Determination of hauptmoduls and construction of abelian extensions of quadratic number fields

Abstract

We obtain Hauptmoduls of genus zero congruence subgroups of the type T-0(+)(p) := T-0(p) + wp, where p is a prime and wp is the Atkin-Lehner involution. We then use the Hauptmoduls, along with modular functions on T, (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.