On a Certain Quotient of the Complex Upper Half Plane

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On a Certain Quotient of the Complex Double

Half Plane

Misha Schram

July 9, 2019

Bachelor thesis Mathematics Supervisor: dr. Arno Kret

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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1. Abstract

This thesis treats the basic notions needed to study Fuchsian groups. Fuchsian groups are one way of defining Shimura curves. Which are analogues of the well studied modular curves. We treat the notion of a quaternion algebra, primarily over number fields. We provide a short introduction to hyperbolic geometry and quotient spaces. Afterwards we turn to a specific example of a Fuchsian group and determine some of it’s properties.

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2. Introduction

The goal of this thesis was to gain an understanding in Shimura curves. A subject which has gained some notoriety amongst mathematicians for being “too difficult”. Shimura curves are analogues of modular curves. These are quotients of the complex upper half plane by certain discrete subgroups” of SL2(Z). These quotients arise through

the natural action of matrix groups on C. More precisely, given a b

c d ∈ SL2(Z) and

z ∈ H = {w ∈ C | Re w > 0} the action is defined as

a b c d · z =

az + b cz + d.

These quotients, also called modular curves, are studied in number theory and algebraic geometry because they are closely related to elliptic curves.

Shimura curves arise in a similar fashion. Given a quaternion algebra D over a number field F (usually assumed to be totally real we can define an embedding of D into M2(C).

We now have a natural action of the quaternion algebra on the complex upper half plane. If we take certain subgroups of D×, so called Fuchsian groups, we obtain Shimura curves as quotient spaces.

One of the main goals in the study of Shimura curves is to generelize results known about the modular curves. This will have a great impact in the Langlands program.

The approach taken in this thesis introduces all the basic notions needed to begin a deeper study into Shimura curves and provides plenty of references to other sources. After these notions are introduced we will study a specific example instead of continuing with abstract theory. This specific example was given to me by Arno Kret in some notes he had written some years ago. The idea was to provide details. The main = is to define a certain Fuchsian group and determine whether or not a certain quotient with this group is compact.

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3. Quaternion Algebras

In this chapter we will define the notion of a quaternion algebra and state some of its basic properties. Throughout this chapter k denotes a field.

We will prove that any quaternion algebra over k has a basis {1, u, v, w} with u2, v2∈ k and the relation uv = −vu. We show that any quaternion algebra can be embedded into M2(F ) for a suitable k ⊂ F . There is an excellent book by Vigneras written about

quaternion algebras [4]. Another, more introductory, text is by Keith Conrad [3]. There is also the, as of yet unfinished, book by John Voight [5].

3.1. Definition and Basic Properties

There are many equivalent definitions of quaternion algebras over fields. We use one which is rather hands on.

Recall that from a finite set of indeterminates {X1, . . . Xn} we can construct set of

words in using X1, . . . Xn. Take n = 5, then examples of words are X12X5X42 or X42X3X46.

In general a word is given by an expression of the form Xe1

s1X e2 s2 · · · X ek sk for suitable sequences {si}i and {ei}i.

Definition 3.0.1. Let R be a commutative ring. For n ∈ Z≥0 let {X1, . . . Xn} be a

set of n indeterminates. The free R algebra in X1, . . . , Xn is the free R-module with as

basis the set of all possible words in {X1, . . . Xn} together with multiplication given by

concatenation: (Xe1 s1X e2 s2 · · · X ek sk) · (X ˆ e1 ˆ s1X ˆ e2 ˆ s2 · · · X ˆ em ˆ sm) = X e1 s1X e2 s2 · · · X ek skX ˆ e1 ˆ s1X ˆ e2 ˆ s2 · · · X ˆ em ˆ sm.

We denote the free R algebra in indeterminates X1, . . . , Xn by RhX1, X1, . . . , Xni.

Remark 3.1.1. The free R algebra in n indeterminates is sometimes also called the noncommutative polynomial ring in n variables.

Definition 3.0.2. Let α, β ∈ k. We define (a, b)k to be the quotient of Rhα, βi with the

ideal generated by the relations

α2 = a, β2 = b, αβ = −βα.

A quaternion algebra over k is a k-algebra which is isomorphic to (a, b)k.

Example 3.1.1. We define H = (−1, −1)R. Elements of the algebra H are called

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Proposition 3.0.1. Let a, b ∈ k and let (a, b)k be a quaternion algebra over k. Then the

images of X 1, α, β, αβ of 1, α, β, αβ in (a, b)kform a basis of (a, b)k over k. Consequently

Dimk((a, b)k) = 4.

Proof. Page 1 in [16].

Since it is tedious to write 1, α, β each time we simply write 1, α, β instead. Similarly we define γ := αβ. The basis {1, α, β, γ} is called the standard basis of (a, b)k.

Lemma 3.0.1. The following maps are k-algebra isomorphisms. 1. (a, b)k→ (b, a)k, i 7→ j, j 7→ i

2. (ax2, b)k→ (a, b)k, i 7→ ix−1, j 7→ j

3. (1, b)k→ k , i 7→ −1 01 0 , j 7→ 0 b1 0.

Proof. Page 1 in [16].

Remark 3.1.2. If we are given a k-algebra D of dimension 4 with basis {1, u, v, w} and relations

u2, v2 ∈ k, uv = w, uv = −vu.

Then (u2, v2)k is isomorphic to D. The isomorphism is induced by

1 7→ 1, u 7→ α, v 7→ β.

Definition 3.0.3. Given a quaternion algebra (a, b)k choose a root

√ a ∈ ¯k of a. Define the map φ : D ,→ M2(¯k), α 7→ √a 0 0 −√a , β 7→ 0 −1 −b 0 .

Proposition 3.0.2. The map φ is an injective morphism of k-algebras. Proof.

We stress that the map φ is not always surjective as the following example shows. Example 3.1.2. We can embed H into M2(C). This is done as follows;

i 7→ i 0 0 −i , j 7→0 −1 1 0 .

This map is not surjective for M(C) is 8 dimensional over R while H is 4 dimensional. As noted in the introduction to this section there are many equivalent definitions of quaternion algebras. For reference we will state them here.

Proposition 3.0.3. If D is a k-algebra of dimension 4 then the following are equivalent. 1. D is a quaternion algebra over k.

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2. D has center k and has no non-trivial two sided ideal (i.e. it is simple). 3. D ⊗kk ∼= M2(k) where k is an algebraic closure of k.

Proof. Proposition 1.2 in [16]. The proof uses the following fact.

Lemma 3.0.2. Given a quaternion algebra (a, b)k and a field extension k ⊂ L then the

mapping (a, b)k⊗kL → (a, b)L given by x ⊗ l 7→ xl is an isomorphism of L-algebras.

Proof. [16]

3.2. Split Quaternion Algebras

In this section we show that all quaternion algebras are either division algebras or isomorphic to M2(F ). We provide conditions when either is the case.

Recall that for a quaternion algebra (a, b)k every element x ∈ (a, b)k can be written

uniquely in the form x = x0+ x1α + x2β + x3γ for certain xi∈ k.

Definition 3.0.4. Let B be a k-algebra. An involution is a k-linear map ι : B → B which satisfies

1. ι ◦ ι = idB;

2. ι(αβ) = ι(β)ι(α) for all α, β ∈ B.

Remark 3.2.1. This definition of an involution is sometimes also called an anti-involution. We simply call it an involution in this text and follow the convention as in the book of involutions [19]

Proposition 3.0.4. Let (a, b)k be a quaternion algebra and let x = a + bα + cβ + dγ ∈

(a, b)k. Define x = x0− x1α − x2β − x3γ. Then the map (a, b)k→ (a, b)k, x 7→ x defines

an involution on (a, b)k.

Proof. We have the morphism φ : khα, βi → khα, βi induced by α 7→ −α, β 7→ −β and αβ 7→ βα. We can compose this with the quotient map to obtain a morphism ψ : khα, βi → (a, b)k. The kernel of this map is contained in the ideal generated by the

relations

α2 = a, β2 = b, αβ = −βα.

. Therefore ψ factorizes over (a, b)k. The map we obtain in this manner the map described

as above.

Definition 3.0.5. Let (a, b)k be a quaternion algebra. The map defined above is called

conjugation on (a, b)k.

Definition 3.0.6. Let (a, b)k be a quaternion algebra and let x ∈ (a, b)k. We define the

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Remark 3.2.2. For a quaternion x = x0 + x1α + x2β + x3γ ∈ (a, b)k we get explicit

formulas for trd(x) and nrd(x). We have

trd(x) = 2x0, nrd(x) = x20+ x21a + x22b + x23ab.

