On elliptic modular foliations, II

TL;DR

This paper constructs a specific one-dimensional foliation in a weighted projective space with algebraic leaves corresponding to modular curves, providing new models and explicit formulas for their defining ideals.

Contribution

It introduces a novel example of a foliation with algebraic leaves linked to modular curves and offers new models with explicit equations, challenging previous beliefs.

Findings

Existence of a foliation with algebraic leaves isomorphic to modular curves.

New models for modular curves with explicit defining equations.

Provides closed formulas for elements in the defining ideals of these curves.

Abstract

We give an example of a one dimensional foliation $\cal F$ of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves $X_0(d),\ d\in\Bbb B$ minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on introducing faster algorithms to compute equations for small values of $d$.