, approach k -1 as N-+oo and are distinct from all other zeros of ZN. In the following a n will be restricted to K[X] ni~ CN). ~t Let ~n be the endomorphism II (I--k~aN) of K[X]n~ CN) (~N=I if a=o). 4=1 Since ~N annihilates the primary components of K[X]n~ (N) relative to a N corresponding to the cigenvalues )'I,N, ---, k,,N, it is clear that ~NIK[X]nEIN)) is the direct sum of the primary components of K[X]r~(N) corresponding to the remaining eigenvalues of a n. Hence if a~' denotes the restriction of a n to ~N(K[XJo~tN)), we can conclude that (2-45) det(I--ta~') = det (I--ta N (I --th.N). Let ~ be a given element of K{X}nL(qx). We must find ~q in the same space such that (I--Z-~a)~+:~= (I--Z-xa)~. We may suppose that ~eL(qx, o). Let ~N----TN~. Since k is not an eigenvalue of a~, there exists BNeK[X]ns such that (2.46) (I --Z -~ aN) ~N~n = ~z~ ~N" Eventually we shall complete the proof by taking the limit of this relation as N-+~. The main problem is the demonstration that ~s may be chosen such that its limit lies in L(q~). We note that ~N~N is uniquely determined by (2.46) and hence ~N is uniquely determined modulo the kernel, WN, of ~N in K[X]n~ <N), a subspace of dimension ~. We shall show that there exists a real number c' independent of N such that ~N can be chosen so as to satisfy the further requirement (2.47) ~NeK[X]~L(qx, c') for an infinite set of integers, N. We first construct a basis of ~N(K[X]~IN)). For each Ue~:N, let Y,,N----~N X~. The set {Y~,N} indexed by ue~n, spans ~N(K[X]o~IN/) but does not (unless ~t =o) constitute a basis of that space. In the proof of Theorem 2.2, it was shown that there exists an infinite subset, 91, of Z+ and a set S of ~z elements of ~5 such that for each Ne91, the kernel, WN, of ~N in (K[X]c~ Is/) has a basis {g,,N}~es consisting of elements of K[X]n~(N)(qx, --e)n~[X] indexed by S such that for each veS the coefficient of X ~ in g~,z~ is the Kronecker 3,,,. (In the previous remark, e= (~ -]- (q-- I) -~) ordk, precisely as in the proof of Theorem 2.2.) Thus for each u e S, we have (N being assumed in the remainder of the proof to lie in 91), (~' .48) g.,~ = X"+ ZE~(w, u)X w, 32 ON THE ZETA FUNCTION OF A HYPERSURFACE We may the sum being over all we3;~--S, and furthermore ord EN(W , u)>qXwo--e. now conclude that for each ueS, since o= ~s(g,,N), that (e-49) --Y.,N = ZEN(w, u)Yw,~, the sum being over all WE~N--S. Thus the set {Yw,~} indexed by ~N--S spans ~N(K[X]ns (N/) and hence must be a basis of that space, since it contains the correct number of elements. We have noted that if ~z~ is a solution of (2.46), then the sum of ~N and any K-linear combination of the g~,s is also solution of (2.46). Equation (2.48) shows that ~qN may be chosen such that the coefficient of X" in ~ is zero for each uES. (In fact these additional conditions uniquely determine ~qN). Thus we may write aqN=ZB~,NX v, the sum being all vE3;N--S. By hypothesis ~EL(qx, o) and we write ~---ZG,X ~, the sum being over all vz%. Thus ~N=TN~=ZG,X *, the sum now being over ZN, and ord G~> qxv 0. Thus ~N~N = y~ GvY~,N = Y~ G~Y~,N + ~] G~Y~,N" Applying (2.49) v~ZN v~ZN--S u~S we now obtain ~N~N----=ZY,,N{G~-- 2 G~EN(V , u)}, the sum being over all ve~s--S. u@8 Thus ~N~=ZGv, NY~,N, the sum being again over all ve3;N--S. Here G~,~ = G,-- Z g~(v, u)G. u@S and hence ord G~,N>q� , c being a real number independent of N. We now determine the matrix of ~' relative to the basis {Yo,N},eZ,,-s of ~N(K[X]ns Since ~N commutes with [~, we have 0~N'Yv, N = 0~N~N xv ~--- ~N~N xv = ~N Z Aqw_vX w. w~N With the aid of equation (2.49), it is easily seen that for W~s--S (2.5 o) a~'Yv, N ---- EASt(w, v)Yw, rr, the sum being over all WeZN--S, where for (w,v)a(3;N--S) ~, A~(w, v) = Aqw_ v- ?~ Ez~(w, u)Aq,,_~. uES It is easily verified that ord A~r(w , v)>� Let A~ be the square matrix indexed by 35N--S whose w, v coefficient is AN(w ,' v). Equation (2.5 o) shows that det(I--X-~A~) = det(I--^'-laN)."' Since ~F(t) = lim det(I--taN) , we conclude from (2.45) that lim det(I--X-~A~) is the value N-o,- oo N~oo assumed at t=X -~ by zr(t)/(I--Xt) ~. This value is not zero since ~z is the multiplicity of X -1 as zero of Zr and hence for N large enough, det(I--X-lA~) is bounded away from zero. Explicitly there exists a rational number c" such that for N large enough, (2.5 I) ord det(I--X-lA~) <c". 33 34 BERNARD DWORK Equations (2.46) and (2.5 o) show that the set {B,,s} indexed by veX~--S is a solution of the system of equations indexed by w~3;~--S Z(8~,~--x-*a~(w, v))B,,N = Gw, z~, (~. 52) the sum being over w3;s--S, it being understood that ~,~ is the Kronecker ~ symbol. To verify equation (2.47), we apply Cramer's rule to equations (2.52) and estimate ordB,,s for each w3;s--S. For each element (w, v) of (3~--S) ~, let((I--X-tA~), (w, v)) be the square matrix obtained from I--x-~A~ by deleting row w and column v. Clearly --1 , B,,~.det(I--X A~)=Z+det((I--X-tA~), (w, v))G~,~ the sum being over all w~3;~--S. In view of (2.5i) it is enough to show that there exists c'" independent of N such that (e.53) ord det((I--X-~A~), (w, v))+ord G~.~>_q� for all (w,v)~(%N--S) 2. Equation (2.53) is however a direct consequence of Lemma 2.2 (ii) and our estimates for ordG~, N and ordA~(w, v). This completes the proof of (2.47). Since K{X}nL(qx, c') is compact, we conclude that the infinite sequence {~N} has a limit point v? in L(qx, c'). Taking the limit of equation (2.46) as N~oo over a suitable infinite subset of Z+, we obtain (I--X-l~)~+tzq =(I--X-t~)~. Thus we have Shown that (I--X-I~)~+~(K{X}nL(q�215 This completes the proof of the theorem. Corollary. -- In the notation of the theorem, let R = K{X}nL(q� and let W be the kernel of (I--X-t~) ~ in R. For each integer j, j>I we have (R=W+ (I--Z-l~)J~ (e 54) tWn(i__Z-~e)jR= (I__X-~)jW. If (I--X-'~)'W={o} then (2.55) (I--X-t~) ~+~R = (I--X-re) ~R- Proof. -- For simplicity let us use the symbol 0 for (I--X-le). The theorem shows that given ~ there exists ~ such that 0~=0~+t~, which shows that 0~(~--0~) =o and therefore ~q~W-t-0~. This shows that ~cW-+-0!~ and hence using the fact that 0WoW we easily see that RcW+0~'R if j>I. This proves the first half of equation (2.54). Writing this with j= I and applying 0" to both sides we obtain 0v!R----0~W-I-0~+t~R, which proves (2.55), since 0~W=o. If ~ER and 0i~EW then 0i+~0~W={o} and using Theorem 2. 4 we see that ~W, which shows that 0~~e0iW. This completes the proof of (2.54). 34 ON THE ZETA FUNCTION OF A t-IYPERSURFACE w 3. Differential Operators. a) Introduction In this section we modify the notation of the previous section so as to facilitate the application of our results to projective varieties. Let Q' and ~ be as before. Let ~,) be a finite extension of Q' in f2, whose absolute ramification is divisible byp -I. Let n>o,d>I be fixed integers as before. Let Z now be the set of all U -==- (U(), Ul, ..., Un+I) EZ~ +2 such that duo=Ux+... +u,+ x. The definitions ofL(b, c), L(b), ~, ~2 (N), PJl(b, c) are now precisely as in w 2 except that the set Z is given a new meaning and furthermore these additive groups now lie in DO{X 0, X1,..., X,+~} instead of ~{X0, X1, ..., X,}. Let S be the set {I,2,...,n+I}. For each subset A of S (including the empty subset), let M A be the monomial l-Ix~, (M o = i) and let iffA LA(b, c), LA(b), ~A, ~A,(NI, ~A,(NI(b ' C) be the subsets of the previously defined sets which satisfy the further condition of divisibility by M t in f~o{X0, Xa, ..., X,}. Let S'~-{o,i,...,n+I}, S"={o,I,...,n}. Let ~0 be the ring of integers in DO and let K be the residue class field of D 0. Let E i be the derivation ~ X~X] ~ of DO{X0, ..., X,+a}, i=o, ,, ..., n+ i~ A homo- geneous form fin D0{X~,..., X,+~} will be said to be regular (with respect to the variables Xa, ..., X,+~) if the images in K[Xt, ..., X,+~] of the polynomials f, E~f, ..., E,+a/ have no common zero in n-dimensional projective space of characteristic p. Let ~ be an element of1~ 0 such that ord ~= ~/(p--I), f a form of degree d in ~)0[X~, ..., X,+I] which is regular with respect to the variables X1, ..., X,+ 1 and let H be an element of L(i/(p--x)) such that HeDO{X} H - ~zX0fmod Xo 2 H~=E,H~L(p/(p--~),--~), i=o, ~, ..., n+,. For i=o, I,..., n+I, let D i be the differential operator ~--~Ei~+~.H~, mapping L (--oo) into itself. It is easily verified that dD 0 = D 1 + ... + D,+~, that D~oD~ = DioD~ and that D~ maps L(b) into itself for b~p/p-- I. The object of this section is the study n+l of the factor space L(b)/Y, D~L(b). To make use of the regularity of f we must recall some well-known facts about polynomial rings. b) Polynomial Ideals. If A is any set and G is an additive group then a set of elements ~. i in G indexed by A� will be said to be a skew symmetric set in G indexed by A if ~.,i=--~i.