Class fields related to the thompson series and certain arithmetic curves

Abstract

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.