보형함수의 특이값에 의한 유체의 생성

학위논문(박사) - 한국과학기술원 : 수학전공, 2002.2, [ vi, 71 p. ]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.한국과학기술원 : 수학전공