Zeta functions of alternate mirror Calabi-Yau families

TL;DR

This paper establishes a link between the zeta functions of Calabi-Yau families with the same dual weights, revealing a common factor related to hypergeometric differential equations, and applies this to K3 surface pencils to explore mirror symmetry.

Contribution

It proves the existence of a common zeta function factor for Calabi-Yau invertible pencils with identical dual weights using Dwork cohomology and relates it to Picard--Fuchs equations.

Findings

Shared zeta function factors for Calabi-Yau families with same dual weights

Connection between zeta function factors and hypergeometric Picard--Fuchs equations

New cases of arithmetic mirror symmetry for K3 surface pencils

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.