Uniformity of fibres of period mappings and the $S$-unit equation

Abstract

In this paper we give a refinement of the method introduced by Lawrence and Venkatesh and thereby showing that their proof of Mordell's conjecture is uniform up to a uniform bound on the number of Galois representations attached to some family of abelian varieties. We are also able to give an unconditional proof of a uniform boundedness statement on the number of solutions to $S$-unit equations, which qualitatively is best possible, recovering a result of Evertse.