QUATERNION ALGEBRAS AND ELLIPTIC CURVES OVER …

The Pennsylvania State University

The Graduate School

The Eberly College of Science

QUATERNION ALGEBRAS AND ELLIPTIC CURVES OVER

FUNCTION FIELDS OF FINITE CHARACTERISTIC

A Dissertation in

Mathematics

by

Ryan T. Flynn

c© 2013 Ryan T. Flynn

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

August 2013

The dissertation of Ryan T. Flynn was reviewed and approved* by the following:

Mihran PapikianAssistant Professor of MathematicsDissertation AdviserChair of Committee

A. Kirsten EisentraegerAssociate Professor of Mathematics

Wen-Ching W. LiDistinguished Professor of Mathematics

Emily GrosholzProfessor of Philosophy

Yuxi ZhengProfessor of MathematicsHead of the Department of Mathematics

* Signatures are on file in the Graduate School.

ii

Abstract

The main objects of study in this work are the function field analogues ofShimura curves arising from indefinite quaternion algebras over Q. We studythese curves and their Jacobians mostly using rigid-analytic techniques. Ourmain results are:

(1) A description of the action of the Hecke operators in terms of the rigid-analytic uniformization.

(2) An analytic construction of elliptic curves associated to harmonic Heckeeigenforms via Tate periods.

(3) The calculation of the degree of the modular parametrization in terms of apairing on harmonic cochains.

(4) Applications to computer calculations.

iii

Contents

List of Notation v

Acknowledgements vi

1 Introduction 1

2 Elliptic Curves and Quaternion Algebras 92.1 Quaternion Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Maximal Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Elliptic Curves over a Field . . . . . . . . . . . . . . . . . . . . . 15

3 Modular Curves of D-Elliptic Sheaves 193.1 Rigid Analytic Upper Half Plane . . . . . . . . . . . . . . . . . . 193.2 The Bruhat-Tits Tree . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Proof of Goldman-Iwahori . . . . . . . . . . . . . . . . . . . . . . 233.4 An Orientation Convention, and the r Map . . . . . . . . . . . . 253.5 ΓΩ as a Mumford Curve . . . . . . . . . . . . . . . . . . . . . . 293.6 Algebraic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Automorphic Forms 364.1 Automorphic Forms and Representations . . . . . . . . . . . . . 374.2 Local Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Global Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Representations Associated to Elliptic Curves . . . . . . . . . . . 464.6 Jacquet-Langlands Correspondence . . . . . . . . . . . . . . . . . 494.7 Quaternionic Uniformization of E . . . . . . . . . . . . . . . . . . 51

5 Analytic Uniformization of E 535.1 Harmonic Cochains for Γ . . . . . . . . . . . . . . . . . . . . . . 535.2 The j Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 The Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.5 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.6 Hecke Equivariance of the Jacobian Sequence . . . . . . . . . . . 765.7 Uniformization of Elliptic Curves . . . . . . . . . . . . . . . . . . 825.8 The Degree of the Parametrization . . . . . . . . . . . . . . . . . 86

iv

6 Computer Applications 916.1 Explicit Hecke Operators . . . . . . . . . . . . . . . . . . . . . . 926.2 Computing Eigenforms . . . . . . . . . . . . . . . . . . . . . . . . 946.3 Computing the Tate Period . . . . . . . . . . . . . . . . . . . . . 956.4 Computing Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . 100

References 102

v

List of Notation

We will use the following notation throughout this work.

k = Fq is the finite field with q = pm elements;

C/k is a smooth, projective, geometrically irreducible curve over k;

F is the field of rational functions on C;

Fv is the completion of F at the place v;

Ov is the ring of integers of Fv;

mv or pv denotes the maximal ideal of Ov;

πv is a uniformizer of mv;

Fv or kv is the residue field Ov/mv;

qv is the order of kv;

ordv is the canonical valuation on Fv;

| · |v is the absolute value on Fv, normalized so that |πv|v = q−1v ;

∞ is a fixed place of F ;

A ⊂ F is the ring of functions on C regular away from ∞;

D is a quaternion algebra over F split at ∞;

R or O is an A-order in D (it should be clear from context whether O is referring

to an order in a quaternion algebra or to the ring of integers in a complete

nonarchimedean field);

A is the ring of adeles of F ;

A× is the group of ideles of F ;

D×(A) and GL(2,A) are the unit groups of D⊗A and M(2, F )⊗A, respectively.

vi

Acknowledgements

This work would not have been possible without the help and support of several

people, only a few of whom it is possible to mention here. I owe a great deal

to my adviser, Mihran Papikian, whose constant encouragement was a major

source of motivation throughout the writing of this thesis. Mihran’s vast knowl-

edge and ease of explanation are second only to his seemingly unlimited supply

of patience, for which I have been very grateful. I would like to thank profes-

sors Dale Brownawell, Kirsten Eisentraeger, Winne Li, and Anna Mazzucato for

several helpful conversations during my graduate career. To my friends, thank

you for keeping me sane these past five years. Last, but by no means least, I

thank my family, whose support and encouragement have always been present.

vii

1 Introduction

Our starting point is the fact, due to Wiles, Breuil, Conrad, Diamond, and

Taylor, that every elliptic curve over Q is modular. In other words, for any el-

liptic curve E defined over Q with conductor N , there is a dominant morphism

fE : X0(N) −→ E, where X0(N) denotes the modular curve of level N . This

theorem has several important arithmetic consequences, the most widely-known

of which is the truth of Fermat’s last theorem. Additionally, this “modularity”

theorem raises new questions and conjectures which have connections to impor-

tant open problems. Among these we merely mention that certain asymptotic

bounds on degfE would imply the ABC-conjecture.

A similar sort of “modularity” theorem is known for quaternionic Shimura

curves. More precisely, let E by an elliptic curve over Q with squarefree

conductor N , and suppose that N is divisible by an even number of primes.

Let D denote the quaternion algebra over Q of discriminant N , O a maximal

Z-order in D, and Γ the image in O× in PGL(2,R). Finally, let XN denote

the quotient ΓH, which can be seen as an algebraic curve over Q. Then it

is a consequence of the Jacquet-Langlands correspondence that there exists a

dominant morphism XN −→ E.

In both cases (quaternionic and congruence subgroup), the modularity theorem

can be re-stated in terms of Jacobians. In short, the above-known uniformiza-

tions give rise to surjective maps of abelian varieties

Jac(X) −→ E

via Albanese functoriality. Conversely, the existence of such a map between

Jacobians implies the existence of a map between curves. In light of this, the

modularity theorem can be rephrased as follows: Let E,N,X0(N) be as above,

and let J0(N) denote the Jacobian of X0(N). Then E appears as an isogeny fac-

1

tor of J0(N). The corresponding theorem for Shimura curves can be rephrased

in exactly the same manner.

In the function field setting, the analagous results were known well before Wiles’

proof. We briefly sketch the important developments here. Let F := Fq(T ) de-

note the function field of the projective line over Fq, and let A := Fq[T ] denote

the subring consisting of functions which are regular on the affine line. Let ∞

be the place of F with respect to which A is discrete, F∞ the completion of F

at ∞, and C∞ a completed algebraic closure of F∞. Drinfel’d ([11]) introduced

the notion of elliptic modules for A, which have properties very analagous to

those of elliptic curves over Q. In short, these elliptic modules come equipped

with torsion points, Tate modules, and continuous actions of the absolute Ga-

lois group of F . Drinfel’d showed that the set of isomorphism classes of elliptic

modules is in bijection with the quotient of a “π-adic upper half plane” by an

action of the group GL(2, A), and that the set of pairs (φ,C) with φ an el-

liptic module and C some torsion data, can be realized as the quotient of the

π-adic upper half plane by a “congruence subgroup” Γ0(n∞) of GL(2, A). This

quotient has the structure of a rigid-analytic curve, denoted X0(n), and can be

made compact after adding finitely many points. Drinfel’d was able to use this

interpretation of X to compare automorphic forms for Γ0(n∞) to factors of the

Jacobian Jac(X0(n)). From here, well-known results of Weil, Deligne, Zarhin,

and Mori may be applied to conclude that any elliptic curve over F with con-

ductor n and multiplicative reduction at ∞ may be realized as a quotient of

Jac(X0(n)).

More recently, a quotient map Jac(X0(n)) −→ E has been explicitly constructed

by Gekeler in the rational function field case ([15]) and by Gekeler and Reversat

in full generality ([17]). In rough detail, their construction proceeds as follows.

