Differential algebras of quasi-Jacobi forms of index zero

TL;DR

This paper investigates the structure of quasi-Jacobi forms of index zero, focusing on their subalgebras and stability under derivations, using algebraic and differential operator techniques.

Contribution

It introduces the concept of double depth to distinguish subalgebras within quasi-Jacobi forms and studies their stability under derivations and bidifferential operators.

Findings

Identification of significant subalgebras within quasi-Jacobi forms

Analysis of stability of these subalgebras under derivations

Development of analogs of Rankin-Cohen brackets for these forms

Abstract

The notion of double depth associated with quasi-Jacobi forms allows distinguishing, within the algebra of quasi-Jacobi singular forms of index zero, certain significant subalgebras (modular-type forms, elliptic-type forms, Jacobi forms). We study the stability of these subalgebras under the derivations of this algebra and through certain sequences of bidifferential operators constituting analogs of Rankin-Cohen brackets or transvectants