Sala Seminari (Dip. Matematica).
On the local-global divisibility and the Tate-Shafarevich group. Abstract: Let k be a global field and let A be a commutative algebraic group defined over k. Consider the following question : Problem. Let P be in A(k) and let q be a positive integer. Suppose that for all but finitely many places v of k, there exists Dv ∈ A(kv) such that P = qDv. Does there exist D ∈ A(k) such that P = qD? This problem is called Local-global divisibility problem by q on A over k. If for every P ∈ A(k) the answer to the Local-global divisibility problem is positive, we say that the Local-global divisibility principle for divisibility by q holds for A over k. In a joint work with Laura Paladino and Evelina Viada, we got a criterion for the Local-global divisibility problem in the particular case when A is an elliptic curve. Very recently, with Florence Gillibert, we generalized the criterion to other families of abelian varieties. In our talk we explain these last results and a link, discovered by Ciperiani and Stix, beteween the local-global divisibility problem and a question of Cassels about the Tate-Shafarevich group.