modular forms and converse theorems for dirichlet series278038/fulltext01.pdf · 2009-11-23 ·...

Examensarbete

Modular forms and converse theorems for Dirichlet series

Jonas Karlsson

LiTH - MAT - EX - - 2009 / 05 - - SE


Applied Mathematics, Linkopings Universitet

Jonas Karlsson

LiTH - MAT - EX - - 2009 / 05 - - SE

Examensarbete: 45 hp

Level: D

Supervisor: M. Izquierdo,Applied Mathematics, Linkopings Universitet

Examiner: M. Izquierdo,Applied Mathematics, Linkopings Universitet

Linkoping: june 2009

Matematiska Institutionen581 83 LINKOPINGSWEDEN

june 2009

x x

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-19446

LiTH - MAT - EX - - 2009 / 05 - - SE


Jonas Karlsson

This thesis makes a survey of converse theorems for Dirichlet series. A converse theo-rem gives sufficient conditions for a Dirichlet series to be the Dirichlet series attachedto a modular form. Such Dirichlet series have special properties, such as a functionalequation and an Euler product. Sometimes these properties characterize the modularform completely, i.e. they are sufficient to prove the proper transformation behaviourunder some discrete group. The problem dates back to Hecke and Weil, and has morerecently been treated by Conrey et.al. The articles surveyed are:

• “An extension of Hecke’s converse theorem”, by B. Conrey and D. Farmer

• “Converse theorems assuming a partial Euler product”, by D. Farmer and K.Wilson

• “A converse theorem for Γ0(13)”, by B. Conrey, D. Farmer, B. Odgers and N.Snaith

The results and the proofs are described. The second article is found to contain anerror. Finally an alternative proof strategy is proposed.

Modular forms, Dirichlet series, converse theorems, Hecke groups, Euler products,elliptic curves

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Abstract

This thesis makes a survey of converse theorems for Dirichlet series. A conversetheorem gives sufficient conditions for a Dirichlet series to be the Dirichlet seriesattached to a modular form. Such Dirichlet series have special properties, suchas a functional equation and an Euler product. Sometimes these propertiescharacterize the modular form completely, i.e. they are sufficient to prove theproper transformation behaviour under some discrete group. The problem datesback to Hecke and Weil, and has more recently been treated by Conrey et.al.The articles surveyed are:

• “An extension of Hecke’s converse theorem”, by B. Conrey and D. Farmer

• “Converse theorems assuming a partial Euler product”, by D. Farmer andK. Wilson

• “A converse theorem for Γ0(13)”, by B. Conrey, D. Farmer, B. Odgersand N. Snaith

The results and the proofs are described. The second article is found to containan error. Finally an alternative proof strategy is proposed.

Keywords: Modular forms, Dirichlet series, converse theorems, Hecke groups,Euler products, elliptic curves

Karlsson, 2009. vii

Acknowledgements

First and foremost, I want to thank my supervisor and examiner MilagrosIzquierdo Barrios, who has been a great source of inspiration and support. Herinvaluable help has by no means been limited to the writing of this thesis. Again,thank you!

I would also like to thank my opponent Johan Hemstrom, who has workedhis way through several rough versions of this thesis and caught several errors.This thesis is written in LATEX, and I will take this opportunity to express mygratitude to Donald Knuth for his creation. Several figures, and the calculationson elliptic curves, have been done with the free mathematics software Sage,and I am most grateful to William Stein and all other contributors in the freesoftware community - for this relief much thanks! I wish to thank my motherfor encouraging me to do whatever I want. Thanks to Erik Aas for extensivecomputer support, intelligent questions, and much else besides. Finally, but noless importantly, 2691 + 1 thanks to Karin Lundengard for distraction.

Karlsson, 2009. ix

Nomenclature

Symbols

N The natural numbers; 0 ∈ NR The real numbersC The complex numbersk∗ The nonzero elements of the field k

k An algebraic closure of kchar(k) The characteristic of kR+ The positive real numbersCP1 The complex projective lineQP1 The rational projective lineC The extended complex plane (line)R The extended real lineH The upper half-planeZp p-adic integersQp p-adic numbersZ/NZ The integers modulo N<z The real part of the complex number z=z The imaginary part of the complex number z⊆ Subset, not necessarily properA \B Set differenceS The interior of the set S[a, b] Closed interval]a, b[ Open interval[a, b[ Half-closed intervalS1 Circle (topological)(a : b) Projective coordinates(m,n) GCD of m and nAn The alternating group on n lettersCn The cyclic group of order nSn The symmetric group on n lettersGL The general linear groupSL The special linear groupPGL The projective general linear groupPSL The projective special linear groupSU The special unitary group

Karlsson, 2009. xi

xii

Aut(X) The automorphism group of the space XStab(X) The stabilizer subgroup of X under a group actiong.x The image of x under the action of g

X The universal covering space of Xtr The trace of an operatorf |kγ Slash operator, see definition on page 63I Identity matrix; identity Mobius transformationϕ Euler’s totient function

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Riemann surfaces 32.1 The Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Mobius transformations . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Transitivity properties . . . . . . . . . . . . . . . . . . . . 82.2.2 Stabilizer subgroups . . . . . . . . . . . . . . . . . . . . . 82.2.3 Classification of Mobius transformations . . . . . . . . . . 9

2.3 Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Groups of Mobius transformations . . . . . . . . . . . . . . . . . 12

2.4.1 Continous groups . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 Finite groups of Mobius transformations . . . . . . . . . . 132.4.3 Infinite discrete groups of Mobius transformations . . . . 13

2.5 Group actions, covering spaces and uniformization . . . . . . . . 152.5.1 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.2 Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . 162.5.3 Quotient surfaces . . . . . . . . . . . . . . . . . . . . . . . 182.5.4 The Uniformization theorem . . . . . . . . . . . . . . . . 19

3 Lattices and Elliptic Curves 233.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . 273.2 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 The most general form of the equation . . . . . . . . . . . 303.3.3 The group structure . . . . . . . . . . . . . . . . . . . . . 313.3.4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.5 Elliptic curves over different fields . . . . . . . . . . . . . 333.3.6 The L-series of an elliptic curve . . . . . . . . . . . . . . . 343.3.7 Moduli spaces of elliptic curves . . . . . . . . . . . . . . . 353.3.8 Complex multiplication . . . . . . . . . . . . . . . . . . . 353.3.9 Galois representations . . . . . . . . . . . . . . . . . . . . 363.3.10 Hyperelliptic curves and Abelian varieties . . . . . . . . . 36

Karlsson, 2009. xiii

xiv Contents

4 Dirichlet series 394.1 The Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Euler products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Modular forms 475.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . 485.2.2 The modular invariant . . . . . . . . . . . . . . . . . . . . 49

5.3 Spaces of modular forms . . . . . . . . . . . . . . . . . . . . . . . 515.3.1 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 The L-series of a modular form . . . . . . . . . . . . . . . . . . . 565.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5.1 Additive number theory . . . . . . . . . . . . . . . . . . . 565.5.2 The modularity theorem . . . . . . . . . . . . . . . . . . . 57

5.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 New converse theorems 596.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . 596.2 Hecke’s original converse theorem . . . . . . . . . . . . . . . . . . 606.3 Weil’s converse theorem . . . . . . . . . . . . . . . . . . . . . . . 616.4 An extension of Hecke’s converse theorem . . . . . . . . . . . . . 62

6.4.1 Introduction and results . . . . . . . . . . . . . . . . . . . 626.4.2 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.5 Converse theorems assuming a partial Euler product . . . . . . . 666.5.1 Introduction and results . . . . . . . . . . . . . . . . . . . 666.5.2 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.5.3 An error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.6 A converse theorem for Γ0(13) . . . . . . . . . . . . . . . . . . . 706.6.1 Introduction and results . . . . . . . . . . . . . . . . . . . 706.6.2 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A Bernoulli numbers 79

List of Figures

2.1 A torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 A loxodromic transformation. . . . . . . . . . . . . . . . . . . . . 102.3 Hyperbolic geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 A hyperbolic triangle . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 An octahedral tesselation. . . . . . . . . . . . . . . . . . . . . . . 172.6 An icosahedral tesselation. . . . . . . . . . . . . . . . . . . . . . . 182.7 A fundamental domain for Γ. . . . . . . . . . . . . . . . . . . . . 192.8 A tesselation induced by Γ. . . . . . . . . . . . . . . . . . . . . . 202.9 A fundamental domain for Γ(2) . . . . . . . . . . . . . . . . . . . 202.10 Klein’s quartic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 A lattice and a sublattice of index 2 . . . . . . . . . . . . . . . . 233.2 The square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 The hexagonal lattice . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Some fundamental domains . . . . . . . . . . . . . . . . . . . . . 263.5 A Dirichlet fundamental domain and its tesselation . . . . . . . . 273.6 Addition on y2 = x3 − x . . . . . . . . . . . . . . . . . . . . . . . 323.7 The curve y2 = x3 − x over GF(101) . . . . . . . . . . . . . . . . 34

5.1 The contour of integration. . . . . . . . . . . . . . . . . . . . . . 52

Karlsson, 2009. xv

xvi List of Figures

Chapter 1

Introduction

1.1 Background

In 1859, Riemann made his famous investigation of the zeta function whichtoday bears his name. Among other things, he proved that it satisfies thefunctional equation

π−s/2Γ(s

2)ζ(s) = π−(1−s)/2Γ(

1− s

2)ζ(1− s), (1.1)

where Γ is the well-known gamma function from complex analysis. In 1921,Hamburger noted that (1.1), together with some growth assumptions, deter-mines ζ(s) up to a multiplicative constant [18]. Later, in 1936, Hecke [20]proved his converse theorem, which proves an analogous result for arbitraryfunctions satisfying a suitable functional equation. In modern vocabulary, heproved a converse theorem for the groups Γ and Γ0(N) for N = 2, 3 and 4,where Γ now denotes the so-called modular group of Mobius transformationswith integer coefficients and determinant equal to 1, and Γ0(N) is the subgroupconsisting of transformations

z 7→ az + b

cz + d(1.2)

where in addition N |c. A converse theorem asserts that some function definedby a Dirichlet series arises from a modular form, which is a function transform-ing in a certain way under a group of Mobius transformations. Which groupis determined by a functional equation satisfied by the Dirichlet series. Theresults obtained by Hecke were the best possible under the assumptions made(growth conditions, Dirichlet series, functional equation). In 1967, Weil stud-ied the problem and managed to generalize the results by imposing additionalassumptions [45]. More precisely, he assumed a functional equation not onlyfor the Dirichlet series itself, but also for “twisted” Dirichlet series, which areconstructed using multiplicative (Dirichlet) characters. In 1995, Conrey andFarmer made another generalization [5], this time by assuming that the Dirich-let series has an Euler product of a certain form. This allowed them to proveconverse theorems for the groups Γ0(N) for 5 ≤ N ≤ 12, 14 ≤ N ≤ 17, andN = 23. In 2005, this was further explored by Farmer and Wilson [15], whoonly assumed a partial Euler product. Finally, in 2006, Conrey, Farmer, Odgers,

Karlsson, 2009. 1

2 Chapter 1. Introduction

and Snaith [6] proved a converse theorem for Γ0(13), which was lacking from[5]. Again, the proof used the assumption of a partial Euler product.

This thesis surveys the theorems described, with emphasis on the last threearticles. This is done in the last chapter. The first four chapters contain thenecessary background material.

1.2 Chapter outline

The thesis is organized as follows:

Chapter 2: Riemann surfaces This chapter introduces Riemann surfaces andtheir automorphisms. For the so-called Riemann sphere, these are theMobius transformations. Mobius transformations are isometries in a modelof hyperbolic geometry. We classify Mobius transformations and studygroups acting on surfacs, quotient surfaces, and uniformization. The mainreferences for this chapter are [14], [23] and [27].

Chapter 3: Lattices and Elliptic Curves This chapter introduces discretegroups acting on the complex plane, so-called lattices, and the correspond-ing compact Riemann surfaces. These are called elliptic curves. They arenot only geometric objects but have a group structure as well. We intro-duce the concept of moduli spaces of curves. The main references are [1],[13], [22], [24], [28], [37], [38], and [39].

Chapter 4: Dirichlet Series This chapter introduces Dirichlet series, the Mellintransform, and Euler products. The modern setting for these classical ob-jects involves the notion of an adele, which is briefly described in the finalsection. The main references are [24], [36], [42] and [43].

Chapter 5: Modular Forms This chapter introduces modular forms, whichare certain functions defined on the moduli space of elliptic curves. By thediscussion in chapter 3, this is a set of lattices, parametrized by the upperhalf-plane. We introduce spaces of modular forms, determine their dimen-sion, and describe the Hecke operators which act on these spaces. Finally,we mention some applications of modular forms. The main references are[1], [13], [36], [39].

Chapter 6: New Converse Theorems The final chapter surveys conversetheorems for Dirichlet series. These theorems give sufficient conditionsfor Dirichlet series to arise from modular forms. This line of research wasinitiated by Hecke in 1936 [20] and continued by Weil in 1967 [45]. In1995, Conrey and Farmer published the article “An extension of Hecke’sconverse theorem” [5], which uses a different approach than Weil’s. In2005 and 2006, the articles “Converse theorems assuming a partial Eulerproduct” [15] by Farmer and Wilson, and “A converse theorem for Γ0(13)”[6] by Conrey, Farmer, Odgers and Snaith further developed thes ideas.The results by Hecke and Weil are mentioned, and the articles by Farmeret.al. are described in some detail.

Chapter 2

Riemann surfaces

Riemann surfaces are of central importance in complex analysis and geometry.Intuitively, they are surfaces which locally look like the complex plane, allowingone to define analytic functions. For the general theory of Riemann surfaces,see [14]. We begin with some definitions.

Definition A surface S is a (second countable) Hausdorff topological space,every point of which has a neighbourhood homeomorphic to an open discin the complex plane C. A pair (φi, Ui) of an open set Ui ⊆ S and ahomeomorphism φi : Ui → C is called a chart. A collection of charts, theopen sets of which cover S, is called an atlas. If for any two charts (φi, Ui)and (φj , Uj), the so-called transition function

φi|Ui∩Uj φj |−1Ui∩Uj

: φj(Ui ∩ Uj) → φi(Ui ∩ Uj)

is analytic, the atlas is called analytic. Two analytic atlases are calledcompatible if their union is again an analytic atlas. Clearly this is anequivalence relation. An equivalence class of analytic atlases is called acomplex structure. A Riemann surface, finally, is a surface equipped witha complex structure.

Note One often requires a Riemann surface to be connected as well. This willbe assumed in the following.

The word “surface” indicates two-dimensionality, and indeed Riemann surfaceshave dimension two over R. It is, however, equally possible to regard them asone-dimensional complex manifolds, that is, as curves. In fact, this point ofview is implicit in the definition above, where a surface was defined by a localidentification with the complex “plane”. A more natural definition from thereal perspective would be to define a Riemann surface as a two-dimensional realmanifold with a conformal structure, that is, a notion of angles between curves.One can show that these definitions are equivalent.

The natural notion of equality between Riemann surfaces is biholomorphic(also called conformal) quivalence: a map f : S1 → S2 between Riemann sur-faces is called holomorphic if the corresponding map f∗ from C to itself is:

Karlsson, 2009. 3

4 Chapter 2. Riemann surfaces

S1f //

φ1

²²

S2

φ2

²²C

f∗ // CWhere f∗ = φ2 f φ−1

1 . If in addition f is bijective, S1 and S2 are calledbiholomorphically equivalent; it turns out that the inverse is automatically holo-morphic as well, so this is an equvalence relation. Next, we give some examplesof Riemann surfaces.

Example 1 The simplest example of a Riemann surface is C itself, which lookslike C not only locally but globally as well. It is non-compact and simplyconnected (has trivial homotopy group).

Example 2 Any open, simply connected, proper subset of C is a Riemann sur-face. By the Riemann mapping theorem, any such set is biholomorphicallyequivalent to the open unit disc z ∈ C : |z| < 1.

Example 3 A punctured open disc, such as z ∈ C : 0 < |z| < 1, is aRiemann surface. This set is not simply connected; instead, the first ho-motopy group is isomorphic to the infinite cyclic group. Thus, it cannot behomeomorphic to C or the unit disc. A fortiori, it is not biholomorphicallyequivalent to either of them.

Example 4 The quotient of C by a discrete group of translations inherits acomplex structure. Thus, for example, the group generated by the trans-lation z 7→ z + 1 gives rise to en equivalence relation

z1 ∼ z2 ⇔ (z1 − z2) ∈ Z, (2.1)

and the quotient C/ ∼ is a Riemann surface (topologically a cylinder,S1 × R).

Example 5 Taking instead the group generated by two translations, z 7→ z +1and z 7→ z + c for some non-real c gives a compact quotient: indeed, anypoint in C is equivalent to one in the parallelogram s + tc : 0 ≤ s, t < 1,and opposite sides are identified, giving the quotient the topology of a torus(S1 × S1). To emphasize the complex structure, one calls it a complextorus. A torus is shown in figure 2.1.

Example 6 Affine curves are sets of the form

C = (z, w) ∈ C2 : p(z, w) = 0 (2.2)

for some polynomial p. Locally, such curves look like C in the followingsense: if (z0, w0) is a point on the curve and if the partial derivatives

∂p

∂z|(z0,w0) and

∂p

∂w|(z0,w0) (2.3)

are not both zero, then by the implicit function theorem one of the vari-ables is a holomorphic function of the other in some neighbourhood of(z0, w0). When this holds at every point of the curve, it is called smoothor nonsingular. Projection onto one of the variables gives a chart, and thecurve is a Riemann surface.

2.1. The Riemann sphere 5

Figure 2.1: A torus [46]

Example 7 Examples number 4 and 5 constructed Riemann surfaces from Cby forming the quotient with a discrete group. It is also possible to startin the two-dimensional complex manifold C2 and quotient by a continousgroup, reducing the complex dimension to 1. The complex projective lineis the set C2 \ (0, 0) of pairs of complex numbers, not both zero, modulothe equivalence relation

(a, b) ∼ (c, d) if (a, b) = (zc, zd) (2.4)

for some non-zero z ∈ C. The equivalence class of (a, b) is written (a : b).They can be thought of as one-dimensional subspaces of C2, namely theline through (a, b) and the origin (0, 0). If b 6= 0, one can represent the class(a : b) = (a/b : 1) by the single complex number a/b. Thus the projectiveline contains a copy of C. In addition, it contains the point (1 : 0) (notethat if b = 0, a is nonzero and can be normalized to 1). Now considera sequence of points (an : bn), normalized so that |an|2 + |bn|2 = 1for every n, and such that bn tends to 0 as n tends to infinity. Then|an/bn| tends to infinity, suggesting that (1 : 0) plays the role of a “pointat infinity”. This intuition can be made precise by means of the Riemannsphere.

