志村曲線上的自守型式

國 立 交 通 大 學

應用數學系

博 士

論

文

志村曲線上的自守型式

Automorphic Forms on Shimura Curves

研 究 生：凃芳婷

指導老師：楊一帆 教授

志村曲線上的自守型式

Automorphic Forms on Shimura Curves

研 究 生：凃芳婷 Student：Fang-Ting Tu

指導教授：楊一帆 Advisor：Yifan Yang

國 立 交 通 大 學

應用數學系

博 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science,

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

PhD

In

Applied Mathematics

September 2013

Hsinchu, Taiwan, Republic of China

志村曲線上的自守型式

研究生：凃芳婷 指導教授：楊一帆 教授

國 立 交 通 大 學

應用數學系

摘 要

在上個世紀，模型式和模曲線在數論的發展上佔了很重要地位。志村的曲線是模曲

線的一個推廣，因此自守型式和志村曲線的算術性質在近代數論的發展也是舉足輕重。

我們的主要目標是研究自守型式的算術性質。這篇論文的工作是研究自守型式算術性質

的一個起點。

根據楊一帆教授最近的結果，我們可以用 Schwarzian 微分方程的解來描述虧格為零的

志村曲線上的自守型式，這提供了我們一個明確的方法來對自守型式作計算並幫助我們

瞭解自守型式的算術性質。因此，如何找到的相關的 Schwarzian 微分方程就成為我們現

在最重要的問題。

在這篇論文中，我們決定了大部分虧格為零志村曲線的 Schwarzian 微分方程。另外，

在學習自守型式的算術性質時，我們有個有趣的發現：

-超幾何函數的代數變換。這

主要的概念是把志村曲線上的自守型式用超幾何函數來表示，並利用自守型式之間的相

等關係，我們就可以看到這些有趣的代數變換。

中華民國一

二年九月

Automorphic Forms on Shimura Curves

Student: Fang-Ting Tu

Advisor: Yifan Yang

Department (Institute) of Applied Mathematics

National Chiao Tung University

Abstract

During the last century, modular forms and modular curves played important roles in the

developments of number theory. Shimura curves are natural generalizations of classical

modular curves. The arithmetic properties of automorphic forms and Shimura curves are

particularly important in modern number theory. Our aim is to study the arithmetic properties

of automorphic forms and automorphic functions on Shimura curves. The work in this

dissertation is a starting point.

Due to the recent work of Yifan Yang, if the Shimura curve is of genus zero, then one can

express its automorphic forms in terms of the solutions of the associated Schwarzian

differential equation. This provides a concrete space of automorphic forms. We then can do

explicit computation on the spaces to study the arithmetic properties of automorphic forms

and functions. Therefore, the main question is how to find the Schwarzian differential

equations.

In this thesis, we determine the Schwarzian differential equations for certain Shimura

curves of genus zero. As a byproduct of study on automorphic forms on Shimura curves, we

also obtain several algebraic transformations of

-Hypergeometric functions. This

discovery is achieved by interpreting Hypergeometric functions as automorphic forms on

Shimura curves.

Acknowledgement

First of all, I would like to express my deepest appreciation to my advisor Professor Yifan

Yang for his enthusiasm, insight, and patience throughout years of guidance in my graduate

school career. Much contents and the results of the dissertation herein are discussed with him.

Without his constant encouragement, patient instruction, and enthusiastic supports, it is

impossible for me today to be in this position. Many thanks are due to Professor Hashimoto of

Waseda University for providing me the opportunity to be a research student under his

supervision. He gives me several enlightening advices during the periods while I stay there.

Part of the results in Chapter 3 included in this thesis was written at that time.

I would also like to show my gratitude to Professor Ming-Lun Hsieh, Professor

Ming-Hsuan Kang, Professor Wen-Ching Winnie Li, and Professor Jing Yu for reviewing my

dissertation and serving on the committee of my thesis defence.

For my student life, I felt an immense gratitude to Professor Winnie Li, Professor Jing Yu,

as well as the government of Republic of China. Because of their organizations and supports,

I have the chances to attend conferences, workshops, and study aboard. Also, I would like to

thank my friends, especially best friends, and those in Japan. Thank you for doing me favors

generously when I got into troubles. Thank you guys for making my student life colorful and

smooth. There are too many persons to whom I want to express my gratitude, so please

forgive me if I do not list all of you.

Finally, I am particularly indebted to my family for their love and support in my entire

life.

TU, Fang-Ting

Department of Applied Mathematics

College of Science

National Chiao Tung University

2013. 9. 12

Contents

中 文 摘 要

i

Abstract

ii

Acknowledgement

iii

Contents

iv

2. Quaternion Algebras and Shimura Curves 5

2.1 Quaternion algebras 5

2.1.1 Quaternion algebras and quadratic forms 5

2.1.2 Automorphism theorem 6

2.2 Orders and Ideals 7

2.3 Quaternion Algebras over Local Fields 9

2.3.1 Quaternion algebra over non-Archimedean local fields 10

2.3.2 Orders in B = (π,e) 10

2.3.3 Orders in M(2,K) 11

2.4 Quaternion Algebras over Number Fields 12

2.4.1 Classification of quaternion algebras over number fields 13

2.4.2 Orders in a quaternion algebra over a number field 14

2.5 Shimura Curves 15

2.6 Signatures of Shimura curves 16

2.7 Automorphic Forms on Shimura Curves 16

3. The Shimura Curves 18

3.1 Eichler orders O(D;N) and Shimura curves 18

3.1.1 The Shimura curves 19

3.1.2 The Atkin-Lehner involutions on 19

3.2 Optimal Embeddings 20

3.2.1 Optimal embeddings of quadratic orders into Q-quaternion algebras 20

3.2.2 Optimal embeddings of quadratic orders into O(D,N) 21

3.3 Complex Multiplication Points on 22

3.3.1 The set of CM-points by an order 23

3.3.2 Fixed points of Atkin-Lehner involutions 24

3.3.3 Fields of definition of CM-points 24

3.5 Cerednik-Drinfeld Theory 27

3.5.1 The Cerednik-Drinfeld theory 27

3.5.2 Dual graph and bad reduction 28

3.6 The Jacquet-Langlands correspondence 29

4. Automorphic Forms in Terms of Solutions of Schwarzian Differential Equations 31

4.1 Automorphic Forms on Shimura Curves and Schwarzian Differential Equations 32

4.1.1 Automorphic forms on Shimura curves of genus zero 33

4.1.2 Hypergeometric functions as automorphic forms on Shimura curves 34

4.1.3 Transformation laws of automorphic derivatives 36

4.2 Shimura Curves of Genus Zero 36

4.3 Coverings of Shimura Curves 39

4.4 Schwarzian Differential Equations Associated to Shimura Curves of Genus Zero 55

5. Applications of the Arithmetic of Automorphic Forms 60

5.1 Hecke Operators on 1 / 61

5.2 Ramanujan-type Formulae 68

6. Algebraic Transformations of Hypergeometric Functions Arising from Theory of Shimura Curves 72 6.1 Preliminaries 74

6.1.1 Triangle groups 74

6.1.2 Automorphic forms on Shimura curves 75

6.1.3 Algebraic transformations of hypergeometric functions 75

6.2 Kummer’s Quadratic Transformations and Automorphic Forms 76

6.2.1 Kummer’s quadratic transformation 76

6.2.2 Automorphic forms on arithmetic triangle groups in Takeuchi’s class II and the associate algebraic transformations 78

6.3 Algebraic Transformations Associated to Class III 81

6.4 Algebraic Transformations Associated to Class VI 91

6.5 Algebraic Transformations Associated to Other Classes 94

6.5.1 Classes II, V, and XII 95

6.5.2 Classes IV, VIII, XI, XIII, XV, XVII 96

Chapter 1

Introduction

During the last century, modular forms and modular curves played important roles in the developments of number theory. A reason of this fact is because of the connection with the moduli space of elliptic curves, and that the elliptic curves, being algebraic curves of the smallest positive genus, are related with many non-trivial Diophantine problems in number theory. For example, the arithmetic properties of elliptic curves are essential in Andrew Wiles’ proof of Fermat’s Last Theorem. Shimura curves are natural generalizations of classical modular curves. Similar to the classical modular curves, Shimura curves are moduli spaces of certain abelian surfaces with quaternionic multiplication. The arithmetic properties of Shimura curves are particularly important in modern number theory. Our aim is to study the arithmetic of automorphic forms and automorphic functions on Shimura curves. The work in this dissertation is a starting point.

