Modular forms for triangle groups

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Modular forms for triangle groups

Modulära former för triangelgrupper

Elisabet Edvardsson

Faculty of Health, Science and Technology Mathematics, Bachelor Degree Project 15 ECTS

Supervisor: Håkan Granath Examiner: Niclas Bernhoff February 2017

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Abstract

Modular forms are important in different areas of mathematics and theoretical physics. The theory is well known for the modular group PSL(2, Z), but is also of interest for other Fuchsian groups. In this thesis we will be interested in triangle groups with a cusp. We review some theory about mapping of hyperbolic triangles in order to derive an expression for the Hauptmodul of a triangle group, and use this to write a SageMath-program that calculates the Fourier series of the Hauptmodul. We then review some of the results presented in [4] that describe generalizations of well known concepts such as the Eisenstein series, the Serre derivative and some general results about the algebra of modular forms for triangle groups with a cusp. We correct some of the mistakes made in [4] and prove some further properties of the generators of the algebra of modular forms in the case of Hecke groups. Then we use the results from [4] to write a SageMath-program that calculates the Fourier series of the generators of the algebra of modular forms for triangle groups with a cusp and that also finds the relations between the generators in the special case of Hecke groups. Using the results from this program, we present some conjectures concerning the generators of the algebra of modular forms for a Hecke group, which, if proven to be true, give us a generalization of some of the Ramanujan equations. We conclude by explicitly calculating the generalized Ramanujan equations for the first few Hecke groups.

Sammanfattning

Modulära former är viktiga inom olika delar av matematik och teoretisk fysik. Teorin är välkänd för den modulära gruppen PSL(2, Z), men är ocks˚a intressant för andra Fuchsgrupper.

Den här uppsatsen inriktar sig p˚a triangelgrupper med en spets. Vi ˚aterger hur avbildningar av hyperboliska trianglar kan beskrivas i syfte att hitta ett uttryck för en Hauptmodul för en triangelgrupp, och använder detta för att skriva ett program i SageMath som beräknar Hauptmodulens Fourierserie. N˚agra av resultaten fr˚an [4], som beskriver generaliseringar av välkända begrepp som Eisensteinserierna och Serrederivatan, samt n˚agra allmänna resultat för algebran av modulära former, ˚aterges. Vi rättar n˚agra av felen som gjorts i [4] och visar att generatorerna till algebran av modulära former har vissa egenskaper i fallet med Heckegrupper.

Resultaten fr˚an [4] används för att skriva ett program i SageMath som beräknar Fourierserierna för generatorerna till algebran av modulära former för en triangelgrupp med en spets samt hittar relationerna mellan generatorerna i fallet med Heckegrupper. Utifr˚an resultaten som f˚as fr˚an programmet, kommer vi fram till och presenterar n˚agra förmodanden ang˚aende generatorerna till algebran av modulära former för en Heckegrupp. Om dessa förmodanden visar sig vara sanna ger de oss en generalisering av n˚agra av Ramanujanekvationerna. Vi avslutar genom att explicit beräkna de generaliserade Ramanujanekvationerna för de första Heckegrupperna.

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1 Introduction

The theory of modular forms for the modular group PSL(2, Z) is well known and used in both mathematics and physics. Somewhat loosely a modular form is a complex function on the upper half plane that transforms in a certain way under M¨obius transformations. More precisely, a modular form f (τ ) of weight k of some group Γ ⊂ SL(2, R) satisfies

f aτ + b cτ + d

= (cτ + d)^kf (τ ), for a b c d

∈ Γ. (1.1)

If k = 0, we call this a modular function. The main importance of modular forms is in number theory. For example, they were integral in the proof of Fermat’s last theorem by Andrew Wiles and Richard Taylor who proved the Taniyama-Shimura conjecture [5, p. 130]. Also, modular forms have Fourier expansions with coefficients that have interesting arithmetic properties, and thus can be used to prove relations that are otherwise very difficult to prove or even realize exist. The applications are not limited to number theory. There is a surprising connection, called Monstrous moonshine [5], between modular forms and the representation theory of the monster group, which is the largest of the sporadic simple groups. This connection is related to a certain vertex operator algebra, which is the structure underlying conformal field theory in physics. Further applications in physics are mostly related to string theory.

In this thesis we will be interested in the theory of modular forms for discrete subgroups of PSL(2, R), also called Fuchsian groups. More specifically we will be interested in modular forms for hyperbolic triangle groups, which are groups that describe how the complex upper half plane is tessellated by hyperbolic triangles.

