Modular equations for some $\eta $-products

Tom 161 / 2013

François Morain Acta Arithmetica 161 (2013), 301-326 MSC: Primary 11F03; Secondary 14G35, 11G18. DOI: 10.4064/aa161-4-1


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.


  • François MorainINRIA Saclay–Île-de-France & Laboratoire d'Informatique (CNRS/UMR 7161)
    École polytechnique
    F-91128 Palaiseau, France

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