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K. Kodaira (1954)
Some Results in the Transcendental Theory of Algebraic VarietiesAnnals of Mathematics, 59
P. Monsky, G. Washnitzer (1964)
THE CONSTRUCTION OF FORMAL COHOMOLOGY SHEAVES.Proceedings of the National Academy of Sciences of the United States of America, 52 6
B. Dwork (1964)
On the Zeta Function of a Hypersurface: IIIAnnals of Mathematics, 80
B. Dwork (1962)
On the zeta function of a hypersurfacePublications Mathématiques de l'Institut des Hautes Études Scientifiques, 12
Formal Cohomology, Part I, to appear
B. Dwork (1960)
On the Rationality of the Zeta Function of an Algebraic VarietyAmerican Journal of Mathematics, 82
(1963)
Some Properties of Formal Schemes, Mimeographed notes of Math
F. Hirzebruch (1956)
Der Satz von Riemann-Roch in Faisceau-theoretischer Formulierung: einige Anwendungen und offene Fragen
Manuscrit regu Ie 21 septembre 1967
Algebraic Curves over Fields with Differentiation
A. Grothendieck (1966)
On the de rham cohomology of algebraic varietiesPublications Mathématiques de l'Institut des Hautes Études Scientifiques, 29
ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES by NICHOLAS M. KATZ TABLE OF CONTENTS PAGES INTRODUCTION .............................................................................. 7 I I. -- The Algebraic Theory ............................................................. 82 Notations and Definitions ............................................................. 82 The (Local) Algebraic 0 Mapping .................................................... 82 Koszul and DeRham Complexes ....................................................... 85 Differentiating Cohomology Classes .................................................... 87 Globallzafion of O ................................................................... 89 The Cohomology of Regular Hypersurfaces ...... ....................................... 9o The Cohomology of the Complement .................................................. 92 Diffcrentlals of the Second Kind ...................................................... 93 II. -- The Analytic Theory ............................................................... 95 Review of Dagger Theory; Estimates .................................................. 95 The Analytic ~) Mapping; Deformations ............................................... 96 The Frobenius; Diagonal Case ........................................................ 97 The Frobenlus; General Case ......................................................... Io2 III. -- The Analytic Theory of the Complement .......................................... io 3 Deformations ........................................................................ io 3 The Frobenius ...................................................................... io 4 The ~ Mapping ..................................................................... io 5 INTRODUCTION Picard-Fuchs Equations. Let X be a non-singular projective curve of genus g, defined over a field K of characteristic zero. Recall that a meromorphic differential is said to be of the second kind if all of its residues are zero, that exact differentials are all of the second kind, and that the quotient space is of dimension 2g over K. (When K = C we may take Yt,.-., Y2g a basis of Ht(X, C), and map this quotient isomorphically to C ~ by o~ ~+ __(Ivlc~ ...,_.Iv~lco);_ thus the quotient is dual to HI(X , C), and so identified with Hi(X, C).) Thus the K-space of differentials of the second kind modulo exact 223 NICHOLAS M. KATZ 7~ differentials provides a cohomology group which is defined over K, and defined in a purely algebraic manner [6 a]. Suppose X depends on certain parameters -- i.e. that the field K admits non-zero derivations. Heuristically, the periods f r are functions of the parameters, and hence Yi are susceptible to differentiation with respect to the parameters. It was first observed by Fuchs that, given co, the 2g periods f r are all solutions of a linear differential Yi equation of degree 2g which has coefficients in K, the Picard-Fuchs equation. This comes about as follows. Let x be a non-constant function, so that the function field of X is a finite extension of K(x); every derivation D of K may be extended to K(x) by requiring that D(x)= o, and thus to a derivation of the function field of X. We call this derivation D~ to indicate the dependence on x. Similarly, dxx is the derivation of K(x) which annihilates K and has value I at x; its extension to all functions is also denoted --. d dx It is easily seen that D~ and ~ commute, since their commutator annihilates both K and x. Finally, D~ acts on differentials by D~(fdg)=D~(f)dg§ or, more simply, by D~(fdx)=D~(f)dx. Because D~(dg)=d(Dzg), D x preserves exactness. The formula res~(D~(o))=D(resp(o~)) insures that D~ preserves the differentials of the second kind. By passage to quotients, D~ acts as a derivation of the differentials of the second kind, modulo exact differentials. On the quotient space, the action of D, is independent of x, because, ify is another non-constant function, (Du-- D~) (fdx) = d(fDu(x)). Thus, the quotient space has a canonical structure of module over the algebra of derivations of the base field K. Hence, for each ~ of the second kind, the diffe- rentials ~a, D~co, D~(o), ..., D~(e0), must be K-dependent modulo the exact differentials, 2g ..., aiD~(~)=dg. Integrating i.e., there are ao, al, a~eK and a function g with ~ 2g i = 0 over the homology class y~ gives the equation Y~ aiD ~ ( ~ = f dg = o. Equivalently, ~=o ~vj v~ let oh, ..., c% be a K-basis of differentials of the second kind modulo exact differentials; 2g each derivation D of K gives rise to the system of equations D~i:5~ a~io~,_ (modulo exact differentials), with a~ieK. The situation for non-singular X of higher dimension is more involved. For a good notion of cohomology over K we must turn to the hypercohomology of the complex ~2 x of sheaves of germs of holomorphic algebraic forms [4]. It should be remarked here that the analogue of Leray's theorem allows the hypercohomology to be obtained as the total homology of the bicomplex (CP(~2 q, 1I)), where ~I={U~} is any covering of X by affine open sets. Let us compute the one-dimensional hypercohomology group when X is a curve. We may take the covering 1I ={U~, Uz} to consists of the complements of two disjoint finite sets of points. The cocycles in C~ 1, ll)| ~ ~t) are the triples, (eoa, t%,f12) , 224 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 73 where each r i is a I-form regular on U~, f12 is a function regular on UtnU2, and e01--oh 4-dflz = o. Thus o) 1 and 0) 2 are differentials of the second kind, whose difference is an exact differential. The coboundaries are those triples (dfl, df2,fl--f~) where each f is a function regular on U~. The mapping (~0~, c%,f~),~ c% establishes the desired identification of this hypercohomology group with the differentials of the second kind modulo exact differentials. We continue with this example, and introduce the action of a derivation D of K. Select functions, x andy, so that at every point peX either x--x(p) or y--y(p) is a uniformizing parameter. Let U 1 be those p where x--x(p) is a uniformizing parameter, and take, for U2, the analogous set fory. It follows [6 a] that D~ and Dy, respectively, are stable on the functions regular on U 1 and U2, respectively, and that both are stable on functions regular on U~ n U 2. Define X : f~l_+f~0, by X(fdx) =fDu(x ). The mapping on i-cochains of the bicomplex, given by (c%, oh,ft2) ~ (D,o~, Dye%, Xo~24-D~f2) , preserves cocycles and coboundaries, and so induces a mapping on the one-dimensional hypercohomology. This mapping gives the action of D on differentials of the second kind, modulo exact differentials. In the higher-dimensional case, analogous formulae will, under restrictive hypo- theses (see (i .6)), endow the bicomplex (CP(f~ q, ~I))with the structure of module over the algebra of derivations of K, and thus allow the differentiation of cohomology classes. This construction is presumably a special case of Grothendieck's " Gauss-Manin connec- tion " (:), but in any case our restrictive hypotheses are satisfied by principal affine open subsets of non-singular hypersurfaces. The differentials of the second kind in higher dimensions no longer give the coho- mology, as they did for curves. Indeed, a closed meromorphic differential e0 on a pro- jective non-singular X is said to be of the second kind if there is an affine open set U, on which co is holomorphic, such that the cohomology class on U determined by co lies in the image of the restriction mapping H*(X)-+H*(U). This mapping is seldom an injection (except for H~(X) -+H~(U)), although for sufficiently nice U, one can determine the kernel (I. i x). Because differentiation of cohomology commutes, whenever it is defined, with the restriction H*(X)~H*(U), the image of H*(X) in H*(U) will be stable under differen- tiation, thus giving rise to a Picard-Fuchs equation for the subspace of H*(U) spanned by differentials of the second