Proposition 3.0.5. Let (a, b)k be a quaternion algebra and x ∈ (a, b)k. Then x is

invertible if and only if nrd(x) 6= 0. In this case x−1 = x/nrd(x). Proof.

Theorem 3.1. Let (a, b)k be a quaternion algebra. Then (a, b)k is either a division

algebra or (a, b)k∼= M2(F ).

Proof. See Theorem 4.20 in Keith Conrad’s notes on Quaternion Algebras [3].

Definition 3.1.1. Let (a, b)k be a quaternion algebra. We say (a, b)k is split if (a, b)k∼=

M2(F ). We say (a, b)k is non-split if it is a division algebra.

Lemma 3.1.1. We have that (a, b)k is split if and only if there exists x, y ∈ F such that

ax2+ by2= 1.

Proof. See Theorem 4.25 in [3].

Example 3.2.1. We have already shown that H is not split in Remark 3.1.2. Using the previous proposition we see this even more easily. There clearly are no solutions to −x2_{− y}2_{= 1 in R.}

3.3. Quaternion Algebras over Number Fields

In this section we restrict our view to quaternion algebras defined over number fields (i.e finite field extensions of Q). For simplicity we view them as a subset of a fixed algebraic closure Q = C of Q. In this case any quaternion algebra embeds into M2(C). In this

section we state the classification of quaternion algebras over number fields through the notion of ramification.

Before we prove this we state some algebraic preliminaries.

Definition 3.1.2. A norm is a map | · | : k → R such that for all x, y ∈ k the following hold:

1. |x| ≥ 0 and |x| = 0 if and only if x = 0 2. |xy| = |x||y|

3. |x + y| ≤ |x| + |y|

Moreover, if a norm also satisfies |x + y| ≤ max{|x|, |y|} we say the norm is non-archimedean otherwise we say the norm is non-archimedean.

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Remark 3.3.1. Given a norm | · | on k one can define a distance function on k by defining d : k × k → R≥0; (x, y) 7→ |x − y|. The function d is a metric and (k, d) is a metric space.

Now k naturally carries a topology induced by d.

Given a norm | · |, there are multiple “equivalent ones”. For example for any real number c ∈ R>0 we have that c| · | is another norm on k. These norms induce the same

topology on k. This leads us to the following definition.

Definition 3.1.3. Two norms | · |1, | · |2 on k are called equivalent if they induce the

same topology on k.

Remark 3.3.2. The above definition turns out to be equivalent to the statement that there exists a nonzero positive real number c ∈ R>0 such that | · |1 = c| · |2.

Example 3.3.1. For a prime p ∈ Z we define the p-adic norm on Q. It is defined as follows: for a/b ∈ Q with a/b 6= 0 write a/b = pnr/q for certain r, q, n ∈ Z with p - rq. Then define |a/b|p= p−n. Also define |0|p= 0. Then the map Q → R≥0, a/b 7→ |a/b|p is

a norm.

Definition 3.1.4. A place v of k is a class of equivalent nontrivial norms. We write S(k) for the set of places of k.

Remark 3.3.3. Places are usually denoted by v because an equivalent definition of a place is as an equivalence class of “valuations”. We will however not need this and instead only work with an equivalence class of norms instead.

It is a theorem by Ostrowski that the p-adic norms together with the “standard” norm on Q (i.e. the standard absolute value function on R restricted to Q) form a complete set of representatives for the the places of Q. Therefore we can naturally identify the places of Q with the set of prime numbers and the point infinity. In this correspondence a prime number p is identified with the p-adic norm and the point infinity is identified with the usual absolute value.

Given a field extension k ⊂ L we have the mapping S(L) → S(k), | · | 7→ | · ||k. This is

well defined.

Definition 3.1.5. Let k ⊂ L be a field extension v ∈ S(L) and w ∈ S(k). We say that v lies above w if v|k = w and denote this by w | v.

Remark 3.3.4. In fact if k ⊂ L is an algebraic extension it is surjective by Theorem 8.1 on pages 161,162 in [1]. So every place of L is obtained as an “extension” of a place of k. In other words, every place of L lies above some place of k.

The proof of Theorem 8.1 Pages 161,162 in [1] uses the notion of a “completion” of a field with a norm. It will take us too far afield to completely prove this theorem. We will however briefly introduce the notion of a completion. For a complete covering of the subject we refer the reader to chapter 8 in Neukirch [1] or to the first few chapters of [2] for the case k = Q.

Given a norm | · |v on k recall that a sequence {xn} ⊂ k is called Cauchy if for all

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independent of the norm chosen to represent a place. We say that a sequence {xn} ⊂ k

converges to x ∈ k if for all > 0 there exists some N ∈ Z>0 such that if n > N we have

that |xn− x|v < . We say that a field k is complete is every Cauchy sequence converges

to some element in k.

Proposition 3.1.1. Let k be a field an |.|v a norm on k. Then there exists a k-algebra

kv with a norm extending |.|v (i.e. the restriction of this norm to k is equal to | · |v) that

is complete and satisfies the following universal property.

For any k-algebra R with a norm | · | extending the norm on k we have that there exists a unique morphism φ

k kv

R

φ

making the diagram commute.

Sketch 1. Write C for the subring of kN _{of all Cauchy sequences and write I for the ideal}

of all Cauchy sequences which converge to 0. We define the set kv = C/I.

We have that k is embedded into kv by mapping x to the sequence (x, x, x, . . .). We

can equip kv with an extended norm. For (x1, x2, . . .) we define |(x1, x2, x3, . . .)|v =

limn→=∞(|xn|v) where we slightly abuse notation by also denoting this norm with | · |v.

Then kv is a complete field with the described universal property.

Definition 3.1.6. The field kv described above is called the completion of k with respect

to | · |v.

Example 3.3.2. Let p be a prime number and | · |p the p-adic absolute value. Let p

denote the place corresponding to | · |p. Then Qp is the completion of Q with respect to

the p-adic norm. This is consistent with the notation Qp reserved for the p-adic numbers.

Theorem 3.2. Let k ⊂ L be a finite seperable field extension and v a place of k. Then the mapping L ⊗kkv →Q_w|vLw given by

a ⊗ b 7→Y

w|v

ab

is an isomorphism.

Proof. Page 164, prop 8.3. in Neukirch [1]

This theorem is useful to determine the places which lie above v as is demonstrated in the following example.

Example 3.3.3. Let v be a place of Q. In order to determine the places above v in Q(i) by Theorem 3.2 it suffices to understand the structure of Q(i) ⊗QQv. We distinguish

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1. Suppose v = ∞. Then we have that

Q(i) ⊗QQv = Q(i) ⊗QR = R(i) = C

Since C is a field Theorem 3.2 implies that we have precisely one place above ∞ in Q(i).

2. Now suppose v = p = 1 mod 4. Then −1 is a quadratic residue modulo p and hence x2+ 1 splits modulo p. By Hensel’s Lemma we get that x2+ 1 also splits in Qp. We obtain that.

Q(i) ⊗QQv = Q(i) ⊗QQp = Qp[x]/(x

2_{+ 1) ∼}

= Qp× Qp.

So that there are precisely 2 places above v.

3. Suppose v = p = 3 mod 4. Then x2_{+ 1 is irreducible. So by the same reasoning as}

above there will be precisely 1 place above v

4. v = 2 then we have that x2+ 1 is irreducible since (x + 1)2+ 1 is irreducible by the generilized Eisenstein criterion. Through Hensel’s Lemma on (x + 1)2+ 1 we obtain that (x + 1)2+ 1 is irreducible in Q2. It follows that there is precisely one

place above 2.

This example will be useful later on when we will be dealing with quaternion algebras over Q(i).

Recall that given a finite extension Qp ⊂ L we have that L is a local field.

Theorem 3.3. If k is a local field and k 6= C then there is exactly one non-split quaternion algebra over k up to isomorphism.

Remark 3.3.5. We can be more precise and give explicit descriptions of this quaternion algebra. This assumes some knowledge on uniformizers and (un)ramified extensions. If F is any finite extensino of Qp this algebra is given by (a, π)k where π is the uniformizer

of k and a ∈ O×_F is such that F (√a) is the quadratic unramified extension of k. Proof. See Theorem 1.5 in the lecture note by Chenevier [16]

Example 3.3.4. Indeed, if we take k = R. Then then up to isomorphism H is the unique quaternion algebra over R by the simple isomorphisms given in 3.0.1. In the case of k = Qp we have that (1, p)Qp is the unique quaternion algebra over Qp up to

isomorphism. The unique quaternion algebra over Q2 is given by (−1, −1)Q2. [16]

Definition 3.3.1. Let F be a number field and D a quaternion algebra over F . For a place v of F we say that D/F ramifies at v if D ⊗F Fv is non-split.

