The Ring of Quasimodular Forms for a Cocompact Group

Details The Ring of Quasimodular Forms for a Cocompact Group The Ring of Quasimodular Forms for a Cocompact Group

2006-03-13

arxiv.org/abs/math/0603268v1

Mathematics

Number Theory

22 pages, 1 figure

Scientific paper

We describe the additive structure of the graded ring $\widetilde{M}_*$ of quasimodular forms over any discrete and cocompact group $\Gamma \subset \rm{PSL}(2, \RM).$ 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 $\widetilde{I}$ are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that $\widetilde{M}_*$ is contained in some finitely generated ring $\widetilde{R}_*$ of meromorphic quasimodular forms with $\dim \widetilde{R}_k = O(k^2),$ i.e. the same order of growth as $\widetilde{M}_*.$

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