Modular forms for triangle groups Modulära former för triangelgrupper Elisabet Edvardsson Faculty of Health, Science and Technology Mathematics, Bachelor Degree Project 15 ECTS Supervisor: Håkan Granath Examiner: Niclas Bernhoff January 2017 Abstract Modular forms are important in different areas of mathematics and theoretical physics. The theory is well known for the modular group PSL(2; Z), but is also of interest for other Fuchsian groups. In this thesis we will be interested in triangle groups with a cusp. We review some theory about mapping of hyperbolic triangles in order to derive an expression for the Hauptmodul of a triangle group, and use this to write a SageMath-program that calculates the Fourier series of the Hauptmodul. We then review some of the results presented in [4] that describe generalizations of well known concepts such as the Eisenstein series, the Serre derivative and some general results about the algebra of modular forms for triangle groups with a cusp. We correct some of the mistakes made in [4] and prove some further properties of the generators of the algebra of modular forms in the case of Hecke groups. Then we use the results from [4] to write a SageMath-program that calculates the Fourier series of the generators of the algebra of modular forms for triangle groups with a cusp and that also finds the relations between the generators in the special case of Hecke groups. Using the results from this program, we present some conjectures concerning the generators of the algebra of modular forms for a Hecke group, which, if proven to be true, give us a generalization of some of the Ramanujan equations. We conclude by explicitly calculating the generalized Ramanujan equations for the first few Hecke groups. Sammanfattning Modul¨araformer ¨arviktiga inom olika delar av matematik och teoretisk fysik. Teorin ¨ar v¨alk¨andf¨orden modul¨aragruppen PSL(2; Z), men ¨arocks˚aintressant f¨orandra Fuchsgrupper. Den h¨aruppsatsen inriktar sig p˚atriangelgrupper med en spets. Vi ˚atergerhur avbildningar av hyperboliska trianglar kan beskrivas i syfte att hitta ett uttryck f¨oren Hauptmodul f¨or en triangelgrupp, och anv¨anderdetta f¨oratt skriva ett program i SageMath som ber¨aknar Hauptmodulens Fourierserie. N˚agraav resultaten fr˚an[4], som beskriver generaliseringar av v¨alk¨andabegrepp som Eisensteinserierna och Serrederivatan, samt n˚agraallm¨anna resultat f¨or algebran av modul¨araformer, ˚aterges.Vi r¨attarn˚agraav felen som gjorts i [4] och visar att generatorerna till algebran av modul¨araformer har vissa egenskaper i fallet med Heckegrupper. Resultaten fr˚an[4] anv¨andsf¨oratt skriva ett program i SageMath som ber¨aknarFourierserierna f¨orgeneratorerna till algebran av modul¨araformer f¨oren triangelgrupp med en spets samt hittar relationerna mellan generatorerna i fallet med Heckegrupper. Utifr˚anresultaten som f˚asfr˚an programmet, kommer vi fram till och presenterar n˚agraf¨ormodanden ang˚aendegeneratorerna till algebran av modul¨ara former f¨oren Heckegrupp. Om dessa f¨ormodanden visar sig vara sanna ger de oss en generalisering av n˚agraav Ramanujanekvationerna. Vi avslutar genom att explicit ber¨aknade generaliserade Ramanujanekvationerna f¨orde f¨orstaHeckegrupperna. 1 Contents 1 Introduction 3 2 Introduction to modular forms 4 2.1 The upper half plane and M¨obiustransformations . .4 2.2 Modular functions and modular forms . .5 2.3 The modular group . .7 3 Mapping of hyperbolic triangles 11 3.1 Definition of triangle groups . 11 3.2 Hauptmoduls for triangle groups with a cusp . 14 4 Algebras of modular forms 26 4.1 Generators for the algebra of modular forms . 28 4.2 The Hilbert series . 29 4.3 Implementation in SageMath . 31 4.4 Results . 34 5 Differential equations for modular forms 35 5.1 Generalization of the Serre derivative . 35 5.2 Generalization of the Ramanujan equations . 37 6 Discussion 41 A SageMath-code 42 A.1 Fourier series for Jt ..................................... 42 A.2 Generator relations for Hecke groups . 42 A.3 Differential equations for modular forms . 