Generation of class fields by the modular functions

Abstract

In this thesis we mainly focus on the generation of class fields over an imaginary quadratic field by singular values of some elliptic modular functions. In particular, as is well-known in the class field theory, the ray class fields over an algebraic number field $K$ correspond to specific congruence subgroups $P_{K,1}$, which are the most extreme cases. In the imaginary quadratic cases, we discovered that these groups $P_{K,1}$ are concerned with the structure of the congruence subgroups $\Gamma_{1}(N)$ of the full modular group $SL_{2}(\mathbb Z)$ and singular value(s) of the generator(s) of the modular function field $K(X_{1}(N))$. \par When the genus of the modular curve $X_{1}(N)$ is zero, i.e. $1\leq N \leq 10$ or $N=12$, $K(X_{1}(N))$ is a rational function field over $\mathbb C$. In these cases, we can generate the ray class field $K_{(N)}$ (resp. $K_{\mathfrak f}$) with modulus $N$ (resp. an ideal $\mathfrak f$ strictly dividing $N$) by one singular value of the generator which generates $K(X_{1}(N))$. However, when the genus of $X_{1}(N)$ is equal to or greater than one, there is certain universal generation of the modular function field $K(X_{1}(N))$, which is generated by two modular functions over $\mathbb C$. In these cases, we can generate ray class fields $K_{\mathfrak f}$ universally by making use of this result.