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

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 $p$.

19 pages; final version, to appear in Compositio Mathematica

References in corpus (1)

• Parabolicity conjecture of $F$-isocrystals

Cited by in corpus (2)

• Boundedness of the p-primary torsion of the Brauer group of products of varieties
• On the Brauer groups of fibrations