A Shimura curve is a quotient space of the upper half plane h = {τ : C : Im(τ ) > 0} obtained by certain quaternion order. More precisely, we let K be a totally real number of degree n and B be a quaternion algebra over K that splits exactly at one infinite place, that is,

B ⊗_QR ' M (2, R) × Hn−1,

where M (2, R) is the algebra of 2 by 2 matrices over R and H is Hamilton’s quaternion algebra. Up to conjugation, there is a unique embedding ι∞ from B into M (2, R).

Given an order O of B, we let O1be the group of the elements of reduced norm 1 of O. Then the image Γ(O) = ι∞(O1) under the embedding ι∞is a discrete subgroup of

SL(2, R), and hence there is a group action of Γ(O) on h by the usual fractional linear transformations a b c d  τ = aτ + b cτ + d, a b c d  ∈ Γ(O).

When B 6= M (2, Q), we denote by X(O) the Riemann surface Γ(O)\h. This is the so-called Shimura curve associated to O. In the case of B = M (2, Q), the compactified curve Γ(O)\(h ∪ P1(Q)) by adjoining cusps is the classical modular curve.

function f : h → C such that f aτ + b cτ + d  = (cτ + d)kf (τ ), ∀ τ ∈ h, a b c d
∈ Γ(O).

For the classical modular forms, i.e., the case of B = M (2, Q), we need additional conditions on cusps.

Even though it is true that many theoretical aspects of classical modular curves can be extended to the case of Shimura curves, to the best knowledge of the author, it is not true for explicit methods. In the case of classical modular curves, many problems about modular curves can be answered using Fourier expansions of modular forms or mod-ular functions involved, and there are many explicit methods for constructing modmod-ular functions, modular forms and computing their Fourier expansions. In fact, because the Fourier coefficients of a normalized Hecke eigenform on congruence subgroups are identical with the eigenvalues of Hecke operators, one can compute the expansions of Hecke eigenforms without actually constructing them. However, unlike their classical counterpart, Shimura curves do not have cusps and hence automorphic forms or auto-morphic functions on Shimura curves do not have Fourier expansions. Because of this, as far as we know, there have been very few explicit methods to construct automorphic forms and automorphic functions on Shimura curves. Also, any method for classical modular curves that uses Fourier expansions can not possibly be extended to the case of Shimura curves. Therefore, the question is how to construct automorphic forms on Shimura curves with Taylor series at a CM-point.

Recently, Yang [33] had a breakthrough for constructing automorphic forms on Shimura curves. In the work of Yang [33], he proposed a new method to study automor-phic forms on Shimura curves of genus zero, in which automorautomor-phic forms are expressed in terms of solutions of Schwarzian differential equations. He then demonstrated how to compute Hecke operators explicitly on these automorphic forms. Moreover, since Schwarzian differential equations that with exactly 3 singularities are essentially hy-pergeometric, this approach leads to many identities among hypergeometric functions by interpreting the hypergeometric functions as automorphic forms on Shimura curves. This was the main theme of my joint paper with Yang [24], the author [22] also gave more examples of algebraic transformations of hypergeometric functions to illustrate the role Shimura curves play in proving these identities.

Due to the results of Yang [33], once the Schwarzian differential equation for a Shimura curve of genus zero is determined, we can study the arithmetic properties of the automorphic forms on this Shimura curve as t-series, where t is a generator of the field of functions on the Shimura curve of genus zero. Because of the importance of Schwarzian differential equations in explicit methods for Shimura curves, one of the main goals is to determine Schwarzian differential equations for as many Shimura curves as possible. Especially, we are most interested in the Shimura curves attached to Eichler orders of the indefinite quaternion algebras over Q and their quotients by Atkin-Lehener involutions.

We denote by X₀D(N ) the Shimura curve obtained by an Eichler order of level N in an indefinite quaternion algebra defined over Q of discriminant D. (When D = 1, the curve X01(N ) is the classical modular curve X0(N ).) Let WD,N be the group

of all the Atkin-Lehner involutions wmof X0D(N ). In this dissertation, let us focus

on the Shimura curves X0D(N )/G, quotient by some subgroup G of WD,N, D > 1.

We will determine the Schwarzian differential equations for certain Shimura curves XD

0 (N )/WD,N of genus zero.

In order to determine the Schwarzian differential equation for a given Shimura curve, we will first compute the defining equations of Shimura curves over Q, and then construct coverings, we can find the coverings between Shimura curves. These rela-tions will help us determine the Schwarzian differential equarela-tions. The key ingredients for determination of the equations of Shimura curves are the ˇCerednik-Drinfeld theory of p-adic uniformization for Shimura curves, and the Jacquet-Langlands correspon-dence. The Jacquet-Langlands correspondence gives a bijection from automorphic representations on X0D(N ) and certain modular representations on X0(DN ). This

tells us the isogeny class of a given Shimura curve which is an elliptic curve defined over Q. The ˇCerednik-Drinfeld theory gives us the information of the bad reductions of Shimura curves, and then we can determine the isomorphism class of the given Shimura curve.

For the rest of this dissertation, we will first say a few words about quaternion al-gebras, Shimura curves and then introduce my recent work of automorphic forms on Shimura curves. In Chapter 2, we introduce quaternion algebras, quaternion orders, Shimura curves, automorphic forms and automorphic functions on Shimura curves. In Chapter 3, we briefly recall some basic and useful properties of the Eichler or-ders of level (D, N ), the Shimura curves XD

0 (N ), automorphic forms on X0D(N ),

the ˇCerednik-Drinfeld theory of p-adic uniformization for Shimura curves, and the Jacquet-Langlands correspondence.

In Chapter 4, we provide the connection between the automorphic forms on Shimura curves and the Schwarzian differential equations. Also, we will work out Schwarzian differential equations for certain Shimura curves X0D(N )/WD,1of genus zero. As

ap-plications of the arithmetic of automorphic forms on Shimura curves of genus zero, in Chapter 5, we compute Hecke operators Tp with prime p on X014(1)/W14,1 and use

numerical computation to obtain Ramanujan-type series for the curve X014(1)/W14,1.

This gives a numerical evidence to Yang’s conjecture in [32].

Finally, in Chapter 6, as a byproduct of the study on arithmetic properties of au-tomorphic forms, we obtain some algebraic transformations of 2F1-hypergeometric

functions.

For the future studies on the arithmetic of automorphic forms on Shimura curves of genus zero, we plan to determine the coordinates of CM-points on Shimura curves. The CM-points on Shimura curves correspond to abelian surfaces with endomorphism algebra equal to a matrix algebra of degree 2 over an imaginary quadratic number field. Another application is related to the Ramanujan-type formulae for Shimura curves. Moreover, a main future work is to generalize Yang’s result. One restriction of Yang’s approach is that the genus of the Shimura curve has to be zero. That is, it is not known how to express automorphic forms on Shimura curves using solutions of Schwarzian differential equations when the genus is positive. We will try to extend Yang’s method to higher genus cases. Elkies [6], Greenberg, Voight [11, 28, 29, 30] also introduced many methods to do computations on the arithmetic of the Shimura curves X0D(N ),

field F , or the Shimura curves arising the arithmetic triangle groups. For instances, they compute CM-points on the Shimura curves, determine the system of Hecke eigenvalues by using the Jacquet-Langlands correspondences. Another furure work is to generalize their results.

Chapter 2

Quaternion algebras and

Shimura curves

In this chapter, we will briefly recall some basic definitions and properties of quaternion algebras, especially quaternion algebras over a local field or number field. Then we will define the Shimura curves. Most of the materials are taken from the references [1, 26]. From now on, we let K be a field with characteristic not 2.

2.1

Quaternion algebras

2.1.1

Quaternion algebras and quadratic forms

A quaternion algebra B over a field K is a central simple algebra of dimension 4 over K, or equivalently, there exist i, j ∈ B and a, b ∈ K∗so that

B = K + Ki + Kj + Kij, i2= a, j2= b, ij = −ji.

In such case, we denote bya,b_Kthe quaternion algebra B, which has canonical K-basis {1, i, j, ij}. Familiar examples are Hamilton’s quaternions H = −1,−1_R
and the matrix algebra M (2, K) ∼= 1,1_K
.

Theorem 2.1.1. If a quaternion algebra B over K has a zero divisor, then it is iso-morphic toM (2, K).

According to Theorem 2.1.1, if a has a square root α in K then the quaternion algebra B has a zero divisor h = α − i, and B is isomorphic to the 2-by-2 matrix algebra. Hence, if K is an algebraically closed field, then the only structure of K-quaternion algebra is the matrix algebra.

Notice that an element h in a quaternion algebra satisfies a monic polynomial over K of degree less than 2. Therefore, any quaternion algebra B is provided with a unique K-linear anti-involution ¯: B −→ B, ¯ h = a0− a1i − a2j − a3ij, if h = a0+ a1i + a2j + a3ij ∈  a, b K  .