kind. Parameters and the Zeta Function. Recall that tile zeta function of a variety, V, defined over GF(q), the field of q elements, is the power series, exp[,~ 1 s )' where N, is the number of points on V (1) (Added in proof.) It is. A general algebraic construction of the Gauss-Manin connection is given by T. Oda and the author in (( On the differentiation of De Rham cohomology classes with respect to parameters )), to appear in Kyoto Journal of Maths. lo 74 NICHOLAS M. KATZ whose coordinates lie in GF(qS). This function, which we will write Z(v, V/GF(q)), is known (Dwork [I]) to be a rational function of %', with rational integral coefficients. Consider now a " variable " variety, Vr, defined over GF(q)[F]. For a special I'0eGF(q a) the special variety Vr. is defined over GF(qe), and thus we may speak of the zeta function Z(-r, Vr./GF(qe)). We ask for its dependence upon F 0. Let us begin by studying the elliptic curve E x given by y2=x(x--I)(X--X) (here p--1 p-t~- t 2 . p+2). Let H(X)-~(--I) ~- Y~ (-fi] V, the Hasse invariant, and denote by N~(Xo) the j=o\J / number of points (including the point at infinity) on Ex~ with coordinates in GF(p~), where it is understood that x0eGF(p~). In this case, it is well known that (mod p) H(Xo)H(XPo)... H(XPo '-') --=I--N~(Xo) and hence :--H(Xo)H(Xg)...H(X~'-')z (modp) Z(-:, Exo/GF(p~)) -= -- I--%" More precisely (I --r (~.0)'17) (I- O) (0)-- INS%-) Z(z, Ez./GF(p'))= (I --%') (I __pS,) where m(Xo)-=H(Xo)...H(X~ '-1) (modp). Clearly, when H(Xo)*O (modp), one reci- procal zero, m(Xo), of Z('r, Ezo/GF(ps)), is distinguished by being a p-adic unit; it has been determined analytically [2 a]. Consider the hypergeometric series, F(I/2, I/2, i, t)= N (-11z)2t j, as function of J20 the p-adic variable t; it is convergent in Ir (here p+2). The function U(t)=F(I/2, I/2, I,t)/F(I/2, 1/2, i, t v) extends analytically to the region [t]<i, [H(t) l--~ {we now regard H as a p-adic function). Let t o be the Teichmtiller repre- sentative of X 0, and suppose I H(t0) I -- x ; then the previously distinguished co(X0) is given ~s-1 s-I by to(Xo)=(--I)--ff-- [I U(t~i). One might say that the hypergeometric series analyti- j=0 cally determines the zeta function of the family E x. Finally, we remark that F( I I ) ~, ~, I, t is annihilated by the differential operator, d 2 d 4t(i--t)~+4(i--2t)~--i, corresponding to the Picard-Fuchs equation, d dx d Y . Dwork's deformation theory generalizes these results to " good " families of hyper- surfaces. Consider a one-parameter family, Xr, of hypersurfaces in pn+l(f~), where [2 is the completion of the algebraic closure of Qp. We envision a defining form F(X, F), 226 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 75 whose coefficients are polynomials in P with ~-integral coefficients, and we suppose these coefficients to reduce (modp) to elements of GF(q), thus defining a family over GF(q) IF]. The specialized hypersurface, Xr0 , corresponding to the special value Foe~ , is O F said to be non-singular, and in general position, if the forms Xc0-~. " (X, F0) , i < i <n -t- 2, have no common zero. By elimination theory, there is a polynomial R(F), with ~-integra] coefficients, such that the specialized Xr0 is non-singular, and in general position, precisely when F 0 is not a zero of R(F). Applying this result in characteristic p, we find that, for ord F0~ o, the specialized Xro and its reduction (mod p) will both be non-singular, and in general position, so long as P o lies in the region ] F o [< I, ]R(Fo) ] = I ; it is in this region that the theory of Dwork applies. This theory associates to each F o with I ro[< I, IR(Po)]--I, a finite dimensional f~-space WS(Fo), and a mapping, a(Fo) : wS(ro)-+ws(r0 ). For ]Po--F1]<I , there is given a canonical isomorphism, C(Fo, Fx) : WS(Po) --->WS(Fx), which gives a commutative diagram .(to) ws(r0) m_+ ws(rg) c(r., r~) c(r. q , r~) + + ws(rd, ,cr,k ws(u ) Observe that when P0r F 0 and IR(r0) l=*, the reduction (modp) of Xro is defined over GF(q0, and the composition, ~(F0r ~(F0q)~(F0), is an endomorphism of Ws(F0). Dwork related the number Ns(F0) of GF(q~)-rational points on the reduction (mod p) of Xr0 to the trace of this composition: I--q s(n+l) (--I) n trace ~(F o )...~(Foe)g(Fo) N~(V0) + i --q~ q~ Further, we may choose simultaneously bases for the WS(F0), in terms of which the matrix coefficients of ~(P) are analytic functions on the region IF]< I, IK(P) I=I. (Recall that a function on such a region is analytic (in the sense of Krasner) if it is the uniform limit of rational functions regular on that domain.) The matrix of C(o, P), on the other hand, has entries which are merely convergent power series for I P[<i. However for I P I < I, the relation = C(o, rq) (o)c(r, o) holds identically in F. It follows, from the analyticity of the matrix entries of a(F), that the matrix entries of C(o, Fq)a(o)C(o, F), which apparently exist only for ]FI<I, 227 7 6 NICHOLAS M. KATZ are, in fact, analytic throughout the region ] F I< i, I R(F) I = i, Further, the uniqueness theorem for analytic functions shows that the relation ~(re)... ~(r%(r) = C(o, r%(o)C(o, r) -1 is valid for [rl<,, IR(r)[=~. Thus, the matrix of C(o, F) provides the analytic continuation of the zeta function from F=o to the region I F]<~, ]R(F)[=I, in much the same way as the hypergeo- metric series analytically determines the zeta function of elliptic curves in the family E x. The analogy goes even deeper, for the matrix of C(o, F) satisfies a differential equation. This arises as follows. The generic space, WS(p)| is, in a natural way, a module for ~-~, by means of an action ~r, which arises formally as the twisting c(r, o) ~r = C(o, r). ~. 0 O where~-~ operates in WS(o)| through the second factor. Because ~ annihilates the f~-space WS(o), ~r annihilates the f~-space C(o, P)(WS(o)). Let us write W 0 and W r for the column vectors whose components are, respectively, the f~-basis for WS(o) and the f~(I')-basis of wS(p), by means of which our matrix representation of C(o, F) is given. As ~r acts on the f)(P)-space, ws(p)| we may write ~r(Wr) =B(F).W r where B(F) is a matrix of rational functions. To avoid confusion between matrices and mappings, let C(P) denote the matrix of the mapping C(o, F); then C(o, F)(W0)=C(P).Wr, and we have the equation oc(r) o = ~(c(r).w~) = 0--b- .wr + c(r)~.w,. /oc(r) = ~--oU + c(r)s(r)) .w,, ec(r) whence -- C(F)B(F). 0F In [3] Dwork computed the Picard-Fuchs equation tbr the family of elliptic curves, X3+Y3+Z3--3FXYZ , and found it to be (in suitable bases) ~P(F) 0r _B(r)P(r) where P(F) is the period matrix of differentials of the second kind modulo exact differentials. It should be remarked that the result is of an algebraic nature: the generic 228 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 77 space W s (P), the operation ~r, and the Picard-Fuchs equations are all defined over f~ (F). The present paper grew out of attemps to identify the differential equation for C(o, P) with a Picard-Fuchs equation, in greater generality. This is achieved by Theorem (x .I9). -- Let X be a projective hypersurface in p~+l, defined over afield, K, of characteristic zero, which is nonsingular and in general position. Denote by X ~ the open subset where no coordinate vanishes, and by H"(X, K) and H"(X ~ K) the n-dimensional cohomology groups of X and X ~ respectively. We regard H"(X, K) and H"(X ~ K) as modules over the algebra of derivations of K, as explained earlier, while the K-space W s, associated to X by Dwork' s theory, admits the derivations of K by means of the ~ operation. There is an isomorphism 0 of W s with the image of H"(X, K) in H"(X ~, K), which is an isomorphism of modules over the algebra of derivations of K. Furthermore, the kernel of H"(X, K) ~ H"(X ~ K) is reduced to zero for n odd, and is one-dimensional for n even (and n>o). We should point out that the matrix C(P) is the transposed inverse of the matrix denoted by the same symbol in Dwork [3, P. 262], which arises by passing from the " dual space " at o to the generic dual space at F; thus that matrix is the transpose of the (suitably normalized) period matrix. (Also the B matrix is the transpose of its analogue there.) Explicit Computations. To explain how Theorem (I. I9) comes about, it is necessary to examine the spaces of Dwork in some detail. We fix integers n>o and d>o, and define ~o to be the K-linear span of monomials Z~"X~l...X~+~ with w~Z, dwo=~Wi>O. contains certain subspaces of interest to us: ~o, where /2)02 I 2,~ where all w~>o s +, is &,~176 .W + ~c~as, where all wi> I. We fix a non-zero constant, % and corresponding to each defining form F, of degree d in X1, ..., Xn+2, we define twisted operators on ~f a 0 r:Z OF Dxl = exp (--r:ZF). X~. exp(~ZF) = Xi--- -t- Dz = exp(--r~ZF) "Z~ =OZ "exp(rcZF) Z ~O + ~ZF. This construction is rational over any field K, which contains ~ and the coefficients of the form F. For each derivation D of K, we define the twisted derivation ~D of ~ as K-space by ~D = exp(--r~ZF). D. exp (r~ZF) = D q- r~ZF D. ,229 78 NICHOLAS M. KATZ The fundamental problem is to construct a generic mapping from ~0 to H"(X~ Corresponding to each element ZX w is the differential form, regular on X ~ given, in local coordinates, by X ~ d(Xl/X,+2 a(X./X,,+~) A...A aF Xl/X,~+2 X,/X,+~ Xn+i OX~+ 1 For the generic F, given by iw~=eAwX'~ with the A~ independent, ~lw is given by =-7-. n u =ZX ~, so that .