Tate Conjecture and Higher Brauer Groups of Abelian Varieties in Characteristic Zero

TL;DR

This paper links the finiteness of higher Brauer groups' torsion to the Tate conjecture for abelian varieties over fields finitely generated over Q, extending computational methods for these groups.

Contribution

It establishes a new criterion for the Tate conjecture based on higher Brauer groups and advances computational techniques for these groups.

Findings

Finiteness of higher Brauer group torsion implies Tate conjecture validity.

Extended computational methods for transcendental and higher Brauer groups.

Provides new insights into the structure of Brauer groups in characteristic zero.

Abstract

Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold. Furthermore, we extend methods for computations of transcendental Brauer groups to higher Brauer groups.