Differential algebras of quasi-Jacobi forms of index zero

TL;DR

This work develops a comprehensive algebraic framework for singular and quasi-Jacobi forms of index zero, identifying a hierarchy of subalgebras (elliptic, quasielliptic-type, and quasimodular-type) and proving their stability or controlled instability under natural derivations. It then constructs explicit formal deformations of these algebras via Rankin-Cohen brackets and transvectants, connecting Jacobi forms to classical modular form theory and invariant theory. Central results include the polynomial description of the full singular-quasi-Jacobi algebra as $\mathbb{C}[\wp,\partial_z\wp,e_4,E_1,e_2]$, detailed differential relations (Oberdieck derivation, Ramanujan-type identities), and dimension formulas for spaces of singular Jacobi forms. The paper thus provides a robust toolkit for deformation-quantization-like approaches to Jacobi-analytic structures and paves the way for applications to elliptic and modular phenomena through explicit generators and relations.

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

Figures (1)

• Figure 1: Remarkable Subalgebras.

Theorems & Definitions (71)

• Definition 1
• Remark 2
• Remark 3
• Definition 4
• Remark 5
• Definition 6
• Remark 7
• Theorem 8
• proof
• Proposition 9