~4 '~ considered as module for the derivations of K, is spanned by its elements of Z-degree one. Thus, generically, there is at most one homomorphism from ~0 to H~(X ~ which is given as above on elements of Z-degree one and which respects the derivations of K. That this map exists, and that it annihilates D z ~-t-~ Dx~ L~~176 is almost the content of Theorem i. We regard H'(X ~ as coming from ~ CP(f~q), for a suitable covering; p+q=n in Theorem I we examine C~ and in Lemma (~ .8) we turn to Cl(g~-l). In this way we obtain a surjection | : .s176 Dx,~~176163 ~ ~ H"(X ~ with @(.oq ~s) lying in the image of H"(X) in H~(X~ We then compute the Betti numbers of X ~ and Pn+~ -,co ~P~+* is the open ~n+2 -z~ k~n+2 subset of P~ + 1 with all X~.=~ o), making strong use of the assumption of general position and the explicit formulas of Hirzebruch [5]. Then we construct an isomorphism, suggested by [9], /(~ I.-ln+l/pn+ 1 3r : s Dxl ~~ -> ** ~-~+2--~ J by defining (Z oXW) = x d(xjx § (--=)~~ F ~~ XjX~+2 ^"" X,~+l/X~+2 " It is clear that ~(_~os) is the image of H'~+I(P n+l X) in ~'+lrP~+~--~c~ -- ** t--n+ 2 ~ 1. OF The regularity condition, that the X.~ have no common zero, insures that ~~176176 of Z-degree one. We explicity compute in local coordinates x~ --X~/X,+~, f- X~+~ x w dx 1 (zx w) - ------A...A -- ...A-- -- x,+ 1 Of xl xn f f xl Xn+l 0Xn+ 1 and hence (9 (ZX w) -= Residue(N(ZXW)). 230 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES The final result is best expressed by a commutative diagram H,~+~(p.+~_X ) ,e,idu8 > H'(X) ~restrietion ~rc striction l-In + 1 {On + 1~ restriction ]~n + 1 {pn + 1 residue O > ~ k--n+2/ > ~ k--n+2--X O) > gn(x O) - > o T ~ o , (ZF) > ~e~163 ~ . ~~163163176 ) ---> o o > (ZF) > s176 ~176 > W=~q~~176176163 ~ > o U U s ~ n ~ Dxis176 +) --+ wS= s176 s n (Dz-g ~ + ~ Dx, Ae ~ +)) The P-adic computation of the Zeta Function. Let F(X)=F(X1, ..., X,+2) be a form of degree d over GF(q). Denote by N~ the number of zeros of F in projective space rational over GF(q~), all of whose coordinates are non-zero; this N~ is easily expressed as a character sum. Take a nontrivial character Z~ : GF(q~) +-+~* of the additive group of GF(q ~) with " p-adie values " For each x=(xl,..., x,+~) rational over GF(q~), q~, if F(x)=o *~ o, if F(x)+o. Hence x ) N;= (qS-- )n+ 2 + :Cz (z (x) ) } where the sum is taken over z, xl, .-., xn+2sGF(qS) *. It remains to construct )8. Fix a non-trivial character )0 of GF(p), and put Z~=zo.tr where "tr" denotes the trace mapping tr :GF(q ~) -+GF(p). Explicitly, for beGF(q*), )~(b)=~ tr(b), where ~ is a fixed p-th root of unity, and tr(b)=~b po, the sum taken over v = o, ..., s logp(q)-- I. With this in mind, we fix an element rc of f~, with rc p-1 =--p, and define a power series 00(Z) by setting 00(Z)=exp(nZ--nZP). Then [ I] r--p I) 00(Z ) has p-integral coefficients, and converges for ord(Z)> p2 9 2) 00(I)----~, a primitive p-th root of unity. 28l 80 NICHOLAS M. KATZ 3) for o~e~ with :r e, the formal identity, V--:[ H O0( P;Z) = 3 0 specializes at Z=I to v--1 H 0 (~pi~__ ~,~+ap+...+~pv-1 j=O Ok ] -- "~ Observe that the left-hand side is the value of 00(Z)0o(ZP)...00(Z pv-1) = exp(~Z--r:Z p') at Z-~:r Thus, if ~ = ~q' is the Teichmfiller representative of beGF(q~), we have x~(b) = the value at Z=~ of exp(~cZ--~Zqs). Hence, if we suppose F(X)=F(X1, ..., Xn+2) to have coefficients which are (q~--i)--st roots of unity in ~, and if :r ~1, ---, ~+2 are (qS 1)--st roots of unity, the value at Z=0r X=~ of Z~(EF(~))=[ exp(r:ZF(X)--rcZCF(Xr where ~=(~1, ..., ~n+2), and :r is reduction (modp). Thus, exp(nZF(X)--nZq F(X ))), where, in the sum, Z, Xl, ..., X,+ 2 vary independently over the (q~--I)--st roots of unity in ~. We next express this sum as a trace. Denote by L(o +) the space of power series Y~A~Z~oX ~ which satisfy n+2 I) dWo=lWl= Z y. Wo>O. ~) For some constants b>o, and c, ord A~>bwo+C. The endomorphism ~q of L(o+) is defined by +q(ZwoXW)= '~ if not.if each wi=qv ~ For each element HeL(o-t-), we write ~bq.H for the endomorphism of L(o+) given by This operator is " of trace class ", and [I, 9] (q-- I)~+3tr(~bq. H) = Y~H(Z, X) where Z, Xl, ..., X~+ 2 are independently summed over the (q--I)--st roots of unity in ~. In particular, this trace formula may be applied to 0~=q~q.Hq, where Hq=exp(r:ZF(X)--r~ZqF(xq)). It is immediate that the s-th iterate ~ of ~ is nothing other than +r162 Hence, combining the above formulae, we have q~N; = (q~-- I)~+1 + (qS I)~+2tr(od). 232 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 8t This is the connection of 0~ with the zeta function. It is convenient to consider a one- sided inverse [5 to ,, given by [5 = exp(r~ZqF(Xq) --=ZF(X)). qb, where ~(ZW'X ~) : Zq~~ qw. Connections with Formal Cohomology. We wish to relate the operators 0c and [5 to certain operators arising in the formal cohomology theory of Washnitzer and Monsky. We begin with a special affine variety over k = GF(q); this is, by definition, an A=k[x,-r]/I-, where ]-= (f(x), i--'~g(x)~): here x=(x~,..., x,,+l ). Fix algebra *~ ga. a complete, discrete valuation ring R, of characteristic zero, whose residue field is k, and denote by R ix, ":] + the subalgebra of R [ ix, -r]] satisfying a certain growth condition. Take liftings, f and g, of f and g, to R[x], and define A +:R[x, ":]+/I, where I=(f, I--'rg~). This algebra is independent of choice ofliftings, up to non-cano- nical isomorphism. Fixing these choices for a moment, every map ~0 : A-+B of special affines may be lifted to a q~+ :A+--~B +. Passing to continuous R-differentials, the induced map, ~0 + :~(A+)|174 is determined, up to homotopy, by ~. In this way, the deRham cohomology (i.e., the homology of the complex i2(A+)| becomes functorial in A. In particular, A + admits an endomorphism Fr, which lifts the q-th power mapping. Fr is an injection and A + is a finite module over Fr(A+). Define an additive mapping, : A+|174 by requiring that the composition, qnFro~, be the trace mapping from A + to Fr(A+). We first consider a special affine subset of an irreducible projective hypersurface, F=o. Namely, let f(xl, ...,x,,+l)=F (. ..,XjX~+2, ...), and consider the locus Of =o, with << plus >> algebra A +. Themapping| I f(x) :o, I --~:x 1, 9 .., x~+l Ox~+l extends by " continuity " to a map of L~ +) to A+NQ, where L~ +) is the subspace without constant term, and our result is the commutativity of the diagram q~> L0(o § L~ +) r~> A+| A+| where Fr is that lifting of the q-th power map with Fr(xi)=x~ for i= I, ..., n+ I. 11 82 NICHOLAS M. KATZ Somewhat more straightforward is the case of the " complement ", i.e. the algebra R[x, v]/(i--.~xa, . .., x,+lf ). Again, we select the Fr with Fr(xl)=x] for i=i, ..., n+I. Our main results are the commutativity of the diagrams L0(o+) ~) L0(o+) L~ q~) L0(o+) A+| q,> A+| " A+| Fry A+| ' Here the mapping ~ is the one explained earlier. The image of~ in A+| consists of those functions regular on the larger open set {xeP "+1, F(x)+o}, while, on the level of differential forms, the image of ~ in f~'~+I(A+)| consists of those forms, meromorphic on {xeP "+1, F(x)+o}, whose only singularities are, at worst, first order poles along the coordinate axes. Working with the form X 1 ... X,+2F(X) would allow a surjection, : L~ ~ A+| but at the cost that the differential operators Dxi for this form are difficult to analyze, even under the most favorable hypotheses on F. Difficulties of this sort prevent the direct application of Dwork's work to prove the finite dimensionality of any ~ plus - cohomology groups. However application of (2.I5) to the form X1... X,+2F(X ) is easily seen to imply the trace formula obtained by Reich and Monsky for the mapping + of A+QQ, namely N= (q-- i)"+ltr(+) where N is the number of points (x), rational over GF(q), where xl...x,+if(x)+o. I wish to thank my teacher, B. Dwork, for so very much, and to acknowledge many helpful discussions with G. Washnitzer. ALGEBRAIC THEORY Notations. -- We work over a field K of characteristic zero, and fix an element ~eK*. Let F(X1, ..., X,+2) be a homogeneous form of degree d over K, defining a non-singular hypersurface X. Denote by X ~ the open subset where no Xq. vanishes, by U(b, i), OF b+i, the open subset where x~.~x~+O , and by U~ i) the intersection U(b, i) nX ~ Any derivation D of K extends to a derivation of each coordinate ring f~~ i)) by requiring D(XflX~.)=o for i+j, j+b over U(b, i); when there is no ambiguity we will denote this derivation also by D. 234 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 8 3 We define ~ to be the K-span of the monomials Z~X~ 1 5(~'+~ (hereafter ..... n+2 written zax ~) for which da = ~bi, a>o, .oq ~~ the subspace " divisible by Z ", i.e. with D 0 OF a> i. (Observe that the b i may be negative.) Define operators x~=X~.=;;-~ +TvZX~. 