~, ~,i=o for all i,j~A. Let JR be a field of arbitrary characteristic, and let a be a homogeneous ideal in R[X]~-R[Xa, ..., X,+J. The ideal a has an irredundant decomposition into 35 3 6 BERNARD DWORK homogeneous primary ideals, a= n q/. The dimension of a is defined to be i=1 Max dim q~ and dimension here is in the projective sense. We recall [5], I. If geR[X] then (a:g)=a if and only if gisqi ,i-=I,2,... r. II. If g is a non-constant homogeneous form then dim (a + (g)) = dim a-~ if g lies in no primary component q~ of maximal dimension, while otherwise dim(a + (g)) = dim a. III. (Unmixedness Theorem): If a= (gl, g2, 9 .., gt), t<n+ i and dim a-~n--t then each primary component of a is of dimension n--t. Lemma 3. x. -- If gl, 9 9 g,+l are non-constant homogeneous forms in R[X1, . .., X,+t] with no common zero in n-dimensional projective space of characteristic equal to that of R and if {P~}~EA is a set of polynomials indexed by a subset a of S={I, ~,..., n+ I} such that Pig/= o then there exists a skew symmetric set ~/,i in ~R[X] indexed by A such that P/= $'. ~%gi for each i~A. Furthermore/f{Pi}iea consists of homogeneous elements such that deg(P/gl) = m is independent of i, then each ~i,j may be chosen homogeneous of degree m--deg(gigi). Proof. -- Let a,. = (g,, 9 gr), I < r < n + I. By hypothesis dim a,+ 1 = -- i, while by II, dimar--I~dimar+t~dimar for r=I,e, ...,n. Also by II, dima,~n--x. These inequalities show that dim ar=n--r for r= i, 2, ..., n-I- i and that dim a,+ t = dim r i for r~n. Hence by III, the primary components of a, are all of dimension n--r and by II, gr+, does not lie in any primary component of ar for r=I, 2, ...,n. Hence by I, (ar:gr+l)=a~" Furthermore since dim%=n--I, we know gl:~~ If a--{I} then Pl=~ and hence we can assume a={I,a, ...,r+I}, r:>x. Since (a, :gr+~)=a~, P,+lea~ and hence there exist polynomials h,, h2, ..., h~ such that P,+~= ~ h/gi. Thus ~ (Pi+higr+,)g~=o. Using the obvious induction i=1 i=l hypothesis on r, there exists a skew symmetric set ~/,i in R[X] indexed by {x, 2, ..., r} such that P~+h~g~+~= ~ ~%g~. for i= I, 2, . .., r. Let ~+~,/=h~, ~i,r+t-~--h~, for i= i= i, 2, ... r and let ~+~,~+~, r=o. The assertion follows directly. The valuation of ~0 can be extended to a valuation of the polynomial ring ~0[X] in the usual way, if g(x) =Za,,x", let ordg=Min ord a,. Lemma 3. ~. -- Let g~ , ..., gn + 1 be non-constant homogeneous forms in s ..., X, + t] whose images in K[X] have only the trivial common zero. Let A be a non-empty subset of S and let g be an element of the ideal Z (g~) of x20[X ]. Then there exist elements {hi},e ~ of ~0[X] such that g = Y~ g/hi and such that ord hi~ ord g for each /cA. i@A Proof. -- We may suppose that g 4= o and hence that ord g = o. By hypothesis g = Y, g/hi, hie~0[X ]. Let e be the absolute ramification of ~0 and let -- b = e. Min ord h;. Clearly b is an integer and we complete the proof by showing that if b>o then there 36 ON THE ZETA FUNCTION OF A HYPERSURFACE exists a set of elements {h~}i~ A indexed by A in f20[X ] such that g-= Y~ g~h~ and such i@A that e. Minordh;~--b+ I. Let II be a prime element of f20. By definition, iEA l-lbhi~s for each j~A and if b>o then ZgiHbh~=Ilbg-omod(II). Let G~ be the image ofg i and let ~ be the image of IPh i in K[X] for each leA. Thus in K[X], o= Y~ Gi~ i and so by Lemma 3. i, there exists a skew symmetric set, {~,i}, in K[X] iEA indexed by A such that ~= Y~ ~,jG i foreach ieA. We now choose a skew symmetric set i@A {~, j} in s indexed by A such that ~, i is the image in K[X] of ~'~ for each (i, j) cA � A. Hence Hbh~ = E ~igimod(II) for each i~A. We now define a set of elements {hl}~E A in f~0[X] by the equations Ilbh~ = 1-Ibh~ + Y, ~',igJ for each icA. Clearly iEA IIbh~=omod(II) and hence e. Minordh~--b+I. On the other hand the skew i EA symmetry of the set ~q~'j shows that g= Y~ gih~ which completes the proof of the lemma. ~A Corollary.- If ga, ...,g,+~ satisfy the conditions of the above lemma and {P~},cA is a set of elements of I20[X ] such that ~ Pig~=o, then the skew symmetric set ~i,i of Lemma 3. may be chosen such that Min ord ~q~,~> Min ord P~. i,i Letfbe the form of degree d in 230{X~, ..., X,,+~} which is regular with respect to the variables Xa, Xz, ...,X,+~. Let f0=f,f=EJ for i=I,2, ...,n+I. Since df0=fl+f~+..--q-f,+a, it is clear (lettingf~ be the image off~ in K[X~, ..., X,+~]) that the regularity off is equivalent to (i) f0,f~, .-.,~ have only the trivial common zero (ii)fl,f~, ...,f,+~ have only the trivial common zero if ptd. Condition (ii) is simpler for most of our applications but will not be used since it would limit our results to the case in which d is prime to p. However we do note that in any case the regularity off implies the triviality of the common zeros in f~ off~,fz, .- 9 ,f,+ ~. Thus Lemma 3. x shows that f~,f~, ...,f,+~ are linearly independent over D~ 0 (and f~). The following lemma refers to ideals in either f~0[X] or in K[X]. To simplify the statement we use the same symbol for f and f~. Lemma 3.3. -- Let B be a non-empty subset of S = { i, ~, ..., n + I }., (i) (M,)n Y~ (f) = Y, (M~f) + X (Msf/X~) i~ A iuA --B /~Ac~B if A is any non-empty subset of S, provided the characteristic does not divide d (i.e. the assertion holds in any case in ~2o[X ] and if ptd in K[X]). (ii) In either characteristic, if A + S then (M,) o ((fo) + X (f)) = (MBJo) + X (M,f) + X (M~f/X~.). i~A i~A--B i~A ('IB unless both AoB=S and A contains n elements. 37 3 8 BERNARD DWORK Proof. -- In both statements the ideal on the right side clearly lies in the ideal on the left side. To prove (i) it is clearly enough to show that if MB.he ~ (f) then IEA (3.,) 2 (f,)+ 2 (f, lX,) IGA--B i~Af~B Let B n A = C, B' = B-- C. Let h' = Mch. We first show that h' E Y~ (f). This is clear iEA if B' is empty, hence we may use induction on the number of elements in B'. If j~B', then letting B"= B'--{j}, h" = Mw,h' then Xjh" = Mwh'e ~ (f) and if we can show that h"e Y, (f~) then by the induction hypothesis we may conclude that the same holds for h'. Thus we consider jCA, Xjh"e Y~ (f) and recall that Xj, {f}~.j,~s is a set ~A of n § I non-constant polynomials with no non-trivial common zero (since the charac- teristic does not divide d) and hence Lemma 3.i shows that h"eZ (f). Hence iEA Mche Y~ (f) as asserted. If d---- I then if C is empty, (3- i) is trivial, while if jeC, i@g then f//Xj is a non-zero constant which again shows that (3. I) is trivial. Hence it may be supposed that d> i, in which case f~ =f/X i is a non-constant form for eachieS. We may assume that C={I,2,...,r},A={i, 2,...,t},r<t<n+I Thus t t X,X2...Xrhe 2 (f) and hence for some polynomial hi, XI(X2...Xfi--f(ht)e 2 (f). i=l /=2 We now apply Lemma 3- I to the n+ i polynomials, Xl,f2 , ...,f,f+l, 9 9 .,f,+t and conclude that X2...Xfl~(f() + ~ (f) (the left side is h if r= I). Now suppose for i=2 some s, I<s< r, X~+lX,+2...XrhE ~ (fi') @ Z (f/). Then there exists a polynomial, i=1 ]=s+l h~+,, such that X~.t(X~+e...X fl h~+tf~'+l)~ (f() + ?~ (.~). The n+I poly- i:= 1 i=s+2 nomials fl' ,f2', 9 9 9 ,f,', X~. t,f, + 2, 9 .. ,J~ + 1 are non constant forms satisfying the conditions s+l t of Lemma 3.~ and hence X~2...Xfle xZ (f') + Y, (f). This completes the proof i=l i=s+2 of (3. I), and hence of the first part of the lemma. (ii) Here it is enough to show that if MBhe (f0) + 2 (f~) then (3.2) 2 z (Z). i@A--B iCA~B Let C and B' be defined as before and let h'= Mch. We first show that (3.3) h' (f0) + z (f,). To show this, it is enough (as before) to show that if ICA and Xth"e Z (f) + (f0) then /@A the same holds for h". By hypothesis B' is empty if A contains n elements and hence for the proof of (3-3) it may be assumed that A does not contain n elements. Thus 38 ON THE ZETA FUNCTION OF A HYPERSURFACE Ata{I} contains at most n elements. Let C' be a subset of S disjoint from {x} which contains A and consists of exactly n-- I elements. The n + I polynomials, f0, X~, {f~}~c' satisfy the conditions of Lemma 3. I and hence h"e 52 (f) + (f0). This proves (3.3)- 4cA If C is empty then (3-2) is trivially true, hence we may assume C not empty. If d= I the f/'=I for each ieAc~BoeD and hence it may again be assumed that d>i. We may now let C={I, 2, ...,r},A={I, 2, ...,t},r<t<n. Since (3.3) now shows that X1.--X,h~(f0)+ }2 (fi), we have for some polynomial, h~, i=t Xl(X2...Xrh--hlft')c(L ) ~- ~a (fi)" i--2 The set of n + ~ polynomials, (f0, X~,f~, ... ,f,) satisfy the conditions of Lemma 3- I and hence X2...X~he(f0)+ (f~')+ Z (f). We now suppose that for some s, I<_s<r, i=1 we have X~+t...Xrh~(fo) § ~] (f()§ Y~ (f). Then for some polynomial i~l i=s+l h~+~,X~+~(X,+2...X~.h--fj+th~)e(fo)+~(f()+ Z (L). The n-k-~ polyno- i=l i=s+2 mials f0,f~',...,fj, X~+l,fs+-~,-..,f~ satisfy the conditions of Lemma 3.i and s+l t hence X~+2...X~he(f0)§ ~] (f')+ Y~ (f) which completes the proof of (3.2) i=1 i=s+2 and hence of the lemma. C) P-adic Directness. Let W be a vector space of dimension N over f~0 which has a <( naturally >~ preassigned basis. For the purpose of the immediate exposition, we may let W be the space all N-tuples, Do s, with coefficients in ~0- However in the applications in the following parts of this section, W will be a subspace of f~0[X] whose << natural ~ basis is a finite set of monomials. Let ~ be the ~)0-module, ~2, in W and let q~ be the natural map of ~ onto the K-space, W*= K N. For each subspace W t of W there exists a subspace, W~ =9 (Wa r~ ~B), of W*. The correspondence WI~W ~ maps the set of all subspaces of W onto the set of all subspaces of W* and preserves dimension. If W t and W~ are subspaces of W then (W 1 nW2)*c W~ n W.~, but equality need not hold If however W~ nW~ = {o }, then equality must hold and hence Wtr~W 2 ={o}. We shall say that W 1 + W 2 is a p-adically direct sum, written W~[+]W2, if W~c~W~={o}. In particular if Wt[+]W2=W then we shall say that Wz is p-adically complementary to W 1 in W. It follows from the above remarks that given a subspace W a of Q, there exists a subspace of W which is p-adically comple- mentary to W 1 in W. The notion of p-adic directness is introduced because of the metric naturally associated with W. If w=(wt, ..., wN) is an element of W then let ord w=Min ord w i. 39 40 BERNARD DWORK If weW'+W", (W' and W" being subspaces of W), then w=w'§ where w'eW', w"EW". Certainly ord w> Min(ord w', ord w"), but if the sum W'+ W" isp-adically direct then ord w=Min (ord w', ord w") and hence ord w'>_ord w. d) General Theory'. Let 9/be the ideal (f0,fx, -..,f,) in ~0[X1, ..., X,+,]. For each integer m~o, let W (m) be the space of forms of degree dm in f~0[X1, ..., X,+I] , let 9/~ =W(m~c~9/ and let V <'~; be a subspace of W (m) p-adically complementary to ~Im in W ('), with respect to the monomial basis of W Ira). Since (f0, fl, ... ,f,) have no common nontrivial zero in ~, 9/ must contain all homogeneous forms of high enough degree and hence there exists an integer, No, such that V Ira) ={o} for m>N 0. We shall show eventually that we may take N o to be n. We note that V~ We now let V ~ ~ X~'V/"), a subspace ofE (N~ and for each pair of real numbers b, c, m=0 let V(b, c)=Vt~L(b, c). It follows from Lemma 3.2, the construction of V and the regularity of the polynomial f that if Q is a homogeneous form in ~0[X1, ..., X,+I] of degree dm, then Q=P+ ~ Pif, where P~V (m), P0, P1, ..., P, each lie in W (m-l) i=0 and ord P>ord O, ord P~ord Q for i=o, i, ..., n. We now proceed with the analysis of the differential operators introduced in part a) of this section. We recall that H~EL(p/(p-- ~), -- I), and that H~ has no constant term. It follows easily that if b<p/(p-- x) then H~L(b, --e), where e= b-- (p-- ~)-~. Lemma 3-4. -- L(b, c) = V(b, c) + ~ H4L(b, c + e) if b<p/(p-- i), e = b-- I/(p-- i). i=0 Proof. -- It is clear that the left side contains the right side. If ~ is an element of L(b, c), we show that for each N~Z+ there exists ~qNeV(b, c)n~ Is/ and a set ~,N-a of elements in L(b, c+e) indexed by iES"={o, I,..., n} such that (3-4) ~ - ~N + ~ H,~,,N-1 mod(X0 N+~) (3.5) i~-1 - ~z~ mod X0 N I~,N-~- ~,N-z mod X0 N-1 for each ieS". Let pl0) be the constant term of ~, then (3.4) holds for N = o if we set ~0 = P;~ b, c) and ~i,_l=o for each ieS". We now suppose N>o and use induction on N. Then ~(N)=~--(~N_ 1+ ~ Hi~.,N_21 lies in L(b, c)and is divisible by X0 s. Let p(N)be the \ / {=0 coefficient of X N in ~(N). Clearly ord P(N)~bN+c and as noted above there exists O~N)~V(N),p~0~-I) ' ...,p~N-1) each in W (N-~) such that P(N)=O~)+E~p~N-')f, where ord O~)~ bN ~- c, ord p~N--1)> bN + c-- (p-- ~)-1 = (b -- I )N ~- c -}- e for each i e S". We now let ~=~N_~+X0nO~N)eV(b, c), and for each ieS" let ~"N-- ID(N--I) ~ I /I~ 40 ON THE ZETA FUNCTION OF A HYPERSURFACE 41 and compute ~-- ~-q- H~ ~,~ t = i 0 = o mod Xo s +' 2 Xo ( ) This completes the proof of (3-4) and (3-5)- The proof is completed by taking weak limits, ~,s-+~eL(b, c+e) for each ieS", ~N-+~eV(b, c) and hence ~=~+ ~ H~. i=0 Lemma 3-5.