Moment zeta functions for toric Calabi-Yau hypersurfaces

arXiv:math/0702679v1 [math.NT] 23 Feb 2007 MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES ANTONIO ROJAS-LEON AND DAQING WAN 1. Introduction Let n≥2 be a positive integer. We consider the following family Xλ:x1+···+xn+1 x1···xn =λ of (n−1)-dimensional toric Calabi-Yau hypersurfaces in Gn mparameterized by λ∈A1. Let P∆be the projective toric variety associated to the Newton polytope of the above Laurent polynomial. The projective closure Yλof Xλin P∆is simply the quotient by G= (Z/(n+ 1)Z)n−1 of the following Dwork family of projective Calabi-Yau hypersurfaces in Pn: Wλ:xn+1 0+···+xn+1 n=λx0···xn. The crepant resolution of the family Yλis the mirror family of Wλ. Let Fqbe a finite field of qelements with characteristic p. In this paper, we are interested in the moment zeta function which measures the arithmetic variation of the zeta function of Xλover Fqas λvaries in Fq. The moment zeta function grew out of the second author’s study [28][29][30] of Dwork’s unit root conjecture. Its general properties were studied in Fu-Wan [11] and Wan [26][31]. Note that the zeta function of Yλdiffers from the zeta function of Xλby some trivial factors. The zeta function of the Dwork family Wλover finite fields had been studied extensively in the literature, first by Dwork [9] and Katz [17], and more recently in connection with arithmetic mirror symmetry by Candelas, de la Ossa and Rodriques-Villegas [3][4], and by Wan [33][34] and Fu-Wan [14]. By [33][34], the zeta function of Xλis the most primitive piece of the zeta function of Wλ. Thus, we shall restrict ourself to the family Xλ. The Hasse-Weil zeta function (but not its higher moment zeta function which would seem to be too hard at the moment) in a similar number field example is studied in a recent paper by Harris, Shepherd-Barron and Taylor [16]. More precisely, for a positive integer d, let Nd(k) denote the number of points on the family Xλsuch that xi∈Fqdk for all 1 ≤i≤nand 1 2 ANTONIO ROJAS-LEON AND DAQING WAN λ∈Fqk. The d-th moment zeta function of the morphism Xλ→λ∈A1 is defined to be Zd(A1, Xλ) = exp( ∞ X k=1 Nd(k) kTk)∈1 + TZ[[T]]. This sequence Zd(A1, Xλ) (d= 1,2,···) of power series gives a simple diophantine reformulation on the arithmetic variation of the zeta function of the family Xλ. It is a rational function in Tfor each d. In the special case n= 2, Xλis a family of elliptic curves and the moment zeta function Zd(A1, Xλ) is closely related to arithmetic of modular forms. In general, Dwork’s unit root zeta functions [10] attached to this family are the p-adic limits of this sequence of moment zeta functions. They are thus infinite p-adic moment zeta functions in some sense. Our aim of this paper is to give a precise study of this sequence Zd(A1, Xλ) and their p-adic variation as dvaries p-adically. One main consequence of our results is a complete determination of the purity decomposition and the trivial factors for the moment zeta function Zd(A1, Xλ) for all d, all nand all pnot dividing n+ 1. This provides the first higher dimensional example for which all higher moment zeta functions are determined. Theorem 1.1. Assume that pdoes not divide n+ 1. Then, the d-th moment zeta function has the following factorization Zd(A1, Xλ)(−1)n−1=Pd(T)Qd(T), where Qd(T)is the trivial factor given explicitly by (1 −qd(n−1) 2T)1+(−1)d+n 2 (1 −qd(n−1) 2+1T)−(−1)n−(−1)n+d 2 [n−2 2] Y k=0 1−qdkT 1−qdk+1T n−1 Y i=0 (1−qdi+1T)(−1)i+1(n i+1), and Pd(T)is the non-trivial factor which has the form Pd(T) = Y a+b=d,0≤b≤n Pa,b(T)(−1)b−1(b−1), where each Pa,b(T)is a polynomial in 1 + TZ[T], pure of weight d(n− 1) + 1, whose degree is given explicitly in Theorem 3.10. Corollary 1.2. Assume that pdoes not divide n+1. Let Nd(k)denote the number of points on the family Xλsuch that xi∈Fqdk for all 1≤i≤nand λ∈Fqk. Then for every positive integer k, we have the estimate |Nd(k)−((qkd −1)n qk(d−1) +1 2(1 + (−1)d)qk(d(n−1) 2+1))| ≤ (D+ 2)qk(d(n−1)+1 2), MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES 3 where Dis the total degree of the rational function Pd(T). Since the first Hodge number h0,n−1(Xλ) = 1, the zeta function of each fibre Xλhas at most one non-trivial p-adic unit root. One deduces the p-adic continuity result: If nm + 1 ≤d1≤d2are positive integers such that d1≡d2(mod (p−1)pm), then Zd1(A1, Xλ)≡Zd2(A1, Xλ) (mod pm+1). For a p-adic integer s∈Zpand a residue class r∈Z/(p−1)Z, let {di}∞ i=1 be a sequence of positive integers in the residue class rmod(p−1), going to infinity as complex numbers but approaching to sas p-adic numbers, then the limit ζr,s(A1, Xλ) = lim i→∞ Zdi(A1, Xλ)∈1 + TZp[[T]] exists as a formal p-adic power series. This limit depends only on sand r, not on the particular chosen sequence {di}∞ i=1. The limit ζr,s(A1, Xλ) is precisely Dwork’s unit root zeta function attached to the family Xλ. It is a p-adic meromorphic function in Tfor every s∈Zpand r∈Z/(p−1)Z, as conjectured by Dwork [10] and proven by Wan [30]. It should be viewed as a two variable p-adic zeta function in (s, T). Similarly, combining p-adic methods in [30] and ℓ-adic methods, we show the limit Pr,s,b(T) = lim i→∞ Pdi,b(T)∈1 + TZp[[T]] exists and is in fact a p-adic entire function for each r,sand b. Taking the limit of the previous theorem, we