Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

  • 237 Accesses

  • 9 Citations

Abstract

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Andre, Y., Baldassarri, F.: De Rham Cohomology of Differential Modules on Algebraic Varietie. Prepublication Institut de Mathematiques de Jussieu 184

  2. 2.

    Andreatta, F., Gasbarri, C.: The canonical subgroup for families of abelian varieties. Compos Math. 143(3), 566–602 (2007)

  3. 3.

    Andreatta, F., Goren, E.Z.: Geometry of Hilbert modular varieties over totally ramified primes. Int. Math. Res. Not. 33, 1785–1835 (2003)

  4. 4.

    Andreatta, F., Iovita, A., Pilloni, V.: p-adic families of Siegel modular forms. Preprint (2012) http://perso.ens-lyon.fr/vincent.pilloni/

  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)

  6. 6.

    Borel, A., Wallach, N.: Continuous cohomology. In: Discrete Subgroups and Representation Theory of Reductive Groups, 2nd edn. American Mathematical Society, Providence (1999)

  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)

  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)

  9. 9.

    Buzzard, K.M., Taylor, R.L.: Companion forms and weight 1 forms. Ann. Math. 149, 905–919 (1999)

  10. 10.

    Chai, C.L., Faltings, G.: Degenerations of Abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 22. Springer, Berlin (1990)

  11. 11.

    Chenevier, G.: Familles p-adiques de formes automorphes pour GL(n). Journal fur die reine und angewandte Mathematik 570, 143–217 (2004)

  12. 12.

    Coleman, R.F.: Classical and overconvergent modular forms. Inventiones Math. 124, 215–241 (1996)

  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)

  14. 14.

    Emerton, M.: On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Inventiones Math. 164(1), 1–84 (2006)

  15. 15.

    Faltings, G.: On the cohomology of locally symmetric Hermitian spaces. In: Lecture Notes in Mathematics, vol. 1029, pp. 55–98 (1983)

  16. 16.

    Goren, E.Z., Kassaei, P.L.: Canonical subgroups over Hilbert modular varieties. Journal fur die reine und angewandte Mathematik (to appear, 2012). http://www.mth.kcl.ac.uk/~kassaei/research/files/cshmv.pdf

  17. 17.

    Goren, E.Z., Oort, F.: Stratifications of Hilbert modular varieties. J. Algebraic Geom 9, 111–154 (2000)

  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)

  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)

  20. 20.

    Hida, H.: Control Theorems of coherent sheaves on Shimura varieties of PEL type. J. Inst. Math. Jussieu 1, 1–76 (2002)

  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)

  22. 22.

    Kassaei, P.L.: p-adic modular forms over Shimura curves over \(\mathbb{Q}\). PhD Thesis. Massachusetts Institute of Technology (1999). http://www.mth.kcl.ac.uk/~kassaei/research.html

  23. 23.

    Kassaei, P.L.: A gluing lemma and overconvergent modular forms. Duke Math. J. 132(3), 509–529 (2006)

  24. 24.

    Kedlaya, K.S.: Finiteness in rigid cohomology. Duke Math. J. 134, 15–97 (2006)

  25. 25.

    Kedlaya, K.S.: Fourier transforms and “ p-adic Weil II”. Compos. Math. 142, 1426–1450 (2006)

  26. 26.

    Kisin, M.: Overconvergent modular forms and the Fontaine-Mazur conjecture. Inventiones Math. 153(2), 373–454 (2003)

  27. 27.

    Kisin, M., Lai, K.F.: Overconvergent Hilbert modular forms. Am. J. Math. 127, 735–783 (2005)

  28. 28.

    Kottwitz, R.: Points on Shimura varieties over finite fields. J. AMS 5:373–444 (1992)

  29. 29.

    Lan, K-W.: Arithmetic compactifications of PEL-type Shimura varieties. Ph.D. thesis, Harvard University (2008)

  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)

  31. 31.

    Laumon, G.: Cohomology of Drinfel’d Modular Varieties I. Cambridge University Press, Cambridge (1996)

  32. 32.

    Lan, K-W., Polo, P.: Dual BGG complexes for automorphic bundles. Preprint (2010) http://www.math.princeton.edu/klan/academic.html

  33. 33.