Theorem 3.4. Let D be a quaternion algebra over a number field F . Then the set Ram(D) ⊂ S(F ) of places v ∈ S(F ) such that D/F ramifies at v is a finite set with an even number of elements.

Conversely for any finite set S ⊂ S(F ) such that S has an even number of elements there is a unique quaternion algebra D over F such that Ram(D) = S.

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Proof. See Theorem 1.7 in [16] or Theorem 14.6 in [5]

Since we have determined all places of Q(i) in 3.3.3 we can now easily define quaternion algebras over Q(i).

3.4. Orders in Quaternion Algebras

In this section we introduce the notion of an order in a quaternion algebra. It is a generelization of the ring of integers in a number field.

Definition 3.4.1. Let R be a domain with field of fractions k and let V be a k-vector space. An R-lattice Λ ⊂ V is a finitely generated R-module that generates V as a k-vector space.

Example 3.4.1. For a quaternion algebra (a, b)_Q with standard basis {1, α, β, γ} we have that Z + αZ + βZ + γZ is a Z order. Indeed, it is finitely generated as a Z module and contains a basis for (a, b)_Q.

Definition 3.4.2. Let R be a domain with field of fractions k and let B be a finite dimensional k-algebra. An R-order is an R-lattice that is also a subring of B. An R-order is called maximal if it is not properly contained in any other R-order.

Example 3.4.2. The lattice described in Example 3.4.1 is also a Z order. More generally we have for any quaternion algebra (a, b)k that R + αR + βR + γR is an R-order. Let

Ok denote the ring of integers of a number field k. Then there are Ok-orders and in

particular Z orders in k.

Example 3.4.3. Let D = (−1, −1)_Q. Then the Hurwitz Order given by Z + iZ + jZ + (1 + i + j + k)/2 Z is, as the name suggests, a Z-order. In fact, it is a maximal Z-order.

See Chapter 11 in [5].

Theorem 3.5. Let R be a domain with field of fractions k and let B be a k-algebra. Then there exists a maximal order in B.

Proof. There is an entire book devoted to maximal orders. It is by Reiner [15]. This result can be found in chapter 2. This result can also be found in [4] chapter 1 paragraph 4 and [5] on page 143.

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4. Complex Upper Half Plane

In this section we introduce hyperbolic geometry. We introduce two models of hyperbolic geometry. The Poincar´e half plane and the Poincar´e unit disk. We define an action of PGL2(R) =

_{a b}

c d ∈ M2(R) | ad − bc = 1 on the complex upper double half plane

H := {z ∈ C | Rez 6= 0} = P1C − P1R.

These actions are called “M¨obius” transformations. We study the action of certain “discrete” subgroups of PSL2(R) and look at qoutient spaces induced from this action. In

the last section we define fundamental domains for these spaces and discuss a constructive way to show these are compact through the notion of the Ford fundamental domain.

4.1. Hyperbolic Geometry

There are many equivalent descriptions of 2-dimensional hyperbolic geometry. We describe two of them. A good treatment of the material we use can be found in [8] and [7].

4.1.1. The Poincar´e Half Plane Model

The first “model” which we describe is the so called “Poincare Half Plane Model”. As a set it is the upper half plane H ∼= {(x, y) ∈ R2 | y > 0}. Let ˜H denote its closure in R2. The following is a rather technical approach, it treats much more than one needs to know about hyperbolic geometry. The rest of thesis can be read from def inition 4.0.2 onwards without too much difficulty.

It can be equipped with the differential

ds = p

dx2_{+ dy}2

y

where dx, dy both denote the standard metric on R. For a curve γ = x + iy : [0, 1] → H its hyperbolic length is given by

||γ|| = Z 1

0

p(dx/dt)2_{+ (dy/dt)}2

y(t) dt. For two points p, q ∈ H their hyperbolic distance is given by

d(p, q) = inf ||γ|| (4.1)

where the infimum is taken over all curves connecting p and q. This defines a metric on H and induces the same topology on H as the “standard” topology. See Theorem 1.3.3 on page 11 in [7].

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Remark 4.1.1. There are explicit formulas for determining the distance between two points. They involve the inverse of the hyperbolic sine and cosine. We will not need them and instead refer to Theorem 1.2.6 in [7] or Theorem 7.2.1 in [8]

Definition 4.0.1. The Poincar´e Half-Plane Model is the upper half-plane equiped with the metric d.

In the hyperbolic geometry there the notion of “geodesics”. A formal definition will lead us too far astray so we instead give an informal definition and again refer to either [8] or Theorem 1.2.1 in [7] for a complete treatment.

Definition 4.0.2. The geodesics in the Poincar´e half-plane model are given by semi-circles and straight lines orthogonal to the real axis.

Theorem 4.1. For two points in the Poincar´e half-plane there is a unique geodesic passing through both.

Remark 4.1.2. A way to think about geodesics is that they are “the shortest” way to walk between two points.

4.1.2. Poincar´e Disk Model

Another “model” of hyperbolic geometry is the Poincar´e disk model. As a set it is the open unit disk U ⊂ C. Let ˜U denote its closure in R2. The map f : ˜H → ˜U defined by f (z) = z − i

z + i is a bijection. In particular, it maps the real line to the Euclidean boundary of U . We use this map and our metric d on H defined in Equation 4.1 to define a new metric d∗ on U . It is given as the pull-back of d under f . Explicitely for p, q ∈ U we have d∗(p, q) := d(f−1p, f−1q).

Definition 4.1.1. The Poincar´e disk model is the open unit disk U equipped with metric d∗.

Remark 4.1.3. The map f is called the Caley-Transform. It is a fractional linear transformation. It is a theorem that these fractional linear tranformations map straight lines and circles onto straight lines and circles and preserve angles. See Theorem 5.2 on page 234 and Theorem 7.1 on page 34 in [17].

Theorem 4.2. The geodesics in the Poincar´e disk model are given by the straight lines and circles orthogonal to S1⊂ C.

Sketch. This is a combination of the previous remark combined with fact that the real line is mapped to the boundary of D2.

In Euclidean geometry polygons are the regions enclosed by a finite number of straight lines segments. In hyperbolic geometry the role of straigt lines is played by the geodesics. Definition 4.2.1. A hyperbolic n-sided polygon in ˜H is a closed set in ˜H bounded by n geodesic segment.

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Remark 4.1.4. We can replace ˜H with ˜U to obtain the definition of hyperbolic polygons in the Poincar´e disk model.

Definition 4.2.2. For a subset A ⊂ H we define the hyperbolic area of A by

µ(A) = Z

A

dxdy y2

if this integral exists. Otherwise the hyperbolic area is infinite.

It is a corrolary of the Gauss - Bonnet formula that the hyperbolic area of an n-sided hyperbolic polygon is given by (n − 2)π − α1− . . . − αn. Where the αi are the angles

made between the geodesic segments in radians (See Theorem 1.4.2 in [7]). These angles are calculated as they are in ordinary Euclidean geometry.. This makes it “easy” to determine the hyperbolic area of hyperbolic polygons.

4.2. PSL

2

(R)

In this section we define a transitive action of PSL2(R) = SL2(R)/R× on H. Through

the Caley-Transform this action also defines an action of a group conjugate to PSL2(R)

on U . We will distinguish three different types of elements in PSL2(R) and define the

notion of discrete and properly discontinuous groups. Lastly we define the notion of a fundamental domain and give an explicit example of a fundamental domain of discrete properly discontinous group without proof.

In the previous section we defined the Poincar´e half-plane. From this point on it will be more natural to insted work with C \ R. The results in the previous section apply mutatis mutandis to the double half plane. We note that the Caley transform z 7→ z − i

z + i maps the complex double half plane C \ R to the space P1C \ S1. We will often alternate between H and C \ R.

Let (_wz) ∈ C2 _and a b

c d ∈ PSL2(R). The natural action of a bc d on ( z

w) is given by

matrix multiplication:

a b

c d (wz) = az+bcz+d .

This action is C-linear. It therefore extends to an action on C2/C×∼= P1C induced by (_wz) 7→ [z, w]. This action is given by

a b

c d · [z, w] = [az + bw, cz + dw].

Since PSL2(R) preserves P1R it will also preserve the complement of P1R in P1C. Proposition 4.2.1. For a b_{c d} ∈ PSL2(R) and z ∈ H the action of PSL2(R) on C \ R

defined by

a b c d · z =

az + b cz + d is a well defined transitive group action.

