Local-global divisibility on algebraic tori

In 2001, Dvornicich and Zannier introduced the so-called Local-Global Divisibility Problem for commutative algebraic groups, as a generalization of a particular case of the Hasse principle for quadratic forms. During the last twenty years, several results have been produced for different algebraic groups. In this talk, after an introduction to the problem, we will present some results in the case of algebraic tori. In particular, we will show a generalization of the Grunwald-Wang Theorem to any algebraic tori with bounded dimension and exhibit a counterexample for the local-global divisibility by any power of an odd prime.