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

We prove that the p(infinity)-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(infinity)-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 & eacute;ron-Severi groups in characteristic p.