Formal deformations of the algebra of Jacobi forms and Rankin-Cohen brackets


The aim of this work is to emphasize the arithmetical and algebraic aspects of the Rankin-Cohen brackets in order to extend them to several natural number-theoretical situations. We build an analytically consistent derivation on the algebra $\widetilde{\mathcal{J}} {\mathrm{ev},\ast}$ of weak Jacobi forms. From this derivation, we obtain a sequence of bilinear forms on $\widetilde{\mathcal{J}} {\mathrm{ev},\ast}$ that is a formal deformation and whose restriction to the algebra $\mathcal{M} {\ast}$ of modular forms is an analogue of Rankin-Cohen brackets associated to the Serre derivative. Using a classification of all admissible Poisson brackets, we generalize this construction to build a family of Rankin-Cohen deformations of $\widetilde{\mathcal{J}} {\mathrm{ev},\ast}$. The algebra $\widetilde{\mathcal{J}} {\mathrm{ev},\ast}$ is a polynomial algebra in four generators. We consider some localization $\mathcal{K} {\mathrm{ev},\ast}$ of $\widetilde{\mathcal{J}} {\mathrm{ev},\ast}$ with respect to one of the generators. We construct Rankin-Cohen deformations on $\mathcal{K} {\mathrm{ev},\ast}$. We study their restriction to $\widetilde{\mathcal{J}} {\mathrm{ev},\ast}$ and to some subalgebra of $\mathcal{K} {\mathrm{ev},\ast}$ naturally isomorphic to the algebra of quasimodular forms.

Comptes Rendus. Mathématique. Académie des Sciences, Paris