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

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We study the Siegel modular variety A g ⊗ F ¯ p \mathscr {A}_g\otimes \overline {\mathbb {F}}_p of genus g g and its supersingular locus S g \mathscr {S}_g . As our main result we determine precisely when S g \mathscr {S}_g is irreducible, and we list all x x in A g ⊗ F ¯ p \mathscr {A}_g\otimes \overline {\mathbb {F}}_p for which the corresponding central leaf C ( x ) \mathscr {C}(x) consists of one point, that is, for which x x corresponds to a polarised abelian variety which is uniquely determined by its associated polarised p 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 g=4 .