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Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

Published:2007-09-01
Printed: Sep 2007
• Hung-Jen Chiang-Hsieh
• Yifan Yang
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Abstract

We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is a prime and $w_p$ is the Atkin--Lehner involution. We then use the Hauptmoduls, along with modular functions on $\Gamma_1(p)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.
 MSC Classifications: 11F03 - Modular and automorphic functions 11G16 - Elliptic and modular units [See also 11R27] 11R20 - Other abelian and metabelian extensions

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