Transgressions of the Godbillon-Vey class and Rademacher functions

Abstract

We construct, out of modular symbols, 1-traces that are invariant with respect to the actions of the Hopf algebra $\Hc_{1}$ on the crossed product $\Ac_{Q}$ 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.