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

TL;DR

This paper investigates the algebraic and arithmetic properties of Rankin-Cohen brackets, demonstrating their role as formal deformations of algebras of Jacobi forms and exploring their connections with modular and quasimodular forms.

Contribution

It introduces a comprehensive algebraic framework for Rankin-Cohen brackets as formal deformations and applies this to weak Jacobi forms and related modular structures.

Findings

Rankin-Cohen brackets act as formal deformations of algebraic structures.

Established links between Jacobi forms and modular/quasimodular forms.

Developed methods for restriction-extension in algebraic deformations.

Abstract

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.