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

François Dumas,François Martin,YoungJu Choie,Emmanuel Royer

doi:10.5802/crmath.193

François Dumas, François Martin + Show 2 more

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https://doi.org/10.5802/crmath.193

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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.

Highlights

Appearing in the late 1950s in the study of modular forms, Rankin–Cohen brackets have undergone considerable development in many related fields, giving rise to a very abundant literature in recent decades
It is impossible to give here complete references on such a vast subject, but it is necessary for our study to mention that the article [6] gives as a corollary of more general results an explicit method to construct from any homogeneous derivation D on a graded algebra A formal Rankin–Cohen brackets which give a deformation on A; this type of brackets correspond to the notion of standard RC algebra in [23]
We prove in Proposition 10 that the two formal deformations of A obtained by these two approaches are not isomorphic

Summary

Introduction

Appearing in the late 1950s in the study of modular forms, Rankin–Cohen brackets have undergone considerable development in many related fields, giving rise to a very abundant literature in recent decades. It is impossible to give here complete references on such a vast subject, but it is necessary for our study to mention that the article [6] gives as a corollary of more general results an explicit method to construct from any homogeneous derivation D on a graded algebra A formal Rankin–Cohen brackets which give a deformation on A; this type of brackets correspond to the notion of standard RC algebra in [23] This general process cannot be applied directly to the algebra M of modular forms because it is not stable by the complex derivative. K whose restriction to M are the classical Rankin–Cohen brackets on M , and determine in Theorem 18 the values of the parameters for which these brackets give deformations of J

Definition and first properties

Isomorphic formal deformations

Example

Connes–Moscovici deformations

Formal Rankin–Cohen brackets

Original problem

A general argument about the restriction of some formal deformations

C Ei4 E6j

Application of the extension-restriction method to modular forms

Serre Rankin–Cohen brackets

Formal deformations on quasimodular forms

The notion of weak Jacobi form