Generation of class fields by the modular functions = 보형함수의 특이값에 의한 유체의 생성

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.
Advisors
Koo, Ja-Kyungresearcher구자경researcher
Publisher
한국과학기술원
Issue Date
2002
Identifier
174556/325007 / 000965429
Language
eng
Description

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

Keywords

유체론; 보형함수; class field theory; modular forms; Automorphic functions; modular functions; 모듈러형식; 유체론; 보형함수; class field theory; modular forms; Automorphic functions; modular functions; 모듈러형식

URI
http://hdl.handle.net/10203/41844
Link
http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=174556&flag=t
Appears in Collection
MA-Theses_Ph.D.(박사논문)
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