When is a polarised abelian variety determined by its -divisible group?

Abstract

We study the Siegel modular variety Ag ⊗ F̅p of genus g and its supersingular locus lg. As our main result we determine precisely when lg is irreducible, and we list all x in Ag ⊗F̅p for which the corresponding central leaf 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 pdivisible 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.