Modular equations for some $\eta $-products
Volume 161 / 2013
Acta Arithmetica 161 (2013), 301-326
MSC: Primary 11F03; Secondary 14G35, 11G18.
DOI: 10.4064/aa161-4-1
Abstract
The classical modular equations involve bivariate polynomials that can be seen to be univariate in the modular invariant $j$ with integer coefficients. 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.
