Brauer groups of abelian varieties over fields of finite characteristic

TL;DR

This paper investigates the structure of the Brauer group of abelian varieties over algebraically closed fields of characteristic p, focusing on p-primary torsion and classifying associated unipotent groups.

Contribution

It determines the dimension and p-exponent bounds of a key unipotent group related to the Brauer group and classifies its isogeny class for low-dimensional abelian varieties.

Findings

Determined the dimension of U_A and established upper bounds for its p-exponent.

Classified the isogeny class of U_A for abelian varieties of dimension ≤ 3.

Computed the p-torsion subgroup dimension for principally polarised abelian varieties using Ekedahl--Oort types.

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.