Anneau des formes quasi-modulaires sur un groupe co-compact

We describe the additive structure of the graded ring ${\stackrel{?}{M}}_1(\Gamma )$ of quasimodular forms over any discrete and cocompact group $\Gamma ?\mathrm{PSL}(2,\mathbb{R})$. We show that this ring is not finitely generated. We calculate the exact number of new generators of weight k (even). This number is constant for k sufficiently large and equals ${\dim }_{\mathbb{C}}I/(I\cap {\stackrel{?}{I}}^2)$, where I and $\stackrel{?}{I}$ are the ideals of modular forms and quasimodular forms, respectively, over Γ of strictly positive weight. To cite this article: N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 343 (2006).