Arithmetic properties of Siegel functions and applications

Abstract

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.