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.
机构:
Univ Barcelona, Dept Matemat & Informat, Gran Via Corts Catalanes 585, Barcelona 08007, Catalonia, Spain
Univ Barcelona, Ctr Recerca Matemat, Gran Via Corts Catalanes 585, Barcelona 08007, Catalonia, SpainUniv Barcelona, Dept Matemat & Informat, Gran Via Corts Catalanes 585, Barcelona 08007, Catalonia, Spain
机构:
Univ Calabria, Dipartimento Matemat, IT-87036 Arcavacata Di Rende, CS, ItalyUniv Calabria, Dipartimento Matemat, IT-87036 Arcavacata Di Rende, CS, Italy
Paladino, Laura
Ranieri, Gabriele
论文数: 0引用数: 0
h-index: 0
机构:
Scuola Normale Super Pisa, Coll Puteano, IT-56100 Pisa, ItalyUniv Calabria, Dipartimento Matemat, IT-87036 Arcavacata Di Rende, CS, Italy
Ranieri, Gabriele
Viada, Evelina
论文数: 0引用数: 0
h-index: 0
机构:
Univ Basel, Dept Math, CH-4051 Basel, SwitzerlandUniv Calabria, Dipartimento Matemat, IT-87036 Arcavacata Di Rende, CS, Italy