Finiteness of the Tate-Shafarevich group over function fields for groups of multiplicative type

Abstract

Let $K=k(X)$ be the function field of a smooth geometrically integral variety $X$ of dimension $\ge 2$ over a field $k$ of characteristic 0 and $V$ be the set of discrete valuations of $K$ associated with the prime divisors on $X$. We show that if $D$ is a $k$-defined group of multiplicative type, then the corresponding Tate-Shafarevich group $Sha(D,V)=\ker \left(H^1(K,D)\to {\prod }_{v\in V}{H}^1(K_v,D)\right)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k)\ne \varnothing$; (2) $k$ is a number field. This complements previous work of Harari and Szamuely, which considered the case where $X$ is a curve.