Skip to main content Accessibility help

Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type

  • Chao Zhang (a1)


For a Shimura variety of Hodge type with hyperspecial level structure at a prime $p$ , Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\,>\,2$ . This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn, and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l\left( w \right)$ (i.e., the length of $w$ ) if it is non-empty. We also determine the closure of each stratum.



Hide All
[1] Borel, A., Linear algebraic groups. Second ed., Graduate Texts in Mathematics , 126, Springer-Verlag, New York, 1991.
[2] Demazure, M. and Grothendieck, A., SGA 3,I, II, III. Lecture Notes in Math. , 151–153, Springer, 19621970.
[3] Ekedahl, T. and van der Geer, G., Cycle classes of the E-O stratification on the moduli of abelian varieties. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math. 269, Birhäuser Boston, Boston, MA, 2009, pp. 567636.http://dx.doi.Org/10.1007/978-0-8176-4745-2J3
[4] Goldring, W. and Koskivirta, J.-S., Strata Hasse invariants, Heche algebras and Galois representations. http://arxiv:1507.05032
[5] Hartshorne, R., Algebraic geometry. Graduate Texts in Mamthematics , 52, Springer-Verlag, New York-Heidelberg, 1977.
[6] Illusie, L., Reports on crystalline cohomology. In: Algebraic geometry (Proc. Symp. Pure Math., 29, Humboldt State Univ., Arcata, Calif. , 1974), American Mathematical Society, Providence, RI, 1975, pp. 459478.
[7] Kisin, M., Integral models for Shimura varieties ofabelian type. J. Amer. Math. Soc. 23(2010), no. 4, 9671012.
[8] Knutson, D., Algebraic spaces. Lecture Notes in Mathematics , 203, Springer-Verlag, Berlin-New York, 1971.
[9] Koskivirta, J.-S. and Wedhorn, T., Generalized Hasse invariants for Shimura varieties of Hodge type. http://arxiv:1406.2178
[10] Kottwitz, R. E., Shimura varieties and twisted orbital integrals. Math. Ann. 269(1984), no. 3, 287300.http://dx.doi.Org/10.1007/BF01450697
[11] Lang, S., Algebraic groups over finite fields. Amer. J. Math. 78(1956), 555563.
[12] Madapusi Pera, K., Integral canonical models for spin Shimura varieties. Compos. Math. 152(2016), no. 4, 769824.http://dx.doi.Org/10.1112/S0010437X1500740X
[13] Milne, J., Introduction to Shimura varieties. In: Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc , 4, American Mathematical Society, Providence, RI, 2005, pp. 265378.
[14] Moonen, B., Models of Shimura varieties in mixed characteristics. In: Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge University Press, Cambridge, 1998, pp. 267350.
[15] Moonen, B., Group schemes with additional structures and Weyl group cosets. Moduli of abelian varieties (Texel Island, 1999), Prog. Math. , 195, Birkhäuser, Basel, 2001, pp. 255298.
[16] Moonen, B., A dimension formula for Ekedahl-Oort strata. Ann. Inst. Fourier (Grenoble) 54(2004), 666698.
[17] Moonen, B. and Wedhorn, T., Discrete invariants of varieties in positive characteristic. Int. Math. Res. Not. 2004, no. 72, 38553903.http://dx.doi.Org/10.1155/S1073792804141263
[18] Mumford, D., Fogarty, J., and Kirwan, F., Geometric invariant theory. Third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete , 34, Springer-Verlag, Berlin, 1994.
[19] Oort, F., A stratification of a moduli space of abelian varieties. In: Moduli of abelian varieties (Texel Island, 1999), Prog. Math. , 195, Birkhäuser, Basel, 2001, pp. 345416.http://dx.doi.Org/10.1007/978-3-0348-8303-0J3
[20] Pink, R., Wedhorn, T. and Ziegler, P., Algebraic zip data. Doc. Math. 16(2011), 253300.
[21] Pink, R., Wedhorn, T. and Ziegler, P., F-zips with additional structure. Pacific J. Math. 274(2015), no. 1,183236.
[22] Vasiu, A., Integral canonical models of Shimura varieties of preabelian type. Asian J. Math. 3(1999), no. 2, 401518.
[23] Vasiu, A., Modp classification of Shimura F-crystals. Math. Nachr. 283(2010), no. 8,10681113.http://dx.doi.Org/10.1002/mana.200714000
[24] Viehmann, E. and Wedhorn, T., Ekedahl-Oort and Newton strata for Shimura varieties of PEL type. Math. Ann. 356(2013), 14931550.http://dx.doi.Org/10.1007/s00208-012-0892-z
[25] Wedhorn, T., Ordinariness in good reductions of Shimura varieties of PEL-type. Ann. Sci. École Norm. Sup. 32(1999), 575618. http://dx.doi.Org/10.1016/S0012-9593(01)80001-X
[26] Wedhorn, T., The dimension of Oort strata of Shimura varieties of PEL-type. In: The moduli space of abelian varieties (Texel Island, 1999), Prog. Math. , 195, Birkhäuser, Basel, 2001, pp. 441471.
[27] Wedhorn, T., De Rham cohomology of varieties over fields of positive characteristic. In: Higher-dimensional geometry over finite fields, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. , 16, IOS, Amsterdam, 2008, pp. 269314.
[28] Wedhorn, T., Bruhat strata and F-zips with additional structure. Münster J. Math. 7(2014), no. 2, 529556.
[29] Wortmann, D., The μ-ordinary locus for Shimura varieties of Hodge type. http://arxiv:1310.6444
[30] Zhang, C., Remarks on Ekedahl-Oort stratifications. http://arxiv:1401.6632
MathJax is a JavaScript display engine for mathematics. For more information see


Related content

Powered by UNSILO

Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type

  • Chao Zhang (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.