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 $X(k)\ne \varnothing$; (2) $k$ is a number field.

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