On the Bruhat–Tits stratification of a quaternionic unitary Rapoport–Zink space

  • Haining WangEmail author


In this article we study the special fiber of the Rapoport–Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat–Tits stratification and is intimately related to the Bruhat–Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.

Mathematics Subject Classification

Primary 11G18 Secondary 20G25 



The author would like to thank Henri Darmon for supporting his postdoctoral studies. He would like to thank Liang Xiao for many helpful conversations regarding to Shimura varieties and beyond. He is grateful to Ulrich Görtz for all the help and his comments on this article. He is inspired by reading many works of Chia-Fu Yu on Siegel modular varieties. He also would like to thank Ben Howard, Eyal Goren, Yichao Tian, Xu Shen and Benedict Gross for valuable discussions related to this article. He would like to thank the referees for carefully reading the article and pointing out all the corrections. While this article is being reviewed, Yasuhiro Oki obtained similar results about the supersingular locus of the quaternionic unitary Shimura variety independently. His method is completely different. He exploited the exceptional isomorphisms between the group \(\mathrm {GU}_{B}(2)\) and the non-split \(\mathrm {GSpin}(3,2)\). Then he embedded the non-split \(\mathrm {GSpin}(3,2)\) in the split \(\mathrm {GSpin}(4,2)\). This allows him to use the results of Howard-Pappas [11] mentioned before. We would like to thank Yoichi Mieda for sending Oki’s work to us.


  1. 1.
    Berthelot, P., Breen, L., Messing, W.: Messing, Théorie de Dieudonné Cristalline. II: Lecture Notes in Mathematics, vol. 930. Springer, Berlin (1982)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bhatt, B., Scholze, P.: Projectivity of the Witt vector affine Grassmannian. Invent. Math. 209(2), 329–423 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cho, S.: The basic locus of the unitary Shimura variety with parahoric level structure and special cycles. arXiv:1807.09997 (Preprint) (2018)
  4. 4.
    Chen, M.-F., Viehmann, E.: Affine Deligne–Lusztig varieties and the action of \(J\). J. Algebraic Geom. 27(2), 273–304 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. (2) 103(1), 103–161 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Görtz, U., He, X.-H.: Basic loci of Coxeter type in Shimura varieties. Camb. J. Math. 3(3), 323–353 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Görtz, U., He, X.-H.: Erratum to: Basic loci in Shimura varieties of Coxeter type. Camb. J. Math. 3(3), 323–353 (2018)zbMATHCrossRefGoogle Scholar
  8. 8.
    Görtz, U., He, X.-H., Nie, S.-A.: Fully Hodge-Newton decomposable Shimura varieties. arXiv:1610.05381 (Preprint) (2016)
  9. 9.
    Görtz, U.: Stratifications of affine Deligne–Lusztig varieties. arXiv:1802.02225 (Preprint) (2018)
  10. 10.
    Helm, D.: Towards a geometric Jacquet–Langlands correspondence for unitary Shimura varieties. Duke Math. J. 155(3), 483–518 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Howard, B., Pappas, G.: On the supersingular locus of the \({\rm GU}(2,2)\) Shimura variety. Algebra Number Theory 8(7), 1659–1699 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Howard, B., Pappas, G.: Rapoport–Zink spaces for spinor groups. Compos. Math. 153(5), 1050–1118 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Helm, D., Tian, Y.-C., Xiao, L.: Tate cycles on some unitary Shimura varieties mod \(p\). Algebra Number Theory 11(10), 2213–2288 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Katsura, T., Oort, F.: Families of supersingular abelian surfaces. Compos. Math. 63(2), 107–167 (1987)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kottwitz, R.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 2, 373–444 (1985)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kottwitz, R.: Isocrystals with additional structure. II. Compos. Math. 109(3), 225–339 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kudla, S., Rapoport, M.: Cycles on Siegel threefolds and derivatives of Eisenstein series. Ann. Sci. École Norm. Sup. (4) 33(5), 695–756 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kottwitz, R.: Isocrystals with additional structure. Compos. Math. 56(2), 201–220 (1985)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Madapusi Pera, K.: Integral canonical models for spin Shimura varieties. Compos. Math. 152(4), 769–824 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Norman, P., Oort, F.: Moduli of abelian varieties. Ann. Math. (2) 112(3), 413–439 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Oki, Y.: On the supersingular loci of Shimura varieties for quaternion unitary groups of degree 2. Master Thesis, The University of Tokyo. arXiv:1907.07026
  22. 22.
    Rapoport, M.: A guide to the reduction modulo \(p\) of Shimura varieties. Automorphic forms. I. Astérisque 298, 271–318 (2005)zbMATHGoogle Scholar
  23. 23.
    Rapoport, M., Zink, T.: Period Spaces for \(p\)-Divisible Droups, Annals of Mathematics Studies, vol. 141, p. xxii+324. Princeton University Press, Princeton (1996)zbMATHGoogle Scholar
  24. 24.
    Rapoport, M., Terstiege, U., Wilson, S.: The supersingular locus of the Shimura variety for \({\rm GU}(1, n-1)\) over a ramified prime. Math. Z. 276(3–4), 1165–1188 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Tits, J.: Reductive Groups Over Local Fields, Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977). American Mathematical Society, Providence (1979)Google Scholar
  26. 26.
    Van Hoften, P.: A geometric Jacquet–Langlands correspondence for paramodular Siegel threefolds. arXiv:1906.04008 (preprint) (2019)
  27. 27.
    Vollaard, I.: The supersingular locus of the Shimura variety for \({\rm GU}(1, s)\). Can. J. Math. 62(3), 668–720 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety of \({\rm GU}(1, n-1)\) II. Invent. Math. 184(3), 591–627 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Wang, H.-N.: On the Bruhat–Tits stratification for GU(2,2) type Rapoport–Zink space: unramified case. arXiv:1909.10902 (preprint) (2019)
  30. 30.
    Wang, H-.N.: On a quaternionic unitary Rapoport–Zink space with parahoric level structure. arXiv:1909.12263 (preprint) (2019)
  31. 31.
    Wang, H.-N.: Level lowering for GSp(4) and vanishing cycles on Siegel threefold. arXiv:1910.07569 (preprint) (2019)
  32. 32.
    Wu, H-F.: The supersingular locus of unitary Shimura varieties with exotic good reduction. PhD thesis, University of Duisburg-Essen. arXiv:1609.08775 (2016)
  33. 33.
    Yu, C.-F.: The supersingular loci and mass formulas on Siegel modular varieties. Doc. Math. 11, 449–468 (2006)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Yu, C.-F.: Geometry of the Siegel modular threefold with paramodular level structure. Proc. Am. Math. Soc. 139, 3181–3190 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Zink, T.: Windows for Displays of \(p\)-Divisible Groups, Moduli of Abelian Varieties (Texel Island, 1999) Progr. Math, vol. 196, pp. 491–518. Birkhäuser, Basel (2001)zbMATHGoogle Scholar
  36. 36.
    Zhu, X.-W.: Affine Grassmannians and the geometric Satake in mixed characteristic. Ann. Math. (2) 185(2), 403–492 (2017)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsMcGill UniversityMontrealCanada

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