The Ring of Quasimodular Forms for a Cocompact Group

Abstract

We describe the additive structure of the graded ring ${\tilde{M}}_{\ast }$ of quasimodular forms over any discrete and cocompact group $Γ\subset \rm{PSL}(2, \RM).$ We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight $k$. This number is constant for $k$ sufficiently large and equals $\dim_{\CM}(I / I \cap \widetilde{I}^2),$ where $I$ and $\tilde{I}$ are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that ${\tilde{M}}_{\ast }$ is contained in some finitely generated ring ${\tilde{R}}_{\ast }$ of meromorphic quasimodular forms with $\dim {\tilde{R}}_k=O(k^2),$ i.e. the same order of growth as ${\tilde{M}}_{\ast }.$

22 pages, 1 figure

Cited by in corpus (4)

• Quasimodular forms, Jacobi-like forms, and pseudodifferential operators
• On Quasi-Modular Forms, Almost Holomorphic Modular Forms, and the Vector-Valued Modular Forms of Shimura
• Meromorphic almost rigid geometric structures
• Shimura's Vector-Valued Modular Forms, Weight Changing Operators, and Laplacians