Let be a prime number and let be a number field. Let be an abelian variety defined over . We prove that if contains an element of order dividing not fixing any non-trivial element of and is trivial, then the local-global divisibility by holds for for every . Moreover, we prove a similar result without the hypothesis on the triviality of , in the particular case where is a principally polarized abelian variety. Then, we get a more precise result in the case when has dimension . Finally, we show that the hypothesis over the order of 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.
Soit un nombre premier et un corps de nombres. Soit une variété abélienne définie sur . Dans cet article nous prouvons le résultat suivant : si contient un élément d’ordre divisant ne fixant aucun élément non nul de et que est trivial, alors satisfait le principe de divisibilité locale globale par pour tout . En outre nous démontrons un résultat similaire sans la condition , dans le cas particulier où est une variété abélienne principalement polarisée. Ensuite nous obtenons un résultat plus précis lorsque est de dimension . Enfin nous démontrons que l’hypothèse sur l’ordre de est nécessaire par un contre-exemple.
Dans l’Appendice, nous expliquons le lien entre nos résultats et une question de Cassels sur la divisibilité du groupe de Tate–Shafarevich, qui fut également étudiée par Ciperiani et Stix ainsi que Creutz.
Revised:
Accepted:
Published online:
Keywords: Local-global, Galois cohomology, abelian varieties, abelian surfaces
Mot clés : Local-global, Cohomologie galoisienne, variétés abéliennes, surfaces abéliennes
@article{AIF_2018__68_2_847_0, author = {Gillibert, Florence and Ranieri, Gabriele}, title = {On the local-global divisibility over abelian varieties}, journal = {Annales de l'Institut Fourier}, pages = {847--873}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {2}, year = {2018}, doi = {10.5802/aif.3179}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3179/} }
TY - JOUR AU - Gillibert, Florence AU - Ranieri, Gabriele TI - On the local-global divisibility over abelian varieties JO - Annales de l'Institut Fourier PY - 2018 SP - 847 EP - 873 VL - 68 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3179/ DO - 10.5802/aif.3179 LA - en ID - AIF_2018__68_2_847_0 ER -
%0 Journal Article %A Gillibert, Florence %A Ranieri, Gabriele %T On the local-global divisibility over abelian varieties %J Annales de l'Institut Fourier %D 2018 %P 847-873 %V 68 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3179/ %R 10.5802/aif.3179 %G en %F AIF_2018__68_2_847_0
Gillibert, Florence; Ranieri, Gabriele. On the local-global divisibility over abelian varieties. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 847-873. doi : 10.5802/aif.3179. http://archive.numdam.org/articles/10.5802/aif.3179/
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