Uniformity of stably integral points on principally polarized abelian varieties of dimension ≤2

Abstract

The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. Most of our argument works in arbitrary dimension and the restriction on the dimension ≤2 is used only at the last step, where we apply Pacelli’s stronger uniformity results for elliptic curves.