On the local-global divisibility over abelian varieties

Resumen

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∈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 Ciperiani and Stix.