Tate's conjecture and the Tate-Shafarevich group over global function fields

TL;DR

This paper establishes a connection between the finiteness of the Brauer group of a variety over a function field and Tate's conjecture and the Tate-Shafarevich group, extending known results from surfaces to higher dimensions.

Contribution

It generalizes a theorem of Artin and Grothendieck by linking the finiteness of the Brauer group to Tate's conjecture and the Tate-Shafarevich group for varieties over global function fields.

Findings

Finiteness of the Brauer group is equivalent to Tate's conjecture and Tate-Shafarevich group finiteness.

Provides a formula relating the orders of these groups when finite.

Extends results from surfaces to higher-dimensional varieties.

Abstract

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and only Tate's conjecture for divisors on $X$ holds and the Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the formula given for a surface.