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

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.