Modular equations for some $\eta$-products

TL;DR

This paper extends classical modular equations to include double eta-quotients, characterizing all parameters for such equations, and provides methods for their computation along with numerical examples.

Contribution

It introduces a comprehensive framework for modular equations involving double eta-quotients, expanding upon prior work with new characterizations and computational techniques.

Findings

Characterization of parameters for double eta-quotient modular equations

Methods for computing these modular equations

Numerical examples illustrating the theory

Abstract

The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $\eta$-quotients and the Weber functions $\gamma_2$ and $\gamma_3$. In the present work, we extend this idea to double $\eta$-quotients and characterize all the parameters leading to this kind of equation. We give some properties of these equations, explain how to compute them and give numerical examples.