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


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.



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”.


  1. 1.
    Björner, A., Brenti, F.: Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231. Springer, Berlin (2005)Google Scholar
  2. 2.
    Berthelot, P., Breen, L., Messing, W.: Théorie de Dieudonné cristalline II, Lecture Notes in Math, vol. 930. Springer, Berlin (1982)Google Scholar
  3. 3.
    Borovoi, M.: Abelian Galois cohomology of reductive groups. Mem. AMS 132, 1–50 (1998)MathSciNetGoogle Scholar
  4. 4.
    Bültel, O., Wedhorn, T.: Congruence relations for Shimura varieties associated to some unitary groups. J. Inst. Math. Jussieu 5, 229–261 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chai, C.-L.: Newton polygons as lattice points. Am. J. Math. 122, 967–990 (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    Chen, M., Kisin, M., Viehmann, E.: Connected components of closed affine Deligne-Lusztig varieties for unramified groups (in preparation)Google Scholar
  7. 7.
    Deng, B., Du, J., Parshall, B., Wang, J.: Finite dimensional Algebras and Quantum Groups, Mathematical Surveys and Monographs, vol. 150. AMS, Providence (2008)Google Scholar
  8. 8.
    Deligne, P., Pappas, G.: Singularités des espaces de module de Hilbert, en les charactéristiques divisant le discriminant. Comp. Math. 90, 59–79 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Grothendieck, A., Dieudonné, J.: Eléments de Géométrie Algébrique. I Grundlehren der Mathematik 166 (1971) Springer, II-IV Publ. Math. IHES 8 (1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967)Google Scholar
  10. 10.
    Ekedahl, T., van der Geer, G.: Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties. In: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, vol. I, pp. 567–636, Progr. Math., 269, Birkhäuser Boston, Inc., Boston (2009)Google Scholar
  11. 11.
    Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5, 41–68 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fargues, L.: Cohomologie des espaces de modules de groupes \(p\)-divisibles et correspondances de Langlands locales, in Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales. Astérisque 291, 1–199 (2004)MathSciNetGoogle Scholar
  13. 13.
    Gashi, Q.: On a conjecture of Kottwitz and Rapoport. Ann. Sci. École Norm. Sup. 43, 1017–1038 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Görtz, U., Haines, Th. J., Kottwitz, R.E., Reuman, D.C.: Dimensions of affine Deligne Lusztig varieties. Ann. Sci. École Norm. Sup. 39, 467–511 (2006)Google Scholar
  15. 15.
    Görtz, U., Haines, Th. J., Kottwitz, R.E., Reuman, D.C.: Affine Deligne-Lusztig varieties in affine flag varieties. Compos. Math. 146, 1339–1382 (2010)Google Scholar
  16. 16.
    Goren, E., Oort, F.: Stratifications of Hilbert modular varieties. J. Algebra Geom. 9, 111–154 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Görtz, U., Wedhorn, T.: Algebraic Geometry I. Vieweg/Teubner, Braunschweig/Stuttgart (2010)zbMATHCrossRefGoogle Scholar
  18. 18.
    Harashita, S.: Configuration of the central streams in the moduli of abelian varieties. Asian J. Math. 13, 215–250 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Harashita, S.: Generic Newton polygons of Ekedahl–Oort strata: Oort’s conjecture. Ann. Inst. Fourier 60, 1787–1830 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    He, X.: The \(G\)-stable pieces of the wonderful compactification. Trans. Am. Math. Soc. 359, 3005–3024 (2007)zbMATHCrossRefGoogle Scholar
  21. 21.
    Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001)Google Scholar
  22. 22.
    Kottwitz, R.E.: Isocrystals with additional structure. Comp. Math. 56, 201–220 (1985)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kottwitz, R.E.: Points on some Shimura varieties over finite fields. J. AMS 5, 373–444 (1992)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kottwitz, R.E., Rapoport, M.: On the existence of \(F\)-crystals. Comment. Math. Helv. 78, 153–184 (2003)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lan, K.-W.: Arithmetic compactification of PEL type Shimura varieties, PhD thesis, Harvard 2008, revised version (2010), London Mathematical Society Monographs (to appear)Google Scholar
  26. 26.
    Lau, E.: A duality theorem for Dieudonné displays. Ann. Sci. École Norm. Sup. 42, 214–259 (2009)Google Scholar
  27. 27.
    Lau, E.: A relation between Dieudonné displays and crystalline Dieudonné theory. arXiv:1006.2720v1Google Scholar
  28. 28.
    Laumon, G., Moret-Bailly, L.: Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebie- te, 3. Folge 39. Springer, Berlin (2000)Google Scholar
  29. 29.
    Lucarelli, C.: A converse to Mazur’s inequality for split classical groups. J. Inst. Math. Jussieu 3, 165–183 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lusztig, G.: Parabolic character sheaves. II. Mosc. Math. J. 4, 869–896 (2004)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Faber, C., van der Geer, G., Oort, F. (eds.): Moduli of Abelian Varieties, Progress in Mathematics, vol. 195. Birkhäuser, Basel (2001)Google Scholar
