Divisibility questions in commutative algebraic groups

Abstract

Let $k$ be a number field, let $\mathcal{A}$ be a commutative algebraic group defined over $k$ and let $p$ be a prime number. Let $\mathcal{A}[p]$ denote the $p$-torsion subgroup of $\mathcal{A}$. We give some sufficient conditions for the local-global divisibility by $p$ in $\mathcal{A}$ and the triviality of $Sha(k,\mathcal{A}[p])$. When $\mathcal{A}$ is an abelian variety principally polarized, those conditions imply that the elements of the Tate-Shafarevich group $Sha(k,\mathcal{A})$ are divisible by $p$ in the Weil-Châtelet group $H^1(k,\mathcal{A})$ and the local-global principle for divisibility by $p$ holds in $H^r(k,\mathcal{A})$, for all $r\ge 0$.