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 $\mathop {\mathrm{Gal}} (k (\mathcal{A}[p]) / k)$ contains an element $g$ of order dividing $p-1$ not fixing any non-trivial element of $\mathcal{A}[p]$ and $H^1 (\mathop {\mathrm{Gal}} (k (\mathcal{A}[p]) / k), \mathcal{A}[p])$ 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 $H^1 (\mathop {\mathrm{Gal}} (k (\mathcal{A}[p]) / k), \mathcal{A}[p])$, in the particular case where $\mathcal{A}$ 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 that the hypothesis over the order of $g$ is necessary, by providing a counterexample.

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 Ciperiani and Stix and Creutz.