Linear relations and arithmetic on abelian schemes

Abstract

We investigate linear relations in Mordell-Weil groups of abelian varieties over finitely generated fields over $\mathbb{Q}$. Based on important and classical results for abelian varieties over these fields and on lifts of abelian varieties to suitable abelian schemes, we prove theorems concerning the reduction maps on torsion and non-torsion elements in Mordell-Weil groups of these varieties. These theorems and the arithmetic of abelian schemes and their endomorphism algebras are our key tools in the solutions of linear relation problems we work with in the last chapter of this paper.