Cohomology of Congruence Subgroups of SU (2, 1)p and Hodge Cycles on Some Special Complex Hyperbolic Surfaces

  • Don Blasius
  • Jonathan Rogawski
Part of the Progress in Mathematics book series (PM, volume 171)


Let G be a semisimple algebraic group over a number field F and set G = ∏ vS Gv, where S is the set of archimedean places of F. As is well-known, the cohomology of a cocompact lattice Γ ⊂ G is expressed in terms of the decomposition
$${L^2}\left( {\Gamma \backslash {G_\infty }} \right) \simeq \hat \oplus m(\pi ,\Gamma )\pi $$


Unitary Group Division Algebra Cusp Form Discrete Series Congruence Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BFG]
    D. Blasius, J. Franke, F. Grunewald, Cohomology of S-arithmetic subgroups in the number field case, Inventiones Math., 116, 1–3 (1994), 75–94.MathSciNetCrossRefMATHGoogle Scholar
  2. [BR1]
    D. Blasius, J. Rogawski, Tate classes and arithmetic quotients of the twoball. In [Mo].Google Scholar
  3. [BR2]
    D. Blasius, J. Rogawski, Zeta functions of Shimura varieties. In: Motives (U. Jannsen, S. Kleiman, J.-P. Serre, eds.), Proc. Symp. Pure Math. 55, Part 2. AMS, 1994.Google Scholar
  4. [Bo]
    A. Borel, Cohomologie de sous-groupes discrets et représentations de groupes semi-simple, Astérisque, 32–34 (1976), 72–112.Google Scholar
  5. [BW]
    A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. of Math. Studies 94 (1980).MATHGoogle Scholar
  6. [Cl]
    L. Clozel, Sur une question d’Armand Borel, C.R. Acad. Sci. Paris, 324 (1997), 973–976.MathSciNetCrossRefMATHGoogle Scholar
  7. [C2]
    L. Clozel, On the cohomology of Kottwitz’s arithmetic varieties, Duke Math. J., 72 (1993), 757–795.MathSciNetCrossRefMATHGoogle Scholar
  8. [GRS1]
    S. Gelbart, J. Rogawski, D. Soudry, On periods of cusp forms and algebraic cycles for U(3), Israel J. Math., 83 (1993), 213–252.MathSciNetCrossRefMATHGoogle Scholar
  9. [GRS2]
    S. Gelbart, J. Rogawski, D. Soudry, Endoscopy, theta-liftings, and period integrals for the unitary group in three variables, Annals of Math., 145 (1997), 419–476.MathSciNetCrossRefMATHGoogle Scholar
  10. [HLR]
    G. Harder, R. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal Flächen, J. Reine. Angew. Math., 366 (1986), 53–120.MathSciNetMATHGoogle Scholar
  11. [K]
    R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Inventiones Math., 108 (1992), 653–665.MathSciNetCrossRefMATHGoogle Scholar
  12. [Ka]
    D. Kazhdan, Some applications of the Weil representation, J. Anal. Math., 32 (1977), 235–248.MathSciNetCrossRefMATHGoogle Scholar
  13. [Kn]
    A. Knapp, Lie Groups: Beyond an Introduction,Birkhäuser, Boston, Basel, Berlin, 1996.MATHGoogle Scholar
  14. [Mo]
    The zeta-functions of Picard modular surfaces (R. Langlands and D. Ramakrishnan, eds.). CRM Pubs., Montreal, 1992.Google Scholar
  15. [RZ]
    M. Rapoport and T. Zink, Über die lokale Zetafunktion von Shimurava-rietäten, Inventiones Math., 68 (1982), 21–101.MathSciNetCrossRefMATHGoogle Scholar
  16. [R1]
    J. Rogawski, Automorphic representations of the unitary group in three variables, Ann. of Math. Studies, 123 (1990).Google Scholar
  17. [R2]
    J. Rogawski, Analytic expression for the number of points mod p. In [Mo].Google Scholar
  18. [S]
    G. Shimura, Automorphic forms and the periods of abelian varieties, J.Math. Soc. Japan, 31 (1979), 561–592.MathSciNetCrossRefMATHGoogle Scholar
  19. [VZ]
    D. Vogan and G. Zuckerman, Unitary representations with non-zero cohomology, Compositio Math 53 (1984), 51–90.MathSciNetMATHGoogle Scholar
  20. [Z]
    G. Zuckerman, Continuous cohomology and unitary representations of real reductive groups, Annals of Math. 107(2) (1978), 495–516.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Don Blasius
    • 1
  • Jonathan Rogawski
    • 1
  1. 1.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA

Personalised recommendations