First, a theorem of Mumford says that Jac(X0(n)) admits a uniformization by

2

a lattice. That is, there exists a positive integer g and a discrete A-submodule

Λ of Cg∞ such that Jac(X0(n)) ' Cg∞/Λ. This is analagous to the fact that

abelian varieties over Q admit uniformizations of the form Cg/Λ, where Λ is

a full lattice in Cg. Next, (at least in principle) work of Manin and Drinfeld

([29]) allows one to make this uniformization explicit. This is accomplished by

constructing theta functions for Γ (from elements of Γ itself) and showing that

the theta-multipliers which arise in this manner form a full lattice in Cg∞, where

g is the free rank of Γab. However, Gekeler and Reversat need to slightly mod-

ify this construction, since Γ contains torsion elements. The result is an exact

sequence (whose terms we intentionally leave vague for the moment)

1 −→ Γ −→ Theta multipliers ⊗ C∞ −→ Jac(X0(n)) −→ 0.

Finally, it is shown that each object in the above diagram comes equipped with

an action of a “Hecke algebra”, and that the maps in this diagram are Hecke-

equivariant. One knows abstractly a given elliptic curve E/F of conductor n ·∞

with split multiplicative reduction is a quotient of Jac(X0(n)). A short diagram

chase then reveals that the Hecke algebra cuts out a one-dimensional factor of

Γ, generated by (say) the class of α ∈ Γ. Then the theta multiplier associated to

α in the previous paragraph, evaluated at a suitable (possibly different) element

β ∈ Γ, gives an element t ∈ C×∞ with |t|∞ < 1, and it is shown that E ' Tate(t).

Summarizing, one realizes the map Jac(X0(n)) as coming from a sublattice of Λ;

additionally, one explicitly realizes E as a Tate curve with a Tate period which

can be (in principal) made explicit.

The main objects of study in this thesis are the function field analogues of

Shimura curves arising from indefinite quaternion algebras over Q. We study

these curves and their Jacobians mostly using rigid-analytic techniques. Our

main results are:

3

(1) A description of the action of the Hecke operators in terms of the rigid-

analytic uniformization.

(2) An analytic construction of elliptic curves associated to harmonic Hecke

eigenforms via Tate periods.

(3) The calculation of the degree of the modular parametrization in terms of a

pairing on harmonic cochains.

(4) Applications to computer calculations.

Now we describe our results and the contents of this thesis in more detail.

Let C be a projective, smooth, geometrically irreducible curve over a finite field

Fq, where q = pm, and let F be its field of rational functions. Fix a closed

point ∞ of C and let A denote the ring of rational functions on C regular away

from ∞. For example, if we take C = P1 and ∞ = 1T , then F = Fq(T ) and

A = Fq[T ]. The place ∞ induces a nonarchimedean absolute value | · |∞ on F ;

let F∞ denote the completion of F with respect to | · |∞. Finally, let C∞ denote

a completed algebraic closure of F∞.

The objects A, F , F∞, C∞ are the function field analogues of Z, Q, R, and

C, respectively. For now we mention only a few similarities: A is a Dedekind

ring whose nonzero ideals have finite norm (= index in A); F is the field of

fractions of A; the places of F are precisely the prime ideals of A together with

an additional “infinite” place; A sits discretely and co-compactly in F∞; F∞ is

a locally compact topological group, hence can be given a Haar measure with

respect to which the quotient F∞/A has volume 1.

Let D be the quaternion algebra of conductor n = p1 · · · p2k over F , and as-

4

sume that D is split at ∞. Let R be a maximal A-order in D and let Γ = R×.

Γ acts on the set P1(C∞)−P1(F∞) =: Ω (the Drinfel’d upper half plane), and the

quotient XΓ := ΓΩ is a smooth projective rigid analytic curve. XΓ is known

to be a coarse moduli scheme classifying isomorphism classes of objects called

D-elliptic sheaves on C; see section 3.6. This is analogous to the classical result

that Shimura curves classify abelian surfaces with quaternionic multiplication.

By work of Manin and Drinfeld, one has an explicit analytic uniformization of

Jac(XΓ) in terms of theta functions. That is, Jac(XΓ) ≈ Cg∞/Λ for some pos-

itive integer g and some lattice Λ ⊂ Cg∞. We remark that in this setting (and

in contrast to the work of Gekeler and Reversat), one does not need to modify

the theta functions of Manin and Drinfeld.

Now let E be an elliptic curve over F with conductor n · ∞ and with split mul-

tiplicative reduction at ∞. The condition at ∞ says that E admits an analytic

uniformization in the same spirit as elliptic curves over C. As a consequence of

several deep theorems, there exists a surjective homomorphism of abelian vari-

eties J := Jac(XΓ)→ E. We briefly describe how one sees this. First, for ` 6= p,

we associate to E its `-adic representation ρ`(E) : Gal(F sep/F )→ V ∨` (E). One

then associates to ρ`(E) a cusp form πE on GL(2) with the same conductor,

L-factors, and ε-factors as ρ`. By the Jacquet-Langlands correspondence, we

can find a cusp form π′E on D× (the newform associated to E) which shares L-

and ε- factors with πE . An important result of Laumon, Rapoport, and Stuhler

then tells us that V ∨` (E) is a quotient of V ∨` (J) as Gal(F sep/F )-modules. Fi-

nally, a theorem of Zarhin says that in this case E is actually a quotient of J .

We now return to the question of analytic uniformization. Modulo a few de-

tails, we can think of π′E as a harmonic cochain in Har(T ,Z)Γ, i.e. as a certain

Z-valued function on the Bruhat-Tits tree T of PGL(2, F∞). The analytic uni-

formization of Jac(XΓ) takes the form

5

1 −−−−→ Γabc−−−−→ Hom

(Γab, C×∞

)−−−−→ JacΓ (C∞) −−−−→ 0yr

Har(T ,Z)Γ

We consider the action of the Hecke algebra on each object in this diagram.

It turns out that π′E is a simultaneous eigenform for all Hecke operators Tv

with v coprime to n. We prove in section 5.6 that the above diagram is Hecke

equivariant (this is the content of Propositions 5.6.3, 5.6.4, and 5.6.5). From

this it follows that our elliptic curve E gives rise to a subset ∆ ⊂ C×∞ of the

form tZ × µd. Thus after raising to the dth power, we obtain a lattice and

an isomorphism E ' C×∞/∆d. Our main theorem, Theorem 5.7.5, may be

summarized as follows.

Theorem 1.0.1. Let F,A,∞, F∞, C∞ denote the usual objects. Let n be a

squarefree ideal of A which is divisible by an even number of prime ideals of

A. Let D denote the quaternion algebra over F of conductor n, O a maximal

A-order in D, and Γ the image of O× in PGL(2, F∞). Denote by XΓ the rigid-

analytic curve ΓΩ. Let E/F be an elliptic curve with conductor n · ∞ and

with split multiplicative reduction at ∞. Then the quaternionic uniformization

Jac(XΓ) −→ E can be made explicit. More precisely, Jac(XΓ) and E fit into a

diagram of the form

1 −−−−→ Λ −−−−→ (C×∞)g −−−−→ Jac(XΓ) −−−−→ 0y y y1 −−−−→ ∆ −−−−→ C×∞ −−−−→ E −−−−→ 0,

and one is able to explicitly describe the lattices ∆,Λ, as well as the vertical

maps, explicitly in terms of theta functions.

Perhaps the main difficulty here, in contrast to the congruence subgroup

case, is that one does not have a good handle on explicit representatives for the

6

Hecke algebra in the quaternionic situation. That is, the double coset space

ΓαΓ/Γ does not have an easy-to-describe set of representatives. This causes

problems when attempting to prove equivariance, and is especially troublesome

when considering the map taking theta functions to divisor classes. We cir-

cumvent this difficulty by using strong approximation to cook up a ‘nice’ set

of double coset representatives; in particular, this collection of coset represen-

tatives has a symmetry which makes it amenable to the sort of manipulations

we need in the proof of equivariance. The details of this construction may be

found in section 5.6.

Now, if E is optimal in its isogeny class with respect to the above uniformization

(meaning that J → E, and there does not exist E′ admitting J → E′ → E),

then E is actually a subvariety of the Jacobian. In other words, one has

E → Jac(XΓ). By a standard argument, this actually gives us a morphism

f : XΓ → E. A small word of warning is in order here: f will in general not

be defined over F , because XΓ has no F -rational points. Our next result is a

formula for the degree of f . In section 5.3 we define a positive-definite Z-valued

bilinear form (·, ·) : Γ × Γ → Z, and by the above line of reasoning we have

assigned to E an element φE of Γ. Let m := min(φE , α) > 0|α ∈ Γ. We prove

in section 5.8 the following

Theorem 1.0.2. Let XΓ and E be as in Theorem 1.0.1. Let f : XΓ −→ E be

a morphism, and assume that f is optimal. Then the degree of f is given by

deg(f) = d·(φE ,φE)m .

