Divisibility questions in commutative algebraic groups

Abstract

Let k be a number field, let A be a commutative algebraic group defined over k and let p be a prime number. Let A[p] denote the p-torsion subgroup of A. We give some sufficient conditions for the local-global divisibility by p in A and the triviality of the Tate-Shafarevich group ▪. When A is a principally polarized abelian variety, those conditions imply that the elements of the Tate-Shafarevich group ▪ are divisible by p in the Weil-Châtelet group H1(k,A) and the local-global principle for divisibility by p holds in Hr(k,A), for all r≥0.