Quasi-modular forms attached to elliptic curves, I

Abstract: 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.