Hasse--Witt matrices and mirror toric pencils

TL;DR

This paper explores the relationship between mirror symmetry and arithmetic properties of toric hypersurfaces, showing how Hasse--Witt matrices can be used to compute point counts over finite fields.

Contribution

It establishes a connection between Picard-Fuchs equations and Hasse--Witt matrices for mirror pairs of toric hypersurfaces, enabling explicit point count computations.

Findings

Number of rational points matches modulo the field size for certain mirror pairs.

Explicit computation of points for K3 surface examples.

Application of hypergeometric functions in point counting.

Abstract

Mirror symmetry suggests unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this paper, we obtain a similar result for certain mirror pairs of toric hypersurfaces. We use recent results by Huang, Lian, Yau and Yu describing the relationship between the Picard-Fuchs equations of these varieties and their Hasse--Witt matrices, which encapsulate information about the number of points. The result allows us to compute the number of points modulo the order of the field explicitly. We illustrate this by computing K3 surface examples related to hypergeometric functions.