    Le Stum, B.: Rigid cohomology. In: Cambridge Tracts in Mathematics, vol. 172. Cambridge University Press, Cambridge (2007)

  34. 34.

    Le Stum, B.: The overconvergent site. To appear in Memoire de la SMF (2012). http://perso.univ-rennes1.fr/bernard.le-stum/Publications.html

  35. 35.

    Loeffler, D.: Overconvergent algebraic automorphic forms. Proc. Lond. Math. Soc. 102(2), 193–228 (2011)

  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)

  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 http://www.jmilne.org/math/articles/

  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). http://www.jmilne.org/math/articles/

  39. 39.

    Mok, C.-P., Tan, F.: Overconvergent family of Siegel-Hilbert modular forms. Preprint (2012). https://www.math.mcmaster.ca/~cpmok/

  40. 40.

    Nakamura, K.: Classification of split trianguline representations of p-adic fields. Compos. Math. 145(4), 865–914 (2009)

  41. 41.

    Pilloni, V.: Prolongement analytique sur les varietes de Siegel. Duke Math. J. 157(1), 167–222 (2011)

  42. 42.

    Pilloni, V., Stroh, B.: Surconvergence et classicite: le cas Hilbert. Preprint (2012) http://perso.ens-lyon.fr/vincent.pilloni/

  43. 43.

    Pilloni, V., Stroh, B.: Surconvergence et classicite: le cas deploye. Preprint (2012) http://perso.ens-lyon.fr/vincent.pilloni/

  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) http://www.cantabgold.net/users/s.sasaki.03/

  45. 45.

    Shin, S.W.: On the cohomology of Rapoport-Zink spaces of EL-type. Am. J. Math. (to appear, 2012) http://math.mit.edu/~swshin/

  46. 46.

    Taylor, R.L., Yoshida, T.: Compatibility of local and global Langlands correspondences. J. Am. Math. Soc. 20–2, 467–493 (2007)

  47. 47.

    Tian, Y.: Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degree. Preprint (2012). http://arxiv.org/abs/1104.4583

  48. 48.

    Tian, Y., Xiao, L.: p-adic cohomology and classicality of overconvergent Hilbert modular forms (2012) http://math.uchicago.edu/~lxiao

  49. 49.

    Tzusuki, N.: On base change theorem and coherence in rigid cohomology. In: Documenta Mathematica, Extra, vol., pp. 891–918 (2003)

  50. 50.

    Urban, E.: Eigenvarieties for reductive groups. Ann. Math. 174(3), 1685–1784 (2011)

  51. 51.

    Yoshida, T.: Betti cohomology of Shimura varieties-the Matsushima formula. Notes. http://www.dpmms.cam.ac.uk/~ty245/2008_AGR_Fall/2008_agr_week2.pdf

Download references

Acknowledgments

The author would like to thank his PhD supervisor Kevin Buzzard for suggesting this problem and for his constant help and encouragement during every aspect of this project. He would also like to thank his second supervisor Toby Gee for valuable advice during the write-up, as well as Wansu Kim, Christopher Lazda, James Newton, Shu Sasaki and Teruyoshi Yoshida for many helpful discussions relating to this work, and Francesco Baldassarri and Bernard Le Stum for answering questions about rigid and overconvergent de Rham cohomology. The author wishes to thank the Engineering and Physical Sciences Research Council for supporting him throughout his doctoral studies. It is also a pleasure to thank the Fields Institute, where part of the write-up of this paper was done, for their support and hospitality as well as for the excellent working conditions provided. Finally the author wishes to thank the anonymous referee for correcting some typos and for insightful comments. In the first version of this paper, \(p\) was assumed to be inert in \(F\). The author wishes to sincerely thank the referee for pointing out that the methods should extend to the case of \(p\) unramified, and for urging the author to investigate the general case.

Author information

Correspondence to Christian Johansson.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Johansson, C. Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C. Math. Ann. 357, 51–88 (2013). https://doi.org/10.1007/s00208-013-0900-y

Download citation

Mathematics Subject Classification (2000)

  • 11F33
  • 11G18
  • 14F30