This map is called the conjugation. The reduced trace, and reduced norm on B are defined by

tr(h) = h + ¯h, and n(h) = h¯h,

respectively. We remark that tr(h) = 2h and n(h) = h2_{, if h lies in the center K. If}

B = M (2, K) then the reduced trace and reduced norm of an element h ∈ B are the trace and the determinant of h. These maps tr and n lead to a nondegenerate symmetric K-bilinear form on B, which is given by tr(x¯y). In other words, the quaternion algebra B is a quadratic space with the quadratic form given by the reduced norm of B.

Recall that a quadratic space with a quadratic form Q is said to be isotropic if there is a non-zero element x so that Q(x) = 0. We have the following facts.

Theorem 2.1.2. For a quaternion algebra B = a,b_K over K, the following are equivalent.

(1) B is isomorphic to M (2, K).

(2) B is not a division quaternion algebra.

(3) B is isotropic as a quadratic space with the reduce norm. (4) The quadratic formax2_{+ by}2_represents1.

(5) IfF = K(√b), then a is an element of NF /K(F ).

Denote B0by the pure quaternion space, B0= {x ∈ B : tr(x) = 0}.

Theorem 2.1.3. Let B and B0be two quaternion algebras overK. Then B is isometric toB0if and only ifB0andB00 are isomorphic. Equivalently, the quaternion algebras



a,b K



,a0_K,b0are isomorphic if and only if the quadratic forms ax2+ by2− abz2

anda0x2+ b0y2− a0_b0_z2

are equivalent overK.

2.1.2

Automorphism theorem

Theorem 2.1.4. (Noether-Skolem Theorem)

LetL, L0 be two commutativeK-algebras over K contained in a quaternion alge-braB over K. Then all K-isomorphism from L to L0 can be extended to an inner automorphism ofB. The K-automorphisms of B are all inner automorphisms. Remark 2.1.5. An inner automorphism of B is an automorphism given by k 7→ hkh−1_{, for some invertible element h of B. Therefore, according to the Theorem}

2.1.4, the automorphism group of the quaternion algebraB, AutK(B), is isomorphic

Corollary 2.1.6. For all separable quadratic algebras F over K contained in B, there exists an elementθ ∈ K×such that

B = F + F u, u2= θ and um = σ(m)u,

whereσ denotes the non-trivial K-automorphism of F . In this case, we use the symbol {F, θ} to denote the quaternion algebra B.

Remark 2.1.7. Let σ : F −→ L be a nontrivial K-automorphism of L. Then there existu ∈ B∗so thatumu−1 = σ(m), for all m ∈ F . The fact t(u) = 0 implies that u2 _{= θ ∈ K. In this way, we realize B as B = {F, θ}, moreover, B =} a,b

K

 = {K(i), b}.

2.2

Orders and Ideals

As the fractional ideals in a number field, there is a similar theory for ideals in a quater-nion algebra. Let R be a Dedekind domain and K be its field of fractions. An R-lattice of a K-vector space V is a finitely generated R-module contained in V . A complete R-lattice Λ of V is an R-lattice Λ of V such that K ⊗RΛ ' V .

Example 2.2.1. We consider the cases in the quaternion algebras and quadratic num-ber fields.

1. LetΛ1= R + Ri and Λ2 = R + Ri + Rj + Rij. Then they are both R-lattice

ofH and Λ2is complete.

2. Given_{R = Z, K = Q. Let V = Q(}√m) and Λ be its number ring, where m is a square-free integer. ThenΛ is a complete lattice.

Definition 2.2.1. An ideal of a quaternion algebra B is a complete R-lattice in B. If an ideal ofB is also a ring with unity, it is called an order. Moveover, we say that I is aleft ideal of O if OI ⊂ I; I is a right ideal of O if IO ⊂ I.

Definition 2.2.2. A maximal order of B is an order that is not properly contained in another order ofB. An intersection of two maximal orders of B is called an Eichler order.

Now if an ideal I is given, we can define two orders associated to I, the left order of I,

O`(I) = {h ∈ B : hI ⊆ I},

and the right order of I,

Or(I) = {h ∈ B : Ih ⊆ I}.

Definition 2.2.3. An ideal I is said to be two-sided if O`(I) = Or(I), said to be

integral if I is contained in both O`(I) and Or(I). If O`(I) and Or(I) are maximal

An element x of a quaternion algebra B is called to be integral over R if R[x] is a R-lattice of B. For instance, the element i in the classical quaternion algebra H = Q + Qi + Qj + Qij is an integral element but i/2 is not. Actually, we have a useful criterion to determine whether if an element is integral or not.

Lemma 2.2.2. An element of a quaternion algebra B is integral if and only if its reduced trace and norm are in the ringR.

Also, we have an equivalently definition of an order of a quaternion algebra. Proposition 2.2.3. Let B be a quaternion algebra over K.

1. O is an order of B if and only if O is a ring of integral elements in B which containsR and K-basis for B.

2. Every order is contained in a maximal order.

The second proposition is followed from the first one and Zorn’s Lemma. From this proposition, we can see that an integral ideal is an ideal whose elements are all integral elements.

There are also the analogue of the norm of an ideal, and the discriminant of an order as in the algebraic number theory. The inverse of I is defined to be

I−1= {h ∈ A : IhI ⊂ I},

which is also an ideal. The norm of I, n(I), is the R-fractional ideal generated by {n(x) : x ∈ I}. The dual I∗_{of I is}

I∗= {h ∈ A : tr(hI) ⊂ R}.

The discriminant of an order O is DO = n(O∗)−1. If I is a left ideal of O, then the

discriminant of I is given by DI = n(I∗)−1n(I).

Proposition 2.2.4. We have the following properties: (1) II−1 ⊆ O`(I) and I−1I ⊆ Or(I).

(2) The square of discriminant ofO, D2

O, is equal to the ideal overR generated by

{det(tr(xixj)) : 1 ≤ i, j ≤ 4, xi, xj ∈ O}.

In particular, ifO has free basis {e1, e2, e3, e4} over R, then D2_Ois the principal

R-ideal det(tr(eiej))R.

(3) If an orderO0_{is contained in the other orderO, then D}

O dividesDO0.

There-fore,DO = DO0is and only ifO = O0.

(3) IfI is a left ideal of an order O, then DI = n(I)2DOand

Example 2.2.5. (1) The discriminant of the orderM (2, R) is R. (2) Consider the two orders

O = Z + Zi + Zj + Zij and

O0_{= Z + Zi + Zj + Z}1 + i + j + ij 2 in the quaternion algebra−1,−1

Q



. It obvious thatO ⊂ O0_and

D_O20 = 4Z ⊃ 16Z = DO.

In the case of the quaternion algebra B = M (2, K). One can identify B with the endomorphsim ring of some vector space over K. To be more precise, let V be a vector space over K with basis {e1, e2}. Then with respect to this basis, M (2, K) is viewed

as End(V ). Given a complete R-lattice Λ in V , we can see that End(Λ) = {α ∈ End(V ) : αΛ ⊂ Λ}

is a maximal order in End(V ). Conversely, for a given order O in End(V ), we can associate an R-module

Λ = {αei: α ∈ O, i = 1, 2},

which is a complete R-lattice, to the order O contained in End(Λ).

Proposition 2.2.6. If R is a principal ideal domain, then each maximal order in M (2, K) is conjugate to the maximal order M (2, R).

2.3

Quaternion Algebras over Local Fields

For a local field K, there are at most 2 non-isomorphic structures of quaternion algebras over K. If K = C, there is only one C-quaternion algebra, namely, the matrix algebra M (2, C). For the Archimedean local field R, a quaternion algebra over R is either isomorphic to M (2, R) or the quaternions of Hamilton H. If K is non-Archimedean, then a quaternion algebra over K is isomorphic to exactly one of M (2, K) or the unique division quaternion algebra over K.

Theorem 2.3.1. (Frobenius Theorem)

Let_{D be a division ring containing R in its center of finite dimension over R. Then D} is isomorphic to H, the Hamiltonian quaternion.

Hence, Frobenius’ Theorem tells us that a quaternion algebra is either isomorphic to M (2, R) or H.

2.3.1

Quaternion algebra over non-Archimedean local fields

For a non-Archimedean local field K, we let R be its ring of integers and π be a fixed uniformizer with respect to the valuation ν.

Theorem 2.3.2. There is a unique division quaternion algebra over K and it is iso-morphic to π,e_K
, where K(√e) is the unique unramified quadratic extension of K.

While h 6= 0 in π,e_K
, the map ω given by ω(h) = 1

2ν(N (h)) defines a discrete

valuation on the division algebra π,e_K
.