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Proof.

The importance of PSL2(R) is made clear in the following theorem.

Theorem 4.3. The group PSL2(R) is a subgroup of index 2 of the isometries of the

Poincar´e half plane.

Proof. Theorem 1.3.1 on page 8 in [7]. Definition 4.3.1. Let a b

c d ∈ PSL2(R). We define the trace as Tr( a bc d) = |a + d|. We

say an element T ∈ PSL2(R) is

1. elliptic if Tr(T ) < 2, 2. parabolic if Tr(T ) = 2, 3. hyperbolic if Tr(T ) > 2.

Remark 4.2.1. Note that the absolute value in the definition of the trace ensures the map is well defined.

We equip PSL2(R) with a topology as follows. We put a topology on M2(R) by viewing

it as R4 with the standard Euclidean topology. Equip SL2(R) ⊂ M2(R) with the subspace

topology. Lastly equip PSL2(R) with the qoutient topology. For S, T ∈ PSL2(R) the

maps

(S, T ) 7→ ST, T 7→ T−1

are continuous since the are given by homogeneous polynomial equations in the four entries. So PSL2(R) is a topological group.

We give PGL2(R) a topology in the same way.

Definition 4.3.2. A subgroup of PGL2(R) is discrete subgroup if it is a discrete

topo-logical space with respect to the subspace topology on PGL2(R).

Definition 4.3.3. Let G be a group acting on a topological space X. We say that G acts properly discontinuously if each point x ∈ X has a neighborhood V such that T (V ) ∩ V 6= ∅ for only finitely many T ∈ G.

Theorem 4.4. Let Γ be a subgroup of PSL2(R). Then Γ is discrete if and only if Γ acts

properly discontinuously on C − R. Proof. Theorem 2.2.6 on page 32 in [7].

The equivalence of these two definitions is usefull because often it is much easier to see that a subgroup is discrete than that it is to see that it acts properly discontinuously as is demonstrated in the following example.

Example 4.2.1. The subgroup PSL2(Z) ⊂ PSL2(R) is discrete. Indeed, the subgroup

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Corollary 4.4.1. Let Γ be a subgroup of PSL2(R) which acts properly discontinuosly on

H. Then the orbit Γ · z is a discrete set for all z ∈ H. Proof. See Corollary 2.2.7 on page 32 in [7]

Definition 4.4.1. Let G be a group acting on a topological space X. Equip G\X = {G · x | x ∈ X} with the qoutient topology under the map x 7→ G · x. The topological space G\X is called the qoutient space of X under G.

The previous corrolary allows us to not only put a topology on the qoutient space Γ\H. We can also equip it with a smooth complex structure. This is done in the following manner.

Equip H with the complex structure inherited from the standard complex structure on C. Let p ∈ H and Γ · p be its orbit under Γ. Then by the previous corrolary we have that Γ · p is discrete. Let V ⊂ H be an open neigborhood of p such that V ∩ (Γ · p) = {p}. Denote V for the image of V under the qoutient map π : H → Γ\H. Define a chart φ : V → C by mapping v ∈ V to the unique v ∈ V such that π(v) = v. The transition maps between two of these charts is given by the action of an element of Γ. This action is a holomorphic map since it is a fractional linear equation. So the collection of these charts form a smooth atlas. Consequently Γ\H is a Riemann surface (i.e. a one-dimensional complex manifold).

Theorem 4.5. Every compact Riemann surface is a complex projective variety. (i.e. can be embedded into a complex projective space and the image is the zero set of a set of homogeneous polynomials.)

Proof. Theorem 1.9 on page 171 in [18].

We informally treat the notion of “curvature” of spaces. We take as examples the Riemann sphere (or equivalently S2 ⊂ R3_{), the complex plane and the hyperbolic plane.}

Remark 4.2.2. Since the complex plane is “flat” we say the complex plane has curvature 0. Note that in a triangle the sum of the interior angles add up to 180 degrees. The Riemann sphere is not “flat” it should therefore have curvature. We say that it has curvature +1. Note that when we draw a triangle on the sphere the interior angles add up to more than 180 degrees. In the hyperbolic plane the interior angles of triangles add up to less than 180 degrees. We define the curvature of the hyperbolic plane to be −1. Again, it is intuitively clear but by no means formal. It will however take us too far astray to formally define curvature.

Remark 4.2.3. The metric of the hyperbolic plane H descends to a quotient space Γ\H of H. Therefore we can talk about hyperbolic area in quotient spaces.

Theorem 4.6. Let Γ be a discrete subgroup of PSL2(R). The quotient space Γ\H is

compact if and only if Γ\H has finite hyperbolic area and Γ contains no parabolic elements. Proof. See Corrolary 4.2.7 in [7].

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Remark 4.2.4. Most of these theorems concern the compactness of Γ\H instead of compactness of Γ\(C \ R). This will not be a problem in the future. For our group in the last chapter will have an element that maps the lower half plane to the upper half plane and vice versa. Therefore compactness of Γ\(C − R) is equivalent to compactness of Γ\H.

We end this section with the definition of a fundamental domain and give an example with a proof sketch of a fundamental domain of a certain quotient of the upper half plane. Definition 4.6.1. Let G be a group acting on a space X through homeomorphisms. A fundamental domain for G\X is a closed set F such that no two points in the interior of F are conjugate to eachother by an element of G and F contains at least one element from each orbit.

Remark 4.2.5. Given a fundamental domain F and some g ∈ G we have that g · F = {g · x | x ∈ F }

is another fundamental domain. Consequently fundamental domains are not unique. Remark 4.2.6. We rarely consider the case X = H (or C \ R). We will often work with X = ˜H (or C) instead. This allows a fundamental domain to intersect the boundary R ⊂ C of H in P1C.

Example 4.2.2. Given the action of Z on ˜H given by multiplication. One can take the upper half of the unit disk as a fundamental domain for G\ ˜H. Another possibility is to take the annalus with radi 1 and 2.

Example 4.2.3. Let PSL2(Z) ⊂ PSL2(R) act on H by the standard action. The group

PSL2(Z) is generated by the matrices (1 1_{0 1}) and 0 −1_{1 0} . We sketch how to construct

a fundamental domain using only these two elements. The inverse of (1 1

0 1) is given by 1 −1

0 1 and −1 00 1 is its own inverse. For z ∈ H we have that

(1 1_{0 1}) · z = z + 1, −1 00 1 · z =

−1 z .

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Indeed, the element (1 1

0 1) can bring z inside the vertical strip between <(z) = −1/2 and

<(z) = 1/2. Suppose we land inside the unit circle. Then the element 0 1

−1 0 brings us

outside the unit circle. We can now translate again with (1 1

0 1) to land inside the strip

again. For a full proof see 1001[13]

Remark 4.2.7. The group PSL2(Z) is called the full modular group and the space

PSL2(Z)\H is a modular curve. The modular group is intimately related to elliptic

curves.

4.3. Ford Fundamental Domain

In the previous section we discussed the action of discrete groups on the upper half plane. In this section we discuss a method to explicitely determine such a fundamental domain. We follow the paper of L.R. Ford [10].

Assume that we have a group G ⊂ {(a c

c a) ∈ SL2(C)} acting on H which has an invariant

circle or line which maps the interior of this circle or a half plane on one side of the line to itself. Assume that there is a point p, not on the invariant line or circle, such that there exists an open neighborhood V of p with G · p ∩ V = {p}. Then there is a map φ which maps p to the origin and the invariant circle or straight line to the boundary S1 of the unit disk U . The group H = φGφ−1 naturally acts on U and has the property that there is a neighborhood of 0 where there are no points congruent to 0 and that S1 is invariant. It suffices to find a fundamental region for the new group.

The next proposition assumes some knowledge about orientation preserving maps. This will not be important to us, we only need the fact that G consists of isometries with respect to the hyperbolic metric.

Proposition 4.6.1. The group of orientation-preserving isometries of U with respect to the hyperbolic metric is given by {(a c

c a) ∈ SL2(C)} . Where the action is given by

(a c

c a) · z = az+caz+c.

Proof. Exercise 1.10 in [7].

Given such transformation T = (a c_{c a}) ∈ H we have that around z ininitesimal Euclidean lengths are multiplied by |T0z| = _|cz+a|1 2 through our transformation T . Similarly

infinitesimal areas are multiplied by T0(z). These distances and areas are invariant under T if and only if |cz + a| = 1. Whenever c 6= 0 (equivalently T is not the identity transformation) this can be described by

|z − a/c| = 1/|c|.

