Arithmetic of some modular functions

Abstract

In this thesis we mainly focus on two subjects: The first is the generation of class fields over an imaginary quadratic fields by special values of modular function with the monster simple group. Based on Hilbert``s 12th problem, that is, the construction of class fields by transcendental method, we will find the modular function $j_{1,8}$ which gives rise to a field generator of function field over the modular curve $X_1(8)$ of genus 0, and with this function we will generate various class fields over an imaginary quadratic field. On the other hand, the "moonshine conjecture" was proposed by Conway-Norton in 1979 and solved by Borcherds in 1992, which suggests a mysterious relationship of modular functions with finite simple groups and infinite dimensional Lie algebras. As a first step to find out this connection, we will derive recursion formulas satisfied by the Fourier coefficients of the normalized generator $N(j_{1,N})$.