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Mathematische Annalen

, Volume 356, Issue 4, pp 1493–1550 | Cite as

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

  • Eva Viehmann
  • Torsten WedhornEmail author
Article

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 \(\ell (w)\) (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. As an interesting application 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. We give a necessary criterion when a given Ekedahl–Oort stratum and a given Newton stratum meet.

Notes

Acknowledgments

We thank E. Lau for providing a proof of Lemma 4.1 which is much simpler than our original one. Further we thank F. Oort for helpful discussions and D. Wortmann and the referee for helpful remarks. The first author was partially supported by the SFB/TR45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” and a Heisenberg fellowship of the DFG and by ERC Starting Grant 277889 “Moduli spaces of local \(G\)-shtukas”. The second author was partially supported by the SPP1388 “Representation Theory”.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.MünchenGermany
  2. 2.PaderbornGermany

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