Brauer groups of abelian varieties over fields of finite characteristic
Livia Grammatica, Alexei N. Skorobogatov, Yuan Yang
TL;DR
This work identifies and analyzes the p-primary part of the Brauer group Br$(A)$ for abelian varieties $A$ over algebraically closed fields of characteristic $p>0$ by introducing the connected unipotent quasi-algebraic group ${\bf U}_A$ attached to the F-crystal ${\rm H}^2_{\rm cris}(A/W)$. It provides a concrete dimension formula ${\rm dim}({\bf U}_A)=\frac{g(g-1)}{2}-\sum_{0\le\lambda<1}(1-\lambda)m_{\lambda}$, tying ${\bf U}_A$ directly to the slopes of crystalline cohomology and to the isogeny class of $A$. The paper establishes a universal bound ${\rm p-exp}({\bf U}_A)\le g-1$ (for $p\neq 2$), proves sharpness by constructing supersingular examples, and, in the threefold case, classifies the isogeny type of ${\bf U}_A$. For principally polarized abelian varieties, it expresses ${\bf U}_A[p]$ in terms of Ekedahl–Oort types, yielding a sharp bound ${\rm dim}({\bf U}_A[p])=\frac{a(a-1)}{2}+\sum_{i=1}^{g-a}(m_i-i)$ where $a$ is the $a$-number and $\varphi$ encodes the EO type. Collectively, these results connect the Brauer group’s $p$-torsion to Dieudonné theory, Newton polygons, and EO stratifications, with consequences for the transcendental Brauer group and the formal Brauer group of $A$.
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.