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

Abstract

We introduce and study iterated primitives of meromorphic quasimodular forms for ${\mathrm{SL}}_2(\mathbb{Z})$, 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 Paşol--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.

18 pages; final version; to appear in Transactions of the AMS

References in corpus (1)

• Magnetic (quasi-)modular forms