Ideaux stables dans certains anneaux differentiels de formes quasi-modulaires de Hilbert

Nesterenko proved, among other results, the algebraic independence over $\QQ$ of the numbers $\pi,e^{\pi},\Gamma(1/4)$. A very important feature of his proof is a multiplicity estimate for quasi-modular forms associated to $\SL_2(\ZZ)$ which involves profound differential properties of certain non-linear differential systems. The aim of this article is to begin the study of the analogous properties for Hilbert modular and quasi-modular forms, especially those which are associated with the number field $\QQ(\sqrt{5})$. We show that the differential structure of these functions has several analogies with the differential structure of the quasi-modular forms associated to $\SL_2(\ZZ)$.