In this paper, we generate the class fields by special values of modular functions at imaginary quadratic arguments, over imaginary quadratic field K.
Thompson series is a Hauptmodul for a genus zero group which lies between $ ＼Gamma_0(N)$ and its normalizer in $PSL_2(\Bbb R)$ ([2]). We construct explicit ring class fields over an imaginary quadratic field $K$ from the Thompson series $T_g$, which would be an extension of [Theorem 3.7.5 (2)]{Chen} by using the Shimura theory.
The function field $K(X_1(N)^*)$ over $X_1(N)^*$ is a rational function field over $\mathbb{C}$ since the modular curve $X_1(N)^* = ＼Gamma _1(N)\backslash \frak{H}^*$ has genus zero exactly for the fourteen case 1 ≤ N ≤ 12, 14 and N=15. We find such a field generator $j^*_{1,N}$ and construct explicit class fields over an imaginary quadratic field K from the modular function $j^*_{1,N}$ and $＼zeta _N+ ＼Gamma_N^{-1}$ by using the Shimura`s reciprocity.