obtain the following result for Dwork’s unit root zeta function. Theorem 1.3. Assume that pdoes not divide n+ 1. Then, Dwork’s unit root zeta function is given by ζr,s(A1, Xλ)(−1)n−1=1−T (1 −qT)n+1 n Y b=0 Pr,s,b(T)(−1)b−1(b−1). In the elliptic family case, the p-adic entire function Pr,s,0(T) (b= 0) is the characteristic power series of the Up-operator acting on the space of overconvergent p-adic modular forms. It would be interesting but apparently difficult to get precise information on the slopes for the entire function Pr,s,b(T). Buzzard [2] had a complicated but explicit conjecture in some elliptic modular cases. We now briefly explain the ideas in proving the above theorems. For a prime ℓ6=p, let Hj(K) denote the relative ℓ-adic cohomology with 4 ANTONIO ROJAS-LEON AND DAQING WAN compact support of the family Xλ. Then, Zd(A1, Xλ) can be expressed in terms of the L-function over A1of the d-th Adams operation of the sheaf Hj(K): Zd(A1, Xλ) = 2(n−1) Y j=0 L(A1,[Hj(K)]d)(−1)j. It is thus a rational function in Tfor each positive integer d. For a prime number ℓwhich may be equal to p, let Fℓbe the nontrivial part of the relative ℓ-adic cohomology with compact support of the family Xλparameterized by λ∈A1. If ℓ6=p, then Fℓis the nontrivial part of the middle dimensional relative cohomology Hn−1(K) and the generic rank of Fℓis n. If ℓ=p, then the generic rank of Fp is 1 as the first Hodge number h0,n−1(Xλ) = 1 and the family Xλis generically ordinary [32]. The d-th moment zeta function is then given up to trivial factors, by the d-th moment L-function: Zd(A1, Xλ)∼L(A1,[Fℓ]d)(−1)n−1, where [Fℓ]ddenotes the d-th Adams operation of the sheaf Fℓon A1. Similarly, the unit root zeta function ζr,s(A1, Xλ) is given up to trivial factors by the unit root L-function: ζr,s(A1, Xλ)∼L(A1, ω(Fp)r⊗(Fp⊗ω(Fp)−1)s)(−1)n−1, where ω(Fp) denotes the Teichm¨uller lifting of the reduction Fp⊗Fp. Fix a prime number ℓ6=p, let Fdenote the ℓ-adic sheaf Fℓ. For non-negative integers aand b, let Ga,b := SymaF ⊗∧bF, which is an ℓ-adic sheaf on A1, vanishing if b > n. Thus, we shall assume that 0 ≤b≤nfrom now on. The generic rank of Ga,b is n+a−1 an b, which goes to infinity as agoes to infinity. The d-th moment L-function is then given [28] by the formula L(A1,[Fℓ]d) = n Y b=0 L(A1,Gd−b,b)(−1)b−1(b−1). Thus, to a large extent, the moment zeta functions are reduced to the study of the L-function L(A1,Ga,b) of the sheaf Ga,b for all non-negative integers aand b. To understand the purity decomposition and the trivial factors of this last L-function, the key is to determine the local and global monodromy of the sheaf F. This is accomplished in Section 2. As a consequence, we obtain MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES 5 Theorem 1.4. Assume that pdoes not divide n+ 1. Let aand bbe non-negative integers with 0≤b≤n. Then, we have the formula L(A1,Ga,b) = Pa,b(T)Q[(a+b)(n−1)/2] k=0 (1 −qkT)αa,b(k) (1 −q(a+b)(n−1)/2T)δa,b (1 −q(a+b)(n−1)/2+1T)δa,b , where Pa,b(T)∈1 + TZ[T]is a polynomial whose degree is explicitly given, Pa,b(T)is pure of weight (a+b)(n−1) + 1,δa,b = 0 or 1is explicitly given by Proposition 3.9, αa,b(k)is the coefficient of xkzbin the power series {(1 −xn)···(1 −xa+n−1) (1 −x2)···(1 −xa)}(1 + z)(1 + xz)···(1 + xn−1z), where the quantity in the bracket is understood to be 1−xnif a= 1, and 1−xif a= 0. Let now s∈Zpbe a p-adic integer and let rbe a residue class modulo (p−1). Choose a sequence of positive integers {di}∞ i=1 in the residue class rmodulo (p−1), going to infinity as complex numbers but approaching to sas p-adic integers. For integers 0 ≤b≤n, we define Lr,s,b(A1, T ) = lim i→∞ L(A1,Gdi−b,b)∈1 + Zp[[T]]. This limit exists as a formal p-adic power series. It depends only on r, s and b, not on the choice of the sequence {di}∞ i=1. It follows from the general result in [30] that Lr,s,b(A1, T) is a p-adic meromorphic function in T. The formula L(A1, ω(Fp)r⊗(Fp⊗ω(Fp)−1)s) = n Y b=0 Lr,s,b(A1, T)(−1)b−1(b−1) shows that the unit root L-function on the left side is also p-adic meromorphic in T. It study is reduced, to a large extent, to the study of the L-functions Lr,s,b(A1, T ) for all r, s and b. Combining the above theorem together with the p-adic limiting argument in [30], we obtain the following more precise result. The proof is similar to the one given in [13] for the Kloosterman family. The key point is that the number of p-adic zeros of the polynomial Pa,b(T) in any fixed p-adic disc |T|p< M (Mfinite) is uniformly bounded for all aand b. This fact holds only for those motive Fwhose first Hodge number is 1, which is the case for Calabi-Yau hypersurfaces. Theorem 1.5. Assume that pdoes not divide n+ 1. Let aand bbe non-negative integers with 0≤b≤n. Then, for each s∈Zpand each 6 ANTONIO ROJAS-LEON AND DAQING WAN residue class r∈Z/(p−1)Z, we have the factorization Lr,s,b(A1, T) = Pr,s,b(T) ∞ Y k=0 (1 −qkT)βb(k), where Pr,s,b(T)∈1 + TZp[[T]] is a p-adic entire function and βb(k)is the coefficient of xkzbin the power series (1 + z)(1 + xz)···(1 + xn−1z) (1 −x2)(1 −x3)···(1 −xn−1). In particular, Lr,s,b(A1, T)is a p-adic entire function with a zero at T=q−kof multiplicity at least βb(k)for each non-negative integer k. It would be interesting to determine the slopes of the polynomials Pa,b(T) and the entire functions