Ekedahl-Oort strata and Kottwitz-Rapoport strata

TL;DR

This paper compares the structures of Ekedahl-Oort and Kottwitz-Rapoport strata in the moduli space of abelian varieties over positive characteristic, providing explicit descriptions and establishing isomorphisms between these stratifications.

Contribution

It explicitly describes Kottwitz-Rapoport strata within the supersingular locus and proves all Ekedahl-Oort strata are isomorphic to parahoric Kottwitz-Rapoport strata.

Findings

Both strata are isomorphic to unions of Deligne-Lusztig varieties.

Explicit descriptions of strata in the supersingular locus are provided.

All Ekedahl-Oort strata are isomorphic to parahoric Kottwitz-Rapoport strata.

Abstract

We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a parahoric Kottwitz-Rapoport stratum.