We define the Hasse invariant of the quaternion algebra B by ε(B) =

(

1, if B ∼= M (2, K), −1, otherwise. In the case of K = Qp, the Hasse invariant of B =

_a,b

Qp



coincides with the Hilbert Symbol (a, b)p, which is given by

(a, b)p=

(

1, if ax2+ by2_{reprents 1,}

−1, otherwise.

Remark 2.3.3. From the Theorem 2.3.2, for p > 2, we have a simple description for the Hilbert symbol(a, b)pwithp - a,

(a, b)p= (1, ifp - a, b,  a p  , _{p - a, p | b,} where_p·is the Legendre symbol.

2.3.2

Orders in B =

For the unique division quaternion algebra B = π,e_K
, it is known that there is a unique maximal order in B, which is the associated valuation ring

O = {h ∈ B : w(h) ≥ 0} = {h ∈ B : N (h) ∈ R} with respective to the valuation w. The ring

P = {h ∈ B : w(h) > 0} is a two-sided prime ideal of O.

Theorem 2.3.4. Let B = e,π_K
, F = K(√e), and O be the unique maximal order in B. Then we have

1. P = Oj is a prime ideal of O and P2= Oπ.

2. O = RF+ RFj, where RFis the ring of integers ofF .

2.3.3

Orders in M (2, K)

If B is isomorphic to M (2, K), then as the consideration in the end of the last section, each maximal order in B is then isomorphic to the maximal order M (2, R). We now let B = M (2, K).

Theorem 2.3.5. 1. A maximal order ofM (2, K) is conjugate to M (2, R) by an element ofGL(2, K).

2. The set of all maximal orders is in one-to-one correspondence with the cosets K∗GL(2, R)\GL(2, K).

The standard coset representatives of K∗GL(2, R)\GL(2, K) are πa _c

0 πb 

,

where a and b are nonnegative integers and c are from R/(π)b_{, subject to the condition}

that v(c) = 0 if a, b > 0. Therefore, we can classify all maximal orders of M (2, K) as πa _c 0 πb −1 M (2, R)π a _c 0 πb  , a, b ≥ 0, c mod πb, and c /∈ πR if a, b > 0.

Also, we can classify the Eichler order of M (2, K). Proposition 2.3.6. (Hijikata)

IfO is an order in M (2, K), then the followings are equivalent. 1. O is an Eichler order.

2. There exists a unique pair of maximal ordersO1andO2such thatO = O1∩O2.

3. There existsn ∈ Z>0such thatO is conjugate to πRn_{R R}R
.

4. The orderO contains R ⊕ R as a subring.

We say that an Eichler order in M (2, K) is of level πn_{R, if it is conjugate to}

R R

πnR R
.

We now introduce the graph of maximal orders of M (2, K). First, let us define the distance between the maximal orders. Let O1, O2be two maximal orders in M (2, K).

If the the Eichler order O = O1∩ O2is of index qnin O1, then the distance between

O1and O2 is d(O1, O2) = n, where q is the cardinality of the residue field R/πR.

Equivalently, the Eichler order O1∩ O2is of level πnR.

Now we define a graph X of maximal orders as follows. The vertices of X are the maximal orders and two vertices are connected by a simple edge if the two correspond-ing maximal orders has distance 1.

Example 2.3.7. Let O0= M (2, R), O1= (π 00 1) −1 O0(π 00 1) = R π −1_R πR R
, and O2= (1 00 π) −1 O0(1 00 π) = R πR π−1R R
. We haveO0∩ O1= (πR RR R), O0∩ O2= (R πRR R ) ∼= (πR RR R) , O1∩ O2= (πR RR πR) ∼= R R π2_{R R}
.

Thus, d(O0, O1) = d(O0, O2) = 1 and d(O1, O2) = 2. The subgraph of these

maximal orders is

O0

O1

O2

Proposition 2.3.8. The graph X is a (q + 1)-regular tree, i.e., a connected graph without cycles, and every vertex has preciselyq + 1 edges connecting to it.

Example 2.3.9. Here is a subtree of maximal orders of M (2, Q2). The matrix α next

to a vertex means that the maximal order isα−1_{M (2, Z}2)α.

(1 0 0 1) (2 0 0 1) (1 0 0 2) (1 1 0 2) (4 0 0 1) (2 10 2) (1 0 0 4) (1 2 0 4) (1 1 0 4) (1 3 0 4)

Remark 2.3.10. We remark that the group PGL(2, K) acts on the coset K∗GL(2, R)\GL(2, K), and hence acts by conjugation on the tree of maximal orders inM (2, K). In particular,

PGL(2, K) acts on the set

L(n)= {(O1, O2) : d(O0, O2) = n}

double transitively.

2.4

Quaternion Algebras over Number Fields

We now recall the classification of quaternion algebras over a number field. Let K be a number field, and R be its ring of integers. Let Kvbe the local field with respect to

2.4.1

Classification of quaternion algebras over number fields

A quaternion algebra B over a number field K is said to be ramified at v if Bv =

B ⊗ Kvis a division algebra. Otherwise, B is unramified or split at v.

Theorem 2.4.1. (Hasse-Minkowski Theoerm)

The quaternion algebraB is isomorphic to M (2, K) if and only if B splits over Kv

for all placesv.

Let Ram(B) denote the set of ramified places of B. The reduced discriminant of quaternion algebra B is the integral ideal of R defined by

DB=

Y

v∈Ram(B)

v.

In the case that R is a principal ideal domain, we identify the ideal DB with its

gen-erator, up to units. That is, DBR =

Y

v∈Ram(B)

v; for a quaternion algebra over Q, its discriminant is an integer.

The structure of the quaternion algebra B is uniquely determined by the reduced discriminant.

Theorem 2.4.2. (1) The cardinality of Ram(B) is finite and even.

(2) Two quaternion algebrasB and B0overK are isomorphic if and only if Ram(B) = Ram(B0).

(3) Given a finite setS of noncomplex places of K such that |S| is even, there exists a quaternion algebraB over K such that Ram(B) = S.

Therefore, if an even number of noncomplex places of K is given, then there exists one and only one K-quaternion algebra that ramifies exactly at these places.

Example 2.4.3. (1) A quaternion algebra over a number fieldK is isomorphic to M (2, K) if and only if DB = R.

(2) The discriminant of the quaternion algebra−1,−1

Q



is2, since the values of the Hilbert symbols are

(−1, −1)p=

(

−1, ifp = ∞, 2, 1, ifp > 2.

For any field F , if B is a quaternion algebra over F and L is a field extension of F . We say that L splits B if L ⊗F B is isomorphic to M (2, L). We now address

the conditions that when a K-quaternion algebra B splits over a quadratic extension field F of K. In particular, one has the conditions for which quadratic fields can be embedded into B. Let L be a finite extension field over K, and w be a place of L.

Proposition 2.4.4. Let B be a quaternion algebra over K. Then B splits over L if and only ifBvsplits overLwfor any placew|v of L. In particular, if L is a quadratic field

overK, then followings are equivalent: (1) The fieldL is a splitting field for B.

(2) The fieldL is K-isomorphic to a maximal subfield of B containing K. (3) There exists an embedding overK form L into B.

(4) Each placev in K that ramifies in B is not totally split in L.

For a totally real number field K, if a quaternion algebra over K is ramified at all the real infinite places, we say that the quaternion algebra is definite; otherwise, it is indefinite. We remark that a quaternion algebra B is definite if and only if the quadratic form given by < x, y >= tr(x¯y) on B is positive definite.

2.4.2

Orders in a quaternion algebra over a number field

Let I be an ideal in a quaternion algebra B over a number field K. Denote Rv the

ring of integers of the localization Kv. Then the localization Iv = I ⊗Z Rv is an

ideal in the quaternion algebra Bv and I = B ∩ (QvIv). As the Hasse-Minkowshi

theorem for quaternion algebras, being a maximal order or an Eichler order satisfied the local-global correspondence.

Proposition 2.4.5. Let Λ be a lattice in a quaternion algebra B over K. For any finite placev in K, we consider a local lattice LvinBv. Assume thatLv = Λv for almost

allv. Then there exists a lattice Λ0inB such that Λ0_v= Lvfor all finite placesv.

This gives us the existence of a global lattice.

Note that if O is a maximal order of B, it is clear that Ovis again an order in Bvand

(DO)v = DOv. We have a criterion for global maximal orders from the information

of the discriminants.

Proposition 2.4.6. An order O is maximal in the quaternion algebra B if and only if its discriminant is equal to the discriminant ofB, i.e, DO = DB.

Example 2.4.7. In the quaternion algebra B =−1,−1

Q



, the orderO = Z + Zi + Zj + Zij is a maximal order with DO = 2 = DB.

Definition 2.4.1. The level of a global Eichler order is the unique integral ideal NOin

R so that NOvis the level of eachOvat each finite place ofK. That is, NO=

Q

vNOv.