Finally, we remark that for a compact Riemann surface S, the field of mero-morphic functions defined on S is a finite extention of the field C(z) of rationalfunctions in a complex variable, so that any two meromorphic functions arealgebraically dependent.

2.1 The Riemann sphere

We now construct the Riemann sphere, which is the Riemann surface hintedat in the last example above. For more on this, see [23]. Identify the complexnumber z = x+ iy with the point (x, y, 0) in R3, and consider the unit 2-sphere:

S2 = (x, y, w) ∈ R3 : x2 + y2 + w2 = 1. (2.5)

By stereographic projection from the “north pole” (0, 0, 1), the sphere (minusthe point of projection) can be identified with the complex plane C. Explicitly,


one has a projection

π1 : S2 \ (0, 0, 1) → C

π1(x, y, w) =x + iy

1− w(2.6)

with an inverse given by

π−11 (z) = (

2<z

|z|2 + 1,

2=z

|z|2 + 1,|z|2 − 1|z|2 + 1

). (2.7)

It is clear that there is nothing special with the point (0, 0, 1): any choiceof projection point will do, and using instead (0, 0,−1), one obtains anotherprojection π2:

π2 : S2 \ (0, 0,−1) → C

π2(x, y, w) =x− iy

1 + w

with inverse given by

π−12 (z) = (

2<z

|z|2 + 1,− 2=z

|z|2 + 1,1− |z|2|z|2 + 1

). (2.8)

The projections are homeomorphisms, the two charts (π1,2, S2 \(0, 0,±1))

cover the entire sphere, and the transition function

π2 π−11 : C∗ → C∗

π2 π−11 (z) =

1z

(2.9)

is holomorphic. Thus the sphere is a Riemann surface, called the Riemannsphere. The point (0, 0, 1) is the “point at infinity” referred to in the previoussection. It is denoted ∞. Under the identification (a : b) ↔ a/b, it corre-sponds to the point (1 : 0). When thought of in this way, the Riemann sphereis the aforementioned complex projective line, often denoted CP1. An equiva-lent description is that of the extended complex plane, obtained by adjoining asingle point ∞ to C, declaring every set of the form z ∈ C : |z| > M to bea neighbourhood of ∞. This embeds C in a compact space, and one speaksaccordingly of a one-point compactification. The space obtained is denoted C.We will favour this notation. The Riemann sphere is the only compact Riemannsurface of genus zero, i.e. a two-sphere admits only one complex structure. Thenext section introduces and classifies the self-mappings of the Riemann sphere.

2.2 Mobius transformations

Given any Riemann surface S, the set of biholomophic mappings from S toitself, so-called automorphisms, forms a group under composition, called theautomorphism group and denoted Aut(S). We now determine Aut(C). First,note that the concepts of holomorphic and meromorphic functions make senseon C. A function f : C → C is said to have a pole (zero) of order n at ∞

2.2. Mobius transformations 7

if f( 1z ) has a pole (zero) of order n at z = 0. Liouville’s theorem (see for

example [23], appendix 1) implies that two meromorphic functions with polesand zeros of the same order at the same places are proportional, because theirquotient is a holomorphic function bounded in all of C (hence a constant). Itfollows that the meromorphic functions from the Riemann sphere to itself areprecisely the rational functions. Bijectivity implies that neither nominator nordenominator can have degree greater than one. Thus the automorphisms of theRiemann sphere are precisely the Mobius transformations, which are functionsof the form

z 7→ az + b

cz + d, where ad− bc 6= 0. (2.10)

The condition on the coefficients assures that the mapping is non-constant.In projective coordinates, the action looks as follows:

(z : w) 7→ (az + bw : cz + dw), (2.11)

so

∞ = (1 : 0) 7→ (a : c) =a

cif c 6= 0,

∞ 7→ ∞ otherwise. (2.12)

This is clearly consistent with what one obtains from limiting procedures, as is

−d

c= (d : −c) 7→ (ad− bc : 0) = ∞. (2.13)

It also shows that Mobius transformations can be conveniently represented by2× 2-matrices, i.e. one has a full (surjective) representation

ρ : GL(2,C) → Aut(C) (2.14)

sending a nonsingular matrix(

a bc d

)to the Mobius tranformation (2.10).

Here, GL(2,C) denotes the general linear group of nonsingular 2 × 2-matriceswith entries in C, under multiplication. Clearly this is a much larger groupthan Aut(C). Indeed, any scalar multiple of a matrix gives the same Mobiustransformation:

awz + bw

cwz + dw=

az + b

cz + d, w 6= 0, (2.15)

so by division by√

ad− bc, one may assume that the matrix has determinantequal to 1. The determinant is a homomorphism from matrices to complexnumbers:

det : GL(2,C) → (C∗, ·), (2.16)

and the group thus obtained is the kernel of this homomorphism. It followsthat it is a normal subgroup. It is called the special linear group and is denotedSL(2,C). The restriction of ρ to SL(2,C) is still not faithful (injective), because

az + b

cz + d≡ z

givesaz + b ≡ cz2 + dz ⇒ b = c = 0, a = d,


so ker ρ = ±I, where I is the 2×2-identity matrix. The quotient SL(2,C)/±Iis called the projective special linear group and is denoted PSL(2,C). It is iso-morphic to the group Aut(C) of Mobius transformations. One can also projec-tivize GL(2,C) directly, forming the projective general linear group PGL(2,C) =GL(2,C)/zI : z ∈ (C∗). It is readily seen to be isomorphic to PSL(2,C). Fi-nally, note that a Mobius transformation can fix at most two different points,so that two Mobius transformations which agree in three points are identicallyequal.

2.2.1 Transitivity properties

Mobius transformations act 3-transitively on C, that is, given any two setsz1, z2, z3 and w1, w2, w3 of nonequal points in C, there is a Mobius trans-formation T such that T (zi) = wi, i = 1, 2, 3. To see this, note that

T1(z) =z − z1

z − z3/z2 − z1

z2 − z3(2.17)

takes z1, z2, z3 to 0, 1,∞. Let T2 be the transformation taking w1, w2, w3to 0, 1,∞. Then T−1

2 T1 is the required Mobius transformation. The expres-sion

z − z1

z − z3/z2 − z1

z2 − z3(2.18)

is called the cross-ratio of z, z1, z2, z3 and is denoted [z, z1, z2, z3]. Cross-ratiosare preserved by Mobius transformations, i.e., for any transformation T one has[T (z), T (z1), T (z2), T (z3)] = [z, z1, z2, z3]. One can show that the four pointsz, z1, z2, z3 lie on a circle if and only if their cross-ratio is a real number (wherecircle is taken to mean circle on the Riemann sphere; these project to ordinarycircles or lines in C, depending on whether or not the circle passes through ∞).It follows that Mobius transformations take circles to circles, and moreover, theaction is transitive, since a circle is determined by three points.

Example To determine a Mobius transformation mapping the real line to theunit circle in C, one can take T1 to be the identity transformation (as 0,1, and ∞ already belong to R), and T2 to be the transformation taking−1,−i, 1 to 0, 1,∞, i.e.

T2(z) =z + 1z − 1

/−i + 1−i− 1

= −iz + 1z − 1

,

and the inverse is given by

T−12 (z) =

z − i

z + i.

One can show that this map sends the upper half-plane to the interior ofthe unit circle, for example by calculating that i 7→ 0, or by noting thatthe mapping is orientation-preserving.

2.2.2 Stabilizer subgroups

Next, we determine some important stabilizer subgroups. For a subset S in C,this is defined as the subgroup of Aut(C) mapping S into S (but not necessarily

2.2. Mobius transformations 9

fixing it pointwise). We denote it Stab(S). By the discussion in the previoussection, any circle (i.e., circle or line in C) can be mapped onto the extended realline R = R∪∞. Thus the stabilizer subgroup of any circle is conjugate to thestabilizer of R, which we now determine. Suppose that T ∈ Stab(R) and takethree different points z1,2,3, none of which maps to ∞. Let T (zi) = wi. Thenall zi, wi are real numbers and the construction in the previous section showsthat T is equal to a Mobius transformation with real coefficients. Conversely, itis easy to see that any such transformation fixes R. Thus Stab(R) is isomorphicto the group PGL(2,R) modulo the group zI : z ∈ R∗. This is not quitethe special linear group, since the determinant equals ±1. Rather, Stab(R) ∼=PSL(2,R) × ±I. The transformations with determinant −1 are orientation-reversing, i.e., they interchange the interior and exterior of a circle. It followsthat the upper half-plane z : =z > 0, denoted H, has stabilizer PSL(2,R),and that the stabilizer of any disc (interior of a circle) is conjugate to this group.Finally, Stab(C) is equal to the group of affine transformations z 7→ az + b.

2.2.3 Classification of Mobius transformations

We now classify all Mobius transformations according to their conjugacy classesin PSL(2,C). First, note that all elements of a conjugacy class must fix thesame number of points. A nonconstant transformation

T : z 7→ az + b

cz + d

fixes ∞ if and only if c = 0. Solving

az + b

cz + d= z

gives (2cz + (a− d))2 = (a− d)2 + 4bc = (a + d)2 − 4, (using ad− bc = 1) so Thas one fixed point if and only if (a+ d)2 = 4, and two otherwise. The quantitya+d is the trace of the matrix representing the transformation; it is determinedup to a sign, so the squared trace tr2(T ) is a well-defined function of the Mobiustransformation. Moreover, it depends only on the conjugacy class:

tr2(UTU−1) = tr2(U−1UT ) = tr2(T ). (2.19)

Now suppose T has only one fixed point. By conjugation, it can be moved to∞, and any Mobius transformation fixing only ∞ has the form z 7→ az + b.By another conjugation, the representative can be taken to be the translationz 7→ z +1. If T has two fixed points, they can be taken to be 0 and ∞, givinga representative of the form z 7→ az, a 6= 0, 1. It is now straightforward to verifythat tr2(T1) = tr2(T2) is a sufficient condition for T1 and T2 to be conjugate,and we have already seen that it is necessary. Thus the squared trace completelyclassifies conjugacy classes of Mobius transformations. A Mobius transformationT is called

Elliptic if tr2(T ) is real and belongs to [0,4[; such a transformation has twofixed points and is conjugate to a rotation, z 7→ eiθz,

Parabolic if tr2(T ) = 4; such a transformation has one fixed point and isconjugate to the translation z 7→ z + 1,


Hyperbolic if tr2(T ) is real and ≥ 4; such a transformation has two fixedpoints and is conjugate to a dilatation, z 7→ rz, r ∈ R,

Loxodromic otherwise. Such a transformation has two fixed points and isconjugate to a transformation z 7→ cz, c nonreal and of absolute value6= 1.

Sometimes hyperbolic transformations are counted as loxodromic as well. Thename loxodromic means “slanting path” and describes how the flow lines of aloxodromic transformation on the Riemann sphere (fixing 0 and ∞) cross themeridians at constant angle. For an elementary (and visually pleasing) accountof this classification, see [32]. Figure 2.2 shows the flow lines of a loxodromictransformation with two finite fixed points.

Figure 2.2: A loxodromic transformation [27]

2.3 Hyperbolic geometry

As noted by Poincare, the upper half-plane H, equipped with the PSL(2,R)-invariant metric

ds2 =dx2 + dy2

y2, (2.20)

is a model of a hyperbolic plane. The geodesics (straight lines) are lines per-pendicular to the real axis, and half-circles meeting the real axis orthogonally;see figure 2.3.

PSL(2,R) acs on this plane as a group of isometries. It follows from thediscussion in section 2.2.1 that its action on the set of geodesics is transitive.Furthermore, there exists a unique geodesic between any two points. Whendoing geometry in this model, it is convenient to adjoin the point at infinity andthe boundary =z = 0. Any lines intersecting at these points meet at an anglezero. Figure 2.4 depicts a hyperbolic triangle.

2.3. Hyperbolic geometry 11

Figure 2.3: Hyperbolic geodesics

Figure 2.4: A hyperbolic triangle

In this example, all angles of the triangle are zero; that the sum is less thanπ characterizes a space of negative curvature. The appropriate notion of areain this model is the Haar measure for SL(2,R):

dµ =dxdy

y2. (2.21)

Remarkably, the area of a triangle depends only on the sum of its angles. Moreprecisely, it is given by the Gauß-Bonnet formula: let ∆ be a triangle withangles α, β, γ. Then

µ(∆) = π − α− β − γ. (2.22)

For a proof, see [23], chapter 5. This immediately shows, for example, thatthe triangle in figure 2.4 has area equal to π. Finally, we remark that anothercommon model of two-dimensional hyperbolic geometry uses the unit disc inC instead of the upper half-plane. The Mobius transformation constructed asan example in section 2.2.1 can be used to translate between the models. Thisalternative model is called the Poincare disc model.


2.4 Groups of Mobius transformations

2.4.1 Continous groups

We now study some groups of Mobius transformations, and their generators.We have already treated PSL(2,C). It is generated by transformations of thefollowing forms:

• Translations: z 7→ z + c, c ∈ C

• Rotations: z 7→ eiθz, θ ∈ R

• Dilatations: z 7→ rz, r ∈ R

• The inversion z 7→ 1z .

This is easily seen by writing

az + b

cz + d=

a

dz +

b

d, c = 0

az + b

cz + d=

a

c− 1

c(cz + d), c 6= 0. (2.23)

Equation (2.23) is valid for real Mobius transformations as well, and in thiscase, on can dispose with the rotations and take the translations to be real.This gives generators for PSL(2,R).

A rotation of the Riemann sphere is clearly an automorphism. Thus it shouldbe possible to interpret SO(3,R), the special orthogonal group, which acts onR3 as rotations, as a group of Mobius transformations. It turns out to be easierto work not with SO(3,R) directly but with the group PSU(2,C), the projectivespecial unitary group. It is well known that the special unitary group SU(2,C)of unitary 2 × 2-matrices with determinant equal to one is a double cover ofSO(3,R)1; the quotient SU(2,C)/±I is isomorphic to SO(3,R). Noting that aMobius transformation is a rotation if and only if it commutes with the antipodalmapping z 7→ −1/z, one finds

bz − a

dz − c= −cz + d

az + b, (2.24)

whence (b −ad −c

)= λ

( −c −d

a b

)(2.25)

for some nonzero λ ∈ C. One must have λ = ±1 (take determinants), and usingad− bc = 1 one finds b = −c, d = a. Furthermore,

(b −a

a b

)(b a−a b

)=

( |b|2 + |a|2 00 |a|2 + |b|2

), (2.26)

and |a|2 + |b|2 = ad− bc = 1. This gives the expected identification.

1Cf. quantum mechanics, where SU(2,C) acts on spinors rather than vectors.

2.4. Groups of Mobius transformations 13

2.4.2 Finite groups of Mobius transformations

Any finite group of Mobius transformations consists only of rotations. We nowclassify these groups, i.e., finite rotation groups of spheres, following [9] and[23]. Let G be such a group, and let |G| = n. Any nonidentity element of G hasexactly two (antipodal) fixed points, which are the points of intersection withthe rotation axis. Such a point is called p-gonal if it belongs to a rotation oforder p (and no higher order). Thus a p-gonal point is fixed by p−1 nonidentitytransformations. Now, count the pairs (z, g) of nonidentity elements of G andtheir fixed points; there are clearly 2(n− 1) such pairs, and counting by pointsin the sphere, one finds

2(n− 1) =∑

z

(|Stab(z)| − 1), (2.27)

since the identity of G has to be excluded from every stabilizer subgroup. Theonly nonzero terms are those from fixed points of G, so the sum is finite. Onthe other hand, every point in an orbit containing a p-gonal point is p-gonal,and by the orbit/stabilizer theorem, the length of such an orbit is n/p. Writingp1, p2, . . . for the occuring p:s (not necessarily distinct), one finds

2(n− 1) =∑

z

(|Stab(z)| − 1) =∑

i

n

pi(pi − 1), (2.28)

whence

2(1− 1n

) =∑

i=1

(1− 1pi

), (2.29)

where ` is the number of orbits, and the terms in the right-hand side are ≤ 12 ,

so there can be at most three terms. In the case ` = 1, n = 1 also, so thegroup is trivial. In the case ` = 2, G is cyclic. Finally, it can be checked that` = 3 gives the solutions p1, p2, p3 = 2, 2, p for an integer p (in which casen = 2p and G is dihedral), p1, p2, p3 = 2, 3, 3, 2, 3, 4 or 2, 3, 5. Thecorresponding orders of G are 12, 24 and 60, respectively, and the groups areA4, S4 and A5, which are the symmetry groups of a tetrahedron, octahedron andicosahedron ([23], [27]). This completely classifies the finite groups of Mobiustransformations.

