On Slope Subspaces of Cohomology of p-adic Verma Modules

  • J. MahnkopfEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 245)


We determine bounds for the dimension of the slope subspaces of cohomology groups of arithmetic subgroups of semi simple algebraic groups \(\mathbf{G}\) with coefficients in p-adic Verma modules.


Cohomology of arithmetic groups Gouvea-Mazur conjecture Verma module 



I am grateful to the referee for pointing out errors and for helpful remarks. I am thankful for the invitation to participate in a conference in honor of Prof. Schwermer’s 66th birthday.


  1. 1.
    Ash, A., Stevens, G.: p-adic deformations of arithmetic cohomology, preprint (2008)Google Scholar
  2. 2.
    Borel, A.: Properties and linear representations of Chevalley groups. Seminar on Algebraic Groups and Related Finite Groups. LNM, vol. 131. Springer, Berlin (1970)CrossRefGoogle Scholar
  3. 3.
    Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1974)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bewersdorff, J.: Eine Lefschetzsche Fixpunktformel für Hecke Operatoren, Bonner Mathematische Schriften, vol. 164 (1985)Google Scholar
  5. 5.
    Brown, K.S.: Cohomology of Groups. GTM, vol. 87. Springer, New York (1982)zbMATHGoogle Scholar
  6. 6.
    Bourbaki, N.: Lie Groups and Lie Algebras, Chap. 1–3. Springer, Berlin (1989)zbMATHGoogle Scholar
  7. 7.
    Buzzard, K.: Families of modular forms. J. de Théorie de Nombres de Bordeaux 13, 43–52 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Buzzard, K., Calegari, F.: A counter-example to the Gouvea–Mazur conjecture. Comptes rendus - Mathématiques 338(10), 751–753 (2004)CrossRefGoogle Scholar
  9. 9.
    Chevalley, C.: Certains schémas de groupes semi simples. Semin. Bourbaki 219, 219–234 (1961)zbMATHGoogle Scholar
  10. 10.
    Gouvêa, F.: p-Adic Numbers. Springer, Berlin (1993)Google Scholar
  11. 11.
    Gouvêa, F., Mazur, B.: Families of modular eigenforms. Math. Comput. 58(198), 793–805 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hida, H.: On p-adic Hecke algebras for GL2 over totally real fields. Ann. Math. 128, 295–384 (1988)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hida, H.: Elementary Theory of Eisenstein Series and \(L\)-Functions. London Mathematical Society. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  14. 14.
    Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1972)CrossRefGoogle Scholar
  15. 15.
    Kuga, M., Parry, W., Sah, C.: Group cohomology and Hecke operators. In: Hano, J., et al. (eds.) Manifolds and Lie groups, pp. 223–266. Birkhäuser, Boston (1981)CrossRefGoogle Scholar
  16. 16.
    Mahnkopf, J.: On truncation of irreducible representations of Chevalley groups. J. Number Theory 133, 3149–3182 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mahnkopf, J.: Local constancy of dimension of slope subspaces of automorphic forms. Pac. J. Math. 289(2), 317–380 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    vanRooij, A.C.M.: Non-archimedean Functional Analysis. Marcel Dekker, New York (1978)Google Scholar
  19. 19.
    Serre, J.-P.: Endomorphismes complétement continus des espaces de Banach p-adiques. Publ. Math. I.H.E.S 12, 69–85 (1962)CrossRefGoogle Scholar
  20. 20.
    Urban, E.: Eigenvarieties for reductive groups. Ann. Math. 174, 1685–1784 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wan, T.: Dimension variation of classical and p-adic modular forms. Invent. Math. 133, 449–463 (1998)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institut fur MathematikUniversitat WienViennaAustria

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