Comparing the Brauer group and the Tate Shafarevich group

TL;DR

This paper establishes a formula linking the Brauer group of a surface over a finite field to the Tate-Shafarevich group of its generic fiber's Jacobian, revealing that the Brauer group is a square if finite.

Contribution

It provides a new explicit formula connecting the Brauer group and the Tate-Shafarevich group for surfaces over finite fields.

Findings

The Brauer group of a finite field surface is a square if finite.

The formula relates the order of the Brauer group to the Tate-Shafarevich group.

Implications for the structure of Brauer groups over finite fields.

Abstract

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.