45 2 1 Introduction The theory of modular forms for the modular group PSL(2; Z) is well known and used in both mathematics and physics. Somewhat loosely a modular form is a complex function on the upper half plane that transforms in a certain way under M¨obiustransformations. More precisely, a modular form f(τ) of weight k of some group Γ ⊂ SL(2; R) satisfies aτ + b a b f = (cτ + d)kf(τ); for 2 Γ: (1.1) cτ + d c d If k = 0, we call this a modular function. The main importance of modular forms is in number theory. For example, they were integral in the proof of Fermat's last theorem by Andrew Wiles and Richard Taylor who proved the Taniyama-Shimura conjecture [5, p. 130]. Also, modular forms have Fourier expansions with coefficients that have interesting arithmetic properties, and thus can be used to prove relations that are otherwise very difficult to prove or even realize exist. The applications are not limited to number theory. There is a surprising connection, called Monstrous moonshine [5], between modular forms and the representation theory of the monster group, which is the largest of the sporadic simple groups. This connection is related to a certain vertex operator algebra, which is the structure underlying conformal field theory in physics. Further applications in physics are mostly related to string theory. In this thesis we will be interested in the theory of modular forms for discrete subgroups of PSL(2; R), also called Fuchsian groups. More specifically we will be interested in modular forms for hyperbolic triangle groups, which are groups that describe how the complex upper half plane is tessellated by hyperbolic triangles. We review some of the work done in [4], an article that describes the theory of modular forms in the case of triangle groups. We will correct several of the errors made in this article, simplify some of the results, and use their results in order to generalize some of the well known theory of modular forms for the modular group PSL(2; Z). In addition, we will write a program in SageMath [12] that calculates Fourier series for the Hauptmodul and the generators for the algebra of modular forms, finds the relations between these generators, and finds differential equations corresponding to the Ramanujan equations for Hecke groups. The outline of the thesis is as follows. In Section 2, we will review some of the standard theory of modular forms and in particular describe some important results for the modular forms of the modular group, PSL(2; Z). This section is meant to provide a context for the rest of the material in the thesis, and thus explanations will be brief, and no proofs will be given. In Section 3 we go on to introduce triangle groups and show how a Hauptmodul, i.e. a function that generates all modular functions, can be calculated for such groups by studying mapping properties of hyperbolic triangles. We also correct some mistakes made in [4] concerning the fundamental domain of a triangle group. We then show how we can calculate the Fourier series for such Hauptmoduls using SageMath. In Section 4 a closer study of the algebra of modular forms for a triangle group is made. We review some of the results presented in [4] and use these results to write a program in SageMath that calculates the relations between the generators of the algebra of modular forms and thus determines the structure of this algebra. We also prove some further results and formulate some conjectures concerning the generators in the case of Hecke groups. In Section 5 we will describe 3 and correct the definitions and results from [4] concerning the generalization of the Eisenstein series and the Serre derivative. We will then use these results together with the results and conjectures about the generators of the algebra of modular forms from Section 4 to suggest a way to generalize some of the Ramanujan equations in the case of Hecke groups. We also write a SageMath program that finds these differential equations. 2 Introduction to modular forms In this section a review of the basic theory of modular forms will be presented. The contents are meant as a background for the theory that is presented in later sections, and thus explanations will be kept brief and most proofs will be left out. The interested reader is referred to [1] and [8] for details. 2.1 The upper half plane and M¨obiustransformations We denote by H the complex upper half plane H = fz 2 C j Im(z) > 0g. The special linear group, SL(2; R) acts on H by M¨obius transformations, namely, for γ 2 SL(2; R) a b az + b γ = : ! ; z 7! γ(z) = : (2.1) c d H H cz + d That this is a group action can be seen by the following considerations: For x 2 H and γ 2 SL(2; R), we have that az + b [a(i Imz + Rez) + b][c(i Imz − Rez) + d]) Im(γz) = Im = Im cz + d jcz + dj2 (2.2) Im(z) = ; jcz + dj2 since (ad − bc) = 1, and thus γz is in H if z is. Furthermore, if we denote the 2 × 2 identity matrix by I2×2, we have that z + 0 I z = = z: (2.3) 2×2 0 + 1 a b a0 b0 Also, if γ = , and γ0 = , we have that c d c0 d0 a0z + b0 a + b a b a0z + b0 0 0 (aa0 + bc0)z + ab0 + bd0 γ(γ0z) = = c z + d = c d c0z + d0 a0z + b0 (ca0 + dc0)z + cb0 + dd0 c + d (2.4) c0z + d0 aa0 + bc0 ab0 + bd0 = z = (γγ0)z: ca0 + dc0 cb0 + dd0 4 Thus the action of SL(2; R) satisfies the properties of a group action.