IfR is a PID, we identify the ideal NOwith its generator, up to units.

Unlike the case of maximal orders, we have no explicit classification of Eichler orders in terms of the discriminant.

Proposition 2.4.8. If O is an Eichler order of level N , then the discriminant of O is DO = DBN .

Lemma 2.4.9. Let I be an ideal in B and its right order O = Or(I) is a maximal

order. Then there exists an elementhv∈ Bv∗so thatIv = hvOv.

Corollary 2.4.10. For an ideal I in B, the right order of I, Or(I), is maximal if and

only if the left order ofI, O`(I), is maximal.

Corollary 2.4.11. If I is a normal ideal in B, then I−1I = Or(I) and II−1= O`(I).

2.5

Shimura Curves

We are now in a position to introduce Shimura curves. In this section, we will focus on the indefinite quaternion algebras over totally real number fields, especially the rational field.

Assume that K is a totally real number field and take a quaternion algebra B over K that splits exactly at one infinite place among all infinite places. That is, B ⊗QR '

M (2, R) × H[K:Q]−1,where H is Hamilton’s quaternions. Notice that we have a natural embedding from B into B ⊗_QR, we now let i∞ : B ,→ M (2, R) be the projection

onto the first factor. Let O be an order of B, O1_{= {γ ∈ O : n(γ) = 1},}

and Γ(O) = i∞(O1).

Then Γ(O) is a discrete subgroup of SL(2, R) and hence it acts on the upper half plane h_{= {τ : C : Im(τ ) > 0} by the usual fractional linear transformations.}

We denote X(O) the quotient space Γ(O)\h (or Γ(O)\h ∪ Q ∪ {∞} if B = M(2, Q)), which has a complex structure as a compact Riemann surface. It is the so-called Shimura curve associated to O. In the case of the classical modular curve, the associated quaternion algebra is the matrix algebra B = M(2, Q) with discriminant D = 1.

Example 2.5.1. (1) Let_{B = M (2, Q). If O = M (2, Z), then Γ(O) = SL(2, Z)} andX(O) is the classical modular curve X(1) = X0(1). For the Eichler order

O = Z Z

N Z Z
, X(O) is the modular curve X0(N ).

(2) LetO be the order Z + Zi + Zj + Z1+i+j+ij2 in the quaternion algebraB =

_−1,3

Q



. The quaternion algebra is ramified at the finite places 2 and 3. An embeddingi∞: B → M (2, R) is given by i 7→0 −1 1 0  , j 7→ √ 3 0 0 −√3  and i∞(O1) =  α β −β α  : αα + ββ = 1, α, β ∈ Z[√3]  .

2.6

Signatures of Shimura curves

Recall that a nonidentity element γ = a b

c d
of SL(2, R) is called parabolic,

hy-perbolic, or elliptic if γ has one fixed point, 2 distinct points of P1(R), or a pair of conjugate complex numbers, respectively. The points τ fixed by γ are the roots of

cτ2+ (d − a)τ − b = 0.

Hence, it can be simplified that γ is parabolic, elliptic, or hyperbolic, corresponding to whether |tr(γ)| = 2, |tr(γ)| < 2, or |tr(γ)| > 2.

Definition 2.6.1. Let γ be an element of Γ(O).

1. The fixed point of a parabolic element is called acusp. We let e = ∞.

2. The pointτ in the upper half-plane fixed by an elliptic element is called an ellip-tic point of order e, where e is the number of elements in Γ(O)/ ± 1 that fixes τ . In other words, e is the order of the isotropy subgroup of τ in Γ(O)/ ± 1. Note that cusps can only appear when the quaternion algebra is M(2, Q). There-fore, if B 6= M(2, Q), the quotient space Γ(O) \ h is a compact Riemann surface; if B = M(2, Q), we compactify the Riemann surface Γ(O) \ h by adjoining cusps. Proposition 2.6.1. If Γ(O) has a parabolic element, then the related quaternion alge-bra must beM(2, Q).

Proof. Let γ ∈ Γ(O) be a parabolic element and h be the associated element in O1. Then tr(h) = 2 or −2, and N (h) = 1. Note that ±1 are elements of O1_{and hence}

±1 − h belong to O. Without loss generality, we may assume that tr(h) = 2. Then 1 − h is an element has reduced trace 0 and reduced norm 0. This means that the quaternion algebra has a zero divisor element 1 − h and hence it is isomorphic to the 2-by-2 matrix algebra over a totally real number field. The only possibility is the Q-quaternion algebra M(2, Q), for which splits at exactly one real place.

For the curve X(O) with genus g, it is well-known that there exist hyperbolic elements a1, . . ., ag, b1, . . ., bg, and elliptic or parabolic elements c1, . . ., cr that

generate Γ(O)/ ± 1 with relations

[a1, b1] . . . [ag, bg]c1. . . cr= 1, where [ai, bi] = aibia−1i b −1 i .

We let (g; e1, . . . , er) be the signature of the curve X(O). The number ej runs over

all Γ(O)-inequivalent cusps and elliptic points. In particular, if a Shimura curve X(O) has signature (0; e1, e2, e3), we say that Γ(O) is an arithmetic triangle group.

2.7

Automorphic Forms on Shimura Curves

Let X(O) = Γ(O) \ h be the Shimura curve associated to the order O in an indefinite quaternion algebra B. In this section, we let k be a non-negative even integer.

Definition 2.7.1. An automorphic form of weight k on Γ(O) is a holomorphic func-tionf : h → C such that

f aτ + b cτ + d 

= (cτ + d)kf (τ )

for allτ ∈ h and all a b

c d
∈ Γ(O).

Iff is meromorphic and k = 0, then f is called an automorphic function. More-over, if the Shimura curve is of genus0, an automorphic function is said to be a Haupt-modul if it generates the field of automorphic functions on Γ(O).

Remark 2.7.1. For the quaternion algebra B = M (2, Q), we also need additional conditions at cusps. However, we do not consider the classical modular curves here, so we need not to consider the cusps. The curves mentioned in the following discussions are always concerned to be the quotient space related the quaternion algebraB 6= M(2, Q) (if not be pointed out).

The automorphic forms of a given weight k form a complex vector space. We denote it by Sk(Γ(O)) or Sk(X(O)). It is easy to see that the weight 0 automorphic

forms on Γ(O) are exactly the constant functions. Using the Riemann-Roch Theorem, one can figure out the dimension formula of Sk(Γ(O)).

Proposition 2.7.2. If the signature of X(O) is (g; e1, . . . , er), then the dimension of

the space of automorphic forms of weightk on Γ(O) is

dim Sk(Γ(O)) =          1, ifk = 0, g, ifk = 2, (g − 1)(k − 1) +X j  k 2  1 − 1 ej  , ifk ≥ 4.

Chapter 3

The Shimura Curves X

₀

D

(N )

In this chapter, we will review some facts about the Shimura curves X0D(N ), which is

obtained by the Eichler order O(D, N ) of level N in an indefinite quaternion algebra over Q with discriminant D. Most of the materials are coming from [1, 4, 5, 15].

3.1

Eichler orders O(D, N ) and Shimura curves X

(N )

Let B be a quaternion algebra over Q of discriminant D. According to the proposition 2.4.5, for each positive integer N with gcd(D, N ) = 1, there exists an Eichler order of level N . We now give a characterizations of Eichler orders in a quaternion algebra over Q.

Proposition 3.1.1. Let O be an order in a Q-quaternion algebra B of discriminant D. LetN be a positive integer relatively prime to D. Then the following conditions are equivalent:

(1) O is an Eichler order of level N .

(2) For each prime number, the localizationOpis maximal ifp - N , and is

isomor-phic to the order 

Zp Zp

N Zp Zp



ifp | N .

(3) For each prime number, the localizationOpis maximal ifp | D, and is

isomor-phic to the order 

Zp Zp

N Zp Zp



ifp - D.

Proposition 3.1.2. Let O be an order in a Q-quaternion algebra B of discriminant D. (1) IfO is an Eichler order with norm NOwithgcd(D, NO) = 1, then its

discrimi-nantDOis equal toDNO.

(3) LetO and O0be orders inB and they are conjugate. Then O is an Eichler order of levelN if and only if O0is an Eichler order of levelN .

Theorem 3.1.3. In an indefinite quaternion algebra over Q, there is only one Eichler order of a given levelN , up to conjugation. Moreover, such an Eichler order contains a unit of norm−1.

We use the notation O = O(D, N ) to indicate the Eichler order of level N in an indefinite quaternion algebra over Q of discriminant D, where D, N are coprime positive integers. In literature, sometimes, the order O(D, N ) is said to be the Eichler order of level (D, N ). We remark that when N = 1, the order O(D, N ) is a maximal order.

3.1.1

The Shimura curves X

(N )

Note that Theorem 3.1.3 implies that the Shimura curve X(O) attached to the Fuchsian group defined from O = O(D, N ) is only dependent on the discriminant D and the level N . The curve X(O) has a canonical model as a projective curve defined over Q (Shimura [19]). Here, we use the notation X0D(N ) to denote the corresponding

Shimura curve.

Theorem 3.1.4. Let O be the Eichler order of level N in an indefinite Q-quaternion algebra_{B with discriminant D. There is a projective algebraic curve X(O) over Q} such that there exists an open immersion of Riemann surfaces

Γ(O) \ h ,→ X(O)(C). WhenD 6= 1, this map is a biregular isomorphism.

Therefore, the curve X(O(D, N )) has a canonical model over Q, we denote it by XD

0 (N ). The notion of such Shimura curves generalizes that of the classical modular

curves X1

0(N ) = X0(N ).

3.1.2

The Atkin-Lehner involutions on X

(N )

Like the theory of the classical modular curve, we can define the Aktin-Lehner group of the curves XD

0 (N ).

For a compact Riemann surface X uniformized by a Fuchsian group Γ, the quo-tient group of the normalizer of Γ in GL(2, R)+ _{by Γ acts as automorphisms on X.}

Here we let O = O(D, N ) and take Γ = Γ(O), for convenience. To obtain such automorphisms, we pullback to the order O in the Q-quaternion algebra B.

For an integer m | DN with gcd(m, DN/m) = 1, we then have an ideal I = xmO = Oxmwith I2 = mO, for some xm ∈ O with n(xm) = m. Since O has a

unit of reduced norm −1, the norm 1 group O1is equal to the conjugation xmO1x−1m.

Hence, xm gives an automorphism wm of X0(D, N ) with w2m = id. This is called

Atkin-Lehner involution associated to m. The Atkin-Lehner group

is an automorphism group of X0D(N ) associated to the the group NB+(O1)/Q∗O1,

where NB+(O1) = {h ∈ B∗ : hO1h−1 = O1, n(h) > 0} is the normalizer of O in

the subgroup of B∗collecting the positive reduced norm elements. The elements wm

of WD,Ncan be taken to be any generator of the only 2-sided ideal of reduced norm m

of O when m 6= 1. Hence the group WD,N is isomorphic to (Z/2Z)r, where r is the

number of prime factors of DN .

3.2

Optimal Embeddings

Let O be an order of a quaternion algebra B over the field K. Let F be a quadratic extension over K, and OF be its ring of integers. For a given order Λ of OF, an

embedding of Λ in O is an embedding from F into B such that φ(Λ) ⊆ O; an optimal embedding of Λ in O is an embedding from F into B such that

φ(F ) ∩ O = φ(Λ).

We let E(O, Λ) = EK(O, Λ) be the set of all optimal embeddings of the given quadratic

order Λ into the order O.

In the following discussion, we are going to consider the optimal embeddings of quadratic orders into a given Eichler order O = O(D, N ).

3.2.1

_{Optimal embeddings of quadratic orders into Q-quaternion}

algebras

We first consider the case when B is a quaternion algebra over Q of discriminant D and F = Q(√dF) is a quadratic extension field over Q of discriminant dF. We

re-call that there is an embedding from F into B if and only if for any prime p in Q so that Qp ⊗ B  M(2, Qp), the prime number p does not completely split in F . In

other words, we have an embedding F ,→ B defined over Q if and only if the Leg-endre symbol



dF

p



6= 1 if p - D. Naturally, we have an action of B∗ _{on the set}

{φ : F ,→ B is an embedding defined over Q} given by φh_{= h}−1_{φh, for any element}

h ∈ B∗.

Proposition 3.2.1. Let φ : F ,→ B be an embedding defined over Q. For any element h ∈ B∗, one hasφ ∈ E (O, Λ) if and only if φh_{∈ E(h}−1_{Oh, Λ).}

The following fact give conditions for the existence of optimal embeddings. Lemma 3.2.2. Let Bp be the division quaternion algebra over Qp, and Op be the

maximal order ofBp. If there exists an embedding fromFpintoBp, we consider an

orderΛpinFp. ThenE(Op, Λp) is nonempty if and only if Λpis a maximal order.

Proposition 3.2.3. Let O be an Eichler order of level N in the Q-quaternion aglebra B. Let F be a quadratic number field such that there is an embedding from F into B, andΛ be an order of conductor m in F . Then

(2) IfN = 1 and B is indefinite, then E (O, Λ) is non-empty if and only if gcd(D, m) = 1.

Moreover, while B is an indefinite quaternion algebra, there is exactly one structure of an Eichler order O = O(D, N ) of level N with gcd(D, N ) = 1. The action of B∗ on field embeddings gives an action of the normalizer of O in B∗on E(O, Λ). Corollary 3.2.4. Let O be an Eichler order in an indefinite Q-quaternion algebra B. LetNB∗(O) be the normalizer of O in B∗ andG be a subgroup of N_B∗(O). Then

the action ofG on E (O, Λ) is an equivalence relation. Here, φ, φ0 ∈ E(O, Λ) are G-equivalent if there is an element h ∈ G such that φ0= h−1φh.

3.2.2

Optimal embeddings of quadratic orders into O(D, N )

In this subsection, we will count the the number of optimal embeddings of quadratic orders into the Eichler order O(D, N ) of an indefinite Q-quaternion algebra of dis-criminant D.

Let Λ = Λ(dF, m) be an order of conductor m in the field F = Q(

√

dF), where

dF is the discriminant of the quadratic field F . Denote by

ν(D, N, dF, m; O∗) := #E (O, Λ)/O∗

the class number of O∗-equivalent optimal embeddings of Λ in O. In the local case, we let νp(D, N, dF, m; O∗) = ] E (Op, Λp)/O∗p denote the corresponding class number,

where Op, Λpare the localization of O and Λ at prime p, respectively.

Theorem 3.2.5. Assume that there is an embedding of F into B and gcd(m, D) = 1. Then

ν(D, N, dF, m; O∗) = h(dF, m)

Y

p|DN

νp(D, N, dF, m; O∗),

whereh(dF, m) is the ideal class number of the order Λ = Λ(dF, m), and the local

class numbers are given by

(1) Ifp | D, then νp(D, N, dF, m; O∗) = 1 −  dF p  . (2) Ifp | N and p2 - N , then νp(D, N, dF, m; O∗) = ( 1 +dF p  , ifp - m, 2, ifp | m (3) AssumeN = pru1, withp - u1,r ≥ 2. Write m = pku2, withp - u2.

(a) Ifr < 2k, then

νp(D, N, dF, m; O∗) =

(

pk/2_{+ p}k/2−1_{, ifk = 0 mod 2,}

(b) Ifr = 2k, then νp(D, N, dF, m; O∗) = pk−1  1 + p +d_p. (c) Ifr = 2k + 1, then νp(D, N, dF, m; O∗) =          2ψp(m), if  dF p  = 1, pk, if dF p  = 0, 0, if dF p  = −1. (d) Ifr > 2k + 1, then νp(D, N, dF, m; O∗) = ( 2ψp(m), if  dF p  = 1, 0, otherwise. Here, the functionψpis a multiplicative function given by

(

ψp(pk) = pk−1(p + 1)

ψp(m) = 1, if gcd(m, p) = 1.

Corollary 3.2.6. If the integer N is square-free, and assume that there exists an em-bedding ofF into B, gcd(m, D) = 1, then the class number of optimal embeddings of Λ into O can be expressed as

ν(D, N, dF, m; O∗) = ( 0, if there exists_{p | N, p - m,}dF p  = −1, h(dF, m)2s+t, otherwise,

wheres is the number of prime factors p of D so that p is inert in F and t is the number of prime factors ofN that splits in F or divides m.

3.3

Complex Multiplication Points on X

(N )

When F is an imaginary quadratic field, and assume that F embeds in the indefinite Q-quaternion algebra B. Then, for any embedding φ : F ,→ B, the image of F∗ in B∗\ Q∗_{under φ has a unique fixed point on the upper half-plane h.}

To be more precise, it is known that two elements γ, γ0∈ GL(2, R) have the same fixed points if and only if there exist real constants λ 6= 0 and µ so that γ0= λγ + µ · 1. Now, if i∞stands for the fixed embedding of the infinite Q-quaternion algebra B into

M(2, R) and φ is an embedding from F into B, we then have precisely one fixed point in h under the action of the set i∞(φ(F∗)). In this case, we denote τφthe fixed point in

h. It is a complex multiplication point (briefly, CM-point) on the associated Shimura curve X.

