Abstract
We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.
Highlights
By Tate’s conjecture, a theorem due to work of Charles [Cha13], Madapusi Pera [Per15], and Kim–Madapusi Pera [KP16], the Neron–Severi rank of a K3 surface X over Fq is equal to one plus the multiplicity of q as a reciprocal root of P (T ) [vanL07, Corollary 2.3], and this rank is even. (The extra “one” corresponds to the hyperplane section, already factored in.) Theorem 5.1.3(b) implies that each X,ψ has Neron–Severi rank over the algebraic closure Fq at least 18+1 = 19, so at least 20 because it is even
Theorem 5.1.3 implies that the subspace in cohomology cut out by the Picard–Fuchs equation is contained in the SL(FA)-invariant subspace and it contains H2,0
As observed by Kloosterman [Kl17], this implies that the SL(FA)-invariant subspace in He2t(XA,ψ) contains the transcendental subspace: one definition of the transcendental lattice of a K3 surface is as the minimal primitive sub-Q-Hodge structure containing H2,0 [Huy[16], Definition 3.2.5]
Summary
X is the exponential generating function for the number of Fqr -rational points, given by. In his 1962 ICM address [Dwo62], Dwork constructed a family of endomorphisms whose characteristic polynomials determined the zeta functions of the hypersurfaces modulo p He identified a power series in the deformation parameter with rational function coefficients that satisfies an ordinary differential equation with regular singular points. Revisiting work of Gahrs [Gah11], we find that invertible pencils whose BHK mirrors are hypersurfaces in quotients of the same weighted projective space have the same Picard–Fuchs equation associated to their holomorphic form. Theorem 1.2.3 relates the zeta functions of many interesting Calabi–Yau varieties: for example, the dual weights are the same for any degree n + 1 invertible pencil composed of Fermats and loops.
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