  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 PSL(2, Q) with values in
Eisenstein series of weight 2, which when coupled with the `period' cocycle
yields a representative of the Euler class.
