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

## TL;DR

This paper investigates the local-global divisibility principle for ${
m GL}_2$-type varieties over number fields, establishing conditions under which divisibility failures imply the existence of rational points of order p over certain cyclic extensions.

## Contribution

It proves that failures of local-global divisibility by a prime power imply the existence of a cyclic extension where the variety gains a rational point of order p, linking to Cassels' divisibility question.

## Key findings

- Failure of divisibility implies existence of a cyclic extension with a rational p-torsion point.
- Bound on the degree of the cyclic extension depends only on the dimension of the variety.
- Results relate to the divisibility of the Tate-Shafarevich group.

## Abstract

Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-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 $\widetilde{k}$ 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 Ciperiani and Stix and Creutz.