Topics in the analytic theory of L-functions and harmonic Maass forms Open Access
Wagner, Ian (Spring 2019)
Abstract
This thesis presents several new results in the theory of $L$-functions, modular forms, and harmonic Maass forms. In particular, we prove results on the $p$-adic properties of modular and Maass forms, about the hyperbolicity of doubly infinite families of polynomials related to the partition function and general $L$-functions, and study Schwartz functions which tie together the field of modular forms and problems like sphere packing and energy optimization.
We prove a general congruence result for mixed weight modular forms using facts about direct products of Galois representations. As an application we prove explicit congruences for the conjugacy growth series of wreath products of finite groups and finitary permutations groups. We then start to answer a question of Mazur's about the existence of an eigencurve for harmonic Maass forms. We begin to answer Mazur's question by constructing two infinite familes of harmonic Maass Hecke eigenforms, and then assemble these forms to produce $p$-adic Hecke eigenlines.
In work with Larson, we make a result of Griffin, Ono, Rolen, and Zagier effective by showing that $p(n)$ satisfies the degree $3, 4,$ and $5$ T\'uran inequalities for all $n \geq 94, 206,$ and $381$ respectively. We also show that $p(n)$ satisfies the degree $d$ T\'uran inequality for all $n \geq (3d)^{24d} (50d)^{3d^2}$.
Griffin, Ono, Rolen, and Zagier recently showed that for any degree $d$ all but at most finitely many of the Riemann zeta Jensen polynomials are hyperbolic. We extended this result to any suitable $L$-function. In order to prove this result, we also obtain improved estimates for the central derivatives of these $L$-functions.
Recently, Viazovska explicitly constructed special functions using modular forms which led to the resolution of the sphere packing problem in dimensions $8$ and $24$. Together with Rolen, we study possible generalizations of Viazovska's work which can be used to attack sphere packing problems in other dimensions and other related problems. We construct a number of infinite families of Schwartz functions using modular forms, which are eigenfunctions of the Fourier transform.
Table of Contents
1 Introduction 1
1.1 Modular forms, harmonic Maass forms, and L-functions . . . . . 1
1.1.1 Classical modular forms .................. 2
1.1.2 Harmonic Maass forms................... 10
1.1.3 L-functions ......................... 15
1.2 Congruences and p-adic modular forms . . . . . . . . . . . . . . 19
1.2.1 Conjugacy growth series for wreath product finitary symmetric groups ........................ 20
1.2.2 Harmonic Maass form eigencurves. . . . . . . . . . . . . 24
1.3 Distributions and Jensen polynomials . . . . . . . . . . . . . . . 33
1.3.1 Hyperbolicity of the partition Jensen polynomials . . . . 34
1.3.2 The Jensen-Polya program for various L-functions . . . . 36
1.4 Schwartz functions ......................... 41
2 Congruences and p-adic modular forms 48
2.1 Conjugacy growth series for wreath product finitary symmetric groups................................ 48
2.1.1 Conjugacy growth series for the finitary alternating wreath product ........................... 49
2.1.2 Preliminaries ........................ 52
2.1.3 Proof of Theorem 1.2.1................... 58
2.1.4 Proof of Theorem 1.2.2................... 61
2.1.5 An example ......................... 73
2.2 Harmonic Maass form eigencurves................. 74
2.2.1 Hecke operators for harmonic Maass forms and results of Zagier ............................ 75
2.2.2 Proof of Theorem 1.2.3................... 78
2.2.3 Proof of Theorem 1.2.5................... 86
3 Distributions and Jensen polynomials 90
3.1 Hyperbolicity of the partition Jensen polynomials . . . . . . . . 91
3.1.1 Hankel determinants and ratios of close partition numbers 91
3.1.2 Proof of Theorems 1.3.1 and 1.3.2 . . . . . . . . . . . . . 95
3.1.3 Bounds for general d .................... 98
3.2 The Jensen-Polya program for various L-functions . . . . . . . . 106
3.2.1 Asymptotics for Ξ(n)(0). .................. 106
3.2.2 Proof of Theorem 1.3.6................... 110
3.2.3 Dirichlet L-functions .................... 114
3.2.4 Modular L-functions .................... 117
3.2.5 Dedekind zeta-functions .................. 120
4 Schwartz functions 128
4.1 Background on modular forms................... 128
4.2 Proof of Theorem 1.4.5....................... 131
4.2.1 The +1 eigenfunction construction . . . . . . . . . . . . 131
4.2.2 The −1 eigenfunction construction . . . . . . . . . . . . 135
4.2.3 The zeros of the Schwartz functions . . . . . . . . . . . . 139
4.3 Proof of Theorem 1.4.2....................... 141
4.3.1 The +1 eigenfunction.................... 141
4.3.2 The −1 eigenfunction.................... 142
A First Appendix 145
B Second Appendix 148
Bibliography 151
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