0 6x~ X~ D z = Z ~-~ + rcZF. Any derivation D of K extends to a derivation ~D of the K-space ~ (or ~o0) by setting ~D=D+rzZF D, where D acts only on coefficients and F D is the result of applying D to the coefficients of F. Formally ~D=exp(--TvZF)D exp(r:ZF), while Dxi=exp(--=ZF)X~.~-~.exp(rcZF), whence each ~v commutes with all Dx~ and Dz, and ~z commutes with ~Z' if D commutes with D'. Theorem (~). -- There exists for every nonsingular F and for each (b, i), b#i, a unique K-linear mapping O : ~Lf ~ ~ ~~ i)) satisfying: a) O is compatible with specializing the coefficients of the form F; b) O. ~D = D. O for every derivation D of K ; OF c) o(zxw)=xw/X ox ; OXk = ~jk if j # b, i and k # b, i " d) ODxj=XJ~xO, for j#b,i, where xj=N~/X~ and O--~ e) | O(Dx,~~ Proof. -- We begin by constructing O for the generic form F =F~AwX w where to the A w are independent variables. To fix ideas we work over US(n+1, n-4-2) and first content ourselves with verifying b) for the special derivations ~ (we write ~A~ for ~0/eA~)- Uniqueness follows from b) and c), since ~,%(zax~)=r:za+lx ~+w, so that every monomial of ~z ~176 is obtained from one of Z-degree i by successive iteration of any ~A~ ; clearly for a fixed Aw, that expression is unique. As to existence, the last remark shows that for each Aw, there is a mapping Ot~ satisfying b) and c) for A~. We first show that OA~ is independent of A~. We have a b--aw d 7~aza+lx b = ~Aw(ZX ) =~(ZX a b--a. ), so we must show, for f=FIX~+2, ...... \ /( ) i.e. that tbr every monomial x c in the x~ and x~ -1 we have (~ta/( ~ft a X' ~= (~ta/( ~]z" ~ a X c ) ...... ,,)',. NICHOLAS M. KATZ For a-=x this is ~Xn+t OXn+l a ( x aX,~+l] c,,+lx< ' a ( xC&,,+tl -x c a~(xn+~) 8A~ OA,~ OA w OA~, Now by induction /i / oz i ozv o\ ,c] "]= ~ I a+l - Oj~ \ 0s ) \ OXn+l / k,5 ) Let F denote the coefficient of X~+ 2 in F, i.e. the constant term of f. As ~r commutes with the Dxi , and ~ commutes with the Xi~x; for i=I, . .., n, we are reduced to showing d) for a monomial ZX b of Z-degree I, say for Dx. 0(Dx,(ZXb))=O(b~ZX')+r:Z'X X1~)2 b ~F = O(blZX~) + ~ O[X~Z ?-~I.X XT+ 0 =_elx+aX,,+l a [ +ax++l~ -- Ixlx ---~ (we write x~= b -~ X Xn+l) OF OF \ Ox 1 ] Ox,,+t 0 (Ox,,+l t Ox,,+, O =-~'x~ or x~'~2~ 7r-e- o< ar (x'x~ ox,\ ~V/=*~ ~ As for e) we first use b) to reduce to showing O(Dz(Xb))=o, but this is f~ O(rcZFX b)-- = o. Similarly it suffices to compute 0f Xn+l OXn+ 1 OF b,+tx b + ~ (x~) O. Dx,+i (ZX b) = 0(b,,+l ZX ~ + ~r (ZXbX,+~ aX,,+--~ X7-~2)) -- of Xn+l OXn+ 1 We now regard this generic definition as providing formulas for O in terms of the coefficients of the defining form. Clearly it remains only to demonstrate that b) holds for all forms F. Consider F(X)§ 2 over K(X) where X is transcendental over K. Extend D by D(X)=o a by OK a and O~ 0X =o, whence D and 0?~ commute, as do ~D and ~x, whence 236 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 85 O~D (Zw0Xw)= 0~{o__l~D ( ZX ~ X_(wo_l)d] (I ~ ] w~ \~~ ,~+2 ]= ~/ |176 while D0(ZwoX '~) = \~ ~] . D. 0(ZXWXn;~~ ; hence we are reduced to Wo= I. f xW ) -o x O'~D(ZXW)=O~),(ZFDXWXn-:2)=~ \Xn+l~f/~xn+ll =~ (Xn+l D(Xn+I) )' while D@(ZXW)=D =--D ) and as D and--commute, both x. 1 ef 1 0x ' ~x Ox. + ~ sides are D -- ~ ~ D(Xn+I). Q.E.D. Xn+l Xn-bl Corollary (x. I ). -- Over each U~ i), 0 has a natural extension to a mapping from the Koszul complex on ~o/(D z ~ + Dx b ~qo0) with operators the Dxj , j + i, b to the de Rham complex a(U~ i)) of differential r gular on U~ i). Proof. -- We define the Koszul complex and give the proof in the following section. Koszul and de Rham complexes. Let ~1,..., % be commuting endomorphisms of a vector space V/K. Write S={I, ..., n}, /kS the exterior algebra of the free K-space with basis the elements of S. On Horn(AS, V) the Koszul boundary may be defined by ieS where zeAS, toeHom(AS, V). Define 9 on AS by linearity and the requirement that for a monomial zeAkS, 9 z is the monomial of An-kS with zA *z = I A 2 A... An, and let 9 act on Hom(/kS, V) by defining .~o(.v) = c0(x). Let L/K be a function field in n variables with separating transcendence basis xl, ..., x,. The monomials in the dxJx i form a basis for f~K(L) as L-space, which is thus isomorphic (dxi/xi~i) with AS~L, i.e. with Hom(AS, L). Thus exterior differentiation d induces a coboundary on Hom(/kS, L), while the xi--, i= i, ..., n provide the Koszul boundary, dxi Proposition ( I. 2 ). -- *. ~= d... Proof. -- For a monomial ~ and ieS, *~----iA *(~ A i) SO long as ~ A i+o. Hence d(*~)(*,)= ~xi~ ( *o~*(~^ i) ) )= ~xi~ (o(, ^ i) )--=(~)(~)-=,(~o~)( .~). Q.E.D. 287 86 NICHOLAS M. KATZ Globalization for n-forms. Lemma (x.3).--Let b+ i, and say {x, ..., n+2}={kl,..., k,,, b, i}. Define (o( b, i) on U(b,i) to be (I, . .., n@2 ) X~ +ld(xkl/xr .^d(Xk,/Xi) sgn kl ' .-., k,,, b, i OF x xb xk, Then on U(b, i) n U(c,j), o~(b, i) =~(c,j). Pro@ -- Suppose first that {b, i}---{c,j}; to fix ideas take b=n+x =j, i=n+2=c, so we are asserting that ) d "=n+2 A d\xn+2] -- n+e 0Xn+x ~ d~x--~+l) 9 Now X~d(X~/X~)=dX~--(X~/X~)dX~, so we want OF OF +X~+ 2 ----] dXl^ ^ dXn-- X~+l 0X~+ 1 ~X~+2/ " . . OF OF X~dX~ a... ^ dX~, ^... ^ dX, -- dX,~+l + dX,~+~ /x (--I) ~-1 and the right side ( as dF = ~ OX--~ ~] OF dXj^ Y~ (--I)~XjX~A...^dX~a ^dX,=-- ~]= X~ dX~^...^dX,; now apply the Euler relations. In general, given (b, i) and (c,j), we compare both with (b,j), and hence, by the first part, we are reduced to comparing (b, i) with (c, i). To fix ideas we compare (n,n+2) with (n+I, n+2), and write x~=XJX,~+2,f(x 1, ..., x,+l)=F(x 1, ..., x,,+x, I), dxl A . . . ^ dx. dxx ^ . . . A dx,_ x ^ dx, + x so that our assertion becomes -- which follows, as n+l ~7c ef Of df= ~ -~ dx~=o. Q.E.D. dx.+i Ox~ = ~ Ox~ Lemma (x .4). -- Let L/K be a separably generated function field in n variables with sepa- rating transcendence basis xl, . .., x, ; let D be any derivation of L trivial on K. Then on f~K(L), D=dX+Xd, where (for uq ..... ik~L) X(uq, ,Jxq^...^dX, k)=U , 'k X (--I)J-XD(xi)dxqA...^dx,j^...^dx, k. ''" I' "'" j=l Proof. -- We readily compute that for r X(co ^ z) = X(r A'~ + (-- I)Jc0 ^ X(Z), whence it follows easily that (dX+Xd) (r ^ v) = (dX + Xd) (co) ^ z -t- r ^ (dX + Xd) (~), so that dX+Xd is a degree zero derivation of f~K(L) which commutes with d, hence is determined by its restriction to L, and (dx+xd)(xi)=D(xi). Q.E.D. 288 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 8 7 Differentiating Cohomology Classes whith respect to Parameters. Let V be a non-singular variety defined over a field of characteristic zero; a theorem of A. Grothendieck [4] gives an algebraic method of computing the cohomology of V as complex manifold. Namely, fix a covering {Vi} of V by affine open sets; denote by f~q the sheaf of germs of regular algebraic differential forms, and by CP(Y~ q, {V/}) the (alternating) p-cochains for the nerve of the covering {V~} with coefficients in the sheaf f~q. When the context is clear, we will simply write CP(f~q). These CP(t2 q) form a double complex, with d: CP(f~ q) -+ CP(f~ q+l) the usual exterior differentiation and 0 : CV(f~ q) -+ CP+I(f~ q) the nerve-coboundary; for co(i0, ..., ip)eCP(f~q), recall (&o)(i0, ..., ip+l)-=~(--I)Jo)(i0, ..., 1~, ..., ip+l). Then the cohomology of V is the homology of the total complex (whose term of degree n is E CP(Y~q)) under the diffe- p+q=n rential which acts on CP(gl q) by 0+(--1)Pd, which we will write A. Now consider a derivation D of the field of definition of V; we suppose chosen for each V/an extending derivation D/at the function field of V in such a way that the coordinate ring of each intersection V/on... nVip is stable under D/o , ...,Dip. We recall (lemma (1.4)) that Di--Dj=dXo+X/jd where X/~(udxi ^ . . . ^ dx,)----u~ (--I)Z(D/--Dj) (x,) dxi ^ . . . ^ dx~^ . . . ^ dx,. Finally we define X: CP(a q)-+ CP+t(a q-l) by (Xo))(i0, ..., ip+l) =X,o,/,(o)(il, ..., ip+l) ) where we have i0<i1<..-<ip+l, and D : CP(~ q) -+ CP(f~ q) by (Do)) (io, ..., i,) = D/.(o)(i0, ..., i,)) where again i0<i1<.-.<ip. Finally we will write D for the operator which is D + (-- I) p + iX on C p(f~q) ; this is " differentiation with respect to a parameter " Lemma (x.5).- The operators D=D+(--I)P+lZ and A=0+(--I)Pd commute. Pro@ -- Let o)eCP(f~ q) ; D(Ao)) = D(Oo) + (-- I )P do)) = D0o) + (-- ~)' Ddo) + (-- I )P'AOo)--~,do) while A(~)o))=A(Do)+(--Q'+~Xo))=ODo)+(--I)'+lOXo)§ Compa- ring components on both sides we must show that Ddo)---:dDo) in CP(~2q+I), that DOo)--Xdo)=ODo)+dXo) in CP+I(Y2q), and that X0o)+OXo)=o in CP+~(~q-1). The first point is clear, as each D~ commutes with d. For the second, (D0o)" 0Do)) (i0, ..., i, +a) = D~.o)(il, ..., i, +1) + +E (--i)JD/oo)(io,...,~, ... ip+l)--Dio)(il,...,ip+i)-- j>l -- ~ (--I)~D~~ ..., ~, .. iv+l)=(Dio--D,~)o)(il, 9 i,+1), ~_>1 "' " "' while (dXo)+ Xdo))(io, ..., iv+l)=d)t/o,qo)(i~, ..., i~+l)+ X/o,/flo)(i~, ..., ip+l) 239 NICHOLAS M. KATZ (where io<...<ip+l). Finally (XO~o+ ~Xco) (io, ..., ip +2) =-X~0, il((&o) (il, 9 9 i, +2)) + 2 (-- i)~ (X~) (io, ..., ~, ..., ip+2) = j>o =Xi,,ilo~(il, . .., ip+z)-~-Xi,,i~j 2(--I)J-l(o(il, .. ., ~j, . . ., ip+2)-}- --~%i,,i oa(i2, ..., ip+2)--Xi,,i oa(i2, ..., ip+z)-~-3~>~ (--i)JXio,Lr ..., ;, ..., ip+2) , and it is clear from the definition that Xio,i ----Xi~ -t-Xi~.