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

Rapinchuk, Andrei; Rapinchuk, Igor

Comptes Rendus. Mathématique, Tome 362 (2024) no. G7, pp. 739-749

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 $K=k\left(X\right)$ 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 $X\left(k\right)\ne \varnothing$; (2) $k$ is a number field.

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 $K=k\left(X\right)$ 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 $X\left(k\right)\ne \varnothing$ ; (2) $k$ est un corps de nombres.

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