Modular Hecke algebras and their Hopf symmetry

We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of “polynomial coordinates” for the “transverse space” of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the “holomorphic tangent space” of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin–Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon–Vey cocycle gives rise to a 1-cocycle on ${\rm PSL}(2,\mathbb Q)$ with values in Eisenstein series of weigh 2, which, when coupled with the “period” cocycle, yields a representative of the Euler class.