On the local-global divisibility over abelian varieties

TL;DR

This paper establishes conditions under which local-global divisibility by powers of a prime holds for points on abelian varieties over number fields, extending previous results and providing counterexamples and special cases.

Contribution

It proves new criteria for local-global divisibility on abelian varieties, including cases with specific Galois group elements and without certain cohomological restrictions.

Findings

Conditions involving Galois group elements ensure divisibility.

Counterexample shows necessity of hypotheses.

Results relate to Cassels' question on Tate-Shafarevich groups.

Abstract

Let $p \geq 2$ 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 ${\rm Gal} ( k ( {\mathcal{A}}[p] ) / k )$ contains an element $g$ of order dividing $p-1$ not fixing any non-trivial element of ${\mathcal{A}}[p]$ 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 $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) / k ) , {\mathcal{A}}[p] )$, 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 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.