- Vn~ H,L(b)={o} /f b<p/(p--i). i=O Proof. -- Let ~ lie in the intersection, then ~ = ~ Hi,i, ~eL(b) for each i~S". ~=0 Let m be the minimal integer such that the coefficient P~) of X~ in ~i is not zero for at least one ieS". For given ~ it may be assumed that t0' 9 9 ~. have been chosen in L(b) such that m is maximal. For m'<m -t- I it is clear the coefficient of X~' in ~ is zero. Let (m) (m + i) O~ ''+l) be the coefficient ~ in 4- Clearly O~"+~)=r~ Y~f~P~ eV ng.Im+l={o}. i=0 It follows from Lemma 3-x that there exists a skew symmetric set {Bi, i} indexed by S" in W tm-~) such that P~m/=r~Y,~B~,j for each ieS". Let ~I,j-=B~,jX~ -x, ~' =~i-- ~ Hj.~.f~L(b), then ~= ~ Hi~=~ ~ ~ H,~ and for each i~S" the coefficient /=0 i=0 i=0 n of X~' in ~ is zero for m'< m and the coefficient of X~ is P!~)-- 7: ]~Bi, i = o, contra- dicting the maximality of m. j=0 Lemma 3.6. -- L(b,c)=V(b,c)+ ~ D,L(b,c+e) i=0 if (p--I)-~<b<p/(p--I), e---b--I/(p--I). Proof. -- Certainly L(b, c) contains the space on the right side. We first prove inclusion in the reverse direction if e>o (i.e. b>i/(p--1)). Given ~eL(b,c) we construct a sequence of elements indexed by reZ+, (~(r), ~(~), ~(0r), ..-, ~))~L(b, c+re)� c+re)� (L(b, c+ (r+ l)e)) "+1 by letting ~(0)= ~ and the following recursive method. Given ~(r)~L(b, c+ re) we choose by Lemma 3.4, ~(r)ev( b, c+re) and ~)~L(b, c+ (r+ i)e) for i=o, I, . .., 7"/ such that ~(')=~(~)+ ~ Hi~ r). We now define ~(r+l) by i=O (3.6) ~(r+t) = ~(r)__@r)___ X Di~Ir). i=o 41 4~ BERNARD DWORK We must show that ~(r+t)eL(b,c+(r+i)e). We note that t(~+')= - ~ E,~)eL(b, c + (r + I)e) i=0 and this establishes our recursion process. Writing equation (3.6) for r=o, I~ . . .~ h and adding, we obtain h h (3"7) ~(h+l)=~(0) X ~(r) ~ D, Y~ ~!~). r=O i=0 ~ =0 For e>o, ~ ~(r/ converges in V(b,c) and ~ ~,1 converges in L(b,c+e) for each r=0 r=0 ieS". Furthermore t(~'+~l-->o as h~oo and thus taking limits as h~oo, equation (3.7) shows that ~ lies in the right side of the equation in the statement of the lemma. We now consider teL(b, c), b = I/(p-- I). If NeZ+, ~>o, s<N then s(s/N+b) +c--e<sb+c and therefore TNtSL/N/(b+~/N, c--e), which shows since b + s/N> i/(p-- i) that there exists ~q(N)cV(e/N + b, c--s), ~INlcL(b + z/N, c--~ + ~/N) for each i~S" such that (3.8) TN~=~ IN/-4- ~ D,~ Ni. i=0 The space V(b,c--e)� c--e)) ~+~ is compact in the weak topology, which shows that the sequence (~(N/, ~N/ ~(N/~ has an adherent point " " "~ ~n /N=0,1,...~ (vl ("), t~), ..., ~)) in that space. Hence taking limits we obtain from equation (3.8), (3.9) t = ~(~/+ ~ D~t~). i=O We now let ~ run through a monotonically decreasing sequence of positive real numbers with limit zero. The use of compactness shows that the sequence (~(~/, ~1, ..., ~/) indexed by e has an adherent point. Restricting our attention to a converging subsequence we conclude that the adherent point (~, t0, ~1, ..-, ~n) lies in V(b, c--e) � (L(b, c--e)) "+1 for each e in an infinite sequence of positive real numbers with limit o. Thus taking limits in equation (3.9) we obtain t =~q + ~ D~t~, "~eV(b, c), ~.eL(b, c) for each ieS". This completes the proof of the lemma, i=0 We defer for the moment the discussion of Vn ~ D~L(b). i=0 Lemma 3-7. -- Let p,c,b be real numbers, b<p/(p--i), N an element of Z+, e=b--I/(p--i), p+e>_c and let A be a proper subset ofS', A#S. Let {~,}~sA be a set of elements in XoN~20{X}nL(b, c) indexed by A such that ~ Hi~eL(b , p). Then there exists iEA a set of elements {~,},SA in (XoN~20[Xa, ...,X,+t])nL(b , p+e) indexed by A, and a 42 ON THE ZETA FUNCTION OF A HYPERSURFACE skew symmetric set ~ in (XN-~Y~o[X~, ..., X,,+~])nL(b, c +e) indexed by A such that if we set (3" IO) iGA ] for each leA then Z H~;eL(b, p) and ~ lies in L(b, c) and is divisible by X~o+:for each /cA. Proof. -- It is quite clear that if the elements ~i are chosen in L(b, p +e) and the ~qi, j are chosen in L(b, c+e) then ~ as given by (3.IO) certainly lies in L(b, c) and 2~ Hi~ ~ =EH~--NH~i~L(b, p). Thus the only important condition to be satisfied by ~ is that of divisibility by X0 ~+t For each leA, let p~N/be the coefficient of X0 ~ in ~ and let Q~S+~/be the coefficient of X(0 N+~I in ~ H~,. Hence ord p!N/> Nb +c, ord O~N+al> (N + I)b + p, O~ s+~/=~ y,f~p~N~. i@A i~A Lemma 3.2 now shows that there exists a set of homogeneous forms of degree dN, {C~}~A such that o~N+~I=~z ~fC~, ord C~>Nb+9+e. Thus o: ~f(C~-P~)) and hence i~A leA by the corollary of Lemma 3-2, there exists a skew symmetric set of forms of degree d(N--I), {B~,i} indexed by A such that for each leA. (3. ") p+N/= C,-t- ~ N B, if and ord B~i> (N-- i) b + c + e (since by hypothesis, p + e >c). We now let ~i,~ = Bi,~X0 ~-' for each (i,j)~A x A and ~,----C,X0 ~ for each leA. It is clear that X0 ~ divides ~ (as given by equation 3. IO), while the coefficient of X0 s in ~ is p~i~)__ C,--~ 2~ B,,ifi = o. This completes the proof of the lemma, iCA Lemma 3.8. -- Let b, c, p be real numbers, b< p/(p-- i), e=b-- I/(p--1), e +p> c. Let A be a proper subset of S', A 4= S and let {~},eA be a set of elements of L(b, c) indexed by A such that Y~ Hi~.eL(b, p). Then there exists a set of elements {:qi} in L(b, p + e) indexed by A iEA and a skew symmetric set ~i~i in L(b, c +e) indexed by A such that ~ = ~ + 2~ Hj~ i iEA for each i~A. Proof. -- Let _.i~(~ for each icA. It is clear that Lemma 3-7 gives a recursive process by which for each NcZ+ we may construct a set {~sl} in 9 ~(N-1)~ (X~g~0[X1, . X,+l])nL(b , p+e) indexed by A and a skew symmetric set t'J~,j in (X0~-lf~0[Xl, ..., X~+t])nL(b, c+e) indexed by A such that for each icA, (3.x,) = X ~* jCA ~-*ij ) ~SlcL(b, c), X0 s divides ~sl, ~2 H,~N/ is divisible by X~ and lies in L(b, p). Let iGA ~ = ~ ~I ~), ~,j = ~ for each i, jcA, convergence being obvious in the weak topology. N =0 N =0 ,~i,~(~lj 44 BERNARD DWORK Clearly ~q~eL(b, p+e), ~r c+e). For reZ+, we write equation (3.i2) for N---=o, i,..., r and add. This gives N =1 i~A N =1 The lemma now follows by taking limits as r--~oo since lim ~!r+l) = o in the weak topology. I. ,,,~ oo Lemma 3.9. -- Let b, c, p be real numbers such that p > c, I/(p-- I) < b <p/(p-- ~) and let e = b-- I/(p-- I). Let A be a proper subset of S', A + S and let ~i be a set in L(b, c) indexed by A such that Y-, Di~ieL(b , p), then there exists a set {:qi} in L(b, p+e) indexed by A and a i~A skew symmetric set {~} in L(b, c+e) indexed by A such that ~=~-t- y~ D~:%~. Proof. -- There exists a unique element N of Z+ such that (N--I)e + c<p<Ne + c. For each integer r, o<r<N we construct a set {~)} in L(b, c+re) indexed by a and, for o~r<N a set {~)} in L(b, c+ (r+ I)e) indexed by A and a skew symmetric set {~!"}} in L(b, c-t- (r-? ~)e) indexed by A such that (letting ~=- ~ D~.) ~= Z D~ 0 for o<r<N, (3. x3) for r<N, (3- I4) ~(r-}- 1) __-- ~r)__ Z ~(r) for r<N, i Djqij iEA and such that ~0) ~i for each leA. Suppose the set f~(r)~ in L(b, c-t-re) I.~i JiCA satisfying (3. I3) is given for some integer r, o~r<N. We then have Z H,~") = ~-- Z E~!')eL(b, p) +L(b, c-t-re) =L(b, c+re). i~h ich Hence by Lemma 3.8, elements ~r)in L(b, c~-e(r+I)) and ~:} in L(b, c-t-(r+I)e) may be chosen such that equation (3.i4) is valid for each i~A. If ~r+l) is now defined by equation (3.I5) then certainly ~= Z D~ r+l) and furthermore, ~/r+l) (r) Y~ Ei~I~EL(b , c+ (r+ i)e). This completes the construction of ~i r) for i ~ ~i -- r----o, I, N, since ~0) is specified, and also of ~r) and ~!r) for r-= o, I, N--I. In particular, ~!meL(b, c+Ne) eL(b, 9) and therefore Z H~)=f-- E E~S)eL(b, p). i@A i@A Since 9 +e_> c + Ne, we may conclude from Lemma 3.8 that there exists a set {W~m} in L(b, p+e) indexed by A and a skew symmetric set f~(N)~ in L(b,c+(N+I)e) tqi, j J" indexed by A such that equation (3. I4) is valid for r= N. If now we define for each i~A, ~i~(~+1) by setting r=N in equation (3.i5) we have ~I~+1)-=~ s)- Z E~)~L(b, p+e) +L(b, c+ (N+ I)e)=L(b, p+e). 44 ON THE ZETA FUNCTION OF A HYPERSURFACE If now we write equation (3.I5) for r=o, ~,..., N and add, we obtain after (!) obvious cancellation, ~lY+~)=~--i~ D~ ~ 0~ ) . The lemma follows directlyby setting p +e) and :C c +e). r=0 Lemma 3.~o. -- If A is a proper subset of S',A4:S; b, c are real numbers, ~/(p--I)<b~p/(p--I) and if {~} is a set in L(b, c) indexed by A such that ~, D~=o then iGA there exists a skew symmetric set {~,i} in L(b, c-+-e) indexed by A such that ~= ~, Dd% i for each i~A. ie~ Proof. -- Let p be any real number, then ~ D~i~L(b, ~) and hence if O>c there iEA exists a set {~,/} in L(b, p+e) indexed by A and a skew symmetric set {._(0)X in ql, ~j L(b, c+e) indexed by A such that (3-x6) ~ ~)+ E D~(~) " i Ai, j jGA for each ieA. Let p run through an infinite sequence of real numbers towards +m, then by the compactness of the cartesian product of copies of L(b, o) indexed by A and of copies of L(b, c + e) indexed by A � A there exists an infinite subsequence such that if p is restricted to the subsequence, then, as p-+ov, ~q~) converges (necessarily to o) and ~ql~ converges to ~iEL(b, c+e). Clearly the set {~i} is skew symmetric and taking limits in equation (3-16) as p-+ov, the assertion is proved. Lemma 3.ii.