Introduction to Arithmetic Mirror Symmetry

TL;DR

This paper explores the computation of period integrals, Picard-Fuchs equations, and zeta functions for Calabi-Yau manifolds, linking complex geometry with arithmetic over finite fields, and providing explicit examples.

Contribution

It introduces methods to derive period integrals and differential equations for Calabi-Yau families and connects these to point counts over finite fields with explicit examples.

Findings

Explicit calculation of point counts using p-adic period integrals

Derivation of Picard-Fuchs equations for Calabi-Yau families

Discussion of zeta function factorizations for mirror manifolds

Abstract

We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics.