Transgressions of the Godbillon-Vey Class and Rademacher functions

  • Alain Connes
  • Henri Moscovici
Part of the Aspects of Mathematics book series (ASMA)


We construct, out of modular symbols, 1-traces that are invariant with respect to the actions of the Hopf algebra H 1 on the crossed product A of the algebra of modular forms of all levels by GL+(2,ℚ) investigated in earlier work. This provides a conceptual explanation for the construction of the Euler cocycle representing the image of the universal Godbillon-Vey class under the characteristic map of noncommutative Chern-Weil theory which we developed in our earlier work. We then refine the construction to produce secondary data by transgression. For the action determined by the Ramanujan connection the transgression takes place within the Euler class and the resulting cocycle coincides with the classical Rademacher function. The actions associated to cusp forms of higher weight produce transgressed cocycles that implement the Eichler-Shimura isomorphism. Finally, the actions corresponding to Eisenstein series give rise by transgression to the Eisenstein cocycle, expressed in terms of higher Dedekind sums and generalized Rademacher functions.


Hopf Algebra Modular Form Eisenstein Series Euler Class Chern Character 
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  1. [1]
    Asai, T., The reciprocity of Dedekind sums and the factor set for the universal covering group of SL(2, ℝ), Nagoya Math. J. 37 (1970), 67–80.MATHMathSciNetGoogle Scholar
  2. [2]
    Barge, J. and Ghys, E., Cocycles d’Euler et de Maslov, Math. Ann. 294 (1992), 235–265.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Connes, A., Noncommutative differential geometry, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985), 257–360.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Connes, A., Noncommutative Geometry, Academic Press, 1994.Google Scholar
  5. [5]
    Connes, A., Cyclic cohomology, quantum group symmetries and the local index formula for SU q(2), J. Inst. Math. Jussieu, 3 (2004), 17–68.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Connes, A. and Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198 (1998), 199–246.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Connes, A. and Moscovici, H., Cyclic cohomology and Hopf algebra symmetry, Letters Math. Phys. 52 (2000), 1–28.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Connes, A. and Moscovici, H., Modular Hecke algebras and their Hopf symmetry, Moscow Math. J. 4 (2004).Google Scholar
  9. [9]
    Connes, A. and Moscovici, H., Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Moscow Math. J. 4 (2004), 111–130.MATHMathSciNetGoogle Scholar
  10. [10]
    Dupont, J., Hain, R., Zucker, S., Regulators and characteristic classes of flat bundles, “The arithmetic and geometry of algebraic cycles” (Banff, AB, 1998), 47–92, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  11. [11]
    Gorokhovsky, A., Secondary classes and cyclic cohomology of Hopf algebras, Topology, 41 (2002), 993–1016.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Hajac, P. M., Khalkhali, M., Rangipour, B., Sommerhäuser, Y., Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris, 338 (2004), 667–672.MATHMathSciNetGoogle Scholar
  13. [13]
    Hall, R. R., Wilson, J. C., Zagier, D., Reciprocity formulae for general Dedekind-Rademacher sums, Acta Arith. 73 (1995), 389–396.MATHMathSciNetGoogle Scholar
  14. [14]
    Hecke, E., Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 199–224; also in Mathematische Werke, Third edition, Vandenhoeck & Ruprecht, Göttingen, 1983.MATHGoogle Scholar
  15. [15]
    Khalkhali, M., Rangipour, B., Cup products in Hopf-cyclic cohomology, to appear in C. R. Math. Acad. Sci. Paris.Google Scholar
  16. [16]
    Kontsevich M. and Zagier, D., Periods, “Mathematics unlimited—2001 and beyond”, 771–808, Springer, Berlin, 2001.Google Scholar
  17. [17]
    Kubota, T., Topological covering of SL(2) over a local field, J. Math. Soc. Japan, 19 (1967), 231–267.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Meyer, C., Die Berechnung der Klassenzahl Abelscher Korper über quadratischen Zahlkorpern, Akademie-Verlag, Berlin, 1957.Google Scholar
  19. [19]
    Meyer, C., Über einige Anwendungen Dedekindsche Summen, J. Reine Angew. Math. 198 (1957), 143–203.MathSciNetGoogle Scholar
  20. [20]
    Nakamura, H., Generalized Rademacher functions and some congruence properties, “Galois Theory and Modular Forms”, Kluwer Academic Publishers, 2003, 375–394.Google Scholar
  21. [21]
    Petersson, H., Zur analytischen Theorie der Grenzkreisgruppen I, Math. Ann. 115 (1938), 23–67.CrossRefMathSciNetGoogle Scholar
  22. [22]
    Rademacher, H., Zur Theorie der Modulfunktionen, J. Reine Angew. Math. 167 (1931), 312–366.Google Scholar
  23. [23]
    Rademacher, H. and Grosswald, E., Dedekind Sums, The Carus Mathematical Monographs, No. 16, Mathematical Association of America, 1972.Google Scholar
  24. [24]
    Sczech, R., Eisenstein cocycles for GL2(ℚ) and values of L-functions in real quadratic fields, Comment. Math. Helvetici, 67 (1992), 363–382.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Siegel, C. L., Bernoullische Polynome und quadratische Zahlkörper, Nachr. Akad. Wiss. Göttingen Math.-physik, 2 (1968), 7–38.Google Scholar
  26. [26]
    Siegel, C. L., Über die Fourierschen Koeffizienten von Modulformen, Nachr. Akad. Wiss. Göttingen Math.-physik, 3 (1970), 15–56.Google Scholar
  27. [27]
    Stevens, G., Arithmetic on modular curves, Progress in Mathematics, 20, Birkhäuser, Boston, MA, 1982.Google Scholar
  28. [28]
    Stevens, G., The Eisenstein measure and real quadratic fields, in “Proceedings of the International Conference on Number Theory” (Quebec, 1987), 887–927, de Gruyter, Berlin, 1987.Google Scholar
  29. [29]
    Zagier, D., Valeurs des fonctions zeta des corps quadratiques réels aux entiers négatifs, Astérisque, 41–42 (1977), 135–151.MathSciNetGoogle Scholar
  30. [30]
    Zagier, D., Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, In: “Arithmetic Algebraic Geometry”, Progr. Math. 89, Birkhäuser (1991), 391–430.MathSciNetGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Alain Connes
    • 1
  • Henri Moscovici
    • 2
  1. 1.Collège de FranceParisFrance
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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