Skip to main content
SpringerLink
Log in

Cohomology of arithmetic groups and periods of automorphic forms

  • Published:
Japanese Journal of Mathematics Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory. Second ed., Grad. Texts in Math., 41, Springer-Verlag, 1990.

  2. Arthur J.: Unipotent automorphic representations: conjectures. Orbites unipotentes et représentations. II, Astérisque 171-172, 13–71 (1989)

    MathSciNet  MATH  Google Scholar 

  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.

  4. N. Bergeron, Propriétés de Lefschetz automorphes pour les groupes unitaires et orthogonaux, Mém. Soc. Math. Fr. (N.S.), 106, Soc. Math. France, 2006.

  5. Bloch S.: Algebraic cycles and higher K-theory. Adv. in Math. 61, 267–304 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  8. Borel A.: Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4), 7, 235–272 (1974)

    MathSciNet  MATH  Google Scholar 

  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.

  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.

  11. K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations, preprint, arXiv:1009.0785.

  12. D.-C. Cisinski and F. Déglise, Triangulated categories of mixed motives, preprint, arXiv:0912.2110.

  13. Cowling M., Haagerup U., Howe R.: Almost L 2 matrix coefficients. J. Reine Angew. Math., 387, 97–110 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Cremona J.E., Whitley E.: Periods of cusp forms and elliptic curves over imaginary quadratic fields. Math. Comp. 62, 407–429 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Franke J.: Harmonic analysis in weighted L 2-spaces. Ann. Sci. École Norm. Sup. (4), 31, 181–279 (1998)

    MathSciNet  MATH  Google Scholar 

  16. S. Galatius and A. Venkatesh, Derived Galois deformation rings, in preparation.

  17. Goresky M., Kottwitz R., MacPherson R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131, 25–83 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  19. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math., 80, Academic Press, New York-London, 1978.

  20. Hiraga K., Ichino A., Ikeda T.: Formal degrees and adjoint \({\gamma}\)-factors. J. Amer. Math. Soc., 21, 283–304 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jannsen U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107, 447–452 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  24. J.S. Milne, Motives—Grothendieck’s dream, www.jmilne.org/math/xnotes/MOT.pdf.

  25. J. Nekovar, Syntomic cohomology and regulators for varieties over p-adic fields, preprint.

  26. K. Prasanna and A. Venkatesh, Automorphic cohomology, motivic cohomology, and the adjoint l-function, in preparation.

  27. Ribet K.A.: Galois representations and modular forms. Bull. Amer. Math. Soc. (N.S.), 32, 375–402 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  29. Scholze P.: On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2), 182, 945–1066 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Venkatesh, Derived hecke algebra, in preparation.

  31. V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not., 2002, 351–355.

  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.

  33. Vogan D.A. Jr., Zuckerman G.J.: Unitary representations with nonzero cohomology. Compositio Math. 53, 51–90 (1984)

    MathSciNet  MATH  Google Scholar 

  34. Waldspurger J.-L.: Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54, 173–242 (1985)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Akshay Venkatesh

Authors
  1. Akshay Venkatesh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akshay Venkatesh.

Additional information

Communicated by: Toshiyuki Kobayashi

This article is based on the 14th Takagi Lectures that the author delivered at the University of Tokyo on November 15–16, 2014.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Venkatesh, A. Cohomology of arithmetic groups and periods of automorphic forms. Jpn. J. Math. 12, 1–32 (2017). https://doi.org/10.1007/s11537-016-1488-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-016-1488-2

Keywords and phrases

Mathematics Subject Classification (2010)

Search

Navigation