Galois representations and cohomology of GL(n, ℤ)

  • Avner Ash
Part of the Progress in Mathematics book series (PM, volume 102)


Since Serre [Se3] first conjectured the possibility of attaching Galois representations to higher weight modular forms for GL(2), and Deligne [D] proved it, this notion has been expanded to apply to a large class of automorphic forms on more general reductive groups. Clozel [Cl1], following Langlands [La], has given a precise conjecture for GL(n), recalled below. I shall refer to it as the “central conjecture”. Further discussion of the history, which should be traced backwards at least to work of Eichler, Shimura, and Weil, may be found in the last section of [Se3] and the introduction to [Cl1]. One should add the remark that Serre [Se4] seems to have been the first to propose that all L-functions of motives might be L-functions of automorphic forms.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. references>[Al] A. Ash. — Farrell cohomology of GL(n, 7), Israel J. 67 (1989), 327–336.Google Scholar
  2. A2] A. Ash. — Galois representations attached to modp cohomology of GL(n, Z),preprint.Google Scholar
  3. [AGG]
    A. Asx, D. Grayson and P. Green. — Computations of cuspidal cohomology of congruence subgroups of SL(3, 7), J. Number Th. 19, (1984), 412–436.CrossRefGoogle Scholar
  4. AM] A. AsH and M. Mc Connell. — Mod p cohomology of SL(n, 7),to appear in Topology.Google Scholar
  5. APT] A. Ash, R. Pinch and R. Taylor. — An A4 extension of Q attached to a nonselfdual automorphic form on GL(3),preprint.Google Scholar
  6. [AS]
    A. Asx and G. Stevens. — Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J.f.d. reine u. angew. Math. 365, (1986), 192–220.Google Scholar
  7. [Br]
    K. Brown. — Cohomology of Groups, Springer, New York, 1982.CrossRefzbMATHGoogle Scholar
  8. [Cil]
    L. Clozel. — Motifs et formes automorphes: applications du principe de fonctorialité, in Automorphic Forms, Shimura Varieties and L-functions, Proceedings of the Ann Arbor Conference, L. Clozel and J.S. Milne eds, Academic Press I, (1990), 77–159.Google Scholar
  9. C12] L. Clozel. — Représentations galoisiennes associées aux représentations auto-morph es au toduales de GL(n),preprint.Google Scholar
  10. Cr] T. Crespo. — Explicit construction of Ate, type fields, J. Alg. 127 (1989), 452461.Google Scholar
  11. [D]
    P. Deligne. — Formes modulaires et représentations l-adiques, Séminaire Bourbaki 1968/69, n° 355, and Lecture Notes in Math., Springer-Verlag 179, (1971), 139–186.MathSciNetGoogle Scholar
  12. [EM]
    B. Eckmann and G. Mislin.–Galois action on algebraic matrix groups, Chern classes, and the Euler class, Math. Ann. 271, (1985), 349–358.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Gr]
    M. Gras.–Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q, J.f.d. reine u. angew. Math. 277, (1975), 89–116.Google Scholar
  14. [JPSS]
    H. Jacquet, I.I. Piaiij.Isky-Shapiro and J. Shalika.–Conducteur des représentations des groupes linéaires, Math. Ann. 256, (1981), 199–214.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [La]
    R. LANGLANDS.–Automorphic representations, Shimura varieties and motives, Ein Märchen, Proc. Symp. Pure Math. 33, part 2, (1979), 205–246.Google Scholar
  16. [Sel]
    J.-P. Serre.–Sur les représentations modulaires de degré 2 de Gal(-0/Q), Duke J. 54, (1987), 179–230.CrossRefGoogle Scholar
  17. [Se2]
    J.-P. Serre.–L’invariant de Witt de la forme Tr(x 2 ), Comm. Math. HeIv. 59, (1984), 651–676.Google Scholar
  18. [Se3]
    J.-P. Serre.–Une interprétation des congruences relatives à la fonction T de Ramanujan, Seminar Delange-Pisot-Poitou 1967/68, n° 14; # 80 in Collected Works, Volume II, 498–511.Google Scholar
  19. [Se4]
    J.-P. Serre.–Résumé des cours de 1966–1967; # 78 in Collected Works, Volume II, 470–471.Google Scholar
  20. [SW]
    G. Shimura. - Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971.Google Scholar
  21. [Sol]
    C. Soule. — The cohomology of SL(3, 7), Topology 17, (1978), 1–22.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [So2]
    C. Soulé. — K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Inv. Math. 55, (1979), 251–295.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Avner Ash
    • 1
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

Personalised recommendations