The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field

Andrei Rapinchuk; Igor Rapinchuk

doi:10.5802/crmath.588

Article de recherche - Géométrie algébrique, Théorie des nombres

The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field
[La finitude du groupe de Tate–Shafarevich sur les corps de fonctions pour les tores algébriques définis sur le corps de base]

Andrei Rapinchuk ¹ ; Igor Rapinchuk ²

¹ Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 739-749.

Résumé
Abstract

Soit $K$ un corps et $V$ un ensemble de valuations de rang un de $K$ . Le groupe de Tate–Shafarevich d’un $K$ -tore T est $Ш (T, V) = ker (H^{1} (K, T) \to \prod_{v \in V} H^{1} (K_{v}, T))$ . Nous montrons que si $K = k (X)$ est le corps de fonctions d’une variété lisse géométriquement intègre quasi-projective définie sur un corps $k$ de caractéristique 0 et que $V$ est l’ensemble des valuations discrètes de $K$ associées aux diviseurs irréductibles de $X$ , alors pour tout tore $T$ défini sur le corps de base $k$ , le groupe $Ш (T, V)$ est fini dans les situations suivantes : (1) $k$ est de type fini sur le corps premier et $X (k) \neq \emptyset$ ; (2) $k$ est un corps de nombres.

Let $K$ be a field and $V$ be a set of rank one valuations of $K$ . The corresponding Tate–Shafarevich group of a $K$ -torus $T$ is $Ш (T, V) = ker (H^{1} (K, T) \to \prod_{v \in V} H^{1} (K_{v}, T))$ . We prove that if $K = k (X)$ is the function field of a smooth geometrically integral quasi-projective variety over a field $k$ of characteristic 0 and $V$ is the set of discrete valuations of $K$ associated with prime divisors on $X$ , then for any torus $T$ defined over the base field $k$ , the group $Ш (T, V)$ is finite in the following situations: (1) $k$ is finitely generated and $X (k) \neq \emptyset$ ; (2) $k$ is a number field.

Reçu le : 2023-07-11
Révisé le : 2023-09-19
Accepté le : 2023-11-10
Publié le : 2024-09-17

DOI : 10.5802/crmath.588

Affiliations des auteurs :

Andrei Rapinchuk ¹ ; Igor Rapinchuk ²

¹ Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Andrei Rapinchuk and Igor Rapinchuk},
     title = {The finiteness of the {Tate{\textendash}Shafarevich} group over function fields for algebraic tori defined over the base field},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {739--749},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.588},
     language = {en},
}

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Andrei Rapinchuk; Igor Rapinchuk. The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 739-749. doi : 10.5802/crmath.588. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.588/

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