Valentijn Karemaker

We will study the Siegel moduli space of abelian varieties in characteristic p and in particular its supersingular locus. We first determine precisely when this locus is geometrically irreducible. Since it was known that the number of components is a class number, this comes down to solving a “class number one problem” or “Gauss problem”.
Next, we will show when a polarised abelian variety is determined by its p-divisible group. This can be viewed as a Gauss problem for central leaves, which are the loci consisting of points whose associated p-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus 4.