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 Gamma(0)(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 T-g (alpha) to calculate its minimal polynomial where alpha(is an element of h) is the quotient of a basis of an integral ideal in K.