About quasi-modular forms, differential operators and Rankin--Cohen algebras

TL;DR

The paper develops an algebraic framework for constructing modular forms from derivatives of quasi-modular forms via Rankin–Cohen brackets. It proves that any canonical RC algebra of type $\frac{1}{N}\mathbb{Z}$ embeds as a sub-RC algebra of a standard RC algebra, using Cohen–Kuznetsov series to relate two RC-structures and to formulate $E_2$-involved modularity results in purely algebraic terms. It further provides an algebraic analogue of Ramanujan-type differential systems (RRC) by showing that RC brackets with a weight-2 element $\mathcal{E}_2$ yield modular forms, and it links these constructions to $\mathfrak{sl}_2(\mathbb{C})$-module structures, including applications to Calabi–Yau quasi-modular forms. The results unify RC-algebra theory with differential operators and provide structural insight into CY modular phenomena, with potential consequences for arithmetic and geometric contexts where RC brackets govern modularity. Overall, the work extends classic RC theory to broader graded settings and clarifies the algebraic underpinnings of modularity arising from derivatives and quasi-modular sources.

Abstract

We establish sufficient conditions, involving Rankin--Cohen (RC) brackets, under which certain combinations of meromorphic quasi-modular forms and their derivatives yield meromorphic modular forms. To achieve this, we adopt an algebraic perspective by working within the framework of RC algebras. First, we prove that any canonical RC algebra, whose underlying graded algebra is of type $\frac{1}{N}\mathbb{Z}$ with $N\in \mathbb{N}$, is a sub-RC algebra of a standard RC algebra. We then present and prove an algebraic formulation of results stating that specific combinations of the quasi-modular form $E_2$ with either other modular forms or itself, along with their derivatives, result in modular forms. Next, we provide equivalent formulations of these results in terms of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ and Ramanujan systems of RC type. Finally, we discuss applications not only to meromorphic quasi-modular forms but also to Calabi--Yau quasi-modular forms.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.1
• Corollary 3.1