Brauer groups of abelian varieties over fields of finite characteristic

Abstract

We study the Brauer group of an abelian variety A over an algebraically closed field of characteristic p focusing on the p-primary torsion, the key part of which is a certain quasi-algebraic unipotent group U_A. We determine its dimension and obtain a sharp upper bound for its p-exponent. The isogeny class of U_A is classified for abelian varieties A of dimension at most 3. For principally polarised abelian varieties we compute the dimension of the p-torsion subgroup of U_A in terms of the Ekedahl--Oort type of A.