On local-global divisibility over ${\rm GL}_2$-type varieties
Volume 193 / 2020
Acta Arithmetica 193 (2020), 339-354
MSC: Primary 11R34; Secondary 11G10.
DOI: 10.4064/aa180404-15-3
Published online: 14 February 2020
Abstract
Let $k$ be a number field and let $\mathcal A $ be a ${\rm GL}_2$-type variety of dimension $d$ defined over $k$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if local-global divisibility by a power of $p$ does not hold for $\mathcal A $ over $k$, then there exists a cyclic extension $\widetilde {k}$ of $k$ of degree bounded by a constant depending on $d$ such that $\mathcal A $ is $\widetilde {k}$-isogenous to a ${\rm GL}_2$-type variety defined over $\widetilde {k}$ that admits a $\widetilde {k}$-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 Çiperiani and Stix and by Creutz.
