Divisibility questions in commutative algebraic groups

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 III(k, A[p]). When A is a principally polarized abelian variety, those conditions imply that the elements of the Tate-Shafarevich group III(k, A) are divisible by p in the Weil-Chatelet group H^1 (k, A) and the local-global principle for divisibility by p holds in H^r (k, A), for all r >= 0.