ON THE LOCAL-GLOBAL DIVISIBILITY OVER ABELIAN VARIETIES

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-trivial element of A[p] and H-1 (Gal(k(A[p])/k), A[p]) is trivial, then the local-global divisibility by p(n) holds for ,A(k) for every n is an element of 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 that the hypothesis over the order of g is necessary, by providing a counterexample. 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 Ciperiani and Stix and Creutz.