It is worth noting that in the process of proving the above results, we were

able to prove some interesting results in the quaternion case which are con-

jectured in the congruence subgroup cases. In section 5.2 we investigate the

properties of the j-map, which associates to elements of Γ certain Γ-invariante

harmonic cochains. In general this map is known to be injective with finite

7

cokernel; however, due to our knowledge of the torsion subgroups of quater-

nionic groups we are able to prove that in our situation the j-map is actually

an isomorphism. This is the content of Theorem 5.2.3. We prove (Theorem

5.3.11, which is analagous to [17, Theorem 5.6.1]) that the j-map factors as the

composition of two maps, one of which relates elements of Γ to theta functions,

the other of which relates theta functions to harmonic cochains. From this and

previously well-known facts, we see that these two maps are also isomorphisms.

This is the content of Corollary 5.3.12.

Finally we turn our attention to computer implementations of the above re-

sults. Butenuth has developed a package for the algebra distribution MAGMA

which computes the graph ΓT as well as a set of generators for Γ. For the

purposes of computing the Hecke action on elements of Har(T ,Z)Γ, one first

needs an algorithm for finding a set of coset representatives for ΓαΓ/Γ. Next,

one needs good bounds on the convergence of Θ-series in order to estimate the

Tate period of Tate(E) and the j-invariant of E. Finally, we would also like to

compute the constant d in a reasonable amount of time. These questions are

addressed in the final chapter of this thesis.

8

2 Elliptic Curves and Quaternion Algebras

2.1 Quaternion Algebras

We recall here some basic facts about quaternion algebras. Detailed proofs can

be found in [61] and [4]. Let F be a field; for simplicity of exposition we assume

that char(F ) 6= 2; however all the results stated and proved here still hold in

characteristic 2 with little or no adjustment. An algebra over F is a ring A

together with an injection F → Z(A), where Z(A) is the center of A. Algebras

over F are also called F -algebras. A is a central F -algebra if F = Z(A). A is

simple if it has no nontrivial ideals and the set a · b | a, b ∈ A 6= 0.

Definition 2.1.1. A quaternion algebra H over F is a central simple F -algebra

of dimension 4.

Let D be a quaternion algebra over F , and let α ∈ D. Then F (α) is either

F itself, or a commutative subalgebra of D, hence a quadratic extension of F .

By this we mean that F (α) has dimension 2 over F ; F (α) need not be a field.

For α /∈ F let ¯ denote the nontrivial automorphism of F (α) fixing F . These

automorphisms, for the various α /∈ F , extend to a well-defined involutive anti-

automorphism¯: D → D, called conjugation. That is, ¯α = α and αβ = βα for

all α, β ∈ D. We obtain two canonical maps, the reduced trace and the reduced

norm:

Trd : D → F , α 7→ α+ α and Nrd : D → F , α 7→ α · α.

Note that for α /∈ F , the minimal polynomial of α is x2 − Trd(α)x+ Nrd(α).

Lemma 2.1.2. ([61], 1.1.1) The invertible elements in D are precisely the el-

ements with nonzero reduced norm. The reduced norm defines a multiplica-

tive function Nrd : D× → F×. The reduced trace is F -linear. The mapping

(α, β) 7→ Trd(αβ) is a nondegenerate bilinear form on D.

9

Keeping with char(F ) 6= 2, let D be a quaternion algebra over F . Then

there exist a, b ∈ F and a basis 1, i, j, ij of D subject to the relations

i2 = a, j2 = b, ij = −ji.

As a shorthand, we write D = (a,bF ). A similar but slightly more complicated

such basis exists in characteristic 2. For α = x + iy + zj + wij ∈ D, we have

that α = x− iy − zj − wij, and

Trd(α) = 2x, Nrd(α) = x2 − ay2 − bz2 + abw2.

We thus see that the reduced norm defines a quadratic form on the F -vector

space D. The properties of this quadratic form (in particular, whether or not it

represents 0 over F ) will be important in what follows.

Example 2.1.3. The basic example of a quaternion algebra is the matrix al-

gebra M(2, F ), where F → M(2, F ) as diagonal matrices. It is easy to check

that this is a quaternion algebra over any field, and it is the only quaternion

algebra over some fields (for example, algebraically closed fields). The canonical

involution on M (2, F ) is given by

a b

c d

7→ d −b

−c a

. The reduced trace

coincides with the matrix trace and the reduced norm coincides with the deter-

minant map. Note that this is consistent with Lemma 2.1.2, because a matrix

over a field is invertible if and only if its determinant is nonzero.

In the above notation, M(2, F ) =(

1,−1F

), and a basis is given by (say)1 0

0 1

, i =

1 0

0 −1

, j =

0 1

−1 0

, ij =

0 1

1 0

.

For A = x+ iy+ zj +wij ∈M(2, F ), we see that n(A) = x2− y2− z2 +w2. In

particular, the norm form for F is isotropic (i.e. represents 0 nontrivially).

Example 2.1.4. [61, Corollary 1.3.2] Conversely, letD =(a,bF

)be a quaternion

algebra whose norm form is isotropic. Then it can be shown that D 'M(2, F ).

10

Example 2.1.5. If D =(a,bF

)has anisotropic norm form, then D is a division

algebra. Indeed, if the norm is nonvanishing on nonzero elements of D, then by

Lemma 1.2 the nonzero elements of D are all invertible. As a particular case

of this, we mention the historical algebra(−1,−1Q

), the Hamilton quaternions

over Q. This algebra has basis 1, i, j, k satisfying relations i2 = j2 = k2 =

−1, ij = −ji, and the norm of a nonzero element x+ yi+ zj + wij is given by

x2 + y2 + z2 + w2, which is clearly anisotropic over Q.

Example 2.1.6. Let F be the completion of a global field at a real or nonar-

chimedean place. Then there are exactly two quaternion algebras over F up to

isomorphism: the 2× 2 matrices and a unique division quaternion algebra over

F . When F = R the division algebra is given by(−1,−1

R).

We record for future use the following lemma, which can be seen from ex-

amples 2.1.4 and 2.1.5

Lemma 2.1.7. Let D =(a,bF

)be a quaternion algebra over F with norm form

n(α) = x2 − ay2 − bz2 + abw2. Then D is a division algebra if and only if n(α)

is anisotropic.

2.2 Localization

Now let F be a global field and let v be a place of F , with completion Fv. Fix

a quaternion algebra D =(a,bF

)over F . We define the completion of D at v to

be Dv = D ⊗F Fv. The algebra Dv is then a quaternion algebra over Fv, and

it is easy to see that Dv =(a,bFv

). We say that D is ramified at v if Dv is a

division algebra over Fv, and that D is split at v if Dv 'M(2, Fv).

By Lemma 2.1.7, we see that D is split at v if and only if the equation x2 −

ay2 − bz2 + abw2 = 0 has a nontrivial solution over Fv. Note that if v is

any nonarchimedean place of F with residue characteristic 6= 2 and such that

v(a) = v(b) = 0, then this equation has a solution by Henselian lifting. Indeed,

11

any diagonal quadratic form in four variables over a finite field always represents

0, as an easy calculation shows (in fact, this is true even for diagonal quadratic

forms in three variables over a finite field), and as long as the residue field has

characteristic ≥ 3 we may use Hensel’s lemma to lift to a solution over Fv. We

obtain as an immediate consequence the

Proposition 2.2.1. Let D =(a,bF

)be a quaternion algebra defined over a global

field F . Then D is split at all nonarchimedean places v such that v(a) = v(b) = 0

and v - 2 . In particular, D ramifies at only finitely many places v.

There is an important strengthening of this proposition coming from class

field theory. We will state it without proof:

Theorem 2.2.2. Let F be a global field and D a quaternion algebra defined

over F . Then D ramifies at an even number of places v of F . Conversely, let

F be any global field and let S = v1, v2, ..., v2k be any finite set of places of F

of even cardinality. Then there exists a (unique up to isomorphism) quaternion

algebra DS ramified exactly at the places in S. The algebra D is a division

algebra if and only if k 6= 0.

2.3 Maximal Orders

Let F be a field, D a F -algebra, and A a subring of F . Then a subring R of

D is called an A-order if R is a finitely generated A-submodule of D, and if

R contains an F -basis of D. We call R a maximal order if it is not properly

contained in any other A-orders. For example, the ring of integers in a number

field L is the unique maximal Z-order in L. An easy example of a maximal

order in a quaternion algebra is M(2,Z) ⊂ M(2,Q). We will be interested in

maximal orders in quaternion algebras over a global function field F . Maximal

orders are easy to describe over local fields, so we do this first. We then recall

the well-known fact that global orders are determined by their localizations.

12

Lemma 2.3.1. ([63, Chapter XI Prop.4 and Thm1])

(1) Let D be a division quaternion algebra over Fv. Let R = β ∈ D : Nrd(β) ∈

Ov,Trd(β) ∈ Ov. Then R is the unique maximal Ov-order of D. Every left-

or right- ideal of R is two-sided. R has a unique maximal ideal m = Rρ = ρR,

where ρ ∈ R is such that ρ2R = πvR, where πv is a uniformizer for Ov. The

ideals of R are exactly the ideals Rρn.

