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The supersingular locus in Siegel modular varieties with Iwahori level structure

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Abstract

We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl–Oort stratification on the former, the Kottwitz–Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case g is even.

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References

  1. Bonnafé C., Rouquier R.: On the irreducibility of Deligne-Lusztig varieties. C. R. Acad. Sci. Paris Sér. I Math. 343, 37–39 (2006)

    Article  MATH  Google Scholar 

  2. Boyer P.: Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples. Invent. Math. 177(2), 239–280 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Clarke R.: A short proof of a result of Foata and Zeilberger. Adv. Appl. Math. 16, 129–131 (1995)

    Article  MATH  Google Scholar 

  4. Clarke R., Steingrímsson E., Zeng J.: New Euler-Mahonian statistics on permutations and words. Adv. Appl. Math. 18, 237–270 (1997)

    Article  MATH  Google Scholar 

  5. 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., vol. 269. Birkhäuser Boston (2009)

  6. Fargues, L., Mantovan, E.: Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales. Astérisque 291 (2004)

  7. van der Geer, G.: Cycles on the moduli space of abelian varieties. In: Faber, C., Looijenga, E. (eds.) Moduli of Curves and Abelian Varieties, Aspects of Math., E33, pp. 65–89. Vieweg (1999)

  8. Genestier A.: Un modèle semi-stable de la variété de Siegel de genre 3 avec structures de niveau de type Γ0(p). Compos. Math. 123(3), 303–328 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Görtz U.: On the flatness of local models for the symplectic group. Adv. Math. 176, 89–115 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Görtz U.: On the connectedness of Deligne-Lusztig varieties. Represent. Theory 13, 1–7 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Görtz U., Haines T., Kottwitz R., Reuman D.: Affine Deligne-Lusztig varieties in affine flag varieties. Compos. Math. 146, 1339–1382 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Görtz, U., Hoeve, M.: Ekedahl-Oort strata and Kottwitz-Rapoport strata. arXiv:0808.2537

  13. Görtz U., Yu C.-F.: Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties. Journal de l’Institut de Math. de Jussieu 9(2), 357–390 (2010)

    Article  MATH  Google Scholar 

  14. Haines T.: The combinatorics of Bernstein functions. Trans. Am. Math. Soc. 353(3), 1251–1278 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Haines T.: Test functions for Shimura varieties: the Drinfeld case. Duke Math. J. 106(1), 19–40 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Haines, T.: Introduction to Shimura varieties with bad reduction of parahoric type. In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, pp. 583–642. Clay Math. Proc., 4. Amer. Math. Soc. (2005)

  17. Harashita S.: Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties. J. Algebraic Geom. 19(3), 419–438 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties. With an appendix by V. G. Berkovich. Annals of Math. Studies, vol. 151. Princeton University Press (2001)

  19. Hoeve M.: Ekedahl-Oort strata in the supersingular locus. J. Lond. Math. Soc. (2) 81(1), 129–141 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Koblitz N.: p-adic variant of the zeta function of families of varieties defined over finite fields. Compos. Math. 31, 119–218 (1975)

    MathSciNet  MATH  Google Scholar 

  21. Kottwitz R.E., Rapoport M.: Minuscule alcoves for GL n and GSp2n . Manuscr. Math. 102, 403–428 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, K.-Z., Oort, F.: Moduli of supersingular Abelian varieties. In: Lecture Notes in Mathematics, vol. 1680. Springer, Berlin (1998)

  23. Moonen, B.: Group schemes with additional structures and Weyl group cosets. In: Moduli of Abelian Varieties (Texel Island, 1999), pp. 255–298. Progr. Math., 195. Birkhäuser (2001)

  24. Moonen B.: A dimension formula for Ekedahl-Oort strata. Ann. Inst. Fourier (Grenoble) 54(3), 666–698 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moonen B., Wedhorn T.: Discrete invariants of varieties in positive characteristic. Int. Math. Res. Not. 72, 3855–3903 (2004)

    Article  MathSciNet  Google Scholar 

  26. Moret-Bailly, L.: Pinceaux de variétés abéliennes. In: Astérisque, vol. 129 (1985)

  27. Ngô B.C., Genestier A.: Alcôves et p-rang des variétés abéliennes. Ann. Inst. Fourier (Grenoble) 52, 1665–1680 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Oort F.: Subvarieties of moduli spaces. Invent. Math. 24, 95–119 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  29. Oort, F.: A stratification of a moduli space of abelian varieties. In: Moduli of Abelian Varieties (Texel Island, 1999). Progr. Math., 195, pp. 345–416. Birkhäuser (2001)

  30. Rapoport, M.: Non-Archimedean period domains. In: Proc. Int. Cong. Math. (Zürich, 1994), pp. 423–434. Birkhäuser (1995)

  31. Rapoport, M.: A guide to the reduction modulo p of Shimura varieties. In: Automorphic forms. I. Astérisque 298, 271–318 (2005)

  32. Wedhorn, T.: The dimension of Oort strata of Shimura varieties of PEL-type. In: Moduli of Abelian Varieties (Texel Island, 1999). Progr. Math., vol. 195, pp. 441–471. Birkhäuser (2001)

  33. Wedhorn, T.: De Rham cohomology of varieties over fields of positive characteristic. In: Higher-Dimensional Geometry Over Finite Fields, pp. 269–314. IOS Press (2008)

  34. Yu C.-F.: Irreducibility of the Siegel moduli spaces with parahoric level structure. Int. Math. Res. Not. 48, 2593–2597 (2004)

    Article  Google Scholar 

  35. Yu C.-F.: Irreducibility and p-adic monodromies on the Siegel moduli spaces. Adv. Math. 218, 1253–1285 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yu C.-F.: Kottwitz-Rapoport strata in the Siegel moduli spaces. Taiwanese J. Math. 14(6), 2343–2364 (2010)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Ellernstr. 29, 45326, Essen, Germany

    Ulrich Görtz

  2. Institute of Mathematics, Academia Sinica, 6th Floor, Astronomy Mathematics Building, No. 1, Roosevelt Rd. Sec. 4, Taipei, Taiwan

    Chia-Fu Yu

  3. NCTS (Taipei Office), Taipei, Taiwan

    Chia-Fu Yu

Authors
  1. Ulrich Görtz
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  2. Chia-Fu Yu
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Correspondence to Ulrich Görtz.

Additional information

U. Görtz was partially supported by a Heisenberg grant and by the SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG (German Research Foundation). C.-F. Yu was partially supported by grants NSC 97-2115-M-001-015-MY3 and AS-98-CDA-M01.

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Görtz, U., Yu, CF. The supersingular locus in Siegel modular varieties with Iwahori level structure. Math. Ann. 353, 465–498 (2012). https://doi.org/10.1007/s00208-011-0689-5

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  • DOI: https://doi.org/10.1007/s00208-011-0689-5

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