Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2005; 42(2): 203-222

Printed March 1, 2005

Copyright © The Korean Mathematical Society.

Class fields from the fundamental Thompson series of level $N=o(g)$

So Young Choi and Ja Kyung Koo

KAIST, KAIST

Abstract

Thompson series is a Hauptmodul for a genus zero group which lies between $\Gamma_{0}(N)$ and its normalizer in $PSL_{2}(\Bbb R)$ ([1]). We construct explicit ring class fields over an imaginary quadratic field $K$ from the Thompson series $T_{g}(\alpha)$ (Theorem $4$), which would be an extension of [3], Theorem $3.7.5~(2)$ by using the Shimura theory and the standard results of complex multiplication. Also we construct various class fields over $K$, over a CM-field $K(\zeta_{N}+\zeta_{N}^{-1})$, and over a field $K(\zeta_{N})$. Furthermore, we find an explicit formula for the conjugates of $T_{g}(\alpha)$ to calculate its minimal polynomial where $\alpha$($\in\mathfrak H$) is the quotient of a basis of an integral ideal in $K$.

Keywords: modular functions, Thompson series, class fields

MSC numbers: 11F11, 11R04, 11R37