On elliptic modular foliations

TL;DR

This paper studies a family of elliptic curves and the associated modular foliation, revealing algebraic properties, transcendence results, and connections to Ramanujan relations and Eisenstein series.

Contribution

It demonstrates that the modular foliation is algebraic, characterizes the transcendence of its leaves, and links the foliation to Ramanujan relations via a uniformization.

Findings

The foliation is algebraic due to the Gauss-Manin connection.

A transcendental leaf contains at most one algebraic point.

The foliation is described by Ramanujan relations between Eisenstein series.

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.