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

Andrei S. Rapinchuk, Igor A. Rapinchuk

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 \i…