We call this the isometric circle belonging to T and denote it by I(T ). It has radius 1/|c| and centre a/c. This allows us to quickly go from a matrix to it’s isometric circle. Since the determinant of T is 1 we have that 1 + 1/|c|2 = |a/c|2. This condition is equivalent to saying that I(T ) is orthogonal to S1. Note that this also implies that 0 is never inside one of these circles. In other words I(T ) is a geodesic in U . We denote ext(T ) for the area outside I(T ) and similarly denote int(T ) for the area inside the circle.

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Remark 4.3.1. This is not a very precise discussion. A precise way to do this work with tangent spaces and tangent maps instead. This is done in Chapter 37 in Voight [5].

The main result of this section is the following theorem. Theorem 4.7. For H as above we have that

F := \

T ∈H

ext(T ) ∩ U

is a fundamental domain for H\U . It is called the Ford fundamental domain. Lemma 4.7.1. The transformation T = (a c

c a) ∈ H increases Euclidean lengths and areas

in int(T ) and decreases euclidean lengths and areas in ext(T ).

Proof. The norm of the derivative of T is given by |T0z| = _|cz+a|1 2. Inside I(T ) we

have that |cz + a| < 1 and consequently |T0z| > 1. Outside we have |cz + a| > 1 so |T0z| < 1.

Lemma 4.7.2. The isometric circles I(T ) and I(T−1) have the same radius and I(T ) is mapped to I(T−1) by T .

Proof. The inverse of T = (a c_{c a}) is given by _{−c a}a −c. The radius of the isometric circle belonging to this transformation is 1/|c|. The same radius as T . The circle I(T ) is mapped to a circle C by T without alteration of Euclidean lenghts. This circle C is mapped back to I(T ) by applying T−1 without alteration of lengths. This precisely means that C is the circle I(T−1).

Lemma 4.7.3. We have that ext(T ) ∩ ext(T−1) ∩ U is mapped bijectively into I(T ) by T−1 and into I(T−1 by T .

Proof. Let z ∈ ext(T ) ∩ ext(T−1) ∩ U . Since z is outside both isometric circles we have that both T and T−1 transform z to point z0 and z00 respectively with a decrease in Euclidean areas and lengths in an infinitesimal neighborhood around z by Lemma 4.7.1. Then the points z0 and z00 must be mapped back to z with an increase in area by T−1 and T respectively. By the same Lemma this means that z0 ∈ I(T−1_{) and z}00 _{∈ I(T ).}

The argument that this is a bijection is the same.

We now show that each orientation preserving isometry of the hyperbolic unit disk has a nice geometric description in terms of isometric circles. We previously described all of these transformations as the set of matrices {(a c

c a) ∈ SL2(C)} .

Definition 4.7.1. Let C be a circle in C with centre k and radius r. Then the inversion in C is the map P1C → P1C defined by sending z (z 6= k, ∞) to the unique point w on the line through z and the centre k of C such that the distance of w to C is 1/|r|. (A picture helps). By convention k gets mapped to ∞ and vice versa.

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Remark 4.3.2. There is a formula for the inversion in a circle with centre k and radius r. It is given by z 7→ k ¯z + r 2_{− |k|}2 z − ¯k . Theorem 4.8. Let T = (a c

c a) be such orientation preserving transformation of U . Then

T is an inversion in I(T ) followed by a reflection in in the perpendicular bisector between the centers of I(T ) and I(T−1).

Proof. We have that I(T ) is described by |z − a/c| = 1/|c|. Suppose c = rei and let Tφ(z) = eφiz. Let S be given by S = Tφ◦ T ◦−1φ . Then S(z) = eφi

ae−φiz+c r+a

= z+r_r+a. The centre of I(S) is therefore given by a/r and it has radius 1/r. The centre of I(S−1) is given by −a/r which is just reflection in the imaginary axis. If we prove that S is an inversion in I(S) followed by a reflection in the imaginary axis we are done. Indeed, in that case S = s ◦ j where s is the reflection and j is the inversion gives us

T = (T_φ−1◦ s ◦ Tφ) ◦ (T_φ−1◦ j ◦ Tφ).

The first part is a reflection in the perpendicular bisector between the centres of I(T ) and I(T−1) and the second part is the inversion in I(T ).

The isometric circle I(S) is parametrized by |z − a/r| = 1/r. The formula for the inversion in I(S) is given by

z 7→ − a rz + 1 r2 − |a_r|2 z +a_r = − a rz + 1 z +a_r = − az + r rz + a.

Where we used the fact that 1/r2+ 12 = |a/r|2 since S1 and I(S) are orthogonal. If we now reflect in the imaginary axis we get

− −az + r rz + a = az + r rz + a = S(z). So indeed, S is an inversion followed by a reflection.

Proposition 4.8.1. For each > 0 there are at most finitely many isometric circles which intersect the disk with radius 1 − .

Proof. See discussion point 5 in [10]

Sketch of proof of Theorem 4.7. To add the full proof will add a further couple pages. It requires the notion of a Dirichlet domain. The most important observation is that a point in the interior lies outside each isometric circle and therefore each transformation T maps this point outside F . Hence no two points in the interior are congruent. For a full proof see Sections 3.2 and 3.3 in Katok [7] or Section 37.2: Ford Domains in John Voight [5].

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We give an explicit example of a Ford Fundamental domain. It uses what is called an arithmatic Fuchsian group. This is a Fuchsian group derived form a quaternion algebra. There are entire chapters devoted to arithmatic Fuchsian groups. See chapter 5 in [7], Chapter 38 in [5] or Chapter 1 in [14]. In short an arithmatic Fuchsian group is derived from an order in a quaternion algebra.

Let B = (2, 5)_Qbe a quaternion algebra. Let O be the Z-order spanned by the standard basis 1, α, β, γ of (2, 5)_Q. We use an embedding of B into M2(R) given by

α 7→ √ 2 0 0 −√2 , β 7→ 0 √5 −√5 0 .

We then use a computer program to find elements with reduced norm (determinant) 1 in this matrix ring. These matrices we obtain turn out to have an action on the complex upper half plane. We conjugate them with the matrix 1 i_{1 −i} to obtain an action on the Poincar´e disk model. We are now in the situation where we can calculate the Ford fundamental domain. From the list of elements we calculate their isometric circles. We obtain the following picture.

Note that this probably isn’t the Ford fundamental domain. We might have missed a few isometric circles. There are algorithms which obtain the exact Ford fundamental domain. One can be found in a paper by John Voight [6]. However one can easily spot that the area enclosed by the circles is compact. The exact Ford fundamental can only become smaller. So every fundamental domain is compact.

At this point we would like to note the reader about the similarity of this picture with the one depicted on the cover. The picture on the cover is what is called a “tiling”. It

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uses a fundamental domain and then uses group elements to translate this domain to obtain another fundamental domain. All areas enclosed by lines on the cover are in fact fundamental domains. If we translated the domain we obtained in our picture above we would obtain a similar picture.

Here is another picture. This one is obtained by taking B = (3, 7)_Q. Note that the isometric circles do not fully cover the unit circle. From this picture we can neither conclude that the quotient is compact or that the quotient is noncompact. We might have missed some elements.

Remark 4.3.3. It is in fact a theorem that all these quotients are compact (See Theorem 5.4.1 in [7]). Therefore we must have missed a couple elements.

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5. On a Certain Qoutient of the

Complex Double Half Plane

In the first two chapters we discussed quaternion algebras. In this section we define a quaternion algebra and determine some of its properties. In particular we will be interested in determining the quotient space induced by an action of this quaternion algebra on C \ S1.

Explicitely we construct a certain topological group G(R) from a quaternion algebra D. We define a transitive action of G(R) on the complex double half plane, or rather P1(C) \ S1 which is isomorphic to it. Our group G(R) also has a “Z-structure” which will allow us to construct a discrete subgroup G(Z) ⊂ G(R) and consider the corresponding quotient space G(Z)\ P1\ S1_{. In a later section we show that this qoutient is compact.}

5.1. A Certain Quaternion Algebra

Let D0:= (−2, −5)Q and Q(i) be the quadratic extension of Q obtained by adjoining a

solution i ∈ C of X2_{+ 1 = 0 to Q. Define D := D}

0⊗QQ(i). Then D∼= (−2, −5)Q(i)as

Q-algebras by Lemma 3.0.2. On pure tensors the isomorphism maps d ⊗ z ∈ D0⊗QQ(i)

to dz ∈ (−2, −5)_Q(i). From here on out we will make no distinction between D and (−2, −5)_Q(i).

