On the local-global divisibility over abelian varieties

Let $p\ge 2$ be a prime number and let $k$ be a number field. Let $\mathcal{A}$ be an abelian variety defined over $k$. We prove that if ${\rm Gal} ( k ( {$\mathcal{A}$}[p] ) / k )$containsanelement$g$oforderdividing$p-1$notfixinganynon-trivialelementof${$\mathcal{A}$}[p]$and$H^1 ( {\rm Gal} ( k ( {$\mathcal{A}$}[p] ) / k ), {$\mathcal{A}$}[p] )$ is trivial, then the local-global divisibility by $p^n$ holds for $\mathcal{A}(k)$ for every $n \in $\mathbb{N}$$. Moreover, we prove a similar result without the hypothesis on the triviality of $H^1 ( {\rm Gal} ( k ( {$\mathcal{A}$}[p] ) / k ) , {$\mathcal{A}$}[p] )$,intheparticularcasewhere${$\mathcal{A}$}$ is a principally polarized abelian variety. Then, we get a more precise result in the case when $\mathcal{A}$ has dimension $2$. Finally we show with a counterexample that the hypothesis over the order of $g$ is necessary. In the Appendix, we explain how our results are related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperani and Stix.