We review some of the work done in [4], an article that describes the theory of modular forms in the case of triangle groups. We will correct several of the errors made in this article, simplify some of the results, and use their results in order to generalize some of the well known theory of modular forms for the modular group PSL(2, Z). In addition, we will write a program in SageMath [12] that calculates Fourier series for the Hauptmodul and the generators for the algebra of modular forms, finds the relations between these generators, and finds differential equations corresponding to the Ramanujan equations for Hecke groups.

The outline of the thesis is as follows. In Section 2, we will review some of the standard theory of modular forms and in particular describe some important results for the modular forms of the modular group, PSL(2, Z). This section is meant to provide a context for the rest of the material in the thesis, and thus explanations will be brief, and no proofs will be given. In Section 3 we go on to introduce triangle groups and show how a Hauptmodul, i.e. a function that generates all modular functions, can be calculated for such groups by studying mapping properties of hyperbolic triangles. We also correct some mistakes made in [4] concerning the fundamental domain of a triangle group. We then show how we can calculate the Fourier series for such Hauptmoduls using SageMath. In Section 4 a closer study of the algebra of modular forms for a triangle group is made. We review some of the results presented in [4] and use these results to write a program in SageMath that calculates the relations between the generators of the algebra of modular forms and thus determines the structure of this algebra. We also prove some further results and formulate some conjectures concerning the generators in the case of Hecke groups. In Section 5 we will describe

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and correct the definitions and results from [4] concerning the generalization of the Eisenstein series and the Serre derivative. We will then use these results together with the results and conjectures about the generators of the algebra of modular forms from Section 4 to suggest a way to generalize some of the Ramanujan equations in the case of Hecke groups. We also write a SageMath program that finds these differential equations.

2 Introduction to modular forms

In this section a review of the basic theory of modular forms will be presented. The contents are meant as a background for the theory that is presented in later sections, and thus explanations will be kept brief and most proofs will be left out. The interested reader is referred to [1] and [8] for details.

2.1 The upper half plane and M¨obius transformations

We denote by H the complex upper half plane H = {z ∈ C | Im(z) > 0}. The special linear group, SL(2, R) acts on H by M¨obius transformations, namely, for γ ∈ SL(2, R)

γ =a b c d

: H → H, z 7→ γz = az + b

cz + d. (2.1)

That this is a group action can be seen by the following considerations:

For x ∈ H and γ ∈ SL(2, R), we have that Im(γz) = Im az + b

cz + d

= Im [a(i Imz + Rez) + b] [c(i Imz − Rez) + d])

|cz + d|²

= Im(z)

|cz + d|²,

(2.2)

since (ad − bc) = 1, and thus γz is in H if z is.

Furthermore, if we denote the 2 × 2 identity matrix by I2×2, we have that I2×2z = z + 0

0 + 1 = z. (2.3)

Also, if γ =a b c d

, and γ⁰ =a⁰ b⁰ c⁰ d⁰

, we have that

γ(γ⁰z) =a b c d

a⁰z + b⁰ c⁰z + d⁰ =

aa⁰z + b⁰ c⁰z + d⁰ + b ca⁰z + b⁰

c⁰z + d⁰ + d

= (aa⁰+ bc⁰)z + ab⁰+ bd⁰ (ca⁰+ dc⁰)z + cb⁰+ dd⁰

=aa⁰+ bc⁰ ab⁰+ bd⁰ ca⁰+ dc⁰ cb⁰+ dd⁰

z = (γγ⁰)z.

(2.4)

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Thus the action of SL(2, R) satisfies the properties of a group action.

We extend the action to the extended upper half-plane¹, ¯H = H ∪ R ∪ {i∞}, by noting that all of the arguments above hold also on the boundary ∂H = R ∪ {i∞} if one defines the special cases

γ −d c

= i∞, (2.5)

and

γ(i∞) = a

c. (2.6)

One classifies M¨obius transformations in the following way: An element γ ∈ SL(2, R) is called [8, p. 7]

1. parabolic if (trγ)² = 4, 2. elliptic if (trγ)² < 4, 3. hyperbolic if (trγ)² > 4, where trγ denotes the trace of γ.

We have the following result [8, p. 7]:

Proposition 2.1. Let γ ∈ SL(2, R). Then γ is

1. elliptic if and only if γ has exactly one fixed-point in H.

2. parabolic if and only if γ has a unique fixed point on the boundary ∂H = R ∪ {i∞}

3. hyperbolic if and only if γ has two distinct fixed points on ∂H.

2.2 Modular functions and modular forms

Consider an element γ ∈ SL(2, R). We have that (−γ)(z) = −az − b

−cz − d = az + b

cz + d = γz. (2.7)

This means that instead of looking at the whole group SL(2, R), one can study the group PSL(2, R) ∼= SL(2, R)/ {I, −I}, see [5, p. 107]. In fact, if we consider the map φ : SL(2, R) → Aut(H), we note that this map has the kernel ker(φ) = {I, −I}, so PSL(2, Z) acts faithfully on H.