- For b)l/(p--I), Vn ~ D,L(b) ={o}. Proof. -- Let ~ be an element of the intersection. It may be assumed that i/@-- i)<b~p/(p--i). With b fixed in this range, let p be chosen such that ~L(b, p), ~r If ~:~o then p certainly exists. Since ~e~D~L(b), Lemma 3.9 i=0 shows that there exist ~0, ~1, ---, ~n in L(b, pWe) such that ~= ~ D~ i. Thus i=0 ~-- ~ H~,---- ~ Edq~L(b , p+e). Lemma 3.4 shows that there exists ~'~V(b, p+e), i=0 i--0 ~o, .-., ~',~ in L(b, 9+2e) such that ~-- ~ H~=~'~- ~ H~ o. This shows that ~--~,' i=0 i=0 lies in Vn ~ HiL(b), and hence by Lemma 3.5, ~--~'=o. Thus ~=~'eL(b, p+e), i--0 which contradicts the choice of 9. Hence ~----o. This completes the (( general )) theory of the differential operators. We note that if b<p/(p--I) then for each subset A of S, the subspace LA(b) of L(b) is invariant under each D i. The action of the differential operators on these subspaces must now be discussed in greater detail. 45 4 6 BERNARD DWORK e) Special Theory. In this section we cannot avoid distinctions (1) depending upon whether or not p divides d. Furthermore some of our results will be valid only if H and the H~ = E~H are subject to further restrictions. To avoid confusion, for each ieS, let H~ =r~'X0fi, and let ~3~ be the mapping ~ Ei~ + ~H~, where r:'e f~0, ord r:' = x/(p-- I). For each subset, A, of S = { I, 2, ..., n -t-I }, let X A be the set of variables {X,}~ A. The ring, f20[XA] , of polynomials in the variables X A with coefficients in f~0, is viewed as a subring of ~0[Xs] = f20[X1, ..., X,+I] and in particular if A is empty then f~0[XA] is the field f~0. Let 3A be the homomorphism of f~0[Xs] onto f~0[XA] defined by ~A(X,)=iXi if icA 1 o if ieA As before W iml denotes, for each meZ+, the space of forms of degree dm in f20[Xa]. For each subset A of S let Wk m/=.~A(W/~/) and for each subset B of A, let W~'I"/=W~"/c~(MB), where (MB) denotes the principal ideal in f~0[Xs] generated by the monomial M B = I-I X~. (Unfortunately, our notation permits the same space to be designated by several symbols. Thus if o is the empty subset of S, then W] "/= W~ '/''l and W(m) _ w(m) _ ~O,(m)~ -- ''S -- ''A 1" For each subset A of S let 23~ '('') be a subspace of W~ '(m) which is p-adically comple- mentary (with respect to the monomial basis of W~ '(m)) in W~ '(') to W~'(m)t~3A(9~). Thus we have (a.,7) = For each subset, A, of S, let (3. xs) = the sum being over all subsets, B, of S which contain A. Lemma 3. x2. -- Let A be a subset of S. (i) Ws A'(m)= Y~ W~ '(m), the sum being over subsets, B, of S which contain A. ADB (ii) WsA'(")r (Kernel of UA) = ~] W~ '(~), the sum being over all subsets, B, of S which ~zx contain but are not equal to A. . (iii) W]'Im!n (~ag.t) = ~a(9~caW~'Iml). Proof. -- The first assertion is trivial. For (ii) we observe that a polynomial, ~, lies in the kernel of g~ if and only if each monomial, X ~, appearing in ~ is divisible by at (x) The theory in the case pld is hampered by the fact that Lemma 3-3 fails to give an explicit basis for the n+l ideal (Ms) N (f~,~ .... ,f~) in K[X]. This ideal contains but is not necessarily equal to (Msa~) + ~] (Msa~/Xi) , i=1 a counter-example being given (for n = 3) by ~ =J~Sx(~4--I)fa +fa~(I--~,)f2--A~,~4A, where for i = t, 2, 3, 4, 81 is the specialization of K[X1, X2, Xa, X4] defined by 8iXi = o, ~iXj = X i for j* i. 46 ON THE ZETA FUNCTION OF A HYPERSURFACE least one variable X/such that iES--A. If X ~' is also divisible by M A then certainly there exists a subset, B, of S containing A properly such that M B divides X ~. For the proof of (iii), we use Lemma 3.3 (i) which shows that as an ideal in ~0[XA], (MA) n~n(9~ ) ----  (MA(~Af/)/X~). Intersecting both sides of this last relation with W(A m), i@A we see W]'(m)n 3A9~ is the set of all homogeneous polynomials of degree dm of the form ~] g~MA(~Af~)/X~, the gi being elements of f~0[XA]. By homogeneity it may be assumed i@A that giMAfJXi is a form of degree dm in f20[Xs] and hence lies in 91nW~ '(~). Thisshows that the left side of (iii) lies in the right side, which completes the proof since inclusion in the reverse direction is trivial. Lemma 3.13. -- Let A be a subset of S (i) ~'(~)=Z[+]~ '(m), the sum being over all subsets B of S which contain A. (ii) W A' (m~ __ ~3A,(m)| S -- S (g[c~W~ '(~)) and the sum isp-adically direct ifp[d. Proof. -- (i) The definition of 2~ '(m) shows that it is enough to prove the p-adic directness. For each set B containing A, let ~B be an element of V~ '(~t such that ord (Z~B)> o. Let C be a minimal subset of S which contains A such that ord ~c <o. Clearly ord ~c ---- ord(,~cZ~) > ord(Z~B) > o, which shows that ord ~ > o for each B. (ii) We first prove this assertion without any claim concerning directness. The assertion is equivalent to equation (3.I7) if A=S. Thus we may assume that A:t:S and use induction on the number of elements in A. By Lemma 3-I2 (i), W~'~m)=W~'/")+ Y, WB s'(~). Equation (3.17) and Lemma 3.i2 (iii) show that BDA W2 '("!=~AmA'Im)+~a(2Ic~W~'(m) ) and since -~a acts like the identity on W2 '!m), we may conclude that W2'(m) c2~2' (r~) -t- 2I n Ws A'(m) + (Kernel ~A) n Ws A'(m). Lemma 3- 12 (ii) now shows that W2'(")c 2~]'('~)+ ?I n Ws a' (~) + ~] Ws B'('~) and it is clear from the previous relations BDA that W2 '(m) also lies in this space. The induction hypothesis now shows that ,-~ *Art (m) W~'(")cf~]'I")+'anvv+ ' + Y~ (~B'(m)-t-!~InWff'(m)). Equation (3.I8)now shows that BDA W~'(m~ cm~'~-,s ('*! ~ ~ 2 r WA's ('*) and equality is clear. To show directness (in the ordinary sense) of the sum, let ~ be an element in ~32'(m)n(2oW~'("/). Equation (3.18) shows that for each set B containing A, there exists ~,e~3 B'('/ such that ~=Z~n. Let C be a minimal set containing A such that ~c+o. Clearly ~c=~c~c2[ and hence ~ce(WcC'(m)n~cg/)c~3 c'(~)c , which shows by equation (3.~7) that ~c=O. This contradiction shows that ~ =o for all B and hence ~ =o. Let ~ be an element of~s ~,(m/ and B an element of W~t'/m)ng/, both in s such that ord (~--~q)>o. To complete the proof of the lemma, we must show (ifpld) that ord ~>o. By definition, for each set B containing A there exists ~e2~ '(") such that ~=Z~u. We show that ord ~B>O for each B. Suppose otherwise, then there exists a minimal set C containing A such that ord ~c=O. Then ord(~c--~c~)>o , 47 4 8 BERNARD DWORK while ord(~c~--~c~)>ord(~--~)>o. Thus ~ce~0[Xc] and ord(~c--~c~q)>o. Let ~c be the image of ~c in K[Xc] under the residue class map. Clearly ~c is divisible by M c and lies in the image in K[Xc] of ~0[Xc] n~cg.I. Using the asterisk to denote images in K[Xc] under the residue class map, we may conclude from Lemma 3.3 (i) (since p t d) that there exist a set of forms of degree din, {g~} in K[Xc] indexed by C such that ~c = y' giMc(~cf~)*/Xi '. Choosing forms G i of degree dm in ~0[Xc] which represent the g~ and setting ~ = ~ GiM c (~cf)/Xi eWc' (")n ~cg.I, we have ord (~c-- 4;) > o. iEC Since ~ce2~ c'("), this contradicts equation (3.17) and so the proof of the lemma is completed. For each subset A of S, let Vs A'(m) = ~sA,(~) if p t d, while otherwise let Vs A, (") be chosen in W~ 'Ira) p-adically complementary to (9.InW~'(m)). (Clearly we may let VsS'(")=~s s'(m) in any case.) It follows from the definitions and Lemmas 3.2, 3.3 and 3-13 that if AcS"nS and PeWs A'(m) then there exists O~")eVs A'(m/ and a set of homogeneous elements {Pi} indexed by S" in f~0[Xs] such that (3- I9) P = (~'~)+ Z P4f~M~/X4 + Z pJ~MA, i@A i~S'" --A ord O~m)>ord P, ord P~>ord P. If A is any subset of S, there exists O~m)e~3s ~'(m) and a set of homogeneous elements {P~} indexed by S in f20[Xs] such that (3-20) P = (~) + Z P~fM~/X~ + Y~ P~fM~, i~A iG S --A but in this situation the previous estimates for ord O~ ~) and ord P~ do not hold unless p does not divide d. Finally let Vs~= ~ Vsa'(m)X~, ~s a = ~ Vs~( b, c) = Vs~nL(b, c). In parti- ng=0 m=O cular the space, V, defined previously, may in our present notation be written V~. We shall write V A (resp. ~) instead of V~ (resp. ~Bs ~) and likewise V (resp. ~3) instead of V ~ (resp: ~B") whenever there is no danger of confusion. In particular ~3 = Z~], the sum being over all subsets A, of S. We note that for each subset A of S, V A and ~3 A lie in ~(s~ and have equal dimension. Lemma 3- 14. -- If b <p/(p-- I), and A is any subset of S n S" then LA(b, c) =VA(b, c) + ~ H~LA-(~)(b, c+e) + Y~ H~LA(b, c+e). i~A i@S" --A If p t d then L s (b, c) = V s (b, c) + Z H~L s- ~ (b, c + e). i@S The proof is a step by step repetition of that of Lemma 3.4 and therefore may be omitted. We note that the statement of Lemma 3.4 is obtained from this lemma by setting A = o. Lemma 3.x5. -- If (p--I)-l<b<p/(p--i) and if A is any subset of SnS" then LA(b, r =VA(b, c) + Y, D~LA-~i~(b, c+e) + 2 D~La(b, c+e) IEA i~S" --A ~smA'('~)Y~"-0, ON THE ZETA FUNCTION OF A HYPERSURFACE If p I d then LS(b, c) =VS(b, c) + Z D, LS-{'/(b, c+e). ~s This generalization of Lemma 3.6 follows from Lemma 3. I4 in precisely the same way that Lemma 3-6 follows from Lemma 3.4. We must now overcome some of the difficulties caused by the incompleteness of Lemma 3. I5. Lemma 3.I6. -- For each subset A of S and each NeZ+, N>N0, s =~3A + X ~2 A-{~)'(N-t~ + y. ~A,I~--ll ~6A iES --A and the equality remains valid if 2~ A is replaced by V A. Proof. -- Since the left side of our assertion clearly contains the right side it is enough to shows that the right side contains the left side. We show this inclusion for each NeZ+. This is trivial for N----o since ~A't0)--~A'(0)-----vA'(0)-----{O} (resp. ~0) if A+o (resp. A=o). We now suppose that N>o and use induction on N. Let ~e~ A't~I and let P be the coefficient of X0 ~ in ~. Let homogeneous forms Qsl, {Pi}i~s be chosen as indicated by equation (3.20) (with m replaced by N). Let ~=X~-~MAP~/X~ for i~A and ~=X~-IMAP~ for i~S--A. Let ~=O~X0~e~ A and n+l then ~-- (~ + Y, ~3~.) e~ A' IN-l/. This shows that i=l ~A, (N) C~B(A) + ~ ~)~A-- {~}, (:~ --I) + y, ~A, :N --I) + ~A, :~ --I~ ~EA iES --A and the assertion now follows from the induction hypothesis. The above argument can be used for ~A replaced by VIA), since O~ N) may be choscn in V A' IN) instead of 2~ A' (S). The following ]emma is a special case of Lemma 3-: : unless p divides d. n+l Lemma 3.x7. -- For b~I/(p--I), ~n ]~ ~31L(b)=o. n+l n+t Proof. -- The previous lemma shows that V C~/l~~ ---- ~ + 2~ ~3~ IN~ c~ + Y~ ~3~L (b). We may assume that b<p/(p--I) and use Lemma 3.6 which shows that n+l n+l L(b) =V + 2~ ~3~L(b) and thus conclude that L(b) =~ + 2~ ~3~L(b). Lemma 3. ~ i shows i=1 i=l n+l n+l that V ~ L(b)/]~ ~3~L(b) ~ 23/(~n Z ~3~L(b)). Since V and ~ are vector spaces of the i=I i=i same (finite) dimension, this completes the proof of the lemma. Our next lemma is a weak form of Lemma 3. I5 of interest only if p divides d. Lemma 3.I8. -- IrA is any subset of S and if ~[(p--x)<b<p/(p--x) then n+l L~(b)c2~A+ Y~ ~),L(b). 