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

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.

Corrected some typos, rewrote some parts of Section 7.2; to appear in Math. Ann

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Cited by in corpus (37)

• Connected components of affine Deligne-Lusztig varieties in mixed characteristic
• Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations
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• Tautological rings of Shimura varieties and cycle classes of Ekedahl-Oort strata
• The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type
• Stratifications and foliations for good reductions of Shimura varieties of Hodge type
• Newton Polygons Arising for Special Families of Cyclic Covers of the Projective Line
• Sections of the Hodge bundle over Ekedahl-Oort strata of Shimura varieties of Hodge type
• Generalized mu-ordinary Hasse invariants
• Theta operators on unitary Shimura varieties
• EKOR strata for Shimura varieties with parahoric level structure
• Bruhat strata and F-zips with additional structures
• Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
• Purity of Zip Strata with Applications to Ekedahl-Oort Strata of Shimura Varieties of Hodge Type
• Note on affine Deligne-Lusztig varieties
• On the p-adic cohomology of the Lubin-Tate tower
• Fundamental elements of an affine Weyl group
• Remarks on Ekedahl-Oort stratifications
• On specializations of minimal $p$-divisible groups
• Geometric and homological properties of affine Deligne-Lusztig varieties
• Motivic cohomology of quaternionic Shimura varieties and level raising
• Foliations on unitary Shimura varieties in positive characteristic
• Ekedahl-Oort stratifications of Shimura varieties via Breuil-Kisin windows
• Hecke eigensystems of automorphic forms (mod $p$) of Hodge type and algebraic modular forms
• Stratifications in the reduction of Shimura varieties
• On the generic part of the cohomology of local and global Shimura varieties
• Bruhat strata for Shimura varieties of PEL type
• On arithmetic of the superspecial locus
• On the cohomology of compact unitary group Shimura varieties at ramified split places
• An alternative construction of zip period maps for Shimura varieties
• On the boundary components of central streams in the two slopes case
• On the Ekedahl-Oort stratification of Shimura curves
• On the boundary components of central streams
• Lifting Chern classes by means of Ekedahl-Oort strata
• EKOR strata on Shimura varieties with parahoric reduction
• Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type