We will be interested in discrete subgroups of PSL(2, R). These groups are called Fuchsian groups, and for such a group Γ, we are interested in functions that have certain invariance properties under the action of the elements γ ∈ Γ.

First we recall some properties of complex functions. A function f : C → C is called holomorphic if it is complex differentiable at every point z ∈ C, and a function f : C → C is called meromorphic if it is complex differentiable at every point z ∈ C except for a finite number of points, where f has a pole, see [16, 17].

1Note that we use the symbol i∞ instead of ∞. The reason for this will be clear later.

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We now define the following types of functions that satisfy different properties under the action of a Fuchsian group:

Definition 2.1. [1, sec. 1] Let Γ be a Fuchsian group. A modular function of Γ is a function f that is meromorphic in H and which for each γ ∈ Γ and τ ∈ H satisfies f (γτ ) = f (τ ).

Definition 2.2. [1, sec. 1] Let Γ be a discrete subgroup of SL(2, R). A weakly holomorphic modular form of weight k ∈ Z of Γ is a function f that is holomorphic on H and that for each γ =a b

c d

∈ Γ satisfies

f (γτ ) = (cτ + d)^kf (τ ) (2.8)

We note that if the group Γ contains the element −I, all modular forms must be of even weight.

This is clear from equation (2.8), since f (τ ) = f (−Iτ ) = (−1)^kf (τ ). For simplicity, we will from now on restrict ourselves to consider only modular form of even weight. With this restriction we can just as well talk about modular forms for Fuchsian groups Γ ⊂ PSL(2, R).

It is clear that the set of modular forms of weight 2k forms a vector space, m2k(Γ). Now suppose we have a modular form g_2k(τ ) of weight 2k and another modular form, g_2l(τ ) of weight 2l. Then the product g_2kg_2l is a modular form of weight 2(k + l), which can be seen by the following calculation:

g_2k(γτ )g_2l(γτ ) = (c + dτ )^2k(c + dτ )^2lg_2k(τ )g_2l(τ ) = (c + dτ )^2(k+l)g_2k(τ )g_2l(τ ). (2.9) The set of all modular forms of a Fuchsian group therefore forms a graded algebra [1, sec. 1], which means it can be written as

m(Γ) =M

k

m_2k(Γ). (2.10)

We will also be interested in the modular properties of the derivative of a modular form, that is in finding the expression for df

dτ

aτ + b cτ + d

. We have that d (f (γτ ))

dτ = df

dτ(γτ )dγ

dτ = 1

(cτ + d)² df

dτ(γτ ) (2.11)

Differentiating equation (2.8), we get that d (f (γτ ))

dτ = ck(cτ + d)^k−1f (τ ) + (cτ + d)^kdf (τ )

dτ , (2.12)

which means that

df

dτ(γτ ) = ck(cτ + d)^k+1f (τ ) + (cτ + d)^k+2df (τ )

dτ , (2.13)

and we see that the derivative of a modular form is not necessarily a modular form, and thus the algebra of modular forms is not closed under differentiation. To create a set that is, which includes the modular forms, we make the following definition [1, p. 58]:

Definition 2.3. Let Γ be a Fuchsian group. A quasi-modular form of weight k and depth p of the group Γ is a function f (τ ) which satisfies that f (γτ )(cτ + d)^−k is a polynomial of degree p in _{cτ +d}^c , i.e. we have that

f (γτ ) = (cτ + d)^k

p

X

r=0

fr(τ )

c

cτ + d

r

, (2.14)

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where f_r(τ ) are holomorphic functions of τ that are independent of γ and f_p6= 0.

We see that a modular form, f (τ ), is a special case of a quasi-modular form where p = 0 and f0(τ ) = f (τ ). Also, from equation (2.13), we see that the derivative of a modular form of weight k is a quasi-modular form of weight k + 2 and depth 1.

