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# Modular Equations and Discrete, Genus-Zero Subgroups of $\SL(2,\mathbb{R})$ Containing $\Gamma(N)$

Published:2002-03-01
Printed: Mar 2002
• C. J. Cummins
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## Abstract

Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains $\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there is a unique normalised generator of the field of $G$-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.
 MSC Classifications: 11F03 - Modular and automorphic functions 11F22 - Relationship to Lie algebras and finite simple groups 30F35 - Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]

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