5.1.1. Ramification of a Certain Quaternion Algebra

We first determine the places where D/Q(i) ramifies. Proposition 5.0.1. For D0 and D as above we have that

1. The places where D0/Q ramifies are 5 and ∞.

2. The places where D/Q(i) ramifies are the two places in Q(i) lying above 5. Lemma 5.0.1. Let p be an odd prime and a, b ∈ (Z/pZ)×. Then there exists x, y ∈ Z/pZ such that ax2+ by2 = 1.

Proof. The proof works via the pigeon hole principle. See [3] for a full proof. Lemma 5.0.2. There do not exist x, y ∈ Q5 such that −2x2− 5y2 = 1.

Proof. For a contradiction suppose there do exist such x, y ∈ Q5. Note we have that

|x|5, |y|5 6= 0 by the strong triangle inequality. Suppose that | − 2x2|5 = | − 5y2|5. Then

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discrete set {5i| i ∈ Z} ∪ {0} and |x|5, |y|56= 0. So we must have that | − 2x2|56= | − 5y2|5.

We now get that 1 = | − 2x2− 5y2_|

5= max{| − 2x2|5, | − 5y2|5} = max{(|x|5)2,1₅(|y|5)2}.

This implies that |x|5, |y|5 ≤ 1. In other words, x, y ∈ Z5. Now we can reduce our

equation −2x2− 5y2 _{= 1 modulo 5 to obtain −2x}2_{= 1 mod 5. This is a contradiction}

since modulo 5 we have −2−1 = 2 and 2 is not a quadratic residue modulo 5. Proof of Proposition 5.0.1. Note that for a place w of Q with v | w we have that

D ⊗_Q(i)Q(i)v ∼= (D0⊗QQ(i)) ⊗Q(i)Q(i)v

∼

= D0⊗QQ(i)v

∼

= (D0⊗QQw) ⊗QwQ(i)v.

Where the third isomorphism holds because Qw ⊂ Q(i)v.

So it makes sense to first study where D0/Q ramifies before we try to determine where

D/Q(i) ramifies. For a place w of Q as above recal that by Lemma 3.1.1 we have that (−2, −5)_Qw is split if and only if there exists x, y ∈ Qw such that −2x

2 _{+ −5y}2 _{= 1.}

For w = ∞ it is trivial: there are none. So we have that D0/Q ramifies at infinity. By

Hensel’s lemma we have that solutions x and y of −2x2− 5y2_{= 1 modulo p for which at}

least one partial derivative doesn’t vanish yield solutions Qp. So Proposition 5.0.1 shows

that for any prime different from 2 or 5 the extension D0/Q is unramified. Since there

are only an even number of places where D/Q can ramify by Theorem 3.4 and D/Q already ramifies over ∞ we must have that it either ramifies over 2 or 5. By Lemma 5.0.2 there do not exist x, y ∈ Q5 such that −2x2− 5y2 = 1. This shows that D/Q ramifies

over 5. Consequently the places where D0/Q ramifies are 5 and ∞.

We have shown above that D ⊗_Q(i)Q(i)v ∼= (D0⊗QQw) ⊗QwQ(i)v for a place v which

lies above w. So by the discussion above we get for any v which lies above a prime p with p 6= 5 that

D ⊗_Q(i)Q(i)v ∼= (D0⊗QQw) ⊗QwQ(i)v ∼= M2(Qw) ⊗QwQ(i)v ∼= M2(Q(i)v).

So D/Q(i) does not ramify over any such v.

Now let v be the place of Q(i) which lies above ∞. Since D is a quaternion algebra over Q(i) and Q(i)v ∼= C we get that D ⊗Q(i)Q(i)v ∼= M2(C) by Theorem 3.0.3.

There are only two options which remain. Either D/Q(i) ramifies over the two places of Q(i) which lie above 5 or it doesn’t ramify at all.

The only quaternion algebra which does not ramify is the split quaternion algebra M2(Q(i)). Since D is not split we must have that it ramifies somewhere by Theorem 3.4.

Therefore it ramifies at the two places above 5.

Remark 5.1.1. Note that we could have just defined D to be the unique quaternion algebra over Q(i) which ramifies at the two places of Q(i) above 5. However in this case we would not have an explicit description of our quaternion algebra.

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5.1.2. Involution on a Certain Quaternion Algebra

Recall that D ∼= (−2, −5)_Q(i) and is generated by α, β which anti commute and are such that α2 = −2, β2 = −5. Now define a map from D to M2(C) as follows

φ : (−2, −5)_Q(i) ,→ M2(C), α 7→ i√2 0 0 −i√2 , β 7→ 0 − √ 5 √ 5 0 .

This is a slighty different map than we used in Proposition 3.0.2. However we still have the following result.

Lemma 5.0.3. The map φ as above is an injective morphism of rings.

Proof. We first define a morphism ψ from the non commutative polynomial ring Q(i)hα, βi

ψ : Q(i)hα, βi → M2(C), α 7→ i√2 0 0 −i√2 , β 7→ 0 − √ 5 √ 5 0 .

We have that ψ(α)2 = −2 0_{0 −2} and ψ(β)2 = −5 0_{0 −5}. So ψ factors over the ideal (α2+ 2, β2+ 5) ⊂ Q(i)hα, βi. We obtain our morphism φ : D → M2(C) as above. The

kernel of this map is a two-sided ideal of D. Since D is a quaternion algebra we get by Theorem 3.0.3 that the kernel must be trivial. So φ is injective.

Remark 5.1.2. Using our embedding φ one can see that the reduced norm of x ∈ (−2, −5)_Q(i)is just the determinant of φ(x) and the reduced trace is the trace of φ(x). Also conjugation of x corresponds to taking the conjugate transpose of φ(x) (i.e. φ(x) = φ(x))t. Lemma 5.0.4. For φ as above. The map D ⊗_Q R → M2(C) given by φ ⊗ R is an

isomorphism.

Proof. The dimension over R of D ⊗QR and M2(C) both equal 8. Since the map is linear

the result follows.

Later it will be essential that the “standard involution” (i.e. conjugate transpose) is positive.

Proposition 5.0.2. Let 0 6= M ∈ M2(C). Then Tr(M M t

) ≥ 0. In other words, the involution M 7→ Mt is positive.

Proof. For M = a b c d

we have that Tr(M Mt) = aa + bb + cc + dd. This is clearly positive is M 6= 0.

We will define another involution on D. We define ∗ : D → D, x 7→ αxα−1. On the standard basis 1, α, β, γ := αβ we have the following identities:

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Another way to define ∗ is using our embedding φ : D → M2(C). Then for

a b c d

∈ Im(φ) we have that ∗ “simplifies” to

a b c d ∗ = i 0 0 −i a b c d t −i 0 0 i .

This can be more useful when computing g∗ in the image of φ which we will be doing later.

We will construct a Hermitian form on D using our involution ∗.

Definition 5.0.1. Let V be a complex vector space (i.e. real vector space V with an endomorphism J which has J2 = 1). A Hermitian form is an R-bilinear map (·, ·) : V × V → C such that for v, w ∈ V we have (Jv, w) = i(v, w) and (v, w) = (w, v). Theorem 5.1. Given a hermitian form (·, ·) = φ − iψ on a complex space V where φ and −ψ denote the real and the imaginary part of (·, ·). Then we have that

1. φ is symmetric, φ(J v, J w) = φ(v, w) 2. ψ is alternating, ψ(J v, J w) = ψ(v, w) 3. ψ(v, w) = −φ(v, J w), φ(v, w) = ψ(v, J w).

Conversely, given φ which satisfies the first and third condition then φ(v, w) + iφ(v, J w) defines a Hermitian form.

Proof. See page 6 in the notes by James Milne on Shimura Varieties [12]. Let v, w ∈ D and define

(v, w) = Tr_D/Q(i)(v∗· w). (5.1) Where v · I · w∗ is computed in the embedding D ,→ M2(C). Furthermore TrD/Q(i) is the

trace on M2(C) restricted to D; this means that we use the embedding D ,→ M2(C) and

restrict to D the trace mapping (a b

c d) 7→ a + d of matrices in M2(C).

Explicitely we have for (a b c d), ( x y z w) ∈ M2(C) that a b c d , ( x y z w) = ax − by − cz + dw. (5.2)

Proposition 5.1.1. The pairing defined in Equation 5.1 is a Hermitian form with respect to the endomorphism J of D given by J (x) = −ix.

Proof. For a b c d , (x yz w) ∈ D we get −i ∗ a b c d , ( x y z w) = i(ax − by − cz + dw).

