Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry

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...

This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their...

This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms.

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the ca...

In this paper, we use the unitary representation theory of $SL_2(\mathbb R)$ to understand the Rankin-Cohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.

We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry in codimension 1. Besides the Hopf algebra found by Connes and the first author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its `Schwarzian' quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalizati...

We generalize the linear relation formula between the square of normalized Hecke eigenforms of weight $k$ and normalized Hecke eigenforms of weight $2k$, to Rankin-Cohen brackets of general degree. As an ingredient of the proof, we also generalize a formula of Zagier on the Petersson inner product of Rankin-Cohen brackets involving Eisenstein series.

For any positive integer $n$, we introduce a quasi-homogeneous vector field $\textsf{D}$ of degree $2$ on a moduli space $\textsf{T}$ of enhanced Calabi-Yau $n$-folds arising from the Dwork family. By Calabi-Yau quasi-modular forms for Dwork family we mean the elements of the graded $\mathbb{C}$-algebra $\widetilde{\mathcal{M}}$ generated by the components of a particular solution of $\textsf{D}$, which are provided with natural weight. Using $\textsf{D}$ we introduce the der...

We give the algebra of quasimodular forms a collection of Rankin-Cohen operators. These operators extend those defined by Cohen on modular forms and, as for modular forms, the first of them provide a Lie structure on quasimodular forms. They also satisfy a ``Leibniz rule'' for the usual derivation. Rankin-Cohen operators are useful for proving arithmetic identities. In particular we give an interpretation of the Chazy equation and explain why such an equation has to exist.

Hopf algebroids are generalizations of Hopf algebras to less commutative settings. We show how the comultiplication defined by Kostant and Kumar turns the affine nil Hecke algebra associated to a Coxeter system into a Hopf algebroid without an antipode. The proof relies on mixed dihedral braid relations between Demazure operators and simple reflections. For researchers new to Hopf algebroids we include additional examples from ring theory, representation theory, and algebraic...