Modular Hecke Algebras and their Hopf Symmetry

By Alain Connes, H. Moscovici. 2003

75 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38c87172bc077daf05c18e65a443314193f92730

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Matilde Marcolli
2. V. Ginzburg
3. Alain Connes
4. I. Gordon

In Geometry and Topology

1. Matilde Marcolli
2. V. Ginzburg
3. Alain Connes
4. I. Losev
5. I. Gordon
6. Michel van den Bergh
7. Pavel Etingof
8. T. Arakawa

In Mathematical Physics

1. Matilde Marcolli
2. V. Ginzburg
3. Alain Connes
4. I. Losev
5. A. Premet
6. I. Gordon
7. Pavel Etingof
8. T. Arakawa

Citations in the corpus

• Lectures on Noncommutative Geometry (2005, background): “The same Hopf algebra was later shown to act on the noncommutative space of modular Hecke algebras [42].”
• Noncommutative Geometry, Quantum Fields and Motives (2007, methodology): “74 that if f1 and f2 are arithmetic elements then (f1 ∗ f2)g,m ∈ F (compare with Proposition 2 of [102]).”