Mathematische Annalen

, Volume 357, Issue 1, pp 51–88 | Cite as

Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C



We study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms. This generalises work of Coleman, and is a first step in an ongoing project to extend the cohomological approach to classicality to higher-dimensional Shimura varieties.

Mathematics Subject Classification (2000)

11F33 11G18 14F30 


  1. 1.
    Andre, Y., Baldassarri, F.: De Rham Cohomology of Differential Modules on Algebraic Varietie. Prepublication Institut de Mathematiques de Jussieu 184Google Scholar
  2. 2.
    Andreatta, F., Gasbarri, C.: The canonical subgroup for families of abelian varieties. Compos Math. 143(3), 566–602 (2007)MathSciNetMATHGoogle Scholar
  3. 3.
    Andreatta, F., Goren, E.Z.: Geometry of Hilbert modular varieties over totally ramified primes. Int. Math. Res. Not. 33, 1785–1835 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Andreatta, F., Iovita, A., Pilloni, V.: p-adic families of Siegel modular forms. Preprint (2012)
  5. 5.
    Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Differential operators on the base affine space and a study of \(\mathfrak{g}\)-modules. In: Lie groups and their representations. Summer School of the Bolyai Janos Mathematical Society, (Budapest, 1971), Adam Hilger Ltd., London (1975)Google Scholar
  6. 6.
    Borel, A., Wallach, N.: Continuous cohomology. In: Discrete Subgroups and Representation Theory of Reductive Groups, 2nd edn. American Mathematical Society, Providence (1999)Google Scholar
  7. 7.
    Boutot, J.F.: Varietes de Shimura: Le probleme de modules en inegale caracteristique. In: Varietes de Shimura et fonctions L. Publ. Math. Univ. Paris VII 6 (1979)Google Scholar
  8. 8.
    Buzzard, K.M.: Eigenvarieties. In: L-functions and Galois Representations, pp. 59–120. London Mathematical Society. Lecture Note Series, vol 320. Cambridge University Press, Cambridge (2007)Google Scholar
  9. 9.
    Buzzard, K.M., Taylor, R.L.: Companion forms and weight 1 forms. Ann. Math. 149, 905–919 (1999)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chai, C.L., Faltings, G.: Degenerations of Abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 22. Springer, Berlin (1990)Google Scholar
  11. 11.
    Chenevier, G.: Familles p-adiques de formes automorphes pour GL(n). Journal fur die reine und angewandte Mathematik 570, 143–217 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Coleman, R.F.: Classical and overconvergent modular forms. Inventiones Math. 124, 215–241 (1996)MATHCrossRefGoogle Scholar
  13. 13.
    Deligne, P., Pappas, G.: Singularites des espaces de modules de Hilbert, en les caracteristiques divisant le discriminant. Compos. Math. 90, 59–79 (1994)MathSciNetMATHGoogle Scholar
  14. 14.
    Emerton, M.: On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Inventiones Math. 164(1), 1–84 (2006)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Faltings, G.: On the cohomology of locally symmetric Hermitian spaces. In: Lecture Notes in Mathematics, vol. 1029, pp. 55–98 (1983)Google Scholar
  16. 16.
    Goren, E.Z., Kassaei, P.L.: Canonical subgroups over Hilbert modular varieties. Journal fur die reine und angewandte Mathematik (to appear, 2012).
  17. 17.
    Goren, E.Z., Oort, F.: Stratifications of Hilbert modular varieties. J. Algebraic Geom 9, 111–154 (2000)MathSciNetMATHGoogle Scholar
  18. 18.
    Gouvea, F.Q.: Continuity properties of Modular forms. In: Elliptic Curves and Related Topics. CRM Proceedings and Lecture Notes, AMS, vol. 4, pp. 85–99 (1994)Google Scholar
  19. 19.
    Harris, M., Taylor, R.L.: The geometry and cohomology of some simple shimura varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001)Google Scholar
  20. 20.
    Hida, H.: Control Theorems of coherent sheaves on Shimura varieties of PEL type. J. Inst. Math. Jussieu 1, 1–76 (2002)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Humphreys, J.E.: Representations of semisimple Lie algebras in the BGG category \({\cal O}\). In: Graduate Studies in Mathematics, vol. 94. American Mathematical Society, Providence (2008)Google Scholar
  22. 22.
