On elliptic modular foliations

Abstract

In this article we consider the three parameter family of elliptic curves $E_t: y^2-4(x-t_1)^3+t_2(x-t_1)+t_3=0, t\in\C^3$ and study the modular holomorphic foliation $\F_{\omega}$ in $\C^3$ whose leaves are constant locus of the integration of a 1-form $\omega$ over topological cycles of $E_t$. Using the Gauss-Manin connection of the family $E_t$, we show that $\F_{\omega}$ is an algebraic foliation. In the case $\omega=\frac{xdx}{y}$, we prove that a transcendent leaf of $\F_{\omega}$ contains at most one point with algebraic coordinates and the leaves of $\F_{\omega}$ corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family $E_t$, we find a uniformization of $\F_{\omega}$ in $T$, where $T\subset \C^3$ is the locus of parameters $t$ for which $E_t$ is smooth. We find also a real first integral of $\F_\omega$ restricted to $T$ and show that $\F_{\omega}$ is given by the Ramanujan relations between the Eisenstein series.