7 5 ~ BERNARD DWORK Proof. -- By Lemmas 3. I5 and 3.16 we have if A# S, n+l n+l n+l LA(b)cVA+ Z ~L(b)ct~A'(N')- b Z ~)~L(b)c~A + Z ~L(b). i=1 i=i i=I To prove the lemma for A=S, let B={I, 2, . .., n} and let ~+l denote the mapping of f~0{Xs} onto f~0{XB} obtained by replacing X,+~ by o. For each ieB let ~ be the mapping ~E~+~,,+~H~ of ~)0{Xn} into itself. For ~en0{Xs} , ieB we have ~,,, l~)i~==~)~nq_l ~, while ~,,+l~)n+l~=O. If ~eLS(b) then from the part of the lemma n+l already proven, there exists ~e~s B such that ~e~ Jr- ~] ~iLs(b). Applying U,+t to this i=1 relation we have o=~,+t~-k ~LB(b ). However equation (3.I8) shows that {=1 ~3~ =~3~-b~ ss andhence ~,+~:qe~3~ and henceliesin ~,e~ ~ ~L~(b), which according i=1 to Lemma 3. ~ 7 (with S replaced by B) is {o} since b> ~ [(p-- ~). Thus ~, +t~ ---- o, which shows that ~e~3s s . This completes the proof of the lemma. f) Exact Sequences. The object of this section is the computation of the dimension of the space V s defined in the previous section. For this purpose we shall need a theorem concerning exact sequences which will be used again in the geometric application of our theory. Let R be a field of arbitrary characteristic and let W be a vector space over R with an infinite family of subspaces indexed by both Z and by the subsets of S={I,2, ...,n+i}. That is, for each t~Z and each subset, A, of S, let W(A,t) be a subspace of W. Let 9a, .-., %+1 be a commutative set of endomorphisms of W with the property (3.2x) 9,W(A, t) cW(Au{i}, t-b i) for each itS, teZ, and each (not necessarily proper) subset, A, of S. For each reZ+ and each pair of subsets A, B of S such that o+A_CB, let ~(t,r;A,B) be the space of all antisymmetric functions g on A r such that g(al, ..., ar)eW(--t--r , B--{ai, a2, ..., at}), it being understood that ~?(t, o; A, B) is to be identified with W(--t,B). For r>i, let 3(t,r;A, B) be the mapping of ~?(t, r; A, B) into ~(t, r--~;A, B) defined by (3.22) (3(t, r; A, B)g)(al, ..., at_l) = Y~ 9ig(a,, ..., ar_l,j) /CA for each ge~(t, r; A, B). This mapping shall be denoted by ~ when no confusion can arise. Theorem 3. x. -- If the sequence 8 8 9 .__). ~(t,r@2,A,B) ~(t,r+~ A,B) ~(t,r;A,B) is exact when r = o for all pairs of subsets A, B of S such that o + AcB then the sequence is exact for all reZ+. 50 ON THE ZETA FUNCTION OF A HYPERSURFACE 5~ Proof. -- We must show that Kernel ~(t, r+ ~; A, B) =Image ~(t, r+2; A, B). We show that the right side is contained by the left side by showing that ~(t,r+i;A,B)3(t,r-b~;A,B)=o. Let g~(t,r+~,A,B), then (~(t, r+ i; A, B)~(t, r+2; A, B)g)(a~, a2, ..., a~) = Y~ %.(8(r+~; A, B)g)(al, ..., a,, j) = ~, ?,?ig(al, ..., a,,j, i)=o iGA ~,iGA by the commutativity of the endomorphism q~ and the skew symmetry of g. To complete the proof we must show: Kernel 3(t, r-t- I; A, B) cImage ~(t, r+2; A, B). This is true by hypothesis for r = o and hence we may assume that r > I. Antisymmetry shows that if A contains just one element then ~(t, r+ I;A, B) =~(t, r+2; A, B) =o for r>_I. The assertion is thus trivial if A contains only one element. We now may assume that A contains at least two elements, that r> I and we use induction on r for all t. Let ge Kernel ~(t, r + i ; A, B). Renumbering the elements of S if neces- sary we may suppose that A={l, 2, ..., s}, s>2 and hence o= ~ q~g(al, ..., at,j) for all (al, ...,a~)~N. With a 1 fixed, say al=I , we consider the mapping (a~, ..., a~+l)-+g(I, a2, ..., at+l) as a function on (A--{I}) ~, indeed as an element of ~(t + I, r; A--{i }, B--{I }) since it is skew symmetric in the <( variables,, a2, ..., a,+ 1 and g(I, a2,...,ar+l)eW(--t--r--I,B--{I}--{a2,...,a~+l}). In this sense the mapping lies in Kernel S(tq- I, r; 1--{i}, B--{I}) and hence by induction on r there exists h'e~(t+ I, r-t- I;A--{I}, B--{I}) such that (~(t-t- I, r-}-I ; A--{I}, B--{I })h')(a2, ..., at+l) =g(I, a2, ..., dr+l) for all (a2, ...,ar+I)E(A--{x}) ~. Let h be the function on {I}� "+1 defined by h(I, a~, ..., a~+z) =h'(a2, ..., a,+2) for all (az, ..., a~+l)e(A--{i}) ~+t. Let a 1 be the set of all (bl, bz, ..., b,+2)eA r+2 such that at least one (( coordinate ,, is i. By anticommutativity, h may be extended uniquely to a mapping (again denoted by h) ofA 1 into W. Furthermore it is easily verified that if (b~, ..., b~+ 1) � A cA I (Le. at least one b,=i) then g(b~, bz, ..., b,+~)= Y~ ~pih(bl, b2, ..., b~,~,j). If (61, ..., br+2) Ea iEA then h(b~, bz, ..., b~ + 2) eW(-- t-- (r + 2), B--{bl, ..., b~+z}) as follows directly from the corresponding property of h'. For each integer m, i<rn<s, let Am={(a~, ...,a~+2)eN+2la~e{~,2, ...,m} for at least one ie{I, 2, ...,r+2}}. Let A~={(aa, ...,ar+l)eA~+llaie{i,2 , ...,m} for at least one iE{I, 2, ..., r + I}}. Suppose (second induction hypothesis) that h has been extended to a skew symmetric function on A,, such that for all (al, ..., a,+2)eA,, and (bl, ..., b,+l)eA ~ wehave h(ax, ..., a,+2)eW(--t--(r+2), B--{al, a.,, ..., ar+~}) and g(b~, ..., b~+~) = Y, ~ih(b~, ..., b~+l,j). If m=s, we are done, i.e. h~(t, r+2; A, B) iGA and 3(t,r+2;A,B)h=g. Hence we may assume that i<m<s. If rn=s--I then 51 BERNARD DWORK since r+ 22 2, g(m+ I, a,~, ..., ar+l) ~---O unless (m+ I, a2, .... ar+l)eA~. Likewise h is defined on A,, and can be extended to an anticommutative mapping of A '+~- into W by letting h map elements of A '+2 not in A,, into o. Thus for (m+ 1, a2, ..., a,+~)eA "+~, g(m+ 2, a2, ..., ar+~) = Z ?~h(m+ I, a2, ..., at+t, i) since this is certainly true if iEA (m + I, a2, 9 9 a,+t) EA~, while otherwise m + i = a2 = 9 9 9 = a,+ t and hence both sides are zero. Thus our second induction hypothesis naay be applied to the case in which i<m<s--I. We know that Y. 9ig(m+ I, a2, ..., a~,j) =o for all (a2, ..., a~)~A ~-1. ~'GA We restrict (a.,,..., at) to (A--{I, 2,..., m}) '-t. For j<m, the second induction hypothesis gives g(m+ i, a,,, ..., a~,j)= ~ 9ih(m+ i, a2, ..., at,j, i) and hence i=1 o= ~ ~ 9ig, h(m + I, a2, . . ., a~,j, i) + X 9ig(m + I, a2, . . ., a,,j). j=li=l /=m+l The anticommutativity of h on A,, shows that o= ~ ~ e~j?ih(m + I, az, ..., a,,j, i) j-ii=l and hence O~ Y~ 9i g(m+l, a2, . . ., a~,j) + 9ih(m + I, a2, . . ., a,, i,j) . i=m+l i=l Since m+ I<S, this last relation may be written o= Y, g'(m+I, a2, ..., a~,j) whereg' i=m+2 is the mapping (a2, ..., ar+l) --> g(m+ I, a.2, ..., a,+l)-- ~ 91h(m+ I, a2, ..., a~+t, i), i=1 of (A--{I, 2, ..., m-t- I}) r into W. It is easily verified that g'ea(t+ 2, r; A--{~, 2, ..., m+ 2}, B--{m+ ~}) and we have just shown that g' lies in the kernel of a(t+I,r;A--{I, 2, . . ., m+ I}, B--{m@ I}) and hence by induction on r, there exists h"e~(t+ r,r + I; A--(I, 2, ...,m+ I),B--{mq- I}) such that a(t+I,r+i;A--{i,2, ...,m+I},B--{m+I})h"=g'. Thus g(m+ 2, a2, ..., a,+l) = k ~h(m+ i, a~, ..., at+, i) + Z ~Y'(a2, ..., a~+,, k) i=1 k=m+2 for all (a2,...,ar+l)e(A--{2,2, ...,I+m}) r. We now define for all (a2, ..., a~+l)e(A--{1, 2, ..., m+ 1}) r+l, h(m+ I, a2, ..., a,+i, a~+2)=h"(a2, ..., at+z) and extend h by antisymmetry to F={(al,..., ar+2)~(A--{I, 2,..., rn}) ~+2 such that at least one a~=m+ i}. (We note that Pt~Am=o while FoAm=Am+t). Thus h is 52 ON THE ZETA FUNCTION OF A HYPERSURFACE now well defined and anfisymmetric on A,,+l. If now (oa,...,ar+l)eA~+ 1 then g(al,..., a,+l)= ]~ ?~h(al,..., ar+l,i ) since this is known by the inttuction hypo- thesis to be true if (aa, ..., a,+x)sA ~ and hence we may assume that (al, ..., ar+l)e(A--{I , 2, ..., m}) ~+t and that at least one of the a i is m + i in which case we may use our relation involving h", our extension of h and the antisymmetry of both h and g. Finally we note that for (aa, ..., a,+ 2)eAm+,, h(a,, ..., a,+~)eW(--t--r--2, B--{a,, ..., at+2} ) since this holds by the induction hypothesis if (al, ..., a,+2)eA =, while otherwise we may suppose a,=m+ I, (a2,..., a,+~)e(A--{I, 2,...,m+ I})" so that h(a , ..., = ..., which lies in the asserted space since h" ~ (t + i, r +i ; A-- { i, 2,..., m +i }, B-- {m + I }). This completes the proof of the theorem. For subsequent applications it is convenient to make available a weaker form of the theorem. Let W now be a vector space over K with an infinite family of subspaces, W(t), indexed by teZ. Let q~l, 9 .-, q0,+t be a commutative set ofendomorphisms of V with the property 9~W(t) cW(t+ I) for each i~S, tEZ. For each r~Z+ and each non-empty subset, A, of S, let ~(t, r; A) be the space of all antisymmetric functions, g, on A ~ such that g(a~, ..., ar)~W(--t--r), it being again understood that ~(t, o; A) is to be identified with W(--t). For r>I, let 3(t, r; A) be the mapping of ~(t, r; A) into ~(t, r-- i ; A) defined as in equation (3. ~). The second corollary follows directly from the theorem. Corollary. -- If the sequence ~(t, r-i- 2; A) ~(t, r+I;A) ~(t, r; A) r~Z+. is exact when r=o for each non-empty subset A of S then it is exact for all For our final result of this section we use the notation of w 3 e. Lemma 3-I9. -- Consider the polynomial, Y"+t(I--Y~-I)"+I/(I __y),+l ____ y~yiyi in one variable, Y. Then for each meZ, dim ~s s' (m) _-- y,,a, and hence dim 23s s ----d-l{ (d-- I) n+l -I-(-- I) n+l . (d--I) }. Proof. -- In the notation of Theorem 3. I, let W----E and let W(t, A)=Ws A'(t) for each tEZ and for each subset A of S. For each i~S, let q~ be the mapping ~-+fi~ of W into itself. It is clear that condition (3-2 I) is satisfied. To apply the theorem we must verify that if o + AcB cS then Kernel ~(t, I ; A, B) = Image ~(t, 2 ; A, B). This is equivalent to the assertion that if h i is a set of elements of W(s -t-t) indexed by A such that h~(MB/XI) and such that Y~ h~f~----o then there exists a skew symmetric set {~ii} 88 54 BERNARD DWORK in W(s -t-2) indexed by A such that h~ = Z ~,js for each ieA and such that ~,.ie(MB/X~X~). iEA This assertion may be proven without difficulty by means of Lemma 3.3 (i), using the fact that the proof of Lemma 3.i shows that (fl, ...,f,):f~+l=(f,,-.