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5.2. The Group G

Our involution ∗ : D → D extends to an involution on D ⊗_QR for any Q algebra R by defining (d ⊗ r)∗ = d∗ ⊗ r. We will make no distinction between ∗ on D and ∗ on D ⊗ R. Note that since D = Q(i) + Q(i)α + Q(i)β + Q(i)γ we have D ⊗Q R ∼=

R(i) + R(i)α + R(i)β + R(i)γ. Therefore ∗ acts in the same way on D ⊗_QR as it does on D (it sends α to −α and acts as the identity on all other basis elements).

Just as “take the additive group” and “take the multiplicative group”, Ga: {Q-algebras} → {Groups} , R 7→ (R, +),

Gm: {Q-algebras} → {Groups} , R 7→ (R, ×),

provide functors from the category of Q-algebras to the category of groups we introduce the following functor.

G : {Q-algebras} → {Groups} , R 7→ {(D ⊗QR)

×_{| g}∗_{g ∈ {1} ⊗ R}×_}.

Lemma 5.1.1. The map G described above is a functor.

Proof. First note that for a Q-algebra R we indeed have that G(R) is a group. The only non-trivial part is to check that if g ∈ G(R) then also g−1 ∈ G(R). For this we note that g−1 = _gnrdgg and nrdg ∈ R×.

Now if φ : R → S is a Q-algebra isomorphism then for g ∈ G(R) the property that g∗g ∈ R×implies φ(g)∗φ ∈ φ(R×) ⊂ S×. Therefore φ descends to a group homomorphism from G(R) to G(S).

In fact G turns out to be “representable” by a Q-algebra. We will however not need this fact. We only need to know that G assigns to each Q-algebra a group G(R) and it does so in a functorial way.

Remark 5.2.1. Given g ∈ G(R) we obtain a map D ⊗ R → D ⊗ R, v 7→ gv. While the hermitian form (·, ·) defined in Equation 5.1 originally is only defined on D it can be extended to D ⊗ R. Denote this extended form by (·, ·) also. It has the property that for v, w ∈ D (gv, gw) = c(g)(v, w). The element c(g) is equal to g∗g ∈ R.

We equip the group G(R) with the following topology. Note that D ⊗QR is a finite

dimensional vector space over the complete field R. All norms over finite dimensional R-vector spaces are equivalent. Therefore there is a unique topology on D ⊗QR which is

compatible with the standard euclidean topology on R. View D ⊗QR as a topological

vector space with respect to this structure. We then have the embedding

(D ⊗_QR)×,→ (D ⊗QR) × (D ⊗QR) (5.3)

given by g 7→ (g, g−1). This allows us to pull back the product topology on (D ⊗_QR) × (D ⊗_QR) back to (D ⊗QR)

×_{. We equip G(R) with the subspace topology.}

In fact one can show that (D ⊗_QR)×⊂ D ⊗QR is open and has the induced structure

from D ⊗_QR. However the way described in Equation 5.3 is the better way because it gives the correct topology when R is replaced with an arbitrary topological ring.

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Lemma 5.1.2. The morphism h : C× → G(R); z 7→ z 0₀ _z_¯

induces a morphism of topological groups C×→ G(R).

Proof. The map is well defined as h(z)∗h(z) is a scalar matrix and hence in G(R). It is clear that h respects the group structure. We need only to show that the map is continuous. The map

˜ h : C×→ (D ⊗_QR) × (D ⊗QR), z 7→ z z , _1/z 1/¯z

is continous when D ⊗_QR is equipped with the topology defined above. Since G(R) has the pull back topology described as in Equation 5.3 the map h is continous also. Definition 5.1.1. We define the topological space X to be the G(R)-conjugacy class of the morphism h. It is a space consisting of morphisms.

We still have to define a topology on X. Note that we have a natural G(R) action on X given by conjugation. Let K∞be the stabilizer of h by this action so that X = G(R)/K∞

by the orbit-stabilizer theorem. Since the image of R× is central under h we have that an element g ∈ G(R) has ghg−1= h if and only if g commutes with h(i) =: I ∈ G(R).

Spelling out what this means gives us that

K∞= {g ∈ M2(C) | a 0_{0 d} , a¯a = d ¯d 6= 0}. (5.4)

Let W ⊂ M2(C) be the subspace of matrices of the form (∗ 0∗ 0). The group G(R) ⊂

GL2(C) has a natural left action on this space given by matrix multiplication. This

action on G(R) is C-linear and therefore it extends to an action on the lines through the origin in W . i.e. G(R) acts on the projective line P(W ) ∼= P1(C). Let = (1 0

0 0) ∈ M2(C)

and define the map

G(R) → P(W ), g 7→ g · C ⊂ W.

We claim that for g ∈ G(R) we have g · C = · C ∈ P(W ) if and only if g ∈ K∞. One

implication is trivial: If g ∈ K∞ then it is of the form g = a 0_{0 d}. Now g = (a 0_{0 0}). Since

a 6= 0 we have indeed that g · · C = · C. Now we check the other implication. We immediately obtain that c = 0. Note that since g ∈ G(R) we have that g∗g = c(g) ∈ R×. We have i 0 −i 0 _{a 0} b d _{−i 0} i 0 _{a b} 0 d = aa ab −ba dd−bb .

Since this must be equal to a scalar matrix c(g)₀ _c(g)0 with c(g) 6= 0 we must have that b = 0. The result follows. We obtain an injection

X = G(R)/K∞,→ P(W ).

We equip X with the structure of a complex manifold through the above injection. We will now give a more explicit description of the image of this injection. We can embed C into P(W ) through the map z 7→ (z 01 0). In this manner we can naturally identify the

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circle S1 in C with a subspace in P(W ). Recall that we have our hermitian form (·, ·) as defined in 5.1. The restriction of this form to W is given by

((a 0

c 0) , (x 0z 0)) = ax − cz.

Therefore this subspace of P1W contains precisely those points z ∈ W such that (z, z) = 0. We denote S1 for this subspace also. We see (z, z) is nonzero if and only if z ∈ P1W \ S1. By Remark 5.2.1 we have that G(R) preserves this form and therefore it will also preserve P1W \ S1. We show that the action is transitive on W \ S1. This is done in two steps. For r ∈ R with r 6= 1 we map (1 00 0) to (r 01 0) by the matrix r −11 r . Note that this matrix

indeed belongs to G(R). Now map the matrix (r 0

1 0) to re φi₀ 1 0 by the matrix e φi₀ 1 0. Again, eφi₀

1 0 is invertible and belongs to G(R). This show that G(R) acts transitively

on W \ S”1.

5.3. A Certain Action on the Complex Upper Half Plane

In the last section of this thesis we show that the quotient of P1C − P1R under the group a special subgroup of G(R) is compact. We do this in the same way as is done in the examples of the Ford fundamental domain.

5.3.1. Compact Quotient

Let O be a maximal Z(i)-order of D. Define the group G0(Z) ⊂ (R) by G0(Z) = {g ∈

O)×| g∗_{g ∈ (Z(i))}. We will show the quotient space P}1

C − P1R under G0(Z) is compact.

To do so we calculate enough elements in G0(Z) and determine their isometric circles.

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What is notable is that there are very few circles, not even enough to determine wether this quotient is compact or not. This is due to the fact that it is very difficult to generate elements of G0(Z). This quotient should be compact. Also note that the circles seem to

come in pairs. For any g ∈ G)(Z) the element g−1 has entries which are “similar in size”

to those of g. Therefore, if the computer can find g, it can also find g−1.

Sadly, this is as far as we will go. We tried very hard to generate more elements but there would always be a flaw that popped up somewhere along the way.

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Bibliography

[1] Neukirch, J¨urgen. “Algebraic Number Theory” Springer (1999)

[2] Gouvea, Fernando. “P-adic Numbers: An Introduction”. Springer (1993)

[3] Conrad, Keith. “Quaternion Algebras”. https://kconrad.math.uconn.edu/blurbs/. 23-01-2019.

[4] Vigneras, Marie-France. “Arithm´etique des alg`ebres de quaternions”. Springer (1980). [5] Voight, John. “Quaternion Algebras”.

https://www.math.dartmouth.edu/ jvoight/quat-book.pdf. 21-05-2019. [6] Voight, John. “Computing fundamental domains for Fuchsian groups”.

https://math.dartmouth.edu/ jvoight/articles/funddom-jtnb-fixederrata.pdf. 03-07-2019.

[7] Katok, Svetlana. “Fuchsian Groups”. The University of Chicago Press (1992). [8] Beardon, Alan F. “The Geometry of Discrete Groups”. Springer (1982).

[9] Shimura, Goro. “Introduction to the Arithmetic Theory of Automorphic Functions”. Princeton University Press (1971).