~ ,. Q.E.D. Lemma ( i. 6). -- Suppose for every derivation D of the field of definition we have chosen the various extensions D~ over the V~ in such a way that for derivations D and D' we have [D, D']I=[D 0 D~]. Suppose further that for each pair D, D' and each i, j, the ratio (D~--D~)(h)/(D~--D~) (h) is independent of the choice of function h (whenever it is defined) and of the choice of (i, j). Then the assignment D ~ ~) is a Lie homomorphism from the ring of derivations of the constants to the ring of additive endomorphisms of the total complex of CP(ft q, {V~}). In particular, if D and D' commute, then I) and I)' commute. Proof. -- Let eo~C'(f2q); we readily compute [D, D']r with ~C'(~q), ,ECP+~(f~ q-~) and yeC'+2(f2q-2); here ~(io, ..., i,)= [D,~ D~.]o~(io, ..., i,), T(i0, ..., ip+l) = (--i)'+t(--X;.,, Dh--D~oX,,,q+X,,,, D~,+D,~ ..., i,+t) , and y(i0, ..., ip+2)=(--Xi.,, X;~,,,+;~~ X,,,i,)(o)(i2, ..., ip+2) ). We first show y=o; write o~(i~, ..., ip+z) as a sum of terms of the form udhl^... Adh r where h 1, . .., h, are functions; it is sufficient if X;.,,.(dh~)X,.,r for every a and b, and as X;.,,,(dha)-=(D~~ this is insured by the hypotheses. Turning to ~, we begin by showing X~,, =X~~176176 ~ is a derivation of degree --I of differentials. As the X and the D are derivations of degrees --I and o respectively, it follows that DioX~o,~--X~o,i D~o and Xio.i D~--D~No,r ~ are derivations of degree --i, whence it suffices that X;.,,~(D,.--D,~)- (D~,--D~)x,.,,, = X;o,,~h0, q d-- dx;~ x,~ ~ be a derivation. Here d is a derivation of degree i, and so it suffices for ),~0,~ X~o,q to be a derivation. However X~~ X~~176 X~~ is a derivation, and hence it suffices if X'~~ Xi.,~ =Xi.,~fi~ which is verified just as in the last paragraph. Finally the ope- rator X;2,q does enjoy dx~s q + X}~, q d= [D, D'],~ D']i~, as an immediate computation shows, and hence X'~s has the proper effect on 1-forms, and thus on all forms. Q.E.D. Application. -- The hypotheses are satisfied by a non-singular hypersurface of equation F = o, if we take the covering from the U(b, i). We will write functions in homogenous coordinates P/Q, and use pi) to denote the result of applying D only to the coefficients of P; then we readily compute QpD pQD 0 O~ + X~~ (P/Q) O~ ) and hence the ratio (D~--Di)(h)/(Ds D'. (--FD/X, D(~,,)(P/Q)= ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 8 9 Globalization of the (9 Mapping in Middle Dimension. We begin with globalization to X ~ where matters are greatly simplified by taking the covering U~ n + 2), b = I, . .., n + i. The proof of theorem I, together with our recent definitions, gives the following Theorem (I.7). -- There is a unique mapping 0 : cr ~ Cv(Y~ v) which is a homo- p+q=n morphism of Lie modules over the ring of derivations of the field of definition, which is compatible with specializing the defining equation, and which assigns to a monomial ZX w the element of C~ n) which is XWco(b, n+2) over U~ n+2). In this way the elements of s of Z-degree one correspond with the algebraic n-forms regular on X ~ As these are closed, it follows that the image of~Cp~ Y, CP(f~ q) p+q=n lies among the cycles (of the total complex), and hence by passage to quotients we have a map (9:~r176176 Further, the aforementioned theorem of Grothendieck [4] states that on the non-singular affine variety X ~ every cohomology class is realized by a regular algebraic form, whence ~ cP~ maps onto H~(X~ Lemma (i.8). -- The kernel of (9 : ~~176 contains Dz~+~Dx4s ~176 Proof. -- The assertion for D z admits the same proof as given in theorem t. To fix ideas consider Dx ~q~~ it suffices to show the kernel contains Dx~(ZX w) by our general reduction procedure. To this end consider the element ~ of C~ ~-1) whose value over (--I)~+l-bx w dx2 dxb dx,~+l --A...A--A...A for U~ U~ n+2) is o for b = i, ~f x2 xb x,~+l X b -- OX b Part d) of theorem t assures us that O(Dx~(ZX*~ and A~ (A=the total eoboundary) agree in their components in C~ it remains only to consider the components in Cx(f~"-~). Consider (9(Dx,(ZX~)) = O(w~ZX '~) + (9 nZ2XWX~ ; we compute by passing to the equation F + rx~+~= a o, 9 then 0 [ ~ ~F ~) (9(Dx, (ZX~)) = (9(w, ZX ~) + ~ | Xl~XT,4 , and the component in C1(~ ~-t) assigns to U(b, n + 2) n U(c, n + 2) (where b<c) the form of xWxl ---- Ox 1 ~xl dx~ dx, + 1 (- t)%,~(- IF +~-~ --^... ^--^...^ _ ~f Xl Xe Xn + 1 X c -- ~x c dxb dxc dx. + l A... A--A . . . A--A . . .A--. - oz Xb Xc Xn + 1 \ Ox / 12 NICHOLAS M. KATZ x w dx~ dx~ dxn+l In case b= i, this is (--i) "+*-c A..,A--A...A-- , as asserted. Similarly Of X2 Xc Xn+t if I<b<v, this is Xc OX~ .... lxW ( oi < + oj e, 3 dx~ dx~ ,Ix.+, A--A...A--A...A af af ', ax b axc ! x2 xb x~+l X b X c -- __ ~x b ~x, (--I)"-~ w dx 2 dxo dXn+ 1 (-- I)n--bxw dx 2 (Ix b dXn+ 1 --A... A--A...A-- Jr" A... A--A...A-- ~f X2 Xc Xn+ 1 Of X 2 X b Xn+ 1 X e -- X b -- Ox b again as asserted. Corollary ( z . 9). -- | maps ~ce~163 a~ onto H"(X~ OF Finally we require a definition. A form F is said to be regular if the forms X~.~X ~ have no common zero; i.e. the locus of F is non-singular, as are all its intersections with the coordinate axes. The Cohomology of Regular Hypersurfaces. Fix in p,+l a system of homogeneous coordinates (Xl, ..., X,+~), and a positive integer d. X" denotes the locus in pn + 1 of a regular form of degree d, Xg the open subset where the first i coordinates are all non-zero, and P"+~ the open subset of P"+I where --n+2 no coordinate vanishes. H i denotes the j-th singular cohomology group, BJ= dim Hi; Hit ) is the j-th group with compact supports, Bi, ) its dimension. /f l<o Theorem (I IO), .--I n __ 9 -- B (X.+2)-- d"+l+n /f /=o [ (n+l'~ /f l>o /f l<o Bn+ l-t(on+ l .5~n ,~ d"+l+n+i /f /=o k~n+2 ~n+21 n+ 2"~ /f l>o. l+l] Proof. -- First notice X~=X~--Xo ~-l, whence the exact sequence Pq H~7*(X~-I) -+ H~c)(X~) -+ H~AXg) -+ H~o)(Xo ~-1) and by the Lefschetz theorem [6, p. 91] pq is an isomorphism for q<n--2, an injection for q-=n--i, and a surjecfion for q>n. As B~,I(X~)=B~,/(P ~) for q4=n by the same theorem, B~cl(X~)=o if q<n or n<q<2n, while 2~ B(cl(Xl) -- I. Similarly for I <i<n+2 we have X"--Y~i-- whence B~c)(X~)<B~-I(X~"--,1)+B~o)(X~_~), ~"/--1--Yn--l"~--l, ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 9 ~ which already shows B(~)(Xi)=o if i> I and q<n. Iterating our last inequality, we have for />i that n+l Bn+ltyn "~<" ~, [n+l"~lJln-j+l {n+l'~__fn+l'~ kn--lJ -- k ]+11" (~) ~-~-,,+~,)~jT'o ~ j )-,- (XI'-~)= To reverse the inequality we turn to the exact sequence of ~-.+2,tP"+ ~ X.+2),- l-ffq [pn+ l__'SIn "~ I-Tel Fpn+ I'~ Now p,,+l is just the (n+I)-fold product of the non-zero complex numbers, so that --n+2 n,~+l+qp~+l~=(.+l) for />o, ofor l<o. This already shows that B(~)(P.+2--X.+2)q n+' n =O a"(c) k--n + 21 -- if q<n, while for l>I we have nn+l+l('~n+l ~n "~ ~ {n+l'~_l_[n+l"i__[n+2"~__{ n+2 (c) \~n+2 "~n+2J--kI+lJlk ~ )--\l+I]--\n+l--U" By Poincar6 duality, the left side is -n"+~-~t~"+l--~"~-.+~ -~.+z). In ~n+t ~ taking ~n+2--*~n+2~ dx~ dx. + l x~ = X~/X, + z,f= F (x~, . .., x, + 1, I ), we claim the n + ~ cohomology classes -- ^... ^-- -~- X 1 Xn+l dx I ^ dx i ^ . ^ -dx" + 1 ("+2) ~^-- ^. . . -- .. , i <i<n + i are linearly independent (hence and the x~ x~ x,~ + 1 df dx 1 dx"+l will give linearly independent cohomology monomials of degree j in , , ..., Xn+~ classes). Observe f xl OF dxn + 1 Xi aXi dxl dxn + 1 df ,tx, &, --A-- A...h--h...A --i ^. 9 .A-- x,, + l F x 1 x,~ + l f x, x i and so by the Euler relation we may assume a relation ai OF Y~ A...A--~O, ~=1 F x~ x~+l We claim a~=o; let y~=Xi/X j if i<j,y~=X~+~/Xj if i~_j; as dyl dy,,+l dx, dx,,+, --^...^-- --i--^. . .^-- Yl y,, + ~ x 1 x,, + 1 ' it suffices to show an+2=o. We rewrite our relations dxl dx,,+l "+lia~--an+~'~df dx~ dxi dx,+l an+~--^...^ ,~ N x~ x,,+l ,=,t ~ t7 x, x~ The regularity of F insures that every term X~ occurs with non-zero coefficient bi, so that when x~, ..., x,+l are all small, f(x) is close to b,+2, hence contains a region {o< 1 x,I <~} x... x {o < I x,,+, l<~-} --n+2pn+l--xx"n''n+2 ^-^'^-^'''^-'x~ NICHOLAS M. KATZ 9~ where ~ is sufficiently small so that log f is a " single-valued " holomorphic function in that region. Integrating our relation over v={[x~l--=lx.,I ..... Ix,~+~l~-~/~} we get , + 1 , a~+2 v a~--a,~+~ ( .[. ,.dx 1 dxi dx,+ 1 -- z, - . aHogJ--A...A--A...A (~:i) ~+~ ~=~ d Jv \ x~ x~ x,~+, , and the right side vanishes. Hence for l>i Rn+~t~xr'~ ~__[,~+l~ g~)+l+qpu+l X, ~__(~,+2~ and for l<o --- , a'(e) k"~xn+2l--kl+l]~ k n+2 n+2l--kl+l), everything is zero, so we must look at Euler characteristics. We readily compute Z/pn+l ~rn ~ __ i'~n+l{Izln+l[pn+l n n n n n I n+2--.Lxn+2]--(-- ] \~'(c) k n+2--Xn+2)--n--I), Z(Xn+2)=(-I) (n(~)(X,,+2)--n), while ~ +t _ *,+ , .rp,,+l X,~ ~ whence it suffices to show Z(Xn+z)=(--I)"d '~+~. But z(X~)=Z/ ~-l)--Zk ~-1) for ~<i<nq-~. Upon I) (n_j)z(X~), whence Y~z(X,,+.)) Z' .... = ~ [ ]Z iteration > ,,-, z x(Xo),--=,. __ n2u ~ j~0 \ I -t- Z,/ But ~z(X~ (I--Z)2(I-~-(d--I)Z) [5, P. 465]. Q.E.D. Corollary ( x. x I ). -- For i> ~ and every q we have a short exact sequence o-~ Hg(X} ~) -+ nq (X~+ 1) -+ Hq- ~ (Xg -~) -+o and thus for i2I , O~rl* .... 