Pr,s,b(T). This seems to be quite difficult in general. The simplest case n= 2 (the elliptic family case) has been studied extensively in connection to slopes of modular forms, over-convergent p-adic modular forms [27], the eigencurve [5] and the Gouvea-Mazur conjectures. The first step may be to get a good explicit lower bound for the p-adic Newton polygon of Pa,b(T) and Pr,s,b(T). Such a good lower bound is already quite non-trivial to obtain. Taking partial derivative with respect to zin the generating function for βb(k) and then setting z=−1, we deduce ∞ X k=0 ( n X b=0 (−1)b−1bβb(k))xk= 1 −x. This together with the previous theorem implies Corollary 1.6. Assume that pdoes not divide n+ 1. Then, the unit root L-function is given by L(A1, ω(Fp)r⊗(Fp⊗ω(Fp)−1)s) = 1−T 1−qT n Y b=0 Pr,s,b(T)(−1)b−1(b−1). It would be of great interest to understand the cancellation nature in the above alternating product of p-adic entire functions. The paper is organized as follows. In Section 2, we determine both the local monodromy and the global monodromy of the sheaf F. These results are then used in Section 3 to calculate the L-function of the sheaf Ga,b and its local factors at bad points. In Section 4, we treat the degenerate case when pdivides n+ 1. Acknowledgements. We thank L. Fu and N. Katz for helpful comments and for providing several relevant references. We were informed by Katz that many of the results in Section 2 were proved indepedently MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES 7 by him in his forthcoming paper [23] on the Dwork family. The second author was partially supported by NSF. The first author was partially supported by MTM2004-07203-C02-01 and FEDER. 2. The monodromy via Fourier transform. Let k=Fqbe a finite field of characteristic p,n≥2 an integer, X⊂ An+1 kthe hypersurface defined by x1···xn+1 = 1, and σ:X→A1 kthe restriction of the sum map (x1,...,xn+1)→x1+...+xn+1 to X. Fix a prime ℓ6=p. We want to study the local monodromy of the non-trivial part of the object K:= Rσ!¯ Qℓ∈ Db c(A1 k,¯ Qℓ), which parameterizes the cohomology of the family described in the introduction. The main results are summarized in the following theorem: Theorem 2.1. The cohomology sheaves Hj(K) = Rjσ!¯ Qℓvanish for j < n −1and j > 2n−2. We have isomorphisms Hj(K)∼ =¯ Q(n j−n+2) ℓ(n−1−j) for n≤j≤2n−2, and an exact sequence 0→¯ Qn ℓ→ Hn−1(K)→ F → 0 where Fis the extension by direct image of a geometrically irreducible smooth sheaf on the dense open set U=A1 k−{(n+ 1)ζ:ζn+1 = 1}, of rank nand punctually pure of weight n−1. It is endowed with a non-degenerate pairing Φ : F ×F → ¯ Qℓ(1 −n), which is symmetric if nis odd and skew-symmetric if nis even. As a representation of the inertia group at infinity, Fis unipotent with a single Jordan block. If pdoes not divide n+ 1,Fis everywhere tamely ramified. The inertia group at each of the n+ 1 singular points x= (n+ 1)ζacts on F¯ηwith invariant subspace of codimension 1. On the quotient F¯η/FIx ¯η, Ixacts trivially if nis even, and through its unique character of order 2if nis odd. If pdivides n+ 1, let n+ 1 = pam, with mprime to p. Then Fis smooth on Gm, and the inertia group at 0acts with invariant subspace of dimension m−1. The action of I0on the quotient F¯η/FI0 ¯ηis totally wild, with a single break 1/(pa−1) with multiplicity m(pa−1) = n−m+ 1. In particular, the Swan conductor at 0is m. The determinant of Fis the geometrically constant sheaf ¯ Qℓ(−n(n− 1)/2) if nis even or pdivides n+1, and the pulled back Kummer sheaf Lχ(λn+1−(n+1)n+1)(−n(n−1)/2) if nis odd and (p, n + 1) = 1, where χis the unique character of order 2of the inertia group I0. 8 ANTONIO ROJAS-LEON AND DAQING WAN The geometric monodromy group of Fis given by                      Sp(n, Φ) if nis even O(n, Φ) if nis odd and (p, n + 1) = 1 SO(n, Φ) if nis odd, p|n+ 1 and (p, n)6= (2,5) or (2,7) G2in its standard 7-dimensional representation if p= 2,n= 7 SL(2) in sym4of its standard representation if p= 2,n= 5 We will deduce most of the properties of the object Kfrom the properties of its Fourier transform L∈ Db c(A1 k,¯ Qℓ) with respect to a fixed non-trivial additive character ψ:k→C⋆∼ →¯ Q⋆ ℓ. The Fourier transform Lis closely related to the Kloosterman sheaf. This connection of the Dwork family with Kloosterman sums was first discovered by Katz [18] (Section 5.5) who uses the properties of the family to get information on certain Kloosterman sums. We will use this connection the other way around and apply Katz’s fundamental results for the Kloosterman sheaf. Recall (cf. [22]) that the Fourier transform is defined by FTψ(K) = Rπ2!(π⋆ 1(K)⊗µ⋆Lψ)[1] where π1, π2:A2 k→A1 kare the projections, µ:A2 k→A1 kis the product map and Lψis the Artin-Schreier sheaf on A1 kassociated to the character ψ. It is an auto-equivalence of the triangulated category Db c(A1 k,¯ Qℓ), and has the following involution property: F T¯ ψFTψ(K) = K(−1). One of the main advantages of this equivalence is that, following Laumon (cf. [24]), the local properties of the object Kcan be read from those of its Fourier transform. This is the method that we will use to deduce most of the results about K. Let us first determine what the Fourier transform of Kis explicitly. Using proper base change on the cartesian diagram X˜π1 ←−−− X×A1 k σ  y  y˜σ A1 k π1 ←−−− A2 k we get π⋆ 1(K) = π⋆ 1(Rσ!