-,f~) for r= I, 2, ..., n. Thus, Theorem 3- i may be applied and denoting 8(--m, r; S, S) by 8~ for fixed m, and ~(--m, r; S, S) by ~, we may conclude that (3.33) Kernel 8 r = Image 8r+ 1 for r= i, 2,... Furthermore ~, being the space of all skewsymmetric functions, g, on S ~ taking values in W (m-') such that g(al, ..., a,) aW s - {" ...... ,,}. (m-,), we easily compute (3.34) dim ~ = ("+~) (~(m--~)+ ~--1) Since Image 8r~-~{~,/Kernel 8, and since ~, is of finite dimension, we have (3.35) dim 3, = dim Kernel 8~ + dim Image 8,. Writing [Im 8~] (resp: [Ker 8~]) for dim Image 8~ (resp: dim Kernel 8~), we now have as power series in Y, ~ Y" dim ~'= ~ Y'[Im 8~]-k- ~ Y'[Ker 8,]. Equation (3.33) now gives , = 1 , = 1 , = t (3.3 6) ~ Y~ dim ~ =Y[Im 8~] + (I --~ y-t) ~ Y'[Im 8~]. Since dim ~, = o for r>n + I, this equation is a relation between polynomials and hence setting Y = -- i in (3.36), we have -- [Im 81] = ~ (-- I)r dim ~r. By definition r=l sS.(m) is isomorphic to the factor space /~ s J and Lemma 3.3 (i) shows that 91nW~d(m)=Image81. Furthermore 30=W~s '(") and hence dim ~3 s'(=) = dim ~o-- [Im at]. We may conclude that dim ~s,(~> = ~ (--I)~ dim ~. (a.37) r=O It is easy to verify with the aid of (3.34) that the right side of (3.37) is the coefficient y~ of Y~ in the polynomial h(Y) =Y"+~(I --Ya-~)"+~/(i _y),,+t _y,+a (i +Y + ... + yd-2),+~. OD Clearly dimes s = 52 y~=d-i~h(o), the sum beingover the d th roots of unity. Clearly I--r d-t =--r and hence h(o~) = (-- 1) "+1 if ~. I, while h(i) = (d-- I) n+l. This completes the proof of the lemma. We now observe that dim V = dim ~ = Y~ dim ~], the sum being over all subsets A of S. In particular for A = o, dim ~o ~ = I and this coincides with the formula of the ~4 WS'(m)/tg-InWS'(m)~s ON THE ZETA FUNCTION OF A HYPERSURFACE previous lemma if we replace n -k- I by o. It is easily verified that dim V ---- d", a result that could have been obtained directly by an argument similar to that of the lemma in which the corollary of Theorem 3. I is used instead of Theorem 3. I. Since the polynomial h in the proof of the previous lemma has the property h (Y) = Y+(" + 1)h(Y-1) it is clear that Ye(n+tt-~=Yi for all jEZ. In particular Y,,~ = Y(n+l-m)a for all m, a result which may be related to the conjectured functional equation of the zeta function. We also note that Ya --= o if and only if d<n -k- I, a fact related to the results of Warning. w 4. Geometrical Theory. The notation of w 3 shall be used whenever possible. In this section q =pa, aeZ+, a> i. The first subsection involves power series in one variable, t, with coefficients in ~2. Such a power series, Xy,+t ~, will be said to lie in L(b, c) if ord ym>mb +c for all m~Z+. a) Splitting functions. In [i] we gave two examples of a power series, 0, in one variable satisfying the conditions (i) OzL(� o),� (ii) O(I) is a primitive pm root of unity. (iii) If yP~-----,( for some integer s, s>o then s-1 8~21~'PJ H 0(v pi) = 0(I) =0 j=0 (iv) The coefficients of 0 lie in a finite extension of Q'. A power series in one variable satisfying these four conditions will be called a splitting function. We shall construct an infinite family of such functions indexed by Z*={+oe}u{s~Zls>i}. Indeed the theory of Newton polygons shows that for each seZ*, the polynomial (or power series), yp /p has a zero, Ys, such that ord y+ = I/(p-- I). While there are p-- I such zeros, we shall suppose one has been chosen for each seZ*. For each s~Z* we now set (4" I ) 0 s (t) = exp ty~) P'/p' . 8g 5 6 BERNARD DWORK Lemma 4. x. -- For each seZ*, 0, is a splitting function. Proof. -- In the following the symboly shall denote a parameter to be chosen in f~ subject to the condition ordy = I/(p-- I). For each seZ+, let gs(t,y) = exp{-- (ty)~/p'~}. It is easily verified that g, eL(as, o), where (4.2) a~ = (p-- I)-l--p-"(s + (p-- I) -t) for seZ+, while a~ is taken to be (p--I) -1 for later use. For seZ+ let G,(t,y)= fi gj(t,y). Since ai+t>a i for eachj~Z+, we conclude /=s+t that G,(t,y)eL(as+t, o). Let E(t) denote the Artin-Hasse exponential series E(t) =exp ! ~ot@P~ I. (4.3) It is well known that E(t) eL(o, o) (4-4) E(t)-=-I +tmodt~Q'{t}. Let h~(t,y)=E(ty) and for seZ+, s> I let h~(t,y) =h| (4.5) and so for seZ* (4.6) hs ( t, y) = exp l i~= o ( tY)Pi [P~ t. Clearly h~(t,y)eL(a| o) and for seZ, s>I, equation (4.5) shows that (4.7) hs(t,y) eL(as+t, o). Since a 2-- (p-- I)/p~>o, we may conclude that hs(t,y ) converges for ord t>o. Further- more equation (4.5) shows that (4.8) hs(t,y ) + (ty)P~+l/p "+t - h~(t,y) mod tl+P~+~ f2{t}. Combining this relation with (4.7) we conclude that for s> I ord(h,(i,y) +yPS+l/ps+x--h~o(I,y) ) > a~+1(i +p~+ ~). (4.9) Since h~(I,y) =E(y), we conclude with the aid of (4-4), and (4.2) that for seZ+, s>I ord(h,(I,y)- I) =I/p-- I (4-io) and (4-4) shows that (4-Io) is valid for all seZ*. Furthermore equation (4.6) shows that for seZ ~ (4. xx) log hs(I,y ) = ZyPJ/p j i=0 and hence loghs(i , 7~) =o. 56 ON THE ZETA FUNCTION OF A HYPERSURFACE Since 08(t)=hs(t,'~8), we conclude from (4.7) that 08eL(a,+~,o), and from (4. II) that 0~ (I) is a p~-- th root of unity for some r while (4. I o) shows that 08 ( i ) is a primitive pt~, r 1 . root of unity, r-~ v ypJ Ify p~ =y where reZ, r~1 then as a power series iny, II hs(yPl, y) = h~(i,y) i =o /=0 as may be seen from equation (4-6). Replacing y by ~'8 we conclude that 08 satisfies condition (iii) in the definition of a splitting function. We have already verified conditions (i) and (i_i). Finally we note that Q'(y~) is a purely ramified extension of Q' of degree p--I, while condition (ii) shows that Q,'(Ys) contains a primitive pth root of unity. We conclude that for each ssZ* the coefficients of 08 lie in the field ofp th roots of unity. This completes the proof of the lemma. If geI +trY{t}, let ~(t)= IIg(tPi), an infinite product which converges in the i=0 formal topology of f~{t}. Clearly g(t)=fi,(t)/~(t p) and if q=p~, a>x then a--1 (4- I2) H g(t pi) = ~ (t)/~ (tq). i=l It follows from the definitions that for each seZ* 0~(t) =exp ,i=078,i , (4. x3) 8-1 tP I where vs, j = Z v '/p (4. x4) i=0 It is worth observing that =(p--~)-lpJ+l--(j-+1) ord u C4. I5) In particular 01 = exp (glt). In the application use will be made only of 0oo and 01. b) Let f(X) be a homogeneous polynomial of degree d in n + I, (n~o) variables, X1, X~, ..., X,+ 1 whose coefficients are either zero or (q-- i)--th roots of unity in f~. We may write (4-i6) f(X) = Z A,M~, i=1 where A~=AI and Mi is a monomial in Xl, ...,Xn+ 1 for i=I, 2, ...,p. Let 6, denote n dimensional projective space of characteristic p and let .~ be the variety in ~ defined over the field k of q elements by the equation f(X)- o mod p. For n=o, extending in the obvious way the usual identifications associated with projective coordinates, 60 consists of just one point which is of course rational over the prime field. In any case .~=6, iffis trivially congruent to zero modp. Iffis not trivial modp then ~ is a hypersurface in 6,, to which we attatch the conventional meaning if n~2, while if n = o then .~ is empty and if n = i then .~ is a set of at most d points on the 57 5 8 BERNARD DWORK projective line which are algebraic over k and closed under field automorphisms which leave :he elements of k fixed. For n>o we say that ~ is a non-singular hypersurface of degree d in ~, if the U gf (mod p) have no common zero in ~,,. For n > 2 this polynonfials f, ~-Xl , ..., ~Xn+ 1 coincides with the usual definition, while for n = I it means that .~ is a set of d distinct points and for n-----o, it means that .~ is empty (i.e. f is not trivial mod p). Let ~(53, t) be ~be zeta function of ~ as variety defined over k and let P(.~, t) be the rational function defined by (4.I7) P(5, t)(-1)'=~(5, t)(I --q"t) ~ (I (~--q't) i=O According to the Well hypothesis, if n> 2 and ~ is a nonsingular hypersurface of degree d in ~, then P(.~, t) is a poly~..,mial of degree d-'{(d--I) "+~ + (--I)"+~(d--I)}. Using the above conventions this hypoJ-,c.sis is easily verified fi r n = o, I as for n = o, -- I I ": ~ is e,~,pty I~) l) (4 9 (l--t) ' if 5=6 while if n = i and .