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

Abstract

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 $Sha(T , V) = \ker\left(H^1(K , T) \to \prod_{v \in V} H^1(K_v , T)\right)$. 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 $Sha(T , V)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \neq \emptyset$; (2) $k$ is a number field.