Brauer groups of torsors under algebraic tori

TL;DR

This paper investigates the Brauer groups of torsors under algebraic tori over global fields, establishing connections with the adèle class group and the Shafarevich-Tate group, thus advancing understanding of their arithmetic properties.

Contribution

It provides new relationships between the Brauer group of torsors and key arithmetic invariants like the adèle class group and Shafarevich-Tate group for algebraic tori.

Findings

Established a link between the Brauer group of torsors and the adèle class group.

Connected the Brauer group to the Shafarevich-Tate group of the torus.

Enhanced understanding of the arithmetic structure of torsors under algebraic tori.

Abstract

Let $T$ be an algebraic torus defined over a global field $K$. For any $K$-torsor $X$ under $T$, we relate the Brauer group of $X$ to the ad\'{e}le class group of $T$ as well as to the Shafarevich Tate group of $T$.