¯ Qℓ) = R˜σ!˜π⋆ 1¯ Qℓ= R˜σ!¯ Qℓ MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES 9 By the projection formula, we have then L= Rπ2!((R˜σ!¯ Qℓ)⊗µ⋆Lψ)[1] = Rπ2!(R˜σ!(˜σ⋆µ⋆Lψ))[1] = R˜π2!(˜µ⋆Lψ)[1] where ˜π1and ˜π2are the projections of X×A1 konto its factors and ˜µ:X×A1 k→A1 kis the map ((x1,...,xn+1), t)7→ t(x1+...+xn+1). Extend the canonical map L→j⋆j⋆Lto a distinguished triangle (1) M→L→j⋆j⋆L→. in Db c(A1 k,¯ Qℓ), where j:A1 k−{0}֒→A1 kis the open immersion. The object Mis punctual supported at 0, since L→j⋆j⋆Lis an isomorphism away from 0. At 0, the object Lis just RΓc(X⊗¯ k, ¯ Qℓ)[1] by proper base change. Since Xis just the product of ncopies of Gm, we have L0= n O i=1 RΓc(Gm,¯ k,¯ Qℓ)[1] From H1 c(Gm,¯ k,¯ Qℓ) = ¯ Qℓ, H2 c(Gm,¯ k,¯ Qℓ) = ¯ Qℓ(−1) and Hi c(Gm,¯ k,¯ Qℓ) = 0 for i6= 1,2, we conclude Hi−1(L)0=¯ Q(n i−n) ℓ(n−i) for n≤i≤2n, and 0 otherwise, so we get a quasi-isomorphism L0∼ = 2n M i=n ¯ Q(n i−n) ℓ(n−i)[1 −i] Away from 0, we have L= R˜π2!(˜µ⋆Lψ)[1], where we now regard ˜π2! as the projection X×Gm→Gm. Consider the automorphism φof An+1 k×Gmgiven by φ((x1,...,xn+1), t) = ((tx1,...,txn+1), t). The image of X×Gmunder φis the variety Ydefined by the equation x1···xn+1 =tn+1, and ˜µ= ˜σ◦φ. Since φis an automorphism, φ⋆= Rφ⋆= Rφ!, and we get j⋆L= R˜π2!(˜µ⋆Lψ)[1] = R˜π2!(φ⋆˜σ⋆Lψ)[1] = = R(˜π2φ)!(˜σ⋆Lψ)[1] = R˜π2!(˜σ⋆Lψ)[1] The stalk of j⋆Lat a geometric point t∈Gm,¯ kis then RΓc({x1···xn+1 = tn+1},Lψ(Pxi))[1]. By [7], Th´eor`eme 7.4, we deduce that Hi(j⋆L) = 0 for i6=n−1, and Hn−1(j⋆L) is the pull-back by the (n+ 1)-th power map of the Kloosterman sheaf given in [7], Th´eor`eme 7.8 and, more generally, in [19], 4.1.1. Therefore we have a quasi-isomorphism j⋆L∼ =[n+ 1]⋆Kln+1(ψ)[1 −n] Denote by Lthe sheaf [n+ 1]⋆Kln+1(ψ) on Gm. It is geometrically irreducible, because it is already irreducible as a representation of the 16 ANTONIO ROJAS-LEON AND DAQING WAN (n+ 1)ζ, with the inertia groups acting via their character χof order two. Therefore, (det F)⊗ˇ Lχ(tn+1−(n+1)n+1)is everywhere unramified, and thus geometrically trivial. So there is some ℓ-adic unit αsuch that det F∼ =αdeg ⊗ Lχ(tn+1−(n+1)n+1). To find the exact value of α, we again evaluate the determinant at t=∞to be qn(n−1)/2using Proposition 2.8. On the other hand, using that Lχ(tn+1−(n+1)n+1)= Lχ(tn+1)⊗Lχ(1+ (n+1)n+1 tn+1 )=Lχ(1+ (n+1)n+1 tn+1 )(since χhas order 2 and n+ 1 is even), we conclude that the Frobenius element at infinity acts trivially on Lχ(tn+1−(n+1)n+1), and therefore α=qn(n−1)/2and det(F) = Lχ(tn+1−(n+1)n+1)(−n(n−1)/2).  Corollary 2.10. Suppose that nis odd and (p, n + 1) = 1, and let t∈Fq. Then the action of a geometric Frobenius element Ftat ton F has χ(tn+1 −(n+1)n+1)q(n−1)/2as an eigenvalue (where χ:F⋆ q→C⋆is the unique character of order 2) and the remaining eigenvalues appear in complex conjugate pairs. Proof. From the previous theorem we know that the product of the eigenvalues is χ(tn+1−(n+1)n+1)qn(n−1)/2. They all have absolute value q(n−1)/2and, given that F((n−1)/2) is self-dual, they are permuted by the map z7→ qn−1/z. So the non-real eigenvalues show up in complex conjugate pairs. There are an odd number of real eigenvalues, all of them necessarily equal to q(n−1)/2or −q(n−1)/2. Grouping them in pairs of identical eigenvalues, we are left with just one, whose sign must be χ(tn+1 −(n+ 1)n+1) (since the product of the other ones is positive). Proposition 2.11. The geometric monodromy group Gof Fis given by                      Sp(n, Φ) if nis even O(n, Φ) if nis odd and (p, n + 1) = 1 SO(n, Φ) if nis odd, p|n+ 1 and (p, n)6= (2,5) or (2,7) G2in its standard 7-dimensional representation if p= 2,n= 7 SL(2) in sym4of its standard representation if p= 2,n= 5 Proof. The connected component G0of Gcontaining the identity is semisimple by [6], 1.3.9. Since Gcontains a unipotent element with a single Jordan block, its Lie algebra gis simple and contains a nilpotent element with a single Jordan block and the representation g→End(F¯η) is faithful and irreducible, by [19], 11.5.2.3. By 2.3, we have an a priori inclusion G⊂Sp(n, Φ) for neven and G⊂O(n, Φ) for nodd. MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES17 Suppose that n+1 is prime to p. Then Gcontains pseudo-reflections (i.e. elements with invariant subspace of codimension 1). Since any element in Gnormalizes g, from [20], Theorem 1.5 we conclude that g=spnif nis even and g=sonif nis odd. Consequently, G=Sp(n, Φ) if nis even and G=SO(n, Φ) or O(n, Φ) if nis odd. But the local monodromies at the points t∈(n+ 1)µn+1(¯ k) contain elements of determinant −1, so Gmust be the full orthogonal group. When pdivides n+ 1, we will make use of the classification theorem in [19], 11.6. According to it, the possibilities for gare: sl2in the (n−1)-th symmetric power of its standard representation, spnif nis even, sonif nis odd and g2in its standard 7-dimensional representation if n= 7. Suppose that g=sl2, and let n+ 1 = pamwith mprime to p. As in the proof of Proposition 2.9 we find a smooth sheaf Gon Gmsuch that F|Gm= [m]⋆G. Since the geometric monodromy group of Fhas finite index in that of G, their Lie algebras are the same. Let G′be the monodromy group of G. The proof of [19],11.5.2.4 shows that we have a faithful representation G′֒→GL(2) if nis even and