(2) The ring M(2,Ov) is a maximal order of M(2, Fv). Any maximal order of

M(2, Fv) is conjugate to M(2,Ov).

(3) Let D be any quaternion algebra over Fv, and let R be a maximal Ov-order of

D. Then any left- or right- ideal of R is principal. One has that Nrd(D) = Fv,

Nrd(R) = Ov, and Nrd(R×) = O×v .

Definition/Theorem 2.3.2. : Let D be a quaternion algebra over a global

(function) field F , and let R be an A-order of D. For each place v of F , D → Dv.

Under this embedding, denote by Rv the closure of R in Dv. Then Rv is an

Av-order in Dv. The adele ring of D is the restricted direct product of the Dv

with respect to the Rv. Similarly, the idele group of D× is the restricted direct

product of the D×v with respect to the R×v . The adele ring of D is denoted D(A)

and the idele group is denoted D×(A). These objects are locally compact with

respect to their canonical topologies, and the natural norm and trace maps to

the adeles and ideles over F are clearly continuous.

The above definition seems to depend on the choice of the A-order R; it

turns out that the above notions of adelization do not depend on this choice.

This is because for any two orders R, S of D, there are only finitely many places

v such that Rv 6= Sv. For a proof see [63].

Lemma 2.3.3. ([63]) Let D be a quaternion algebra over F , and R an A-order.

(1) There exists a maximal A-order of D containing R. R is maximal if and

only if Rv is maximal for all places v 6=∞.

(2) For almost all places v, Dv is isomorphic to M(2, Fv) and Rv is maximal.

13

(3) Suppose that for each place v 6= ∞, an order R′v of Dv is given, and that

R′v = Rv for almost all places v. Then there exists an A-order S of D such that

Sv = R′v for all places v.

(4) R = (D∞ ·∏v 6=∞

Rv) ∩D.

(5) Let Nrd : AD → AF be the norm map sending an adele (βv)v ∈ AD to the

adele (NDv (βv))v ∈ AF . Put

A(1)D = β ∈ A×D : |Nrd(β)| = 1,

where |(αv)v| is given by∏v|αv|v. If D is a division algebra, then D×A(1)

D is

compact.

2.4 Approximation

We recall the approximation theorem for quaternion algebras over global fields.

The strong approximation theorem for the reduced norm-1 groups of central

simple algebras over number fields is due to Kneser ([26]); the analagous result

over function fields is due to Prasad ([39]). An elementary proof (in the case

of quaternion algebras) can be found in [31]. We also mention in passing that

a much simpler proof exists when D = M(2, F ), which uses the approxima-

tion theorem for the adeles of F and the fact that SL2 decomposes as upper-

triangular unipotents times lower-triangular idempotents over any field; see [6]

for details.

Theorem 2.4.1. Assume D is indefinite - that is, D∞ ' M(2, F∞). Fix a

finite set S of places of F distinct from ∞, as well as elements α ∈ A and

N ∈ Z>0. For each v ∈ S, let βv ∈ Rv be such that

αNDv (βv) ≡ 1 mod mNv Av.

Then there exists a global element β ∈ D such that

(1) Nrd(β) = α

14

(2) β ∈ R

(3) β ≡ βv mod mvRv for all v ∈ S.

It is sometimes more convenient to use the following equivalent formulation

of the approximation theorem.

Theorem 2.4.2. Suppose that D is indefinite and that the class number of A

is 1. Suppose further that R is an order of D such that NDv (R×v ) = A×v for all

places v 6= ∞ (for example, any maximal order has this property). Then one

has the equality of sets

A×D = D× · (D×∞ ×∏v 6=∞

R×v ).

The following well-known result gives an illustration of the use of approxi-

mation in local-global arguments.

Corollary 2.4.3. Let D be an indefinite quaternion algebra over F , and assume

that A has class number 1. Then all maximal A-orders of D are conjugate.

Proof. : We already know that all maximal orders of a quaternion algebra over

a local field are conjugate. Let R and S be two maximal orders of D. For each

place v, let βv ∈ D×v be such that Sv = βvRvβ−1v . Put β = (βv)v, where we

understand β∞ = 1. Since β ∈ A×D, by theorem 2.4.2 there exists α ∈ D×,

u ∈ GL(2, F∞)×∏v 6=∞

R×v , so we have that Sv = αRvα−1 = (αRα−1)v.

Then S = αRα−1, because an order is completely determined by its local-

izations.

2.5 Elliptic Curves over a Field

We briefly recall the basic facts about elliptic curves. Our main references are

[53] and [54].

15

An elliptic curve over a field F is a nonsingular projective curve E of genus

one with a choice of F -rational point O. Such a curve can be described by a

Weierstrass equation

Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X

2Z + a4XZ2 + a6Z

3, ai ∈ F .

For any field L containing F , let E(L) denote the set of L-valued points of

E. Concretely these are given by the tuples (0, 0, 0) 6= (x, y, z) ∈ L3 which

satisfy a fixed Weierstrass equation for E. Then E(L) forms an abelian group

whose addition law and inverse law are given by morphisms + : E × E → E

and ·−1 : E → E defined over F . Let E[n] denote the set of n-torsion points

in E with respect to this group law. Then for n coprime to p = char(F ), one

has E[n] ' (ZnZ)⊕2. For an integer n let [n] : E → E denote the “multi-

plication by n” map. Then for any other positive integer k, [n] induces maps

E[k]→ E[ k(n,k) ] on torsion subgroups. For a fixed prime ` let T`(E) := lim

←E[`n]

be the inverse limit taken with respect to these maps. By the above, when ` 6= p

one has that T`(E) ≈ Z⊕2` .

Note that since addition on E is given by a morphism over F , all the tor-

sion elements of E are defined over an algebraic closure F of F . It is trivial to

check that Gal(FF ) acts on the `n-torsion points of E compatibly with the

maps [`n], so that Gal(F , F ) acts on T`(E) as well. This action, or the induced

action on V ∨` = Hom(T` ⊗Z` Q`,Q`), will be called the `-adic representation

attached to E.

Now let E1, E2 be two elliptic curves over F . A nonconstant morphism φ :

E1 → E2 which sends O1 to O2 is called an isogeny; it is easy to see that iso-

genies must be surjective. Isogenies are automatically group homomorphisms,

so that φ(E1[n]) ⊆ E2[n]. Therefore, if φ is an isogeny defined over L, then

16

φ induces a homomorphism of G := Gal(F /L)-modules T`(E1) → T`(E2). We

thus obtain an injective homomorphism

Hom(E1, E2)⊗ Z` → HomG(T`(E1), T`(E2)).

More precisely, this homomorphism maps a pair f ⊗ a to the G-module ho-

momorphism b ∈ T`(E1) 7→ f(ab) ∈ T`(E2). This homomorphism is injective,

since if f ⊗ a maps to zero, then f vanishes on an infinite subset of E1[`], which

implies that f vanishes everywhere. In fact, this map is an isomorphism when

the field F is a finite field (Tate), a number field (Faltings), or a global function

field (Zarhin, Mori).

Given an elliptic curve E over a field L containing F , one can associate a

quantity j(E) ∈ L, the j − invariant of E. Then j(E1) = j(E2) if and only if

E1 ' E2 over L. Suppose now that E is defined over a global field F , and let v

be a nonarchimedean place of F . If F is a global function field let O denote the

Dedekind ring consisting of functions regular away from some fixed place∞ 6= v

of F ; if F is a number field then let O denote the ring of integers of F . Then

Fv = Omv is a finite field, and E := E×F Fv is a (possibly singular) curve over

Fv. If E is nonsingular, then E is said to have good reduction at v. Otherwise

let Ens denote E minus its singular points. Then Ens is isomorphic either to

the multiplicative group scheme or the additive group scheme over Fv; in either

case E is said to have bad reduction at v. In the first case we say that E has

multiplicative reduction at v, and in the second case we say that E has additive

reduction at v. Given a Weierstrass equation f(x, y) = 0 for E with coefficients

in O, one can associate a quantity called the discriminant, which is an element

of O as well. Two different Weierstrass equations for the same curve may have

different discriminants; fix a Weierstrass equation for E whose discriminant D

has the smallest nonnegative v-adic valuation. Then E has good reduction at

v if and only if v(D) = 0. There are conditions on the discriminant and the

17

coefficients of f(x, y) which determine the reduction type of E at v when the

reduction type is bad. In particular, the reduction is additive if E is a cuspidal

curve, and multiplicative if E is a nodal curve. Finally, if E has multiplicative

reduction with nodal point P ∈ E(Fv), then the reduction is said to be split if

the tangent lines of E at P have Fv-rational slopes, and is nonsplit otherwise.

