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

Abstract

Thompson series is a Hauptmodul for a genus zero group which lies between <TEX>$\Gamma$</TEX>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 <TEX>$T_g$</TEX>(<TEX>$\alpha$</TEX>) (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 (<TEX>${\zeta}N + {\zeta}_N^{-1}$</TEX>), and over a field K (<TEX>${\zeta}N$</TEX>). Furthermore, we find an explicit formula for the conjugates of Tg (<TEX>$\alpha$</TEX>) to calculate its minimal polynomial where <TEX>$\alpha$</TEX> (<TEX>${\in}{\eta}$</TEX>) is the quotient of a basis of an integral ideal in K.