Local-global questions for divisibility in commutative algebraic groups

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–Châtelet 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 Cassels’ question, which improves all the results published before about that problem. This answer is best possible over Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Q}$$\end{document}. We also describe some links with other similar questions, for example the Support Problem and the local-global principle for existence of isogenies of prime degree in elliptic curves.