Definition 3.3.1. Let Λ be an order of discriminant dΛ = m2dF in the imaginary

quadratic fieldF . A point τ ∈ X₀D(N ) is said to be a CM-point by Λ or CM-point of discriminant dΛ if it is fixed byi∞(φ), i.e. τ = τφonX0D(N ), for an optimal

embeddingφ in E (O(D, N ), Λ).

Remark 3.3.1. A point on X₀D(N ) is elliptic if and only if it is a CM-point by the ring of integers Z[√−1] or Z[(1 +√−3)/2].

3.3.1

The set of CM-points by an order

It is clear that there are many CM-points on the curve X0D(N ). However, for a given

order Λ, the number of CM-points by Λ is related to the number of non-equivalent optimal embeddings of Λ into the order O(D, N ) and it is finite.

Proposition 3.3.2. Let φ, φ0 ∈ E(O(D, N ), Λ). Then τφ = τφ0 under the action of

Γ(O(D, N )) if and only if φ is O(D, N )1_{-equivalent toφ}0 _or−φ0_{, where−φ is the}

embedding defined by(−φ)(√dF) = −φ(

√ dF).

Note that −φ(F ) = φ(F ) and −φ(Λ) = φ(Λ), hence φ and −φ have the same fixed point in h. Also, they are either simultaneously optimal or not.

Proof. May assume that φ is equivalent to φ0. Suppose that h ∈ O(D, N )1 _{is the}

element such that h−1φ(α)h = φ0(α), for all α ∈ F∗\ Q∗_{. Fixing α ∈ F \ Q, let}

γh, γ, and γ0 in Γ(O(D, N )) be the corresponding elements to h, φ(α), φ0(α). Then

γ_h−1γγh = γ0 and hence τφ0 = γ−1

h τφ, which is Γ(O(D, N ))-equivalent to the point

τφ.

Conversely, suppose that there exists γh ∈ Γ(O(D, N )) so that γh−1τφ = zφ0.

Write h ∈ O(D, N )1 as the associated element to γh. Now, we choose α ∈ F \ Q

with trF_Q(α) = 0. Then both of φ0(α) and h−1φ(α)h fix the point τφ0. Considering the

elements γ = i∞(φ(α)) and γ0 = i∞(α), one has the identity

γ_h−1γγh= λγ0+ µ · 1, λ 6= 0, µ ∈ R.

By the assumption of trF_Q(α) = 0, we can get that the constant µ must be 0, since the trace is Q-linear and preserved by conjugation. The relation between determinants,

N_QF(α) = det(γ) = λ2det(γ0) = λ2N_QF(α) and N_QF(α) 6= 0,

implies that γ−1_h γγh = ±γ0. That is, the embedding φ0is O(D, N )1-equivalent to φ

or −φ.

Lemma 3.3.3. If φ is an embedding from F into B, then φ is not O1-equivalent to−φ, for any orderO in B.

Proof. Suppose that φ is O1_{-equivalent to −φ. For a fixed α ∈ F −Q with tr}F

Q(α) = 0,

there is an element γ ∈ SL(2, R) such that

γ−1i∞(φ(α))γ = −i∞(φ(α)).

Note that if we choose the element α with trace not 0 then the lemma hold by the prop-erties of trace. Now we consider the associated quadratic forms. Since det(i∞(φ(α))) =

NF

Q(α) > 0, we will get a contradiction.

From above results, to count the number of CM-points by the order Λ is equiva-lently to count the number of the non-equivalent class E(O(D, N ), Λ)) under the action of O(D, N )∗. We now let CM(dΛ) denote the set of CM-points of discriminant dΛ,

up to O(D, N )∗-equivalence. Also, we use the same notation CM(dΛ) or CM(Λ)

to indicate the set of the in-equivalent optimal embeddings of Λ into O(D, N ). In the stance, the optimal embedding corresponding to a point τ , means that the O(D, N )-equivalent optimal embedding which fixes the point τ ∈ h.

Theorem 3.3.4. Fix Λ = Λ(dΛ) an order of index m in the the imaginary quadratic

fieldF which has discriminant dF.

#CM(dΛ) = #CM(Λ) = ν(D, N, dF, m; O(D, N )∗),

the class number ofO(D, N )∗_{-equivalent optimal embeddings ofΛ in O(D, N )}

men-tioned in Section 3.2.2.

3.3.2

Fixed points of Atkin-Lehner involutions

Regarding Atkin-Lehner involutions acting on X0D(N ) as optimal embeddings,

CM-points arise in a natural way as fixed CM-points of Atkin-Lehner involutions on X0D(N ).

For a given involution wm, we let h be its corresponding element in the order

O = O(D, N ) with Oh = hO, n(h) = m. Assume that P ∈ XD

0 (N )(C) is a

fixed point of wmon the curve X0D(N ) and τ ∈ h representing for P . Then we have

hτ = uτ , for some u ∈ O1_{. (Here, we use the notation hτ to simplify the action of γ} h

on τ ∈ h with γh∈ SL(2, R).) Therefore, we may assume that hτ = τ and tr(h) ≥ 0,

by replacing −h by h if necessary. Since h fixes a pair of conjugate complex numbers τ and ¯τ , the field Q(h) containing h and Q is an imaginary quadratic field.

Observe that the conjugation ¯h of h generated the same principal ideal Oh = O¯h, n(h) = m, and tr(h) ∈ Q. One has that ¯h = uh, for some u ∈ O1 _{∩ Q(h). In}

particular, u =      ζ4, if m = 2, ζ3, if m = 3, −1, else.

Now let Λ be the quadratic order O(D, N ) ∩ Q(h). It is clear that Λ contains the ring Z[h]. Then for a given fixed point P ∈ X0D(N ) of wm, we can associated 2 optimal

embeddings of R into O(D, N ), corresponding to h and ¯h. Consider an embedding u = γ−1hγ, which is O(D, N )∗-equivalent to h. If n(γ) = 1, then u fixes γτ , which represents the same point P ; if n(γ) = −1, then u fixes the point γ(¯τ ) associated to the point ¯P , the complex conjugate point on the Shimura curve X0D(N ). We can see

that P is a real point (i.e. P = ¯P ) if and only if h is O(D, N )∗-equivalent to ¯h. Proposition 3.3.5. (Ogg [15]) Assume that m > 1 is a square-free exact divisor of DN . Then the set of the fixed points of an Atkin-Lehner involution wmonX0D(N ) is

     CM(−4) ∪ CM(−8), ifm = 2, CM(−m) ∪ CM(−4m), ifm = 3 mod 4, CM(−4m), else.

We remark that in the case m is not square-free, the description of the fixed points is complicated. In general, they will be a proper sunset of ∪f2_|4mCM(−4m/f2).

3.3.3

Fields of definition of CM-points

Let Λ be an order with discriminant d in the imaginary quadratic field F = Q(√−s). Set I(Λ) be the group of the fractional invertible ideal classes of Λ, and HΛbe the ring

class field of Λ. By class field theory, we have the Artin isomorphism from I(Λ) to HΛ

by [p] 7→ Frobp, for all primes p of F unramified in HΛ. Denote Q(P ) be the number

field generated by the coordinates of the CM-point P ∈ CM(d) on XD

0 (N ). Then we

have fundamental result due to Shimura, which is the so-called Shimura’s reciprocity law.

Theorem 3.3.6. [19](Shimura’s reciprocity law) Let Φ be the natural uniformization map h→ Γ(O(D, N )) \ h, τ ∈ h so that Φ(τ ) = P has CM by the order Λ. Then

(1) HΛ = F · Q(P ).

(2) Letφ be the embedding Λ ,→ O(D, N ) corresponding to the point τ . Assume that a∈ I(Λ) and σais the Artin symbol attached to a. Then action of the Galois

groupGal(HΛ/F ) ' Pic(Λ) is given by

σa(P ) = Φ(α−1τ ),

whereα is some element in O(D, N ) with n(α) > 0 satisfying the identity φ(a)O(D, N ) = αO(D, N ).

3.4

Signatures

Recall that the genus of a Shimura curve X is given by

g(X) = 1 +Vol(X) 2 − 1 2 r X i=1  1 − 1 ei  ,

where the sum runs through all elliptic points with ei being their respective orders.

Considering a normalizationR R dxdy/y2_{π for the hyperbolic area, from [17], the}

formulae for the area (volume) and the genus of XD 0 (N ) are Vol X₀D(N )
=DN 6 Y p|D  1 − 1 p  Y p|N  1 +1 p  and g(X₀D(N )) = 1 + Vol(X D 0 (N )) 2 − 1 2 X ei  1 − 1 ei  .

In particular, the total number of elliptic points of order 2 and 3, say v2and v3, are

given by v2=      Y p|D  1 − −4 p  Y p|N  1 + −4 p  , if 4 - N, 0, if 4 | N,

and v3=      Y p|D  1 − −3 p  Y p|N  1 + −3 p  , if 9 - N, 0, if 9 | N.