2.4.3 Infinite discrete groups of Mobius transformations

Finally, we introduce infinite discrete groups. For the background, see [27].[10] gives a general treatment of group presentations and generators. A discretesubgroup 2 of PSL(2,C) is called Kleinian; for example, the group PSL(2,Z+iZ)is Kleinian. A Kleinian group having an invariant disc (which by conjugationcan be taken to be H) is called Fuchsian. Thus a Fuchsian group is, up toconjugacy, a discrete subgroup of PSL(2,R). The most important example isPSL(2,Z), the Mobius transformations with integer coefficients and determinantequal to 1. This group is called the modular group and is traditionally denotedΓ or Γ(1); the second notation will be explained shortly, but first, we determine

2In the metric topology on PSL(2,C).


generators and a presentation for Γ. It is convenient to start with its doublecover SL(2,Z), and introduce the matrices

S =(

0 −11 0

), T =

(1 10 1

), (2.30)

U =(

1 01 1

)V =

(0 −11 1

). (2.31)

It is straightforward to verify that

T = S−1V = S3V, U = S−1V 2 = S3V 2, (2.32)

and henceV = T−1U, S = T−1UT−1, (2.33)

so T and U generate SL(2,Z) if and only if S and V do. To see that this is the

case, let M =(

a bc d

)∈ SL(2,Z). Clearly

TλM =(

1 λ0 1

)(a bc d

)=

(a + λc b + λd

c d

)(2.34)

and

UλM =(

1 0λ 1

) (a bc d

)=

(a b

c + λa d + λb

), (2.35)

so by application of the matrices T and U , M can be brought on one of theforms (

a′ b′

0 d′

)or

(0 b′

c′ d′

). (2.36)

In the first case, a′d′ = 1 so the matrix equals ±Tλ, and can be written Tλ orS2Tλ. In the second case, one has −b′c′ = 1 so the matrix equals STλ or S3Tλ

for some λ. Thus SL(2,Z) is generated by S and V . They satisfy the relations

S2 = V 3, S4 = I. (2.37)

Next, we show that there are no other relations. Suppose to the contrary thatsome word in S and V is equal to the identity. Clearly, S2 commutes witheverything, so we may assume that the relation has the form

S · S3V αS3V β · · ·Sω = I (2.38)

orS3V αS3V β · · ·Sω = I, (2.39)

where the V -exponents are equal to 1 or 2, and using T = S3V and U = S3V 2,this becomes

S ·W (T, U) · Sω = I (2.40)

orW (T, U) · Sω = I (2.41)

2.5. Group actions, covering spaces and uniformization 15

for some word W in T and U . To see that this is impossible, we let the matrices

act on some matrix A =(

a bc d

)satisfying c > 0 and d > 0, according to

M.A = MT AM. (2.42)

Note that this really is an action, i.e., that N.(M.A) = (NM).A. It is straight-forward to verify that tr(S.A) = tr(A), tr(T.A) = c + tr(A) and tr(U.A) =d + tr(A), so the only word in T and U which is equal to the identity is theempty word. The relation then simplifies to Sω = I for some ω, and clearly theonly such relation is already implied by (2.37). Thus (2.37) defines SL(2,Z), andsince the image of S in Γ has order two rather than four, the defining relationsfor the modular group are:

S2 = I, V 3 = I. (2.43)

Finally, we introduce so-called congruence subgroups (of Γ). For an integerN , reduction modulo N is a ring homomorphism from Z to Z/NZ, which extendsto a homomorphism from Γ to PSL(2,Z/NZ). The kernel is a normal subgroupin Γ, called the principal congruence subgroup of level N . It has finite index inΓ. This explains the notation Γ(1) for Γ. Thus Γ(N) consists of transformations

z 7→ az + b

cz + d, a, b, c, d integers, (2.44)

where a ≡ d ≡ ±1 (mod N) and b ≡ c ≡ 0 (mod N). A larger congruencesubgroup is the set of Mobius transformations of the form (2.44) such that N |c.This group is denoted Γ0(N). One sometimes defines an analogous group Γ0(N)of transformations satisfying N |b, and Γ0

0(N) where N divides both b and c, see[4]. Furthermore, one denotes by Γ1(N) the group of Mobius transformationswith a ≡ d ≡ ±1 (mod N) and c ≡ 0 (mod N). Summarizing, one has

Γ(N) ⊂ Γ1(N) ⊂ Γ0(N) ⊂ Γ. (2.45)

In general, a congruence subgroup of Γ is a subgroup containing Γ(N) for someN , and the smallest such N is called the level of the group. 3 See [24], chapter9.

Example The principal congruence subgroup of level 2, Γ(2), has index 6 in Γ;the quotient Γ/Γ(2) is isomorphic to PSL(2,Z/2Z), which can be shownto be isomorphic to S3, the symmetric group on three letters. This groupcan be identified as the group permuting the tree non-identity elements ofthe 2-torsion subgroup of an elliptic curve; see [28], chapter 4 for more onthis. The group Γ(2) is generated by the transformations S : z 7→ −1/zand T 2 : z 7→ z + 2.

2.5 Group actions, covering spaces and uniformiza-tion

2.5.1 Group actions

Group actions are ubiquitous in algebra and geometry and are described in anytextbook on these subjects. In general, an action of a group G on a space X is

3It is a nontrivial fact that not every subgroup of finite index in Γ is a congruence subgroup.


a pair (G,X) together with a homomorphism G → Aut(X). For example, Xcould be a set, and Aut(X) permutations of its elements. As another example,X could be a topological space, in which case the elements of Aut(X) are home-omorphisms from X to itself. This will be assumed in the following. One oftenidentifies the elements of G with their images in Aut(X), and writes g.x for theaction of g ∈ G on x ∈ X. The set g.x : g ∈ G is called the orbit of x (underG). G is said to act freely on X if every point of X has a neigbourhood U suchthat no two of its translates g.U overlap. If at most finitely many translatesoverlap in each orbit, the action is said to be properly discontinous. This isclearly a (strictly) weaker assumption.

The orbits of G partition X, because “belonging to the same orbit” is anequivalence relation, as is readily verified. The set of orbits is denoted 4 X/G,and it inherits a topology from X, as follows: let π denote the canonical pro-jection X → X/G, which takes a point in X to its orbit under G. Define a setS ⊆ X/G to be open if and only if π−1(S) is open in X. The topology thusinduced is called the quotient topology. It is the coarsest topology in which πis continous.

2.5.2 Covering spaces

Again, let X be a topological space. A covering space of X is a topologicalspace Y together with a map π : Y → X such that

1. Every point x ∈ X has a neighbourhood V such that π−1(V ) = ∪αUα isthe union of pairwise disjonts sets Uα in Y ; and

2. the restriction of π to any such neighbourhood is a homeomorphism.

The preimage π−1(x) of a point is called a fiber. Clearly, if X is connected,every fiber has the same cardinality. The significance of free group actionsis the following: the canonical projection is a covering if and only if the groupaction is free. We will also need the concept of a fundamental domain for a groupaction. It is essentially a choice of representative 5 in X for each orbit under G;in practice, one requires it to be connected. Sometimes the closure of such a setis called a fundamental domain. We will not follow this convention. In the caseof discrete groups acting on H, the transformations are isometries (trivially,because of the choice of measure and line element), and the translates of afundamental domain tesselate the space. This clearly works for other discretegroups as well, for example the finite groups classified in section 2.4. Figures2.5 and 2.6 respectively, show tesselations of C induced by the groups S4 andA5.

One can show (see, for example, [36], [39]) that a fundamental domain forthe action of Γ(1) on H is given by the set

z : −1/2 ≤ z ≤ 0, |z| ≥ 1 ∪ z : 0 < z < 1/2, |z| > 1, (2.46)

see figure 2.7. The translation τ 7→ τ + 1 identifies the left and right boundarylines, and the inversion τ 7→ −1/τ identifies the arcs on the unit circle from

4Actually, what we have defined is called a left action and the quotient ought to be writtenG\X; however, we will not deal with double cosets, so no confusion should occur.

5i.e., a global section of the fibration π : X → X/G


Figure 2.5: An octahedral tesselation [27]

(−1 + i√

3)/2 to i and from (1 + i√

3)/2 to i, respectively. Figure 2.8 shows thetesselation induced by this choice of fundamental domain, that is, all translatesof it by elements of Γ. The action of Γ is not free: the mapping τ 7→ −1/τ fixesi and τ 7→ −1/(τ + 1) fixes (−1 ± i

√3)/2, and these points are identified by

the translation τ 7→ τ + 1. The corresponding stabilizer groups are isomorphicto C2 and C3, respectively (the cyclic groups of order 2 and 3). One can showthat Γ posesses no other fixed points in the fundamental domain (2.46), see forexample [36], chapter 7. These points are actually an obstacle when constructingthe quotient. Intuitively, the fundamental domain only sees one half point at iand one sixth of a point at each of (−1 ± i

√3)/2, for a total of one third of a

point. We will gloss over this subtelty in the following.Continuing the final example in section 2.4.3, figure 2.9 shows a fundamental

domain for Γ(2), which is given by the union of the fundamental domain (2.46)for Γ(1) and its image under the translation T : z 7→ z + 1.

There is a useful analogy between covering spaces and Galois theory. Todescribe it, we need the concept of a deck transformation. Suppose that Ycovers X by the projection π, and consider (invertible) functions f : Y → Ysuch that π f = π; that is, functions acting on Y in such a way that theimage in X is unchanged. Such a function f is called a deck transformation.Deck transformations evidently form a group under composition. This shouldbe compared to the Galois group Gal(k′/k) of a field extension k′ ≥ k, consistingof morphisms k′ → k′ which fix k. The algebraic closure of k, denoted by k,in which every every polynomial in k[x] splits completely, corresponds to theso-called universal covering space of X. We denote it here by X. Is is simplyconnected, so the homotopy group π1(X, x0) is trivial (here x0 ∈ X is somebase point; this choice is immaterial, as different choices give the same groupup to conjugacy for connected spaces; for more general spaces, on the other


Figure 2.6: An icosahedral tesselation [27]

hand, one cannot define homotopy groups at all, but only groupoids). It is alsounique up to isomorphism, due essentially to its definition as initial object in thecategory of spaces covering X. For details, see any book on algebraic topology,such as [30]. There is a bijective correspondence between covering spaces of Xand conjugacy classes of subgroups of π1(X, x0), and the cardinality of a fiberof such a covering equals the index of the corresponding homotopy group inπ1(X,x0).

2.5.3 Quotient surfaces

Continuing the discussion in section 2.5, the quotient of a Riemann surface bya freely acting, discrete group is again a Riemann surface. Is is compact if andonly if the group contains no parabolic elements. Even if the group does not actfreely, the quotient is almost a Riemann surface. The more general notion oforbifold can handle the milder types of singulariries which occur, for example,with the modular group. See [40] for more on orbifolds (and much else).

The quotient of H by Γ(N) is denoted Y (N); it is non-compact, becauseΓ(N) contains the parabolic element

τ 7→ τ + N, (2.47)

which fixes only ∞. The compactification can be obtained by letting Γ(N)act on H∪Q = H∪Q∪∞ instead; it is denoted X(N) and is called a modularcurve. The points X(N)\Y (N) are called the cusps of X(N) (and by extensionof Γ(N)); there are only finitely many. For example, Γ(1) has only one cusp,∞, which is equivalent to all real rational numbers at the boundary of H: tosee this, let a/c ∈ Q be arbitrary. We may suppose that (a, c) = 1, in whichcase there are integers b and d such that ad − bc = 1, and hence a Mobiustransformation

τ 7→ aτ + b

cτ + d(2.48)


Figure 2.7: A fundamental domain for Γ

in Γ taking ∞ to a/c. Thus Γ identifies ∞ and all rational numbers atthe boundary of H. Examining the fundamental domain in figure 2.7 and itsidentifictions shows that Y (1) is a sphere with one point missing, so that X(1)is a sphere. It is also a Riemann surface, so by the discussion in 2.1, thereshould be a biholomorphic function from X(1) to C. Such a function doesindeed exist; it will be introduced in section 5.2.2. One can similarly constructsurfaces X0(N) and X1(N), which are the compactifications of the quotientsH/Γ0(N) and H/Γ1(N). In section 3.3.7, we will interpret these surfaces asmoduli spaces of elliptic curves. Finally, note that the groups Γ(N) are normalin Γ, so the action of Γ descends to an action on H/Γ(N). This symmetry groupis isomorphic to Γ/Γ(N) ∼= PSL(2,Z/NZ). We give an example (of considerablehistorical interest):

Klein’s quartic The quotient H/Γ(7) is known as the Klein quartic. In pro-jective coordinates, it has equation X3Y + Y 3Z + Z3X = 0. The groupPSL(2,Z/7Z), also written PSL(2, 7), acts as symmetries of the surface.This group has 168 elements. A theorem by Hurwitz shows that a Rie-mann surface of genus g ≥ 2 has at most 84(g − 1) automorphisms. TheKlein quartic has genus 3, and 168 = 84 · 2, so the maximum is attained.For more on Klein’s quartic, see [26].

2.5.4 The Uniformization theorem

For the background for this section, see [14], [23]. We have seen that the Rie-mann sphere C, the complex plane C and the upper half-plane H are Riemannsurfaces, as are quotients of the last two by discrete groups. The uniformizationtheorem by Poincare shows that, in fact, all Riemann surfaces arise in this way.It states that the universal covering space of any Riemann surface is one of C,


Figure 2.8: A tesselation induced by Γ [46]

Figure 2.9: A fundamental domain for Γ(2)

C and H. A different formulation is that any Riemann surface admits a metric,giving the surface constant curvature, which can be taken to be −1, 0 or 1.Surfaces are accordingly classified as elliptic, parabolic or hyperbolic. The onlycompact elliptic Riemann surface is C itself, and the only compact parabolicRiemann surfaces are the complex tori, C/(Z + τZ). All other compact Rie-mann surfaces are uniformized by H, and so are intrinsically hyperbolic.


Figure 2.10: Klein’s quartic [27]

Chapter 3

Lattices and Elliptic Curves

3.1 Lattices

In this chapter, we introduce the compact Riemann surfaces uniformized by C.The material is standard and is described in for example [1], [22], [23], [24].Consider the additive group (C,+), i.e., the complex numbers under addition.By a lattice in C, we shall mean a free subgroup of rank two. Any such groupΛ has a two-element basis (λ1, λ2), such that the elements in the lattice areprecisely the complex numbers of the form λ = mλ1 + nλ2, where m and nare integers. Thus, a lattice is also a free Z-module on two generators. Toavoid degenerate cases, we will assume that λ2/λ1 is non-real. When we wishto emphasize the basis, we write Λ = Λ(λ1, λ2). The lattice also acts on C asa group of translations in the obvious way: λ corresponds to the translationz 7→ z + λ. Figure 3.1 depicts a lattice and a sublattice of index 2 (in biggerdots). The arrows show a possible basis.

Figure 3.1: A lattice and a sublattice of index 2

The symmetry group of the lattice (seen as a point set in a Euclidean plane)

Karlsson, 2009. 23

24 Chapter 3. Lattices and Elliptic Curves

always contains a translation group generated by the translations λ 7→ λ + λ1

and λ 7→ λ + λ2; it is isomorphic to Z2. It also contains the central inversionλ 7→ −λ. Sometimes the symmetry group is larger, containing rotations as well.Consider a rotation around a lattice point, which may be taken to be the origin.It corresponds to multiplication by a complex number of magnitude 1: λ 7→ cλ,and since the group is finite (the point set being discrete), c is a root of unity. Itis easy to show that the number of points with a minimal distance to the originmust be 4 or 6; hence c equals i or ρ = e2πi/3 = −1+

√3i

2 . Figure 3.2 and 3.3show these lattices, together with a basis for each; they consist of, respectively,the Gaussian and Eisensteinian integers: these are Z + iZ and Z + ρZ. Theyare Euclidean domains, with unique factorization (up to units). They are alsocalled the square and hexagonal lattices. The hexagonal lattice gives the densestpacking of spheres (i.e. circles) in two dimensions. Note that the parametersfor these lattices are precisely the fixed points of Γ, and that the order of thestabilizer groups at these points is reflected in the rotation groups of the lattices.Lattice geometry is described in [9].

Figure 3.2: The square lattice

A central concept is that of a fundamental domain for the lattice (also calledfundamental region or cell). By this, one means a fundamental domain forthe action of the lattice on C as a translation group, see 2.5. We repeat thedefinition for convenience: a fundamental domain is defined as a domain (simplyconnected set) Ω ⊆ C, containing precisely one element from each orbit of Cunder the action of Λ. (Care must be taken at the boundary; the domain can beneither open nor closed. It is not uncommon to ignore this, working instead withthe closure or interior of a domain). The translates of a fundamental domainform a tesselation of C. It follows in particular that each copy of the domaincontains precisely one lattice point. This requirement does not determine afundamental domain uniquely, but it fixes its area. Figure 3.4 shows somepossible fundamental domains for a lattice. A fundamental domain which isalso a polygon is called a fundamental polygon.

For any basis (λ1, λ2), the set sλ1 + tλ2 : 0 ≤ s, t < 1 is a fundamentaldomain, shaped like a parallelogram. The domains in figure 3.4 are of this kind.

Another alternative, known as the Dirichlet fundamental domain, is to let

3.1. Lattices 25

Figure 3.3: The hexagonal lattice

Ω (the interior of Ω) be the set of points closer to a fixed lattice point thanto any other. In other contexts, this is known as a Voronoi tesselation. It isshown in figure 3.5. In both cases, the fundamental domain is a convex polygon,which furthermore is centrally symmetric because the lattice is invariant underthe central inversion λ 7→ −λ. It is easy to show that a Dirichlet fundamentalpolygon always has four or six sides. Less obvious, but nontheless true, is thatthe same restriction holds for any convex fundamental polygon, see [9], chapter4.

Clearly, the basis is not uniquely determined by the lattice. Suppose thatΛ(λ1, λ2) = Λ(λ′1, λ

′2). Then, since λ1, λ2 is a basis, there are integers a, b, c, d

such that (λ′1λ′2

)=

(a bc d

)(λ1

λ2

), (3.1)

and since λ′1, λ′2 is also a basis, there are integers a′, b′, c′, d′ such that(

λ1

λ2

)=

(a′ b′

c′ d′

)(λ′1λ′2

). (3.2)

Putting these together and taking determinants, one finds

det(

a′ b′

c′ d′

)· det

(a bc d

)= 1, (3.3)

so the determinants, being integers, must be equal to ±1. On the other hand,any such matrix has an inverse with integer entries and hence is a valid changeof basis. Thus the automorphism group of the lattice, seen as a Z-module, isequal to SL(2,Z) × 1,−1. The matrices with determinant equal to 1 areorientation-preserving, while those with determinant equal to −1 change thehandedness of the basis. In either case, the area of the fundamental region isunchanged. The factor 1,−1 arises from regarding the basis as an orderedpair, rather than a set.

With the assumption above, the quotient C/Λ is a torus, and all such toriare topologically equivalent. They are, however, not in general equvivalent as


Figure 3.4: Some fundamental domains

Riemann surfaces (with the complex stucture inherited from C). For Riemannsurfaces, the natural concept of equality is that of conformal equivalence. Thecorresponding equivalence relation for lattices is called homothety (or similar-ity): two lattices related by multiplication by a non-zero complex number (ge-ometrically, dilatations and rotations around the origin) are called homothetic.In the following, “lattice” will be taken to mean “equivalence class of homotheticlattices”. One may therefore take one basis element to equal 1, and by suitablynumbering the basis elements, one may suppose that λ2/λ1 lies in the upperhalf-plane H. The quotient λ2/λ1 is called the modulus and will be denoted τ .The corresponding lattice is written Λτ .