~ consists of d distinct points, then ~ ~:- ;~ union oi e disioint sets of points, the i th subset consisting of b/points conjugate ovm ,, ,,,,d each point generating an extension of k of degree bi. In this case d = ~ b i and i=1 (4" I9) ~(~, t)= fi (I--tb/) -1 /=1 Thus if ~ is a non-singular hypersurface of degree d in 6, then ) i/~[I1 ()) if n=o (4.20) P(5, t) = i--tbi /(r--t) if n= I, which is precisely the Well hypothesis in these trivial cases. We know from [I] that the zeta function of~ is related to the linear transformation +oF, where p a--I (4.2I) F(X) = H II 0((XoA, MjPi), i-lj -o O being any splitting function. If 0 is defined as before then since A~ = Ai, (4.22) F(X) = P (X)/P (Xq) where fI 6(A, XoM;) i=1 58 ON THE ZETA FUNCTION OF A HYPERSURFACE ~9 then F takes the If we take the splitting function to be 0s, s= i, 2, ..., +o% form oxply0 I where -~ is the Frobenius automorphism over Q' of a sufficiently large, unramified extension field. Since 0~eL(as+l, o), equation (4-12) shows that Os(t)/O,(tq)eL(pa~+l/q, o). It follows without difficulty that F/X) = F~(X)/~'8(Xq ) eL(pa~+:/q, o) in the sense of w 3, a~+ 1 being given by (4.2). We now recall and clarify the geometrical significance of the characteristic series, ZF, where F is given by (4.21) If gea{t}, let g~' be the power series g(qt) and if g~: +tn{t}, let g~ be the power series gl-~=g(t)/g(qt). n+l If ~' is the << hypersurface )), l-I Xi=o in ~, then by [i, equation (2I)] i=l (recalling that although F now involves a total of n + 2 variables, we are now counting points in projective rather than affine space) (4"2~) ~(5--5', q~):)~F-(--~)n+l(I-t) -( ~)n For each non-empty subset A of S={I, 2, ...,.n+I}, let I+m(A) be the number of elements in A and let ~a be the variety in ~,~(AI defined by the equation in X A, ~Af --= O mod p and let ~[ be the hypersurface II Xl = o in ~,,(AI" Let A A be the power series in one variable defined by ~A , ~ - (- ~)t + re(A) t. __ t) - (- ~),~(A) (4" 25) ~(-~A---~A, qt) = a A ' ~, The precise formulation of A A as a characteristic series in the sense of w 2 does not concern us here, except that we observe that 5s=5, As =ZF. To simplify notation let PA(t) denote P(SA, t) as defined by (4. I7), so that m(A) (4.26) PA(t) (- 1)re(A) ~--- ~(SA, t)(I --q"(A)t)--: l-[ (I --fit). /=0 If B is a non-empty subset of S then (4-27) "~" =A~, (~A--Si), a disjoint union indexed by all non-empty subsets, A, of B. We may conclude with the aid of (4.25) that ~(5~, qt)= I] ~(SA--~A, qt)= 1-] {A~-(-8)l +'n(A)(I--t)--(--8)m(A)}. AcB ACB But an elementary computation gives }2 --(--S)"(A)=8-:(qo:+m(~)--i) and hence BCA ('1-28) ~(~I~, qt) = (I --t) ~ --1 ($1+m(B)_ 1) l-I A2(--~)l+m:h) , AcB 59 60 BERNARD DWORK while equation (4.26) shows that (4" 29) ~ (SB, qt) = { PB (t) (- 1)re(B) ( I --t) -- ~ -1(1 -- ?re(B))}~ comparing (4.28) and (4.29) we obtain (4.3 o) (I--t)PB(t)~(--1)m(B)~--- II A~-(-8)t+m(A) ACB Relations such as (4.3 ~ can easily be inverted by an analogue of the MSbius inversion formula. Explictly if A-+(5 a is a mapping of subsets of S into a mulfiplicative abelian group and if for each subset B of S (4.3 I) G B = I] (5 a, A CB the product being over all subsets, A, of B, then (4.3 2) (~B= 1-[ G(-1) ~(B)-m(AI ACB The inductive proof of (4.3 2) may be omitted since it depends entirely on the well known fact that ~ (~)(--i)~=--~ for each integer r~ I. Applying this to (4.3 o) i=1 and letting B=S, we obtain As(-~)l+"=I-[{Pa(t)~(-1)"Ia)(z--t)}(-1)"-"IAI. Since ZF = As we obtain A 9 s~+" -- (I--t) II PA(qt), (4- 33) LF -- the product being over the non-empty subsets, A of S. (A similar formula appeared in an earlier work [6, equation 2I].) We believe this equation is quite significant since XF is entire even if ~ is singular. Since ~(~A, t) is rational, PA is also rational and hence (4-33) shows that the zeros of ZF and the (q--I)p roots of unity generate a finite extension, ~0, of O.'. With this choice of ~0, the results of w 2 show that the zeros of ZF are explained by the action of ~oF as linear transformation of L (q� if F ~ L (� o). We now fix s~Z*, let F=F~ so that � F=exp H, where s--1 H = Y~ "(s, iX0P~J(XPJ). We shall assume unless otherwise indicated that f is a regular i=0 polynomial (1). Equation (4. I5) shows that H satisfies the conditions of w 3. It follows from (4.22) that a=~oF, may be written (4.34) = while with this choice of H, the mappings D i of w 3 are simply ~-+F-1Ei(~F ). Since qEio~b=~oEi, we conclude for i=o, i, ..., n+I that (4.35) 0coD~ ----- qD, ooc. (1) Thi~ condition on f is equivalent to the condition that ~A is non-singular for each non-empty subset, A, of S. It will be shown that this condition involves no essential loss in generality. 60 ON THE ZETA FUNCTION OF A HYPERSURFACE 6i If X is any non-zero element of Do, let W z be defined as in Theorem 2.4, i.e. Wx={o } if X -~ is not a zero of X~, while Wx=Kernel of (I--X-~) ~ in L(qx) if z- ~ is a zero of multiplicity ~. We note that � ~, F, the D~, H, and the spaces W x depend upon our choice of s. The maximum value of qx is p/(p--i) and corresponds to s=~. The minimum value of qx is (p--I)[p and this exceeds I/(p--~) unless p ~ 2. This minimum value of qx corresponds to s = I. It is assumed in the following that qx> I/(p-- I). Lemma 4.2. -- If A is any subset of S and o<b<qx then Wxta Z DiL(b ) = Z DiWz/q icA leA Proof. -- Let {~} be a set of elements in L(b)indexed by A such that Y~ D,~. = ~cW x. lea Let ~=max{dimW z, dim Wx/q}. It follows from the corollary to Theorem ~.5 that for each icA there exists ~cWz/~ and ~cL(b) such that ~i = ~, + (I-- (X/q) -~ 0~)~;. Thus (I--X -t ~)~' Y, D~ = ~] D~-- Y, D~ = ~-- Y, Di~, which lies in Wx by hypothesis, icA icA icA icA choice of the ~ and equation (4.35). We may now conclude from equation (2.54) that (I--X-I~) ~ ~] Di~q~cWxta (I--?~-~)~L(b) = (I--k-la)~Wx={o}. This shows that leA ~-- ~ BiB i ---- o and hence ~ ~] D~Wx/q. Thus Wxn ~ D~L(b) c Y, DiWz/q and equality iGA ~cA i@A i~A follows without difficulty. Lemma 4.3. -- If A is a non-empty subset of S and {~}icA is a set of elements in W x such that ]~ D~. = o then there exists a skew symmetric set {~%} in Wx/q indexed by A such that icA ~= ]~ D~q~ i for each icA. iCA Proof. -- Let A={I, 2, ..., r}, I <r<n-~- I. Ifr=I then D1~1=o , ~lcL(qx) and hence Lemma 3.IO shows that ~1=o. We may therefore assume r>I and use induction on r. Lemma 3. io shows that there exist ~i, .-., ~:-1 in L(qx) such that r--1 ~, = Z D~. i=1 Since ~rcWx, the previous lemma shows that the ~ may be chosen in Wx/q. Hence r--1 o = Z Di(~+D~.' ) and since ~i+D~'cW x for i= i, 2, ..., r--l, the inductionhypo- i=l thesis shows the existence of a skew symmetric set {~,i} in Wx/q indexed by { i, 2, ..., r- I } such that for i= I, 2, . . ., r--I r--1 ~ + D,~.' = ~] Di~ j i=1 61 6~ BERNARD DWORK We now extend the skew symmetric set by defining ~ql, ~ = --~ = --~%.~ for i = I, 2, . . ., r-- I and ~r.~ = o. It is readily seen that the ~i.j satisfy the conditions of the lemma. Let X be an eigenvalue of ~. We now compute the dimension (as vector space In+ 1 over ~0) of the factor space Wz/i~lDiWz/q. [ In+l \ n+l Lemma 4.4. -- DimtWx/,EtDiWx'q)'= ' = r=oX ('+l) (-- l)r dim Wk/qr. Proof. -- In the statement of the Corollary of Theorem 3. i, let W = L(qx) and for each t~Z, let W(t)=Wxqt , q01=D ~ for i= I, 2, ..., n+ I. The previous lemma shows that the sequence of the Corollary is exact when r = o and hence the Corollary may be applied. In this application ~(o, r; S) is the space of all skew symmetric maps of S' into W(--r)=Wx/qr and hence dim~(o, r; S)= (n+l) dimWx/qr. The corollary may be used to obtain an identity similar to equation (3.36), where ~,=~?(o, r; S), 8r =8(0, r; S) and the assertion follows without difficulty since [ / '~+1 \ dim |Wz/,~t \ - -'= DiWx~q ' ] / = dim ~~ [Im 3'] = ~=0 (--I)r dim ~r" 31+n We can now show that ZF is a polynomial. n+l Theorem 4. x. -- For each X~f~o, let b x = dim Wz/.= * Z t DiWx,q, then z~rl+" = II(i_xt)b~ the product being over all eigenvalues X of ~. Proof. -- Let X be an eigenvalue of ~ with the property that X/q" is not an eigenvalue for any r>I. For each eigenvalue, X', of~, there exists an eigenvalue X with this property such that X'=qiX for some i~Z+. Let aj=dimWzq~ for each j~Z+. The factors of ZF corresponding to terms of type (I--Xqrt), r~Z+ may be written ai cp i Hx(t) = fi (I--t~.qi)ai=(I--tX) i=~ i=O The previous lemma shows that n+J. bxqi = Z (-- I)i("+J)ai_r j=0 and hence (I--~) n+l ~a ai(p i= ~ bxqie~'. i=0 i=O It follows that b~qi (~ i Hx(t) ~n+l _-- (i--~kt)/=0 This completes the proof of the theorem. 62 ON THE ZETA FUNCTION OF A HYPERSURFACE 63 Equations (4.26) and (4-33), together with the known rationality of zeta functions, show that ZSF t+" is a rational function. The theorem shows that the function is entire in the p-adic sense and hence it must be a polynomial. n+, Let ~ be the factor space L(q� DiL(qn). For qx> I/(p--I), we have shown n+, i=' in w 3 that dim ~B =d ". Since 5", D~L(qx) is a subspace of L(qx) which is invariant i--1 under e, there exists an endomorphism Y of g8 deduced from e by passage to quotients. Theorem 4.2. ~,-I-n ZF = det (I--tl), provided qx> I/(p-- I). Proof. -- It is quite clear that the characteristic equation of ~ is independent of D 0 and hence it may be assumed that f20 contains the zeros of det (I--t~). For each non-zero element X of f~0, let ~03 x be the primary component of X in ~ with respect to e. To prove the theorem it is enough in view of Theorem 4. I to show that / n+l \ (4.3 6) dim ~Bx = dim (Wx/~ D,Wx,,,q) Under the natural mapping, J, of L(q� onto 2B, W x is mapped into ~3 x with kernel n+l n+l Wxn ~] D~L(qx), which by Lemma 4.2 is ~] DiWzi q. This shows that dim~3x is at i--1 i--1 least as large as the right side of (4.36) 9 To complete the proof it is enough to show that ~x is the image ofW x under J. To prove this let ~'~, hence there exists r~I such that (I--x-l~)r~'=O. Let ~ be a representative of ~' in L(qx), then n+l (I--X-~)'~ ~ D~L(q� Hence there exists elements ~qt, .-., ~+~ in L(qx) such that i=1 n+, (I--X-~)~ = 2] Di~ ~. i--1 Let ix be the multiplicity of (X/q)-' as zero of )~, then n+I (I--X-'~)r+~ = ~] D~(I--qX-~)~;. i=i i ! Theorem 2.5 shows that there exist ~1, ..., ~,+1 in L(q� such that for i = I, 2, . .., n ~- I (i_qX-,~)~+ r~q~ = The last two displayed formulas show that / n+l ,\ q =o. n-l-I This shows that ~Wx+ ]~ DiL(qx) and hence ~' =J(~)aJ(Wx), which completes the i=1 proof of (4.36 ) and hence of the theorem. 63 6 4 BERNARD DWORK Theorem 't-3. -- The mapping, ~, is a non-singular endomorphism of !lB (and hence X~ ~+" is a polynomial of degree d'~). Proof. -- It is enough to show that K(~B) = 2B, which by Lemma 3.6 is equivalent to the assertion that n+l (4.37) ~Vq- Z D,L(qx)~V. i=l We recall that 0~ depends upon the choice of s~Z* in our construction of F=F~, but .$1+, is clearly independent of s and Theorem 4.2 therefore shows that the degree of/~r dim K(~) is independent of s provided q� I/(p--I). Since dim ~ is also independent of s (subject to the same condition) we conclude that if equation (4.37) holds when s = oo then it holds for all s such that q~> I/(p--I). We may suppose in the remainder of the proof that s ~ oo. Let -: be an extension, which leaves fixed a primitive pth root of unity, to f20 of the Frobenius automorphism over Q' of the maximal unramified subfield of gl 0. Our proof is based on the fact that while F(X)/F(X q) lies in L(p/q(p-- I), o), F(X)/F'(X p) lies in L(I/(p-- i), o). Let +p denote the mapping + with q replaced by p, (i.e. d? = d?