2IA ...... ~ >__~H~(Xg)___>Hn(X~,)) Proof. -- The second assertion for i--i is part of the Lefschetz theorem, and follows for i> i from the first assertion. The first sequence is certainly exact without the end zeros, and our computation showed the alternating sum of the dimensions to be zero. The Cohomology of the Complement. o o ~+1 ~ It is Define a mapping ~ : ~ -+f2 (P~+2--Xn+~) by ~(Z~Xb) - (a--i). X b (__~)a-1 F a O ~(Dz(ZaXb))=a(I F\ a b easily verified that N.Dxi=X~.2, and that --~) (Z X), so that Dz ~~ is precisely the kernel. Write xi=XJX~+~ " and take monomials in the dx~ as a free basis for otP "+1 ~" c)0rp-+ 1 "x:,~ ~ We obtain xr ..k.n+2--~n+2) over ~. \~n+2--~n+2). Theorem (x. x2 ). -- The mapping ~, together with the ,-operation provides, an isomor- p. + 1 5(~ with the Koszul complex on 5#~ ~,0 with phism of the de Rham complex on .,+2--~.,+2 operators Dx~, 9 9 ", Dx,+l; in particular an isomorphism of ~T4'~+ ~ + ~-X~+2)" with 5e~ Lf ~ q- ~Dx s176176 Corollary (I. I3). -- In the case of a regular F, dim s176176 +n+ I. Corollary (I. x4). -- For regular F, dim ~~ + Y~Dxi~~ + n. ~-~+2~P'~ ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 93 Proof. -- As Dz=~Dxi , Dz~ c~Dxi~LP ~ D z (elements of Z-degree o). If X b has Z-degree zero, but Xb:~i, say bl~:o , then so Dz CXDx, ~ But ~(ZF) dxl ^ . . dx"+l ---- . A--, a non-zero cohomology class, whence ZF$~DxiSe~ , so x~ Xn + 1 the drop in dimension is precisely one. Q.E.D. Remark. -- Define ~f+=~Lfn K[Z, X1, . .., X,+2], oW~163176 s It is clear that ~f~176176 , whence s176 ~ is a quotient of s 5r + ~Dx;~f0, +). In the regular case, the latter space has dimension ~d'+l+n, as dim ~q+/~DxioW + 0+ 0+ l+n is d 1 ~" [2, p. 55]- Similarly, dim 5r ' /~DxioLf' ~d +n+ I in the regular case. Corollary (I. I5). -- ~ establishes an isomorphism of ~r176 xi~~ with H,,+t(p,,+l x-o~ which maps WS=s163 ~176 isomorphically onto the image of Hn+l(P~+l--X) in VIn+I(pn+I nZ~ We write W for s176162 +~Dxi~f~ Theorem (x. I6). -- Let F be a regular form. The mapping | : W-+H"(X ~ # an isomorphism of modules over the ring of derivations of the field of definition. Differentials of the " second kind ". Consider now the complete variety X, and the open cover of all the U(b, i). Writing ~9 ~ for the subspace of ~o+ divisible by all the variables, it is easily seen, following the proof of theorem (I-7), that there is a unique map | : s _+ E CP(~ q) with the p+q=n proper effect on elements of Z-degree one, which is a homomorphism of Lie modules, and whose image lies among the " cycles " of the total complex. Thus, letting W s denote the image of ~fs in W, we have that the image of W s in Hn(x ~ under O lies in the image of Hn(X) in Hn(X~ Finally we recall that a closed algebraic differential form X is " of the second kind " if for some affine open set U on which it is regular, the cohomology class it determines on U is the restriction to U of a cohomology class on X. Theorem (X.XT). -- Let F be a regular form. The isomorphism | Of W with Hn(x ~ maps W s isomorphically onto the image Of Hn(X) in H~(X~ i.e. onto the space of n-forms of the second kind holomorphic on X ~ modulo exact such. For any derivation D of the field of definition of F, the equations of deformation of Dwork (the action of ~9 on W s) are identified with the Picard-Fuchs equations on X ~ (the action of D on the image of H"(X) on H'~(X~ "24; NICHOLAS M. K. ATZ Proof.- By (i. I I), the image of Hn(X) in Hn(X ~ has dimension B"(X)--B"(P"-I), and explicit formulas ([2, p. 54] and [5, P. 455]), show this is the dimension of ~s/(zSn~DxiZ+) for n~I. It is enough, then, toshow 5r176162 ~ OF s ~ o or, that ff~a~ZXi~w-~ e~ ~ +~Dx~q a'n, then all a~=o. To show a~+~=o, we follow 9 Ox'~. i i (I. IO) and apply OF a~X~-- Z OX~ dx 1 dx. + 1 ~ . ^...^ me restriction of a class on P"+*--X. i F x~ Xn+I The cycle y in the proof of (I. IO) is homologous to zero in P"+~--X. Inte- grating over 7 thus annihilates the right hand side while the left hand side gives an+2 o. Q.E.D. (2~i) n+l Residues [8 a]. Let B be a nonsingular subvariety, of codimension one, of a nonsingular variety A, in characteristic zero. The exact sequence of cohomology with compact supports Hic~-l(A) ~Hic~-l(B ) ~Hi~)(A--B) ~Hi~)(A) ~Hic)(B) gives by duality an exact sequence of de Rham cohomology HZ- 2(B) -~HZ(i) ~ H*(A-- B) -~H*-I(B) -~H~ + 1 (A) The map H~(A--B) ~ Hz-I(B) is the residue map. When B is given by an equation -- on the level of g=o, the residue map, roughly speaking, extracts the coefficient of dg differential forms, g Theorem ( I. I8). -- O=residue :~, so that we have a commutative diagram, residue "[_]'n + 1/pn + 1" I I._Tn4-1[IJn+l __ XO~ o > "" ~-~+2J > "" ~ ,,+2 J -+ H'(X ~ > o >0 o---- > (ZF) > &~176176 ) > s176176247 L~a ~ + o+ o > (ZF) > 5(,0,+/(~Dxi5r > ~0, /(~]D.:.~,. i "*' +Dz~q~+) > 0 OF Proof. -- By assumption the forms Xi~ " have no common zero, and hence every form of sufficiently high degree lies in the ideal they generate. Momentarily 246 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 77 space Ws(p), the operation ~r, and the Picard-Fuchs equations are all defined over f~(F). The present paper grew out of attemps to identify the differential equation for C(o, P) with a Picard-Fuchs equation, in greater generality. This is achieved by Theorem (x .9). -- Let X be a projective hypersurface in p~+l, defined over afield, K, of characteristic zero, which is nonsingular and in general position. Denote by X ~ the open subset where no coordinate vanishes, and by H"(X, K) and Hn(X ~, K) the n-dimensional cohomology groups of X and X ~ respectively. We regard H~(X, K) and H"(X ~ K) as modules over the algebra of derivations of K, as explained earlier, while the K-space W s, associated to X by Dwork' s theory, admits the derivations of K by means of the ~ operation. There is an isomorphism | of W s with the image of H'(X, K) in H~(X ~, K), which is an isomorphism of modules over the algebra of derivations of K. Furthermore, the kernel of H'(X, K) ~ H~(X ~ K) is reduced to zero for n odd, and is one-dimensional for n even (and n>o). We should point out that the matrix C(P) is the transposed inverse of the matrix denoted by the same symbol in Dwork [3, P. 262], which arises by passing from the " dual space " at o to the generic dual space at F; thus that matrix is the transpose of the (suitably normalized) period matrix. (Also the B matrix is the transpose of its analogue there.) Explicit Computations. To explain how Theorem (I. I9) comes about, it is necessary to examine the spaces of Dwork in some detail. We fix integers n>o and d>o, and define ~ to be the K-linear span of monomials Zw"X~l...X~+~ with w~Z, dwo-=~w~>o. X' contains certain subspaces of interest to us: ~0, where /2)02 I s162 where all wi>o ~q~0, +, is s176176 s162 ~s, where all wi> I. We fix a non-zero constant, % and corresponding to each defining form F, of degree d in X1, ..., Xn+2, we define twisted operators on ~r X 0 ~Z OF Dxl = exp(--nZF). X~. exp(~ZF)= i~q- Xi0X i Dz = exp(--rcZF ) . Z~oz. exp@ZF) = Z~O + nZF. This construction is rational over any field K, which contains ~ and the coefficients of the form F. For each derivation D of K, we define the twisted derivation ~B of ~ as K-space by ~D = exp(--r~ZF). D. exp (r~ZF) = D + r~ZF v. ,229 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 73 where each o~ i is a I-form regular on Ur f12 is a function regular on UlnU2, and r162 q-dflz = o. Thus o h and r are differentials of the second kind, whose difference is an exact differential. The coboundaries are those triples (dfl, df2,fl--f2) where each f is a function regular on U~. The mapping (c01, r r176 establishes the desired identification of this hypercohomology group with the differentials of the second kind modulo exact differentials. We continue with this example, and introduce the action of a derivation D of K. Select functions, x andy, so that at every point peX either x--x(p) or y--y(p) is a uniformizing parameter. Let U 1 be those p where x--x(p) is a uniformizing parameter, and take, for Uz, the analogous set fory. It follows [6 a] that D~ and Dy, respectively, are stable on the functions regular on Uz and U2, respectively, and that both are stable on functions regular on U 1N U~. Define X : f~l_+f~0, by X(fdx) =fDu(x ). The mapping on i-cochains of the bicomplex, given by (~01, c%,ft2) ~ (D~oh, Dye%, Xoh-t-D~f2) , preserves cocycles and coboundaries, and so induces a mapping on the one-dimensional hypercohomology. This mapping gives the action of D on differentials of the second kind, modulo exact differentials. In the higher-dimensional case, analogous formulae will, under restrictive hypo- theses (see (I .6)), endow the bicomplex (C~(f~ ~, 11)) with the structure of module over the algebra of derivations of K, and thus allow the differentiation of cohomology classes. This construction is presumably a special case of Grothendieck's " Gauss-Manin connec- tion " (1), but in any case our