The Brauer group of an abelian variety

The importance of the Brauer group for arithmetic geometry was highlighted by Manin in his celebrated 1970 ICM address. In this talk I will discuss the structure of 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 $U_A$. I will present results on the dimension and the $p$-exponent of $U_A$ based on the classical Manin-Dieudonné theory, leading to the determination of $U_A$ up to isogeny for abelian varieties $A$ of small dimension. This is joint work with Livia Grammatica and Yuan Yang.