Let p ≥ 2 be a prime number and let k be a number field. Let A be an abelian variety defined over k. We prove that if Gal(k(A[p])/k) contains an element g of order dividing p − 1 not fixing any non-is trivial, then the local-global divisibility by p n holds for A(k) for every n ∈ N. Moreover, we prove a similar result without the hypothesis on the triviality of H 1 (Gal(k(A[p])/k), A[p]), in the particular case where A is a principally polarized abelian variety. Then, we get a more precise result in the case when A has dimension 2. Finally we show with a counterexample that the hypothesis over the order of g is necessary.In the Appendix, we explain how our results are related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperani and Stix [C-S].

Let p be a prime number and let k be a number field. Let E be an elliptic curve defined over k. We prove that if p is odd, then the local-global divisibility by any power of p holds for the torsion points of E. We also show with an example that the hypothesis over p is necessary.We get a weak generalization of the result on elliptic curves to the larger family of GL 2 -type varieties over k. In the special case of the abelian surfaces A/k with quaternionic multiplication over k we obtain that for all prime p, except a finite number depending on A, the local-global divisibility by any power of p holds for the torsion points of A.The converse of Proposition 1 is not true. However, in the case when the group H 1 loc (Gal(k(A[p n ])/k), A[p n ]) is not trivial, we can find an extension L of k such that L and k(A[p n ]) are linearly disjoint over k, and such that the local-global divisibility by p n over A(L) does not hold (see [DZ3, Theorem 3] for the details).An abelian variety defined over a number field k is said to be of GL 2 -type if there is an embedding φ : E ֒→ End k (A) ⊗ Q, where E is a number field such that [E : Q] = dim(A) (see [R, Chapter 2, Chapter 5]). Let O E be the ring of the algebraic integers of E.is an order of O E which is isomorphic to some subring of End k (A) via φ. Following [R, Chapter 2], we say that a prime number is good for A if it does not divide the index [O E : R] (the notion of good depends also on E). We prove the following result, generalizing the result on elliptic curves.

Let k be a number field and let A be a GL 2 -type variety defined over k of dimension d. We show that for every prime number p satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of p does not hold for A over k, then there exists a cyclic extension k of k of degree bounded by a constant depending on d such that A is k-isogenous to a GL 2 -type variety defined over k that admits a k-rational point of order p.Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Proposition 3. Suppose that A is a GL 2 -type variety defined over a number field k and let E be a totally real field that embeds to End k (A) ⊗ Q. Moreover suppose that End k (A) = End k (A). Then det(ρ P ) = χ p , where χ p : Gal(k/k) → Z * p is the p-adic cyclotomic character.

Let p and q be two distinct prime numbers, and X pq /w q be the quotient of the Shimura curve of discriminant pq by the Atkin-Lehner involution w q . We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if p and q satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if p is large enough compared to q, then the quotient X pq /w q has no rational point, except possibly special points. RésuméSoient p et q deux nombres premiers distincts et X pq /w q le quotient de la courbe de Shimura de discriminant pq par l'involution d'Atkin-Lehner w q . Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si p et q satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si p est assez grand par rapport à q, alors le quotient X pq /w q n'a pas de point rationnel non spécial.

We give a positive answer to a conjecture by Bhargava, Kane, Lenstra Jr., Poonen and Rains, concerning the cohomology of torsion subgroups of elliptic curves over global fields. This implies that, given a global field k and an integer n, for 100% of elliptic curves E defined over k, the nth Selmer group of E is the intersection of two direct summands of the adelic cohomology group H1false(boldA,E[n]false). We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.