Picard-Fuchs equations of families of QM abelian surfaces

TL;DR

This paper presents an algorithm to compute Picard-Fuchs equations for families of abelian surfaces, applies it to Kummer surfaces over Shimura curves, and explores their parameter correspondences.

Contribution

The paper introduces a new algorithm for computing Picard-Fuchs equations and applies it to specific geometric families, establishing connections with Shimura curves.

Findings

Successfully computed Picard-Fuchs equations for several families.

Established correspondences between parameter spaces and Shimura curves.

Some correspondences proved rigorously.

Abstract

We describe an algorithm for computing the Picard-Fuchs equation for a family of twists of a fixed elliptic surface. We then apply this algorithm to obtain the equation for several examples, which are coming from families of Kummer surfaces over Shimura curves, as studied in our previous work. We use this to find correspondenced between the parameter spaces of our families and Shimura curves. These correspondences can sometimes be proved rigorously.