We conclude this section with a brief discussion of Tate curves. Let F be a

global field, v be a nonarchimedean place of F , Fv the completion of F at v,

and q ∈ Fv such that |q|v < 1. Let Cv denote a completed algebraic closure

of Fv. Tate showed that one can give the set C×v qZ the structure of an el-

liptic curve over Fv. We sketch the motivating idea here. If E is an elliptic

curve over C, then E ' CZ ⊕ Zτ for some τ ∈ H. The exponential map

exp : C → C×, z 7→ e2πiz, sends 1 to 1 and τ to some element q of C with

|q| < 1, and identifies Tate(q) := CqZ with E(C). In this setting, the group

operation is multiplication modulo qZ. In particular, the n-torsion in Tate(q)

is generated by the classes of (1) a primitive nth root of unity in C, and (2) an

nth root of q in C. After some minor modifications the same idea can be made

to work over Fv. We record here the important facts about Tate curves which

will be needed later.

Fact 2.5.1. (See for example [54, Chapter 5].)

(1) j(Tate(q)) = 1q+ (power series in q). In particular, v(j(Tate(q))) = −v(q).

(2) Tate(q) has multiplicative reduction at v.

(3) Conversely, let E be a curve over F with multiplicative reduction at v. Then

E ×F Fv ' Tate(q) over Fv, for some q. This isomorphism can be taken to be

over Fv if the reduction is split.

(4) The map E ' Tate(q) is Galois-equivariant. That is, one may study the

action of G = Gal(Fv/Fv) on E by studying the action of G on F×v .

18

3 Modular Curves of D-Elliptic Sheaves

3.1 Rigid Analytic Upper Half Plane

We assume some familiarity with rigid analysis; our basic references are [20],

[5], and [38].

Definition 3.1.1. The Drinfeld upper half plane is (set-theoretically) given by

Ω = P1(C∞) \ P1(F∞) = C∞ \ F∞.

Since P1(F∞) is compact, Ω has the structure of a rigid analytic space. We

now describe a pure covering of Ω. First, let D0 denote the subset of z ∈ C∞

which satisfy

|π| ≤ |z| ≤ 1, and

|z − c| ≥ 1, |z − cπ| ≥ |π| for all c ∈ F×q → F×∞

and let D(n,x) = πn ·D0 + x. Then D(n,x) is clearly an affinoid space over F∞,

and it is easy to check that D(n,x) does not depend on the choice of uniformizer

π. The ring of holomorphic functions on D(n,x) is given by

A(n,x) = F∞

⟨π−n(z + x), π

n+1

z+x ,1

πn(z+x)c ,z+x

πn+1−c(z+x) |c ∈ F×q

⟩

where the brackets 〈· · · 〉 denote the power series in · · · whose coefficients tend

to zero. The nonarchimedean norm | · | on F can be extended to a norm on

A(n,x) via ||f || := maxz∈D(n,x)

|f(z)|. There is a way to rephrase this in terms of the

coefficients of the power series expansion of f(z); see for example Gerritzen-van

der Put.

One easily sees that the canonical reduction D(n,x) of D(n,x) is isomorphic to

the union of two projective lines M,M ′ over Fq meeting transversally at a Fq-

rational point, with all other rational points deleted:

19

D(n,x) = M ∪M ′ − (M(Fq) ∪M ′(Fq)) ∪ (M ∩M ′) (*)

Note that D(n1,x1) = D(n2,x2) if and only if n1 = n2 and |x1−x2| ≤ |π|n+1. Let

I = (n, x)|n ∈ Z, x ∈ F∞πn+1O∞, so that by the above we have Ω =⋃i∈I

Di.

Given i ∈ I, there are only finitely many i′ 6= i ∈ I such that Di ∩Di′ 6= ∅. In

this case, Di ∩Di′ is isomorphic to a ball with q holes, i.e. P1(C∞) with q + 1

holes. The reduction of Di ∩Di′ is then

(Di ∩Di′) = M −M(Fq)

where M is a projective line over Fq. This follows either by considering the

Tate algebra, or by considering the possible intersections Di ∩ Di′ and taking

(*) into account. We obtain a rigid analytic structure on Ω by gluing the Di

along their intersections. By definition the Di give us a pure covering of Ω. Let

R : Ω → Ω denote the corresponding analytic reduction. Then Ω is a scheme

over Fq which is locally of finite type. Each irreducible component M of Ω is

isomorphic to P1Fq, and meets exactly q+ 1 other components - one for each

rational point of M . We will work with the dual graph of Ω. That is, we form a

graph T whose vertices are the irreducible components M of Ω, with adjacency

relation M adjacent to M ′ if and only of M ∩M ′ 6= ∅. It is well-known that T

is a (q + 1)-regular tree.

Notation 3.1.2. Put M∗ = M − M(Fq) and for adjacent M,M ′, (M ′)∗ =

M ∪M ′ − (M(Fq) ∪M ′(Fq)) ∪ (M ∩M ′). Then there exist i, i′ ∈ I such that

R−1(M∗) = Di ∩Di′ and R−1((M ∪M ′)∗) = Di.

This tells us that the reduction map R induces a bijection between the edges

of T and the index set I. Finally, note that GL(2, F∞) acts on Ω by fractional

linear transformations. This action sends discs centered in F∞ with F∞-rational

radius to discs of the same type, and therefore induces an action on the covering

(Di)i∈I , hence on T .

20

3.2 The Bruhat-Tits Tree

Definition 3.2.1. : An O∞-lattice in F 2∞ is a rank-two O∞-submodule of F 2

∞.

Two O∞-lattices L1 and L2 are equivalent if there exists x ∈ F×∞ such that L1 =

xL2. Write [L] for the equivalence class of the lattice L. Two classes [L1], [L2]

are adjacent, or neighbors, if there exist representatives L′1, L′2 ∈ [L1], [L2] such

that L′1 ⊂ L′2 and L′2L′1 has length one as an O∞-module. This adjacency

relation is symmetric. We obtain a combinatorial graph T by taking the vertex

set X(T ) of T to be the set of lattice classes [L], and forming the edge set Y (T )

via the adjacency relation.

It turns out ([49]) that T is a (q+1)-regular tree. To get an idea of why this

graph is (q + 1)-regular, consider the O∞-lattice L = O∞ ⊕ O∞ ⊂ F∞ ⊕ F∞.

Write v1 = (1, 0) and v2 = (0, 1) for the standard O∞-basis vectors of L. Let π∞

be a uniformizer for O∞, and let ai ∈ O∞ \m∞, i = 1, ..., q∞−1, be representa-

tives of k×∞ = F×q . Then the neighbors of [L] in T are given by the classes of the

lattices Li =< πv1 + aiv2, πv2 > for i = 1, ..., q∞ − 1, Lq∞ =< v1, πv2 >, and

by L0 =< πv1, v2 >. One shows that these lattices are pairwise inequivalent;

similar considerations work for arbitrary lattices in F 2∞.

GL(2, F∞) acts via matrix multiplication on the right on the set of O∞-lattices

in K2∞, and this action preserves lattice equivalence. We define a left action via

γ ?L := Lγ−1. One easily checks that this defines a transitive action on the set

of lattice classes [L] which preserves adjacency. The stabilizer of the standard

vertex L = O∞ ⊕O∞ is the subgroup Z(F∞) ·GL(2,O∞), where Z(F∞) is the

center of GL(2, F∞). We thus obtain the description

GL(2, F∞)GL(2,O∞) · Z(F∞)−→X(T )

γ 7→ γ ? [L].

Similarly, GL(2, F∞) acts transitively on the set of (oriented) edges of T . Let

e = ([L], [L0]) be the standard edge of T , where L0 = πO∞⊕O∞. The stabilizer

21

in GL(2, F∞) of e is easily seen to be Z(F∞) · I, where

I =

a b

c d

∈ GL(2,O) : c ≡ 0 mod ∞

is the Iwahori subgroup. One thus obtains

GL(2, F∞)I · Z(F∞)−→Y (T ),

γ 7→ γ ? ([L], [L0]).

With these descriptions in hand, the natural map Y (T ) → X(T ) induced

by the inclusion I → GL(2,O) associates to an oriented edge e its terminus

t(e) ∈ X(T ).

It is useful to give explicit representatives in GL(2, F∞) for the vertices of T .

Lemma 3.2.2. : (1) Every vertex of T , viewed as a class of GL(2, F∞)I ·

Z(F∞), has a unique representative of the form

πn g

0 1

, where n ∈ Z and

g ∈ F∞πnO∞.