restrictive hypotheses are satisfied by principal affine open subsets of non-singular hypersurfaces. The differentials of the second kind in higher dimensions no longer give the coho- mology, as they did for curves. Indeed, a closed meromorphic differential e0 on a pro- jective non-singular X is said to be of the second kind if there is an affine open set U, on which co is holomorphic, such that the cohomology class on U determined by co lies in the image of the restriction mapping H*(X)-+H*(U). This mapping is seldom an injection (except for Hi(X) -+HI(U)), although for sufficiently nice U, one can determine the kernel (I. 1 I). Because differentiation of cohomology commutes, whenever it is defined, with the restriction H*(X)->H*(U), the image of H*(X) in H*(U) will be stable under differen- tiation, thus giving rise to a Picard-Fuchs equation for the subspace of H*(U) spanned by differentials of the second kind. Parameters and the Zeta Function. Recall that the zeta function of a variety, V, defined over GF(q), the field of q elements, is the power series, exp[~ 1 s )' where N~ is the number of points on V (1) (Added in proof.) It is. A general algebraic construction of the Gauss-Manin connection is given by T. Oda and the author in << On the differentiation of De Rham cohomoIogy classes with respect to parameters )>, to appear in Kyoto Journal of Maths. 10 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 97 X.+2=I. The | mapping~ for F is easily computed; pass to F+FX~+2, 0 I (O~w'-~[ x w \ so ~r----~-~+=ZX~+2 whence e(Zw~ [ --Of /I , hence of | (Z w~ ~) is a polynomial in Xl, 9 9 x~+ 1, i/x,+ 10x,+ 1 all of whose coefficients have --log w 0 ordinal>ord(w0--I ) !--ord ~w~ 1 Hence we have -- log p Theorem (2.4). -- 19 naturally extends to a mapping @:L~ A+| where A*=~[x, v]/I, I= x), I--'rxl...x~+lO . Deformations. Explicit Construction. Let f, hzO[x],fr=f+Ph, Ar=d~[x, F,-r]/I, I= fr, I--X~+lOxn+,] and define ArO~ = hm<" Pir/(P,* P)'~A r.* Our previous estimates show that, defining ~-~ on A re~ by ( ) Oxi oF--o, i=I,., n, the series Y~ ---P-"l--ln converges to an endomorphism f) of A~ "' n>o n! \OF/ (a homomorphism by Leibniz rule). Clearly f) - identity mod(P), f)(1-')=o so Ox f)(rA~)=o, and f)(x)=x if ~=o. Hence f) induces f)(0,r):A~/rA?-+A~. Write, for ord ~t>o, K~=2/r/(F--~), composing with f)lo, r the specialization F--->~ then A r co provides D(0,~,l : A~ --->AN, a map reducing to the identity mod p and fixing Xl, ..., x n. Here 13 is the maximal ideal of 0. Lemma (2.5). -- Suppose F and H are firms of degree d, and let | for ord ~t>o denote the (9 map for F+ ~H. Then we have the commutative diagram L0(o+) ox,i- Hl L0(o+) A~| , A+| The Frobenlus (Diagonal Case). Denote by ~ the endomorphism of L~ -) given by wo w q q qwo qw [~(Z X )=exp(~Z F(X)--r~ZF(X))Z X , 13 9 8 NICHOLAS M. KATZ andby Fr the endomorphism of A* where A*= d)[x, x]/I, I= (,, x), i--'rx 1 ...Xn+lOx~-+l which reduces to the q-th power mapping mod p, and maps x i~x] for i= i, ..., n. We will prove Theorem (~. x2). -- For F irreducible mod p, qOo~=Fro| We begin with a special case. Theorem (~,. 6). -- Let F(X1, ..., X,+z) be a form of degree e such that upon affinization by X,+ 2 we have f(xl, ..., Xn+l)=g(xl, .. ", X,)--X,+ d 1 where d is prime top. Then over U"~n+r, nq-2) we have a commutative diagram LO(o+) q~, LO(o+) Fr A+| , A+| Proof.- Write A(x)=g(xq)--g(x)q; then Fr(x,+ i) a = g(x q) = g(x) q q- A(x) = x~q~+ l(i -~- A(X)Xn~dl) whence Fr(xn+l)----%q+i~ (lld~ A ( x ) m B* ,mJ ~. Let = d~ [x, -r] /I where "*n+l I -=(g(x)q--xq, d+l,I--ZXl.., x,+l), this is not special affine, but clearly there is a restriction mapping res : B+-->A +, and our formula shows that we may interpret Fr as factoring A + Fr B+ res A+" Now we return to the 19 mapping over UO(nq - I, n+2) for G(X)--X~+~X~+~. We introduce the family G(X)_Sra .L,+i.~,+2tPX~+2, y,-a _1_ and we then have O(ZWoXW)__(~ t~t_WoC~wo_ltTywye(l_woh , I 0t~ ( XW ) r=0 --vPk'~ "~P k ..... n+2 l] I'~o 7~w~176 --dxna+l w I w n ~W 0 -- 1 W W 1 W.+ 1 ...... (~W,+ 1-a~ " but the write x =x 1 ...Xn+I, SO O(ZWoX w) x 1 ...x, _d Wo_l OFWo-l Wn+i j I'=0 relation x,+ 1 =g(xl, 9 9 x,) +I', ~=o Oxl if i<n, shows oWo -1 /W"'t-I__I~ wn+l_Wo d O-'-~g--~_l(g'~-+ll--d)=(W0--I)' i ~O__i ]~n+l ' so that finally o(zWoxW]:--(W0--I)'[~--I~,~, 1 " ~Wn~.wn+l--Wod J drcwo-1 ~]\ WO--I "'~n ~n+l " 250 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 99 We note that if we replace d by qd and w.+ 1 by qw.+~ then our answer is " essentially " integral; so define L(r to be the space of series W.AwZ~~ w with the usual growth condi- n+2 w tion but with homogeneity condition qewo=~=w~, w0~> I, and such that each w.+~ is divisible by q. Then let | mean the | mapping for over U~ n+2); we have computed that | maps L(q)(o+) to B+| Finally define @(1) and (I) (2) by @~176 Z~oX ~'~, @(~)(Z'~ ~) = Z~w~ w, define H ---= exp(~ZX~ + t X~+<[--r;Z~Xt~+ tX}~+-~ a) + r~ZqG(X)~--r~ZG(X))~L~ 0G and G = exp(~ZG(X ~) --~ZG(X) ~) ~L(~)(o+), as -- = o. 0X. + Clearly now ~(Z~'X~)=Ho@(2)oGo~ (1), and so it suffices if both the following diagrams are commutative. qGor ~ It| ~(,) L (q) o +) L <q) o +) L~ L~ og Oq O Fr rss A+| A+OO B+| B+| > The first diagram. -- Take ~ = ZwoXW; then I 9 9 "Xn+l ----eaO k ) __ Fr 0~)= -(w0-i),d. ~ -- z[ d / \ m\ m ]x.+, {(w.+,ld)-q ,',(x)" --'~I 9 " " "~n+l -- L--~g--I..-~E an x.+ 1 ~ m! " \ m+Wo--I " i x.+l q~" The other way, q|174 ~!(ZG(Xq)--ZG(X)q)'~Zw0Xq~ . The expres- sion under the ~ is a sum ofmonomials whose Z-degree is m +Wo, and whose X.+l-degree is qWn+l; we have qOq G@ (~) (0) = q ~ _ qdrc" + ~o- ~ m ! (m + Wo-- i ) ! \ m+wo-~ I x~ +~" The second diagram. -- Say L(q)(o+)~=Z%X~ '. .... ~r176 to compute 0(H@(2)~), first define constants A. and B n by (2.7) ~A.X" = exp(~Xq--~X), ~B.X" = exp(~X--~Xq). -~n+1-=.+2~'~ "~.+~'~.+~Yq(~-a), ~..c'-t"cq~--Yqa: ioo NIGHOLAS M. KATZ Then | @(Y, ~ ~-~] n " #~ {w O to 1 ,~tOtly~ c ~w.,~ For = B.Am(ZXn+~X.+2) (ZG(X)) Z X, ..... . "~.+z".+z," .,m fixed n, m the expression under the Z is a sum of monomials of Z-degree n + m + qwo, whose X.+z-degree is nd+qc, so that --BnAm(n+m+qwo--X)! ~ (qc/a)~-n-- I I x.+~g x~ . .. X,, x~+~ O(HaP(~)(~))=.~,~ dn-+m+q~ofZ / -- -- / - --gZ4;5--q-~0~ \n~-m-t-qWo--W x.+a but g---x.+l, so O(Hr )= 2~ --B.Am(n+m+qw0--I)l (qc/d)+n-- z x)...x. Xn+ ~ 2r ,~,m drc"+m+q~:o -~ \n+m+qwo_i] x~+l ) w 1 Wn qc (WO--I)! (c/d)--I X 1 ...X n Xn+ 1 3~% - , whence we are reduced to showing Finally Oiql(~ ) = __qdrc%-i \ Wo--Z x.+t , (w0--I)! ((c/d)--I] I -g,,A m ((qc/d)+n--I] = ~o X'-2~-g(n+m+qwo--I) ! qTr. wO \ W O- I ] xq o., m ~ ~ \n + m + qw o- I ] Lemma (2.8). -- Let a be a strictly positive integer, and b a rational p-adic integer. Then qrc a(1-q) a--I .,m~ ~ \qa+n+m--I " ;. p-- I ord(Bj)_>>j ; for fixed a, both sides represent Proof. First ord(Aj)~ 3 ~ , Pq / continuous functions ofb on the rational p-adic integers, and so we may assume that b is a positive rational integer. Both sides vanish unless b>a, and in that case the right side B. )[ A m :I The second factor is the coefficient becomes ~-~(qb-+-n--i ~--~ (q(b--a)--m)!" of X#-a' in exp(X)~ AmXm ( Xq ) - ~-~ =exp(X)exp( Xq.--XI\r~ q-I ] =exp ~ , so the right side is ~ I ~ngnn (b_a) [ ~(~_tl(b_a I (qb +n--I)! and we are reduced to showing Z~ ~Iq-l"(b--z) ' (qb+n--I)!= t Define = X qb+n-1, z p--I and set g(X)--Z d'~n(f(X)) , easily seen to convergent for ord X>p_i pq ' --~_>0dX ---- dg I p--I and satisfy g--dX =f(X). As the only solution converge for ord X>p_ I Pq , of the homogenous equation is a constant multiple of exp(X), converging only 252 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES lOl for ordX~p_---~, it follows that g(X) is the unique power series solution of dg =Xqb_~ex p _ which converges for ordX> I g--dX ~ p-- I pq In fact there is a solution of the form H(Xq)exp ( X--=~=i_, . ,We must have _va-,aq~1) H(Xq)___~H(Xq)=xqb_l, or, multiplying by X, H(X q) -- ( I -- -- q XqH(X q) --X~--~ H(X q) = X qb, T~q-- i which is equivalent with ~ xH(X)--qX d H(X)= Xb; so g(x)=(i_gT2q-ld~)-l(g~q-lxb-1 ) b~.al/ d \J[7~ q-1 , , 9 r:(q -*)b ( xq) In particular H(o)-- (b--I)1, and by uniqueness g(X)=H(Xq)exp X---~i_ ~ , rciq- a)b q Bn so g(o)=--~(b--I)!, and g(o)=~-~(n+qb--i)!. Q.E.D. We pass to the general case by a sort of analytic continuation. Lemma (2.9). ' Let A* be an (9 algebra with A"/pA* a domain, and p not a zero divisor. p is not a zero divisor Suppose xsA*, xCpA*, xA*/pA**A*/pA*, and that for every aep, in A*/(x--a)A*. Then ~9~(x--a)A~=o. Proof. -- We proceed in short steps. (I) x is not a zero divisor in A*/p'A*. Proof.