Geodesic
Modular Equations and Discrete, Genus-Zero Subgroups of SL(2, R) Containing Γ(N)
Canadian mathematical bulletin, Tome 45 (2002) no. 1, pp. 36-45

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Let $G$ be a discrete subgroup of $\text{SL}\left( 2,\,\mathbb{R} \right)$ which contains $\Gamma \left( N \right)$ for some $N$ . If the genus of $X\left( G \right)$ 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.
DOI : 10.4153/CMB-2002-004-6
Mots-clés : 11F03, 11F22, 30F35 
Modular Equations and Discrete, Genus-Zero Subgroups of SL(2, R) Containing Γ(N). Canadian mathematical bulletin, Tome 45 (2002) no. 1, pp. 36-45. doi: 10.4153/CMB-2002-004-6
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     journal = {Canadian mathematical bulletin},
     pages = {36--45},
     year = {2002},
     volume = {45},
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     doi = {10.4153/CMB-2002-004-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2002-004-6/}
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