On elliptic modular foliations ✩

In this article we consider the three parameter family of elliptic curves Et: y2 − 4(x − t1 )3 + t2 (x − t1) + t3 = 0, t ∈ ℂ3, and study the modular holomorphic foliation ℱω, in ℂ3 whose leaves are constant locus of the integration of a l-form ω over topological cycles of Et. Using the Gauss—Manin connection of the family Et, we show that ℱω is an algebraic foliation. In the case , we prove that a transcendent leaf of ℱω contains at most one point with algebraic coordinates and the leaves of ℱω corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family Et, we find a uniformization of ℱω in T, where T ⊂ ℂ3 is the locus of parameters t for which Et is smooth. We find also a real first integral of ℱω. restricted to T and show that ℱω is given by the Ramanujan relations between the Eisenstein series.