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arXiv · 2504.20230

Finitude du groupe de Tate-Shafarevich pour les groupes de type multiplicatif constants sur des corps des fonctions

Melvyn El Kamel-Meyrigne
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Abstract

Let $k_0$ be a number field, $K$ be a finite extension of $k_0(\!(x_1,...,x_n)\!)$ and let $R$ be the integral closure of $k_0[[x_1,...,x_n]]$ in $K$. Consider a group of multiplicative type $G$ defined over $K$. We study the Tate-Shafarevich group given by elements of $H^1(K,G)$ locally trivial at completions of $K$ with respect to the points of codimension $1$ of $Spec(R)$. We show the finiteness of the Tate-Shafarevich group when $G$ comes from a group of multiplicative type $G_{k_0}$ defined over $k_0$ provided that two technical conditions are satisfied. We then prove that the Tate-Shafarevich group is trivial when the ring of integers $R$ is regular. -- Soient $k_0$ un corps de nombres, $K$ une extension finie de $k_0(\!(x_1,...,x_n)\!)$ et soit $R$ la cl\^oture int\'egrale de $k_0[[x_1,...,x_n]]$ dans $K$. Soit $G$ un groupe de type multiplicatif d\'efini sur $k_0$. On \'etudie le groupe de Tate-Shafarevich donn\'e par les \'el\'ements de $H^1(K,G)$ localement triviaux aux compl\'etions de $K$ par rapport aux points de codimension $1$ de $Spec(R)$. On \'etablit la finitude du groupe de Tate-Shafarevich lorsque $G$ provient d'un groupe de type multiplicatif $G_{k_0}$ d\'efini sur $k_0$ sous r\'eserve que deux hypoth\`eses techniques soient satisfaites. On montre que le groupe de Tate-Shafarevich est trivial lorsque l'anneau des entiers $R$ est r\'egulier.

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Melvyn El Kamel-Meyrigne. 2025-04-28. Finitude du groupe de Tate-Shafarevich pour les groupes de type multiplicatif constants sur des corps des fonctions. https://arxiv.org/abs/2504.20230

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