By the discussion above, different moduli τ = λ2/λ1 and τ ′ = λ′2/λ′1 giverise to equivalent tori if they correspond to different bases for the same lattice,i.e. if there are integers a, b, c, d such that ad− bc = ±1 and

(λ′2λ′1

)=

(a bc d

)(λ2

λ1

), (3.4)

whence

τ ′ =λ′2λ′1

=aλ2 + bλ1

cλ2 + dλ1=

aτ + b

cτ + d. (3.5)

Thus moduli of different tori correspond to orbits of elements in H under theaction of Γ; this is what motivates the name modular group. The quotient X(1)(see section 2.5.3) is called the moduli space of elliptic curves1. We will returnto this in section 3.3.7, having introduced elliptic curves.

1Technically, a coarse moduli space

3.2. Elliptic functions 27

Figure 3.5: A Dirichlet fundamental domain and its tesselation

3.1.1 Eisenstein series

Given a lattice, one defines the Eisenstein series of weight 2k, k ≥ 2 an integer,as the sum

G2k(Λ) =∑

λ∈Λ\0

1λ2k

, (3.6)

or, as a function of a complex variable τ ∈ H, G2k(τ) = G2k(Λτ ). Note that forodd exponents, the sum is zero by the central inversion symmetry of the lattice;k ≥ 2 assures uniform convergence. See [1], [36],[39].

3.2 Elliptic functions

We now introduce elliptic functions. See [23], chapter 3 for a more completeaccount. Lattices, as defined above, are period modules of doubly periodic func-tions, generalizing the ordinary periodic functions. We will consider only mero-morphic functions periodic with respect to some lattice Λ(ω1, ω2). Such func-tions are called elliptic with respect to Λ. The reason for this is historical:elliptic functions first arose as inverses of elliptic integrals, which in turn arosein attempts to calculate the arc-length of an ellipse. Another similar problemasks the same question about lemniscates, which are curves with equations ofthe form (x2+y2)2 = 2c2(x2−y2). The problem was studied by Jakob Bernoulli,but in fact the lemniscate is a special case of the Cassini ovals. Other contribu-tors are Fagnano, Euler and Gauß. The special case of the lemniscate leads tointegrals of the form ∫

dx√1− x4

, (3.7)

and more generally, one considers integrals∫

R(x, y)dx, (3.8)


where R is a rational function and y2 = p(x) for some polynomial p of degreethree or four without repeated roots. By suitable changes of variables, suchintegrals can be reduced to a small canonical set. Several conventions exist,such as those of Jacobi, Legendre, Weierstraß and more recently Carlson. Thatof Weierstraß is the cleanest and will be described here. Apart from those as-pects of elliptic functions relevant here, the theory has numerous applicationsin geometry, mechanics and astronomy. Indeed, Jacobi commented on the re-markable unity of mathematics evidenced by the occurence of elliptic functionsboth in the study of both number theory and the pendulum.

The poles of a meromorhic function are isolated, hence countable. Thus, wemay assume that the boundary of the fundamental polygon contains no pole.Applying Cauchy’s residue theorem to the path encircling the boundary, oneobtains zero, the contributions from opposing sides cancelling by periodicity.It follows that the residues sum to zero. The sum of the orders of the polesin the fundamental polygon is called the order of the elliptic function. Anelliptic function of order zero is bounded everywhere and must be constant byLiouville’s theorem. Elliptic functions of order one would have simple poles inthe fundamental polygons and hence the sum of the residues would be nonzero.Thus elliptic functions of order one do not exist. Weierstraß constructed anelliptic function of order two, as follows:

℘(z) = ℘(z, Λ) =1z2

+∑

λ∈Λ\0

(1

(z − λ)2− 1

λ2

). (3.9)

℘ is an even elliptic function of order two, and its derivative ℘′ is odd and hasorder three. The easiest way to verify the Λ-periodicity is to note that thederivative

℘′(z) = −2∑

λ∈Λ

1(z − λ)3

(3.10)

is periodic (termwise differentiation being allowed by uniform convergence).Next, take a λ ∈ Λ and integrate

℘′(z + λ)− ℘′(z) = 0 (3.11)

to obtain℘(z + λ)− ℘(z) = C(λ), (3.12)

where C(λ) is independent of z but may depend on λ. Finally, take z = −λ/2:

℘(λ/2)− ℘(−λ/2) = C(λ) = 0 (3.13)

since ℘ is even. Clearly, the elliptic functions with respect to some fixed latticeform a field, and since all constant functions are trivially elliptic, this field canbe seen as en extention of C. It is in fact a finite extention, because two ellipticfunctions with poles and zeroes of the same order and at the same places arenecessarily proportional (their quotient being an elliptic function without poles).This can be used to express any given even elliptic function with prescribedzeroes and poles as a rational function of ℘, so the subfield of even ellipticfunctions equals C(℘). Splitting an arbitrary elliptic function in an even and anodd part and using that ℘′ is odd, one obtains an expression which is a rationalfunction of ℘ and ℘′. Thus the field of elliptic functions equals C(℘, ℘′).

3.3. Elliptic curves 29

Next, we expand ℘ in a Laurent series:

℘(z) =1z2

+ 3G4z2 + 5G6z

4 + 7G8z6 + · · · , (3.14)

giving

℘′(z) = − 2z3

+ 6G4z + 20G6z3 + 42G8z

5 + · · · ,

℘′(z)2 =4z6− 24G4

z2− 80G6 + (36G2

4 − 168G8)z2 + · · · ,

℘(z)3 =1z6

+9G4

z2+ 15G6 + (21G8 + 27G2

4)z2 + · · · ,

℘′(z)2 − 4℘(z)3 = −60G4

z2− 140G6 − (72G2

4 + 252G8)z2 + · · · ,

℘′(z)2 − 4℘(z)3 + 60G4℘(z) + 140G6 = (108G24 − 252G8)z2 + · · · .

Note that the last expression is an elliptic function without poles, hence a con-stant, and letting z = 0 shows the constant to be zero. Thus Weierstraß’ ellipticfunction satisfies the differential equation:

(℘′)2 = 4℘3 − g2℘− g3, (3.15)

whereg2 = 60

∑

λ∈Λ\0

1λ4

= 60G4 (3.16)

andg3 = 140

∑

λ∈Λ\0

1λ6

= 140G6 (3.17)

are called the elliptic invariants. We note in passing that the vanishing of thecoefficients in the right-hand side gives relations between the Eisenstein series:

G8 =37G2

4,

G10 =511

G4G6, . . .

These identities will later be interpreted as equalities between modular forms.

3.3 Elliptic curves

We now introduce elliptic curves. References for this section are [13], [22],[24], [33], [37], [38], and [39]. An elliptic curve is a curve of genus 1, whichis simultaneously a group; over a field with characteristic 6= 2, 3, it is givenby an equation of the form y2 = p(x) where p is a polynomial of degree threeor four without repeated roots. As a Riemann surface, this is a torus, whichis uniformized by C, and the group structure of (C,+) descends to a groupstructure on the torus. However, over other fields, the group structure can bedescribed both algebraically and geometrically. First, we give some historicalbackground.


3.3.1 History

We give two examples of problems giving rise to elliptic curves, one ancient andone more recent.

Example 1 A congruent number is a number which is the area of some righttriangle with rational side-lengths. The problem of determining whetheror not a given number is congruent has been studied at least since antiquityby Arabian and Greek mathematicians; Fibonacci treated the problem aswell. For a more detailed account, see [44]. Denote by h the hypothenuseof the triangle; then the integer D is congruent if and only if (h/2)2 ±D)are both squares (of rational numbers), i.e. if and only if the system

u2 + Dv2 = w2 (3.18)

u2 −Dv2 = z2 (3.19)

is solvable in rationals. A change of variables gives the equation y2 =x3 − D2x, which is an elliptic curve. One can show that the originalsystem is solvable precicely when there are infinitely many rational pointson the elliptic curve. In 1983, Tunnell made a connection with the theoryof modular forms and obtaind a sufficient condition for an integer to benon-congruent; the other direction relies on the Birch/Swinnerton-Dyerconjecture and remains unsolved.

Example 2 In 1982, Frey realized that a non-trivial solution (a, b, c) to theFermat equation xn + yn = zn gives rise to an elliptic curve y2 = x(x −an)(x + bn), which can be shown to be non-modular. This would con-tradict the conjecture by Taniyama and Shimura that every elliptic curveis modular; thus Taniyama-Shimura (actually the weaker special case ofsemistable curves) implies “Fermat’s last theorem”. For more on this, see[41].

3.3.2 The most general form of the equation

Consider a cubic polynomial in two variables over some field k, say p(x, y). Itdefines a curve, namely the locus of points with p(x, y) = 0. This curve is anaffine variety; by homogenization, it gives rise to a projective curve. Assumethat the projective curve has a k-rational point O; take the tangent at thatpoint as Z-axis in projective coordinates. Take as X-axis a point of intersectionbetween the Z-axis and the cubic, and as Y -axis any other line through O.Dehomogenizing with respect to Z (x = X/Z, y = Y/Z) gives an equation ofthe form

xy2 + axy + by = cx2 + dx + e,

and multiplication by x and the change of variables y = xy gives

y2 + (ax + b)y = cx3 + dx2 + ex.

If char(k), the characteristic of k, does not equal 2, the square in the left handside can be completed, and if in addition char(k) 6= 3, the cubic in the righthand side can be depressed. The most general form of a cubic in this case is

y2 = cx3 − g2x− g3,


and finally taking c = 4, one obtains the Weierstraß normal form. An equationy2 = p(x) with p of degree 4 can be transformed to this form via rational changesof variables, sending one of the roots of p to infinity. An alternative standardform is

Eλ : y2 = x(x− 1)(x− λ),

which is called the Legendre normal form. From this form, it is appearent thatthe parameter λ is not uniquely determined by the curve, because the roots ofthe original cubic can be permuted before being sent to 0, 1, λ. Thus, oneexpects a group isomorphic to S3, the symmetric group on 3 letters, to act onλ. Indeed, suppose that Eλ and Eµ are isomorphic elliptic curves. Then thechange of variables

x = u2x′ + r, y = u3y′

(which is the most general possible if the form of the equation and its intersectionwith the line at infinity are to be preserved, see [38]) gives

x(x− 1)(x− λ) =(x +

r

u2

) (x +

r − 1u2

)(x +

r − λ

u2

),

so µ is one of λ, 1− λ,

1λ

,1

1− λ,

λ

λ− 1,λ− 1

λ

,

and this set of Mobius transformations is a group permuting 0, 1,∞, henceisomorphic to S3.

3.3.3 The group structure

As will be described below, one may consider one and the same equation overseveral different fields. We write E(k) for the solution set in the field k, andcalls it a curve over k. For curves over R, the group structure can be describedgeometrically. First, note that for p of degree three, a chord through two pointson the curve generically intersects it in a third point as well. By adjoining apoint at infinity, vertical lines also have three point of intersection (ignoring,for the moment, tangent lines)2. If P and Q are points on the curve, denoteby P ∗Q the third point of intersection. The binary operation ∗ does not makethe curve a group, as it lacks an identity element. Any point on the curve canbe choosen as the identity; suppose the choice has been made, and denote thepoint O. Next, define an operation + by P + Q = (P ∗ Q) ∗ O. Clearly thismakes O an identity: P + O = (P ∗ O) ∗ O = P for any P , and any P hasan inverse, denoted by −P and given by −P = P ∗ O. The operation ∗ iscommutative, hence so is +. The only thing remaining to check is associativity,which however is somewhat involved to do geometrically. It is common practiceto take the point at infinity as the identity; the “∗O”-operation then correspondsto a reflection in the x-axis. Finally, a line tangent to the curve at the point Pcan be thought of as the limit of the family of lines through P and Q as Q → P ;this allowes one to add poins to themselves. To summarize, the operation “+”defined above makes the points of E(R) an abelian group. This is in fact thesame structure as the addition in C, described intrinsically. The next section

2In projective coordinates, Bezout’s theorem can be applied directly


describes the operation algebraically. Figure 3.6 shows addition of the points

P1 = (− 1√3,− 4

√427 ) and P2 = (0, 0) on the curve y2 = x3 − x.

Figure 3.6: Addition on the elliptic curve y2 = x3 − x

3.3.4 Algebra

We now derive formulæ for the group operation. Consider the curve y2 =x3 + ax2 + bx + c, and let P1 = (x1, y1) and P2 = (x2, y2) be two points on it.The line through P1 and P2 has equation

y = λx + ν, λ =y2 − y1

x2 − x1, ν = y1 − λx1,

and substitution in the equation gives, after rearrengement,

x3 + (a− λ2)x2 + (b− 2λν)x + (c− ν2) = 0.

The roots of this equation are the x-coordinates of P1, P2 and P1 ∗P2; call thislast coordinate x3. The left-hand side factors as

(x− x1)(x− x2)(x− x3) = x3 − (x1 + x2 + x3)x2 + · · · ,


and comparing x2-coefficients gives

x3 = λ2 − a− x1 − x2 = (y2 − y1

x2 − x1)2 − a− x1 − x2.

This is the x-coordinate of P3 = P1 + P2. The y-coordinate equals

y3 = −(λx3 + ν),

where the minus sign accounts for the reflection in the x-axis. To add a pointto itself, the tangent is used instead:

λ =3x2

1 + 2ax + b

2y1.

The inverse, finally, is given simply by −(x, y) = (x,−y). This constructionis easily extended to more general equations (i.e. arbitrary fields), and it re-mains true that the group operations are expressible as rational functions of thecoordinates.

Example The curve y2 = x3 + 17 contains the rational points P1 = (2, 5) andP2 = (−2, 3). Using the formulæ above, one finds:

2P1 = (−6425

,59125

),

P1 + P2 = (14,−33

8),

2P1 − P2 = (43, 282),−P1 + 3P2 = (52, 375).

All linear combinations of these points have rational coordinates, but thenumerators and denominators grow large quickly: for example,

5P1 = (279124379042111229587121

,21246408827070452537096290830311831

).

3.3.5 Elliptic curves over different fields

Over C, an elliptic curve is simply a torus with a choice of origin for the groupoperation. The torus is the quotient of C by a lattice Λ, and the group operationarises from addition in C. The lattice gives rise to Eisenstein series and henceto an equation y2 = 4x3 − g2x − g3, and conversely, for any such nonsingularequation, a corresponding lattice can be found. Then the Weierstraß ellipticfunctions parametrize the curve. Over other fields, the picture is more compli-cated. The cases of curves over R and Q have already been discussed. One canalso consider curves over number fields (finite-dimensional extensions of Q) andover finite fields. As noted above, for some fixed equation, one denotes by E(k)the elliptic curve defined over the field k.

In the early 1900s, Poincare pointed out that the group E(k′) is a subgroup ofE(k) when k′ is a subfield of k, and raised the question whether the group E(Q)is finitely generated. That this is indeed the case was proved by Mordell in 1922.Later, in his doctoral disseration, Weil extended the result to arbitrary abelian


Figure 3.7: The curve y2 = x3 − x over the finite field with 101 elements

varieties over number fields (finite field extentions of Q). Abelian varieties arebriefly discussed in section 3.3.10 below.

The group E(Q), being a finitely generated abelian group, is isomorphic tothe direct product of its torsion subgroup E(Q)tor and Zr for some integer r; ris called the rank of the group, and, by extention, of the curve:

E(Q) ∼= E(Q)tor × Zr. (3.20)

Work by Mazur (1977) shows that E(Q)tor is isomorphic to the cyclic group CN

with 1 ≤ N ≤ 10 or N = 12, or the direct product C2 × C2N with 1 ≤ N ≤ 4.The rank is much more difficult to understand. It is not even known (but widelybelieved to be true) that there exists curves with arbitrarily high rank. Theelliptic curve with the (currently) highest known rank is the following, found byElkies in 2006:

y2 + xy + y = x3 − x2−20067762415575526585033208209338542750930230312178956502x+344816117950305564670329856903907203748559443593191803 . . .

. . . 61266008296291939448732243429. (3.21)

It has trivial torsion and rank at least 28. The Birch and Swinnerton-Dyer con-jecture asserts that the rank is equal to the order of vanishing of the associatedL-function L(E, s) at s = 1 (see the next section). The conjecture is supportedby numerical evidence, but partial proofs exist only for rank ≤ 1.

3.3.6 The L-series of an elliptic curve

Let E be an elliptic curve over Q and p a prime. If reducing the equation modulop gives a non-singular elliptic curve, E is said to have good reduction at p. LetNp be the number of points on E over Fp (excluding the point at infinity) and let


ap = p−Np. One defines the local L-series as Lp(T ) = 1− apT + T 2 for primeswith good reduction; for the (finitely many) primes with bad reduction, anotherlocal series is used. The local data is then assembled into a global L-series:

L(E, s) =∏p

1Lp(p−s)

.

The product has analytic continuation to the entire complex plane: this followsfrom the modularity theorem. Define a function

G(s) =

(√N

2π

)s

Γ(s)L(E, s), (3.22)

where N is an integer called the conductor of the curve; its prime factors arethe primes of bad reduction. It satisfies a functional equation:

G(s) = ±G(2− s). (3.23)

The Birch/Swinnerton-Dyer conjecture asserts that the rank of the curve Eequals the order of vanishing of L(E, s) at s = 1. This is a specific exampleillustrating a general principle: algebraic properties of an object imply analyticproperties of a suitable associated L-series.

3.3.7 Moduli spaces of elliptic curves

Recall from 2.5.3 that any point in X(1) corresponds to a lattice, and conversely,any lattice corresponds to precisely one point in X(1). Moreover, the geometryof X(1) reflects the properties of sets of lattices: lattices that are “close” cor-respond to “close” points in the moduli space. Since the (equivalence classesof) lattices are in corresponence with elliptic curves, this construction actu-ally parametrizes elliptic curves. Thus, one calles X(1) the moduli space ofelliptic curves. More generally, the modular curve X(N) parametrizes ellipticcurves with a preferred basis for their N -torsion subgroup. Similarly, the sur-face X0(N) parametrizes elliptic curves with a chosen cyclic subgroup of orderN , and X1(N) parametrizes elliptic curves with a chosen point of order N ; see[22]. Note that it is intuitive that elliptic curves with extra structure have largermoduli spaces, which are obtained by forming quotients by smaller groups.

