CLASS FIELDS FROM THE FUNDAMENTAL THOMPSON SERIES OF LEVEL N = o(g)

Thompson series is a Hauptmodul for a genus zero group which lies between $\Gamma$ o(N) and its normalizer in PSL2(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 Tg ( $\alpha$ ) to calculate its minimal polynomial where $\alpha$ ( ${\in}{\eta}$ ) is the quotient of a basis of an integral ideal in K.