(2) Denote a vertex by its representative matrix. Then the vertices adjacent

to

πn g

0 1

are given by

πn−1 g

0 1

,

πn+1 g

0 1

, and

πn+1 g + aiπn

0 1

,

where ai runs through a system of representatives for k×∞ = F×q .

Thus, as we move along a path P (without backtracking) in T , we obtain a

sequence of matrices Mi =

πni gi

0 1

for which the upper-left entries either

(a) diverge, or (b) converge to 0.

In case (a), the only possibility is that our sequence eventually looks like

π−n 0

0 1

,

with n tending to infinity. In this case, we say that the limit point in P1(F∞)

of the path P is ∞ (This is the point at infinity of the projective line, not to be

22

confused with the valuation ∞ of F ).

In case (b), letting Mi =

πni gi

0 1

denote our sequence of matrices, it turns

out that the sequence gii converges in F∞ to some point g ∈ F∞ and we say

that the limit point in P1(F∞) of the path P is g.

Definition 3.2.3. A half -line in T is a sequence [Li] of vertices of T such

that [Li] and [Li+1] are adjacent for all i = 1, 2, ..., and such that no two [Li]

are equivalent. An end of T is an equivalence class of half-lines, where one says

that two half-lines are equivalent if they differ in a finite graph.

The above identifications then show that the ends of T are in canonical

bijection with the points of P1(K∞). If one wants to think of T as an analogue

of the upper half plane, then the ends of T are to be thought of as the rationals,

i.e. as the cusps. We conclude this section with a theorem of Goldman and

Iwahori. Note that T and T are abstractly isomorphic because both are (q+1)-

regular trees.

Theorem 3.2.4. (Goldman-Iwahori) [21, Section 3] There is a canonical GL(2, F∞)-

equivariant isomorphism of graphs b : T → T .

3.3 Proof of Goldman-Iwahori

The Goldman-Iwahori map is constructed by matching up the points on T and

T to similarity classes of norms on a 2-dimensional vector space. We follow [21].

Recall the conventions set in 3.1.2.

Definition 3.3.1. : A non-archimedean norm on a F∞-vector space V is a map

ν : V → R such that

(i) ν(v) ≥ 0 with equality if and only if v = 0

(ii) ν(cv) = |c|∞ν(v)

(iii) ν(v + w) ≤ supν(v), ν(w)

23

for all v, w ∈ V and all constants c ∈ F∞. Two norms are similar if one is

a real multiple of the other. GL(V ) acts on V on the right, which yields a left

action on the set of similarity classes of norms ν, given by (γν)(v) = ν(vγ).

We are interested in the case V = F 2∞.

Step 1: Vertices of T ←→ similarity classes of norms.

Let [L] ∈ X(T ). Here we view all lattices L as living inside V . We associate to

[L] the norm

νL(v) := inf|c|∞ : c ∈ F∞, v ∈ cL.

For example, if L = O∞ ⊕ O∞ is the standard lattice, and v = (v1, v2) ∈ V is

some vector, then νL(v) is just max|v1|∞, |v2|∞. Thus the unit ball in V with

respect to νL is L, and this is true for any νL′ constructed in this way from a

lattice. It is immediate to check that if two lattices are similar then so are their

associated nonarchimedean norms.

Step 2: Points on an edge of T ←→ similarity classes of norms.

Let P ∈ T (R) belong to the edge ([L], [L′]), with πL′ ⊂ L ⊂ L′; viewing

this edge as a unit interval, say P = (1− t)[L] + t[L′], some 0 ≤ t ≤ 1.

Define b(P ) to be the class of the norm νP given by

νP (v) := supνL(v), qt∞νL′(v).

All norms constructed in this way are pairwise inequivalent, and a straightfor-

ward verification shows that (on vertices of T ) this correspondence isGL(2, F∞)-

equivariant.

24

Step 3: Points of Ω ←→ norms on V .

The points on T have now been identified with similarity classes of norms on

V = F 2∞. Let λ : Ω→ T (R) be the map which associates to z ∈ Ω the similarity

class of the norm [νz], where

νz((v1, v2)) := |v1z + v2|

This is a norm; here it is important that z ∈ Ω, since if z were an element of

F∞ then νz would vanish at nonzero vectors of V . It is not hard to verify (i)

λ(Ω) = T (Q), and (ii) λ is GL(2, F∞)-equivariant.

Step 4: T ←→ T .

All we need to do is identify the vertex sets of T and T in such a way that

adjacent vertices correspond to adjacent vertices. Let M ∈ X(T ) be an ir-

reducible component of Ω. Then R−1(M∗) = Ui ∩ Ui′ for some indices i, i′.

There exists a unique [L] ∈ X(T ) such that λ−1([L]) = R−1(M∗); this is the

Goldman-Iwahori map. It is GL(2, F∞)-equivariant because the intermediate

steps are.

3.4 An Orientation Convention, and the r Map

For i = (n, z) the affinoid Ui has reduction Ui = (M ∪M ′)∗. Let e = (M,M ′) =

([L′], [L]) ∈ Y (T ) be an oriented edge of T , with e(R) and e0(R) = e(R) −

M,M ′ the corresponding closed and open edges of T (R). Also put U0i for the

subset of Ui defined by strict inequalities:

U0i = x ∈ C∞ : |π|n+1 < |x− z|i = |z − x| < |π|n

Let also Ci = x ∈ C∞ : |x − z|i = |z − x| = |π|n, so that Ci is the “outer

boundary circle” of Ui (i.e. the outer boundary circle minus those discs which

25

were deleted in the formation of Ui). Then via the Goldman-Iwahori maps, we

have the relationships

λ−1(e(R)) = Ui = R−1((M ∪M ′)∗) and

λ−1(e0(R)) = U0i = R−1(M ∩M ′)

Depending on the orientation of the edge e = ([L′], [L]), one of two things hap-

pens:

Case 1: λ−1([L]) = C(n,z) = R−1(M∗) and λ−1([L′]) = C(n+1,z) = R−1(M′∗)

Case 2: λ−1([L]) = C(n+1,z) = R−1(M∗) and λ−1([L′]) = C(n,z) = R−1(M′∗)

That is, starting with an oriented edge e of T , one has its preimage in Ω equal

to some Ui. As we traverse the edge from its origin to its terminus, we imagine

in Ω going from an inner circle of Ui to an outer circle in case 1, and from

an outer circle to an inner circle in case 2. We are now able to describe an

important map which takes invertible holomorphic functions on Ω and outputs

integer functions on T .

Let f ∈ OΩ(Ω)× be a rigid holomorphic function on Ω without zeros on Ω.

Let e ∈ Y (T ) be an oriented edge corresponding to M ∩M ′ in Ω, where M ′

corresponds to o(e) and M corresponds to t(e). Let i = (n, x) be an index such

that R−1((M ∪M ′)∗) = Ui. We wish to define a function r(f) : Y (T )→ Z; to

this end, it is sufficient to say what r(f)(e) is.

r(f)(e) := log (||f ||spR−1(M∗) ||f ||spR−1(M ′∗)

)

= log (||f ||spR−1(M∗))− log (||f ||spR−1(M ′∗)

) (**)

That is, r measures how much a function grows in absolute value going

from the “boundary circle” of Ui corresponding to o(e) to the “boundary circle”

26

corresponding to t(e). Recall that any rigid analytic function can be written

locally (i.e. on rational subsets of Ω) as a function with constant absolute

value uniformly on that affinoid times a polynomial times a function whose only

zeros/poles are at some prescribed point outside that rational subset. In the

situation that f ∈ OΩ(Ω)×, this reasoning yields that there exists a constant

c ∈ C×∞, an integer k ∈ Z, and g ∈ OΩ(Ω)× such that

f(z) = c(z − x)kg(z)

with |g(z)| constant equal to 1 on Ui. Then, since |g(z)| is constant on Ui and

since the c term appears in the numerator and denominator of (**), r(f)(e) is

given by

r(f)(e) = k in case 1

r(f)(e) = −k in case 2

Example 3.4.1. : The simple function f(z) = z is certainly holomorphic in

any sense of the word. We compute the function r(f) on Y (T ). Consider

first the affinoids Un,0 - i.e. the affinoids “centered at 0”. Then one can write

f(z) = 1 ·(z−0)1 ·1 on U . Consider the line in T going from 0 to∞, i.e. the line

corresponding to the sequence of affinoids centered at 0. Then r(f) associates

to oriented edges on this line the value 1 if they point towards infinity and the

value -1 if they point to 0.

Now consider an affinoid U(n,x) where x ∈ K∞/πnO∞ is not in the same class

as 0. Looking at the defining relations of Un, one sees that the inequalities

|z| ≤ |π|n and |z| ≥ |π|n both appear. So, |z|∞ is constant and equal to |π|n

on U(n,x). We write f(z) = πn · (z − x)0 · zπn , thus r(f) takes on the value 0

identically outside the line connecting 0 to ∞.

