  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 relation with the Rankin-Cohen brackets on
modular and quasimodular forms.