G′֒→SO(3)×µn⊂GL(3) if nis odd. Let Hbe the corresponding sheaf. As a representation of the wild inertia group P0at 0, the breaks of Gare 0 and 1/(n+1−m), so the breaks of Hare at most 1/(n+1−m). In particular, the Swan conductor of Has a representation of P0is ≤2/(n+1−m) if nis even (≤3/(n+1−m) if nis odd). If n+1−m > 3 (or >2 if nis even), this automatically implies that His tame at zero as a representation of π1(Gm,¯ k) (since the Swan conductor is an integer) and therefore if factors through the abelian tame fundamental group of Gm. In particular, the monodromy group would be finite, which contradicts the assumption that g=sl2. This rules out the possibility g=sl2for all cases except (p, n) = (2,3), (2,5) or (3,2). Therefore the classification theorem forces g=spnif nis even and g=sonif nis odd as long as (p, n)6= (2,3), (2,5), (2,7) or (3,2). So in that case G=Sp(n, Φ) if nis even, and G=SO(n, Φ) if nis odd (since the determinant of Fis geometrically trivial by Proposition 2.9). If (p, n) = (2,3), (2,7) or (3,2), n+ 1 is a power of p, so Fis totally wild at 0 with Swan conductor 1. By [19], Theorem 8.7.1, applied to the sheaf ι⋆F(where ι:Gm→Gmis the inversion map), ι⋆Fis just a translation of a Kloosterman sheaf on Gm, so it has the same geometric monodromy group. Using [19], Theorem 11.1, we conclude that G=Sp(n, Φ) if (p, n) = (3,2), G=SO(n, Φ) if (p, n) = (2,3) and G=G2if (p, n) = (2,7). 18 ANTONIO ROJAS-LEON AND DAQING WAN For the remaining case p= 2, n= 5, we have two possibilities, g=so5or g=sl2in the fourth symmetric power of its standard representation. In the first case, Gwould be SO(5), since the determinant is trivial. We will rule out this possibility by computing the third moment of Fover F216 . Suppose that G=SO(5), and let Vbe the stalk of F at the generic point of A1, viewed as a representation of SO(5). The altertating square of ∧2Vof Vis irreducible, and the symmetric square sym2Vcontains the trivial representation and another irreducible factor W. So V⊗Vdecomposes as ∧2V⊕1⊕W. None of these irreducible factors is isomorphic to V, so V⊗V⊗V∼ =HomG(V⊗V, V ) (since Vis self-dual) does not contain the trivial representation. Therefore H2 c(Gm,¯ k,F⊗3) vanishes, being the dual of (V⊗V⊗V)G= 0. Since F⊗3does not have punctual sections, its H0 cvanishes too, and then the trace formula gives X t∈k⋆ Tr(Ft|Ft)3 =|Tr(F|H1 c(Gm,¯ k,F⊗3))| ≤ dim H1 c(Gm,¯ k,F⊗3)q6+ 1 2 since F⊗3is pure of weight 12. Now F⊗3has rank 125, it is smooth on Gm, tamely ramified at infinity and all its breaks at 0 are ≤1 (since the only breaks of Fat 0 are 0 and 1). Therefore its Swan conductor at 0 is at most 125, and then the Euler-Poincar´e formula gives dim H1 c(Gm,¯ k,F⊗3) = −χ(Gm,¯ k,F⊗3) = Sw0(F⊗3)≤125 so X t∈k⋆ Tr(Ft|Ft)3≤125 ·q6+ 1 2. Now using the explicit formula given in Proposition 4.1, we find for k=F216 that X t∈k⋆ Tr(Ft|Ft)3≃5.48857 ·1033 >2.5353 ·1033 ≃125 ·216(6+ 1 2) in contradiction with the inequality above. So g=sl2in sym4of its standard representation, and therefore G0=SL(2) in sym4of its standard representation. G0is normal in G, being its identity component. For every g∈G, conjugation by ggives an automorphism of G0. But every automorphism of SL(2) is inner, so there is an element g0∈G0 such that gg−1 0is in the centralizer of G0. Now the centralizer of G0 in GL(5) is the set of scalar matrices (a matrix commuting with all matrices of the form sym41a 0 1 and sym41 0 a1 MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES19 must already be a scalar). But G⊂SO(5), and the only scalar matrix in SO(5) is the identity. Therefore, g=g0∈G0, and G=G0=SL(2) in sym4of its standard representation.  3. L-functions of symmetric and alternating powers of F Throughout this section we will assume that n+ 1 is prime to p. We will describe the L-function of the smooth sheaf SymaF ⊗∧bFon the set U=A1 k− {(n+ 1)ζ:ζn+1 = 1}. For simplicity, we will assume that k=Fq, with (n+ 1)|(q−1), which is always true after a finite extension of the base field. Proposition 3.1. The L-function of Fon Uis given by L(U, F, T) = (1 −T)P(T)n+1 where P(T)∈1 + TZ[T]is a polynomial of degree n−1. If nis odd, all reciprocal roots of P(T)have absolute value q(n−1)/2. If nis even, P(T) = (1 ±q(n−2)/2T)P1(T), where all reciprocal roots of P1(T)have absolute value q(n−1)/2. Proof. Since Fis smooth, geometrically irreducible and not geometrically constant on U,L(U, F, T) = det(1 −F·T|H1 c(U⊗¯ k, F)). If j:U→P1is the inclusion, the Euler-Poincar´e formula gives χ(P1 ¯ k, j⋆F) = 1 + n−(n+ 1) = 0. Therefore, Hi(P1 ¯ k, j⋆F) = 0 for all i, and we get an isomorphism H1 c(U⊗¯ k, F)∼ =(M ζn+1=1 FI(n+1)ζ)⊕FI∞ A similar argument gives FI∞∼ =H1 c(A1 ¯ k,F). By Proposition 2.8, we have then L(U, F, T) = (1 −T)Y ζn+1=1 det(1 −F·T|FI(n+1)ζ) But the isomorphism F∼ =[ζ]⋆Fimplies that P(T) = det(1 −F· T|FI(n+1)ζ) is independent of ζ. The absolute values of the reciprocal roots of Pare given by Proposition 2.6.  