Abstract
We give an explicit formula for the Hauptmodul \(\left( \frac{\eta (\tau )}{\eta (13 \tau )}\right) ^2\) of the level-13 Hecke modular group \(\Gamma _0(13)\) as a quotient of theta constants, together with some related explicit formulas. Similar results for primes \(p=2, 3, 5, 7\) (the other p for which \(\Gamma _0(p)\) has genus zero) are well known, and date back to Klein and Ramanujan. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan’s modular equation of degree 13, but with different kinds of modular parameterizations.
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Acknowledgments
The author thanks Pierre Deligne for his comments on an earlier version of this paper. The author thanks George Andrews and Bruce Berndt for their encouragement on the present paper. In particular, the author thanks the referee very much for his helpful comments.
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Yang, L. The Dedekind \(\eta \)-function, a Hauptmodul for \(\Gamma _0(13)\), and invariant theory. Ramanujan J 42, 689–712 (2017). https://doi.org/10.1007/s11139-016-9800-6
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DOI: https://doi.org/10.1007/s11139-016-9800-6