If the Fuchsian group contains an element1 h 0 1

with h 6= 0, we get that f (τ + h) = f (τ ), which means that the function is periodic. Because of this, a modular form of this group will have a Fourier expansion of the form

f (τ ) =

∞

X

n=−∞

anqⁿ, (2.15)

where q = e^{2πiτ /h}. More generally, around a cusp, say z, such an expansion will have the form [4]

f (τ ) = j_z(k; τ )

∞

X

n=−m

a_nq_zⁿ, (2.16)

where the finite lower bound on n comes from requiring meromorphicity at the cusp and j_z(k; τ ) =

(1 if z = i∞,

τ_z^k if z ∈ R. (2.17)

Furthermore one lets q_z= e^{2πiτ /h}^z, where h_z is a parameter depending on z to be defined below in Section 3. We will be mostly interested in expansions around z = i∞, which means that we will study Fourier expansions of the form

f (τ ) =

∞

X

n=−m

a_nqⁿ_z. (2.18)

It turns out that the modular functions form a field and when the geometric genus of the Riemann surface H/Γ is zero, this field can be expressed as the rational functions C(f ) of some generator f . Such a generator is called a Hauptmodul. The Hauptmodules are mapped to one another by M¨obius transformations of elements in SL(2, C) and a Hauptmodul is therefore determined by three complex parameters, see [4], which e.g. can be the values of the function at three given points.

2.3 The modular group

The most studied Fuchsian group is PSL(2, Z) ∼= SL(2, Z)/ {I, −I}, the modular group, and to put our later discussions into context, we will now give a short review of the classical theory of modular forms for this group and point out the most important results. For simplicity we will continue to denote elements of PSL(2, Z) as matrices just as for SL(2, Z), even though, to be precise, one should denote the elements as cosets.

It is well known that PSL(2, Z) is generated by the matrices [1, p. 6]

S =0 −1

1 0

and T =1 1 0 1

. (2.19)

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Figure 1: A fundamental domain for the action of PSL(2, Z) on H. Reprinted from [18] under the Creative Commons Attribution/Share-Alike License

We see that up to a sign (remember that we are in PSL(2, Z))

S² = I and (ST )³= I. (2.20)

Letting g1 = S and g2 = ST , the group has the following presentation in terms of generators and relations [5, p. 127]:

g₁, g₂ | g₁² = g³₂ = 1 , (2.21) so it is isomorphic to the free product Z2∗ Z3.

Letting the matrices S and T act on τ ∈ H, cf. eq. (2.1), one gets that S(τ ) = −1

τ, (2.22)

and

T (τ ) = τ + 1. (2.23)

A common choice of fundamental domain for the action of PSL(2, Z) on H is the set

τ ∈ H | |Re τ | < 1

2, |τ | > 1

. (2.24)

This fundamental domain is illustrated in Figure 1.

Important modular forms for the modular group are the Eisenstein series. The classical Eisenstein series are given by [5, p. 2]

G2k(τ ) = X

m,n∈Z (m,n)6=(0,0)

1

(mτ + n)^2k, (2.25)

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for k ≥ 2, where k is an integer. The classical Eisenstein series are commonly normalized by the Riemann zeta function [5, p. 127]:

E_2k(τ ) = 1

2ζ(2k)G_2k(τ ). (2.26)

Remembering the Bernoulli numbers, Bn, defined by a generating function in the following way [14]

x e^x− 1 =

∞

X

n=0

Bnxⁿ

n! , (2.27)

and the divisor functions

σm(n) =X

d|n

d^m, (2.28)

for m ∈ N, it can be shown that for k ∈ Z and k ≥ 2 [5, p. 127]

E_2k(τ ) = 1 − 4k B_2k

∞

X

n=1

σ_2k−1(n)qⁿ, (2.29)

where q = e^2πiτ. The divisor function is important in number theory, which makes this result important since, as we will see below, it can be used to find properties of the divisor function which are otherwise hard to both prove and to find.

The first few Eisenstein series are given by [1, p. 17]

E4(τ ) = 1 + 240q + 2160q²+ · · · = 1 + 240(q + 9q²+ . . . ), (2.30) E₆(τ ) = 1 − 504q − 16632q²− · · · = 1 − 504(q + 33q²+ . . . ), (2.31) E8(τ ) = 1 + 480q + 61920q²+ · · · = 1 + 480(q + 129q²+ . . . ). (2.32) One can see from equation (2.25), that if we were to let k = 1, the series would not converge.

However, it is still possible to extend the definition of the Eisenstein series to include also the case k = 1, by simply letting k = 1 in equation (2.29), i.e.

E2(τ ) = 1 − 4 B2

∞

X

n=1

σ1(n)qⁿ

= 1 − 24q − 72q²− . . . .

(2.33)

In this case, however, the series is no longer a modular form. In [1, p. 19] it is proven that E₂ aτ + b

cτ + d

= (cτ + d)²E₂(τ ) − 6ic

π (cτ + d), (2.34)

which means that E₂ in fact is a quasi-modular form of depth 1 and weight 2.

