On the local-global divisibility over ${\mathrm{GL}}_2$-type varieties

Abstract

Let $k$ be a number field and let $\mathcal{A}$ be a ${\mathrm{GL}}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of $p$ does not hold for $\mathcal{A}$ over $k$, then there exists a cyclic extension $\tilde{k}$ of $k$ of degree bounded by a constant depending on $d$ such that $\mathcal{A}$ is $\tilde{k}$-isogenous to a ${\mathrm{GL}}_2$-type variety defined over $\tilde{k}$ that admits a $\tilde{k}$-rational point of order $p$. Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperiani and Stix and Creutz.

22 pages