Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety

TL;DR

This paper proves that the $p^ olinebreak^ ext{infty}$-torsion in the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded, addressing a question by Skorobogatov and Zarhin.

Contribution

It establishes a boundedness result for the $p^ olinebreak^ ext{infty}$-torsion of the transcendental Brauer group and proves a flat Tate conjecture for divisors in characteristic $p>0$.

Findings

Boundedness of the $p^ olinebreak^ ext{infty}$-torsion in the transcendental Brauer group.

Existence of infinitely $p$-divisible classes outside the transcendental Brauer group.

Relation between $p$-divisible towers and failure of surjectivity of specialization maps.

Abstract

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a "flat Tate conjecture" for divisors. In the text, we also study other geometric Galois-invariant $p^\infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$-divisible. We explain how the existence of these $p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of N\'eron--Severi groups in characteristic $p$.