We now turn to the study of the L-function of the sheaf Ga,b := SymaF ⊗∧bF, which is smooth of rank n+a−1 an band pure of weight (a+b)(n−1) on U. Let us find the bad factor of the L-function at infinity first. The local monodromy of Ga,b it infinity is clearly unipotent, since that of Fis. By Proposition 3.1, the eigenvalues of the geometric Frobenius element at infinity acting on Ga,b are qi1+···+ia+j1+···+jbfor 20 ANTONIO ROJAS-LEON AND DAQING WAN all possible choices of integers 0 ≤i1≤i2≤ ··· ≤ ia≤n−1 and 0≤j1< j2<···< jb≤n−1. Let Nn,a,b,k be the number of such possible choices with i1+···+ia+j1+···+jb=k, that is, Nn,a,b,k = #{(i1,...,ia, j1,...,jb) : 0 ≤i1≤i2≤ ··· ≤ ia≤n−1, 0≤j1< j2<···< jb≤n−1, i1+···+ia+j1+···+jb=k} It is clear that Nn,a,b,k =Nn,a,b,(a+b)(n−1)−k(just change il7→ n−1− ia+1−land jl7→ n−1−jb+1−l) and Nn,a,b,k = 0 for k < b(b−1)/2 and k > (a+b)(n−1) −b(b−1)/2. Proposition 3.2. The dimension of the invariant subspace GI∞ a,b is Nn,a,b,c where c=⌊(a+b)(n−1) 2⌋and Nn,a,b,c is the coefficient of xczbin the expansion of the power series (1 −xn)···(1 −xa+n−1) (1 −x)···(1 −xa)(1 + z)(1 + xz)···(1 + xn−1z). If (a+b)(n+ 1) is even, all Jordan blocks for the action of I∞on Ga,b have odd size, and the number of blocks of size 2k+ 1 is Nn,a,b,c−k− Nn,a,b,c−k−1for all k≥0. If (a+b)(n+ 1) is odd, all Jordan blocks for the action of I∞on Ga,b have even size, and the number of blocks of size 2k+ 2 is Nn,a,b,c−k−Nn,a,b,c−k−1for all k≥0. Proof. This is just a translation of [6], 1.8.4 and [19], 7.0.7 to this particular situation, considering that Ga,b is pure of weight (a+b)(n−1) and all Frobenius eigenvalues of Ga,b at infinity are integral powers of q(that is, they have even weight). In fact, the multiplicity Nn,a,b,0of the minimun Froebnius eigenvalue q0is equal to the number of Jordan blocks with length (a+b)(n−1) + 1. Removing these blocks, then the multiplicity Nn,a,b,1−Nn,a,b,0of the minimun remaining Frobenius eigenvalue qis equal to the number of blocks with length (a+b)(n− 1)−1. By induction, for 0 < k ≤c, one deduces that Nn,a,b,k −Nn,a,b,k−1 is equal to the number of blocks with length (a+b)(n−1)−2k+1 and with minimun Frobenius eigenvalue qk. The dimension of the invariant subspace GI∞ a,b is simply the total number of Jordan blocks: c X k=0 (Nn,a,b,k −Nn,a,b,k−1) = Nn,a,b,c.  Corollary 3.3. The local L-function of j⋆Ga,b at infinity has degree Nn,a,b,c and is given by det(1 −F∞·T|GI∞ a,b ) = c Y k=0 (1 −qkT)αa,b(k) MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES21 where αa,b(k) = Nn,a,b,k −Nn,a,b,k−1. We can construct a generating function for α(k) in the following way. Let Cn,a,k = #{(i1,...,ia) : 0 ≤i1≤i2≤ ··· ≤ ia≤n− 1, i1+···+ia=k}= #{(h0,...,hn−1) : 0 ≤hi, h0+···+hn−1= a, h1+2h2+···+ (n−1)hn−1=k}(to check that both numbers agree, just let hjbe the number of l= 1,...,a such that il=j). By [13], Theorem 3.1, we have X k≥0 (Cn,a,k −Cn,a,k−1)xk={(1 −xn)···(1 −xn+a−1) (1 −x2)···(1 −xa)}, where the quantity in the bracket is understood to be 1 −xnif a= 1, and 1 −xif a= 0. Let Bn,b,j = #{(j1,...,jb) : 0 ≤j1<···< jb≤n−1, j1+···+jb=j}. It is the coefficient of xjzbin the expansion of (1 + z)(1 + xz)···(1 + xn−1z). Then Nn,a,b,k = k X j=0 Cn,a,k−jBn,b,j, and thus αa,b(k) = Nn,a,b,k −Nn,a,b,k−1= k−1 X j=0 (Cn,a,k−j−Cn,a,k−j−1)Bn,b,j +Bn,b,k. Therefore αa,b(k) is the coefficient of xkzbin the expansion of {(1 −xn)···(1 −xa+n−1) (1 −x2)···(1 −xa)}(1 + z)(1 + xz)···(1 + xn−1z). In particular, the number Nn,a,b,c is the coefficient of xczbin the expansion of the power series (1 −xn)···(1 −xa+n−1) (1 −x)···(1 −xa)(1 + z)(1 + xz)···(1 + xn−1z). We now look for the bad factors of the L-function at the finite singular points t= (n+1)ζwith ζn+1 = 1. Suppose that nis even. Then the local monodromy at tis unipotent, with a Jordan block of size 2 and all other blocks of size 1. The Frobenius eigenvalues on FItare ǫq(n−2)/2, with ǫ= 1 or −1, and (n−2)/2 pairs of conjugate complex numbers α1,...,α(n−1)/2,¯α1,...,¯α(n−1)/2of absolute value q(n−1)/2. That is, as a representation of It,F∼ =U2⊕1n−2, where Umdenotes the unique (up to isomorphism) non-trivial unipotent tame representation 22 ANTONIO ROJAS-LEON AND DAQING WAN of Itof dimension mwith a single Jordan block. Therefore, we get isomorphisms SymaF∼ = a M i=0 SymiU2⊗Syma−i1n−2= a M i=0 U(n−3+a−i n−3) i+1 ∧bF∼ =∧b1n−2⊕(U2⊗∧b−11n−2)⊕∧b−21n−2∼ =1(n−2 b−2)+(n−2 b)⊕U(n−2 b−1) 2. Lemma 3.4. Let Vand Wbe vector spaces of dimensions n≥2and 2 respectively over an algebraically closed field kof characteristic 0, and let T:V→Vand U:W→Wbe unipotent endomorphisms with a single Jordan block. Then T⊗U:V⊗W→V⊗Wis unipotent with two Jordan blocks of sizes n+ 1 and n−1. Proof. Let {x,y}be a basis for Wsuch that U(x) = xand U(y) = x+y. We claim that the invariant subspace of T⊗Uis the subspace of elements that can be written as v⊗x+ (v−T(v)) ⊗yfor v∈ Ker((T−IV)2), which has dimension 2 by hypothesis: (T⊗U)(v⊗x+ (v−T(v)) ⊗y) = T(v)⊗x+T(v−T(v)) ⊗(x+y) =T(v)⊗x+ (v−T(v)) ⊗(x+y) = v⊗x+ (v−T(v)) ⊗y. Conversely, if (T⊗U)(v⊗x+w⊗y) = v⊗x+w⊗y, we get T(w) = w and T(v)+T(w) = v, so w=T(w) = v−T(v) and (T−IV)2(v) = 0. This shows that T⊗Uhas precisely two Jordan blocks. From T⊗U−I⊗I= (T−I)⊗(U−I) + I⊗(U−I) + (T−I)⊗I we get that (T⊗U−I⊗I)n+1 is a sum of terms (T−I)α⊗(U−I)βwith α+β≥n+ 1 and therefore equal to 0, since (T−I)n= (U−I)2= 0. So the Jordan blocks of T⊗Uhave size ≤n+ 1. Finally, if v∈V is a vector such that w:= (T−I)n−1(v)6= 0 and x,y∈Ware as above, the same expression shows that (T⊗U−I⊗I)n(v⊗y) = (n−1)(T−I)n−1(v)⊗(U−I)(y) = (n−1)w⊗x6= 0, so v⊗y generates a Jordan block of size n+ 1, and the other block must have size 2n−(n+ 1) = n−1.  