~). Let ~p be the mapping X~-+X p" of no{X } onto itself. Let Co, ~o be the Q'-linear mappings of Do{X } into itself defined by We note that ~0 and [~0 are endomorphisms of DO{X} as Q'-space, not (necessarily) as f20-space. In view of our previous remarks we easily verify since F (X)/F'(X v) 9 L( I [(p--I) ) that i ~oL(p/(P-- I)) cL(I/(p-- i)) (4.38) ~0L(iI(p - i)) r i)) and since +poOr= I, we conclude trivially that (4.39) ~oO~o = I. Since .:a leaves F invariant, the definitions show that (4.40) = = Equations (4.38) and (4-39) give L(p/ (p-- I ) ) =~o~oL(p/ (p-- i ) ) c%L(i/(p--i)) r (p-- i ) ) which shows that %L(I/(p-- i)) = L(p[(p-- I)). (4-4 I) Furthermore, the definitions show that for i = o, i, , , ,, n + I (4" 42) ~0oD~ =pD, o%. 64 ON THE ZETA FUNCTION OF A HYPERSURFACE 65 n+l Lemma 3.6 shows that L(I/(p-- i)) =V + ~ DiL(i/(p--I)) ; applying % to both sides of i=l this relation and applying (4.41 ) and (4.42) we find n+l (4.43) L(p/(p-- i)) c~0V + Y, D,L(p/(p-- I)). i=1 [n+l ) n+l (4.44) 0 ,XlD, L(p/<p--i)) c X D,L(p/Ip--II). i=1 Since VcL(p/(p--I)), we may conclude that for j=o, i, ..., a--i n+l a~Vc0@tV + Z D,L(p/(p-- i)) i=l an elementary consequence of which is n+l Vc~V+ Y~ D~L(p/(p--I)). i=1 Since L(p/(p--I)) is stable under 9 and V may be assumed to have been constructed so as to be stable under v a, equation (4.4 o) and this last relation give n+l Vc~V+ ~] D~L(p/(p--i)), i=1 which is the form taken by (4- 37) when s = oo. This completes the proof of the theorem. We have thus shown that iff is a regular polynomial then (I--t)IIPA(qt) (the product being over all non-empty subsets, A, of S) is a polynomial of degree d"; and if s is chosen such that q� (p--I) -t then this polynomial is simply the characteristic equation of ~. Since � =pa~+t/q, equation (4- 2) shows that qx certainly exceeds (p-- I) t if s>i (resp. s>3) when p>2 (resp. p=2). We now propose to investigate the factor Ps(qt) under the restriction that the hypersurface is of odd degree if the characteristic is 2. To do this we now specialize s. Ifpdividesdlets= I. Ifpdoesnotdividedletsbesolargethat qx>I/(p--i) (says=m). For each subset A of S, a ring homomorphism, ~A of f~0[Xs] onto D~0[Xa] was defined in w 3. We now use the same symbol to denote the extension of this homo- morphism to one of f~0{X0, Xs} onto a0{X0, Xa} which is defined by ~A(X0)-----X0. For each subset, A (including the empty subset) of S and for each subset B of A and each real number b, let LA(b ) = ~AL(b) L](b) ={~ELA(b ) such that M B divides ~}. For ieau{o}, let D,, A be the mapping ~--->~AD,~ of f~0{X0, XA} into itself. Let "A be the mapping ~--->~A(,~) of LA(qx ) into itself. Using an obvious analogue of equation 4.35, the subgroup ~ Di, ALA(qx) of LA(qx ) is mapped into itself by % and hence by passage to quotients we define an endomorphism 0c- A of the factor space ~BA = LA(q~)/i~A Di'A LA (q~)" (Thus in the notation of Theorem 4.2, ~B = ~gs, 0~ = 5s)- 9 65 BERNARD DWORK Now let ~B] be the image in 213 A of L](qx). We note that L](qx) is mapped into itself by % and hence YA maps ~] into itself. Let Y] be the restriction of ~A to ~]. For the empty subset, ~, of S, we have ~AF=I, Lo(qx ) =S0, D0,oLo(q~ ) ={o}, O,.~, ~o=~3o~f~0, % is the mapping ~--->+~ of f~0 into itself. Clearly % operates as the identity mapping on f20 and hence det(I--t~) = i--t. Theorem 4.4. det(I--t~) = II det(I--t~), the product being over all subsets, A, of S. Proof. -- Lemmas 3. i i, 3. :5, 3. :7, 3. :8 show that under the natural mapping of LA(qx ) onto ~3A, ~ is mapped isomorphically onto ~I~]. The proof oflemma 3.17 shows that ~BmN =Z~] and here the isomorphism is given by the natural map of L s (~) = L(qx) onto ~3. For each subset A of S, let ~3A be a basis of N] and let ~ = u ~3:. Lemma 3-I3 shows that ~3 is a basis of ~. We use this basis to construct a matrix corresponding to Y. For each coe~3 we may write (by virtue of Lemmas 3.15 and 3. i8) (4.'t5) s(co)e X 9X(co, co')co'+ 2 D,L(qx), r /GS where 9~(~, r 0. It follows from Lemmas 3.1I and 3-~7 that this relation uniquely determines 9~(r r If M A divides ~ then ~(~o)eL~(q� and hence by Lemmas 3- :5, 3- 18, ~: ~(r r A, which shows that 9~(r r =o unless M~ divides ~'. We now order the elements of ~ so that the elements of ~ preceed those of ~B if the number of elements in B exceed the number in A and such that for A 4= B no element of ~B lies between two elements of ~. Let ~ be the matrix indexed by � ~ with general coefficient 9~(r co') and with the elements of ~ ordered as indicated. Let 9~ be the submatrix obtained from 9~ by restricting (co, ~') to ~ � ~. It is clear that 9~A is a square matrix, its diagonal lies along the diagonal of ~ and the coefficients of 9~ lying below 9~ are zero since these coefficients are of type 9~(r ~o'), where ~o'e~ and r is divisible by M B for some B not contained by A. It now follows that (4.`16) det (I--tg"J~) = II det(I--tgX~), the product being over all subsets, A, of S. It follows from (4.45) that det (I--tE)= det (I--t93l). For r e ~A if we apply ~A to both sides of equation (4.45) we obtain s](co) =%(o~) =..~(~z~)~ Y, 9cJl(% o~')~'+ 2~ D,,ALA(q� ). co' G ~3A i GA Since ~A is a set of representatives of a basis of ~B], this shows that for each subset A det(I-- tgJlA) = det(I--t~]). The theorem now follows from (4.46) 9 66 ON THE ZETA FUNCTION OF A HYPERSURFACE Corollary. Ps(qt) = det (I--t~s s) deg Ps = d-l{( d-I)n+~ + (d--i)(--i )n+ 1} Proof. -- Theorem 4-2, equation 4.33 and Theorem 4.4 show that for each non-empty subset B, of S, H det (I--t~) = H PA(qt) A A the products being over all non-empty subsets A of B. This system of relations can be solved for P~(qt)/det (I--t~) by means of equation (4.32). This gives the first assertion of the corollary. The assertion concerning the degree follows from the compu- tation of dim ~3s s (Lemma 3.19) and the proof (Theorem 4-4) that Y (and hence ys) is non-singular. c) Let k (as previously) be the field of q elements and let us extend the notion of regularity (in the obvious way) to polynomials in k[X1, ..., X,+I]. We have verified a part of the Weil hypothesis for a non-singular hypersurface, 9, in ~ defined over k provided d is odd if p = 2 and provided the defining polynomial f ek[X] of ~ is regular. (f = image forf under the residue class map). We now consider the situation in which f is not necessarily regular. Let A = (%) be an (n + i) � (n + i) matrix whose coefficients are algebraically independent over k[X1, ..., X,+L]. We consider the coordinate transformation n+l X~= Z %Yj, j= I, 2, ..., n + I i = 1 and consider f as a polynomial in Yt, ..., Y,+ 1 with coefficients in k(%, ..., a, + 1,, + 1). We easily compute n+l Of rT_vOf -- ~ Xt~,%A1Jdet A i,t=l i where Aij is the cofactor of % in A. Our problem is to specialize the matrix A subject to the conditions (i) det A 4= o (2) f, (det A)f~', ..., (det A)f,'+l have no common zero in ~n. Let U be the set of all A with coefficients in the algebraic closure of k which fail to satisfy these conditions, i.e. U is the set of all A such that either detA=o or J~ (det a)j-~, ..., (det a)f-~+ 1 have a common zero in ~,. It follows from elimination theory that U is an algebraic variety in ~m, where m = (n + i)2 i. On the other hand it is known ([7], Chap. VIII, prop. 13) that the generic hyperplane section of a non- singular variety is non-singular and therefore U 4= ~m. Hence the dimension of U is at most m--I (and hence must in fact be m--i). Thus if k~ is the field of qr elements, the number of points of U rational over k r is no greater than b(qr(m-li--I)/(q~--I) for 67 68 BERNARD DWORK some fixed real number b. On the other hand there are (q"--I)/(q~--I) points in ~,, rational over k,. Thus there exists an integer r 0 such that for each r>ro, there exists a point of~ m rational over k~ but not in U. This means that for each r>r o there exists a coordinate transformation rational over k, such that ~ is defined by a regular polynomial with respect to the new coordinates. For each integer r, let ~, be the zeta function of as hypersurface over k~ and let P, be the rational function defined by n--1 P~(t)(-1)"----~r(t) 1-[ (I--q'it). i=O It follows that for each r~Z, r_) L(t) = II P~(vt~/'), vr=l the product being over all r th roots of unity, ,~. Furthermore if r>r o then P~ is a polynomial of a certain predicted degree m'. If 1)1 is a polynomial then clearly it must also be of degree m', and hence to complete our treatment of Pt it is enough to show that P~ is a polynomial. Since t)1 is a power series with constant term I, we may write V,(t) = fi (I--b,t) (I--blt) i=1 where the b; are distinct from the b~. Consider b~. If Pr is a polynomial then there must be an r th root of unity, v, such that b~v----b~ for some integer i, i < i< c. Let r run through c q- i distinct primes each greater than r 0. By the pigeon hole principle there exists one integer i such that b~v'= b~-----b~v", where v' (resp. ~") is a p'-th (resp. p"-th) root of unity, p', p" being distinct prime numbers. It is clear that v'=v"= t and b~ =bl, contrary to hypothesis. It is now clear that for the treatment of a non-singular hypersurface, the hypothesis that the defining polynomial is regular is no essential restriction. REFERENCES [I] B. DWORK, On the rationality of the zeta function of an algebraic variety, Amer. 07. Math., vol. 82 (t96o), pp. 631-648. [2] J.-P. SERRE, Rationalit3 des fonctions z~ta des varidtds algdbriques, S6minaire Bourbaki, x9592x96o, n ~ i98. [3] A. Wsm, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., vol. 55 (1949), PP. 497-5 o8, [4] E. ARTm, Algebraic numbers and algebraic functions, Princeton University, New York University, i95o-i951 (Mimeographed notes). [5] W. GR6BNER, Moderne Algebraische Geometric, Wien, Springer, i949. [6] B. DWORK, On the congruence properties of the zeta function of algebraic varieties, .7. Reine angew. Alath., vol. 23 096o), PP. I3O-I42. [7] S. LANO, Introduction to algebraic geometry, Interscience Tracts, n ~ 5, New York, I958. Refu le 15 aodt 1961. The Johns Hopkins University.

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Publications mathématiques de l'IHÉSSpringer Journals

Published: Aug 4, 2007

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