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

We prove that the p^∞ -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. We also study other geometric Galois-invariant p^∞ -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 .