3.3.8 Complex multiplication

Let E be an elliptic curve. The map from E to itself, given by P 7→ nP is clearlya endomorphism for any (real) integer n. An elliptic curve with endomorphismsnot arising in this way is said to admit complex multiplication. Here, we willonly deal with elliptic curves over C. Consider an morphism between ellipticcurves φ : E → E′. It lifts to an endomorphism φ of the group (C, +), whichfixes 0 and hence must be of the form φ(z) = cz for some c ∈ C:

Cφ //

π

²²

C

π

²²C/Λ

φ // C/Λ′


For the diagram to commute, one must have cΛ ⊆ Λ′; equality correspondsto a surjective map (automorphism). The only real c for which the conditionis satisfied are real integers (the surjective ones being c = ±1, corresponding tothe identity mapping and the map sending an element to its inverse, which is anautomorphism since the group is Abelian). These are the inner endomorphisms.All other endomorphisms arise from non-real c, motivating the name complexmultiplication. For example, the curve y2 = x3 − x is the quotient of C by thesquare lattice Z + iZ and has an automorphism arising from the map z 7→ iz.In terms of coordinates, it is (x, y) 7→ (−x, iy), see [43], chapter 3.

As for any abelian group, the elements of a fixed order n in a curve E oversome field k form a subgroup, called the n-torsion group, or in this contextthe n-division points. This group is denoted E[n]. Addition being expressibleby rational functions, the equation nP = O has solutions with coordinateswhich are algebraic over k. Considering the group structure in C shows thatE[n] ∼= (Z/nZ)2.

3.3.9 Galois representations

Galois representations are homomorphisms from some Galois group of interestto GL(n,R) for some integer n and ring R. The most interesting and generalquestions concern the “absolute” Galois group G = Gal(Q/Q) for some algebraicclosure Q of Q. This object is however very large and only defined up to innerautomorphisms; this is what necessitates the study of its representations instead.We now describe how to get Galois representations from elliptic curves. Theconstruction uses the so-called Tate module. Let E be an elliptic curve overa field k and n ≥ 2 an integer coprime to char(k) (if finite). The equationnP = O is an algebraic equation, so the elements of Gal(k/k) map the groupE[n] of n-division points to itself, respecting the group structure. Hence thereis a homomorphism ρ : Gal(k/k) → Aut(E[n]) ∼= GL(2,Z/nZ). This alreadycaptures some information about the Galois group but is not entirely satisfactorysince Z/nZ has finite characteristic. It is natural, then, to study representationson GL(2,Z`n) for primes ` and then pass to the inverse limit (n →∞), obtaininga representation on GL(2,Z`) instead. One has a system of projections

E[`n+1] π`−→ E[`n], (3.24)

given by P 7→ `P = P + P + . . . + P ,where there are ` terms. The Tate moduleis the inverse limit:

T`(E) = lim←−n

E[`n], (3.25)

which is a Z`-module with a natural topology. The action of Gal(k/k) is con-tiuous and commutes with the maps π`, giving a continuous action on T`(E) aswell; if ` 6= char(k), T`(E) ∼= Z2

` , so Aut(Z2`) ∼= GL(2,Z`).

3.3.10 Hyperelliptic curves and Abelian varieties

Allowing the polynomial in y2 = p(x) to have higher degree than four (whilestill requiering the roots to be distinct) gives a so-called hyperelliptic curve. Forp of degree n, the genus of the curve equals bn−1

2 c.


Another generalization of elliptic curves is the concept of an Abelian variety:for an arbitrary integer g ≥ 1, form the quotient of Cn by a lattice of rank 2g.The resulting space is torus of complex dimension g, with a group structureinherited from Cn. If in addition the torus is a projective variety over C, it iscalled an Abelian variety. For g = 1, any complex torus is an Abelian variety,but this does not hold for higher genera.

To any non-singular complex curve, on associates an Abelian variety, calledthe Jacobian of the curve, containing the curve as a subvariety. The Jacobian,as a group, is generated by the curve. See [43], chapter 2.

Chapter 4

Dirichlet series

A Dirichlet series is a series of the form

F (s) =∞∑

n=1

ane−λns, (4.1)

where an are complex numbers and λn are real numbers tending to infinity withn. The variable s is allowed to take complex values. For the general theory ofsuch series, see [19]. Taking λn = n gives a power series in e−s. Often, onetakes instead λn = ln n (so-called normal Dirichlet series), which have the form

F (s) =∞∑

n=1

an

ns. (4.2)

Such series were studied by Dirichlet; see, for example, [25]. If |an| = O(nc)for some c, the series is absolutely convergent in the half-plane <s > c + 1.Similarly, the convergence is uniform in some half-plane. These half-planesneed not coincide.

Examples include an = 1 ⇒ F (s) = ζ(s) (the Riemann zeta function). Thenext example requires the notion of a multiplicative character. Fix an integerN and consider a group homomorphism χ from the additive group Z/NZ to themultiplicative group (C∗, ·) (nonzero complex numbers; note that, Z/NZ beingfinite, the image must actually consist of roots of unity). Extend χ periodicallyto a function χ : Z→ C∗. Such a function is called a (multiplicative) charactermodulo N . A series of the form

L(s, χ) =∞∑

n=1

χ(n)ns

(4.3)

is called a Dirichlet L-series. They were used by Dirichlet to prove that everyarithmetic progression an + b : n ∈ N, (a, b) = 1 contains infinitely manyprimes.

Dirichlet series satisfy( ∞∑

n=1

f(n)ns

)( ∞∑n=1

g(n)ns

)=

∞∑n=1

(f ∗ g)(n)ns

, (4.4)

Karlsson, 2009. 39

40 Chapter 4. Dirichlet series

where the sums are absolutely convergent, and where

(f ∗ g)(n) =∑

d|nf(d)g(

n

d) (4.5)

is a Dirichlet convolution. This implies, for example, identities such as

1ζ(s)

=∞∑

n=1

µ(n)ns

, (4.6)

where µ is the Mobius function, and

ζ2(s) =∑n=1

d(n)ns

, (4.7)

where d(n) is the number of divisors of n. More generally, it suggests that aDirichlet series should be thought of as some sort of transform of the sequencean; cf. the convolution theorem for Fourier transforms. Indeed, a Dirichletseries is a special case of the Laplace-Stieltjes transform, defined by

∫ ∞

0

e−stdα(t), (4.8)

which for dα(t) = α′(t)dt gives the ordinary Laplace transform of α′, and for αa step function with jumps a(n) at λn gives the general Dirichlet series. Thiswill be developed further in the next section.

4.1 The Mellin transform

The Mellin transform of a function f is defined as

Mf(s) =∫ ∞

0

f(t)tsdt

t, (4.9)

so that, for example, the Γ function is the Mellin transform of the function e−t:

Γ(s) =∫ ∞

0

e−ttsdt

t. (4.10)

The Mellin transform is intimately connected to the Fourier transform, defined(essentially) by

Ff(ξ) =∫ ∞

−∞f(t)e−2πiξdt. (4.11)

To see this, note that

Ff(e−t)(ξ) =∫ ∞

−∞f(e−t)e−2πiξdt =

∫ ∞

−∞f(y)y2πiξ dy

y(4.12)

after the change of variable e−t = y. The measure dtt is invariant under dilata-

tions instead of translations, i.e., it is the Haar measure for the multiplicative

4.1. The Mellin transform 41

group (R+, ·). Note that ts are the characters of this group. A key example isthe following. Define the theta function

θ(z) =∑

n∈Zeπin2z. (4.13)

Then

π−sΓ(s)ζ(2s) =∫ ∞

0

12(θ(it)− 1)ts

dt

t, (4.14)

because12(θ(it)− 1) =

∞∑1

e−πn2t, (4.15)

giving

∞∑1

∫ ∞

−∞e−πn2tts

dt

t=

∞∑1

1(πn2)s

∫ ∞

−∞e−yys dy

y= π−sΓ(s)

∞∑1

1n2s

. (4.16)

This works for any Fourier series, sending it to a Dirichlet series times a gammafactor: if

f(τ) =∞∑

n=1

ane2πinτ , (4.17)

(note the omitted constant term), one has

Mf(iτ) = (2π)−sΓ(s)∞∑

n=1

an

ns(4.18)

after an analogous calculation (see, for example [43], chapter 4). The thetafunction above satisfies

θ(i

t) = θ(− 1

it) =

√t θ(it), (4.19)

which implies a functional equation for ζ. First, we prove this so-called thetainversion formula. Consider the gaussian

g(x) = e−πtx2, t > 0, (4.20)

and its Fourier transform

Fg(ξ) = g(ξ) =1√te−πξ2/t. (4.21)

Poisson’s summation formula,∑

n∈Zg(n) =

∑

n∈Zg(n), (4.22)

gives in this case

θ(it) =∑

n∈Ze−πtn2

=1√t

∑

n∈Ze−πn2/t =

1√tθ(

i

t), (4.23)


proving the assertion. Next, introduce the auxiliary function

G(s) = π−s/2Γ(s

2)ζ(s) =

∫ ∞

0

12(θ(it)− 1)ts/2 dt

t, (4.24)

and split the integral as∫∞0

=∫ 1

0+

∫∞1

. The interval [0, 1] is sent to [1,∞] bythe substitution u = 1/t:

∫ 1

0

12(θ(it)− 1)ts/2 dt

t=

12

∫ 1

0

θ(it)ts/2 dt

t− 1

s=

12

∫ 1

0

θ(− 1it

)t(s−1)/2 dt

t− 1

s=

12

∫ ∞

1

θ(iu)u(1−s)/2 du

u− 1

s=

∫ ∞

1

12(θ(iu)− 1)u(1−s)/2 du

u−

(1s

+1

1− s

),

giving the representation

G(s) =∫ ∞

1

12(θ(it)− 1)(ts/2 + t(1−s)/2)

dt

t−

(1s

+1

1− s

). (4.25)

The function θ(it) tends to 0 rapidly as t →∞, so the integral defines a functionwhich is analytic everywhere. Thus G(s) is meromorphic everywhere, withsimple poles at s = 0, s = 1 and no other places. The integral representationmakes it obvious that G(s) = G(1− s), which is the functional equation for theRiemann ζ function. Explicitly,

π−s/2Γ(s

2)ζ(s) = π−(1−s)/2Γ(

1− s

2)ζ(1− s). (4.26)

Finally, we note that the factor π−s/2Γ( s2 ) is nowadays interpreted as corre-

sponding to a “prime at infinity”, rather than something extrinsic to the zetafunction.

4.2 Euler products

For the history of Euler products, see [42], chapter 6. The archetypal Eulerproduct is Euler’s formula for the classical zeta function, found in 1737:

ζ(s) =∏p

11− 1

ps

, (4.27)

where p runs over all primes. The formula can be thought of as expressing thefundamental theorem of arithmetic: expand the factors in geometric series:

11− 1

ps

= 1 +1ps

+1

p2s+

1p3s

+ · · · , (4.28)

and notice that in the product, every positive integer will occur as denominatorexactly once. Euler then notes that the harmonic series diverges, so

∏

p≤N

(1− 1

p

)→∞ as N →∞. (4.29)

4.3. Adeles 43

Thus the sum ∑p

1p

(4.30)

diverges as well. This was arguably the first time an arithmetic fact was provedby analytic methods, and the calculation marks the birth of analytic numbertheory. Euler went further, considering products of the form

∏p

1

1− f(p)ps

, (4.31)

where the function f is fully multiplicative, i.e.

f(mn) = f(m)f(n) (4.32)

for all integers m, n. A similar argument now gives

∞∑n=1

f(p)ns

=∏p

(1− f(p)

ps

)−1

, (4.33)

which can be used to prove analogous results for primes belonging to prescribedresidue classes modulo fixed integers. A later generalization of the classicalEuler products is to consider products of the form

∏p

P (p−s)−1, (4.34)

where P is a polynomial with constant term equal to 1. The degree of thepolynomial is called the degree of the product. Many such products (which,too, are called L-functions) have been constructed in arithmetic and geometry.An example is the L-function of an elliptic curve, to be defined in the nextchapter. Another example is the L-function of a modular form (see 5.4). Theseobjects encode arithmetic and geometric data as analytic propertes of the L-function, allowing one to translate problems from one area to another. Theclassical reciprocity laws can be understood in this way as equalities betweenL-functions. The corresponding Euler products are of degree one. The currentchallenge is the obtain a similar theory for products of higher degree. Therecent proof of the modularity theorem (chapter 5.5.2) is a partial result inthis direction. Another description of this line of research is as search for a“nonabelian class field theory”. The classical class field theory of Kroneckerand Weber shows that any field extention k of Q with abelian Galois group iscontained in a cyclotomic extention, say Q(e2πi/N ). The Langlands program isa set of conjectures extending this to higher-dimensional representations (thenon-abelian case). See [17] for an introduction.

4.3 Adeles

The topics in this chapter are classical; much of the material has been reformu-lated in more modern terms during the 20th century. The notion of an adele, inparticular, is central in current thinking. This section gives a brief introduction.


In arithmetic, a natural notion of size should reflect the multiplicative be-haviour of primes. This lead Kurt Hensel (in 1897) to introduce p-adic numbers.First, given a prime p, define the order of a rational number r, denoted by vp(r),as the unique integer n such that

r = pn s

t, (4.35)

where s and t are integers not divisible by p; in addition, let vp(0) = ∞ forevery p. Clearly vp satisfies

vp(qr) = vp(q) + vp(r),vp(q + r) ≥ min(vp(q), vp(r)),vp(q) = ∞ if and only if q = 0.

Any such map from a field to R ∪ ∞ is called a valuation. Next, define thep-adic norm (or absolut value) as

|r|p = p−vp(r).

This is indeed a norm, satisfying

|0|p = 0,

|1|p = 1,

|qr|p = |q|p|r|p,|q + r|p ≤ |q|p + |r|p.

One considers equivalence classes of norm, where two norms are consideredequivalent if one is a power of the other; the equivalence classes are calledplaces. In fact, by virtue of the inequality satisfied by a valuation, the p-adicnorm satisfies a stronger condition, namely

|q + r|p ≤ max(|q|p, |r|p). (4.36)

Such a norm gives rise to a so-called ultrametric. Conversely, it is easy to seethat any ultrametric norm arises from a valuation. The corresponding placesare called finite (or nonarchimedean). By contrast, the usual absolute value iscalled infinite (archimedean). It is accordingly written |.|∞. This explains theremarks on a “prime at infinity” made in section 4.1.

The p-adic numbers, denoted Qp, are the completion of Q with respect tothe p-adic norm. A general p-adic number can be written

∞∑n=m

anpn,

where m is an integer and an belongs to 0, 1, . . . , p − 1; the sum convergeswith respect to the norm |.|p. The p-adic numbers for which m ≥ 0 form asubring, called p-adic integers and denoted Zp. Note that several conventionsfor the representation of p-adics exist; Teichmuller used instead the pth roots ofunity as coefficients an. The notion of Witt vector encompasses both of theseconventions.

4.3. Adeles 45

The p-adics are a natural framework for the local study of equations. Indeed,if an equation has a solution in integers (or rationals), the solution descends to asolution modulo every prime; counterpositively, if some equation lacks solutionslocally (“at one place”), this “local obstruction” shows that no solution existsglobally. Unfortunately, the converse does not hold in general.

Example As shown by Thue (see [37]), the equation 3x3 + 4y3 + 5z3 = 0 hassolutions modulo every prime but no solution over Z.

In any case, one wants to complete Q with respect to all places simultane-ously; this motivates the study of the profinite completion:

Z = lim←−Z/nZ, (4.37)

which by the chinese remainder theorem factors as

Z =∏p

Zp. (4.38)

The real significance of this setup becomes clear with the introduction oftools from harmonic and functional analysis, and topological groups. In par-ticular, Tate’s celebrated thesis, written under the direction of Artin, has beenmost influential. Tate succeeded in generalizing the Fourier (Mellin) transformand zeta functions to so-called adele rings. This is now seen as the proper set-ting for these arithmetic constructions. See [42], chapter 6, and [43], chapter9.

Chapter 5

Modular forms

5.1 Definitions

The theory of modular forms is treated in [1], [11], [13], [36], [39] and [43],chapter 4. Modular forms can be approached in several different ways. Webegin by defining them as functions on lattices.

First definition A modular form f is a complex-valued function on lattices,satisfying the following conditions:

1. For a fixed λ1, f(Λ(λ1, λ2)) is a holomorphic function of λ2.

2. There is a constant k, such that f(zΛ) = f(Λ(zλ1, zλ2)) = z−kf(Λ)for any nonzero z.

3. f is bounded from above if the smallest nonzero element of Λ isbounded from below.

The constant k is called the weight of the form. Next, we define modular forms

as functions on H, by putting f(τ) = f(Λτ ). Suppose that(

a bc d

)∈ Γ.

Then one has:

f(aτ+bcτ+d ) = f(Λ(1, aτ+b

cτ+d )) =

(cτ + d)kf(Λ(cτ + d, aτ + b)) = (cτ + d)kf(τ),

motivating the following definition:

Second definition A modular form for Γ is a holomorphic function f : H → C,bounded as τ → i∞, satisfying

f(γτ) = f(aτ + b

cτ + d) = (cτ + d)kf(τ) (5.1)

for all γ ∈ Γ.

Modular forms for subgroups of Γ are defined analogously, with the requirementthat f be holomorphic at the cusps (recall the definition of cusps from section

Karlsson, 2009. 47

48 Chapter 5. Modular forms

2.5.3). If in addition f vanishes at the cusps, it is called a cusp form. Since thetranslation τ 7→ τ + 1 is contained in Γ, modular forms are 1-periodic:

f(τ + 1) = f(τ), (5.2)

and can be written as functions of q = e2πiτ , defined in the punctured open unitdisc z : 0 < |z| < 1. The point q = 0 corresponds to τ → i∞; boundednessand the Riemann removability theorem then implies that f is holomorphic atq = 0 as well, and can be expanded in a Taylor series:

f(q) =∞∑

n=0

anqn =∞∑

n=0

ane2πinτ . (5.3)

5.2 Examples

5.2.1 Eisenstein series

The Eisenstein series, defined in section 3.1, are modular forms; the propertransformation under the action of Γ is readily verified, and the growth conditioncan be checked by inspection of their Fourier series, which we now determine.Start with the identity

π cot πτ =1τ

+∞∑

n=1

(1

τ + n+

1τ − n

)(5.4)

and note that

π cot πτ = πiq + 1q − 1

= πi

(1− 2

∞∑n=0

qn

). (5.5)

Equate the right-hand sides and differentiate k−1 times with respect to τ . Thisgives

∑

m∈Z

1(τ + m)k

=(−2πi)k

(k − 1)!

