Some results on modular forms : valence formulas, Eisenstein series, vector-valued L-functions

Abstract

This thesis is divided into three parts, studying modular forms in three directions: valence formula, Eisenstein series and vector-valued modular form.

There are four chapters.

Chapter 1 provides the basic definitions and background of modular forms and related topics.

Chapter 2 studies the valence formula. It is well-known in the classical theory of modular forms that there exists a formula, named as the valence formula, relating the zeroes and poles of a modular form for congruence subgroups. In this chapter, a valence formula for Fuchsian groups of the first kind is found. In particular, we obtain a simple alternative proof for the valence formula for the group $\Gamma_0(N)^+$. The (first) proof is published in a paper of Choi and Kim in 2014. A salient point of the method here is that it illustrates the relation between the constant $v_{N,k}$ in the formula and the underlying Fuchsian group.

Chapter 3 is related to the real-analytic Eisenstein series $E(z,s)$ for some congruence group $\Gamma$. A famous property of $E(z,s)$ is its functional equation: $E(z,1-s) = \Phi(s)E(z,s)$ where $\Phi(s)$ is called the scattering matrix. Apparently $\Phi(s)$ plays an important role. In this chapter we confine to the case $\Gamma=\Gamma_0(p^m)$ and compute explicit formulas for the entries of $\Phi(s)$.

Chapter 4 focuses on the $L$-function of a vector-valued modular form. We evaluate the functional equation of a vector-valued $L$-function with or without twist by an additive character $e(\frac{u}{r})$. Moreover, we give an application on the functional equations for the twisted $L$-function of a (scalar-valued) integral weight cusp form on the congruence group $\Gamma_0(4)$.