Divisibilità locale-globale in alcuni gruppi algebrici commutativi

In this thesis we study a local-global principle in algebraic groups that was first introduced by Dvornicich and Zannier in 2001, called the local-global divisibility problem. The problem is the following: given a commutative algebraic group G defined over a number field k and given q a positive integer, if a point P in G(k) is such that for all but finitely many places v of k there exists a q-divisor of P in G(k_v); can we conclude that there exists a q-divisor of P also in G(k)? In the first part of the thesis, we give a complete answer when q is any power of an odd prime, in the case of algebraic tori. In the second part, we study the 7-division fields of some families of elliptic curves with CM. The study of these fields is related to the local-global divisibility problem.