On the local-global divisibility over abelian varieties

Gillibert, Florence; Ranieri, Gabriele

Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 847-873.

Soit $p\ne 2$ un nombre premier et $k$ un corps de nombres. Soit $𝒜$ une variété abélienne définie sur $k$. Dans cet article nous prouvons le résultat suivant : si $\mathrm{Gal}\left(k\right(𝒜\left[p\right])/k)$ contient un élément $g$ d’ordre divisant $p-1$ ne fixant aucun élément non nul de $𝒜\left[p\right]$ et que $H^1(\mathrm{Gal}(k\left(𝒜\left[p\right]\right)/k),𝒜\left[p\right])$ est trivial, alors $𝒜\left(k\right)$ satisfait le principe de divisibilité locale globale par $p^n$ pour tout $n\in \mathbb{N}$. En outre nous démontrons un résultat similaire sans la condition $H^1(\mathrm{Gal}(k\left(𝒜\left[p\right]\right)/k),𝒜\left[p\right])=0$, dans le cas particulier où $𝒜$ est une variété abélienne principalement polarisée. Ensuite nous obtenons un résultat plus précis lorsque $𝒜$ est de dimension $2$. Enfin nous démontrons que l’hypothèse sur l’ordre de $g$ 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.

Let $p\ge 2$ be a prime number and let $k$ be a number field. Let $𝒜$ be an abelian variety defined over $k$. We prove that if $\mathrm{Gal}\left(k\right(𝒜\left[p\right])/k)$ contains an element $g$ of order dividing $p-1$ not fixing any non-trivial element of $𝒜\left[p\right]$ and $H^1(\mathrm{Gal}(k\left(𝒜\left[p\right]\right)/k),𝒜\left[p\right])$ is trivial, then the local-global divisibility by $p^n$ holds for $𝒜\left(k\right)$ for every $n\in \mathbb{N}$. Moreover, we prove a similar result without the hypothesis on the triviality of $H^1(\mathrm{Gal}(k\left(𝒜\left[p\right]\right)/k),𝒜\left[p\right])$, 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.

Classification : 11R34, 11G10
Keywords: Local-global, Galois cohomology, abelian varieties, abelian surfaces
Mot clés : Local-global, Cohomologie galoisienne, variétés abéliennes, surfaces abéliennes

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