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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 apolarised abelian variety which is uniquely determined by its associatedpolarised $p$-divisible group. The first problem translates to a class numberone problem for quaternion Hermitian lattices. The second problem alsotranslates to a class number one problem, whose solution involves massformulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata ingenus $g=4$.

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