Ekedahl-Oort and Newton strata for Shimura varieties of PEL type

TL;DR

This paper investigates the structure and relationships of Ekedahl-Oort and Newton strata in Shimura varieties of PEL type, providing new criteria for their non-emptiness and intersection, and generalizing previous results in the Siegel case.

Contribution

It introduces a group-theoretical framework for Ekedahl-Oort strata, determines their closure relations, and proves conjectures on Newton strata non-emptiness and intersections.

Findings

All Ekedahl-Oort strata are smooth, quasi-affine, and have dimension l(w).

The paper establishes criteria for when Ekedahl-Oort and Newton strata intersect.

It proves conjectures by Fargues and Rapoport regarding non-empty Newton strata.

Abstract

We study the Ekedahl-Oort stratification for good reductions of Shimura varieties of PEL type. These generalize the Ekedahl-Oort strata defined and studied by Oort for the moduli space of principally polarized abelian varieties (the "Siegel case"). They are parameterized by certain elements w in the Weyl group of the reductive group of the Shimura datum. We show that for every such w the corresponding Ekedahl-Oort stratum is smooth, quasi-affine, and of dimension l(w) (and in particular non-empty). Some of these results have previously been obtained by Moonen, Vasiu, and the second author using different methods. We determine the closure relations of the strata. We give a group-theoretical definition of minimal Ekedahl-Oort strata generalizing Oort's definition in the Siegel case and study the question whether each Newton stratum contains a minimal Ekedahl-Oort stratum. We give necessary criteria when a given Ekedahl-Oort stratum and a given Newton stratum meet. We determine which Newton strata are non-empty. This criterion proves conjectures by Fargues and by Rapoport generalizing a conjecture by Manin for the Siegel case.