The Eisenstein series are important since E4 and E6 generates the algebra of modular forms, m(PSL(2, Z)) = C [E4, E6], so knowing the Eisenstein series allows us to generate all other modular

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forms. Furthermore, the algebra of quasi-modular forms is generated by E₂, E₄and E₆and is given by C [E2, E₄, E₆], see [1, p. 15, 48].

As we have seen earlier, the algebra of modular forms is not closed under differentiation, since one gets a quasi-modular form after differentiation. Since the algebra of quasi-modular forms is generated by E2, E4 and E6, we see that if we introduce a new differential operator that somehow depends on E2 one could get that the algebra of modular forms is closed under this action. Indeed, the Serre derivative, D_2k : m_2k → m_2k+2, acting on a modular form of weight 2k, is defined by [1, p. 48]

D2k = 1 2πi

d dτ −2k

12E2 = q d dq −2k

12E2. (2.35)

That the action of D_2k on a modular form of weight 2k gives us a new modular form can be seen by comparing this expression with the one for E₂ and equation (2.13). As we will see below in Section 5, one can show that the Serre derivative satisfies the product rule

D_2k+2l(f g) = D_2k(f )g + f D_2l(g), (2.36) where f and g are modular forms of weights 2k and 2l respectively.

The space of modular forms of weight 2k is finite dimensional, which means that we can find relations between different modular forms by comparing a finite number of coefficients in their Fourier expansions. For example, from equations (2.30) and (2.32), we see that the constant term in the Fourier series of both E₄ and E₈ is 1. This means that E₄², which is of weight 8, also has constant term 1. As we will see below, it can be shown that the dimension of m8(PSL(2, Z)) is 1.

From this one immediately gets that

E₈ = E₄². (2.37)

Combining this with equation (2.29), one gets that [1, p. 18]

n−1

X

m=1

σ₃(m)σ₃(n − m) = σ₇(n) − σ₃(n)

120 . (2.38)

For other relations between the Eisenstein series, one obtains other relations between the σi(n), and as mentioned earlier, these relations are hard to prove otherwise, so results like these are important for number theoretical purposes.

In a similar fashion differential equations for the modular forms can be derived. Since E2, E4

and E₆ generate the ring of quasi-modular forms, it is enough to determine the derivative of these functions in order to get enough information to find the derivative of all modular forms using the product rule for the Serre derivative in equation (2.36). One gets the following three differential equations by looking at the first few terms of the Fourier expansions of the products of the relevant Eisenstein series [1, p. 49]:

ϑE2= E₂²− E₄

12 , (2.39)

ϑE₄= E2E4− E₆

3 , (2.40)

ϑE6 = E2E6− E₄²

2 , (2.41)

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where we have defined ϑ = 1 2πi

d dτ = q d

dq. These are called the Ramanujan equations.

As mentioned earlier, the Hauptmodul of a group is an important function, and for PSL(2, Z) a Hauptmodul can be expressed using the Eisenstein series, namely

j(τ ) = 1728E₄(τ )³

E4(τ )³− E₆(τ )² = q⁻¹+ 744 + 196884q + . . . (2.42) This is the standard choice for a Hauptmodul, and it is called the j-function [5, p. 3].

In other situations it can be preferable to choose another Hauptmodul, in particular when one studies the so called Monstrous moonshine [5, ch. 0] one is more interested in

J (τ ) = j(τ ) − 744. (2.43)

Monstrous moonshine is the connection between the coefficients of J (τ ) and the dimensions of the smallest irreducible representations of the Monster finite simple group, denoted by M, which is the largest of the exceptional finite simple groups. The dimensions of the first few irreducible representations of M are 1, 196883, 21296876 and 842609326, and we get the following equalities

196884 = 196883 + 1, (2.44)

21493760 = 21296876 + 196883 + 1, (2.45)

864299970 = 842609326 + 21296876 + 2 · 196883 + 2 · 1, (2.46) where we recognize the left hand sides as the first coefficients in J (τ ). This is a very interesting result since it connects very different areas of mathematics. In particular it is explained by studying a certain vertex operator algebra, which is the structure underlying conformal field theory, and thus also connects this to theoretical physics.

We have now reviewed some of the basics of the modular group, and will now turn to other Fuchsian groups. The groups we will be interested in are triangle groups, which can be seen as a natural generalization of the modular group.

3 Mapping of hyperbolic triangles

The aim of this section is to introduce a special type of Fuchsian groups called triangle groups and to study some of the basic modular properties of these. This will serve as a base for the rest of the text.

3.1 Definition of triangle groups

Consider a triangle in Euclidean geometry with angles π/k, π/l and π/m, with k, l, m ∈ Z⁺. We have that

1 k +1

l + 1

m = 1. (3.1)

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It is easy to check that there are only a few integer solutions to this equation, namely (2, 3, 6), (2, 4, 4) and (3, 3, 3), up to permutation. These are precisely those triangles that can tessellate the Euclidean plane by successive reflections in the sides of the triangles. This is illustrated in Figure 2.

