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

TL;DR

This work proves that over a finitely generated field of characteristic $p>0$, the $p^ fty$-torsion in the transcendental Brauer group of an abelian variety is bounded, yielding a direct sum of a finite group and a finite-exponent $p$-group for $ ext{Br}(A_{k_s})^{k}$, with finiteness under a Witt-cohomology finiteness condition. A flat variant of the Tate conjecture for divisors is established via a cycle-class isomorphism between $ ext{NS}(A)_{Z_p}$ and the fixed part of $igl(H^2_{ ext{fppf}}(A_{ar{k}},mu_{p^ullet})igr)^{k}$, using the $p$-divisible group of $A$ to bridge crystalline and fppf cohomology. The paper also identifies exceptional $p$-divisible Brauer classes beyond the transcendental part, relates them to failures of specialization of Néron–Severi groups in characteristic $p$, and provides counterexamples to conjectures on Galois-fixed Brauer elements. Overall, the results unify cohomological, crystalline, and abelian-variety techniques to bound and describe $p$-primary Brauer phenomena in positive characteristic with explicit structural consequences for NS specialization and Brauer finiteness.

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 Néron--Severi groups in characteristic $p$.

Theorems & Definitions (42)

• Theorem 1.1: Theorem \ref{['fini-Brau:t']}
• Proposition 1.2: Proposition \ref{['counter:p']}
• Theorem 1.3: André, Ambrosi, Christensen
• Theorem 1.4: Theorem \ref{['NS:t']}
• Theorem 1.5: Theorem \ref{['fppf-tate:t']}
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3
• proof