On Rationally Parametrized Modular Equations

TL;DR

This paper derives numerous rationally parametrized elliptic modular equations from families of elliptic curves associated with genus-zero congruence subgroups, connecting classical Ramanujan theories to modern algebraic and differential frameworks.

Contribution

It introduces new algebraic transformations of special functions derived from elliptic curves and clarifies Ramanujan's elliptic integral theories within a modern mathematical context.

Findings

Derived modular equations from elliptic curve families

Connected Ramanujan's elliptic integrals to algebraic transformations

Extended Ramanujan's signature 6 theory into a Gauss-Manin framework

Abstract

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup $\Gamma_0(N)$, as an algebraic transformation of elliptic curve periods, parametrized by a Hauptmodul (function field generator). The periods satisfy a Picard-Fuchs equation, of hypergeometric, Heun, or more general type; so the new modular equations are algebraic transformations of special functions. When N=4,3,2 they are modular transformations of Ramanujan's elliptic integrals of signatures 2,3,4. This gives a modern interpretation to his theories of integrals to alternative bases: they are attached to certain families of elliptic curves. His anomalous theory of signature 6 turns out to fit into a general Gauss-Manin rather than a Picard-Fuchs framework.