Corollary 3.5. Suppose that nis even. As a representation of It, Ga,b = SymaF ⊗∧bFis isomorphic to a M i=0 U(n−3+a−i n−3)[(n−2 b−2)+(n−2 b)] i+1 ⊕U(n−3+a−i n−3)(n−2 b−1) i⊕U(n−3+a−i n−3)(n−2 b−1) i+2 = a+2 M i=1 Ud(i) i where d(i) = n−3+a−i n−3+n−1+a−i n−3n−2 b−1+n−2+a−i n−3n−2 b−2+n−2 b. MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES23 Corollary 3.6. Suppose that nis even. The local L-function of j⋆Ga,b at t,det(1 −Ft·T|GIt a,b)has degree Dn,a,b := Pa+2 i=1 d(i) = n−3+a n−2+n−1+a n−2n−2 b−1+n−2+a n−2n−2 b−2+n−2 b. For every i= 1,...,a+ 2, it has d(i)roots which are pure of weight (a+b)(n−1) −(i−1). For nodd, the situation is much simpler. In that case, as a representation of It,F∼ =χ2⊕1n−1, where χ2:It→¯ Q⋆ ℓis the unique character of order 2. Therefore, we get isomorphisms SymaF∼ = a M i=0 Symiχ2⊗Syma−i1n−1∼ = a M i=0 ieven 1(n−2+a−i n−2)⊕ a M i=0 iodd χ(n−2+a−i n−2) 2 ∧bF∼ =∧b1n−1⊕(χ2⊗∧b−11n−1)∼ =1(n−1 b)⊕χ(n−1 b−1) 2. SymaF ⊗∧bF∼ =1α(n−1 b)+β(n−1 b−1)⊕χα(n−1 b−1)+β(n−1 b) 2 where α= a X i=0 ieven n−2 + a−i n−2, β = a X i=0 iodd n−2 + a−i n−2 Corollary 3.7. Suppose that nis odd. The local L-function of j⋆Ga,b at t,det(1 −Ft·T|GIt a,b)has degree Dn,a,b := n−1 ba X i=0 ieven n−2 + a−i n−2+n−1 b−1a X i=0 iodd n−2 + a−i n−2. All its roots are pure of weight (a+b)(n−1). Consider the sheaf j⋆Ga,b on P1. The Tate-twisted sheaf Ga,b((n− 1)(a+b)/2) is self-dual, so Poincar´e duality gives a perfect pairing of Gal(¯ k/k)-modules Hi(P1 ¯ k, j⋆Ga,b)×H2−i(P1 ¯ k, j⋆Ga,b)→¯ Qℓ((a+b)(1 −n)−1) for i= 0,1,2. Since Ga,b is smooth on U, the zeroth cohomology group H0(P1 ¯ k, j⋆Ga,b) corresponds to the maximal geometrically constant subsheaf of Ga,b. Since Ga,b is pure of weight (n−1)(a+b) and all Frobenius eigenvalues of j⋆Ga,b at infinity are integral powers of q, such a subsheaf must be a direct sum of copies of ¯ Qℓ((1 −n)(a+b)/2). Incidentally, this shows that H0(P1 ¯ k, j⋆Ga,b) = 0 if (n−1)(a+b) is odd. Therefore, we have: 24 ANTONIO ROJAS-LEON AND DAQING WAN Proposition 3.8. The L-function of j⋆Ga,b on P1has the form L(P1, j⋆Ga,b) = Pa,b(T) (1 −q(a+b)(n−1)/2T)δa,b (1 −q(a+b)(n−1)/2+1T)δa,b where δa,b = dim H0(P1 ¯ k, j⋆Ga,b), and Pa,b(T)is a polynomial that satisfies the functional equation Pa,b(T) = ±Trq((a+b)(n−1)+1)r/2Pa,b(1/q(a+b)(n−1)+1T) where r= deg(Pa,b). Proof. We have just seen that H0(P1 ¯ k, j⋆Ga,b) = ¯ Qℓ((1 −n)(a+ b)/2)δa,b , and Poincar´e duality implies that H2(P1 ¯ k, j⋆Ga,b) = ¯ Qℓ((1 − n)(a+b)/2−1)δa,b . This gives the denominator. The numerator is Pa,b(T) = (1−α1T)···(1−αrT), where α1,...,αr are the Frobenius eigenvalues of H1(P1 ¯ k, j⋆Ga,b). By Poincar´e duality, these eigenvalues are permuted by α7→ q(a+b)(n−1)+1/α. In particular, (Qαi)2=q((a+b)(n−1)+1)r. We have Pa,b(1/q(a+b)(n−1)+1T) = (1 −1 α1T)···(1 −1 αrT) =1 α1···αrTr(α1T−1) ···(αrT−1) = (−1)r ±Trq((a+b)(n−1)+1)r/2Pa,b(T) and the functional equation follows.  To find the dimension of H0(P1 ¯ k, j⋆Ga,b) we will use the knowledge of the global monodromy of F, as in [21]. Let Vbe the geometric generic fibre of F, regarded as a representation of π1(U⊗¯ k). We know that the Zariski closure Gof the image of π1(U⊗¯ k) in GL(V) is Sp(n) if nis even and O(n) if nis odd. The dimension we are looking for is the dimension of the invariant subspace dim(Syma(V)⊗∧b(V))G= dim HomG(Syma(V),∧b(V)) (since Vis self-dual as a representation of G). Suppose n= 2mis even. The representations of G=Sp(n) are in one to one correspondence with the representations of the Lie algebra g=spn. If L1, . . ., Lmare generators of the weight lattice for g, then SymdVis the irreducible representation with maximal weight dL1, and the kernel of the natural contraction map ∧dV→ ∧d−2Vis the irreducible representation of maximal weight L1+...+Ldfor 1 ≤d≤m. ([15], ch.17) Therefore we have ∧bV∼ =W(L1+...+Lb)⊕W(L1+...+Lb−2)⊕...⊕V if b≤mis odd and ∧bV∼ =W(L1+...+Lb)⊕W(L1+...+Lb−2)⊕...⊕1 MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES25 if b≤mis even and ∧bV∼ =∧n−bVfor m≤b≤n. So SymaV⊗∧bV contains exactly one copy of the trivial representation if a= 0 and b≤nis even or if a= 1 and b≤nis odd, and does not contain the trivial representation otherwise. Suppose n= 2m+ 1 is odd. The representations of SO(n), the connected component of Gcontaining the identity, are in one-to-one correspondence with the representations of the Lie algebra g=son contained in the tensor algebra of the standard representation. Each of them gives rise to two different representations of O(n) (given one of them, the other one is obtained by tensoring with the determinant). If L1,...,Lmare generators of the weight lattice for g, then ∧dVis the irreducible representation with maximal weight L1+...+Ldfor d≤m,∧dV∼ =∧n−dVfor m+ 1 ≤d≤n, and the kernel of the natural contraction map SymdV→Symd−2Vis the irreducible representation of maximal weight dL1(cf. [15], ch.19). Therefore we have SymaV∼ =W(aL1)⊕W((a−2)L1)⊕...⊕V if ais odd and SymaV∼ =W(aL1)⊕W((a−2)L1)⊕...