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, Volume 151, Issue 1–2, pp 19–48 | Cite as

A trace formula approach to control theorems for overconvergent automorphic forms

Article

Abstract

We present an approach to proving control theorems for overconvergent automorphic forms on Harris–Taylor unitary Shimura varieties based on a comparison between the rigid cohomology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris–Taylor version of the Langlands–Kottwitz method. We also prove a higher level version, generalizing work of Coleman.

Mathematics Subject Classification

11F33 11G18 14F30v 

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References

  1. 1.
    Andreatta F., Iovita A., Pilloni V.: p-adic families of Siegel modular forms. Ann. Math. 181, 1–75 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bellaiche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Asterisque 324, 314 (2009)Google Scholar
  3. 3.
    Bijakowski, S.: Classicité de formes modulaires surconvergentes. http://arxiv.org/abs/1212.2035
  4. 4.
    Bijakowski, S.: Formes modulaires surconvergentes, ramification et classicité. http://arxiv.org/abs/1504.07421
  5. 5.
    Bijakowski, S.: Analytic continuation on Shimura varieties with \({\mu}\)-ordinary locus. http://arxiv.org/abs/1504.07423
  6. 6.
    Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups and Representation Theory of Reductive Groups, 2nd edn. Amer. Math. Soc. Providence (1999)Google Scholar
  7. 7.
    Buzzard K.M., Taylor R.L.: Companion forms and weight 1 forms. Ann. Math. 149, 905–919 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Casselman, W.: Introduction to the theory of admissible representations of \({p}\)-adic reductive groups. http://www.math.ubc.ca/~cass/research/publications.html
  9. 9.
    Coleman R.F.: Classical and overconvergent modular forms. Invent. Math. 124, 215–241 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Coleman R.F.: Classical and overconvergent modular forms of higher level. Journal des théories des nombres de Bordeaux 9(2), 395–403 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Faltings, G.: On the Cohomology of Locally Symmetric Hermitian Spaces. Lecture Notes in Mathematics, vol. 1029, pp. 55–98 (1997)Google Scholar
  12. 12.
    Faltings, G., Chai, C-L.: Degenerations of abelian varieties. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 22. Springer Berlin (1990)Google Scholar
  13. 13.
    Fujiwara K.: Rigid geometry, Lefschetz–Verdier trace formula and Deligne’s conjecture. Invent. Math. 127(3), 489–533 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Goldring, W., Nicole, M-H.: The \({\mu}\)-ordinary Hasse invariant of unitary Shimura varieties. https://sites.google.com/site/wushijig/
  15. 15.
    Harris, M., Lan, K-W., Taylor, R., Thorne, J.: On the Rigid Cohomology of certain Shimura varieties. http://www.math.ias.edu/rtaylor/
  16. 16.
    Harris, M., Taylor, R.L.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Math. Studies 151. Princeton Univ. Press Princeton (2001)Google Scholar
  17. 17.
    Hida H.: Control theorems of coherent sheaves on Shimura varieties of PEL type. J. Inst. Math. Jussieu 1(1), 1–76 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hida, H. : p-adic Automorphic forms on Shimura varieties. Springer Monographs in Mathematics Springer, New York (2004)Google Scholar
  19. 19.
    Humphreys, J.E.: Representations of Semisimple Lie Algebras in the BGG Category \({\mathcal{O}}\). Grad. Stud. Math., 94, Amer. Math. Soc. (2008)Google Scholar
  20. 20.
    Johansson C.: Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C. Math. Annalen 357(1), 51–88 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kassaei P.L.: A Gluing Lemma And Overconvergent Modular Forms. Duke Math. J. 132(3), 509–529 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Katz, N.: Serre-Tate Local Moduli. Lecture Notes in Mathematics, vol. 868, pp. 138–202Google Scholar
  23. 23.
    Kleiman, S.L.: Algebraic cycles and the Weil conjectures. In: Dix esposés sur la cohomologie des schémas. North-Holland, Amsterdam, pp. 359–386 (1968)Google Scholar
  24. 24.
    Kottwitz R.E.: On the \({\lambda}\)-adic representations associated to some simple Shimura varieties. Invent. Math. 108, 653–665 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lan, K.W., Polo, P.: Dual BGG complexes for automorphic bundles. http://www.math.umn.edu/kwlan/academic.html
  26. 26.
    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)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lan K.W., Stroh B.: Relative cohomology of cuspidal forms on PEL-type Shimura varieties. Algebra Number Theory 8(8), 1787–1799 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Laumon, G.: Cohomology of Drinfeld Modular Varieties. Part II: Automorphic Forms, Trace Formulas and Langlands Correspondences. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge (2009)Google Scholar
  29. 29.
    Le Stum, B.: Rigid Cohomology, vol. 172. Cambridge Tracts in Mathematics Cambridge University Press, Cambridge (2007)Google Scholar
  30. 30.
    Mantovan E.: On the cohomology of certain PEL type Shimura varieties. Duke Math. J. 129(3), 573–610 (2005)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mieda, Y.: Cycle classes, Lefschetz trace formula and integrality for p-adic cohomology. Algebraic Number Theory and Related Topics 2007, RIMS Kokyoroku Bessatsu B12, pp. 57–66 (2009)Google Scholar
  32. 32.
    Milne, J. S.: Canonical Models of (Mixed) Shimura varieties and Automorphic Vector Bundles. In: Automorphic Forms, Shimura Varieties, and L-functions. Proceedings of a Conference held at the University of Michigan, Ann Arbor, July 6–16, pp. 283–414 (1988)Google Scholar
  33. 33.
    Pilloni, V., Stroh, B.: Surconvergence et classicité : le cas Hilbert. http://perso.ens-lyon.fr/vincent.pilloni/
  34. 34.
    Pilloni, V., Stroh, B.: Surconvergence et classicité : le cas deploye. http://perso.ens-lyon.fr/vincent.pilloni/
  35. 35.
    Scholl, A.: Classical motives. In: Jannsen, U., Kleiman, S., Serre, J.-P. (eds) Motives. Seattle 1991. Proceedings of the Symposium in Pure Mathemathics, vol. 55, part 1, pp. 163–187 (1994)Google Scholar
  36. 36.
    Shin S.W.: Counting points on Igusa varieties. Duke Math. J. 146(3), 509–568 (2009)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Shin S.W.: Stable trace formula for Igusa varieties. J. Inst. Math. Jussieu 9, 847–895 (2010)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Shin S.W.: Galois representations arising from some compact Shimura varieties. Annals of Math. 173, 1645–1741 (2011)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Shin S.W.: On the cohomology of Rapoport–Zink spaces of EL-type. Am. J. Math. 134, 407–452 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Taylor R.L., Taylor R.L., Taylor R.L.: Compatibility of local and global Langlands correspondences. J. Am. Math. Soc. 20(2), 467–493 (2007)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Tian, Y.: Classicality of overconvergent Hilbert eigenforms: Case of quadratic residue degree. http://arxiv.org/abs/1104.4583
  42. 42.
    Tian, Y., Xiao, L.: p-adic cohomology and classicality of overconvergent Hilbert modular forms. To appear in Astérisque.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA

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