Stratifications on moduli spaces of abelian varieties and Deligne-Lusztig varieties

This thesis is about the relation between three different stratifications on moduli spaces of abelian varieties: the Ekedahl-Oort stratification, the Newton polygon stratification, and the Kottwitz-Rapoport stratification. The first part concerns Ekedahl-Oort strata that are completely contained in the supersingular locus. Building on work of Harashita I show that each such stratum is isomorphic to a disjoint union of Deligne-Lusztig varieties. In particular I determine the number of connected components of each stratum. The second part, which I wrote together with Ulrich Görtz, relates the results in the first part to work of Görtz and Yu on Kottwitz-Rapoport strata contained in the supersingular locus. We show that each Ekedahl-Oort stratum is isomorphic to a parahoric Kottwitz-Rapoport stratum, and that for supersingular strata this isomorphism is compatible with the descriptions of both strata in terms of Deligne-Lusztig varieties. We also extend several results of Görtz and Yu on supersingular Kottwitz-Rapoport strata from moduli spaces with Iwahori level structure to moduli spaces with general parahoric level structures. The third part analyzes how Kottwitz-Rapoport strata and Ekedahl-Oort strata behave under forgetful morphisms, morphisms between moduli spaces of abelian varieties with parahoric level structures that forget some of the abelian varieties in the chain. I give an algorithm to determine the image of every stratum, and the fibre over every point in the image. The algorithm breaks up the fibre into a number of simple building blocks: the affine line, the affine line minus the origin, and Deligne-Lusztig varieties.