Bianchi's elliptic quintic curves and several modular function fields of level ten

TL;DR

This work gives an explicit link between Bianchi's elliptic quintic $E_\phi$ and Ramanujan's theta/auxiliary functions by describing two $2$-division points on $E_\phi$ in terms of Ramanujan functions $\phi(\tau)$ and the cubic roots $g_i(\tau)$. It then develops a unified framework to generate and relate modular function fields of level $10$ for subgroups between $\Gamma(5)$ and $\Gamma(10)$, using $\phi(\tau)$, the $g_i(\tau)$, and the discriminant $\delta(\tau)$. The paper provides explicit equations for the principal field $A_0(\Gamma(10))$ (genus 13) and for its genus-0, genus-1, genus-4, and genus-5 subfields, yielding concrete models of Bring's curve and related modular-curve structures. The results deepen explicit modular-function-field theory by tying division points on a classical elliptic curve to modular-curve function fields via Ramanujan’s and theta-function identities, with potential applications in computational and arithmetic aspects of modular curves and associated surfaces.

Abstract

We provide an explicit description of two torsion points on the classical Bianchi elliptic quintic curve in terms of Ramanujan's functions. As a byproduct, we describe generators and defining equations of several modular function fields of level 10 using those functions.

Figures (1)

• Figure 1: Function fields between $A_0(\Gamma_0(5))$ and $A_0(\Gamma(10))$.

Theorems & Definitions (20)

• Proposition 3.1
• Remark 3.2
• Lemma 3.3
• Theorem 4.1
• proof
• Remark 4.2
• Theorem 5.1
• proof
• Proposition 5.2
• proof