Inventiones mathematicae

, Volume 184, Issue 3, pp 629–682 | Cite as

Special cycles on unitary Shimura varieties I. Unramified local theory

  • Stephen Kudla
  • Michael Rapoport


The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme \(\mathcal{N}\). In the case when p is an inert prime, we define special cycles \({\mathcal{Z}}({\bold x})\) in \(\mathcal{N}\), associated to collections \({\bold x}\) of m ‘special homomorphisms’ with fundamental matrix T∈Herm m (O k ). When m=n and T is nonsingular, we show that the cycle \({\mathcal{Z}}({\bold x})\) is either empty or is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient condition, in terms of T, for \({\mathcal{Z}}({\bold x})\) to be irreducible. When \({\mathcal{Z}}({\bold x})\) is zero dimensional—in which case it reduces to a single point—we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.


Modulus Space Eisenstein Series Fundamental Matrix Hermitian Form Jordan Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    ARGOS Seminar on intersections of modular correspondences. Astérisque 312 (2007) Google Scholar
  2. 2.
    Deligne, P., Milne, J., Ogus, A., Shih, K.-y.: Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900. Springer, Berlin (1982) zbMATHGoogle Scholar
  3. 3.
    Gross, B.: On canonical and quasi-canonical liftings. Invent. Math. 84, 321–326 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Hironaka, Y.: Local zeta functions for hermitian forms and its application to local densities. J. Number Theory 71, 40–64 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Keating, K.: Lifting endomorphisms of formal A-modules. Compos. Math. 67, 211–239 (1988) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge Tracts in Math. vol. 106. Cambridge University Press, Cambridge (1993) zbMATHCrossRefGoogle Scholar
  7. 7.
    Kottwitz, R.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 5, 373–444 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kudla, S.: Intersection numbers for quotients of the complex 2-ball and Hilbert modular forms. Invent. Math. 47, 189–208 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kudla, S.: Central derivatives of Eisenstein series and height pairings. Ann. Math. 146, 545–646 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kudla, S., Millson, J.: The theta correspondence and harmonic forms, I. Math. Ann. 274, 353–378 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kudla, S., Millson, J.: Intersection numbers for cycles in locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several variables. Publ. Math. IHÉS 71, 121–172 (1990) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kudla, S., Rapoport, M.: Arithmetic Hirzebruch-Zagier cycles. J. Reine Angew. Math. 515, 155–244 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kudla, S., Rapoport, M.: Height pairings on Shimura curves and p-adic uniformization. Invent. Math. 142, 153–223 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kudla, S., Rapoport, M.: Cycles on Siegel threefolds and derivatives of Eisenstein series. Ann. Sci. Ecole Norm. Super. (4) 33, 695–756 (2000) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kudla, S., Rapoport, M., Yang, T.: On the derivative of an Eisenstein series of weight one. Int. Math. Res. Not. 7, 347–385 (1999) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kudla, S., Rapoport, M., Yang, T.: Modular Forms and Special Cycles on Shimura Curves. Annals of Mathematics Studies, vol. 161. Princeton University Press, Princeton (2006) zbMATHGoogle Scholar
  17. 17.
    Nagaoka, S.: An explicit formula for Siegel series. Abh. Math. Semin. Univ. Hamb. 59, 235–262 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Pappas, G.: Local structure of arithmetic moduli of PEL Shimura varieties. J. Algebr. Geom. 9, 577–605 (2000) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pappas, G., Rapoport, M.: Local models in the ramified case. I: The EL-case. J. Algebr. Geom. 12, 107–145 (2003) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Pappas, G., Rapoport, M.: Local models in the ramified case. II: Splitting models. Duke Math. J. 127, 193–250 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pappas, G., Rapoport, M.: Local models in the ramified case. III: Unitary groups. J. Inst. Math. Jussieu 8, 507–564 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Rapoport, M.: Deformations of formal groups, in [1], pp. 139–170 Google Scholar
  23. 23.
    Rapoport, M., Zink, T.: Period Spaces for p-divisible Groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996) Google Scholar
  24. 24.
    Shimura, G.: On Eisenstein series. Duke Math. J. 50, 417–476 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Terstiege, U.: Antispecial cycles on the Drinfeld upper half plane and degenerate Hirzebruch-Zagier cycles. Manuscr. Math. 125, 191–223 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Terstiege, U.: Intersections of arithmetic Hirzebruch-Zagier cycles. Math. Ann. (2010). doi: 10.1007/s00208-010-0520-8 Google Scholar
  27. 27.
    Terstiege, U.: Intersections of special cycles on the Shimura variety for GU(1, 2). arXiv:1006.2106
  28. 28.
    Vollaard, I.: The supersingular locus of the Shimura variety for GU(1, s). Can. J. Math. 62, 668–720 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Vollaard, I.: Endomorphisms of quasi-canonical lifts, in [1], pp. 105–112 Google Scholar
  30. 30.
    Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety for GU(1,n−1) II. Invent. Math. (2010). doi: 10.1007/s00222-010-0299-y zbMATHGoogle Scholar
  31. 31.
    Wedhorn, T.: Ordinariness in good reductions of Shimura varieties of PEL-type. Ann. Sci. Ecole Norm. Super. (4) 32, 575–618 (1999) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wewers, S.: Canonical and quasi-canonical liftings, in [1], pp. 67–86 Google Scholar
  33. 33.
    Wewers, S.: An alternative approach using ideal bases, in [1], pp. 171–178 Google Scholar
  34. 34.
    Zink, T.: Windows for displays of p-divisible groups. In: Moduli of Abelian Varieties Texel Island, 1999. Progr. Math., vol. 195, pp. 491–518. Birkhäuser, Basel (2001) Google Scholar
  35. 35.
    Zink, T.: The display of a formal p-divisible group. In: Cohomologies p-adiques et applications arithmétiques, I. Astérisque, vol. 278, pp. 127–248 (2002) Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Mathematisches Institut der Universität BonnBonnGermany

Personalised recommendations