∞∑n=0

nk−1qn. (5.6)

Next, rewrite (note that k is even) as

Gk(τ) =∑

(m,n)6=(0,0)

1(nτ + m)k

= 2∞∑

m=1

1mk

+ 2∞∑

n=1

∑

m∈Z

1(nτ + m)k

, (5.7)

which, using (5.6), becomes

Gk(τ) = 2ζ(k) + 2(−2πi)k

(k − 1)!

∞∑n=1

σk−1(n)qn, (5.8)

where σk(n) denotes the sum of the kth powers of the divisors of n:

σk(n) =∑

d|ndk. (5.9)

Finally, using Euler’s formula relating values of the ζ function for even integerarguments to the Bernoulli numbers (see appendix A),

ζ(2k) = − (2πi)2k

2(2k)!B2k, (5.10)

5.2. Examples 49

one obtains

G2k(τ) = 2ζ(2k)

(1− 4k

B2k

∞∑n=1

σ2k−1(n)qn

). (5.11)

It is sometimes useful to normalize the constant term to one, putting

E2k(τ) =G2k(τ)2ζ(2k)

. (5.12)

Thus, for example,

E4(τ) = 1 + 240∞∑

n=1

σ3(n)qn,

E6(τ) = 1− 504∞∑

n=1

σ5(n)qn. (5.13)

Theta functions

Historically, a source of modular forms are the theta functions. These are not ingeneral invariant under the action of Γ, but transform in such a way that certaincombinations of them are modular forms. The literature on theta functions isvast, and we will only give a short example. More on theta functions can befound in [8], [11] and [31]. The Jacobi theta function is defined as follows:

ϑ(z, τ) =∑

n∈Zeπin2τ+2πinz, (5.14)

for z ∈ C and τ ∈ H. The sum can be shown to converge absolutely anduniformly on compact sets ([31]). One can show ([31],[11]) that θ(τ) = ϑ00(0, τ)k

satisfies the equations

θ(−1/τ) = (−iτ)k/2θ(τ), θ(τ + 2) = θ(τ), (5.15)

making it a modular form for Γ(2) of weight k/2.

5.2.2 The modular invariant

The discriminant of the cubic polynomial 4x3−g2x−g3 is, up to constant factors,∆ = g3

2 − 27g23 ; by construction, it vanishes when the roots of the polynomial

are not all distinct. Since g2 has weight 4 and g3 has weight 6, the discriminantis a modular form of weight 12. It is related to the Dedekind eta function, givenby:

η(τ) = q1/24∞∏

n=1

(1− qn). (5.16)

Namely, one has∆ = (2π)12η24, (5.17)

giving

∆ = (2π)12q∞∏

n=1

(1− qn)24. (5.18)


It is evidently a cusp form, as expected from its definition. The Fourier coeffi-cients of η24 are integers; one defines the Ramanujan function τ(n) by

q∞∏

n=1

(1− qn)24 =∞∑

n=1

τ(n)qn. (5.19)

It has been widely studied and satisfies numerous identities, such as

τ(m)τ(n) =∑

d|(m,n)

d11τ(mn

d2), (5.20)

τ(n) ≡ σ11(n)(mod 691), and (5.21)

n610τ(n) ≡ σ1231(n)(mod 36) when 3 - n. (5.22)

Note that both ∆ and g32 are modular forms of weight 12. Thus, their quotient

g32/∆ is invariant under the modular group. It is not a modular form because

it fails to be bounded at the cusp i∞; instead, it is called a modular function.It is convenient to introduce a normalizing factor, defining

j = 123 g32

∆(5.23)

It has a simple pole at infinity. All its Fourier coefficients are integers. Theexpansion begins:

j(τ) =1q

+ 744 + 196884q + 21493760q2 + . . . . (5.24)

Liouville’s theorem shows that any modular function, i.e., meromorphic func-tion invariant under the modular group, is a rational function of j; hence, it iscalled the modular function. The Fourier coefficients have remarkable proper-ties, related to the so-called monster group. More precicely, the coefficients are(simple) linear combinations of the dimensions of irreducible representations ofthe monster, viz. 196884 = 1 + 196883. See [16].

Another remarkable fact is that j( 1+√−D2 ), where D is a squarefree natural

number, is an integer when the ring Z[ 1+√−D2 ] is a unique factorization domain;

the largest D for which this happens is 163, giving that

−eπ√

163 + 744− 196884e−π√

163 + · · ·

is an integer; eπ√

163 is large, so the series is well approximated by the first twoterms. Thus eπ

√163 should be close to an integer, and indeed

eπ√

163 ≈ 262537412640768743.99999999999925 . . . .

This is described in, for example, [13].The modular function can also be expressed directly in terms of the co-

efficients of the equation of the elliptic curve. For the Legendre curve y2 =x(x− 1)(x− λ), one has:

j(E) = 28 (λ2 − λ + 1)3

λ2(λ− 1)2, (5.25)

5.3. Spaces of modular forms 51

and a similar but more complicated formula exists for elliptic curves over fieldswith characteristic equal to 2 or 3. j(E) is a complete invariant in the followingsense: if E and E′ are elliptic curves, then j(E) = j(E′) if and only if E and E′

are equivalent. We now show this (following [13]). By the discussion in section3.3.2, it suffices to show that j(λ) = j(λ′) precisely when λ′ is one of λ, 1− λ,1/λ, 1/(1 − λ), λ/(λ − 1) or (λ − 1)/λ. To see that this holds, consider thepolynomial

p(x, λ) = (x−λ)(x−(1−λ))(

x− 1λ

)(x− 1

1− λ

)(x− λ

λ− 1

)(x− λ− 1

λ

),

and put k = j/27 for convenience. A straightforward calculation shows that

p(x, λ) = x6 − 3x5 − (k − 6)x4 + (k − 7)x3 − (k − 6)x2 − 3x + 1, (5.26)

so k, hence also j, is invariant under the Mobius transformations in question.For the other direction, suppose that j(λ) = j(λ′). Then p(x, λ) = p(x, λ′)holds identically, so that in particular p(λ, λ′) = p(λ, λ) = 0. It follows thatλ and λ′ are related by a Mobius transformation of the desired kind, provingthe assertion. It is also possible to construct elliptic curves with the desiredj-invariant: one finds that the curve E given by

y2 = x3 − 12x2 − 36

j − 1728x− 1

j − 1728(5.27)

satisfies j(E) = j; also y2 = x3 − 1 has j-invariant 0 and y2 = x3 − x hasj-invariant 1728.

5.3 Spaces of modular forms

Modular forms of a fixed weight k obviously form a vector space over C. Infact, these spaces are all finite-dimensional, a consequence of the requirementof boundedness at the cusps. Acting on these spaces is an algebra of operators,introduced by Hecke. First, we determine the dimensions. This expositionmosty follows [36].

Let f(τ) be a modular form of weight k. Denote by vp(f) the order ofvanishing of f at a point p, i.e., the unique integer n such that

f(τ) = (τ − p)ng(τ) (5.28)

for some analytic function g, which is nonzero in p. Define v∞(f(τ)) as v0(f(q)).By assumption, f is holomorphic, so there are no poles. Also, the zeroes areisolated, so there is some neighbourhood of q = 0 containing no other zero. Inthe fundamental region in the upper half-plane, this neighbourhood correspondsto some number M such that all zeroes except the one at i∞ have imaginarypart ≤ M . The intersection of the closure of the fundamental domain with theset τ : =τ ≤ M is compact, and hence contains only finitely many zeroes. LetN(f) be this number, counted with multiplicity. Enclose this region by arcs asin figure 5.1, and suppose that f has no zeroes on the arcs. Recall from 2.5.2that the points i and (−1 + i

√3)/2 are the fixed points of Γ in the canonical

fundamental domain. The stabilizer subgroups have order 2 and 3, respectively.Then, N(f) is given by


Figure 5.1: The contour of integration

N(f) =1

2πi

∫

A

df

f, (5.29)

where A is the contour of integration. The integrals along A1 and A4 cancel(note that f(τ + 1) = f(τ)). Letting the radii of the circles enclosing ρ, i and∞ tend to zero, one obtains

12πi

∫

A5

→ −v∞(f), (5.30)

and similarly1

2πi

∫

C1

→ −16vρ(f), (5.31)

12πi

∫

C2

→ −16vρ(f), (5.32)

12πi

∫

C3

→ −12vi(f). (5.33)

Finally, the involution S : τ 7→ − 1τ takes the arc A3 to A2, but f is not invariant

under S. Rather, f(Sτ) = zkf(τ), giving

df(S(τ))f(S(τ))

= kdτ

τ+

df

f, (5.34)

so1

2πi

∫

A2∪A3

df

f=

12πi

∫

A2

(df

f− k

dτ

τ− df

f

)→ k

12. (5.35)


Taken together, this gives

N(f) + v∞(f) +13vρ(f) +

12vi(f) =

k

12. (5.36)

In the case that f has zeroes on the contour, indentions have to be made. Itturns out that formula (5.36) is valid in these cases as well. All the variableson the left-hand side are nonnegative integers, so k must be even. The fraction2/12 = 1/6 is not expressible in this way, so k 6= 2. On the other hand, theEisenstein series give modular forms for every even integer ≥ 4. Next, note thatfor a cusp form, v∞(f) ≥ 1, giving k ≥ 12. The modular discriminant ∆ givesan example of a cusp form of weight 12. Now take f = E4:

N(E4) +13vρ(E4) +

12vi(E4) =

13, (5.37)

giving vρ(E4) = 1, vi(E4) = 0 and N(E4) = 0. Similarly one finds vρ(E6) = 0,vi(E6) = 1 and N(E4) = 0. Furthermore,

N(∆) + v∞(∆) +13vρ(∆) +

12vi(∆) = 1, (5.38)

and v∞(∆) ≥ 1 forces N(∆) = vρ(∆) = vi(∆) = 0, v∞(∆) = 1. Denote byMk the C-vector space of level one modular forms of weight k, and by Sk thesubspace of cusp forms. Further, put mk = dimCMk and sk = dimC Sk andconsider the evaluation homomorphism φ : f 7→ f(i∞). Then, whenever thereexists non-cusp forms of weight k, i.e. when k ≥ 4, the following sequence isexact, where the second arrow is inclusion:

0 // Sk// Mk

φ // C // 0.

It follows that in these cases, mk = sk + 1. The Eisenstein series Ek belongs toMk\Sk, so for k ≥ 4,

Mk = Sk ⊕ CEk. (5.39)

Also, multiplication by the discriminant ∆ ∈ S12 clearly gives an isomorphismfrom Mk to Sk+12 for every k, invertible becauce ∆ only has a simple zero atinfinity. Thus Mk

∼= Sk+12 and mk = sk+12. Knowing that sk = 0 for k < 12gives mk ≤ 1, and the Eisenstein series show that in fact mk = 1 for 4 ≤ k ≤ 10,k even. This detemines the dimensions for every k. The following table showssome results:

k mk sk Basis for Mk

0 1 0 12 0 0 -4 1 0 E4

6 1 0 E6

8 1 0 E24

10 1 0 E4E6

12 2 1 E26 ,∆

14 1 0 E24E6

All basis elements shown are polynomials in E4 and E6 (Recall that ∆ =g32 − 27g2

3). This holds in general:


Theorem Every element of Mk is expressible as∑

4m+6n=k

am,nEm4 En

6 ,

where the coefficients are complex numbers and the indices are nonnega-tive integers.

Proof For k ≤ 12, this is clear from the table. If f ∈ Mk for some k > 12,subtract some form f(i∞)Em

4 En6 of the correct weight, giving a cusp form.

Then (f − f(i∞)Em4 En

6 )/∆ is a modular form of weight k − 12. Theassertion follows by induction.

Theorem All monomials Em4 En

6 of equal weight are algebraically independent.

Proof Assume they are not. Dividing the equation with one of the termswould give a polynomial equation in E3

4/E26 ; thus this function would be

constant. This is not the case, and the assertion follows.

Theorem The dimension mk is given by

mk = b k

12c if k ≡ 2 (mod 12),b k

12c+ 1 if k 6≡ 2 (mod 12).

Proof One way is to check the formula for k ≤ 12, and then note that equalitymust continue to hold because the formulæ give mk+12 = mk +1. Anotherway is to count the number of nonnegative integers satisfying 4m+6n = k,which by our earlier theorems is equal to the dimension.

The fact that all the spaces Mk are finite-dimensional, implies that a set ofmore than mk forms is linearly dependent. This in turn gives relations betweenthe Fourier coefficients of the forms. For example, M8 has dimension one, andboth E8 and E2

4 belong to this space. Since the constant terms are equal, onehas E8 = E2

4 , and equating coefficients yields the relation:

σ7(n) = σ3(n) + 120n−1∑

k=1

σ3(k)σ3(n− k), (5.40)

which would be rather more difficult to discover and prove using more elemen-tary methods.

5.3.1 Hecke operators

Hecke introduced a set of linear operators acting on modular forms. They canbe thought of as periodizing the forms. In terms of lattices, the nth Heckeoperator, acting on Mk, is defined by (see [24], [39])

Tk(n)f(Λ) =∑

[Λ:Λ′]=n

f(Λ′), (5.41)

where the sum is taken over all sublattices of index n. For example, when n = 2,there are three such sublattices. If the lattice Λ is spanned by λ1 and λ2, these


sublattices are spanned by 2λ1, λ2, λ1, 2λ2 and λ1 +λ2, λ1−λ2. The lastsublattice is the one depicted in figure 3.1. It is readily seen that Tk(n) mapsMk into itself for every n and k. It is also clear that for m and n relativelyprime, one has

Tk(m)Tk(n) = Tk(mn), (5.42)

there being only one lattice Λ′ satisfying [Λ : Λ′] = m and [Λ′ : Λ′′] = n. Lessclear, but true, is that for any m, n:

Tk(m)Tk(n) =∑

d|(m,n)

dk−1Tk(mn

d2). (5.43)

For functions f(τ) = f(Λτ ), one defines

Tk(n)f(τ) = nk−1∑

ad=n, a≥00≤b<d

d−kf

(aτ + b

d

), (5.44)

which corresponds to the previous definition except for the factor nk−1. Theaction on the Fourier coefficients looks as follows:

Tk(n)

( ∞∑m=0

a(m)qm

)=

∞∑m=0

∑

d|(m,n)

dk−1a(mn

d2)

qm, (5.45)

where (0, n) is interpreted as n. Requiering

a(m)a(n) =∑

d|(m,n)

dk−1a(mn

d2) (5.46)

amounts to asking thatTk(n)f = a(n)f, (5.47)

i.e. that f is an eigenform for the operator Tk(n). Clearly the Hecke operatorstake cusp forms to cusp forms, so all cusp forms of weight 12, 16, 18, 20, 22and 26 are automatically eigenforms for all Hecke operators (these spaces beingone-dimensional). In particular it holds for ∆, proving the equation

τ(m)τ(n) =∑

d|(m,n)

d11τ(mn

d2) (5.48)

satisfied by Ramanujan’s function. In general, multiplicativity implies

∞∑m=1

a(m)ms

=∏p

(1 +∞∑

n=1

a(pn)p−ns), (5.49)

and Hecke showed that

a(p)a(pn) = a(pn+1) + pk−1a(pn−1), (5.50)

for primes p, whence

∞∑n=1

a(n)ns

=∏p

11− a(n)p−s + pk−1−2s

. (5.51)


Thus the factors in the Euler product arise from f being an eigenform forHecke operators; a form which is a simultaneous eigenform for all T (p) (whichgenerate the algebra of Hecke operators) will have a full Euler product. TheHecke operators have been generalized by Atkin and Lehner: see [2]. Theseoperators will be used in the final chapter. Finally, note that for forms f , g ofweight k, the measure

f(z)g(z)yk dxdy

y2(5.52)

is invariant and hence passes to a well-defined measure on H/Γ. One definesthe (Petersson) inner product on Sk by

〈f, g〉 =∫∫

H/Γ

f(z)g(z)yk dxdy

y2. (5.53)

The integral converges, f and g being cusp forms. This inner product makes Sk

a Hilbert space. The Hecke operators are Hermitian with respect to this innerproduct, and since they commute, they can be simultaneously diagonalized.Thus Sk has an orthonormal basis of eigenforms.

5.4 The L-series of a modular form

Analogously to the L-series of an elliptic curve (recall section 3.3.6), one definesthe L-series of a modular form

f(τ) =∞∑

n=0

anqn (5.54)

as

L(f, s) =∞∑

n=1

an

ns. (5.55)

As in section 4.1, the mapping from f to L(f, s) need not be treated purelyalgebraically, as it is essentially the Mellin transform. See [43], chapter 4.

5.5 Applications

5.5.1 Additive number theory

With hindsight, the applications of modular forms to problems in additive num-ber theory arguably goes back as far as to Jacobi, who in 1834 determined thenumber of ways to express an arbitrary integer as a sum of squares (althoughnaturally, the solution was not formulated in that way). Let rk(n) denote thenumber of ways to express n as a sum of k squares of integers, where two repre-sentations are considered different if the order differs (i.e., rk(n) is the numberof k-tuples (x1, x2, . . . , xk) ∈ Zk such that

∑ki=1 x2

i = n). For example, one hasr2(25) = 12. Diophantus had claimed that every natural number is expressibleas a sum of four squares, i.e. that r4(n) > 0 for all n ∈ N. This was proved byLagrange in 1770. Jacobi succeeded in determining r4(n) explicitly. The most

5.5. Applications 57

natural solution uses theta functions, so will will only state the result. Notethat

ϑ3(0, q)4 =

(∑

n∈Zqn

)4

=∞∑

n=0

r4(n)qn. (5.56)

One can show that

ϑ3(0, q)4 = 1 + 8∞∑

n=1

(∑

d|n4-d

d)qn, (5.57)

so r4(n) equals eight times the sum of the divisors of n which are not divisibleby four. The partition function also has connections to modular forms. Thepartition function p(n) is defined as the number of ways to express n as a sumof positive integers (regardless of the order). As noted already by Euler, thegenerating function factors as

∞∑m=0

p(m)qm =∞∏

n=1

11− qn

. (5.58)

Recall from section 5.2.2 that Dedekind’s eta function is given by

η(τ) = q1/24∞∏

n=1

(1− qn). (5.59)

and that∆ = (2π)12η24 (5.60)

where ∆ is the modular discriminant, which is a modular form. Expressing theproduct in (5.58) in terms of this form and using its transformation properties,Rademacher obtained an explicit formula for p(n); for this calculation, see [1],chapter 5. Modular forms also arise as weight enumerators for lattices andlinear codes; for these topics, see [8], [12]. For other applications, such as theconstruction of Ramanujan graphs, see [34].

