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This work is devoted to the algebraic and arithmetic properties ofRankin-Cohen brackets allowing to define and study them in several naturalsituations of number theory. It focuses on the property of these brackets to beformal deformations of the algebras on which they are defined, with relatedquestions on restriction-extension methods. The general algebraic resultsdeveloped here are applied to the study of formal deformations of the algebraof weak Jacobi forms and their relation with the Rankin-Cohen brackets onmodular and quasimodular forms.

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