(a) (2, 3, 6) (b) (3, 3, 3) (c) (2, 4, 4)

Figure 2: Illustration of the tilings obtainable from the three Euclidean triangle groups. Reprinted from [19]

under the Creative Commons Attribution/Share-Alike License

We will, however, not be interested in Euclidean triangles. Instead we will focus on triangles in hyperbolic geometry, so called hyperbolic triangles. We will denote such a triangle with angles π/k, π/l and π/m by ∆(k, l, m). The angles of a hyperbolic triangle satisfy

1 k +1

l + 1

m < 1, (3.2)

and there are infinitely many such combinations of k, l, m ∈ Z⁺. Consider the hyperbolic triangle

∆(k, l, m) and denote the length of the side opposite to the angle π/k by a. Then we have from [6, p. 16] that

cosh(a) = cos(π/l)cos(π/m) + cos(π/k)

sin(π/l)sin(π/m) , (3.3)

which means that the lengths of the sides in a hyperbolic triangle are determined by the angles of the triangle. This in turn implies that all hyperbolic triangles with the same angles are congruent.

Also the area of this triangle is determined by the angles and is given by [5, p. 107]

π

1 − 1

k −1 l − 1

m

. (3.4)

As in the Euclidean case, successive reflections in the sides of a triangle ∆(k, l, m) give a tessellation of the hyperbolic plane. This is shown in [9, ch. VI, sec. 5]. As is discussed in [5, p. 107], the group of isometries of H, Isom(H), consists of all M¨obius transformations in PSL(2, R) together with the transformation σ(τ ) = −¯τ , i.e.

Isom(H) ∼= PSL(2, R) ∪ σPSL(2, R). (3.5)

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This group contains all isometries, including reflections, which means that the elements are not necessarily orientation preserving. It can be shown that the group of orientation preserving isometries of H, i.e. Aut(H) is given by

Aut(H) ∼= PSL(2, R). (3.6)

Now consider the hyperbolic triangle ∆(k, l, m). As we mentioned, H can be tessellated by suc- cessively reflecting this triangle in its sides. This can be described by the discrete subgroup G_(k,l,m)⊂ Isom(H) generated by the three reflections in the sides of the triangle. The subgroup of G_(k,l,m) that only contains orientation preserving elements is given by

Γ_(k,l,m) = G_(k,l,m)∩ PSL(2, R). (3.7)

This group is called the triangle group of the triangle ∆(k, l, m). A fundamental domain of this group is given by the triangle ∆(k, l, m) together with the triangle ∆⁰(k, m, l) obtained by reflecting

∆(k, l, m) in one of its sides. We note that the triangle group of ∆(k, l, m) is the same as the triangle group of ∆⁰(k, m, l). This together with the fact that any hyperbolic triangle with angles π/k, π/l and π/m can be mapped to either ∆(k, l, m) of ∆⁰(k, m, l) using an element of PSL(2, R) means that a triangle group is determined up to conjugation by an element of PSL(2, R).

A triangle group Γ_(k,l,m) has an abstract representation D

g1, g2| g^k₁ = g₂^l = (g1g2)^m= 1 E

. (3.8)

Hyperbolic triangles can have one or more cusps, i.e. one or more of k, l or m can be infinity. In these cases we simply remove the relation containing the infinite parameter, e.g.

Γ_(k,l,∞)=D

g₁, g₂| g₁^k= g^l₂= 1E

. (3.9)

Another common presentation of a triangle group, Γ_(k,l,m), using three generators γ1, γ2 and γ3, is the following [7]:

D

γ₁, γ₂, γ₃ | γ^k₁ = γ₂^l = γ₃^m= γ₁γ₂γ₃ = 1E

. (3.10)

That these representations describe the same group can be seen by noting that

γ₃ = (γ₁γ₂)⁻¹, (3.11)

which means that

1 = γ₃^m= (γ₁γ₂)^−m, (3.12)

and therefore

(γ1γ2)^m= 1, (3.13)

and thus if we let γ₁ = g₁ and γ₂= g₂ we see that the representations describe the same group.

The reason for using the second representation involving three generators is that it gives a more intuitive description of how the triangle group acts on a triangle. The action of the group elements γ_i on a hyperbolic triangle ∆(k, l, m) is illustrated in Figure 3. Here we see that each of the elements γ_i represents a clockwise rotation of one of the white triangles into one of the other white triangles, yielding the relation γ1γ2γ3 = 1.

