When is a polarised abelian variety determined by its $\mathbf{p}$-divisible group?

Abstract

We study the Siegel modular variety ${\mathcal{A}}_g\otimes {\overline{\mathbb{F}}}_p$ of genus $g$ and its supersingular locus ${\mathcal{S}}_g$. As our main result we determine precisely when ${\mathcal{S}}_g$ is irreducible, and we list all $x$ in ${\mathcal{A}}_g\otimes {\overline{\mathbb{F}}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$.

Published version, 41 pages