    Kassaei, P.L.: p-adic modular forms over Shimura curves over \(\mathbb{Q}\). PhD Thesis. Massachusetts Institute of Technology (1999).
  23. 23.
    Kassaei, P.L.: A gluing lemma and overconvergent modular forms. Duke Math. J. 132(3), 509–529 (2006)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kedlaya, K.S.: Finiteness in rigid cohomology. Duke Math. J. 134, 15–97 (2006)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kedlaya, K.S.: Fourier transforms and “ p-adic Weil II”. Compos. Math. 142, 1426–1450 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Kisin, M.: Overconvergent modular forms and the Fontaine-Mazur conjecture. Inventiones Math. 153(2), 373–454 (2003)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Kisin, M., Lai, K.F.: Overconvergent Hilbert modular forms. Am. J. Math. 127, 735–783 (2005)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Kottwitz, R.: Points on Shimura varieties over finite fields. J. AMS 5:373–444 (1992)Google Scholar
  29. 29.
    Lan, K-W.: Arithmetic compactifications of PEL-type Shimura varieties. Ph.D. thesis, Harvard University (2008)Google Scholar
  30. 30.
    Lan, K.-W., Suh, J.: Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties. Duke Math. J. 161(6), 1113–1170 (2012)Google Scholar
  31. 31.
    Laumon, G.: Cohomology of Drinfel’d Modular Varieties I. Cambridge University Press, Cambridge (1996)Google Scholar
  32. 32.
    Lan, K-W., Polo, P.: Dual BGG complexes for automorphic bundles. Preprint (2010)
  33. 33.
    Le Stum, B.: Rigid cohomology. In: Cambridge Tracts in Mathematics, vol. 172. Cambridge University Press, Cambridge (2007)Google Scholar
  34. 34.
    Le Stum, B.: The overconvergent site. To appear in Memoire de la SMF (2012).
  35. 35.
    Loeffler, D.: Overconvergent algebraic automorphic forms. Proc. Lond. Math. Soc. 102(2), 193–228 (2011)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Milne, J.S.: Points on Shimura varieties mod p. In: Proceedings of Symposia in Pure Mathematics, vol. 33, part 2, pp. 165–184 (1979)Google Scholar
  37. 37.
    Milne, J.S.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic Forms, Shimura Varieties, and L-functions, pp. 283–414. Proceedings of a Conference Held at the University of Michigan, Ann Arbor, July 6–16, 1988. Also available at
  38. 38.
    Milne, J.S.: Introduction to Shimura varieties. In: Arthur, J., Kottwitz, R. (eds.) Harmonic Analysis, the Trace Formula and Shimura Varieties. AMS (2005).
  39. 39.
    Mok, C.-P., Tan, F.: Overconvergent family of Siegel-Hilbert modular forms. Preprint (2012).
  40. 40.
    Nakamura, K.: Classification of split trianguline representations of p-adic fields. Compos. Math. 145(4), 865–914 (2009)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Pilloni, V.: Prolongement analytique sur les varietes de Siegel. Duke Math. J. 157(1), 167–222 (2011)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Pilloni, V., Stroh, B.: Surconvergence et classicite: le cas Hilbert. Preprint (2012)
  43. 43.
    Pilloni, V., Stroh, B.: Surconvergence et classicite: le cas deploye. Preprint (2012)
  44. 44.
    Sasaki, S.: Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representation, and weight one forms. Preprint (2012)
  45. 45.
    Shin, S.W.: On the cohomology of Rapoport-Zink spaces of EL-type. Am. J. Math. (to appear, 2012)
  46. 46.
    Taylor, R.L., Yoshida, T.: Compatibility of local and global Langlands correspondences. J. Am. Math. Soc. 20–2, 467–493 (2007)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Tian, Y.: Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degree. Preprint (2012).
  48. 48.
    Tian, Y., Xiao, L.: p-adic cohomology and classicality of overconvergent Hilbert modular forms (2012)
  49. 49.
    Tzusuki, N.: On base change theorem and coherence in rigid cohomology. In: Documenta Mathematica, Extra, vol., pp. 891–918 (2003)Google Scholar
  50. 50.
    Urban, E.: Eigenvarieties for reductive groups. Ann. Math. 174(3), 1685–1784 (2011)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Yoshida, T.: Betti cohomology of Shimura varieties-the Matsushima formula. Notes.

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations