MSc Mathematics

Master Thesis

Shimura Curves and Period Domains

Author: Supervisor:Juultje Kok prof. dr. L.D.J. Taelman

Examination date:May 24, 2016

Korteweg-de Vries Institute forMathematics

Abstract

We use quaternion algebras over totally real number fields to construct embeddingsfrom compact Shimura curves into quotients ΓzDpΛq. Here DpΛq is the period domainassociated with a lattice Λ, and Γ is a subgroup of the orthogonal group OpΛq. Weshow that such embeddings exists for lattices of signature p2, nq and totally real numberfields of degree d ď 1

3pn´ 1q. This construction is applied in the context of polarized K3surfaces, where ΓzDpΛq is closely related to the moduli space of polarized K3 surfaces.

Title: Shimura Curves and Period DomainsAuthor: Juultje Kok, juultje.kok@gmail.com, 10025820Supervisor: prof. dr. L.D.J. TaelmanSecond Examiner: dr. M.D. ShenExamination date: May 24, 2016

Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamScience Park 105-107, 1098 XG Amsterdamhttp://kdvi.uva.nl

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Contents

Introduction 4

1 Quadratic forms 91.1 Quadratic modules over a ring . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Quadratic forms over Qp and Q . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Period domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 K3 surfaces 192.1 K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Moduli space of polarized K3 surfaces . . . . . . . . . . . . . . . . . . . . 22

3 Quaternion Algebras 253.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Quaternion algebras as quadratic spaces . . . . . . . . . . . . . . . . . . . 27

4 Fuchsian Groups 314.1 The action of PSL2pRq on the upper half-plane . . . . . . . . . . . . . . . 314.2 The action of Fuchsian groups on the upper half-plane . . . . . . . . . . . 334.3 Fuchsian groups derived from quaternion algebras . . . . . . . . . . . . . . 36

5 Embedding of Shimura curves 415.1 Orthogonal groups of quaternion algebras . . . . . . . . . . . . . . . . . . 415.2 Embedding of H˘ into period domains . . . . . . . . . . . . . . . . . . . . 435.3 Embedding of Shimura curves into ΓzDpΛq . . . . . . . . . . . . . . . . . 455.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Popular summary 52

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Introduction

The goal of this thesis is to construct embeddings of compact Shimura curves into certainquotients of period domains. In order to give a precise formulation of this goal, we firstdiscuss the definitions of period domains and compact Shimura curves.

Period domains and Shimura curves

A lattice is free Z-module of finite rank together with a non-degenerate symmetric bilin-ear form x¨, ¨y : ΛˆΛ Ñ Z. The signature of Λ is the signature of the R-lineair extensionon ΛR :“ Λb R of the bilinear form x¨, ¨y.

Definition. Let Λ be a lattice. The period domain associated with the lattice Λ is thecomplex manifold given by

DpV q :“ tx P PpV bR Cq | xx, xy “ 0, xx, xy ą 0u.

The orthogonal group OpΛq of a lattice consist of the automorphisms of Λ which pre-serve the bilinear form. Note that OpΛq has a natural action on the period domain DpΛq.In this thesis we will consider quotients ΓzDpΛq of period domains DpΛq by torsion freesubgroups Γ Ď OpΛq for lattices Λ with signature p2, nq. In that case, ΓzDpΛq is acomplex manifold.

The group PSL2pRq acts on the upper half-plane H :“ tz P C | Impzq ą 0u viaMobius transformations. Shimura curves are examples of quotients of the upper half-plane by discrete subgroups Γ of PSL2pRq. These quotients have a canonical structureof a Riemann surfaces. For Shimura curves, this discrete subgroup Γ comes from aquaternion algebra over a totally real number field. A quaternion algebra over a field Fis an associative F -algebra A with basis 1, i, j, k satisfying the following relations:

i2 “ a,

j2 “ b,

ij “ ´ji “ k,

where a, b are some elements in F ˚. The reduced norm nrdpxq of an element x “x0 ` x1i` x2j ` x3k in A is given by nrdpxq “ x2

0 ´ ax21 ´ bx

22 ` abx

23.

A totally real number field is a finite field extension F of Q of degree d such thatF bQ R – Rd. If F is a totally real number field with rF : Qs “ d and A is a quaternionalgebra over F , then there are positive integers r, s with r ` s “ d such that

AbQ R “M2pRqr ˆHs,

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where H denote the Hamiltonian quaternions. If r “ 1, then we can associate in thefollowing way a discrete subgroup of PSL2pRq with such quaternion algebras.

Let A be a quaternion algebra over a totally real number field F of degree d. An orderin A is a subring O Ď A such that O – Z4d as Z-modules and O bZ Q – A as vectorspaces over Q. The projection onto the factor M2pRq of the subgroup

O˚,1 :“ tx P O˚ | nrdpxq “ 1u

then induces a discrete subgroup ΓO Ď PSL2pRq.

Definition. A compact Shimura curve is a compact quotient Γ1zH of the upper half-plane by a finite index subgroup Γ1 of ΓO.

Main theorem

In this thesis we will construct embeddings of Shimura curves into the complex manifoldΓzDpΛq for lattices Λ with signature p2, nq and torsion free subgroups Γ of OpΛq of finiteindex. More precisely, we want to prove the following:

Theorem 1. Let Λ be a lattice of signature p2, nq and let Γ be a torsion free subgroupof OpΛq of finite index. Let F be a totally real number field of degree d and let A be aquaternion algebra over F such that

AbQ R –M2pRq ˆHd´1.

Let O be an order in A. Assume that d ď 13pn ´ 1q and that A is a division algebra.

Then there exists an embedding

Γ1zH ãÝÑ ΓzDpΛq

of complex manifolds for some finite index subgroup Γ1 of ΓO. Moreover, the quotientΓ1zH is a compact Shimura curve.

Quaternion algebras over totally real number fields play a crucial role in the construc-tion of the embedding Γ1zH ãÑ ΓzDpΛq. If F is totally real number field of degree dand A is quaternion algebra which satisfies the condition of Theorem 1, then we canassociate a natural quadratic space VA with A, such that PGL2pRq is a subgroup of theorthogonal group OpVAq.

The main idea of the proof is to first construct an PGL2pRq-equivariant isomorphismof complex manifolds

H˘ „ÝÑ DpVAq,

where H˘ “ P1pCqzP1pRq. Then the proof is reduced to constructing an ‘PGL2pRq-equivariant’ embedding of period domains

DpVAq ãÝÑ DpΛq.

This will give an embedding of H into the period domain DpΛq, which will induce anembedding Γ1zH ãÝÑ ΓzDpΛq for a suitable choice of Γ1 Ď ΓO.

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Remark. Both Γ1zH and ΓzDpΛq are examples of Shimura varieties, so we constructembeddings of Shimura curves into Shimura varieties. Since the subject of Shimuravarieties is very wide and difficult, we will not give further details on this subject.

Remark. We can apply Theorem 1 in the context of K3 surfaces. A K3 surface is aconnected, complete and smooth algebraic surfaceX over C, such thatX is diffeomorphicto the zero locus of X4

0 `X41 `X

42 `X

43 in P3. A polarized K3 surface is a K3 surfaces

together with an ample line bundle. It turns out that there is a lattice ΛpK3 and a finiteindex subgroup ΓpK3 Ď OpΛpK3q such that the moduli space M of polarized K3 surfacesmaps injectively into the complex manifold ΓpK3zDpΛpK3q. Moreover, the image of themoduli space M is a dense and open subset of ΓpK3zDpΛpK3q, which makes this quotientΓpK3zDpΛpK3q particular interesting.

In this case the signature of Λ is p2, 19q, so we can apply Theorem 1 and constructembeddings of compact Shimura curves into ΓpK3zDpΛpK3q. One could use Brieskorn-resolutions to modify the image of a compact Shimura curve in ΓpK3zDpΛpK3q to obtainfamilies of smooth K3 surfaces over compact curves.

Structure of this thesis

Chapter 1: Quadratic forms

This chapter is a short summary of the basic definition and results of quadratic formsand quadratic spaces over Z and Q. We describe the classification of quadratic spacesover Qp and Q. In Proposition 1.2.8 we use this classification to show that if V and Eare two quadratic spaces over Q with dimpV q´dimpEq ě 3, then there exists a quadraticspace W such that V – E ‘W as quadratic spaces. This proposition is crucial for theproof of Theorem 1. In the next section we consider lattices and give a classificationof even, unimodular and definite lattices. We end with a description of period domainsassociated with lattices.

Chapter 2: K3 surfaces

First we give a short overview of the basic definitions of K3 surfaces. We describe theHodge structure on the integral cohomology of a K3 surface and introduce the K3 lattice.In the next section we describe the period domain DpΛpK3q of polarized K3 surfaces andthe moduli space M of polarized K3 surfaces. We end with the statement that thismoduli space maps injectively into ΓpK3zDpΛpK3q with an open and dense image. Thischapter is meant as a summary of the theory, so we give references instead of proofs.

Chapter 3: Quaternion Algebras

We start with a short overview of the basic definition and results on quaternion algebras.In the next section we explain how one can associate quadratic spaces with quaternionalgebras. This will be used to construct the quadratic space VA for the proof of Theorem1.

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Chapter 4: Fuchsian groups

We first study the action of PSL2pRq via Mobius transformation on H and then theaction of discrete subgroups of PSL2pRq, which are called Fuchsian groups. We thengive the quotient of the upper half-plane by a Fuchsian group a natural structure of aRiemann surface.

In section 4.3 we consider quaternion algebras over totally real number fields withA bQ R – M2pRq b Hd´1. We explain how one can use orders to associate a Fuchsiangroup ΓO to such quaternion algebras. The most important result in this chapter isTheorem 4.3.6, which states that if A is division algebra, then the quotient ΓOzH is acompact Riemann surface.

Chapter 5: Embedding of Shimura curves

In this chapter we work towards the proof of Theorem 1. In the first section we studyorthogonal groups of quadratic spaces associated with quaternion algebras in more detail.In the next section we construct a PSL2pRq-equivariant embedding of the upper half-plane into the period domain DpVAq. In the last section we show that the period domainDpVAq is a complex submanifold of the period domain DpΛq. In Theorem 5.3.5 we thenshow that Γ1 :“ ΓO XΓ has a finite index in the Fuchsian group ΓO. After that we haveproven enough to be able to give the proof of Theorem 1.

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Acknowledgments

Finishing a thesis is never easy and therefore I would like to express my gratitude towardsa couple of people. First of all I would like to thank my supervisor Lenny Taelman, forall his time every week despite his ridiculous agenda, for reading many versions of thisthesis and giving a lot of feedback, for being an enthusiastic lecturer, and most of all,for helping me to find (and get!) a nice PhD-position. I am really glad he came to theUvA, I couldn’t ask for a more dedicated supervisor.

I would also like to thank Ben Moonen, because without him I would have never evenconsidered pursuing a PhD. In every conversation we have, he always make me realizewhy I put so much effort in understanding mathematics.

Then I would like to thank my fellow students, for the nice conversation and discus-sions, from the best way to cut oranges, to the bio-industry and general thesis frustra-tions, during lunch breaks in the first semester. I especially thank Simon, for all thediscussions about Leuven versus Edinburgh, and Didier for cooking nice meals.

Lastly, I would like to express my gratitude towards Tim, for being there.

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1 Quadratic forms

The aim of this chapter is to give a short summary of the basic definitions and results ofquadratic forms. We therefore do not give a lot of detailed proofs, except for Proposition1.2.8. This proposition is not very difficult, but we need it in Chapter 5 and it is usuallynot proven in literature. A good reference for a more complete overview on quadraticforms is Serre’s book ‘A course in arithmetic’, [Ser73].

In the first section we give a general setup of quadratic forms over an arbitrary ringR. The next section works towards a classification of quadratic forms over Q and Qp

and it ends with the proof of Proposition 1.2.8. The next section covers the case whenR “ Z, and also gives a classification. In the last section we give a description of perioddomains associated with lattices.

1.1 Quadratic modules over a ring

Definition 1.1.1. Let V be a free module over a commutative ring R of rank n. Wecall a map q : V Ñ R a quadratic form if

1) qprxq “ r2qpxq for all r P R and x P V ,

2) the map x¨, ¨y : V ˆ V Ñ R given by xx, yy “ qpx ` yq ´ qpxq ´ qpyq is a bilinearform.

We refer to the pair pV, qq as a quadratic module over R. The map x¨, ¨y : V ˆ V Ñ R iscalled the associated bilinear form with q.

Remark. Sometimes, when the quadratic form is clear from the context, we omit q inthe notation and refer to V as a quadratic module. If R is a field, we call pV, qq also aquadratic space.

Proposition 1.1.2. Let R be a commutative ring and let V be a free module over R offinite rank n. If 2 is a unit in R, then the map

tquadratic forms on V u ÝÑ tsymmetric, bilinear forms on V u

q ÞÝÑ xx, yy “ qpx` yq ´ qpxq ´ qpyq

is a bijection.

Proof. Note that we have xx, xy “ qp2xq ´ 2qpxq “ 2qpxq. So if 2 is a unit in R, weindeed have a bijection.

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Definition 1.1.3. Let pV, qq and pV 1, q1q be quadratic modules over R. A morphism ofR-modules φ : V Ñ V 1 is called a metric morphism if q1 ˝ φ “ q. If a metric morphismφ : V Ñ V 1 is bijective, we call the quadratic modules pV, qq and pV 1, q1q equivalent.

Definition 1.1.4. Let pV, qq be a quadratic module over R. We call the quadratic formq non-degenerate if the associated bilinear form is non-degenerate. In other words, q isnon-degenerate if the map V Ñ HomRpV,Rq given by v ÞÑ xv, ¨y is injective.

Definition 1.1.5. Let V be a free R-module of finite rank n and let te1, . . . , enu be abasis for V . Let q be a quadratic form on V . Then the matrix M given by

Mij :“ pxei, ejyqij

is called the associated matrix with q, with respect to the basis teiu. Note that the asso-ciated bilinear form is given by xx, yy “ xtMy. The discriminant dpqq of the quadraticform q is then defined by

dpqq :“ detpMq P RR˚,2,

where R˚,2 denote all the squares in R˚.

Remark. Note that a change of basis only changes the discriminant by a square, hencedpqq does not depend on the choice of basis for V . In other words, it is an invariant ofthe quadratice module pV, qq.

Remark. If R is an integral domain, then the quadratic form q is non-degenerate if andonly if dpqq ‰ 0.

Definition 1.1.6. Let pV, qq and pV 1, q1q be quadratic modules over R. The orthogonalsum of V and V 1 is the quadratic module given by the direct sum V ‘V 1, together withthe quadratic form

q ` q1 : V ‘ V 1 ÝÑ R

px, x1q ÞÝÑ qpxq ` q1px1q.

An orthogonal basis for V is a basis te1, . . . , enu for the module V such that xei, ejy “ 0for all i ‰ j.

Remark. Let pV, qq be a quadratic module with basis te1, . . . , enu. Define the numbersai :“ xei, eiy and consider the quadratic modules pRei, qiq with qipxq “ aix

2. Then theset te1, . . . enu is an orthogonal basis for V if and only if

V – Re1 ‘ ¨ ¨ ¨ ‘Ren

as quadratic modules.

Proposition 1.1.7. Let F be a field with charpF q ‰ 2. Then every quadratic spacepV, qq over F has an orthogonal basis. In other words

pV, qq – pFn, a1x21 ` . . .` anx

2nq

for some ai P F .

Proof. See [Ser73, thm.VI.1.1, p.30].

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1.2 Quadratic forms over Qp and Q

We denote by V the collection of all prime numbers p together with 8 and we defineQ8 :“ R. Recall the definition of the Hilbert symbol pa, bqv of two elements a, b P Q˚v :

pa, bqv “

#

1 if z2 ´ ax2 ´ by2 “ 0 has a non-trivial solution in Q3v,

´1 otherwise.

Note that the number pa, bqv does not change when a and b are multiplied by squares inQ˚v . So the Hilbert symbol can also be seen as map from Q˚vQ

˚,2v ˆQ˚vQ

˚,2v to t˘1u.

Definition 1.2.1. Let pV, qq be a quadratic space over Qv of rank n and let te1, . . . , enube an orthogonal basis for V . Denote by ai the number ai :“ xei, eiy. We define thenumber εpqq by

εpqq :“ź

iăj

pai, ajqv.

Proposition 1.2.2. Let pV, qq be a non-degenerate quadratic space over Qv of rank nand let te1, . . . , enu be an orthogonal basis for V . Then the number εpqq does not dependon the choice of orthogonal basis, hence εpqq is an invariant of pV, qq.

Proof. See [Ser73, thm. IV.2.5, p.35]

We can classify all non-degenerate quadratic spaces over Qp using this epsilon invariantand the discriminant, as the following theorem shows.

Theorem 1.2.3. Let p be a prime number. Two non-degenerate quadratic spaces pV, qqand pV 1, q1q over Qp are equivalent if and only if they have the same rank, same discrim-inant and same invariant ε.

Proof. See [Ser73, thm. IV.2.7,p.39]

Remark. We can classify (non-degenerate) real quadratic spaces pV, qq using only oneinvariant. Namely, by Proposition 1.1.7 we have q “ a1x

21 ` . . .` anx

2n for some ai P R.

Moreover, we can perform a change of basis such that ai “ ˘1 for all i, so the quadraticform q is equivalent to

x21 ` . . .` x

2r ´ y

21 ´ . . .´ y

2s ,

for some integers r, s. Sylvester’s law of inertia then states that this pair pr, sq is aninvariant of pV, qq.

Definition 1.2.4. Let pV, qq be a real non-degenerate quadratic space with q equivalentto the form x2

1 ` . . .` x2r ´ y

21 ´ . . .´ y

2s . The pair pr, sq is called the signature of pV, qq.

We call q definite if r “ 0 or s “ 0, otherwise we call q indefinite.

We will now explain how non-degenerate quadratic spaces over Q can be classified.

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Definition 1.2.5. Let pV, qq be a quadratic space over Q. Consider the vector spaceVQv over Qv given by

VQv :“ V bQ Qv.

The Qv-linear extension of the polynomial q : V Ñ Q defines a quadratic form q : VQv ÑQv, so pVQv , qq is a quadratic space over Qv.

We denote the discriminant of VQv by dvpqq, and the epsilon invariant by εvpqq. Notethat dvpqq equals the image of dpqq under the map

Q˚Q˚,2 ÝÑ Q˚vQ˚,2v .

We call dvpqq, εvpqq and the signature pr, sq the local invariants of the quadratic spacepV, qq.

Theorem 1.2.6. Let pV, qq and pV 1, q1q be non-degenerate quadratic spaces over Q withsignatures pr, sq respectively pr1, s1q. Then the following statements are equivalent:

(i) pV, qq – pV 1, q1q,

(ii) all the local invariants of pV, qq and pV 1, q1q are the same,

(iii) dpqq “ dpq1q, pr, sq “ pr1, s1q, and εvpqq “ εvpq1q for all v P V.

Proof. See [Ser73, thm. VI.3.9, p.44] and [Ser73, cor. VI.3, p.44].

Proposition 1.2.7. Let pV, qq be a non-degenerate quadratic space over Q of rank n.Then the local invariants of pV, qq satisfy the following relations:

1) εvpqq “ 1 for almost all v P V andś

vPV εvpqq “ 1,

2) if n “ 1, then εvpqq “ 1,

3) if n “ 2 and dvpqq “ ´1 in Q˚vQ˚2v , then εvpqq “ 1,

4) r, s ě 0 and r ` s “ n,

5) d8pqq “ p´1qs in R˚R˚,2,

6) ε8pqq “ p´1qsps´1q2 in R˚R˚,2.

Moreover, if d, εv and pr, sq satisfy the relations (1) to (6) above, then there existsa quadratic space pV, qq of rank n over Q with invariants dpqq “ d, εvpqq “ εv andsignature pr, sq.

Proof. See [Ser73, rmk. IV.3, p.44] and [Ser73, prop. IV.3.7, p.44].

We end this section with a sufficient condition for when there exists an orthogonalcomplement of a quadratic space over Q. This statement will be used later on in Chapter5.

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Proposition 1.2.8. Let pV, qV q and pE, qEq be non-degenerate quadratic spaces over Qsuch that dimpV q ě dimpEq ` 3. Then there exists a quadratic space pW, qW q over Qsuch that

V – E ‘W

as quadratic spaces.

For the proof of this proposition we need the following lemma:

Lemma 1.2.9. Let pE, qq and pE1, q1q be non-degenerate quadratic spaces over Q withlocal invariants dpqq “ d, pr, sq and εvpqq “ εv, respectively dpq1q “ d1, pr1, s1q and εvpq

1q “

ε1v. Then the local invariants of the orthogonal sum pE ‘ E1, q ` q1q are given by

dpq ` q1q “ d ¨ d1,

prE‘E1 , sE‘E1q “ pr ` r1, s` s1q,

εvpq ` q1q “ εv ¨ ε

1v ¨ pd, d

1qv.

Here we use the image of d and d1 under the map Q˚Q˚,2 Ñ Q˚vQ˚,2v in pd, d1qv.

Proof. Choose orthogonal bases te1, .., enu for pE, qq, and ten`1, .., en`mu for pE1, q1qand define the numbers ai :“ xei, eiy. Note that the discriminant of q is then given byd “ ra1 ¨ . . . ¨ ans in Q˚Q˚,2, and d1 “ ran`1 ¨ . . . ¨ an`ms in Q˚Q˚,2.

By definition of the orthogonal sum, the matrix associated with E ‘ E1 with respectto the basis teiu

n`mi“1 is given by the diagonal matrix M “ diagpa1, . . . , an`mq. So it is

immediately clear that we have:

dpq ` q1q “ d ¨ d1,

sE‘E1 “ s` s1, and

rE‘E1 “ r ` r1.

The episilon invariant is given by εvpq ` q1q “ś

iăjpai, ajqv. We can split the productup into three parts:

εvpq ` q1q “

ź

iăj1ďiďn1ďjďn

pai, ajqvź

iăjn`1ďiďn`mn`1ďjďn`m

pai, ajqvź

iăj1ďiďn

n`1ďjďn`m

pai, ajqv.

Note that the first and second factor of the product are just the invariants εv andε1v. Using the property paa1, bqv “ pa, bqv ¨ pa

1, bqv of the Hilbert symbol (see [Ser73,prop.III.1.2, p.20]), we can rewrite the last factor in the following way:

ź

pai, ajqv “ pa1, an`1qv ¨ pa1, an`2qv ¨ . . . ¨ pan, an`mqv

“ pa1, an`1 ¨ ¨ ¨ an`mqv ¨ . . . ¨ pan, an`1 ¨ ¨ ¨ an`mqv

“ pa1 ¨ ¨ ¨ an, an`1 ¨ ¨ ¨ an`mqv

Therefore we indeed have εvpq ` q1q “ εv ¨ ε

1v ¨ pd, d

1qv.

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We also need the following property of the Hilbert symbol:

Lemma 1.2.10. Let a, b P Q˚. Then pa, bqv “ 1 for almost all v P V andś

vPVpa, bqv “1.

Proof. See [Ser73, thm. III.2.3, p.23]

Proof of Proposition 1.2.8. Let t “ dimpV q and n “ dimpEq and denote the local in-variants of pV, qq respectively pE, q1q by dpqV q “ dV , prV , sV q, and εvpqV q “ εvpV q,respectively dpqEq “ dE , prE , sEq, and εvpqEq “ εvpEq. Define the following numbers:

d :“dVdE,

s :“ sV ´ sE ,

r :“ t´ n´ s, and

εv :“εvpV q

εvpEqpdE , dqv.

We will now verify that these numbers satisfy the relations given in Proposition 1.2.7.

First we verify relation (1). Since both V and E are quadratic modules, there are onlyfinitely many v P V such that εvpV q, εvpW q ‰ 1 by Proposition 1.2.7. Furthermore, byLemma 1.2.10, we have pdE , dqv “ 1 for almost all v P V. Therefore εv “ 1 for almostall v P V. By construction we now have

ź

vPVεvpV q “

ź

vPVεv ¨ εvpEq ¨ pdE , dqv “

ź

vPVεv

ź

vPVεvpEq

ź

vPVpdW , dqv.

By Proposition 1.2.7 and Lemma 1.2.10 we haveź

vPVεvpV q “

ź

vPVεvpEq “

ź

vPVpdW , dqv “ 1,

so we can indeed conclude thatś

vPV εv “ 1. Hence relation p1q is satisfied.

Next we verify relation (5). By Proposition 1.2.7 both dV,8 and dE,8 represent thesame class in R˚R˚2 “ t˘1u if and only if sV ” sE pmod 2q. Therefore d8 is given by:

d8 “dV,8dW,8

“

#

1 if sV ” sE pmod 2q

´1 otherwise.

In other words, we have d8 “ p´1qsV ´sE “ p´1qs, so relation (5) is satisfied.

We now verify relation (6). Let a, b P R, then pa, bq8 “ 1 if a or b is greater than zero,and pa, bq8 “ ´1 if both a and b are less than zero (see [Ser73, thm.III.1.1, p.20]). Inparticular we have:

pdE , dq8 “

#

1 if dE,8 ą 0 or d8 ą 0

´1 if dE,8, d8 ă 0.

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Recall that d8 ă 0 if and only if sV ı sE pmod 2q. Furthermore we have dE,8 “

p´1qsE ă 0 if and only if sE ” 1 pmod 2q. Therefore we have pdE , dq8 “ ´1 if and onlyif both sE ” 1 pmod 2q and s ” 1 pmod 2q. This gives us that pdE , dq8 “ p´1qsEs.

Using Proposition 1.2.7 we can now conclude the following:

ε8 “ ε8pV q ¨ ε8pEq´1 ¨ pdW , dq

´18

“ p´1qpsE`sqpsE`s´1q2 ¨ p´1q´sEpsE´1q2 ¨ p´1q´sEs

“ p´1qsps´1q2.

Hence relation (6) is also satisfied.

Note that r`s “ t´n ě 3, so relation (2) and p3q are immediately satisfied. Relationp4q is satisfied by construction. Hence we can use the second part of Proposition 1.2.7 toconclude that there exist a quadratic space pW, qW q over Q of rank t´n with invariantsdW “ d, prW , sW q “ pr, sq and εvpW q “ εv. Moreover, by construction the invariants ofthe orthogonal sum E‘W are equal to the invariants of V (see Lemma 1.2.9). Theorem1.2.6 now gives us that V –W ‘ E as quadratic spaces.

1.3 Lattices

Definition 1.3.1. A lattice Λ is a free module over Z of finite rank n, together with anon-degenerate symmetric bilinear form

x¨, ¨y : Λˆ Λ Ñ Z.

Remark. If Λ is a lattice, then the map q : Λ Ñ Z given by qpxq “ xx, xy is a (non-degenerate) quadratic form. In other words, pΛ, qq is a quadratic module over Z. Wedenote by dpΛq the discriminant of pΛ, qq.

Definition 1.3.2. Let Λ and Λ1 be two lattices. We call Λ and Λ1 equivalent if they areequivalent as quadratic modules over Z.

Definition 1.3.3. Let Λ be a lattice. We call Λ a unimodular lattice if the map

Λ ÝÑ HomZpΛ,Zqx ÞÝÑ xx, ¨y

is an isomorphism. Note that a lattice Λ is unimodular if and only if dpΛq “ ˘1.

Example 1.3.4. Let Λ “ Z and consider the symmetric bilinear form xx, yy “ 2xy.The map Λ Ñ HomZpΛ,Zq – Z is then just multiplication by 2. Hence the bilinear formis non-degenerate, but Λ is not a unimodular lattice.

Definition 1.3.5. A lattice Λ is called even if the associated quadratic form q : Λ Ñ Ztakes value in 2Z.

15

Definition 1.3.6. Let Λ be a lattice. Then the vector space ΛR :“ ΛbZR together withthe R-linear extension q : ΛR Ñ R of the associated quadratic form of Λ is a quadraticspace over R. If pΛR, qq has signature pr, sq, then the integer τpΛq :“ r ´ s is called theindex of Λ.

This section will be ended with a classification of all even unimodular indefinite lat-tices. In order to do that, we first give some important examples of unimodular lattices.

Example 1.3.7. The hyperbolic plane U is the lattice given by eZ‘fZ with the bilinearform given by xe, fy “ 1 and xe, ey “ xf, fy “ 0. The associated quadratic form is givenby qpx1e ` x2fq “ 2x1x2, so the lattice U is even. The discriminant of U is given bydpUq “ ´1 and the index of U is τpUq “ 0.

Example 1.3.8. The E8-lattice is given by Z8 together with the bilinear form definedby the matrix

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

2 0 ´1 0 0 0 0 00 2 0 ´1 0 0 0 0´1 0 2 ´1 0 0 0 00 ´1 ´1 2 ´1 0 0 00 0 0 ´1 2 ´1 0 00 0 0 0 ´1 2 ´1 00 0 0 0 0 ´1 2 ´10 0 0 0 0 0 ´1 2

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

The discriminant of E8 is given by dpE8q “ 1, so the lattice is unimodular. The latticeE8 is also even and the index is given by τpE8q “ 8.

The matrix above is the Cartan matrix of the root system associated with the Dynkindiagram E8. The other Dynkin diagrams of ADE-type give also rice to lattices, butonly E8 gives rise to a unimodular lattice.

Example 1.3.9. Let Λ be a lattice and let m be an integer. The twist Λpmq of Λ is thelattice given by Λ together with the bilinear form

xx, yyΛpmq :“ m ¨ xx, yyΛ.

Theorem 1.3.10. Let Λ be an indefinite, even unimodular lattice of signature pr, sq.Then τpΛq “ 0 pmod 8q and we have:

(i) If τpΛq ě 0, then Λ – U‘s ‘ E‘ τ

88 .

(ii) If τpΛq ă 0, then Λ – U‘r ‘ E8p´1q‘´τ8 .

Proof. See [Ser73, cor.V.2.1, p.53] and [Ser73, thm.V.2.5, p.54].

16

1.4 Period domains

Definition 1.4.1. Let pV, qq be a real quadratic space. The period domain associatedwith V is the following complex manifold:

DpV q :“ tx P PpV bR Cq | xx, xy “ 0, and xx, xy ą 0u Ď PnpCq.

Let Λ be a lattice and define ΛR :“ Λ bZ R. The period domain associated with thelattice Λ is the complex manifold given by DpΛRq.

Definition 1.4.2. Let pV, qq be a quadratic module over a ring R. The orthogonal groupof V is then defined as:

OpV q :“ tg P AutpV q : qpgxq “ qpxq for all x P V u.

We denote by SOpV q the following subgroup of the orthogonal group of V :

SOpV q “ tg P OpV q : detpgq “ 1u.

Remark. If V is real quadratic space of signature pr, sq, then the orthogonal group of Vis sometimes denoted by Opr, sq, as V is completely determined by its signature. Notethat for a definite real quadratic space, the orthogonal group Opnq coincides with theusual definition of Opnq.

Let Λ be a lattice. Then the orthogonal group OpΛq is a subgroup of the real orthogonalgroup OpΛRq. Note that the group OpΛRq has a natural action on the period domainDpΛRq associated with Λ. The following proposition gives another description of theperiod domain associated with Λ using orthogonal groups.

Proposition 1.4.3. Let Λ be a lattice of signature pr, sq. Then the associated perioddomain DpΛRq is isomorphic to

Opr, sqpSOp2q ˆOpr ´ 2, sqq

Proof. Denote by Grpop2,ΛRq the submanifold of the Grassmanian Grp2,ΛRq given by

Grpop2,ΛRq :“ tplanes P Ď ΛR | P is oriented, x¨, ¨y|P is positive definiteu

Let rzs “ rx`iys be a point in DpΛRq. Then we have xz, zy “ xx, xy´xy, yy`2ixx, yy “ 0.So x and y are orthogonal to each other and xx, xy “ xy, yy. From the condition xz, zy ą 0it then follows that xx, xy “ xy, yy ą 0. Hence Pz :“ Rx‘ Ry is a point in Grpop2,ΛRq.

On the other hand, if P “ Re1‘Re2 is a point in Grpop2,ΛRq, then one easily checksthat re1 ` ie2s is a point in the period domain DpΛRq. This gives us that

DpΛRq – Grpop2,ΛRq

as complex manifolds.

17

Now choose an isomorphism ΛR – Rr`s. The orthogonal group Opr, sq has a naturalaction on Grpop2,Rr`sq – Grpop2,ΛRq and it is easy to see that this action is transitive.Let v1 and v2 be the first two basis vectors of Rr`s. Then we have

StabOpr,sqpRv1 ‘ Rv2q “ tg P Opr, sq | gpviq P Rv1 ‘ Rv2 and detpgq “ 1u

“ SOp2q ˆOppRv1 ‘ Rv2qKq

“ SOp2q ˆOpr ´ 2, sq

Therefore we have Grpop2,ΛRq – Opr, sqpSOp2qˆOpr´2, sqq, and we can conclude thatDpΛRq is isomorphic to Opr, sqpSOp2q ˆOpr ´ 2, sqq.

Let Λ be a lattice and let Γ be a subgroup of OpΛq. The following proposition showswhen the quotient ΓzDpΛq is a complex manifold, which will be used in Chapter 5.

Proposition 1.4.4. Let Λ be a lattice of signature p2, nq. Let Γ be a torsion free subgroupof OpΛq. Then the action of Γ on the period domain DpΛq is free and the quotient ΓzDpΛqcarries a natural structure of a complex manifold.

For the proof of this proposition we need the following lemma:

Lemma 1.4.5. Let G be a locally compact group. Let H be a subgroup of G and K be acompact subgroup of G. Then the action of H on the quotient GK has discrete orbitsand finite stabilizers if and only if H is a discrete group of G.

Proof. See [Wol11, lem.3.1.1,p.98]

Proof of Proposition 1.4.4. By Proposition 1.4.3 we have

DpΛRq – Op2, nqpSOp2q ˆOpnqq.

Note that both SOp2q and Opnq are compact, so SOp2qˆOpnq is a compact subgroup ofOp2, nq. The group Γ is a subgroup of OpΛq, so in particular it is a discrete subgroup ofthe real orthogonal group Op2, nq. Thus by Lemma 1.4.5 the action of Γ on DpΛRq hasdiscrete orbits and finite stabilizers. Since Γ is torsion free, every stabilizer is trivial, sowe can conclude that the action of Γ is free.

The orbits of Γ are discrete, so every point x has an open neighbourhood Ux such thatΓxX Ux “ txu. Therefore gUx X Ux “ H for all g ‰ id in Γ. These neighbourhoods Uxcan be used as holomorphic charts for the quotient ΓzDpΛRq. See [Mil16, prop.3.1, p.32]for a more detailed explanation.

18

2 K3 surfaces

The goal of this chapter is to give a description of the moduli space of (polarized) K3surfaces. This chapter is meant as a summary, so we do not give detailed proofs. Themain reference we use is the book ‘Lecutures on K3 surfaces’ by Huybreghts [Huy16].

In the first section we explain the notion of a K3 surface, the Hodge structure of aK3 surface and we introduce the K3 lattice. In the next section we motivate why wewant to study polarized K3 surfaces. We then define the period domain of polarized K3surfaces and give a description of the moduli space of polarized K3 surfaces.

2.1 K3 surfaces

Definition 2.1.1. Let k be a field. A complete non-singular variety X over k of dimen-sion 2 such that Ω2

Xk – OX and H1pX,OXq “ 0 is called an (algebraic) K3 surface.Here we mean by a variety X over k a seperated, geometrically integral scheme of finitetype over k.

Definition 2.1.2. A complex K3 surface is a compact, connected complex manifold Xof dimension 2 such that Ω2

X – OX and H1pX,OXq “ 0.

Remark. Via the GAGA principle one can associate to every variety X over C a complexmanifold Xan, whose set of points are just the closed points in X. Moreover, with everycoherent sheaf F on X we can associate a sheaf Fan on Xan such that for exampleOanX – OXan and Ωan

XC – ΩXan . If X is projective, we also have

H˚pX,Fq – H˚pXan,Fanq.

Since every smooth complete variety of dimension 2 is projective, every algebraic K3surface is projective. Hence all algebraic K3 surfaces over C give rise to a complex K3surface Xan. In this way every projective complex K3 surface can be obtained. However,it is important to keep in mind that are also non-projective complex K3 surfaces. Sothere are more complex K3 surfaces than algebraic K3 surfaces over C.

Example 2.1.3. Let k be a field and let X be a smooth quartic in P3k. By the adjunction

formula (see [Har77, prop.II.8.20, p.182]) we have

Ω2X – Ω2

P3 b LbOX ,

where L is the associated invertible sheaf with X (seen as a divisor on P3). The canonicalsheaf on P3 is given by Ω2

P3 – OP3p´4q (see [Har77, exa.II.8.20.1, p.182]). Since X is a

19

quartic surface, we have L – OP3p4q. Therefore we can conclude that

Ω2X – OP3p´4q bOP3p4q bOX

– OX .

Denote by i the closed immersion of X into P3. The ideal sheaf of X is given byIX – L_ – OP3p´4q. So the following sequence

0 ÝÑ OP3p´4q ÝÑ OP3 ÝÑ i˚OX ÝÑ 0

is short exact, which induces a long exact sequence in cohomology:

. . . ÝÑ H1pP3,OP3q ÝÑ H1pP3, i˚pOXqq ÝÑ H2pP3,OP3p´4qq ÝÑ . . .

Since i : X Ñ P3 is a closed immersion, we have

H1pP3, i˚pOXqq – H1pX,OXq.

Moreover, we have H2pP3,OP3p´4qq “ 0 and H1pP3,OP3q “ 0 (see [Har77, thm.II.5.1,p.225]). Therefore we can conclude that H1pX,OXq “ 0, and hence every non-singularquartic surface in P3 is a K3 surface.

Let V be a free module over Z of rank r. Recall that a Hodge structure of weight non V is a decomposition of the complex vector space VC :“ V bZ C given by:

VC “à

p`q“n

V p,q,

such that V p,q “ V q,p. An important example of Hodge structures is the cohomology ofcompact kahler manifolds, given by

HnpX,Cq “à

p`q“n

Hp,qpXq,

where Hp,qpXq is the Dolbeault cohomology HppX,ΩqXq, see [Voi02, cor.6.10, p.142]. It

turns out that every complex K3 surface is a Kahler manifold, see [Huy16, sec.7.3.2,p.131].

Proposition 2.1.4. Let X be a complex K3 surface. Define the numbers hp,q :“dimCH

ppX,ΩqXq. Then the Hodge diamond of X is given by

h0,0 1

h1,0 h0,1 0 0

h2,0 h1,1 h0,2 “ 1 20 1

h1,0 h0,1 0 0

h0,0 1

20

Proof. See [Huy16, sec.1.2.4, p.12]

By Serre-duality we have H2pX,OXq – H0pX,OXq and since X is compact this givesus that H2pX,OXq – C. So the short exact sequence of sheaves 0 Ñ Z Ñ OX Ñ O˚Xinduces on cohomology the exact sequence

0 ÝÑ PicpXq ÝÑ H2pX,Zq ÝÑ C.

Both C and the PicpXq are torsion free (see [Huy16, rem.1.2.5, p.12]), so it follows thatH2pX,Zq is also torsion free. Therefore, using Proposition 2.1.4, we can conclude thatH2pX,Zq – Z22. Note that the toplogical intersection form (given by the cupproduct)is a symmetric bilinear map

x¨, ¨y : H2pX,Zq ˆH2pX,Zq ÝÑ H4pX,Zq – Z,

which gives H2pX,Zq the structure of a lattice.

Proposition 2.1.5. Let X be a complex K3 surface. Then the integral cohomologytogether with the intersection form is equivalent to the lattice

H2pX,Zq – U ‘ U ‘ U ‘ E8p´1q ‘ E8p´1q.

See Example 1.3.7 and Example 1.3.8 for the definition of the lattices U and E8p´1q.

Proof of Proposition 2.1.5. The lattice H2pX,Zq is even, unimodular and has signaturep3, 19q, see [Huy16, prop.1.3.5, p.17]. Together with Theorem 1.3.10 we can then con-clude that these lattices are equivalent.

Definition 2.1.6. We denote by ΛK3 the lattice U ‘U ‘U ‘E8p´1q‘E8p´1q and callit the K3 lattice. Let X be a complex K3 surface. A marking of X is an isomorphismφ : H2pX,Zq „

ÝÑ ΛK3 of lattices. A pair pX,φq is called a marked K3 surface if φ is amarking of X.

Remark. Let X be a complex K3 surface. Then the C-linear extension of the intersectionform is compatible with the Hodge structure on H2pX,Zq in the following sense. Letσi P H

pi,qipXq, then we have:

xσ1, σ2y ‰ 0 if and only if p1 ` q1 “ p2 ` q2 “ 2.

In particular we see that H1,1pXq is the orthogonal complement of H2,0pXq ‘H0,2pXq.Moreover, for every non-zero σ P H2,0pXq we have xσ, σy ą 0, see [Huy16, Exa.6.1.3,p.100]. This property of the Hodge structure of a K3 surface will be used in the nextsection.

The following theorem shows the importance of Hodge structures on the integralcohomology of K3 surfaces.

Theorem 2.1.7 (Global Torelli). Let X and Y be complex K3 surfaces. Then X isisomorphic to Y if and only if there exists a Hodge isometry H2pX,Zq – H2pY,Zq.

21

Here we mean by a Hodge isometry an isomorphism of Hodge structures φ : H2pX,Zq ÑH2pY,Zq, which is compatible with the intersection form. So xφpσq, φpτqyY “ xσ, τyXfor all σ, τ P H2pX,Cq.

Proof of Theorem 2.1.7. See [Huy16, thm.7.5.3, p.135]

2.2 Moduli space of polarized K3 surfaces

In view of the Global Torelli theorem, it is natural to look at the following space:

N :“ tpX,φq | X a marked K3 surfaceu „

where pX,φq „ pX 1, φ1q if and only if there exists an isomorphism f : X Ñ X 1 suchthat φ1 ˝ f˚ “ φ. For a complex K3 surface X there are several choices for an isometryφ : H2pX,Zq Ñ ΛK3. So in order to parametrize all isomorphism classes of complex K3surfaces, we want to look at the quotient of N by the orthogonal group OpΛK3, qK3q.

However, it turns out that N is not Hausdorff. Moreover, the action of OpΛK3, qK3q

has dense orbits in N (see [Huy16, prop.7.1.3, p.126]). So this does not give us a ‘nice’moduli space. If we look at K3 surfaces with extra structure (polarized K3 surfaces), wecan avoid these problems. In this section we will describe the moduli space of polarizedK3 surfaces.

Definition 2.2.1. Let X be a projective complex K3 surface and L P PicpXq. A pairpX,Lq is called a polarized K3 surface, if L is an ample line bundle on X. We call pX,Lqa polarized K3 surface of degree 2d, if L is primitive in PicpXq and L2 “ 2d.

Before we can give the description of the moduli space of polarized K3 surfaces ofdegree 2d, we need a couple of definitions.

Definition 2.2.2. Let λ be a primitive vector in ΛK3 with λ2 “ 2d ą 0. We denote byNλ the following set:

Nλ :“

#

pX,φ,Lq

ˇ

ˇ

ˇ

ˇ

ˇ

pX,φq is a marked K3 surface,

L is ample, and φpc1pLqq “ λ

+

„

where c1 : PicpXq Ñ H2pX,Zq denotes the first chern class. A triple pX,φ,Lq is equiv-alent to another triple pX 1, φ1,L1q if and only if there exists an isomorphism f : X Ñ X 1

such that φ1 ˝ f˚ “ φ, and f˚L1 – L.

Remark. The set Nλ carries a natural structure of a 20-dimensional complex mani-fold. One has to use the fact that for any K3 surface X0 there exists a smooth uni-versal deformation DefpX0q (see [Huy16, cor.6.2.7, p.107]. It turns out that all triplespDefpX0q, φ,L0q can be glued along the intersections DefpX0q XDefpX 10q, which inducesa global complex structure on Nλ. Moreover, Nλ is a Hausdorff space, in contrast to N .See [Huy16, sec.6.3.4, p.112] for a more detailed explanation.

22

Definition 2.2.3. Let λ be a primitive vector in ΛK3 with λ2 “ 2d ą 0. Denote by Dλ

the following complex manifold

Dλ :“ tx P PpΛK3 bZ Cq | xx, xy “ 0, xx, xy ą 0, and xλ, xy “ 0u.

We call Dλ the period domain of polarized K3 surfaces of degree d.

Let pX,φ,Lq be a polarized K3 surface in Nλ. Since c1pLq P H1,1pXq, we havexc1pLq, σy “ 0 for all σ in H2,0pXq. Moreover, all σ in H2,0pXq satisfy xσ, σy “0 and xσ, σy ą 0. In other words, φpH2,0pXqq defines a point in Dλ.

Definition 2.2.4. The map Pλ : Nλ Ñ Dλ given by

Pλ : Nλ ÝÑ Dλ

pX,φ,Lq ÞÝÑ φpH2,0pXqq

is called the period map.

Proposition 2.2.5. Let λ be a primitive vector in ΛK3 with λ2 “ 2d ą 0. Then theperiod map Pλ : Nλ Ñ Dλ is an open immersion.

Proof. See [Huy16, thm.6.3.4, p.113] and [Huy16, rem.6.3.6, p.113].

Definition 2.2.6. Let λ be a vector in the K3 lattice. We denote by Γλ the stabilizerof λ in the orthogonal group of ΛK3:

Γλ :“ StabOpΛK3qpλq

The group Γλ has a natural action on Nλ given by g ¨ pX,φ,Lq “ pX, g ˝ φ,Lq and onthe period domain Dλ. The period domain of polarized K3 surfaces is diffeomorphic to

Dλ – Op2, 19qpSOp2q ˆOp19qq,

see Proposition 1.4.3. Since SOp2q ˆ Op19q is compact, the action of Γλ on the perioddomain of polarized K3 surfaces (and hence also on Nλ) has discrete orbits and finitestabilizers, see Lemma 1.4.5. Therefore the quotients ΓλzNλ and ΓλzDλ are not nec-essarily complex manifolds, as they can have singularities. However, they are complexanalytic spaces.

Definition 2.2.7. Let λ be a primitive vector in ΛK3 with λ2 “ 2d ą 0. The modulispace of polarized K3 surfaces of degree 2d is the complex analytic space given by

Mλ :“ ΓλzNλ.

The period map is equivariant with respect to the action of Γλ on Nλ and Dλ andholomorphic, hence it induces a holomorphic map Pλ : Mλ ÝÑ ΓλzDλ.

23

Theorem 2.2.8. Let λ be a primitive vector in ΛK3 with λ2 “ 2d ą 0. Then

Pλ : Mλ “ ΓλzNλ ÝÑ ΓλzDλ

is an open immersion of complex anlytic spaces, and the image is a dense subset ofΓλzDλ.

Proof. See [Huy16, thm.6.3.4, p.113] and [Huy16, rem.6.3.7, p.114].

Remark. One can choose a torsion free subgroup Γ Ď Γλ of finite index such that ΓzNλ

and ΓzDλ are complex manifolds (see [Huy16, prop.6.1.11, p.103]). Note that ΓzNλ isa finite cover of the moduli space Mλ. The embedding ΓzNλ ãÑ ΓzDλ is then an openimmersion of complex manifolds. See also [Huy16, sec.6.4.2, p.117].

24

3 Quaternion Algebras

In this chapter we will first give some basic definitions and results about quaternionalgebras, and in the next section we explain how one can construct a quadratic spaceusing a quaternion algebra. This construction will be used later on in the next twochapters.

A good reference for quaternion algebras are the lecture notes by Voight [Voi16]. Anice short introduction can be found in Conrad’s expository paper [Con16]. For moredetails on the connection with quadratic spaces, we refer to Chapter III of Lam’s book‘Introduction to quadratic forms over fields’ [Lam05].

3.1 Definitions and examples

Definition 3.1.1. Let F be a field. A quaternion algebra is an F -algebra A such thatA is a central simple algebra and dimF pAq “ 4.

Example 3.1.2. Let F be a field with charpF q ‰ 2 and let a, b be elements in F ˚. Con-sider the associative F -algebra A “ pa, bqF with basis t1, i, j, ku, and with the followingrelations:

i2 “ a,

j2 “ b,

k “ ij “ ´ji.

By construction we have dimF pAq “ 4. It is an easy verification to show that the centreof A is given by ZpAq “ F . One can also directly show that every non-trivial ideal I ofA must be equal to A itself. In other words, A “ pa, bqF is a quaternion algebra.

Lemma 3.1.3. Let F be a field with charpF q ‰ 2 and let a, b, u and v be elements inF ˚. Then we have the following isomorphisms:

(i) pa, bqF – pau2, bv2qF ,

(ii) pa, bqF – pb, aqF – pa,´abqF – pb,´abqF .

Proof. Consider the morphism pa, bqF Ñ pau2, bv2qF given by 1 ÞÑ 1, i ÞÑ ui, j ÞÑ vjand k ÞÑ uvk. It is an easy verification that this map is an isomorphism of quaternionalgebras. The isomorphisms in (ii) are given by permutating the basis vectors i, j andk.

Theorem 3.1.4. Let F be a field with charpF q ‰ 2 and let A be a F -algebra. Then thefollowing statements are equivalent:

25

(i) A is a quaternion algebra,

(ii) A is isomorphic to pa, bqF for some a, b P F ˚.

Proof. See [Voi16, prop.2.5.23, p.23].

Example 3.1.5. The ring of quaternions H is isomorphic to the quaternion algebrap´1,´1qR.

Example 3.1.6. Let b be some number in R˚. The ring M2pF q is isomorphic to thequaternion algebra p1, bqF via the map defined by

ˆ

1 00 1

˙

ÞÑ 1,

ˆ

1 00 ´1

˙

ÞÑ i,

ˆ

0 1b 0

˙

ÞÑ j,

ˆ

0 1´b 0

˙

ÞÑ k.

Corollary 3.1.7. Let A be quaternion algebra over R. Then A is either isomorphic toM2pRq, or A is isomorphic to H.

Proof. Use Lemma 3.1.3 and the above examples.

Note that the H is a division algebra (every non-zero element has an inverse), but theelement p 1 1

1 1 q P M2pF q is not invertible, so M2pF q is not a division algebra. These arein fact all the possibilities for a quaternion algebra, as the following theorem shows.

Theorem 3.1.8. Let A be a quaternion algebra over a field F . Then A is either adivision algebra, or A – M2pF q.

Proof. By Artin-Wedderburns theorem (see [Kna07, thm.II2.4, p.86]) we have A –

MnpDq for some division algebraD. Since dimF pAq “ 4 and dimF pMnpDqq “ n2 dimF pDq,there are two possibilities: either n “ 2 or n “ 1. In the first case we have dimF pDq “ 1.Then D – F and hence we have A – M2pF q. In the second case, we have A – D.Therefore we can conclude that A is either a division algebra, or A – M2pF q.

Remark. Even if A is a division algebra, it can always be viewed as an F -subalgebra ofM2pKq, for some field extension K of F , in the following way. Choose an isomorphismA – pa, bqF . Using Lemma 3.1.3 we have pa, bqF p

?aq – pa, bqF p

?aq, and together with

Example 3.1.6 this gives us the following morphism:

pa, bqF ÝÑ M2pF p?aqq

x “ x0 ` x1i` x2j ` x3k ÞÝÑ gx :“

ˆ

x0 ` x1?a x2 ` x3

?a

bpx2 ´ x3?aq x0 ´ x1

?a

˙

.

One can easily show that gx`y “ gx ` gy and gxy “ gxgy, so A is indeed a subalgebra ofthe quaternion algebra M2pF p

?aqq.

26

3.2 Quaternion algebras as quadratic spaces

In order to explain how a quaternion algebra can be given the structure of a quadraticspace, we first need the notion of an involution, which is explained in the next lemma.

Lemma 3.2.1. Let A be a quaternion algebra. There is an unique morphism ˚ : AÑ A,called involution, satisfying the following properties:

(i) 1˚ “ 1,

(ii) pxyq˚ “ y˚x˚ for all x, y P A,

(iii) px˚q˚ “ x and xx˚ P F for all x P A.

Proof. See [Voi16, cor.2.2.15, p.7].

Example 3.2.2. For A “ pa, bqF the involution is given by

x˚ “ px0 ` x1i` x2j ` x3kq˚ “ x0 ´ x1i´ x2j ´ x3k.

For the quaternion algebra M2pF q the involution is given by the adjoint of a matrix, so

if x “

ˆ

a bc d

˙

, then x˚ “

ˆ

d ´b´c a

˙

.

Definition 3.2.3. The reduced norm is the map from A to F defined by:

nrd : A ÝÑ F

x ÞÝÑ xx˚.

The reduced trace is the map form A to F defined by:

trd : A ÝÑ F

x ÞÝÑ x` x˚.

Note that px` 1qpx` 1q˚ “ xx˚ ` x` x˚ ` 1 is an element of F , hence x` x˚ is alsoan element in F . Thus the reduced trace is indeed a map from A to F .

The reduced trace is clearly a linear map and the reduced norm is a multiplicativemap. Furthermore, if x is an element in A˚, then we have nrdpxq nrdpx´1q “ nrdp1q “ 1,so nrdpxq ‰ 0. On the other hand, if nrdpxq ‰ 0, then we have x x˚

nrdpxq “ 1, so x P A˚.

Hence A˚ is given by the elements with non-zero reduced norm.

Remark. One could also define a trace and norm map on A in a different way. For anelement x in A consider the map lx : A Ñ A given by left multiplication with x. Thenlx is an element EndF pAq – M4pF q. So one can use the trace for matrices to definetrpxq :“ trplxq. Similar we can define Npxq :“ detplxq. It is an easy calculation to showthat trpxq “ trplxq “ 2 trdpxq and Npxq “ detplxq “ nrdpxq2. Hence the name ‘reduced’.

27

Remark. If a is an element A˚, then the map from A to A given by x ÞÑ a´1x˚a alsosatisfies the conditions in Lemma 3.2.1. Since the involution is unique, this gives us thatx˚ “ a´1x˚a. Therefore we have

paxa´1q˚ “ a´1pa´1q˚x˚a˚a “ a´1pa´1q˚a˚ax˚ “ x˚.

This gives us that paxa´1q˚ “ ax˚a´1. From this it follows that the reduced norm andreduced trace are invariant under conjugation by elements of A˚.

Example 3.2.4. Using Example 3.2.2, we see that for an element x “ x0`x1i`x2j`x3kin A “ pa, bqF the reduced norm and reduced trace of x are given by:

nrdpxq “ x20 ´ ax

21 ´ bx

22 ` abx

23,

trdpxq “ 2x0.

So given an isomorphism A – pa, bqF , the formulas above determine the reduced normand trace (as the involution is unique).

Example 3.2.5. Using Example 3.2.2, we see that the reduced norm of a matrix x inM2pF q is given by its determinant. The reduced trace of a matrix x “

`

a bc d

˘

is given byx` x˚ “ pa` bq id, hence the reduced trace of x is just the trace of the matrix x.

We can now explain how quadratic spaces can be constructed from quaternion alge-bras.

Proposition 3.2.6. Let F be a field with charpF q ‰ 2. Let A be a quaternion algebraand let λ P F ˚. Then the map qA : A Ñ F given by qApxq :“ λ nrdpxq is a quadraticform on the F -vectorspace A, with associated bilinear form xx, yy “ 1

2λ trdpxy˚q.

Proof. Take an element α P F ˚. Then we have λαxpαxq˚ “ λα nrdpxqα˚ “ α2λnrdpxq,so we indeed have qApαxq “ α2qApxq. Note that we have

qApx` yq ´ qApxq ´ qApyq “ λpnrdpx` yq ´ nrdpxq ´ nrdpyqq

“ λppx` yqpx˚ ` y˚q ´ xx˚ ´ yy˚q

“ λpxx˚ ` yy˚ ` xy˚ ` yx˚ ´ xx˚ ´ yy˚q

“ λpxy˚ ` pxy˚q˚q

“ λ trdpxy˚q.

Hence the associated form x¨, ¨y is given by xx, yy “ 12λ trdpxy˚q. This form is by

definition symmetric, and it is easily verified that it is also bilinear. Hence the mapqApxq “ λ nrdpxq gives the quaternion algebra A the structure of a quadratic space.

For the two real quaternion algebras, this construction gives the following quadraticspaces.

Lemma 3.2.7. Let A be a quaternion algebra over R and let qA be the quadratic formgiven by qApxq “ λ nrdpxq for some λ P R˚. Then we have:

28

(i) If A – M2pRq, then qA has signature p2, 2q.

(ii) If A – H and λ ą 0, then qA has signature p4, 0q.

(iii) If A – H and λ ă 0, then qA has signature p0, 4q.

Proof. Choose an isomorphism A – pa, bqR. One can easily show that the matrix of thequadratic form qA with respect to the basis t1, i, j, ku is given by:

¨

˚

˚

˝

2λ 0 0 00 ´2λa 0 00 0 ´2λb 00 0 0 2λab

˛

‹

‹

‚

Using Lemma 3.1.3 and Example 3.1.5, we can conclude that A – H if and only if botha and b are less than zero. This gives us immediately that the sign of λ determines thesignature of qA. Furthermore, if A – M2pRq, then it is also clear that the signature isgiven by p2, 2q.

Let F be a field with charpF q ‰ 2 and A a quaternion algebra over F . The quadraticspace pA, qAq then has a canonical quadratic subspace of dimension 3, as we will nowexplain.

Definition 3.2.8. Let A be a quaternion algebra. We define the subalgebra A0 of A by

A0 :“ ta P A : trdpaq “ 0u.

The elements in A0 are called the pure quaternions.

Example 3.2.9. For the quaternion algebra A “ pa, bqF the pure quaternions are givenby the elements x “ x0 ` x1i ` x2j ` x3k such that x0 “ 0. For M2pF q the purequaternions are given by the matrices with zero trace.

Lemma 3.2.10. Let F be a field with charpF q ‰ 2 and let A be a quaternion algebraover F . Let λ P F and consider the quadratic space pA, qAq with qApxq “ λ nrdpxq. Then

A – F ‘A0

as quadratic spaces.

Proof. Note that for any element a in A we have trdpa´ 12 trdpaqq “ trdpaq´ trdpaq “ 0.

So every element a P A can be written as a “ x` y with x P F and y P A0.Moreover, if trdpyq “ 0, then y˚ “ ´y. Thus for any x P F and y P A0 we have

qApx` yq “ λpx` yqpx´ yq “ λx2 ` qApyq.

In other words, the decomposition A – F ‘ A0 is an orthogonal decomposition withrespect to quadratic form qApxq “ λ nrdpxq.

29

We end this chapter with an example of a quadratic space induced by a quaternionalgebra over a totally real number field. This example will be used in Chapter 5 and inChapter 4 we will also study quaternion algebras over totally real number fields.

Definition 3.2.11. A field F is called a totally real number field of degree d if F is afield extension of Q with rF : Qs “ d and F bQ R – Rd.

Remark. An equivalent condition for F to be a totally real number field is that F is givenby F – Qrxspfq, where f is a polynomial in Zrxs which has only real roots. Anotherequivalent condition is that every embedding F ãÝÑ C sends F into R.

Let F be a totally real number field. Then an element a P F defines a Q-linearendomorphism la of F via lapbq “ ab. Using this, we can define a trace map trF Q fromF to Q by trF Qpaq :“ trplaq. This allows us to construct a quadratic space over Q froma quaternion algebra over F .

Example 3.2.12. Let F be a totally real number field of degree d and let λ P F ˚. LetA be a quaternion algebra over F . Using Proposition 3.2.6 it is easy to see that thefollowing map gives A the structure of quadratic space over Q:

qA : A ÝÑ Qx ÞÝÑ trF Qpλnrdpxqq.

Since F bQ R – Rd, the algebra AbQ R splits in the following way:

AbQ R – A1R ˆ . . .ˆA

dR,

where AiR are quaternion algebras over the real numbers. By Corollary 3.1.7 we eitherhave AiR – M2pRq or we have AiR – H. So using Lemma 3.2.7, we can easily calculatethe signature of the quadratic space pA, qq.

Let p, q, r, s be positive integers such that p` q “ d and r` s “ p. Now suppose thatAiR – H for i “ 1, . . . , p and AiR – M2pRq for i ą p. Suppose also that λ1, . . . , λr ą 0and λr`1, . . . , λp ă 0. Then the signature of pA, qq is given by

p2q ` 4r, 2q ` 4sq.

30

4 Fuchsian Groups

In this chapter we study discrete subgroups of PSL2pZq, which are called Fuchsian groups.These Fuchsian groups act on the upper half-plane H by Mobius transformation andwe are interested in the quotient of H by Fuchsian groups. We first study the actionof PSL2pRq on the upper half-plane and in the next section we look at the action ofFuchsian groups on H. In Theorem 4.2.5 we prove that the action of a Fuchsian groupon H has discrete orbits and finite stabilizers. This allows us to give the quotient of theupper half-plane by a Fuchsian group a natural structure of a Riemann surface.

In the last section we return to quaternion algebras over totally real number fields.We introduce the notion of an order O in a quaternion algebra and construct a Fuchiangroup ΓO using this order. The most important result of this chapter is Theorem 4.3.6,which gives a sufficient condition for when the quotient ΓOzH is compact.

For more details on Fuchsian groups we refer to the book ‘Fuchsian Groups’ by Katok[Kat92]. The book by Beardon, ‘The Geometry of Discrete Groups’, is also a goodreference.

4.1 The action of PSL2pRq on the upper half-plane

Recall that the group PGL2pCq act on the Riemann sphere P1pCq via Mobius transfor-mations, for g “

“

a bc d

‰

the action is given by

z ÞÑ g ¨ z :“az ` b

cz ` d.

Such a Mobius transformation is a biholomorfic map, and in fact we have PGL2pCq –AutCpP1pCqq. One can show that for any element g “

“

a bc d

‰

in PGL2pRq we have|cz ` d|2Impg ¨ zq “ detpgqImpzq. So the elements of PSL2pRq preserve the upper half-plane

H :“ tz P C | Impzq ą 0u Ď P1pCq,

and it turns out that we have

PSL2pRq – AutCpHq.

The upper half-plane can be equipped with a hyperbolic metric

ds “

a

dx2 ` dy2

y,

where we use coordinates z “ x ` iy. Under this metric the upper half-plane is aRiemannian surface of constant negative curvature ´1. One can show that the elements

31

in PGL2pRq are also isometries with respect to this hyperbolic metric. So the orientationpreserving isometries of H are given by

PSL2pRq – Isom`pHq.

In other words, the holomorphic and the geometric automorphism group of the upperhalf-plane are the same.

Definition 4.1.1. Let g be an element in PSL2pRq. If | trpgq| ă 2, then g is calledelliptic. If | trpgq| “ 2, then g is called parabolic and if | trpgq| ą 2, then g is calledhyperbolic.

Remark. Note that the characteristic polynomial of g is given by ppλq “ λ2´ trpgqλ` 1,so g is elliptic if and only if g has two complex conjugate eigenvalues, g is parabolic ifand only if g has a double (real) eigenvalue, and g is hyperbolic if and only if g has twodistinct real eigenvalues.

The following lemma gives some properties of elliptic, hyperbolic and parabolic ele-ments.

Lemma 4.1.2. Let g ‰ rids be an element of PSL2pRq. Then the following statementsare equivalent:

(i) g is elliptic, resp. parabolic, resp. hyperbolic,

(ii) g has 2 complex (conjugate) fixed points in P1pCq, resp. g has 1 fixed point inP1pRq, resp. g has 2 distinct fixed points in P1pRq,

(iii) g is conjugate to”

cospθq ´ sinpθqsinpθq cospθq

ı

for a unique θ P r0, 2πq, resp. g is conjugate to

˘r 1 10 1 s, resp. g is conjugate to

”

eθ 00 e´θ

ı

for a unique θ P R.

Proof. Let g ““

a bc d

‰

be an element in PSL2pRq. Then z P P1pCq is a fixed point of g ifand only if cz2 ` pd ´ aqz ´ b “ 0. From this equation one can easily verify that piq isequivalent to piiq.

Now suppose that g is a hyperbolic element with fixed points z1 and z2. Let h bethe unique element in PSL2pRq with hpz1q “ 0, hpz2q “ 8 and hpiq “ i. Then we havehgh´1p0q “ 0 and hgh´1p8q “ 8. Therefore the Mobius transformation hgh´1 is givenby hgh´1pzq “ a

dz for some a, b P R, so we have

hgh´1 “

„

eθ 00 e´θ

with θ “ logpadq P R. Note that θ is uniquely determined by g P PSL2pRq.On the other hand, suppose that g is conjugate to

”

eθ 00 e´θ

ı

for some θ P R˚. Then we

have trpgq “ eθ ` e´θ, hence | trpgq| ą 2. So g is indeed a hyperbolic element.The proof for the cases when g is parabolic or g is elliptic are analoguous to the case

when g is hyperbolic.

32

4.2 The action of Fuchsian groups on the upper half-plane

Definition 4.2.1. A Fuchsian group is a discrete subgroup of PSL2pRq.

An obvious example is the discrete subgroup PSL2pZq. The following lemma gives twoother types of examples:

Lemma 4.2.2. Let Γ “ xgy be a cyclic subgroup of PSL2pRq. Then we have:

(i) If g is hyperbolic or parabolic, then Γ is a Fuchsian group.

(ii) If g is elliptic, then Γ is a Fuchsian group if and only if Γ is finite.

Proof. Suppose g is hyperbolic. By Lemma 4.1.2 there is a unique θ in R such that g is

conjugate to”

eθ 00 e´θ

ı

. This gives us the following morphism of groups:

φ : Γ ÝÑ Rgn ÞÝÑ nθ

Note that this map is continuous. Since every cyclic subgroup of R is discrete, this givesus that Γ is indeed a discrete subgroup of PSL2pRq. A similar argument can be used forthe case when g is parabolic.

In the case when g is elliptic, one can define a (continuous) map from Γ to the circleS1. Since any discrete subgroup of S1 is a finite cylic group, this gives us that Γ isFuchsian group if and only if Γ is finite.

The action of a Fuchsian group Γ on the upper half-plane is in general not free.However, we will prove in Theorem 4.2.5 that Fuchsian groups always act ‘properlydiscontinuously’ on H.

Definition 4.2.3. Let G be a group acting on a space X. The action of G is calledproper discontinuous if for all x P X the orbit Gx is discrete, and the stabilizer Gx isfinite.

In the literature one can find several other, sometimes different, notions of properlydiscontinuous actions. The following proposition shows some notions that are equivalentto the definition above.

Proposition 4.2.4. Let G be a group acting by isometries on a metrizable locally com-pact space X. Then the following statements are equivalent:

(i) G acts properly discontinuously on X.

(ii) For all points x in X there exists an open neighbourhood Ux of x such that the settg P G | gUx X Ux ‰ Hu is finite.

(iii) For any compact subset K Ď X and any x P X the set tg P G | gx P Ku is finite.

(iv) For any compact subset K Ď X the set tg P G | gK XK ‰ Hu is finite.

33

Proof. We first show that piq implies piiq. Take a point x in X. Since the orbit Gx isdiscrete, there is an ε ą 0 such that the intersection of the open ball Bεpxq and the orbitGx only consists of the point x. Now define Ux to be the open ball Bε2pxq. Note thatfor every g P G we have gpUxq Ď Bε2pgxq, since G acts by isometries on X. Thus ifgx ‰ x, we have gpUxq XUx “ H. In other words, if gUx XUx ‰ H, then gx “ x. Sincethe stabilizer of x is finite, we can conclude that piq holds.

We now show why piiq implies pivq. Suppose there exists a compact subset K and ainfinite sequence gi of distinct elements in G with giK XK ‰ H. So there are points xiin K with yi :“ gixi P K. Since K is compact, there are converging subsequences of xiand yi. So we may assume without loss of generality that the sequence xi converges tosome x and the sequence yi to some y. Since G acts by isometries on X, we have

||gix´ y|| ď ||gix´ gixi|| ` ||gixi ´ y|| “ ||x´ xi|| ` ||yi ´ y||.

Therefore the sequence gix converges to y. But then every neighbourhood U of y meetsinfinitely many gix, wich contradicts the assumption that piiq holds. Hence we can con-clude that piiq implies pivq.

Now suppose pivq holds. Let K be a compact subset of X and let x be a point inX. Suppose that hx P K for some h P G. Then we can map the set tg P G | gx P Kuinjectively into the set tg P G | gK XK ‰ Hu via g ÞÑ gh´1. By assumption there areonly finitely many g P G such that gK XK ‰ H, so there are only finitely many g P Gwith gx P K. Thus pivq implies piiiq.

Finally we prove that piiiq implies piq. Suppose the orbit Gx is not discrete for somex in X. So there exists a sequence of distinct gi in G such that the sequence gix hasa limit y. Then every neighbourhood U of y meets infinitely many gix. Since X islocally compact, there is a neighbourhood V of x with compact closure. However, theset tg P G | gV X V ‰ Hu then also contains infinitely many elements, which givesa contradiction with piiiq. Furthermore, txu is a compact set, so by assumption thestabilizer of x is finite. Hence piiiq implies piq.

Theorem 4.2.5. A subgroup Γ of PSL2pRq is a Fuchsian group if and only if Γ actsproperly discontinuously on the upper half-plane H.

In order to prove to prove this theorem, we need the following lemma.

Lemma 4.2.6. Let G be a subgroup of PSL2pRq acting properly discontinously on theupper half-plane H. Let z be a fixed point of some element in G. Then there exists aneighbourhood Wz of z such that z is the only fixed point in Wz.

Proof. Let z be a point in H and suppose gz “ z for some g P G. Suppose there is afixed point in every neighbourhood of z. Then we can find a sequence zn converging toz such that gnzn “ zn for some gn P G. The subset B3εpzq is compact, so by Proposition

34

4.2.4, there are only finitely many g in G such that gz P B3εpzq. Hence we can choose nlarge enough such that gnz R B3εpzq and zn P Bεpzq. Then we have:

|gnz ´ z| ď |gnz ´ gnzn| ` |gnzn ´ z| “ |z ´ zn| ` |zn ´ z| ă 2ε,

were we used for the equality that G acts by isometries on H. This gives a contradictionwith |gnz´ z| ą 3ε. Hence there exists a neighbourhood Wz of z such that z is the onlyfixed point in Wz.

Proof of Theorem 4.2.5. We first prove that a Fuchsian group acts properly discontinu-ously on H. Let K Ď H be a compact set and z a point in K. The set tg P PSL2pRq |gz P Ku is a compact subset of PSL2pRq (see [Kat92, lem.2.2.4, p.30] for a detailedexplanation). Since Γ is discrete, this gives us that the set

tg P Γ | gx P Ku “ ΓX tg P PSL2pRq | gx P Ku

is finite. So by Proposition 4.2.4 the action of a Fuchsian group Γ is indeed properdiscontinuous.

Now suppose that a subgroup Γ of PSL2pRq acts properly discontinously on H and Γis not discrete. Then there exists a sequence gn of distinct elements in Γ such that gnconverges to the identity. Using Lemma 4.2.6, one can choose a point z in H which isnot fixed by any element in Γ other than the identity. Then gnz is a sequence of distinctpoints in H, which converges to idpzq “ z. This shows that the orbit of z is not discrete,which gives a contradiction. Hence Γ is a discrete subgroup of PSL2pRq.

Let Γ be a Fuchsian group. Recall that the fixed points of hyperbolic and parabolicelements in PSL2pRq are contained in P1pRq. So if a point x in H is a fixed point forsome g P Γ, then g must be an elliptic element.

Lemma 4.2.7. Let Γ be a Fuchsian group. Then the fixed points of elliptic elements inΓ do not accumulate in H.

Proof. By Theorem 4.2.5 the group Γ acts properly discontinuously on H. Let z be apoint in H and suppose that gz “ z for some g P Γ. Using Lemma 4.2.6 we can find aneighbourhood Wz such that z is the only fixed point in Wz. Therefore the fixed pointsof elliptic elements can not accumulate.

Note that Lemma 4.2.6 shows that the action of Γ near a fixed point is given bymultiplication by e2πian for some integers a and n. Moreover, by the lemma above,these fixed points do not accumulate. One can use this to give the quotient ΓzH acanonical structure of a Riemann surface.

Theorem 4.2.8. Let Γ be a Fuchsian group. Then the quotient ΓzH has a canonicalstructure of a Riemann surface.

Proof. See [Bea83, thm.6.2.1, p.118].

35

4.3 Fuchsian groups derived from quaternion algebras

Definition 4.3.1. Let F be a totally real number field of degree d and let A be aquaternion algebra over F . An order in A is a subring O Ď A such that O – Z4d asZ-modules and O bZ Q – A as vector spaces over Q.

Example 4.3.2. Consider a quaternion algebra A “ pa, bqQ. Using Lemma 3.1.3 we canassume without loss of generality that a, b P Z. Therefore the set

O “ tx0 ` x1i` x2j ` x3k P A : xi P Zu

is a subring of A. The subring O is an order in A. Namely, we have O – Z‘Zi‘Zj‘Zkas Z-module, and since 1, i, j and k are elements in O, we also have O bZ Q – A.

We will now explain how one can associate Fuchsian groups to certain quaternionalgebras. Let F be a totally real number field of degree d and let A be a quaternionalgebra over F such that

AR :“ AbQ R – M2pRq ˆHd´1

as R-algebras. The factor M2pRq is canonical, and we will denote it with AR,mat Ď AR.The term canonical here means that if A Ñ B is an isomorphism, then the inducedisomorphism AR Ñ BR will map AR,mat onto BR,mat. Note that there also is a canonicalsurjective R-algebra homomorphism π : AR AR,mat. Denote by ρ the morphism fromA to AR,mat given by

A AbQ R

AR,mat

i

ρ π

Lemma 4.3.3. The map ρ : AÑ AR,mat defined above is injective.

Proof. If d “ 1, then AR – M2pRq, so ρ is injective. Now if d ě 2, then A is a divisionring (otherwise A would be isomorphic to M2pF q, and hence AR – M2pRqd). Thus, sinceρ is not the zero map, the morphism ρ : AÑ AR,mat is injective.

Definition 4.3.4. Let A be a quaternion algebra over a totally real number field F ofdegree d such that AR – M2pRq ˆHd´1. Let O be an order in A and denote by

O˚,1 “ tx P O˚ : nrdpxq “ 1u

the group of units in O of reduced norm 1. Then we define the group ΓO by

ΓO :“ O˚,1t˘1u.

Remark. Note that the image of O˚,1 under ρ is contained in SL2pRq via an isomorphismM2pRq – AR,mat, so we can view ΓO as a subgroup of PSL2pRq. Furthermore, themorphism ρ : AR Ñ AR,mat is injective, thus we have

ΓO – ρpO˚,1qt˘ idu Ď PSL2pRq.

36

Proposition 4.3.5. Let A be a quaternion algebra over a totally real number field Fof degree d such that AR – M2pRq ˆHd´1 and let O be an order in A. Then the groupΓO Ď PSL2pRq is a Fuchsian group.

Proof. Since O is an order in A we have O – Z4d. This gives us the following commu-tative diagram:

O A AR

Z4d Q4d R4d

– – –

Hence O is discrete in AR and therefore O˚,1 is discrete in A˚,1R . Note that H˚,1 isisomorphic to the compact group SUp2q – S3, so we have

A˚,1R – SL2pRq ˆ SUp2qd´1.

The group SL2pRq is locally compact, so there exist an open neighbourhood V of theidentity in SL2pRq with compact closure. Since O˚,1 is a discrete subgroup of A˚,1Rand SUp2qd´1 is compact, the set pV ˆ SUp2qd´1q X O˚,1 is finite. Therefore the setπppV ˆ SUp2qd´1q XO˚,1q “ V X ρpO˚,1q is a finite subset of SL2pRq and hence ρpO˚,1qis discrete in SL2pRq. Since t˘1u is compact, we can use the same argument to showthat ρpO˚,1qt˘1u is discrete in PSL2pRq and hence ΓO is a Fuchsian group.

The quotient ΓOzH has a canonical structure of a Riemann surface by Theorem 4.2.8.The following theorem states when this quotient is a compact Riemann surface. Thistheorem will be used in Chapter 5.

Theorem 4.3.6. Let A be quaternion algebra over a totally real number field F of degreed such that AR – M2pRq ˆ Hd´1. Let O be an order in A and let Γ be a finite indexsubgroup of the Fuchsian group ΓO. Suppose that A is a division algebra. Then thequotient space ΓzH is a compact Riemann surface.

Remark. If d ě 2, then every quaternion algebra A over F with AR – M2pRqˆHd´1 is adivision algebra. Otherwise A – M2pF q (see Theorem 3.1.8), and hence AR – M2pRqd.Remark. If d “ 1, then the condition that A is a division algebra, is necessary. Namely,let A be the quaternion algebra over F “ Q given by A “ M2pQq. Let O be the orderfrom Example 4.3.2. Then ΓO – PSL2pZq. It is a well-known fact that the quotientPSL2pZqzH is isomorphic to C, and hence non-compact.

We will explain how Theorem 4.3.6 follows from a theorem in Shimura’s book ‘In-troduction to the Arithmetic Theory of Automorphic functions’. In order to state thistheorem, we first introduce some notation. Let F be a totally real number field of degreed and let A be a quaternion algebra over F such that AR – M2pRqˆHd´1. Let R eitherbe the ring R, or Q, or the adele ring A. We define the following two groups:

GpRq :“ A˚ bQ R,

G1pRq :“ tx P GpRq | nrpxq “ 1u,

37

where nr denotes the map from A to Q given by nrpxq “ ´ trF Qpnrdpxqq. This is apolynomial with coefficients in Q, so we can extend it to a map nr: GpRq Ñ R. We cannow state the result from which Theorem 4.3.6 will follow.

Theorem 4.3.7. Let F be a finite field extension of Q. Let A be quaternion algebraover F . Then we have:

(i) G1pQq is a discrete subgroup of G1pAq.

(ii) If A is a division algebra, then the quotient G1pQqzG1pAq is compact.

Proof. See [Shi71, thm.9.1, p.242].

Let O be an order in A and denote by Af the non-archimedean part of A. We denoteby KO the following subgroup of G1pAf q:

KO :“ tx Pź

p

O bZ Zp | nrpxq “ 1u.

From the fact that Zp is open and compact in Qp, it follows that O bZ Zp is open andcompact in pAbQQpq

˚, and hence KO is an open and compact subring of G1pAf q. Nowconsider the following open subgroup of G1pAq:

ΓpOq :“ pKO ˆG1pRqq XG1pQq.

Note that G1pQq is diagonally embedded in G1pAq, so every element in ΓpOq is of theform pγ, γ, . . .q. Denote by Γ8pOq the projection of ΓpOq Ď G1pAq onto G1pRq. Fromthe fact that Z “ p

ś

p Zp ˆ Rq XQ, where the intersection is taken in A, it follows that

Γ8pOq – O˚,1 Ď G1pRq.

Proposition 4.3.8. Let F be a totally real number field of degree d and let A be aquaternion algebra over F such that AR – M2pRq ˆ Hd´1. Let O be an order in A.Suppose that A is a division algebra. Then the quotient space Γ8pOqzG1pRq is compact.

For the proof of Proposition 4.3.8 we need the following statement.

Proposition 4.3.9. Let G be a locally compact group. Let K be a compact subgroup ofG and let Γ be a discrete subgroup of G. Then ΓzG is compact if and only if ΓzGK iscompact.

Proof. See [Shi71, prop.1.1.8, p.4]

Proof of Proposition 4.3.8. Consider the following continuous map:

φ : Γ8pOqzG1pRq ÝÑ G1pQqzG1pAqKO

rg8s ÞÝÑ rpg8, 1, 1, . . .qs

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We first show that φ is a well-defined map. Suppose that g8 “ γ ¨g18 for some γ P Γ8pOq.Then we have

pγ, γ, . . .q ¨ pg18, 1, 1, . . .q ¨ pγ´1, γ´1, . . .q “ pg8, γ, γ, . . .qpγ

´1, γ´1, . . .q

“ pg8, 1, 1 . . .q,

and hence φ is a well-defined map.Now suppose that φprg8sq “ φprg18sq. Then there exist a h P G1pQq and a pk2, k3, . . .q

in KO such that pg8, 1, 1, . . .q “ hpg18, 1, 1, . . .qpk2, k3, . . .q. So we have hkp “ 1, andhence h is an element of KO. In other words, h is an element of ΓpOq, and hencerg8s “ rg

18s in Γ8pOqzG1pRq. Therefore the map φ is injective.

Now consider the projection map

π : G1pQqzG1pAqKO Ý G1pQqzG1pAqpG1pRq ˆKOq

Note that the image of φ is mapped under π onto rp1, 1, 1, . . .qs. Moreover, suppose thatπprg8, g2, g3, . . .qs “ rp1, 1, 1, . . .qs. Then there exists a h P G1pQq and pk8, k2, k3, . . .q inG1pRq ˆKO such that phg8k8, hg2k2, . . .q “ p1, 1, 1, . . .q. This gives us that

rpg8, g2, g3, . . .qs “ rphg8, 1, 1, . . .qs

in G1pQqzG1pAqKO. Hence π´1ptrp1, 1, 1, . . .qsuq is also contained in the image of φ.Since φ is injective, we can now conclude that the quotient ΓzG1pRq is homeomorphicto the inverse image of rp1, 1, 1, . . .qs under π.

By Theorem 4.3.7 and Proposition 4.3.9 the quotient G1pQqzG1pAqK is compact.Since π´1ptrp1, 1, 1, . . .qsuq is closed, this gives us that Γ8pOqzG1pRq is also compact.

Before we can explain how Theorem 4.3.6 follows from Proposition 4.3.8 (and hencefrom Theorem 4.3.7), we need the following proposition.

Proposition 4.3.10. Let G1, and G2 be locally compact groups. Let Γ be a closedsubgroup of G1ˆG2 and let Γ1 be the projection of Γ to G1. Suppose that G2 is compact.The quotient ΓzpG1 ˆG2q is compact if and only if Γ1zG1 is compact.

Proof. See [Shi71, prop.1.1.10, p.4].

Proof of Theorem 4.3.6. Consider the action of SL2pRq on the upper half-plane by Mobiustransformations. The stabilizer of i is then given by

StabSL2pRqpiq “ t`

a bc d

˘

P SL2pRq : ai` b “ ´c` diu“ t

`

a b´b a

˘

P M2pRq : a2 ` b2 “ 1u

“ SOp2q – S1

Since SL2pRq acts transitively on H, the upper half-plane is isomorphic to SL2pRqSOp2qas real Lie group. So by Proposition 4.3.9 it suffices to prove that ρpO˚,1qzSL2pRq iscompact.

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By Proposition 4.3.8 the quotient O˚,1zG1pRq is compact. Recall that ρ : AÑ M2pRqis given by ρ “ π˝i, where i is the inclusion from A into GpRq, and π is the projection ontoM2pRq. So we have ρpO˚,1q “ πpΓ8pOqq. Furthermore, the group H˚,1 is isomorphicto S3, and hence compact. So using Proposition 4.3.10 we can now conclude thatρpO˚,1qzSL2pRq is compact. Therefore ΓOzH is a compact Riemann surface.

Now suppose that Γ is a subgroup of ΓO with rΓO : Γs “ n. Then this gives us aholomorphic map ΓzH ÝÑ ΓOzH between Riemann surfaces of degree n. Since ΓOzHis a compact Riemann surface, this gives us that ΓzH is also compact.

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5 Embedding of Shimura curves

In this chapter we prove Theorem 1 stated in the Introduction, so we will constructembeddings of compact Shimura curves into complex manifolds of the form ΓzDpΛqwith Γ Ď OpΛq, see Definition 1.4.1.

In the first section we orthogonal groups of quadratic spaces pA0, qAq associated witha quaternion algebra A are studied in more detail. In the next section we consider thequaternion algebra A “ M2pRq and explain that PGL2pRq is a subgroup of the orthogonalgroup OpM2pRqtr“0q, and hence PGL2pRq acts on the period domain DpM2pRqtr“0q. Wethen construct a PGL2pRq-equivariant isomorphism from H˘ “ P1pCqzP1pRq to theperiod domain DpM2pRqtr“0q.

In the last section we consider quaternion algebras over a totally real number field ofdegree d such that AR – M2pRq ˆ Hd´1. We show when there exists an ‘equivariant’embedding

H ãÝÑ DpM2pRqtr“0q ãÝÑ DpA0Rq ãÝÑ DpΛq.

This embedding is used for the proof of Theorem 1. We end this section with an appli-cation in the context of K3 surfaces.

5.1 Orthogonal groups of quaternion algebras

Let F be a field with charpF q ‰ 2 and let A be a quaternion algebra over F . We considerA as quadratic space via the following quadratic form (see Proposition 3.2.6):

qApxq “ ´ nrdpxq,

with the associated bilinear form:

x¨, ¨y : AˆA ÝÑ F

px, yq ÞÝÑ ´1

2trdpxy˚q.

In chapter 2 we have seen that the reduced norm of a quaternion algebra is invariantunder conjugation. So in particular the quadratic form qA is invariant under conjugationwith elements in A˚. Denote for an element a in A˚ by ca : A Ñ A the map given byconjugation with a. This gives us the following morphism of groups:

c : A˚ ÝÑ OpAq

a ÞÝÑ ca.

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Note that ca “ id if and only if a is an element of the centrum of A. So the kernel ofthe map c is given by kerpcq “ ZpAq “ F ˚. Hence we can view A˚F ˚ as a subgroup ofthe orthogonal group OpAq.

Recall that the pure quaternions of A are given by A0 “ tx P A : trdpxq “ 0u. Sincethe reduced trace is invariant under conjugation, A0 is an invariant subspace under theaction of A˚F ˚ on the quadratic space A.

Proposition 5.1.1. Let F be a field with charpF q ‰ 2 and let A be a quaternion algebraover F . Consider the quadratic space pA0, qAq with qApxq “ ´ nrdpxq. Then the map

c : A˚F ˚ ÝÑ SOpA0q

a ÞÝÑ ca

is an isomorphism of groups.

Proof. We first show that the image of the map c : A˚F ˚ Ñ OpA0q is indeed containedin SOpA0, qAq. Denote by la : A Ñ A the map given by left multiplication with a, andby ra˚ : AÑ A the map given by right multiplication with a˚. One can easily show thatdetplaq “ detpra˚q. Since a´1 “ nrdpaq´1a˚, this gives us the following:

detpcaq “ detplaqdetpra´1q “ nrdpaq21

nrdpaq4nrdpaq2 “ 1.

So the image of the map c : A˚F ˚ Ñ OpA0q is indeed contained in SOpA0q.

Recall that there is a canonical decomposition A – F ‘A0 as quadratic spaces. Fromthis it follows that an element a P A˚ commutes with all x P A0 if and only if a commuteswith all y P A. In other words, the map c : A˚F ˚ Ñ SOpA0, qAq is injective.

Now it remains to show that the map c is also surjective. Let y be an element in A0

with nrdpyq ‰ 0, then the reflection τy associated with y is given by

τypxq :“ x´2xx, yy

xy, yyy.

Note that y˚ “ ´y for any element y in A0, so in this case we have:

τypxq “ x´2 ¨ 1

2 trdp´xy˚q12 trdp´yy˚q

y “ x´xy ` yx

y2y “ ´yxy´1 “ ´cypxq.

By the Cartan-Dieudonne theorem (see [Lam05, thm.1.7.1, p.18]) the orthogonal groupOpA0q is generated by reflections. Since the determinant of a reflection is -1, the groupSOpA0q is generated by products τy1τy2 of reflections. Note that we have:

τy1τy2pxq “ τy1p´y2xy´12 q “ y1y2xy

´12 y´1

1 “ cy1y2pxq.

This shows that the group SOpA0q is contained in the image of c, so the map c is indeedan isomorphism of groups.

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Corollary 5.1.2. The group PGL2pRq is isomorphic to the orthogonal group SOp2, 1q.

Proof. Consider the quadratic space pM2pRq, qq with qpxq “ nrdpxq. Note that PGL2pRq “M2pRq˚R˚, so Proposition 5.1.1 gives us that PGL2pRq is isomorphic to SOpM2pRq0, qq.It follows from Lemma 3.2.7 that the signature of pM2pRq0, qq is given by p2, 1q. So weindeed have PGL2pRq – SOp2, 1q.

5.2 Embedding of H˘ into period domains

Let V be a real 2-dimensional vector space and let A be the real quaternion algebra

A “ EndRpV q – M2pRq.

Consider the real quadratic space pA0, qAq, where qApρq “ ´ nrdpρq “ ´ detpρq. Notethat EndRpV qbRC “ EndCpVCq, so the period domain associated with A0 (see Definition1.4.1) is in this case given by

DpA0q “ trρs P PpEndCpVCqq | trpρq “ detpρq “ 0, xρ, ρy ą 0u.

By Corollary 5.1.2 the group PGLpV q is isomorphic to SOpA0q. Hence PGLpV q actson DpA0q by conjugation. On the other hand, PGLpV q also has a natural action onH˘ – PpVCqzPpV q via g ¨ rvs “ rgvs.

Theorem 5.2.1. Let V be a real 2-dimensional vector space and let A “ EndRpV q. LetpA0, qAq be the quadratic space with qApxq “ ´nrdpxq. Then for every rρs in DpA0q thekernel is a 1-dimensional subspace of VC and kerpρq lies in PpVCqzPpV q. Moreover, themap

φ : DpA0q ÝÑ PpVCqzPpV qrρs ÞÝÑ kerpρq

is an isomorphism of complex manifolds and equivariant with respect to the action ofPGLpV q on both spaces.

In order to see why φ is a well-defined map, we need the following lemma.

Lemma 5.2.2. Let VC be a 2-dimensional complex vector space and let ρ be an endo-morphism of VC. Then impρq “ kerpρq if and only if ρ ‰ 0 and trpρq “ detpρq “ 0.

Proof. First note that the characteristic polynomial of an endomorphism ρ P EndpVCq isgiven by ppλq “ λ2 ´ trpρqλ` detpρq. So the condition trpρq “ detpρq “ 0 is equivalentto the condition that ρ has a double eigenvalue λ “ 0.

Now suppose impρq “ kerpρq. Then we clearly have ρ ‰ 0. Let λ be an eigenvalue ofρ and v the corresponding eigenvector, so ρpvq “ λv. Then v is also in the image of ρ,so by assumption λv “ 0. Hence all eigenvalues of ρ are zero.

Conversely, if ρ ‰ 0 and zero is a double eigenvalue of ρ, then the kernel of ρ is 1-dimensional. So the image of ρ is also a 1-dimensional subspace of VC. Now supposev P impρq and v R kerpρq, then we have ρpvq “ λv ‰ 0. This gives a contradiction, as ρhas only the eigenvalue λ “ 0.

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Proposition 5.2.3. Let V be a real 2-dimensional vectorspace. Let C be given by

C “ trρs P PpEndCpVCqq : detpρq “ 0, trpρq “ 0u.

Then the map from C to the complex projective line given by

φ : C ÝÑ PpVCqrρs ÞÝÑ kerpρq

is an isomorphism of complex manifolds. Furthermore, the map φ is equivariant withrespect to the action of PGLpV q on both spaces.

Proof. First note that Lemma 5.2.2 gives that kerpρq “ impρq for every rρs P C, hencekerpρq is a 1-dimensional subspace of VC. So φ is a well-defined map. Furthermore, themap φ is clearly equivariant with respect to the action of PGLpV q.

Now take a point L in PpVCq. We can associate to L a point rρLs in C in the followingway. Choose an isomorphism α : V L – L and define ρL,α via the following commutativediagram:

V V

V L L

ρL,α

α

Note that a different choice of α only changes ρL,α by a constant in C˚. So this gives us awell-defined point rρLs in PpEndCpVCqq. By construction we have impρLq “ kerpρLq “ L,so Lemma 5.2.2 gives us that trpρLq “ detpρLq “ 0. Hence rρLs is indeed a point in C.

Denote by ψ : PpVCq Ñ C the map given by ψpLq “ rρLs. By construction we haveφ ˝ ψ “ idPpVCq. From Lemma 5.2.2 it also follows that ψ ˝ φ “ idC , so φ is indeed abijection. Furthermore, if we choose a basis for VC, the coordinate functions of both φand ψ are just polynomials. Hence both φ and ψ are holomorphic maps, which provesthe proposition.

Before we can give the proof of Theorem 5.2.1, we need the following lemma.

Lemma 5.2.4. Let V be a real 2-dimensional vector space and let φ : C Ñ PpVCq be theisomorphism from Proposition 5.2.3. Then the subset

trρs P C : detpρ` ρq “ 0u Ď C

is mapped under φ onto PpV q.

Proof. Note that ρpvq “ ρpvq for any v P VC and rvs “ rvs in PpVCq if and only ifrvs P PpV q. Thus kerpρq “ kerpρq and we have in particular that kerpρq “ kerpρq if andonly if kerpρq P PpV q.

Let rρs be a point in C such that detpρ`ρq “ 0. If ρ`ρ “ 0, then for any v in kerpρq wehave ρpvq “ ´ρpvq “ 0. So v is also in the kernel of ρ, which gives us that ker ρ “ ker ρ.

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If ρ` ρ ‰ 0, then rρ` ρs is a point in C. So the kernel of ρ` ρ is 1-dimensional subspaceof VC, which implies that kerpρq “ kerpρq. Thus in both cases the kernel of ρ is a pointin PpV q. In other words, if detpρ` ρq “ 0, then φprρsq is a point in PpV q.

Conversely, if L is a point of PpV q, then we clearly have ρL P EndRpV q. Hence ρL “ ρL,which gives us that detpρL ` ρLq “ detp2ρLq “ 0. Therefore we can conclude that

φptrρs P C : detpρ` ρq “ 0uq “ PpV q.

Proof of Theorem 5.2.1. First note that the condition xρ, ρy ą 0 is equivalent to thecondition detpρ` ρq ă 0, since we have

2xρ, ρy “ qApρ` ρq ´ qApρq ´ qApρq

“ ´detpρ` ρq.

Let rρs be a point on the curve C. Then ρ has double eigenvalue zero and by Lemma5.2.2 we have kerpρq “ impρq. Now suppose that detpρ` ρq ‰ 0. Then ρ` ρ is injective,so kerpρq ‰ kerpρq. Let v be an eigenvector of ρ, so ρpvq “ 0. Then v is an eigenvectorof ρ and linear independent of v. Hence ρpvq “ av for some nonzero a P C. Sinceρpwq “ ρpwq for any w P VC, we have

pρ` ρqpvq “ ρpvq “ ρpvq “ av

Thus with respect to the basis tv, vu, the endomorphism ρ ` ρ is given by the matrixp 0 aa 0 q. Therefore we have detpρ` ρq “ ´aa ă 0. In other words, detpρ` ρq ď 0 for everyrρs in C.

Using Proposition 5.2.3 and Lemma 5.2.4 we can now conclude that the map φ restrictsto an isomorphism DpA0q – PpVCqzPpV q, which is equivariant for the action of PGLpV q.

5.3 Embedding of Shimura curves into ΓzDpΛq

Let A be a quaternion algebra over a totally real number field F of degree d. We canview A as a quadratic space over Q (see Example 3.2.12) via

qA : A ÝÑ Qx ÞÝÑ ´ trF Qpnrdpxqq.

This form induces on AR :“ AbQR the structure of a quadratic space over R via R-linearextension.

The group GpRq :“ pA bQ Rq˚pF bQ Rq˚ acts by isometries on the quadratic spacepAR, qAq via conjugation. Note that we have a canonical decomposition as quadraticspaces

AR – pF bQ Rq ‘ pA0 bQ Rq,

45

where A0 “ tx P A | trdpxq “ 0u are the pure quaternions in A (see Lemma 3.2.10).The subspace A0

R :“ A0 bQ R is clearly invariant under the action of GpRq. Thereforewe can use the same arguments as in Proposition 5.1.1 to show that

GpRq Ď SOpA0Rq.

Remark. Note that if d ą 1, this inclusion is strict and no longer an isomorphism as inProposition 5.1.1. The dimensions are for example no longer the same.

Now assume that A is a quaternion algebra over a totally real number field F of degreed such that

AR – M2pRq ˆHd´1.

Recall that this gives us a canonical subspace AR,mat Ď AR with AR,mat – M2pRq (seeDefinition 4.3.4). Note that the pure quaternions in AR,mat are given by

A0R,mat :“ tx P AR,mat : trpxq “ 0u “ A0

R XAR,mat.

Therefore the subspace A0R,mat Ď A0

R is invariant under the action of GpRq, so the group

GpRq also acts on the period domain DpA0R,matq. The inclusion of period domains

DpA0R,matq ãÝÑ DpA0

Rq

is clearly equivariant with respect to the action of GpRq on both spaces. Moreover, thecanonical surjective map π : AR Ý AR,mat induces a surjective map

π : GpRq Ý A˚R,matR˚.

By Corollary 5.1.2 we have A˚R,matR˚ – PGL2pRq, so the group GpRq acts on H˘ via the

projection π. This makes the PGL2pRq-equivariant isomorphism from H˘ to DpA0R,matq

given in Theorem 5.2.1 also an GpRq-equivariant isomorphism. The above discussion issumarized in the following diagram:

H˘ DpA0R,matq DpA0

Rq

PGL2pRq GpRq SOpA0Rq

„

ü ü ü

Ď

We will now expand this diagram and construct an embedding of the upper half-planeinto period domains DpΛRq, where Λ is a lattice with signature p2, nq.

Lemma 5.3.1. Let Λ be lattice of signature p2, nq and let A be a quaternion algebraover a totally real number field F of degree d such that

AR – M2pRq ˆHd´1.

Let pA0, qAq be the quadratic space over Q with qApxq “ ´ trF Qpnrdpxqq. Assume that

d ď 13pn´ 1q. Then there exists a quadratic space pW, qW q over Q such that

ΛQ – A0 ‘W

as quadratic spaces over Q.

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Proof. Using Example 3.2.12, we see that the signature of the quadratic form qA is givenby

p2, 3d´ 2q.

The signature of Λ is p2, nq, so if d ď n´13 , we have dimQpΛQq ´ dimQpA

0q ě 3. ThenProposition 1.2.8 gives us that there exists a quadratic space pW, qW q over Q such thatΛQ – A0 ‘W as quadratic spaces over Q.

Lemma 5.3.2. Let Λ, F , A, qA and d satisfy the assumptions as in Lemma 5.3.1. ThenGpRq is a subgroup of SOpΛRq. Moreover, the period domain DpA0

R,matq is a subspace ofDpΛRq and this inclusion is GpRq-equivariant.

Proof. By Lemma 5.3.1 there exists a quadratic space pW, qW q over Q such that ΛQ –

A0 ‘W as quadratic spaces. This isomorphism induces an inclusion of the orthogonalgroup OpA0q into OpΛQq via

OpA0q ãÝÑ OpA0 ‘W q – OpΛQq

g ÞÝÑ g ‘ idW .

The same map also gives an inclusion of the real orthogonal group OpA0Rq into OpΛRq.

Therefore we can view GpRq Ď SOpA0Rq as a subgroup of SOpΛRq.

The isomorphism of quadratic spaces in Lemma 5.3.1 also induces an inclusion ofperiod domains

DpA0Rq ãÝÑ DpA0

R ‘WRq – DpΛRq.

This inclusion is clearly invariant under the action of GpRq on both spaces.

Corollary 5.3.3. Let Λ, F , A, qA and d satisfy the same assumptions as in Lemma5.3.1. Then the isomorphism in Theorem 5.2.1 induces an GpRq-equivariant embedding

H˘ ãÝÑ DpΛRq.

Proof. This follows directly from Lemma 5.3.2 together with the remark that the iso-morphism in Theorem 5.2.1 is also GpRq-equivariant.

The corollary above shows that we can expand the diagram as follows:

H˘ DpA0R,matq DpA0

Rq DpΛRq

PGL2pRq GpRq SOpA0Rq SOpΛRq

„

ü ü ü ü

Ď Ď

In the rest of this section we will show how we can use this GpRq-equivariant embed-ding of H˘ into DpΛRq, to construct embeddings of compact Shimura curves into thecomplex manifolds ΓzDpΛRq, where Γ is some subgroup of OpΛq.

47

Let Λ be a lattice of signature p2, nq. Let A be a quaternion algebra over a totallyreal number field F of degree d with AR – M2pRq ˆHd´1 and let O be an order in thequaternion algebra A. Recall that the Fuchsian group ΓO is given by O˚,1t˘1u (seeDefinition 4.3.4). Now note that for x in F ˚ we have nrdpxq “ x2, thus O˚,1XF ˚ “ t˘1u.So there is an inclusion of groups

O˚,1t˘1u Ď GpQq Ď GpRq,

where GpQq :“ A˚F ˚. Therefore ΓO can be viewed as a subgroup of SOpA0Rq, and

hence as a subgroup of SOpΛRq. So if Γ is a subgroup of OpΛq, then it makes sense toconsider the intersection ΓO X Γ, since both ΓO and Γ are subgroups of OpΛRq.

Theorem 5.3.4. Let Λ be a lattice of signature p2, nq and let Γ be a torsion free subgroupof OpΛq of finite index. Let A be quaternion algebra over a totally real number field Fof degree d such that

AR – M2pRq ˆHd´1.

Let O be an order in A and define the subgroup Γ1 of PSL2pRq by

Γ1 :“ ΓX ΓO.

Assume that d ď 13pn ´ 1q and that A is a division algebra over Q. Then Γ1zH is

a compact Shimura curve and the embedding of H˘ into DpΛRq from Corollary 5.3.3induces an embedding

Γ1zH ãÝÑ ΓzDpΛRq.

of complex manifolds.

Note that Γ1 is by construction a subgroup of Γ, so it follows directly that the GpRq-equivariant embedding in Corollary 5.3.3 will induce an embedding from Γ1zH intoΓzDpΛRq. In order to prove the claim that Γ1zH is a compact Shimura curve, we willfirst prove the following theorem.

Theorem 5.3.5. Let Λ be a lattice of signature p2, nq and let Γ be a finite index subgroupof OpΛq. Let A be a quaternion algebra over a totally real number field F of degree dsuch that

AR – M2pRq ˆHd´1.

Let O be an order in A. Assume that d ď 13pn´ 1q. Then the group Γ1 :“ ΓO X Γ has a

finite index in the Fuchsian group ΓO.

Before we can give the proof of the above theorem, we need some results about finiteindex subgroups of orthogonal groups.

Proposition 5.3.6. Let Λ be a lattice and Λ1 be a sublattice of Λ such that ΛQ – Λ1Q.Then the group OpΛqXOpΛ1q has a finite index in the orthogonal group OpΛ1q. Here theintersection is taken in OpΛQq – OpΛ1Qq.

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Proof. First note that Λ1 has a finite index in Λ. So we can choose an integer n P N suchthat

nΛ1 Ď nΛ Ď Λ1.

Note that the orthogonal groups OpnΛq and OpΛq are isomorphic. So we can describethe group OpΛ1q XOpΛq in the following way:

OpΛq XOpΛ1q “ tg P OpΛ1q | g preserves nΛu.

The orthogonal group OpΛ1q acts by left multiplication on the following finite set:

X :“ tsubgroups of Λ1nΛ1u.

Since nΛ1 is a sublattice of nΛ, an element g in OpΛ1, q1q preserves the lattice nΛ if andonly if g preserves nΛnΛ1 Ď Λ1nΛ1. Therefore we have:

StabOpΛ1qpnΛnΛ1q – tg P OpΛ1q | g preserves nΛu

“ OpΛ1q XOpΛq.

As X is a finite set, the OpΛ1q-orbit of the element nΛnΛ1 is finite. Therefore thestabilizer of nΛnΛ1 has a finite index in the group OpΛ1q. In other words, the subgroupOpΛq XOpΛ1q has indeed a finite index in the group OpΛ1q.

The following basic fact will often be used.

Lemma 5.3.7. Let G be a group and H and K subgroups of G such that rG : Ks ă 8.Then H 1 “ H XK has a finite index in H.

Corollary 5.3.8. Let Λ be a lattice over Z, and let Λ0 ‘ Λ1 be a sublattice of Λ suchthat ΛQ – Λ0

Q ‘ Λ1Q. Then the group OpΛ0q XOpΛq has a finite index in the orthogonal

group OpΛ0q.

Proof. This follows directly from Proposition 5.3.6 and Lemma 5.3.7, together with thefact that OpΛ0q is a subgroup of OpΛ0 ‘ Λ1q via g ÞÑ g ‘ id.

Proposition 5.3.9. Let Λ, Γ, F , A, O and d satisfy the assumptions of Theorem 5.3.5.Then the group SOpO0q X Γ has a finite index in the group SOpO0q.

Proof. Since Γ has a finite index in OpΛq, Lemma 5.3.7 with H “ OpO0q X OpΛq, G “OpΛq and K “ Γ gives us that:

rOpO0q XOpΛq : OpO0q X Γs ă 8. (5.1)

By Lemma 5.3.1 there exists a quadratic space pW, qq over Q such that ΛQ – A0‘W .Now define WZ to be the lattice given by WZ :“ Λ XW . Note that the rank of WZ isthe same as the rank of W , so the lattice O0 ‘WZ has a finite index in the lattice Λ.Corollary 5.3.8 now gives us the following:

rOpO0q : OpO0q XOpΛqs ă 8.

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This shows, together with (5.1), that the following inclusions are both of finite index:

OpO0q X Γ Ď OpO0q XOpΛq Ď OpO0q.

Hence OpO0qXΓ has a finite index in the orthogonal group OpO0q, and therefore (usingLemma 5.3.7) the index of SOpO0q X Γ in the group SOpO0q is also finite.

We can now prove Theorem 5.3.5 and Theorem 5.3.4, the goal of this thesis.

Proof of Theorem 5.3.5. This follows from Proposition 5.3.9 together with Lemma 5.3.7with G “ SOpO0q, K “ SOpO0q X Γ and H “ ΓO.

Proof of Theorem 5.3.4. Proposition 1.4.4 gives us that ΓzDpΛRq is a complex manifold.From Theorem 5.3.5 together with Theorem 4.3.6 it follows that Γ1zH is a compactShimura curve.

By construction Γ1 is a subgroup of Γ, so the GpRq-embedding ψ : H ãÝÑ DpΛRq fromCorollary 5.3.3 induces an embedding

ψ : Γ1zH ãÝÑ ΓzDpΛRq.

of complex manifolds.

Remark. If we omit in Theorem 5.3.4 the condition that Γ is a torsion free subgroup ofOpΛq, then we still can construct an embedding Γ1zH ãÝÑ ΓzDpΛRq in the same way.In that case Γ acts properly discontinuous on DpΛq (see Lemma 1.4.5), and it can havenon-trival (finite) stabilizers. So the quotient ΓzDpΛRq will not be a complex manifoldin general, as this quotient can have singularities.

We can apply Theorem 5.3.4 in the context of (polarized) K3 surfaces and use it toembed compact Shimura curves into ΓλzDλ (see Definition 2.2.6).

Let λ be a vector in the K3 lattice ΛK3 with λ2 ą 0. Note that the period domainDλ of polarized K3 surfaces (see Definition 2.2.3) is the period domain DpλKRq associatedwith the lattice λK. Recall that the signature of the lattice λK is given by p2, 19q.

Note that λZ ‘ λK is a finite index sublattice of the K3 lattice. This gives anotherway to describe the group Γλ, namely:

Γλ :“ StabOpΛK3qpλq “ th P OpλKq : g “ h‘ id preserves ΛK3u

“ OpλKq XOpΛK3q.

Thus by Corollary 5.3.8 the index of Γλ in OpλKq is finite. Therefore we can concludethe following:

Corollary 5.3.10. Let A be quaternion algebra over a totally real number field F ofdegree d such that

AR – M2pRq ˆHd´1.

Let O be an order in A and define the subgroup Γ1 of PSL2pRq by

Γ1 :“ Γλ X ΓO.

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Assume that d ď 6 and that A is a division algebra over Q. Then Γ1zH is a compactShimura curve and the embedding of H˘ into DpλKRq from Corollary 5.3.3 induces anembedding

Γ1zH ãÝÑ ΓλzDλ.

5.4 Discussion

Let Λ be a lattice of signature p2, nq and Γ Ď OpΛq a torsion free subgroup of finiteindex. Let A be a quaternion algebra over a totally real number field F of degree d suchthat AR – M2pRqˆHd´1. If the degree of F is given by d “ n`2

3 , then we can not applyTheorem 5.3.4.

However, in this case the signature of the quadratic space pA0, qAq is given by p2, nq,see Example 3.2.12. Thus the quadratic space ΛR is isomorphic to A0

R, and hence theperiod domain DpΛRq is isomorphic to DpA0

Rq. So we can still construct the followingdiagram:

H˘ DpA0R,matq DpA0

Rq DpΛRq

PGL2pRq GpRq SOpA0Rq SOpΛRq

„

ü ü

„

ü ü

Ď –

Therefore it makes sense to define the group Γ1 :“ ΓO X Γ and the construction of thelast section gives an embedding

ψ : Γ1zH ãÝÑ ΓzDpΛRq.

of complex manifolds.For the proof of Theorem 5.3.5 (the statement that Γ1 has a finite index in ΓO) we

need the decomposition ΛQ – A0 ‘WQ of quadratic spaces over Q. Therefore we canno longer use Theorem 4.3.6 to conclude that the quotient Γ1zH is a compact Shimuracurve, we only know that Γ1zH is a Riemann surface.

In other words, for the case when d “ n`23 , the question is still open whether it is

possible to construct embeddings of compact Shimura curves into ΓzDpΛRq. In view ofthe discussion above, this question can be reduced to the question whether

A0 – ΛQ

as quadratic spaces over Q.In the last section we considered A0 as a quadratic space over Q with the quadratic

form qApxq “ ´ trF Qpnrdpxqq. However, we could also change qA by an element λ P F ˚

and consider A0 as quadratic space with the form q1Apxq “ trF Qpλ nrdpxqq. So given alattice Λ and a quaternion algebra A over a totally real number field F , one could lookfor a suitable choice of λ P F ˚ for which A0 – ΛQ.

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Popular summary

The group PSL2pRq :“ SL2pRqt˘1u acts on the upper half-plane

H :“ tz P C | impzq ą 0u

via so-called Mobius transformations:

“

a bc d

‰

¨ z :“az ` b

cz ` d

These Mobius transformations have a lot of nice properties. For example, they preserveangles, map every straight line to a straight line or circle, and map every circle to astraight line or circle. If we consider discrete subgroups Γ of PSL2pRq, e.g. Γ “ PSL2pZq,then we can give the orbit-space HΓ a natural geometric structure of a Riemann surface.This means that HΓ locally looks like C.

These discrete subgroups Γ of PSL2pRq also act on certain higher dimensional spacesDpΛq. For these higher dimensional spaces the orbit-space DpΛqΓ also has a naturalgeometric structure (they look locally like Cn). In this thesis we construct embeddings

HΓ ãÝÑ DpΛqΓ.

The reason why this is interesting is classification. In mathematics it is an importantquestion to classify objects we find interesting. If there is a class of objects and anotion when two such objects are equivalent (for example: if you study groups, youregard isomorphic groups as the ‘same’), we would like to know how many differentequivalence classes there exists. If the objects have a geometric nature, then we wouldlike to capture this geometric structure also in the classification. So when two differentequivalence classes of objects are geometrically very similar, we want to recognize thisin the classification.

A moduli space is a space X with a geometric structure, which parametrizes all dif-ferent equivalence classes of a certain kind of objects. In other words, each point of Xcorresponds to a different equivalence class, and equivalence classes that look geomet-rically similar should be somehow close to each other in X. Such a moduli space givesyou more insight in the objects you study and how the different equivalence classes arerelated to each other.

In our case the space DpΛqΓ is closely related to the moduli space of ‘polarizedK3 surfaces’. So an embedding of HΓ gives a one-dimensional family of polarized K3surfaces. In this thesis we give conditions for when there exists embeddings of HΓ intoDpΛqΓ, and when HΓ is compact. Both these quotients HΓ and DpΛqΓ are examplesof Shimura varieties, which are important geometric objects (which is another reasonwhy such embeddings are interesting).

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