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On Rationally Parametrized Modular Equations

Robert S. Maier

TL;DR

The paper presents a unified Gauss–Manin perspective for rationally parametrized modular equations tied to genus-zero $X_0(N)$, introducing canonical Hauptmoduln $t_N$ and weight-1 modular forms $ rak{h}_N$ to encode modular transformations as pullbacks of Picard–Fuchs/Gauss–Manin equations. It derives explicit parametrizations and transformations for genera-zero levels, giving hypergeometric (for $N=2,3,4$) and Heun-type (for $N=5$–$9$) equations and connecting them to Ramanujan’s elliptic integrals $K_r$, including new Ramanujan-type identities and a modular interpretation of signature $2,3,4$ (and a nonclassical Gauss–Manin case for signature 6). The framework yields a rich catalog of canonical and alternative modular forms via eta-products, with detailed recurrences and eta-expansions, and provides a coherent route to modular equations for elliptic families beyond the classical $j$-invariants. The results illuminate Ramanujan’s theories as natural specializations within a modern modular-differential system, offering new transformations and a path toward a broader Gauss–Manin-based modular transformation theory.

Abstract

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup $Γ_0(N)$, 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.

On Rationally Parametrized Modular Equations

TL;DR

The paper presents a unified Gauss–Manin perspective for rationally parametrized modular equations tied to genus-zero , introducing canonical Hauptmoduln and weight-1 modular forms to encode modular transformations as pullbacks of Picard–Fuchs/Gauss–Manin equations. It derives explicit parametrizations and transformations for genera-zero levels, giving hypergeometric (for ) and Heun-type (for –) equations and connecting them to Ramanujan’s elliptic integrals , including new Ramanujan-type identities and a modular interpretation of signature (and a nonclassical Gauss–Manin case for signature 6). The framework yields a rich catalog of canonical and alternative modular forms via eta-products, with detailed recurrences and eta-expansions, and provides a coherent route to modular equations for elliptic families beyond the classical -invariants. The results illuminate Ramanujan’s theories as natural specializations within a modern modular-differential system, offering new transformations and a path toward a broader Gauss–Manin-based modular transformation theory.

Abstract

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup , 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.

Paper Structure

This paper contains 14 sections, 32 theorems, 92 equations, 19 tables.

Table of Contents

  1. Introduction
  2. Context and overview
  3. Detailed overview
  4. Preliminaries: Fixed Points
  5. Hauptmoduln and Parametrized Modular Equations
  6. Canonical Hauptmoduln
  7. Coverings and parametrized modular equations
  8. Parametrized Modular Equations at Higher Levels
  9. The Canonical Weight-$1$ Modular Form $\mathfrak{h}_N(\tau)$ for $\Gamma_0(N)$
  10. The Modular Form $\mathfrak{h}_N(\tau)$: Explicit Computations (I)
  11. The Modular Form $\mathfrak{h}_N(\tau)$: Explicit Computations (II)
  12. Modular Equations for Elliptic Families
  13. Ramanujan's Modular Equations in Signatures $2,3,4$
  14. A Modular Approach to Signatures $2,3,4$ (and $6$)

Key Result

Proposition 4.1

For all $1<d\mid{N}$ listed in Table tab:intermediate, the Hauptmodul $t_d$ is a polynomial (rather than merely rational) function of the Hauptmodul $t_N$ iff all primes that divide $N$ also divide $d$. This occurs if $\Gamma_0(N)$ is a normal subgroup of $\Gamma_0(d),$ though the converse does not

Theorems & Definitions (84)

  • Remark
  • Proposition 4.1
  • proof
  • Definition 5.1
  • Definition 5.2
  • Definition 5.3
  • Theorem 5.1
  • Theorem 5.2
  • Corollary 5.2.1
  • proof
  • ...and 74 more