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Figure 3: Illustration of the action of the generators γ1, γ2 and γ3 on a hyperbolic triangle.

It is clear from the symmetry between k, l, m in the presentation of a triangle group and the fact that a triangle group is uniquely determined up to conjugation by an element of PSL(2, R) that we can restrict ourselves, without loss of generality, to triangle groups Γ_(m₁_,m₂_,m₃₎ for which 2 ≤ m₁ ≤ m₂≤ m₃ ≤ ∞.

In the rest of the text we will be interested in triangle groups for which m3 = ∞, so all groups will be determined by the two parameters m₁ and m₂. A special case that will be of interest is when m1 = 2, in which case the triangle group is called a Hecke group.

3.2 Hauptmoduls for triangle groups with a cusp

In this section we will show how to obtain a Hauptmodul, i.e. a generator for the field of modular functions, for a triangle group Γt with t = (m1, m2, ∞).

That there exists a Hauptmodul for the triangle group Γ_t, where t = (m₁, m₂, m₃) is proven in [9, ch. VI, sec. 5], and we will denote such a Hauptmodul for the triangle group by Jt. In this section we will determine a Hauptmodul for a triangle group Γ_(m₁_,m₂_,∞). Before we do this, we introduce some notation and describe the results obtained in [4].

A point on the boundary ∂H is called a cusp of Γtif it is fixed by some parabolic element in Γt. We now let HΓ denote the upper half plane together with the cusps of Γ. An important property of a cusp z is the cusp width, which we denote by hz. For i∞ the cusp-width h∞ is the smallest h > 0 such that1 h

0 1

∈ Γ_t. For x ∈ R being a cusp, the cusp-width hx is the smallest h > 0 such that

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0 −1 1 −x

−1

1 h 0 1

0 −1 1 −x

∈ Γ_t. Using the cusp-width, we can now define local coordinates around a cusp z by qz = e^2πiτ^z^/h^z, where τz ∈ HΓ is given by

τz =







−1

τ − z if z ∈ R, τ if z = i∞.

(3.14)

As we have seen, a triangle group Γt is unique up to conjugation with an element of PSL(2, R) and is thus determined by three real parameters. We fix the triangle group by fixing its fundamental domain. In [4] they do this by choosing the locations of the corners of one of the triangles to be

ζ1 = −e^−πi/m¹, ζ2= e^πi/m², ζ3= i∞, (3.15) and then letting the fundamental domain be the union of this triangle and its image under τ 7→ τ +^h₂³ (where h₃ is to be defined below). From the definition of triangle groups and the description of how to obtain a fundamental domain of a triangle group given in Section 3.1, it is clear that the way to construct a fundamental domain described in [4] must be wrong. Instead of translating the triangle by ^h₂³, we want to reflect it in the line Re τ = Re ζ₂. This is illustrated in Figure 4. However, as we will see, the choice of ζ₁ and ζ₂ that is made in [4], is actually inconsistent with the results they present in their first theorem. The correct choice is

ζ1 = −e^−πi/m¹ + cos(π/m1), ζ2= e^πi/m² + cos(π/m1), ζ3 = i∞. (3.16) This triangle, together with its mirror image when mirrored in the line Re τ = Re ζ₂ forms a fundamental domain for the triangle group, which is illustrated in Figure 5.

In Figure 5 we see that the width of the fundamental domain of the triangle group Γ_(m₁_,m₂_,∞), which also is the cusp width of i∞, is h₃ = 2cos(π/m₁) + 2cos(π/m₂). In [4] they argue that the group is generated by

γ₁⁰ =2cos(π/m₁) 1

−1 0

, γ₂⁰ =

0 1

−1 2cos(π/m₂)

, γ₃⁰ =1 h₃

0 1

, (3.17)

which is true for the triangle group corresponding to the triangle in Figure 4. To get the triangle group for the triangle in Figure 5, we must conjugate by the element

γ₀ =1 cos(π/m1)

0 1

. (3.18)

So the desired group is generated by [4]

γ1 = γ0

2cos(π/m₁) 1

−1 0

γ₀⁻¹, γ2= γ0

0 1

−1 2cos(π/m₂)

γ₀⁻¹, γ₃ = γ₀1 h₃

0 1

γ₀⁻¹ =1 h₃

0 1

,

(3.19)

which satisfy

γ1γ2γ3 = γ₁^m¹ = γ₂^m² = I2×2, (3.20)

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Figure 4: Illustration of the fundamental domain used in [4] for the triangle group Γ_(m₁_,m₂_,∞).

and we remember that we are in PSL(2, R). Using the definition of the local coordinates, we see that the local coordinates around the point ζ₃ are given by

q3 = e^{2πiτ /h}³. (3.21)

