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

TL;DR

This paper proves the finiteness of the Tate-Shafarevich group for algebraic tori over certain function fields, extending known results to new cases involving fields of characteristic zero and specific geometric conditions.

Contribution

It establishes the finiteness of the Tate-Shafarevich group for algebraic tori over function fields under new conditions involving base fields and geometric properties.

Findings

Finiteness of Sha(T, V) when the base field is finitely generated with a rational point.

Finiteness of Sha(T, V) over number fields.

Applicable to tori over function fields of smooth geometrically integral varieties.

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.