  In the present 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. 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.