A Hauptmodul for a triangle group Γtis determined by three independent complex parameters and in [4] they fix it by requiring that

Jt(ζ1) = 1, Jt(ζ2) = 0, Jt(ζ3) = ∞. (3.22) As we will see, this choice is what makes the definition of ζi in equation (3.15) wrong in [4]. This Hauptmodul is called the normalized Hauptmodul for Γt. The series expansion of Jt around the point ζ₃ is calculated in [4] in the following way:

Theorem 3.1. Consider the triangle group Γ_(m1,m2,∞), and introduce coordinates ˜q3= α3q3, where

α3= b⁰d⁰

b⁰−1

Y

k=1

2 − 2cos

2πk

b⁰

−¹

2cos

2π^ka0_b0

d⁰−1

Y

l=1

2 − 2cos

2πl

d⁰

−¹

2cos

2π^lc0_d0

, (3.23)

where a⁰/b⁰ = ¹₂(1 + 1/m₁− 1/m₂), c⁰/d⁰ = ¹₂(1 + 1/m₁+ 1/m₂) and gcd(a⁰, b⁰) = gcd(c⁰, d⁰) = 1.

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Figure 5: Illustration of the correct choice of fundamental domain for the triangle group Γ_(m₁_,m₂_,∞).

The normalized Hauptmodul is now given by Jt= ˜q₃⁻¹+

∞

X

k=0

c_kq˜₃^k, (3.24)

with coefficients c_k determined by

− 2...

JtJ˙t+ 3 ¨J_t² = ˙J_t⁴ 1 − (1/m₂)²

J_t² +1 − (1/m1)²

(Jt− 1)² +(1/m1)²+ (1/m2)²− 1 Jt(Jt− 1)

, (3.25) for z = ζ₃ and where a dot denotes ˜q₃_{d ˜}^d_q

3.

In [4] they determine the coefficients in the Fourier expansion by solving the third order non-linear differential equation (3.25) recursively. This, however, is an inefficient way of doing it, and we will now develop another method which then will be implemented in SageMath. This is partly done in [11], in the special case of Hecke groups, but we will attempt to generalize this.

In [9, ch. VI, sec. 5] it is proven that a Hauptmodul exists and in accordance with this proof we begin by studying the mapping of hyperbolic triangles, assuming that the Hauptmodul maps the interior of the hyperbolic triangle ∆(m1, m2, m3) to H and its boundary to R. In the upper half plane a hyperbolic triangle is bounded by geodesics in hyperbolic geometry, which means that we study triangles bounded by circular arcs. The mapping of such triangles is extensively described in [2, ch. 2], and we will follow the outline of that book.

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We begin by quite general arguments that apply to all hyperbolic triangles, ∆(m₁, m₂, m₃) and then in the end we restrict ourselves to the case ∆(m₁, m₂, ∞). That the Hauptmodul maps

∆(m1, m2, m3) to the upper half plane, means that we can choose a local branch of the inverse map that maps H to the interior of the triangle.

To simplify notation, we define vi = 1/mi, for i = 1, 2, 3.

We consider the mapping

u = u(z), (3.26)

that maps H onto a hyperbolic triangle and normalize it in such a way that the vertices of the triangle are the images of the three points z = 0, 1 and ∞.

Now, using a M¨obius transformation, we create a new transformation w = w(z) according to w = γu = au + b

cu + d, (3.27)

where γ ∈ SL(2, R). It can be shown that w also maps H onto the interior of a hyperbolic triangle with vertices w(0), w(1) and w(∞).

The aim is now to determine the mapping function, w, by finding a differential equation for w that is independent of γ.

By differentiating equation (3.27) with respect to z, we get that dw

dz = ad − bc (cu + d)²

du

dz. (3.28)

Taking the logarithmic derivative of this expression, we obtain that d

dzlog dw dz

= d

dzlog du dz

− 2c cu + d

du

dz. (3.29)

In order to simplify notation, we let

W = d

dzlog dw dz

, (3.30)

and

U = d

dzlog du dz

. (3.31)

This way we get that

dU dz −dW

dz = 2c cu + d

d²u

dz² − 2c² (cu + d)²

du dz

2

. (3.32)

Combining this with equation (3.29) gives us that dU

dz −dW

dz = U (U − W ) − 1

2(U − W )² = 1

2U²−1

2W². (3.33)

Rearranging this we get d²

dz²log(w⁰) −1 2

d

dzlog(w⁰)

2

= d²

dz²log(u⁰) −1 2

d

dzlog(u⁰)

2

, (3.34)

Modular forms for triangle groups