U.U.D.M. Project Report 2021:13 Modular Forms and Related Topics Linnea Rousu Examensarbete i matematik, 30 hp Handledare: Wolfgang Staubach Examinator: Magnus Jacobsson Juni 2021 Department of Mathematics Uppsala University ABSTRACT This thesis is a study of modular forms and related topics. It begins with some necessary background information on the modular group, SL2(Z), and the fundamental domain. Following that, modular functions are presented. These functions are invariant under the action of the modular group and are then generalized to modular forms. As an example of modular forms, the Eisenstein series is derived. Thereafter, Hecke operators are discussed. They are defined as averaging operators over a suitable collection of double cosets with respect to a group. This chapter contains a section about Hecke operators on modular forms. The final part of the thesis covers Dirichlet series, which are series of arithmetic functions. An interesting connection to modular forms is discovered. Acknowledgements First and foremost I would like to thank my supervisor, Wulf Staubach for all his help and guidance. I am very grateful that he has taken the time to supervise me. I would also like to thank Alexander Söderberg for his support and the discussions we have had throughout the process. Additionally I would like to thank my family for all the support throughout the years. Furthermore I am grateful to all the teachers who have inspired me to study mathematics. I would also like to thank my classmates for making the years at university memorable. 3 Contents 1 Introduction4 2 Modular group5 3 Fundamental domain7 4 Modular functions9 5 Modular forms 12 5.1 Eisenstein series...................... 17 6 Hecke operators 23 6.1 Introduction........................ 23 6.2 Hecke operators...................... 23 6.3 Hecke operators on periodic functions.......... 25 6.4 Hecke operators on modular forms............ 28 7 Dirichlet series 32 7.1 Introduction........................ 32 7.2 Dirichlet series and modular forms............ 33 7.3 The half plane of absolute convergence.......... 35 7.4 The function defined by a Dirichlet series........ 36 7.5 Multiplication of Dirichlet series............. 38 7.6 Euler products....................... 40 7.7 The half-plane of convergence of a Dirichlet series.... 42 7.8 Analytic properties of Dirichlet series.......... 44 7.9 Dirichlet series with non-negative coefficients...... 45 7.10 Analytic continuation................... 46 7.11 Mean value formulas for Dirichlet series......... 48 7.12 An integral formula for the coefficients of a Dirichlet series 49 4 1 Introduction Historically the study of modular functions (Swe: "modulära funktioner") started in 19th century where they first appeared in the theory of elliptic functions, more specifically as elements of the function field of an elliptic curve. The term goes back to Peter Gustav Lejeune Dirichlet, although the functions also occurred in the works of Carl Friedrich Gauss, Niels Henrik Abel and Carl Gustav Jacobi (in connection to his work on theta functions). Later, they also played a significant role in the works of Leopold Kronecker, Gotthold Eisenstein and Karl Weierstrass. Towards the end of 19th centry, modular functions became a crucial source of inspiration for Felix Klein and Henri Poincaré in the development of the theory of automorphic functions. Indeed the theory of Riemann surfaces became an important tool in this context, and it was Klein who used the term "Modulform" for the first time. A sys- tematic study of modular forms on SL2(Z) and its congruence subgroups, was made by Erich Hecke in 1925, and this established the modular forms as an independent discipline within function theory and analytic number theory. Modular forms are used in several mathematical topics due to their geometrical, arithmetical and topological properties. For example topological modular forms is currently a topic of big interest in research. We will now present some applications of modular forms, based on [8]. In the proof of Fermat’s last theorem, Andrew Wiles uses modular forms exten- sively. Additionally, from this proof new techniques were developed to solve certain diophantine equations. These developments relied on having access to tables or software for computing modular forms. Modular forms are actually used in cryptography and coding theory. More specif- ically, to construct elliptic curve cryptosystems one wants to count the number of points on the elliptic curves. In order to do so there are point counting algorithms which use modular forms. Furthermore, algebraic forms associated to modular forms are used in certain error-correcting codes. This essay