The Ring of Quasimodular Forms for a Cocompact Group

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 \(\widetilde{M}_*.\)