Iterated primitives of meromorphic quasimodular forms for $\operatorname{SL}_2(\mathbb Z)$

TL;DR

This paper extends the theory of iterated primitives to meromorphic quasimodular forms for SL2(Z), establishing an algebraic isomorphism with a shuffle algebra and deriving algebraic independence criteria.

Contribution

It introduces a new framework for iterated primitives of meromorphic quasimodular forms and connects it to explicit shuffle algebra structures, generalizing prior holomorphic results.

Findings

Algebra of iterated primitives is isomorphic to a shuffle algebra

Derived an algebraic independence criterion for primitives

Generalized results to meromorphic modular forms with restricted poles

Abstract

We introduce and study iterated primitives of meromorphic quasimodular forms for $\operatorname{SL}_2(\mathbb Z)$, generalizing work of Manin and Brown for holomorphic modular forms. We prove that the algebra of iterated primitives of meromorphic quasimodular forms is naturally isomorphic to a certain explicit shuffle algebra. We deduce from this an Ax--Lindemann--Weierstrass type algebraic independence criterion for primitives of meromorphic quasimodular forms which includes a recent result of Pa\c{s}ol--Zudilin as a special case. We also study spaces of meromorphic modular forms with restricted poles, generalizing results of Guerzhoy in the weakly holomorphic case.