Zeta functions of alternate mirror Calabi–Yau families

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.

Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

“…The main aim of this paper is to generalize and to comment on a recent result of Doran, Kelly, Salerno, Sperber, Voight and Whitcher [8] on the zeta function of certain pencils of Calabi-Yau hypersurfaces. For a more extensive discussion on the history of this particular result we refer to the introduction of [8].…”

“…In a recent preprint Doran, Kelly, Salerno, Sperber, Voight and Whitcher [8] showed the following result (using Dwork cohomology and some results on the Picard-Fuchs equation):…”

“…To describe the main results from [8], fix a matrix A := (a i,j ) 0≤i,j≤n with nonnegative integral coefficients and nonzero determinant. Then with A we can associate the polynomial Assume that the entries of A −1 (1, .…”

“…, 1)A −1 are proportional. Then for any ψ ∈ F q such that X A,ψ and X A ,ψ are smooth and nondegenerate we have that the polynomials Z(X A,ψ , T ) For a precise definition of nondegenerate we refer to the paper [8]. The condition (1, 1, .…”

Hypergeometric decomposition of symmetric K3 quartic pencils

“…Here we write µ n for the group of nth roots of unity. In previous work [DKSSVW17], we showed that these five pencils share a common factor in their zeta functions, a polynomial of degree 3 associated to the hypergeometric Picard-Fuchs differential equation satisfied by the holomorphic form. (Kloosterman [Klo17] has recently proven a more general version of this theorem, with a different approach.)…”