[10] Ford, L.R. “The Fundamental Region for a Fuchsian Group”. ??? (1925).

[11] Atiyah, M.F.; Macdonald, I.G. “Introduction to Commutative Algebra”. Addison-Wesley Publishing Company (1969).

[12] Milne, James. “Introduction to Shimura Varieties”. https://www.jmilne.org/math/CourseNotes/index.html. 21-05-2019.

[13] Milne, James. “Elliptic Curves”. https://www.jmilne.org/math/CourseNotes/index.html. 28-05-2019.

[14] Alsina, Montserrat. Bayer, Pilar. “Quaternion Orders, Quadratic Forms, and Shimura Curves”. The American Mathematical Society (2000).

[15] Reiner, Irving. “Maximal Orders”. London Mathematical Society (1975).

[16] Chenevier, Ga¨etan. http://gaetan.chenevier.perso.math.cnrs.fr/coursihp.html. 21-5-2019.

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[18] Miranda, Rick. “Algebraic Curves and Riemann Surfaces”. American Mathematical Society (1995).

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Populaire samenvatting

The field of algebraic geomtery is concerned with studying objects in space “cut out” by polynomial equations. The most simply are y = ax + b or x2+ y2 = 1. When viewed over the real numbers these correspond to a line and a circle. It turns out that beyond a degree 1 equation, y = ax + b, or a degree two equation, x2+ y2 = 1, this description becomes a lot more complicated. A step in the right direction would be to study a specific type of third degree equation. One of the form y2 = x3+ ax + b. This equation is called an “elliptic curve”. It turns out that an elliptic curve has a group law. The curve with this group law is a “topological group” which is “the same” as the group described in the picture below.

Two points in the blue area are viewed as complex numbers and added accordingly. When you land outside the blue area you can go back by moving ω1or ω2steps horizontally

or diagonally. If we rotate the blue area 90 degrees we should end up with “the same” group. It turns out that two of these groups are the same if we have that the are transformed through in a certain way. It is given by

z 7→ az + b cz + d

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where a, b, c, d are integers such that ac − bd = 1. If we study the group { a b

c d | a, b, c, d ∈

Z, ac − bd = 1} we take a step in the right direction of classifying all elliptic curves. This is done in the study of modular curves. The topic of this thesis are Shimura curves, which are a generalization of modular curves.

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6. Conclusion

First of all I would like to take this oppurtunity to thank Arno Kret, my advisor in this project. Who, despite of his busy schedule, still managed to find the time to meet weekly. These discussions made this thesis so much more interesting.

Secondly, since this is the first time I have ever done some form of independent research. I feel it is important to reflect on this process and the subsequent result. At the start of this project I found it rather hard to figure out what sources were best. This made it very difficult to begin writing, I had simply read too much to condense in a 30 page thesis. Consequently I am not very content with the written result. This does not mean I am dissapointed in the entire thesis. On the contrary, I greatly enjoyed it. I learned so much more than I previously could have imagined. Discovering the connection between hyperbolic geometry and algebraic number theory blew my mind. So much so that I intend to pursue a PhD after completing my masters degree. For that, I am grateful.

The first goal of this thesis was to determine whether a certain qoutient was compact. Though this ultimately failed we have discovered methods that can find Ford fundamental domains. Most notably an algorithm, which we did not treat, by John Voight [6]. Along the way I discovered that another motive Arno Kret gave me this topic to study further was one assumption on the quaternion algebras from which we derive our Fuchsian groups. In all texts treating Shimura curves all quaterion algebras are assumed to be totally real. We did not make this assumption. What was notable but we coud not yet explain was that the Fuchsian group obtained in the process had peculiar elements. A priori the elements are of the form u + aα + bβ + cγ with u, a, b, c ∈ Z(i) and {1, α, β, γ} a standard basis for the quaternion algebra (−2, −5)_Q(i). We however found that only two cases were found. Either u, a are purely imaginary and b, c are real or the converse. This gave us the idea that the Shimura curve we obtained when taking the quaternion algebra (−2, −5)_Q(i) might be a covering of the “classical” one obtained from (2, 5)_Q. We did not

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A. Appendix

A.1. Small exerpt of list of elements of G

0

(Z).

These should be read as follows. For a quaternion algebra (−2, −5)_Q(i) with standard basis {1, α, β, γ}. The entries u, a, b, c are the coefficients of our group element with respect to this basis. That is g = u + aα + bβ + cγ.

u=-1+0.i a=-3+0.i b=0+2.i c=0+0.i u=-1+0.i a=0+0.i b=0+0.i c=0+0.i u=-2+0.i a=0+0.i b=0+-1.i c=0+0.i u=-2+0.i a=5+0.i b=0+-3.i c=0+-1.i u=-2+0.i a=5+0.i b=0+-3.i c=0+1.i u=-2+0.i a=5+0.i b=0+3.i c=0+-1.i u=-2+0.i a=5+0.i b=0+3.i c=0+1.i u=-3+0.i a=0+0.i b=0+0.i c=0+-1.i u=-3+0.i a=0+0.i b=0+0.i c=0+1.i u=-3+0.i a=5+0.i b=0+-2.i c=0+-2.i u=-3+0.i a=5+0.i b=0+-2.i c=0+2.i u=-3+0.i a=5+0.i b=0+2.i c=0+2.i u=-4+0.i a=0+0.i b=0+-1.i c=0+-1.i u=-4+0.i a=0+0.i b=0+1.i c=0+-1.i u=-4+0.i a=0+0.i b=0+1.i c=0+1.i u=-6+0.i a=-3+0.i b=0+-3.i c=0+-1.i u=-6+0.i a=-3+0.i b=0+-3.i c=0+1.i u=-6+0.i a=-3+0.i b=0+3.i c=0+-1.i u=-6+0.i a=-3+0.i b=0+3.i c=0+1.i u=-6+0.i a=5+0.i b=0+-3.i c=0+2.i u=-6+0.i a=5+0.i b=0+3.i c=0+-2.i u=-6+0.i a=5+0.i b=0+3.i c=0+2.i u=-6+0.i a=8+0.i b=0+-1.i c=0+-4.i u=-6+0.i a=8+0.i b=0+-1.i c=0+4.i u=-6+0.i a=8+0.i b=0+-5.i c=0+-2.i u=-6+0.i a=8+0.i b=0+1.i c=0+4.i u=-6+0.i a=8+0.i b=0+5.i c=0+-2.i u=-6+0.i a=8+0.i b=0+5.i c=0+2.i u=0+-1.i a=0+-2.i b=0+0.i c=-1+0.i u=0+-1.i a=0+-2.i b=0+0.i c=1+0.i u=0+-1.i a=0+-3.i b=-2+0.i c=0+0.i u=0+-1.i a=0+-3.i b=2+0.i c=0+0.i

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u=0+-1.i a=0+2.i b=0+0.i c=-1+0.i u=0+-1.i a=0+2.i b=0+0.i c=1+0.i u=0+-1.i a=0+3.i b=-2+0.i c=0+0.i u=0+-1.i a=0+3.i b=2+0.i c=0+0.i u=0+-2.i a=0+-1.i b=1+0.i c=0+0.i u=0+-2.i a=0+-5.i b=-3+0.i c=-1+0.i u=0+-2.i a=0+-5.i b=-3+0.i c=1+0.i u=0+-2.i a=0+-5.i b=3+0.i c=-1+0.i u=0+-2.i a=0+0.i b=-1+0.i c=0+0.i u=0+-2.i a=0+0.i b=1+0.i c=0+0.i u=0+-2.i a=0+1.i b=-1+0.i c=0+0.i

A.2. Mathematica code

This are some functions defined in Mathematica used to generete the circles.

Candidates is a list like the one above, LinComb interprets this list and outputs Mathematica expressions. FirstElements is a list that consists of group elements.

FirstElements = LinComb @@@ Candidates

Selects suitable elements to apply the procedure of the Ford fundamental domain.

SecondElements := Select[FirstElements, Det[#] == 1 && F[#][[1, 1]] == 1 &] Conjugates the group with a suitable transformation. We are now in the situation where the procedure of the isometric circles begin.

Conj[A_, B_] := FullSimplify[1/Det[{A, B}]*MatT.{A, B}.Inverse[MatT]] Elements = Conj @@@ SecondElements We now generate the isometric circles.

NewElements := Select[Elements, #[[2, 1]] /= 0 &] Circles := Table[Circle[ReIm[Simplify[-NewElements[[i]][[2, 2]]/NewElements[[i]][[2, 1]]]], 1/Abs[NewElements[[i]][[2, 1]]]], {i, Length[NewElements]}]

Now we draw them.