Elliptic modular foliations

www.impa.br/∼hossein 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 1-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 ω = xdxy, 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.