  32. 32.
    Moonen, B.: Group schemes with additional structures and Weyl group cosets. In: Faber, C., van der Geer, G., Oort, F. (eds.) Moduli of Abelian Varieties, Progress in Mathematics, vol. 195, pp. 255–298. Birkhäuser, Basel (2001)Google Scholar
  33. 33.
    Moonen, B.: A dimension formula for Ekedahl-Oort strata. Ann. Inst. Fourier 54, 666–698 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Moonen, B.: Serre-Tate theory for moduli spaces of PEL type. Ann. Sci. École Norm. Sup. 37, 223–269 (2004)Google Scholar
  35. 35.
    Moonen, B., Wedhorn, T.: Discrete invariants of varieties in positive characteristic. IMRN 72, 3855–3903 (2004)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Nicole, M.-H., Vasiu, A., Wedhorn, T.: Purity of level \(m\) stratifications. Ann. Sci. École Norm. Sup. 43, 927–957 (2010)Google Scholar
  37. 37.
    Oda, T.: The first de Rham cohomology group and Dieudonné modules. Ann. Sci. École Norm. Sup. 2, 63–135 (1969)Google Scholar
  38. 38.
    Oort, F.: A stratification of a moduli space of abelian varieties. In: Faber, C., van der Geer, G., Oort, F. (eds.) Moduli of Abelian Varieties, Progress in Mathematics, vol. 185, pp. 435–416. Birkhäuser, Basel (2001)Google Scholar
  39. 39.
    Oort, F.: Newton polygon strata in the moduli space of abelian varieties. In: Faber, C., van der Geer, G., Oort, F. (eds.) Moduli of Abelian Varieties, Progress in Mathematics, vol. 195, pp. 417–440. Birkhäuser, Basel (2001)Google Scholar
  40. 40.
    Oort, F.: Foliations in moduli spaces of abelian varieties. J.A.M.S. 17, 267–296 (2004)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Oort, F.: Minimal \(p\)-divisible groups. Ann. Math. 161, 1021–1036 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Oort, F.: Simple \(p\)-kernels of \(p\)-divisible groups. Adv. Math. 198, 275–310 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory, English translation. Academic Press, London (1994)Google Scholar
  44. 44.
    Pink, R., Wedhorn, T., Ziegler, P.: Algebraic zip data. Doc. Math. 16, 253–300 (2011)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Rapoport, M.: A guide to the reduction modulo \(p\) of Shimura varieties. Astérisque 298, 271–318 (2005)MathSciNetGoogle Scholar
  46. 46.
    Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Compos. Math. 103, 153–181 (1996)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Rapoport, M., Zink, T.: Period spaces for \(p\)-divisible groups, Annals of Math. Studies, vol. 141. Princeton University Press, Princeton (1996)Google Scholar
  48. 48.
    Demazure, M., et al.: Schémas en groupes, I, II, III, LNM 151, 152, 153. Springer, Berlin (1970)Google Scholar
  49. 49.
    Serre, J.-P.: Galois Cohomology, Springer Monographs in Mathematics, Springer, Berlin (2002)Google Scholar
  50. 50.
    Tits, J.: Reductive groups over local fields. In: Automorphic Forms, Representations and \(L\)-Functions (Proc. Sympos. Pure Math. XXXIII), pp 29–69, Am. Math. Soc., Providence, RI (1979)Google Scholar
  51. 51.
    Vasiu, A.: Crystalline boundedness principle. Ann. Sci. École Norm. Sup. 39, 245–300 (2006)Google Scholar
  52. 52.
    Vasiu, A.: Level \(m\) stratifications of versal deformations of \(p\)-divisible groups. J. Algebraic Geom. 17, 599–641 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Vasiu, A.: Mod \(p\) classification of Shimura \(F\)-crystals. Math. Nachr. 283, 1068–1113 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Vasiu, A.: Manin problems for Shimura varieties of Hodge type. J. Ramanujan Math. Soc. 26, 31–84 (2011)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Viehmann, E.: Truncations of level 1 of elements in the loop group of a reductive group. preprint,
  56. 56.
    Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety of \({\rm GU} (1, n-1)\) II. Invent. Math. 184, 591–627 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Wedhorn, T.: Ordinariness in good reductions of Shimura varieties of PEL-type. Ann. Sci. École Norm. Sup. 32, 575–618 (1999)Google Scholar
  58. 58.
    Wedhorn, T.: The dimension of Oort strata of Shimura varieties of PEL-type. In: Faber, C., van der Geer, G., Oort, F. (eds.) Moduli of Abelian Varieties, Progress in Mathematics, vol. 195, pp. 411–471. Birkhäuser, Basel (2001)Google Scholar
  59. 59.
    Wedhorn, T.: Flatness of the mod p period morphism for the moduli space of principally polarized abelian varieties, unpublished note. arXiv:math/0507177 (2005)Google Scholar
  60. 60.
    Wedhorn, T.: Specialization of \(F\)-zips, unpublished note. arXiv:math/0507175 (2005)Google Scholar
  61. 61.
    Yatsyshyn, Y.: Purity of \(G\)-zips. arXiv:1210.8396 (preprint)Google Scholar
  62. 62.
    Yu, C.-F.: Leaves on Shimura varieties. Proc. ICCM II, 630–661 (2007)Google Scholar
  63. 63.
    Zink, T.: A Dieudonné Theory for \(p\)-divisible groups. In: Class Field Theory, Its Centenary and Prospect, pp. 1–22. Advanced Studies in Pure Mathematics, Tokyo (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.MünchenGermany
  2. 2.PaderbornGermany

Personalised recommendations