⊕1 if ais even. So SymaV⊗∧bV(as a representation of g) contains exactly one copy of the trivial representation if ais even and b= 0 or n, or if a is odd and b= 1 or n−1, and does not contain the trivial representation otherwise. For Gitself, since the determinant becomes trivial only in even tensor powers of the standard representation, we get that SymaV⊗∧bV contains exactly one copy of the trivial representation and no copies of the determinant representation if ais even and b= 0, or if ais odd and b= 1, it contains exactly one copy of the determinant representation and no copies of the trivial representation if ais even and b=nor if ais odd and b=n−1, and it does not contain the trivial or the determinant representations otherwise. Therefore we get: Proposition 3.9. The dimension δa,b = dim H0(P1 ¯ k, j⋆Ga,b)is if nis even 1if a= 0 and b≤nis even or a= 1 and b≤nis odd 0otherwise if nis odd 1if ais even and b= 0 or ais odd and b= 1 0otherwise Putting everything together, we get the following expression for the L-function of Ga,b: 32 ANTONIO ROJAS-LEON AND DAQING WAN Corollary 4.3. 1) The rank of the sheaf Fat 0is m−1. 2) The local L-function of the sheaf Fat 0is given by Y d≥1Y σ∈Sm,d 1−TdGn+1 pd(σ(pd−1)) pd!. Proof. From the given formula for the L-function, we see that the degree of the non-trivial part is given by |Sm|=m−1. . References [1] Beilinson, A., Bernstein, J. and Deligne, P. Faisceaux pervers, in Analyse et topologie sur les espaces singuliers (I). Ast´erisque 100. Societ´e Math´ematique de France (1982). [2] Buzzard, K. Questions on slopes of modular forms, Ast´erisque 298 (2005), 1-15. [3] Candelas, P., de la Ossa, X., Rodriques-Villegas, F., Calabi-Yau manifolds over finite fields, I. http://xxx.lanl.gov/abs/hep-th/0012233. [4] Candelas, P., de la Ossa, X., Rodriques-Villegas, F., Calabi-Yau manifolds over finite fields, II. Fileds Inst. Commun., 38(2003). [5] Coleman.R and Mazur. B. The Eigencurve, in Galois Representations in Arithmetic Geometry(Durham, 1996), 1-113, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press. (1998). [6] Deligne, P. La Conjecture de Weil II. Inst. Hautes ´ Etudes Sci. Publ. Math. 52 (1980) 137–252. [7] Deligne, P. Application de la formule des traces aux sommes trigonom´etriques, in Cohomologie ´ Etale (SGA 4 1/2) (Springer-Verlag 1977), 168–232. [8] Deligne, P. and Katz, N. Groupes de Monodromie en G´eom´etrie Alg´ebrique (SGA 7 II). Lecture Notes in Mathematics 340, Springer-Verlag (1973). [9] Dwork, B. p-adic Cycles. Publ. Math. IHES, 37 (1969), 27-115. [10] Dwork, B. Normalized period matrices II. Ann. Math., 98 (1973), 1-57. [11] Fu, L. and Wan, D. Moment zeta functions, partial L-functions and partial exponential sums. Math. Ann., 328 (2004), 193-228. [12] Fu, L. and Wan, D. L-functions for symmetric products of Kloosterman sums. J. Reine Angew. Math. 589 (2005), 79–103. [13] Fu, L. and Wan, D. Trivial factors for L-functions of symmetric products of Kloosterman sheaves. arXiv:math.AG/0610228 v1 6 Oct 2006. [14] Fu, L. and Wan, D. Mirror congruence for rational points on Calabi-Yau varieties. Asian J. Math. 10 (2006), no. 1, 1-10. [15] Fulton, W and Harris, J. Representation Theory. Graduate Texts in Mathematics, 129. Springer-Verlag (1991). [16] Harris, M., Shepherd-Barron, N. and Taylor, R. A family of Calabi-Yau varieties and potential automorphy. Preprint, 2006. [17] Katz, N. Algebraic solutions of differential equations (p-curvature and the Hodge filtration. Invent. Math., 18 (1972), 1-118. [18] Katz, N. Sommes Exponentielles. Asterisque 79(1980), 209pp. [19] Katz, N. Gauss Sums, Kloosterman Sums and Monodromy Groups. Annals of Mathematics Studies 116 (Princeton University Press 1987). MOMENT ZETA FUNCTIONS FOR TORIC CALABI-YAU HYPERSURFACES33 [20] Katz, N. Exponential Sums and Differential Equations. Annals of Mathematics Studies 124 (Princeton University Press 1990). [21] Katz, N. Frobenius-Schur Indicator and the Ubiquity of Brock-Granville Quadratic Excess. Finite Fields Appl. 7(2001), 45–69. [22] Katz, N. and Laumon, G. Transformation de Fourier et Majoration de Sommes Exponentielles. Inst. Hautes ´ Etudes Sci. Publ. Math. 62 (1985) 361–418. [23] Katz, N., Another look at the Dwork family, preprint, 2007. [24] Laumon, G. Transformation de Fourier, constantes d’´equations fonctionelles et conjecture de Weil. Pub. Math. IHES 65 (1987), 131–210. [25] Koblitz, N. The number of points on certain family of hypersurfaces over finite fields. Compositio Math., 48 (1983), no.1, 3-23. [26] Wan, D. Partial zeta functions of algebraic varieties over finite fields. Finite Fields & Applications, 7(2001), 238-251. [27] Wan, D. Dimension variation of classical and p-adic modular forms. Invent. Math., 133 (1998), 449-463. [28] Wan, D. Dwork’s conjecture on unit root zeta functions. Ann. Math., 150 (1999), 867-927. [29] Wan, D. Higher rank case of Dwork’s conjecture. J. Amer. Math. Soc., 13 (2000), 807-852. [30] Wan, D. Rank one case of Dwork’s conjecture. J. Amer. Math. Soc., 13 (2000), 853-908. [31] Wan, D. Geometric moment zeta functions. In Geometric Aspects of Dwork Theory, Walter de Gruyter, 2004, VolII, 1113-1129. [32] Wan, D. Variation of p-adic Newton polygons for L-functions of exponential sums. Asian J. Math., 8(2004), 427-474. [33] Wan, D. Arithmetic mirror symmetry. Pure Appl. Math. Q., Vol. 1(2005) no. 2, 369-378. [34] Wan, D. Mirror symmetry for zeta functions. In Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, Vol. 38, 2007. Department of Mathematics, University of California, Irvine, CA 92697-3875, USA E-mail address:[email protected] Department of Mathematics, University of California, Irvine, CA 92697-3875, USA E-mail address:[email protected]