Differential modular forms,Elliptic curves and Ramanujan foliation

Abstract

In this article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. Then we describe the relation between the dynamics of a foliation in $\C^3$ induced by the Ramanujan relations, with vanishing of elliptic integrals. The fact that a complex manifold over the Moduli of Polarized Hodge Structures in the case $h^{10}=h^{01}=1$ has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.