Quasi-modular forms attached to elliptic curves, I

Movasati, Hossein

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 2, pp. 307-377.

Dans ce texte, on donne une interprétation géométrique des formes quasimodulaires en utilisant les modules des courbes elliptiques avec un point marqué dans leurs cohomologies de de Rham. De cette façon, les équations différentielles des formes modulaires et quasimodulaires sont interprétées comme des champs de vecteurs de ces espaces de modules. Elles peuvent être établies grâce à la connection de Gauss-Manin de la famille universelle de courbes elliptiques correspondante. Pour le groupe modulaire, on calcule une telle équation différentielle qui apparaît être celle de Ramanujan qui relie entre elles les séries d’Eisenstein. On explique aussi la notion de périodes construites à partir des intégrales elliptiques. Elles apparaissent comme le pont entre la notion algébrique de forme quasimodulaire et la définition en terme de fonction holomorphe sur le demi-plan de Poincaré. De cette façon, nous obtenons aussi une autre interprétation, essentiellement due à Halphen, de l’équation différentielle de Ramanujan en termes de fonctions hypergéométriques. L’interprétation des formes quasimodulaires comme sections de fibrés des jets et des problèmes de combinatoire énumérative sont aussi présentés.

In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the Ramanujan differential equation between Eisenstein series. We also explain the notion of period map constructed from elliptic integrals. This turns out to be the bridge between the algebraic notion of a quasi-modular form and the one as a holomorphic function on the upper half plane. In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented.

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