Bounding integral points on the Siegel modular variety A_2(2)

TL;DR

This paper establishes explicit upper bounds for the stable Faltings height of abelian surfaces with S-integral points on the Siegel modular variety A_2(2), using Runge's and Baker's methods.

Contribution

It provides the first explicit bounds for these heights in the context of A_2(2), improving upon previous general results with a novel higher-dimensional Baker's method.

Findings

Runge's method yields uniform bounds for |S|<3.

Baker's method provides bounds for |S|<10.

The explicit Baker's method improves previous general bounds.

Abstract

We determine two explicit upper bounds for the stable Faltings height of principally polarised abelian surfaces over number fields corresponding to S-integral points on the Siegel modular variety A_2(2). One upper bound, using Runge's method, is uniform in S as long as |S|<3; the other, using Baker's method, is not uniform but allows |S|<10. Our application of a higher-dimensional Baker's method is completely explicit and improves upon the general case due to Levin.