These can be obtained equivalently by counting the number of optimal embeddings from the maximal order in the fields Q(√−4) and Q(√−3) into the quaternion order O(D, N ).

Note that the ramification points of this covering X0D(N ) −→ X0D(N )/hwmi are

the exact fixed points of wm on the curve X0D(N ). Therefore, from the

Riemann-Hurwitz formula, we can deduce that the genus of the quotient curve X0D(N )/hwmi is

equal to (g + 1)/2 − Bm/2, where g is the genus of X0D(N ), and Bmis the number

of the fixed points of wmon X0D(N ).

From Proposition 3.3.5, it is easy to determine the number of elliptic points on XD

0 (N )/G for any subgroup G of WD,Nsuch that m is squarefree for any wmin G.

Lemma 3.4.1. [23] Let G be a nontrivial subgroup of the group WD,Nof Atkin-Lehner

involutions onXD

0 (N ) such that m is squarefree for any wm ∈ G. Then the only

possible orders of elliptic points onXD

0 (N )/G are 2, 3, 4, and 6.

1. Ifw2∈ G, then the number of elliptic points of order 2 on X0D(N )/G is

2 |G|        X wm∈G,m6=1 (#CM(−4m) + #CM(−m)) − #CM(−3) , if w3∈ G, X wm∈G,m6=1 (#CM(−4m) + #CM(−m)) , if w3∈ G./

Ifw26∈ G, then the number is (#CM(−4) + 2A)/|G|, where A is

       X wm∈G,m6=1 (#CM(−4m) + #CM(−m)) − #CM(−3) , if w3∈ G, X wm∈G,m6=1 (#CM(−4m) + #CM(−m)) , if w3∈ G./

(If−m is not a discriminant, we simply set #CM(−m) = 0.)

2. Ifw3∈ G, then there are no elliptic points of order 3 on X0D(N )/G. If w36∈ G,

then the number of elliptic points of order3 is #CM(−3)/|G|.

3. Ifw26∈ G, then there are no elliptic points of order 4 on X0D(N )/G. If w2∈ G,

then the number of elliptic points of order4 is 2#CM(−4)/|G|.

4. Ifw36∈ G, then there are no elliptic points of order 6 on X0D(N )/G. If w3∈ G,

then the number of elliptic points of order6 is 2#CM(−3)/|G|.

Proof. The fact that only 2, 3, 4, and 6 can be the orders of elliptic points on X0D(N )/G

is well-known.

Let wm ∈ G. By Proposition 3.3.5, the fixed points of wm consist of CM(−4),

or CM(−m) are fixed only by wmand every other Atkin-Lehner involution other than

w1 permutes them. Thus, there are totally |G|/2 points in CM(−4m) or CM(−m)

that are mapped to the same point in the covering XD

0 (N ) → X0D(N )/G. For points

in CM(−4), which constitute elliptic points of order 2 on XD

0 (N ), they are also fixed

by w2. Thus, if w2 ∈ G, then there are 2#CM(−4)/|G| elliptic points of order 4

on XD

0 (N )/G. If w2 6∈ G, points in CM(−4) contribute another #CM(−4)/|G|

elliptic points of order 2 on XD

0 (N )/G. For points in CM(−3), which are elliptic

points of order 3 on XD

0 (N ), they are also fixed by w3. If w3∈ G, then they become

elliptic points of order 6 on XD

0 (N )/G and there are 2#CM(−3)/|G| such points. If

w3 6∈ G, then they remain elliptic points of order 3. There are #CM(−3)/|G| such

points. Summarizing, we get the lemma.

3.5

Cerednik-Drinfeld Theory

ˇ

In this section, we will review the ˇCerednik-Drinfeld theory of the p-adic uniformiza-tion for Shimura curves, which gives a descripuniformiza-tion of the bad reducuniformiza-tion of Shimura curves X0D(N ). In the following, for fixed integers D and N , we will use X to denote

the Shimura curve X0D(N ).

Due to the moduli interpretation of Shimura curves, the curve X admit a canonical model over Q. Following from the work of Morita, ˇCerednik, and Drinfeld, there exists a proper integral model M = M (D, N )/Z of X which extends the moduli interpretation to arbitrary schemes over Z and it is smooth over Z[DN1 ]. It is known that

the curve X has good reduction only at the prime numbers p with p - DN . For a prime divisor p of D, the curve X/Qpdefined over Qpis a Mumford curve. By Mumford’s

theory, the curve X has a p-adic uniformization expressing it as a quotient of the p-adic upper half plane hpby the action of a discrete subgroup Γ of PGL(2, Qp). The theory

of ˇCerednik-Drinfeld provides an explicit description of this p-adic uniformization. It describes X × Qpas a quadratic twist of Γ \ hpover Qp.

In the following, we will also describe the connection between Brandt matrices and the bad reductions of X from the theory of ˇCerednik-Drinfeld. Let us fix the nota-tions Kp, Kunrp , and Zunrp , as the unramified quadratic extension of Qp, the maximal

unramified extension of Qp, and the ring of integers of Kunrp , respectively.

3.5.1

The ˇ

Cerednik-Drinfeld theory

Let p be a prime with p | D, and O = O(D/p, N ) be an Eichler order of level N in a definite quaternion algebra B0defined over Q of discriminant D/p. Let Z(p)be the set Z[1p] and O (p) = O ⊗ Z(p)_{. Define eΓ} 0= O(p) ∗ and e Γ+= n x ∈ eΓ0: Ordp(n(x)) ≡ 0 mod 2 o . Also, we let Γ0= eΓ0/Z(p) ∗ and define Γ+= eΓ+/Z(p) ∗ .

Identifying the quaternion algebra B0⊗ Qpwith the quaternion algebra M (2, Qp), the

groups eΓ0and eΓ+can be considered as discrete compact subgroups of GL(2, Qp)

con-taining the element p 0_{0 p}
, and Γ0and Γ+can be viewed as discrete compact subgroups

of PGL(2, Qp). Then the quotients Γ0\ hpand Γ+\ hpexist. Moreover, let Γ = Γ0or

Γ+, there exists a unique scheme PΓ proper over Zp such that the formal completion

of PΓ along its closed fibre is canonically the quotient Γ \ hpover Zp. Note that the

scheme PΓis projective over Zp, and its generic fibre XΓ is a smooth curve defined

over Qp.

Theorem 3.5.1. ( ˇCerednik-Drinfeld) There is an isomorphism over Zpsuch that

X × Qp2 ' X_Γχ +,

whereχ is the character χ : Gal(Kp/Qp) −→ Aut(XΓ+⊗ Kp) defined by F rob 7→

wp, andX_Γχ₊is the quadratic twist ofXΓ+byχ.

3.5.2

Dual graph and bad reduction

Let ∆ be the Burhat-Tits tree of SL(2, Qp), ie., PGL(2, Qp)/PGL(2, Zp), on which

PGL(2, Qp) acts in the usual manner. According to the ˇCerednik and Drinfeld’s result,

the special fiber of M ⊗ Zp is determined by a quadratic twist by the finite graph

G = Γ+\ ∆ with lengths. Geometrically, a vertex v of the graph G is corresponding

to the irreducible rational component Cvof Mp, where Mpis the closed fiber of M at

the prime p. An edge e of length `(e) connecting vertices v and v0is corresponding to an intersection point x of the component Cvand Cv0locally at which

Mx× cZp

unr

' SpecZcp

unr

[X, Y ]/XY − p`(e). Now, let us see some properties of the graph G.

We first consider the finite graph G0 = Γ0\ ∆ with lengths. Let I1, I2, . . ., Ihbe

a completely representatives of the left ideals of O, and let Oibe the right order of Ii,

i = 1 . . . h. The vertices of the graph G0form the set Ver(G0) = V , where V collects

the right orders Oi. The vertices v1 and v2 are linked by an edge if and only if the

intersection of the corresponding orders O1and O2is an Eichler order O(D/p, N p),

up to conjugation. Observe that the group Γ+is a subgroup of index 2 of the group Γ0,

and the quotient group Γ0/Γ+is generated by γpΓ+, where γpis corresponding to an

element of O with reduced norm p. We can construct the graph G with lengths from the graph G0.

The vertices of the graph G are the set Ver(G) = V ∪ V0, where V0 = γpV with

v0 = γpv. There are no edges in G connecting 2 vertices from the same set V or V0.

Let `(v) be the weight of the a vertex v, and `(e) be the length of an edge e. One has the following facts.

Proposition 3.5.2. For a given vertex v ∈ V , let v0= γpv ∈ V0.

1. The weight`(vi) of the vertex viis equal to the half of the number of the units in

the corresponding orderOi. That is ,

`(vi) =

#O∗ i