-- For r=I,A*/pA* isadomain, and xCpA*. Using induction if xyepr+lA" then (y)=p'z, whence xzepA*, whence zepA*. (~) (prA*/pr +IA *) n (x'~A*/p'+*A*)=x"p'A*/p'+*A *. Proof. -- Let a, beA* with x~a=-p'b mod p'+~; then (a)=p~d, so (x~a)=x'~p~d. (3) ,O0(x"h*/p~A*) = o. Pro@ -~ For j= I, xA*/pA* is a proper ideal in a domain. Now let yeA*, with and so by (2) mod pJef;'l>0(x"A*/pJA') ; by induction we may suppose y-o mod p~-l, y mod pJ~,90(x"pJ-lA*/pJA*) ,~fl> 0(x"A*/pA* ) = o. (4) f'l (x--a)A ~-= o. tl~p Write Pro@ -- Let yeag~(x--a)A~; let a,,..., a, be distinct elements of p. whence y=(x--al)~(a,) and reduce mod(x--a~); then (a=--al)~(a,)=o mod(x--aa) ~(al)=(x-:a2)~(a,, a2)as p, and hence a2--a,, is not a zero divisor mod(x--a~). : 253 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 77 space WS(p), the operation ~r, and the Picard-Fuchs equations are all defined over f2(F). The present paper grew out of attemps to identify the differential equation for C(o, F) with a Picard-Fuchs equation, in greater generality. This is achieved by Theorem (x.x9). -- Let X be a projective hypersurface in p~+l, defined over afield, K, of characteristic zero, which is nonsingular and in general position. Denote by X ~ the open subset where no coordinate vanishes, and by Hn(X, K) and H"(X ~, K) the n-dimensional cohomology groups of X and X ~ respectively. We regard H"(X, K) and Hn(X ~ K) as modules over the algebra of derivations of K, as explained earlier, while the K-space W s, associated to X by Dwork' s theory, admits the derivations of K by means of the ~ operation. There is an isomorphism 0 of W s with the image of H"(X, K) in H"(X ~, K), which is an isomorphism of modules over the algebra of derivations of K. Furthermore, the kernel of H'(X, K) -+ H"(X ~, K) is reduced to zero for n odd, and is one-dimensional for n even (and n>o). We should point out that the matrix C(F) is the transposed inverse of the matrix denoted by the same symbol in Dwork [3, P. 262], which arises by passing from the " dual space " at o to the generic dual space at P; thus that matrix is the transpose of the (suitably normalized) period matrix. (Also the B matrix is the transpose of its analogue there.) Explicit Computations. To explain how Theorem (I. I9) comes about, it is necessary to examine the spaces of Dwork in some detail. We fix integers n>o and d>o, and define 5 ~ to be the K-linear span of monomials Zw"X~l...X~+g with w~Z, dwo=~Wi2O. 50 contains certain subspaces of interest to us: 500, where /./)02 I 50+, where all w~>o ~o0, +, is 50~ 50+ 50s, where all wi> I. We fix a non-zero constant, % and corresponding to each defining form F, of degree d in X~, ..., Xn+2, we define twisted operators on 50 {9 0 OF Dxl = exp(--r:ZF). Xi~ ~ . exp(=ZF)= X,}~ -t- =ZX, O O =zsg+ D z = exp (--r:ZF). Z~. exp (:zZF) ~ZF. This construction is rational over any field K, which contains ~ and the coefficients of the form F. For each derivation D of K, we define the twisted derivation ~D of 50 as K-space by ~D = exp(--r~ZF). D. exp (~ZF) = D -t- r:ZFD. .229 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES IO 3 The middle box commutes by assumption, and the end boxes arise from deformations, hence commute. Finally it is clear that F.=f)(.,0)oF0of)(~q 0), as both sides have the same reduction mod l~, and agree on xa,..., x.. Corollary (2. x x ). -- Under the assumptions of the lemma, we have a commutative diagram for F(X, i) LO(o+) ~(~), LO(o+) Fr0b. A~+| , A~*| Theorem (2. x2). -- Let F(X) be a form of degree d whose reduction modulo I~ is irreducible. Then q0o~=Fro0. Proof. -- Over U~(n -t- i, n + 2), to fix ideas, write f(x) = xn+ I- i +g(x) (we may supposefhas degree >2) and consider the family f(x, P)=xn+l--I +Pg(x). Asf(x, I) is irreducible mod p, it is not divisible by xn+t--I, hence neither is glx), whencef(x, r) is irreducible, and remains so mod p. Hence Ar and At/pAr are domains; clearly PCpA r and I" generates a proper ideal rood p. Finally for a~p,f(x, a) is congruent to x,+l--x modp, so thatp is not a zero divisor in any Ar/(r--a)A r. Thus we may apply (u.ii). Q.E.D. ANALYTIC THEORY OF THE COMPLEMENT Let F be a form of degree d (non-trivial modp), f the affinization X~+2=I; define C;----[x, v]/I, I----(I--x~... x,+l'cf). Deformations. -- Here C~ and C~- clearly depend only onfmodulo p (if f---g mod p, g-t =f-'(i +f-l(g_f))-~ =Z(g_f).f-.-1). Lemma (2. x3). -- Suppose F-Gmod p; then we have a commutative diagram L0(o+) ~xp(,,z(~-~))> LO(o+) id. c/- , 255 ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES 9 x which already shows B(~)(X~)--o if i> I and q<n. Iterating our last inequality, we have for l>i that .+I n+l[yn "~<~ ~ [n+l~FIn--j __{n+t~__fn+l'~ To reverse the inequality we turn to the exact sequence of 2, ~ X~+2)," "[-Tq I'P n+l n "[.~q ['pn+l'~ Now o "+~ is just the (n+I)-fold product of the non-zero complex numbers, so that --n+2 ~q []Dn+ 1 yn "~ n.~+~+~(l)-+~ (.+x) for />o, o for l<o. This already shows that ._,(~)~-.+~--.~.+~=o a"(c) ~,~n+ 21 = -- if q<n, while for l>I we have B(n?l+Upn+l__~x rn h ~ [n+lh..-l-[nWl'l--[n+2"l--[ n+2 "~ \~n+2 ":~n+2]-- \/+1]/\ l ]--kl+I/--kn+l--U" By Poincar6 duality, the left side is --Rn+t--t(pn+i--yn~,--n+2 -~. +z). In -.+~--.~,~+z,P"+t .. taking &, dx. + , xi=X~/X,~+z,f=F(x~, . .., x,~+~, I), we claim the n+2 cohomology classes --A... A-- ..... x 1 x,~+~ df dx~ dx~ dx. + ~ and ~ ^ --x i ^'" "^--^'xr " "A---, ~ <i<n + I are linearly independent (hence the (,+a)a Xn+l df dx~ , - 9 *, dx"+l will give linearly independent cohomology monomials of degree j in Xn+l f' Xl classes). Observe ~F ,ix. + i X~ aXi dxl dx~ + uf dx, &, --A-- h...h--h...h-- =d- A...A-- x~+ ~ F x 1 x.+ ~ f x, x~ and so by the Euler relation we may assume a relation a i OF 2~ A...h NO. i=1 F x 1 x~+ 1 We claim at=o; let y~=Xi/X ~ if i<j,y~=X~+l/X j if i>j; as dyl dy,,+l dxl dx,~+l --A. . .A --:]:--A. . .A--, Yl .Y,~ + 1 xl x,~ + i it suffices to show a,+~=o. We rewrite our relations dx 1 dx~+ i ,~+i t'n __,~ \ a.+~--^...^ ~=~\ ~ /~^--^...^--^...^-- Xl Xn + 1 Xl Xi Xn + I The regularity of F insures that every term X~ occurs with non-zero coefficient bi, so that when xl, .. ., xn+ 1 are all small, f(x) is close to b,+2, hence --,+2pn+l--N'n''n+2 contains a region {o< I x,I <=} x... x {o < Ix.+, I<~-} ~-~+;P~+ ON THE DIFFERENTIAL EQUATIONS SATISFIED BY PERIOD MATRICES m5 The ~b Mapping. Define on C1 + a linear mapping + by setting, for geCt + +(g)(X)=q,+l Z gQy). tg/=x Clearly +(Fr(g)h)---- g~b(h), and in particular + =f(x)" where ~b(x~)=o unless al,..., a,+, are all divisible by q, in which case +(x")=x "/q. Again there is an evident " factorization " of C~ id.y + id.> r Ct~(~ Ct(+~) > Ct +. Recall the endomorphism a of L(o+) defined by ~boexp(~ZF(X)--~ZqF(Xq)), where +(z~xb)----O unless a and bl,..., b,+ 2 are divisible by q, in which case ~(Z~Xb)=Z~/qz bn. Define +(1):L(o+) ~ LIq)(o+) by +(1)(Z~Xb)=Z~/qxb if qla, o otherwise, and +(2~: L(q)(o+) __~ L0(o+) by +(2)(zaxb)=zax b]q if qlbl, ...,qlb,+2, and o otherwise. i.e. the following diagram commutes. Theorem (2. x5). -- q~tO0~=+o~t, id. id. . CL~ ) , Cf C I > Ctq+(~) G-t L0(o+) ---~ L(q)(o-]-) L(q)(o+) ,a)> +) Proof. -- Here it is clear that the right-most and central boxes commute. Write H -1 = ~B.Z"F"; n+a @)H-~(Z.X b)= ~ B.Z--~F.X b= >~ Bq._~Z.Fq--~X b, . + a =O(q) whence (n--I)! B Xb ~tqt,',IllH-ltZ"X b~- Z B (n--I)! ffa-"xb Y~ (~ q._.f~, t ~v ~ J/--.>1 q---(__~).-----~l ff---~ =._>1 (a--~)~ I- Z (~--~)! B~ o. whence we are reduced to showing (_~:).-1--q._>1(_~).-1 - Consider the space L of all those power series in one variable 5] a.x" with n>O ord a.>bn+c with b>o, L ~ the subspace vanishing at x=o, D the operator x~x +r~x, 14 76 NICHOLAS M. KATZ are, in fact, analytic throughout the region [ P l~ < ~, l R(r) I = ~. Further, the uniqueness theorem for analytic functions shows that the relation <re)... ~(r~)~(r)= C(o, r,%(o)C(o, r) -1 is valid for [rl_<,, IR(r) t=~. Thus, the matrix of C(o, I') provides the analytic continuation of the zeta function from r=o to the region I r/<i, IR(r) I=~, in much the same way as the hypergeo- metric series analytically determines the zeta function of elliptic curves in the family E x, The analogy goes even deeper, for the matrix of C(o, F) satisfies a differential equation. This arises as follows. The generic space, Ws(p)| is, in a natural way, a module for -~, by means of an action ~r, which arises formally as the twisting . c(r, o) ~r = C(o, r). gr 0 0 where~-~ operates in WS(o)| through the second factor. Because ~ annihilates the f~-space WS(o), ~r annihilates the f2-space C(o, F)(WS(o)). Let us write W 0 and W r for the column vectors whose components are, respectively, the f~-basis for WS(o) and the f~(r)-basis of wS(p), by means of which our matrix representation of C(o, F) is given. As ~r acts on the f~(P)-space, ws(p)Nf~(F), we may write ~r(Wr) =B(F).W r where B(r) is a matrix of rational functions. To avoid confusion between matrices and mappings, let C(P) denote the matrix of the mapping C(o, F); then C(o, F)(W0)=C(P).Wr, and we have the equation oc(r) o = ~r(C(r).Wr)-- Or -Wr + C(I')~r.Wr (oc(r) + c(r)B(r)) wr = ~-- ec(r) whence -- C(F)B(F). 0F In [3] Dwork computed the Picard-Fuchs equation tbr the family of elliptic curves, X3+Y3+Z3--3FXYZ , and found it to be (in suitable bases) ~P(F) ~r -B(r)P(r) where P(F) is the period matrix of differentials of the second kind modulo exact differentials. It should be remarked that the result is of an algebraic nature: the generic
Publications mathématiques de l'IHÉS – Springer Journals
Published: Aug 30, 2007
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