is a compendium of modular forms and related topics. Our goal is to make it as self-contained as possible and to include the details which explain the theory. In order to read this essay one should be familiar with abstract algebra, Fourier series, functional analysis and theory of integration on a basic level, as well as complex analysis on an advanced level. We will begin this essay with section2, covering the modular group, which is the integer subgroup of the special linear group. Most of the mathematics in the following will be done on this group, or on some subgroup. In section3 we present the fundamental domain; given a topological space and a group Γ0 acting on it, a fundamental domain is a subspace (of the topological space) containing exactly one point of each Γ0−equivalence class. Section4 covers modular functions which are functions invariant under the modular group and meromorphic on H [ f1g. The 5 modular functions are then generalized to modular forms in section5. These forms are, in the most basic case, modular functions holomorphic on H [ f1g. Following that, we derive a famous example of modular forms, called Eisenstein series in 5.1. In section6 we discuss the properties of the so called Hecke operators. These act as averaging operators over a certain collection of double cosets with respect to a group. They arose from Hecke’s theory on classifying the modular forms having multiplicative Fourier coefficients. In this chapter we also discuss the properties of Hecke operators on modular forms. Section7 covers the Dirichlet series, which are series of arithmetic functions. These series are central in the theory of analytic num- ber theory. For example, the famous Riemann zeta function is actually a Dirichlet series. The chapter begins with an interesting connection between modular forms and Dirichlet series. We show that there is a one-to-one correspondence between certain modular forms and Dirichlet series satisfying a specific functional equation. 2 Modular group The following chapter is based on [5] and [7]. The special linear group SL2(R) = a b : a; b; c; d 2 ; ad − bc = 1 acts on the complex upper half plane = c d R H az+b fz 2 C : Im(z) > 0g by linear fractional transformations as γz = cz+d for a b γ = 2 SL2( ): The action is indeed a group action. Firstly, since if c d R a b γ = 2 SL2( ) and z 2 , then c d R H az + b (az + b)(cz + d) γz = = cz + d (cz + d)(cz + d) (bcz + adz) + (aczz + bd) = (1) (cz + d)(cz + d) acjzj2 + bd + 2bc Re(z) + z = : jcz + dj2 Therefore the imaginary part of γz is ad − bc Im (z) Im (γz) = Im (z) = > 0: (2) jcz + dj2 jcz + dj2 Thus SL2(R) × H ! H. Secondly, for the identity element I we have that 1 · z Iz = = z: 1 6 Thirdly, the action also satisfies the compatibility condition, i.e. (aa~ + bc~)z + (a~b + bd~) (γγ~)z = = γ(~γ(z)) (~ac + dc~)z + (~bc + dd~) a b a~ ~b for all γ = ; γ~ = 2 SL2( ) and z 2 : The transformations de- c d c~ d~ R H scribed above are also called Möbius transformations, which one may recognize from complex analysis. A particular discrete subgroup of SL2(R) plays a fundamental role in various branches of mathematics and physics and is called the modular group. Definition 2.1. The modular group Γ(Swe: "den modulära gruppen") is the group a b of all matrices for a; b; c; d 2 such that ad − bc = 1. c d Z We also note that changing the sign of γ 2 Γ does not change the action. Since, a b if γ = 2 Γ, then we have c d −az − b az + b −γz = = = γz: (3) −cz − d cz + d 0 1 Theorem 2.2. The modular group Γ is generated by S = and T = −1 0 1 1 ; i.e every A 2 Γ can be expressed as A = T n1 ST n2 S : : : ST nk for n 2 0 1 j Z; j = 1; : : : ; k. Proof. Since S; T 2 Γ we have that hS; T i ⊆ Γ where hS; T i is the span of S and T . 1 x It remains to show that Γ ⊆ hS; T i: We have T x = 2 hS; T i for x 2 : 0 1 Z 1 x 1 0 So N := ; x 2 g ⊆ hS; T i: Since S−1T −yS = we also have 0 1 Z y 1 1 0 a b that N := ; y 2 ⊆ hS; T i: Now let be an arbitrary element of y 1 Z c d a b a b Γ. We want to show that 2 hS; T i as well. First note that S−1 S = c d c d d −c ; and so without loss of generality we may assume that jaj jdj : −b a 6 a b 1 x a ax + b Next observe that = . We may therefore assume c d 0 1 c cx + d that 0 6 b < jaj : 1 0 a b a b We also have that = : Hence we may assume y 1 c d ay + c by + d that 0 6 c < jaj : 7 a b Since 2 Γ we have that ad and bc are integers such that ad − bc = 1: c d 2 2 By the assumptions on the matrix elements we have that jadj > a and jbcj < a : a b 1 0 Therefore we must have ad = 1 and bc = 0; which means that = ± 2 c d 0 1 N ⊆ hS; T i: Thus an arbitrary element of Γ is in hS; T i; so Γ ⊆ hS; T i: Now we would like to study the induced maps of the generators of Γ.