Example 3.4.2. : Let f(z) = z−ωz−ω′ , where ω, ω′ ∈ Ω, and where neither of ω, ω′

lies in C0 or C1. We compute for later use the quantity log(|| z−ωz−ω′ ||spC0|| z−ωz−ω′ ||

spC1

).

27

We introduce some notation. Let e0 denote the standard edge of T . Recall

that U0 = C0 ∪ U00 ∪ C1, with λ(C1) = o(e0) and λ(C0) = t(e0). Here the

orientation of e0 is understood to fall into “Case 1” of this section. The graph

T −e0, e0 = T + tT − breaks into two connected components: one containing

the end corresponding to 0, the other containing the end corresponding to ∞.

Say, o(e0) ∈ T − and t(e0) ∈ T +. Then the preimages in Ω are given by

Ω+ = λ−1(T +(R)− t(e0)) = z ∈ Ω : |z| > 1 or |z|i < |z| = 1

Ω− = λ−1(T −(R)− o(e0)) = z ∈ Ω : |z| ≤ |π| and |z|i < |π|

with “closures”

Ω+ = λ−1(T +(R)) = Ω+ ∪ C0 = z ∈ Ω : |z| > 1 or |z|i ≤ |z| = 1

Ω− = λ−1(T −(R)) = Ω− ∪ C1 = z ∈ Ω : |z| ≤ |π| and |z|i < |π|, or |z| =

|z|i = |π|

Then the following useful observation is immediate:

|z − ω| = |ω| if z ∈ U0, ω ∈ Ω+ or z ∈ U00 , ω ∈ Ω+

|z − ω| = |z| if z ∈ U0, ω ∈ Ω− or z ∈ U00 , ω ∈ Ω−

Thus, we have that

log(|| z−ωz−ω′ ||spC0|| z−ωz−ω′ ||

spC1

) = 1,−1, 0

in the cases (i) ω ∈ Ω−, ω′ ∈ Ω+, (ii) ω ∈ Ω+, ω′ ∈ Ω−, (iii) ω, ω′ both belong

to either Ω+ or both belong to Ω−, respectively.

Observation 3.4.3. : Let f ∈ OΩ(U0)×. Then

||f(z)||spC0||f(z)||spC1

= limz∈U0

0 ,|z|→1||f(z)|| lim

z∈U00 ,|z|→|π|

||f(z)||.

28

3.5 ΓΩ as a Mumford Curve

Fix a quaternion algebra D over F which is split at ∞, R a maximal A-order,

Γ0 = R× the group of units in R, and Γ := Γ0F×q the group of units in

R modulo the constants. We think of Γ0 as a subgroup of D× and Γ as a

subgroup of PGL(2, F∞). We will sometimes abusively write Γ for Γ0, but

only in situations where Γ is acting on some set with its center acting trivially.

When D is the trivial quaternion algebra, Γ is conjugate to PGL(2, A). When

D is a division algebra, Γ is still identified with a subgroup of PGL(2,K∞)

which acts discontinuously on Ω. A point z ∈ C∞ is called a limit point of

Γ is there exists some w ∈ C∞ such that Γ · w accumulates at z. By strong

approximation, we know that the double coset space D×(F )D×(A)Z(F∞)

is compact. Therefore its image under

D×(F )D×(A)Z(F∞) −→ ΓGL(2, F∞)Z(F∞)

D×(F ) · (αv)v · Z(F∞) 7→ Γ · α∞ · Z(F∞)

is compact. Since GL(2,O∞) is open in GL(2, F∞), we see that

ΓGL(2, F∞)Z(F∞)GL(2,O∞)

is a finite set. But this set is the vertex set X(ΓT ), so we see that ΓT is

a finite graph. It follows that the set of limit points in C∞ of Γ is equal to

P1(F∞). This implies that Ω is equal to Ω(Γ) in the sense of [12]. Thus one has

the

Theorem 3.5.1. Let Γ ⊂ PGL(2, F∞) be a group coming from a divison quater-

nion algebra D over F . Then ΓΩ is a compact rigid analytic curve. ΓΩ is

algebraizable; that is, there exists an algebraic curve XΓ defined over a finite

extension of F∞ whose analytification is isomorphic to ΓΩ as rigid spaces.

From now on we denote the curve ΓΩ by XΓ. As will follow from the

results of section 3.6, XΓ is defined over F∞. As a rigid space, XΓ admits a

29

pure covering U with the following properties: (1) The spectral norm on O(U)

has values in |F∞|∞ for all U ∈ U , and (2) the reduction r : XΓ → XΓ induced

by the covering U is a finite union of projective lines over Fq which intersect

transversally in Fq-rational points. These conditions imply that XΓ is a totally

split curve over F∞. This implies, in turn, that there exists a Schottky group

Λ ⊆ PGL(2, F∞) such that XΓ = Ω(Λ)Λ, where Λ may be different from Γ.

Here Ω(Λ) is defined in [12]. Such curves are known as Mumford curves, and

have the following important property.

Theorem 3.5.2. ([20, Chapter 6]), [3, Section 1] Let X be a Mumford curve

defined over F∞. Then the Jacobian variety of X, Jac(X), admits an analytic

uniformization. That is, there exists a torus T ' (Gm)g and a full lattice L ⊂ T

admitting a positive-definite Z-valued inner product, such that as rigid analytic

group varieties one has Jac(X) ' TL.

We recall the basic skeleton of the proof here. For a smooth, projective,

geometrically irreducible curve X over F∞ of genus g ≥ 1, we say that a model

X of X over O∞ is semi-stable if (i) X is flat and proper, and (ii) the closed

fibre Xk is reduced and has only double points as its singularities. We say that

X has degenerate reduction if X admits a semi-stable model X over O∞ such

that the normalizations of all irreducible components of Xk are isomorphic to

P1k. We say that X has split degenerate reduction if, further, all double points of

Xk are k-rational with two k-rational branches. The curves with split degener-

ate reduction are precisely the Mumford curves; our curve ΓΩ can be directly

seen to have split degenerate reduction by looking at the intersection graph of

the pure reduction of Ω.

Now let A be an abelian variety, and let A denote its Neron model. Let Ak

denote the closed fibre of A, and A0k the connected component of the identity

of A. Then we have an exact sequence

30

0 −→ A0k −→ Ak −→ ΦA −→ 0

where ΦA is the component group of Ak. It is a fact, then, that A0k is an ex-

tension of an abelian variety B by a connected affine group T × U , where T is

a torus and U is a unipotent algebraic group. One says, then, that A has

good reduction if U = T = 1

semi-abelian reduction if U = 1

toric reduction if U = B = 1

split toric reduction if U = B = 1 and T ' Gdm(k) is a split torus.

Theorem 3.5.2 is then a consequence of the following two facts.

Fact 3.5.3. ([3, Section 1]) An abelian variety A admits a uniformization of

the form A = T/L if and only if A has split toric reduction.

Fact 3.5.4. ([10, Theorem 2.4]) X has split degenerate reduction if and only if

Jac(X) has split toric reduction.

It is an interesting problem to describe XΓ explicitly as a Mumford curve.

It turns out that as long as the conductor of D is divisible by at least one place

of even degree, Γ is already a Schottky group. In general, however, Γ may have

torsion, hence will not be a Schottky group.

3.6 Algebraic Theory

In this section we review the moduli interpretation of the rigid-analytic curve

ΓΩ = XΓ. We first briefly recall the more well-known characteristic zero

case. Let D be a division quaternion algebra over Q which is split at the

archimedean place. Fix a maximal order OD in D as well as an isomorphism

D ⊗Q R ' M(2,R). Let Γ denote the image of O×D in PGL(2,R) and H the

complex upper half-plane. Γ is known to act discontinuously on H with compact

quotient. Hence by GAGA, XΓ is isomorphic to the analytic space associated

to an algebraic curve, also denoted XΓ, defined over Q. It turns out that XΓ

31

parametrizes abelian surfaces with multiplication by OD. That is, the C-points

of XΓ correspond to abelian varieties A over C of dimension 2 whose endomor-

phism rings admit an injection OD → End(A).

We now return to the characteristic p situation. As further motivation, we

outline the relevant concepts from the theory of Drinfel’d modules. Let F , A,

∞, etc. be as usual.

Definition 3.6.1. The ring of twisted polynomials C∞τ is the ring of formal

sumsN∑i=0

aiτi, with multiplication rule given by aτ i · bτ j = abq

i

τ i+j .

Here, τ should be thought of as the qth-power Frobenius endomorphism of

C∞; thus C∞τ acts as additive endomorphisms of C∞. In other words, one

associates to a twisted polynomial∑aiτ

i the Fq-linear endomorphism of C∞

x 7→ f(x) = a0x+ a1xq + · · ·+ adx

qd .

