Japanese Journal of Mathematics

, Volume 12, Issue 1, pp 1–32 | Cite as

Cohomology of arithmetic groups and periods of automorphic forms

Article

Abstract

We recall some unusual features of the cohomology of arithmetic groups, and propose that they are explained by a hidden action of certain motivic cohomology groups.

Keywords and phrases

motivic cohomology arithmetic groups automorphic forms 

Mathematics Subject Classification (2010)

11F75 19E15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory. Second ed., Grad. Texts in Math., 41, Springer-Verlag, 1990.Google Scholar
  2. 2.
    Arthur J.: Unipotent automorphic representations: conjectures. Orbites unipotentes et représentations. II, Astérisque 171-172, 13–71 (1989)MathSciNetMATHGoogle Scholar
  3. 3.
    A.A. Beĭlinson, Higher regulators and values of L-functions, In: Current Problems in Mathematics, 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238.Google Scholar
  4. 4.
    N. Bergeron, Propriétés de Lefschetz automorphes pour les groupes unitaires et orthogonaux, Mém. Soc. Math. Fr. (N.S.), 106, Soc. Math. France, 2006.Google Scholar
  5. 5.
    Bloch S.: Algebraic cycles and higher K-theory. Adv. in Math. 61, 267–304 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, In: The Grothendieck Festschrift. Vol. I, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400.Google Scholar
  7. 7.
    A. Borel, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg. XV, Actualités Scientifiques et Industrielles, 1341, Hermann, Paris, 1969.Google Scholar
  8. 8.
    Borel A.: Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4), 7, 235–272 (1974)MathSciNetMATHGoogle Scholar
  9. 9.
    A. Borel and H. Jacquet, Automorphic forms and automorphic representations, In: Automorphic Forms, Representations and L-functions. With a supplement “On the notion of an automorphic representation” by R.P. Langlands, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, OR, 1977, Part 1, Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, RI, 1979, pp. 189–207.Google Scholar
  10. 10.
    A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Second ed., Math. Surveys Monogr., 67, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  11. 11.
    K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations, preprint, arXiv:1009.0785.
  12. 12.
    D.-C. Cisinski and F. Déglise, Triangulated categories of mixed motives, preprint, arXiv:0912.2110.
  13. 13.
    Cowling M., Haagerup U., Howe R.: Almost L 2 matrix coefficients. J. Reine Angew. Math., 387, 97–110 (1988)MathSciNetMATHGoogle Scholar
  14. 14.
    Cremona J.E., Whitley E.: Periods of cusp forms and elliptic curves over imaginary quadratic fields. Math. Comp. 62, 407–429 (1994)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Franke J.: Harmonic analysis in weighted L 2-spaces. Ann. Sci. École Norm. Sup. (4), 31, 181–279 (1998)MathSciNetMATHGoogle Scholar
  16. 16.
    S. Galatius and A. Venkatesh, Derived Galois deformation rings, in preparation.Google Scholar
  17. 17.
    Goresky M., Kottwitz R., MacPherson R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131, 25–83 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry. Reprint of the 1978 Original, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.Google Scholar
  19. 19.
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math., 80, Academic Press, New York-London, 1978.Google Scholar
  20. 20.
    Hiraga K., Ichino A., Ikeda T.: Formal degrees and adjoint \({\gamma}\)-factors. J. Amer. Math. Soc., 21, 283–304 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jannsen U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107, 447–452 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    R.P. Langlands, On the classification of irreducible representations of real algebraic groups, In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Math. Surveys Monogr., 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170.Google Scholar
  23. 23.
    C. Mazza, V. Voevodsky and C. Weibel, Lecture notes on motivic cohomology, Clay Math. Monogr., 2, Amer. Math. Soc., Providence, RI; Clay Math. Inst., Cambridge, MA, 2006.Google Scholar
  24. 24.
    J.S. Milne, Motives—Grothendieck’s dream, www.jmilne.org/math/xnotes/MOT.pdf.
  25. 25.
    J. Nekovar, Syntomic cohomology and regulators for varieties over p-adic fields, preprint.Google Scholar
  26. 26.
    K. Prasanna and A. Venkatesh, Automorphic cohomology, motivic cohomology, and the adjoint l-function, in preparation.Google Scholar
  27. 27.
    Ribet K.A.: Galois representations and modular forms. Bull. Amer. Math. Soc. (N.S.), 32, 375–402 (1995)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    A.J. Scholl, Classical motives, In: Motives, Seattle, WA, 1991, Proc. Sympos. Pure Math., 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163–187.Google Scholar
  29. 29.
    Scholze P.: On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2), 182, 945–1066 (2015)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    A. Venkatesh, Derived hecke algebra, in preparation.Google Scholar
  31. 31.
    V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not., 2002, 351–355.Google Scholar
  32. 32.
    D.A. Vogan, Jr., Cohomology and group representations, In: Representation Theory and Automorphic Forms, Edinburgh, 1996, Proc. Sympos. Pure Math., 61, Amer. Math. Soc., Providence, RI, 1997, pp. 219–243.Google Scholar
  33. 33.
    Vogan D.A. Jr., Zuckerman G.J.: Unitary representations with nonzero cohomology. Compositio Math. 53, 51–90 (1984)MathSciNetMATHGoogle Scholar
  34. 34.
    Waldspurger J.-L.: Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54, 173–242 (1985)MathSciNetMATHGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2016

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations