# Local-global questions for divisibility in commutative algebraic groups

## TL;DR

This survey explores the Hasse principle for divisibility in commutative algebraic groups, highlighting recent advances, deep connections between related problems, and implications for classical questions like Cassels' problem.

## Contribution

It provides a comprehensive overview of the long-standing and recent results on local-global divisibility questions, establishing new links and answering classical problems over .

## Key findings

- Recent developments imply an optimal answer to Cassels' question over .
- Connections between divisibility in algebraic groups and Tate-Shavarevich groups are clarified.
- The survey links divisibility questions to other local-global principles in elliptic curves.

## Abstract

This is a survey focusing on the Hasse principle for divisibility of points in commutative algebraic groups and its relation with the Hasse principle for divisibility of elements of the Tate-Shavarevich group in the Weil-Ch\^{a}telet group. The two local-global subjects arose as a generalization of some classical questions considered respectively by Hasse and Cassels. We describe the deep connection between the two problems and give an overview of the long-established results and the ones achieved during the last twenty years, when the questions were taken up again in a more general setting. In particular, by connecting various results about the two problems, we describe how some recent developments in the first of the two local-global questions imply an answer to Cassel's question, which improves all the results published before about that problem. This answer is best possible over $\mathbb{Q}$. We also describe some links with other similar questions, as for examples the Support Problem and the local-global principle for existence of isogenies of prime degree in elliptic curves.