5.5.2 The modularity theorem

The modularity theorem has its roots in conjectures made by Taniyama andShimura in 1955-1957; it has been referred to as the Taniyama-Shimura-Weilconjecture and several variations thereof. Work by Frey, Serre and Ribet inthe 1980s showed that the conjecture implies Fermat’s “last theorem” (rather,conjecture), prompting Wiles to attack the problem. With help from Taylor,the semistable case was settled in 1995, proving Fermat’s conjecture. The fullmodularity conjecture was proved by Breuil, Conrad, Diamond and Taylor in1999. It is now seen as part of the even more general Langlands program.

The modularity theorem asserts that all elliptic curves over the rationalnumbers are modular ; that is, parametrized by the modular curve X0(N) (forsome N , assumed to be minimal) via a rational map with integer coefficients:X0(N) → E(Q). This implies that the L-series attached to E equals the L-seriesof a (new-)form of weight two of level N .


Example The equation y2 + y = x3− x2 is parametrized by the modular form

η(τ)2η(11τ)2 = q

∞∏n=1

(1− qn)2(1− q11n)2 =

q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − 2q9−2q10 + q11 − 2q12 + 4q13 − q15 − 4q16−2q17 + 4q18 + 2q20 + 2q21 − 2q22 − . . . (5.61)

and, for example, the equation has 1 + 7 − (−2) = 10 solutions over thefield with 7 elements, as predicted (counting a solution at infinity). As insection 3.3.6, we write Np for the number of points on the curve over thefinite field with p elements. The following table shows p, Np and ap forsome small p:

p 2 3 5 7 11 13 17 19 23 29 97Np 4 4 4 9 10 9 19 19 24 29 104ap −2 −1 1 −2 1 4 −2 0 −1 0 −7

5.6 Generalizations

Several generalizations of modular forms exist; one can consider forms of severalvariables, for more general groups, p-adic forms etc. The forms studied in thischapter are called elliptic modular forms, because of their connection to themoduli space of elliptic curves. We mention also that the term modular issometimes reserved for forms for the full modular group, Γ(1). Forms for othergroups are then called automorphic.

Chapter 6

New converse theorems

6.1 Background and motivation

As we have seen in section 5.4, modular forms give rise to Dirichlet series, andit is natural to ask when a given Dirichlet series arises in this way. That is,starting with the series

F (s) =∞∑

n=1

an

ns, (6.1)

under what conditions is the function

f(τ) =∞∑

n=1

ane2πinτ , (6.2)

a modular form for some Fuchsian group? Recall that f is said to be modularwith respect to the group G if it holomorphic, remains bounded at the cusps ofH/G and satisfies

f(aτ + b

cτ + d) = (cτ + d)kf(τ) (6.3)

for every transformation τ 7→ aτ+bcτ+d in G, and for some fixed integer k, called

the weight of the form. The Mellin transform, defined in 4.1, is linear and sendsforms to Dirichlet series; explicitly, the function

e−2πnτ

is sent to

(2π)−sΓ(s)1ns

.

The transformation behaviour of the form f under the group G translates tospecial properties of the function F . A converse theorem gives sufficient con-ditions on F for f to be a modular form. This line of research can be said tohave started with Hamburger’s proof [18] that the functional equation satisfiedby Riemann’s zeta function, together with some growth conditions, does in factdetermine the zeta function up to constant factors. The Riemann zeta functionis the Mellin transform of a well-known theta function related to a cusp form

Karlsson, 2009. 59

60 Chapter 6. New converse theorems

of weight 4 for the principal congruence subgroup Γ(2). The conditions on thetheta function expressing the proper transformation under Γ(2), namely

θ(− 1iτ

) =√

τθ(iτ)

θ(τ + 2) = θ(τ) (6.4)

imply the functional equation

π−s/2Γ(s

2)ζ(s) = π−(1−s)/2Γ(

1− s

2)ζ(1− s). (6.5)

In [35], Selberg defines what is now known as the Selberg class of Dirich-let series: the functions in this class have Euler products and satisfy certainfunctional equations. This class includes the classical zeta and L-functions, aswell as those from modular forms. Selberg conjectured that all these functionssatisfy a generalized Riemann hypothesis, and they are all believed to emergefrom modular forms. See [7] for more on this. These conjectures motivate thecurrent work on converse theorems. They are also loosely related to the Lang-lands program. Recall from section 4.2 that the degree1 of the polynomial inthe local factors of an Euler product is called the degree of the product, and, byextension, of the L-function. The modern work in this area, to be described insections 6.4 - 6.6, can be seen as an attempt to extend known results concerningdegree 1 to degree 2.

6.2 Hecke’s original converse theorem

In his article ”Uber die Bestimmung Dirichletscher Reihen durch ihre Function-algleichungen” [20], published in Matematische Annalen in 1936, Hecke provesa converse theorem for Dirichlet series. First, we need some definitions. Let k,N and γ be constants, let F be an analytic function and let

G(s) =(

N

2π

)s

Γ(s)F (s). (6.6)

Hecke asks the question when F is defined by a Dirichlet series

F (s) =∞∑

n=1

an

ns, (6.7)

satisfies a functional equation of the form

G(s) = γG(k − s) (6.8)

and furthermore is such that (s − k)F (s) is an entire function. In Hecke’sterminology, a function F satisfying these requirements is of type N, k, γ.One wishes the determine for which N, k, γ such functions exist, and whenthey do, how many linearly independent F exist. By a Mellin transformation,

1More precicely, the maximal degree of all polynomials, which may be different at differentprimes.

6.3. Weil’s converse theorem 61

this translates to a question about the existence of functions automorphic underthe group generated by

τ 7→ τ + N (6.9)

andτ 7→ −1

τ. (6.10)

The equation expressing the transformation behaviour under the transformation(6.10) is equivalent to the functional equation (after a Mellin transformation).Hecke shows that the group generated by (6.9) with N real, and (6.10), isdiscrete precisely when

1 N ≥ 2, or

2 N = 2 cos(π/q), where q ≥ 3 is an integer.

These groups are called Hecke groups. N = 1 gives the modular groupΓ. Hecke goes on to determine the dimensions of the spaces of functions inquestion. It is infinite-dimensional precisely when N > 2. Hecke also recoversHamburger’s theorem. Hecke groups are studied in their own right, see forexample [3], [4]. Hecke’s result, as far as we are concerned, reads as follows(Γ(s) is the gamma function):

Hecke’s converse theorem Suppose that the function F , defined as in (6.7),is such that Γ(s)F (s) is an entire function of a complex variable which isbounded in vertical stripes, and that F satisfies a functional equation ofthe form (6.8). Then, if N = 1, 2, 3 or 4, F is the L-function of a modularform for the group Γ0(N).

Here, Γ0(1) is simply the full modular group Γ. The proof will be described aspart of the proof in section 6.4 below.

6.3 Weil’s converse theorem

In 1967, Weil published the article ”Uber die Bestimmung Dirichletscher Reihendurch Funktionalgleichungen” [45]. As will be described in 6.4 below, Hecke’stheorem does not generalize to higher congruence subgroups without additionalassumptions. Weil introduces the idea to study not only the ordinary Dirichletseries

F (s) =∞∑

n=1

an

ns, (6.11)

but also so-called twisted Dirichlet series:

Fχ(s) =∞∑

n=1

anχ(n)ns

, (6.12)

where χ(n) is a multiplicative character (recall the definition from chapter 4).The auxiliary function (6.6) is modified acordingly:

Gχ(s) =(

N

2π

)s

Γ(s)Fχ(s). (6.13)


Thus G(s) corresponds to the trivial character. Weil also needs Gauß sums ofthe form

g(χ) =∑

r

χ(r)e2πir/m, (6.14)

where m is a fixed integer and r runs over the residue classes modulo m. Weil’sresult is the following:

Weil’s converse theorem Suppose that Gχ(s) is entire, bounded in verticalstripes and satisfies a functional equation of the form

Gχ(s) = ±χ(−N)g(χ)g(χ)

Nk/2−sGχ(k − s) (6.15)

for every m such that (m,N) = 1 and every character χ modulo m. ThenF (s) is a modular form of weight k for the group Γ0(N), and if the Dirichletseries converges absolutely, it is a cusp form.

Actually, an additional character has been hidden in the ± - sign. Weil ends thearticle by commenting on the Taniyama - Shimura conjecture (now theorem),which asserts that all elliptic curves over Q are parametrized by modular forms.By the time of writing, this was already known for some special cases. Weilremarks that the full conjecture appears “problematic at present”, and suggestsit as an exercise for the interested reader.

6.4 An extension of Hecke’s converse theorem

6.4.1 Introduction and results

In 1995, Brian Conrey and David Farmer published the article “An extensionof Hecke’s converse theorem” [5]. In this article, Conrey and Farmer extendHecke’s results to some congruence subgroups of higher level, namely Γ0(N) for5 ≤ N ≤ 12, 14 ≤ N ≤ 17, and N = 23. In the case considered by Hecke,the corresponding spaces of functions have infinite dimension, so additionalassumptions are required to get a converse theorem. Unlike Weil, Conrey andFarmer assume that the Dirichlet series has an Euler product

F (s) =∏p

1Lp(p−s)

(6.16)

with local factors of the form

Lp(p−s) = 1− app−s + pk−1−2s, p - N (6.17)

Lp(p−s) = 1− pk/2−1−s, p|N but p2 - N (6.18)

Lp(p−s) = 1, p2|N, (6.19)

in addition to a functional equation and a growth assumtion. Namely, put

F (s) =∞∑

n=1

an

ns(6.20)

6.4. An extension of Hecke’s converse theorem 63

and

f(τ) =∞∑

n=1

ane2πinτ , (6.21)

and introduce the auxilliary function

G(s) =

(√N

2π

)s

Γ(s)F (s). (6.22)

The functional equation of level N and weight k takes the form

G(s) = ±(−1)k/2G(k − s). (6.23)

In the cases treated by Hecke, the growth condition (F in an entire function,bounded in vertical strips) and the functional equation alone force f to beautomorphic under Γ0(N). This no longer holds when N > 4, so additionalinformation coming from the Euler product has to be used. Of course, it is stillsufficient to check that f transforms correctly on a generating set for Γ0(N).

6.4.2 The proof

First, note that the functional equation (6.23) is equivalent to

f(− 1Nτ

) = ±Nk/2τkf(τ), (6.24)

and the Euler product is equivalent to f being an eigenfunction of the Heckeoperators T (p) when p - N , and the Atkin-Lehner operators U(p) when p|N . Itis convenient, at this point, to work with matrices and introduce the notation

(f |kγ)(z) = (det γ)k/2(cz + d)−kf(az + b

cz + d) (6.25)

for a matrix

γ =(

a bc d

)(6.26)

in GL(2,R). If the matrix has determinant equal to 1, the factor (det γ)k/2

is clearly redundant; it is included only to allow for writing, for example, thetransformation2

F (N) : τ 7→ − 1Nτ

(6.27)

as the matrix (0 −1N 0

)(6.28)

rather than (0 −1/

√N√

N 0

)(6.29)

as would otherwise have been necessary. Thus the “|k”-operator can be usedfor any Mobius transformation with positive determinant. The group of such

2It is actually the well-known Fricke involution, hence the notation; we will not deal withits other properties here.


transformations, denoted GL+(2,R), is a normal subgroup in GL(2,R), of index2. The proper transformation behaviour under a Mobius transformation γ isthen expressed as f |k = ±f . Note also that the transformation (6.27) does notbelong to Γ0(N), but writing T for the translation τ 7→ τ + 1, one has

F (N)(F (N)T )−1 : τ 7→ 1Nτ + 1

, (6.30)

and this transformation does belong to Γ0(N). In fact, together with T , itgenerates Γ0(N) when N ≤ 4. The “|k”-operator is extended linearly to thegroup ring C[GL+(2,R)]:

f |∑

i

ciγi =∑

i

cif |γi, (6.31)

where we suppress the weight, and the ci are complex constants. Clearly the setof Mobius transformations annihilating f are a (right) ideal. It will be denotedΩf . All congruences between Mobius transformations (or their matrices) in thefollowing will be modulo this ideal. This is natural, as the potential form f doesnot “see the difference” between congruent transformations.

In terms of matrices, the Hecke operators (recall section 5.3.1) become

T (p) =(

p 00 1

)+

p−1∑

i=0

(1 i0 p

), (6.32)

and the Atkin-Lehner operators ([2], [5])

U(p) =p−1∑

i=0

(p i0 p

)(6.33)

respectively. The asumptions on F translate to

f |kT (p) = apf, p - N, (6.34)

f |kU(p) = f |k(

p 00 1

), p|N but p2 - N (6.35)

f |kU(p) = 0, p2|N. (6.36)

The proof breaks down in several cases: first, for N = 5, 7, 9, one has gener-ators

Γ0(5) = 〈(

1 10 1

),

(1 05 1

),

(2 15 3

)〉 (6.37)

Γ0(7) = 〈(

1 10 1

),

(1 07 1

),

(2 17 4

)〉 (6.38)

Γ0(9) = 〈(

1 10 1

),

(1 09 1

),

(2 19 5

)〉 (6.39)

and the first two in each case express 1-periodicity (cf (6.21)) and the functionalequation. It remains to check the last one, which has the general form

M2(N) =(

2 1N N+1

2

). (6.40)

6.4. An extension of Hecke’s converse theorem 65

Modulo Ωf , one has from (6.32)

T (2) =(

2 00 1

)+

(1 00 2

)+

(1 10 2

)≡ a2

(1 00 1

), (6.41)

and pre- and postmultiplication by(

0 −1N 0

)gives

(1 00 2

)+

(2 00 1

)+

(2 0N 1

)≡ a2

(1 00 1

), (6.42)

whence (2 0N 1

)≡

(1 10 2

), (6.43)

and postmultiplication by(

1 1/20 1/2

)gives

(2 1N N+1

2

)≡

(1 10 1

)≡

(1 00 1

), (6.44)

which proves the theorem for N = 5, 7, 9. Next, for N = 6, one has generators

Γ0(6) = 〈(

1 10 1

),

(1 06 1

),

(7 −312 −5

)〉 (6.45)

where again the first two are free and the third one has the form A(6)−1

(1 06 1

)A(6),

with

A(N) =( −2 1

N −N+22

)≡

( −1 00 −1

). (6.46)

The congruence in (6.46) follows from the Atkin-Lehner operator U(2), whichmust be used since 2|6. One has, from (6.33) and (6.35),

U(2) =(

1 00 1

)+

(2 10 2

)≡

(2 00 1

), (6.47)

and(

0 −12N 0

)=

(0 −1N 0

)(2 00 1

)≡ ±

(2 00 1

). (6.48)

It follows that (2 10 2

)(0 −1

2N 0

)≡ ∓

(2 10 2

). (6.49)

Cancelling and premultiplying by(

0 −1N 0

)gives (6.46), proving the theorem

for N = 6. The case N = 10 follows as well, as all matrices in a generating setcan be constructed from matrices already examined, and the case 22|N , finally,


gives a proof for N = 8, 12 and 16. Namely, writing

F (N) =(

0 −1N 0

), (6.50)

W (N) =(

1 0N 1

), (6.51)

B =(

2 10 2

), (6.52)

T =(

1 10 1

), (6.53)

one has

Γ0(10) = 〈T, W (10), (W (10)A(10))2,

F (10)(W (10)A(10))2F (10), A(10)−1W (10)−1A(10)T−1〉 (6.54)

and

Γ0(8) = 〈T, W (8), (W (10)A(10))2, B−1W (8)B〉 (6.55)

Γ0(12) = 〈T, W (12), BW (12)−1B,

F (12)B−1W (12)B−1F (12), BF (12)BW (12)−1BF (12)B〉 (6.56)

Γ0(16) = 〈T, W (16), BW (16)−1B, (BF (16))4, (B−1F (16))4〉. (6.57)

Clearly

B =(

2 10 2

)≡

( −1 00 −1

)(6.58)

when U(2) ≡ 0, i.e. when 4|N . This proves the theorem for N = 8, 10, 12 and16. The remaining cases, N = 11, 14, 15, 17 and 23, require more work. ForN = 11 and 17, the authors use some general properties of Hecke operators, butthe proof method is feasible only when a certain parameter n satisfies ϕ(n) = 2,so only n = 3, 4 and 6 can be used. This limits the applicability of the generalmethod to the aforementioned cases N = 11 and 17. The cases N = 14, 15 and23 are outright ad hoc. Finally, it is comparatively easy to verify that f mustvanish at the cusps of Γ0(N) in all cases considered.

6.5 Converse theorems assuming a partial Eulerproduct


The article [5] assumes that F (s) has a full Euler product:

F (s) =∏p

1Lp(p−s)

, (6.59)

where p runs over all primes. However, the proof for the cases N = 5, 6, 7, 8, 9,10, 12 and 16 focuses on the local factor at p = 2. It seems unlikely that any oneprime plays a special role, and indeed the information extracted from this local

6.5. Converse theorems assuming a partial Euler product 67

factor does not suffice to prove a converse theorem for all the groups Γ0(N).There are two obvious ways to proceed. One is to treat all factors symmetrically,hoping to obtain general results. Another is to relax the assumption of an Eulerproduct at all primes. The 2005 article “Converse theorems assuming a partialEuler product” by David Farmer and Kevin Wilson [15] takes the second route.More precisely, the authors assume that the Dirichlet series factors as

F (s) = (1− a22−s + 2k−1−2s)−1∑

2-n

an

ns. (6.60)

This is called a partial Euler product. Recall from 5.3.1 the connection betweenEuler products and Hecke operators. Under certain reasonable (but unproved)assumptions, the authors show that this implies that f is automorphic underthe group

Γ0,2(N) =(

a bc d

)∈ Γ0(N) : a ≡ 2j (mod N) for some integer j

.

(6.61)This is a smaller group than Γ0(N), but still larger than (i.e., containing) Γ1(N)(recall the definitions from page 15). This seems to be the strongest resultpossible under these assumptions.

