2016/11/30 by Gillibert, Florence, Ranieri, Gabriele · 1 citation
#11G10 #11R34 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1612.00058
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 \rm Gal ( k ( A[p] ) / k ) contains an element g of order dividing p-1 not fixing any non-trivial element of A[p] and H1 ( \rm Gal ( k ( A[p] ) / k ), A[p] ) is trivial, then the local-global divisibility by pn holds for A ( k ) for every n ∈ ℕ. Moreover, we prove a similar result without the hypothesis on the triviality of H1 ( \rm 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.