Arithmetic properties of Siegel functions and applications = 지겔함수의 산술성과 응용

In this thesis we mainly focus on the generation of class fields by the singular values of Siegel functions. Siegel functions are modular functions which have zeros and poles only at the cusps. We investigate basic transformation formulas of Siegel functions and give a criterion to determine whether a product of Siegel functions is integral over $\mathbb{Z}[j]$, where $\It{j}$ is the elliptic modular function. To accomplish our goal we make a change of variables of an elliptic curve and obtain a new $\It{y}$-coordinate. The $\It{y}$ -coordinate as a function on the complex upper half plane generates a family of modular functions of level $\It{N}\ge3$ which constitute the field of modular functions of level $\It{N}\ge3$ together with the elliptic modular function. These functions in fact are quotients of Siegel functions and play the role of classical Fricke functions. Based on the Shimura`s reciprocity law which connects the theory of modular function fields and class field theory we construct primitive generators of ray class fields over imaginary quadratic fields by utilizing the singular values of the $\It{y}$ -function. Moreover, the conjugates of high powers of the singular values form normal bases of ray class fields over imaginary quadratic fields. This result simplifies that of Ramachandra who completely settled down the Hilbert 12th problem for the case of construction of class fields of imaginary quadratic fields.
Koo, Ja-Kyungresearcher구자경researcher
Issue Date
418764/325007  / 020065086

학위논문(박사) - 한국과학기술원 : 수리과학과, 2010.2, [ iii, 60 p. ]


elliptic curves; 복소곱정리; class fields; Siegel functions; Shimura`s reciprocity law; complex multiplication; 유체; 타원곡선; 지겔함수; 시무라 상호법칙

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