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

29 juin 2021·
YoungJu Choie
,
François Dumas
,
François Martin
Emmanuel Royer
Emmanuel Royer
· 0 min. de lecture
Résumé
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 J~ ev,\widetilde{\mathcal{J}} {\mathrm{ev},\ast} of weak Jacobi forms. From this derivation, we obtain a sequence of bilinear forms on J~ ev,\widetilde{\mathcal{J}} {\mathrm{ev},\ast} that is a formal deformation and whose restriction to the algebra M \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 J~ ev,\widetilde{\mathcal{J}} {\mathrm{ev},\ast}. The algebra J~ ev,\widetilde{\mathcal{J}} {\mathrm{ev},\ast} is a polynomial algebra in four generators. We consider some localization K ev,\mathcal{K} {\mathrm{ev},\ast} of J~ ev,\widetilde{\mathcal{J}} {\mathrm{ev},\ast} with respect to one of the generators. We construct Rankin-Cohen deformations on K ev,\mathcal{K} {\mathrm{ev},\ast}. We study their restriction to J~ ev,\widetilde{\mathcal{J}} {\mathrm{ev},\ast} and to some subalgebra of K ev,\mathcal{K} {\mathrm{ev},\ast} naturally isomorphic to the algebra of quasimodular forms.
Type
Publication
Comptes rendus mathématiques de l’Académie des sciences de Paris 359, No. 4, 505-521