6.5.2 The proof

The article [15] considers not only ordinary modular forms but the more generalforms with multiplicative (or Dirichlet) characters. Recall that a multiplicativecharacter χ for a group G is a homomorphism χ : G → (C\0, ·) from G to themultiplicative group of nonzero complex numbers; a Dirichlet character moduloN is a Dirichlet character from the cyclic group Z/NZ, extended periodicallyto Z. A modular form f for Γ0(N) of weight k and with character χ satisfies

f(aτ + b

cτ + d) = χ(d)(cτ + d)kf(τ) (6.62)

for every Mobius transformation τ 7→ aτ+bcτ+d in Γ0(N). Ordinary modular forms

correspond to the so-called trivial character χ ≡ 1. For these more generalforms, the Hecke operators become (cf. (6.32))

T (p) = χ(p)(

p 00 1

)+

p−1∑

i=0

(1 i0 p

), (6.63)

and the Atkin-Lehner operators are, as before,

U(p) =p−1∑

i=0

(p i0 p

). (6.64)

The auxilliary function becomes

G(s) =

(√N

2π

)s

Γ(s +k − 1

2)F (s), (6.65)


a full Euler product would be

F (s) =∏p

(1− app−s + χ(p)pk−1−2s)−1, (6.66)

and (6.26) is modified to

(f |kγ)(z) = χ(a)(det γ)k/2(cz + d)−kf(az + b

cz + d), (6.67)

after noting that N |c so that ad − bc = 1 becomes ad ≡ 1 (mod N) and soχ(d)−1 = χ(d−1) = χ(a). As before (cf. (6.23),(6.24)), the functional equation

G(s) = ±(−1)k/2G(k − s) (6.68)

becomes, after an inverse Mellin transform, f |kFN = ±f . The partial product(6.60) becomes T (2)f = a2f with χ(2) = 1. This is what necessitates theintroduction of the group (6.61), as will be described shortly. Recall equation(6.44) (from the article [5]), which says that

(2 1N N+1

2

)≡

(1 00 1

)(6.69)

in the group ring. Noting that(

2 −1−N N+1

2

)= W (N)−1

(2 1N N+1

2

)T−1, (6.70)

(2 1−N −N+1

2

)= W (N)−1

(2 1N N+1

2

), (6.71)

and (2 −1N −N+1

2

)=

(2 1N N+1

2

)T−1, (6.72)

it follows that (2 α

βN αβN+12

)≡

(1 00 1

)(6.73)

whenever |α| = |β| = 1. Next, one whishes to prove by induction that(

2n αβN ∗

)≡

(1 00 1

)(6.74)

for every integer n, where |α|, |β| ≤ 2n−1 and ∗ is such that the determinantequals 1. To achieve this, the authors are forced to make a pairing assuption,namely that if A + B ≡ C + D, then either A ≡ C and B ≡ D, or A ≡ D andB ≡ C for a certain set of matrices A, B, C and D. Under this assumption,the induction works, showing that all matrices of the form (6.74) are congruentto the identity (recall that this is another way of expressing the appropriatetransformation behaviour of f). Next, one wants to show that the matrices(6.74), together with the usual matrices T and W (N), generate the group (6.61).

To this end, the authors also have to assume that if (d, bN) = 1, there areintegers n and k such that 2nb ≡ 1 (mod d + kbN). Assuming this, take anelement

γ =(

a bc d

)∈ Γ0,2(N) (6.75)

6.5. Converse theorems assuming a partial Euler product 69

where b 6= 1, and take a matrix δ of the form (6.74), so that

δW (N)kγ =( ∗ 2nb + α(d + kbN)∗ ∗

). (6.76)

Now, det γ = 1 and N |c forces d and bN to be relatively prime, so by assumptionn and k can be choosen so that 2nb+α(d+kbN) = 1. Thus, in (6.75), one maytake b = 1. Then γ takes the form

γ =(

a 1ad− 1 d

), (6.77)

where a is congruent to 2j modulo N for some integer j, say a+kN = 2j . Then

γW (N)k =(

2j 1ad− 1 + kdN d

), (6.78)

so the matrices (6.74) and W (N) do indeed generate (6.61). The authors endthe article by commenting on the pairing assumption made. The assumptionis clearly not true for general matrices A, B, C and D. The authors note,however, that similar pairing results hold in the context of Weil’s theorem. Theother part of the proof, the existance of the integers n and k in (6.76), is notmentioned in the concluding remarks. It appears to be incorrect, as we describenext.

6.5.3 An error

The authors make an attempt to derive the existence of the integers n andk from Artin’s conjecture on primitive roots, which is unproved; they remarkthat the result needed for this theorem is much weaker, which is clearly thecase. However, to the present author the deduction from Artin’s conjectureappears incorrect altogether. Firstly, they assume that one can get b ≡ 2n

(mod d + kbN) rather than 2nb ≡ 1 (mod d + kbN), which seems to be what isrequired as one wants to have 2nb+α(d+kbN) = 1. We now need the definitionof a primitive root:

Definition An integer a is called a primitive root modulo a prime p if theresidue class of a modulo p is a generator of the multiplicative group((Z/pZ)∗, ·).

If a is a primitive root modulo p, then, for every b not divisible by p, thereis an integer n such that an ≡ b (mod p). In particular, taking a = 2, therewould be an integer n such that 2n ≡ b (mod p). Taking instead b−1 ( whichworks becauce p does not divide b), and assuming that p can be written p =d+kbN , one can obtain 2nb ≡ 1 (mod d+kbN), as needed. Apart from Artin’sconjecture, the authors seem to use Dirichlet’s theorem. We state both forconvenience.

Dirichlet’s theorem If α and β are relatively prime, positive integers, thearithmetic progression α + nβ, n ∈ N, contains infinitely many primes.In fact, for a fixed β, the primes are equidistributed among the possibleresidue classes modulo β, so that the primes of the form α + nβ havedensity 1/ϕ(β) among the primes, where ϕ is Euler’s totient function.


By assumption, (d, bN) = 1, so Dirichlet’s theorem applies, and there areinfinitely many primes of the form d + kbN .

Artin’s conjecture, weak version Any integer which is not a square and notequal to −1 is a primitive root modulo infinitely many primes.

Artin did in fact conjecture more, namely that the set of primes for whicha fixed a is a primitive root has positive density among the primes, and thatwhen a 6≡ 1 (mod 4), this density equals

∏p

(1− 1

p(p− 1)

)≈ 0.37395581 . . . , (6.79)

where p runs over the primes. In particular, 2 is a primitive root for approxi-mately 37 % of all primes. Now, the argument seems to be that one can finda k such that d + kbN is a prime for which 2 is a primitive root. Then theexistence of the number n follows immediately. Of course, there seems to be noreason why the sequence d + kbN should not contain such a prime, or indeed apositive fraction of all primes. This is an unnecessarily strong result, becauseone does not need d + kbN to be prime or 2 to be a primitive root. But itstill does not follow from Artin’s conjecture that it must happen: note that thedensity results allow the two sets of primes to avoid each other completely (onehas ϕ(d + kbN) ≥ 2 so 0.3739 . . . + 1/ϕ(d + kbN) < 1). We remark that Artin’sconjecture has still not been proved for any single number. However, there istheoretical support for it: see [21]. For an introductory survey, see [29].

6.6 A converse theorem for Γ0(13)


Recall from 6.4 that the article [5] proved converse theorems for 5 ≤ N ≤ 12,14 ≤ N ≤ 17, and N = 23. Thus N = 13 remained. This case was settled in2006, in the article “A converse theorem for Γ0(13)” by Conrey, Farmer, Odgers,and Snaith [6]. This article proves a converse theorem for Γ0(13) by assuming, inaddition to holomorphicity of f in H, convergence of the corresponding functionF in some right half-plane, and a functional equation

G(s) = ±G(k − s), (6.80)

that F is expressible as

F (s) =(

1− a2

2s+

2k−1

22s

)−1 (1− a3

3s+

3k−1

32s

)−1 ∑

(n,6)=1

an

ns, (6.81)

that is, that F has a partial Euler product (at 2 and 3). Γ0(13) is generated byfour known matrices, the first three of which are easily dealt with using the samemethods as in the previous articles. The fourth generator requires an analyticargument. No unproved assumptions are made in this article.

6.6. A converse theorem for Γ0(13) 71

6.6.2 The proof

The group Γ0(13) is generated by the matrices

T =(

1 10 1

), (6.82)

W (13) =(

1 013 1

), (6.83)

γ2 =(

2 −113 −6

), (6.84)

γ3 =(

3 −113 −4

), (6.85)

and as before, periodicity and the functional equation imply the correct trans-formation behaviour under the first two. The proof for γ2 uses the local factorsat p = 2 and p = 3. Namely, (6.81) gives

(2 00 1

)+

(1 00 2

)+

(1 10 2

)≡ 21−k/2a2

(1 10 1

)(6.86)

and(

3 00 1

)+

(1 00 3

)+

(1 10 3

)+

(1 20 3

)≡ 31−k/2a3

(1 10 1

), (6.87)

and pre- and postmultiplying by

F (13) =(

0 −113 0

), (6.88)

one finds(

1 00 2

)+

(2 00 1

)+

(2 0−13 1

)≡ 21−k/2a2

(1 10 1

)(6.89)

and(

1 00 3

)+

(3 00 1

)+

(3 0−13 1

)+

(3 0−26 1

)≡ 31−k/2a3

(1 10 1

).

(6.90)Subtract (6.86) from (6.89) to obtain

(2 0−13 1

)≡

(1 10 2

)(6.91)

and note that

γ2 = W (13)(

2 0−13 1

)(1 10 2

)−1

. (6.92)

This shows that γ2 ≡ I. Finally, the generator γ3 requires an analytic treatment.Assuming k > 0, one can show (after deriving appropriate identities satisfiedby γ3) that if f were not invariant under |γ3, the function f(τ)|(I − γ3) wouldbe invariant under a non-discrete subgroup of PSL(2,R). This does not proveit to be identically zero, but the authors succeed in describing all pathologieswhich can occur and are able to conclude that f(τ)|(I − γ3) must vanish, i.e.that γ3 ≡ I. This proves the converse theorem for Γ0(13).

Conclusions

This thesis has made a survey of some converse theorems for Dirichlet series,after introducing the necessary machinery for this. A converse theorem assertsthat a Dirichlet series having certain properties must be the L-series of a mod-ular form. The original work by Hecke [20] used only the functional equationsatisfied by the function defined by the Dirichlet series. Weil’s contribution [45]was to introduce Dirichlet characters, thereby generalizing Hecke’s work. Analternative approach was pioneered by Conrey and Farmer, who assumed thatthe Dirichlet series has an Euler product [5]. This led to a renewed interest inthe problem and some new results in 2005 [15] and 2006 [6]. However, thereseems to be no obvious way to generalize the proofs to congruence groups ofhigher level. Additionally, the article [15] is forced to make some unproved as-sumptions, which are not believed to be strictly necessary. The proofs use theassumptions that a certain Dirichlet series satisfies some growth conditions, afunctional equation and factors, at least partially, in an Euler product, to showthe proper transformation behaviour of the potential modular form under everytransformation in a generating set for the group in question. This obviouslyleads one to use as few generators as possible, which however makes it difficultto construct proofs that work for many groups simultaneously. Intuitively, min-imizing the generating set destroys the inherent symmetry, and one is forcedto make a detailed study of generating sets with few discernible regularities.It seems worthwhile, therefore, to explore more symmetric ways to generatethe groups, and develop methods that work for arbitrary places, rather thanlocalizing at specific primes.

The motivation for these problems are the conjectured properties of theSelberg class of Dirichlet series [7]. These, in turn, are believed to elucidate theRiemann hypothesis and its various generalizations. In a yet wider perspective,this is a part of the extensive Langlands program.

Karlsson, 2009. 73

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[2] A. O. L. Atkin, J. Lehner: Hecke operators on Γ0(m), Math. Annalen 185,134-160, 1970

[3] I. N. Cangul, D. Singerman: Normal subgroups of Hecke groups and regularmaps, Math. Proc. Camb. Phil. Soc. 123, 59-74, 1998

[4] C. Campbell-Grant: Hecke groups and their subgroups of low index, M.Phil. thesis, University of Southampton, 2007

[5] J. B. Conrey, D. W. Farmer: An extension of Hecke’s converse theorem,arXiv:math/9502209v1 [math.NT]

[6] J. B. Conrey, D. W. Farmer, B. E. Odgers, N. C. Snaith: A conversetheorem for Γ0(13), arXiv:math/0601549v1 [math.NT]

[7] J. B. Conrey, A. Ghosh: On the Selberg class of Dirichlet series: smallweights, arXiv:math/9204217v1 [math.NT]

[8] J. H. Conway, N. J. A Sloane: Sphere packings, lattices and groups, thirded., Grundlehren der Mathematichen Wissenschaften 290, Springer-Verlag,New York, 1999

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[10] H. S. M. Coxeter, W. O. J. Moser: Generators and Relations for DiscreteGroups, fourth ed., Ergebnisse der Mathematik und ihrer Grenzgebiete 14,Springer-Verlag, Berlin Heidelberg, 1984

[11] I. Dolgachev: Modular forms, online lecture notes, available athttp://www.math.lsa.umich.edu/~idolga/lecturenotes.html, last re-trieved 2009-06-21

[12] W. Eberling: Lattices and Codes, Vieweg & Sohn, Braun-schweig/Wiesbaden, 1994

[13] T. Ekedahl: One Semester of Elliptic Curves, EMS Series of Lectures inMathematics, 2006

[14] H. M. Farkas, I. Kra: Riemann Surfaces (2nd ed.), GTM 71, Springer-Verlag, New York, 1992

Karlsson, 2009. 75

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[15] D. W. Farmer, K. Wilson: Converse theorems assuming a partial Eulerproduct, arXiv:math/0408221v2 [math.NT]

[16] T. Gannon: Postcards form the edge, or Snapshots of the theory of gener-alized Moonshine, arXiv:math/0109067v1 [math.QA]

[17] S. Gelbart: An elementary introduction to the Langlands program, Bulletin(new series) of the AMS, vol. 10, number 2, 1984

[18] H. Hamburger: Uber die Riemannsche Funktionalgleichung der ζ-Funktion,Mathematische Zeitschrift 10, number 3-4, 240-254, 1921

[19] G. H. Hardy, M. Riesz: The General Theory of Dirichlet’s Series, Cam-bridge Tracts in Mathematics and Mathematical Physics, No. 18, Cam-bridge University Press, 1915

[20] E. Hecke: Uber die Bestimmung Dirichletscher Reihen durch ihre Funk-tionalgleichung, Math. Ann. 112, number 1, 664-699, 1936

[21] C. Hooley: On Artin’s conjecture, J. Reine Angew. Math. 225, 209-220,1967

[22] D. Husemoller: Elliptic Curves, Springer-Verlag, New York, 1987

[23] G. A. Jones, D. Singerman: Complex Functions, an algebraic and geometricviewpoint, Cambridge University Press, 1987

[24] A. W. Knapp: Elliptic Curves, Mathematical Notes 40, Princeton Univer-sity Press, Princeton, 1992

[25] E. Landau: Handbuch der Lehre von der Verteilung der Primzahlen, B.G.Teubner Verlag, Leipzig und Berlin 1909

[26] S. Levy (ed.): The Eightfold Way: The Beauty of Klein’s Quartic Curve,Cambridge University Press, Cambridge, 1999

[27] W. Magnus: Noneuclidean tesselations and their groups, Academic Press,New York and Londeon, 1974

[28] H. McKean, V. Moll: Elliptic Curves - Function theory, geometry, arith-metic, Cambridge University Press, 1999

[29] P. Moree: Artin’s primitive root conjecture - a survey - ,arXiv:math/0412262v1 [math.NT]

[30] J. R. Munkres: Topology (2nd ed.), Prentice Hall, Upper Saddle River,New Jersey, 2000

[31] D. Mumford: Tata Lectures on Theta I, Progress in Mathematics Vol. 28,Birkhauser, Boston, 1983

[32] D. Mumford, C. Series, D. Wright: Indra’s Pearls - The Vision of FelixKlein, Cambridge University Press, 2002

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[34] P. Sarnak: Some applications of modular forms, Cambridge Universitypress, 1990

[35] A. Selberg: Old and new conjectures and results about a class of Dirichletseries, Collected papers, volume 2, Springer-Verlag, Berlin Heidelberg NewYork, 1991

[36] J-P. Serre: A Course in Arithmetic, GTM 7, Springer-Verlag, New York1973, originally published as Cours d’Arithmetique by Presses Universi-taires de France, Paris, 1970

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[46] Wikipedia: pictures of a torus and a tesselation induced by the modulargroup, available as

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respectively.

78 Bibliography

Appendix A

Bernoulli numbers

The Bernoulli numbers, used in section 5.2.1, have many applications in numbertheory. Unfortunately, several notational conventions exist. The one used hereis the following: the Bernoulli numbers Bn are defined by

z

ez − 1=

∞∑n=0

Bn

n!zn. (A.1)

Thus, noting thatz

ez − 1+

z

2=

z

2coth

z

2(A.2)

is an even function, it follows that B1 = − 12 , and that all other Bernoulli

numbers with odd index are zero. One sometimes omits those numbers, so thatall numbers are nonzero. Furthermore, as defined here, the Bernoulli numbersalternate in sign, which is sometimes hidden by using another definition. Next,we derive the formula (5.10) used in section 5.2.1. Note that

πz cot πz = πizeπiz + e−πiz

eπiz − e−πiz= πiz

(1 +

2e2πiz − 1

). (A.3)

Using (A.1), one finds

πz cot πz =∞∑

n=0

B2n

(2n)!(2πiz)2n. (A.4)

On the other hand, the product expression for the sine function [39],

sinπz = πz

∞∏n=1

(1− z2

n2

)(A.5)

gives, after a logarithmic differentiation,

πz cot πz = 1− 2∞∑

n=1

ζ(2n)z2n, (A.6)

whence

ζ(2n) = − (2πi)2n

2(2n)!B2n, (A.7)

Karlsson, 2009. 79

80 Appendix A. Bernoulli numbers

which is equation (5.10). Evidently, all Bn are rational numbers. The first fewnonzero Bernoulli numbers are:

B1 = −12,

B2 =16,

B4 = − 130

,

B6 =142

,

B8 = − 130

,

B10 =566

,

B12 = − 6912730

.

LINKÖPING UNIVERSITY

ELECTRONIC PRESS

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Karlsson, 2009. 81

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