Transgressions of the Godbillon-Vey Class and Rademacher functions

Abstract

we used an ad hoc pairing with modular symbols to convert the Godbillon-Vey class [δ 1 ] ∈ HC 1 (H 1 ) into the Euler class of GL + (2, Q).In this paper we provide a completely conceptual explanation for the above pairing and at the same time extend it to the higher weight case. This is achieved by constructing out of modular symbols H 1 -invariant 1-traces that support characteristic maps for certain actions of H 1 on A Q , canonically associated to modular forms. Moreover, we show that the image of the Godbillon-Vey class through these characteristic homomorphisms, obtained by the cup product between [δ 1 ] and the invariant 1-traces, transgresses to secondary data. For the projective action determined by the Ramanujan connection the transgression takes place within the Euler class, in a manner that ressembles the K-homological transgression in the context of SU q (2) [5], and leads to the classical Rademacher function [23]. For the actions associated to cusp forms of higher weight the transgressed classes implement the Eichler-Shimura isomorphism. The actions corresponding to Eisenstein series give rise by transgression to higher Dedekind sums and generalized Rademacher functions ([28, 20]), or equivalently to the Eisenstein cocycle of [28].Generalized Dedekind sums have been related to special values of L-functions in the work of C. Meyer on the class-number formula [18,19], and higher Dedekind sums appear in the work of Siegel [25,26] and Zagier [29] on the values at non-positive integers of partial zeta functions over real quadratic fields. Eisenstein cocycles were employed by Stevens [28] and by Sczech [24] in order to compute these values more efficiently. The fact that these notions can be interpreted as secondary invariants is reminiscent of the secondary nature of the regulator invariants (cf. e.g. [10]) that are involved in the expression of the special values at non-critical points of L-functions associated to number fields (see e.g. [16], [30]).

Noncommutative Geometry, Quantum Fields and Motives

“…The presence of symmetries that are given by endomorphisms corresponds to the fact that the field K may have non-trivial class number. We also discuss, towards the end of this chapter, some further generalizations of these systems to the case of Shimura varieties [159] and to function fields [172], [104], and the relation of the GL 2 -system to the modular Hecke algebras of Connes and Moscovici [93], [94], [95]. The comparative properties of the three systems are illustrated in the table below and will be explained in detail in the rest of this chapter.…”

The Riemann-Roch strategy, Complex lift of the Scaling Site

“…5, the space of points of C Q fibers over Q × \A Q with fiber the almost periodic compactification G of R. The effect of the almost periodic compactification is occurring purely in the transversal direction and it thus suggests that the ∂ operator associated to the complex structure should be viewed as a K-homology class in the relative type II set-up. The results of [14,15,16] on the transverse structure of modular Hecke algebras should then be brought into play.…”

Cup products in Hopf cyclic cohomology via cyclic modules

“…Here δ is the character on H defined in [4]. If φ ∈ C 1 (A, C δ ) is a Hopf cyclic cocycle, then the above cup product formula φ ∪ (1 ⊗ δ 1 ) coincides with the cocycle gv constructed in [3], which is constructed via the cup product defined in [7]. These two coincidences hint to us that there should be another term in the general formula which makes the above a b + B cocycle and disappears in special cases such as the above.…”

“…The characteristic map was generalized by Crainic [5] and Gorokhovsky [7] by using the fact that a cyclic cocycle on an algebra can be seen as a trace on the universal DG algebra of that algebra. This generalization is used in [3], where it is shown how a cup product is applicable in case the algebra under question possesses no invariant trace; as a replacement one takes advantage of an invariant cyclic cocycle to realize Hopf cyclic cocycles as cyclic cocycles on the algebra. Hopf cyclic cohomology was generalized to study Hopf-(co)module (co)algebras and coefficients (partially in [12] and completely in [8,9]).…”

“…As an intermediate step, characteristic maps via higher traces of Crainic [5] and Gorokhovsky [7] can be thought of as cup products with trivial coefficients. In [3] it is beautifully disclosed how the idea of cup product is applicable in case that (or even in general) the algebra under question possesses no invariant trace; as a replacement one takes advantage of an invariant cyclic cocycle to realize Hopf cyclic cocycles as cyclic cocycles on the algebra.…”