Multiplication in C∞τ then corresponds to composition of endomorphisms.

Definition 3.6.2. A Drinfel’d A-module of rank d is an Fq-algebra homomor-

phism

φ : A −→ C∞τ

such that φ(a) =d·deg(a)∑i=0

ciτi, with c0 = a and cd·deg(a) 6= 0 for all a ∈ A. With

this definition, it is not clear that nontrivial Drinfel’d modules even exist.

The analytic construction of Drinfel’d modules proceeds as follows. An A-lattice

Λ ⊂ C∞ of rank d is a discrete, finitely generated, projective A-submodule of

C∞ of projective rank d. Given such an A-lattice Λ, we define the “exponential

function”

expΛ(z) := z∏

0 6=λ∈Λ

(1− z

λ

), z ∈ C∞.

32

It turns out that expΛ(z) = z +∑i≥1

bizqi . That is, expΛ(z) is Fq-linear. It is a

fact that nonconstant entire functions of one variable are always surjective in

the p-adic setting. These observations, together with a small amount of analysis

which we omit, imply that for α ∈ A one has a functional equation

expΛ(az) = φA,a(expΛ(z))

where φΛ,a(z) = a · z +d·deg(a)∑i=1

cizqi . In this manner one obtains a Drinfel’d

module for each A-lattice of rank d in C∞. One may show, conversely, that

given a Drinfel’d module φ of rank d, there exists a lattice Λφ which gives rise

to φ in the above manner. In addition, one has the notion of a morphism of

Drinfel’d modules; isomorphic Drinfel’d modules give rise to homothetic lattices,

and homomorphisms of Drinfel’d modules correspond to inclusions of lattices.

We remark in passing that the collection of (isomorphism classes of) rank-two

Drinfeld modules is parametrized by the curve GL(2, A)Ω, where Ω is the

usual Drinfel’d upper half plane.

We now return to our curve XΓ. In this case, it was shown in [28] that XΓ is

the moduli space classifying (isomorphism classes of) certain algebro-geometric

objects called D-elliptic sheaves. As the definition and properties of D-elliptic

sheaves would take us too far afield for our purposes, we will instead focus our

attention on an equivalent category. Fix a 2-dimensional vector space V over

C∞ and an isomorphism D ⊗F C∞ 'M(2, C∞) ' End(V ).

Definition 3.6.3. A lattice Λ in V is a discrete and projective A-submodule of

V . By discrete, we actually mean that any ball in V only contains finitely many

points of Λ. Λ is a full lattice if it contains a C∞-basis for V . The rank of Λ is

its rank as an A-module. A morphism from Λ1 to Λ2 is an element α ∈ GL(V )

such that αΛ1 ⊆ Λ2.

In contrast to the characteristic zero case, V admits lattices of arbitrarily

large rank. Call Λ an OD-lattice if OD · Λ ⊆ Λ. It is easy to show that OD-

33

lattices are always full. In [55] it is shown (Cor 3.6, Prop 2.13) that the category

of D-elliptic sheaves is anti-equivalent to the category of OD-lattices Λ in V.

This is done in roughly three steps. First, D-elliptic sheaves are related to An-

derson t-motives (see [1] for the definition). Second, the category of t-motives is

anti-equivalent to the category of t-modules ([1] Thm 1). A certain subclass of

the t-modules admits uniformization by lattices in a similar fashion to the uni-

formization of Drinfel’d modules of generic characteristic. Finally, it is shown

([55] Props 3.3 and 3.5) that the t-modules arising from D-elliptic sheaves pre-

cisely correspond to the isomorphism classes of OD-lattices in V .

It remains to be shown that our Shimura curve classifies such lattices. Re-

call that we have fixed a two-dimensional vector space V over C∞ as well as

an isomorphism D ⊗ C∞ ' End(V ). Let Λ be an OD-lattice, and let FΛ de-

note its F -span. Then FΛ is a free module of rank 1 over D in V , and the

choice of generator marks a point on P1(C∞) and identifies FΛ with D. Indeed,

let a, b ∈ Λ. Then OD · a ⊆ Λ is of full rank. Therefore there exists some

f ∈ F such that f · b ∈ OD · a, which implies that D · a = FΛ. The marked

point can not lie in P1(F∞), since if it did then Λ would not be discrete. This

is because F∞ splits D, and D is dense in M(2, F∞). Therefore OD is dense

in some conjugate of M(2,O∞), which then implies that ODΛ accumulates at 0.

Thus each point z ∈ Ω determines a lattice Λ ⊂ C⊕2∞ : to be precise, Λ := OD ·z.

Two points z1, z2 ∈ Ω give rise to the same lattice if and only if there exists

α ∈ O×D such that α · z1 = z2. We therefore see that the OD-lattices in V are

parametrized by XΓ = ΓΩ.

In fact, more is true. There is a notion of “level” for D-elliptic sheaves, and

in [28] it is shown that the functor mapping a C∞-algebra B to the set of

isomorphism classes of D-elliptic sheaves for B (here we are being a bit imprecise

34

with our terminology) is representable by a C∞-scheme of dimension 1. Taking

the quotient of this scheme by a finite group, one obtains a coarse moduli scheme

E``X,D,A classifying D-elliptic sheaves without level structure.

Theorem 3.6.4. ([55, Theorem 4.6]) There is a natural isomorphism of rigid-

analytic spaces

E``X,D,A(C∞)an ' D×Ω×D×(Afin)(OD ⊗A A)×.

35

4 Automorphic Forms

In this chapter we briefly recall parts of the theory of automorphic forms on

adele groups, which gives a way of comparing cusp forms and certain Galois

representations. The theorems mentioned here were developed to a large extent

by Tate, Jacquet, Langlands, Deligne, Drinfeld, Laumon, Rapoport, Stuhler,

and others. The main result of this chapter is a uniformization of a certain class

of nonisotrivial elliptic curves as quotients of Jacobians of curves arising from

quaternion algebras.

The proof proceeds in the following manner. Let E be an elliptic curve over F

with split multiplicative reduction at the place 1T =∞, multiplicative reduction

at an even number of finite places, and good reduction elsewhere. Then one has

that the conductor of E is n · ∞, where n is a squarefree ideal of Fq[T ] which

is the product of an even number of prime ideals. Let D be the (unique up to

isomorphism) quaternion algebra over F ramified exactly at n. Let XD be the

corresponding curve ΓΩ, and let JD denote its Jacobian.

First, to E we can associate an `-adic representation ρE : Gal(F /F )→ Aut(V`(E))

in the obvious manner. Next, due to Deligne, there is a cuspidal automorphic

representation πE of GL(2,A) associated to ρE in Langlands’ sense. This means

that one can associate local L− and ε− factors to both ρE and πE , and that

these factors agree everywhere. It is important that the conductor nE is equal

to the conductor nπE . Jacquet-Langlands associates to πE an automorphic form

on D× with the same Hecke eigenvalues, L-factors, and ε-factors. The theory

developed in Laumon-Rapoport-Stuhler then implies that ρE appears in the

semisimplification of V`(JD). Finally, a theorem of Zarhin in the characteristic

≥ 3 case and Mori in the characteristic 2 case says that there is a nontrivial

map JD → E. We will make this map explicit in chapter 5.

36

4.1 Automorphic Forms and Representations

Recall that a quasicharacter χ of a group H is a homomorphism χ : H → C×.

A quasicharacter χ is called a character (or unitary character) if |χ(γ)| = 1 for

all γ ∈ H. If H is a topological group then we require all quasicharacters to

be continuous. This condition places a significant constraint on the possible

characters of a nonarchimedean field.

Indeed, let K be a complete nonarchimedean local field. Then K and GLn(K)

both have a basis of neighborhoods of the identity consisting of compact open

subgroups. It is well-known that sufficiently small neighborhoods of 1 ∈ C× do

not contain nontrivial subgroups, from which it follows that any quasicharacter

of K or K× must contain an open subgroup in its kernel. Let O denote the ring

of integers of K and p its maximal ideal. For a nontrivial additive quasicharac-

ter ψ of K there is a unique integer m such that ψ is trivial on pm but not on

pm−1. Then we call n(ψ) := pm the conductor of ψ.

Similarly, if χ is a multiplicative quasicharacter of K× which is nontrivial on

O, then there is a largest ideal pm such that χ is trivial on 1 + pm. We call

n(χ) = pn the conductor of χ. If χ is trivial on O×, then we define the conduc-

tor of χ to be n(χ) = O. Note that in this case χ depends only on χ(ρ) where

ρ is a uniformizer for p. We say that an additive or multiplicative character is

unramified if its conductor is O.

One important class of characters is the Hecke grossencharacters of F , i.e. the

continuous characters of A×F×. Any such character χ can be uniquely de-

composed as χ = χ1 · | · |λ where χ1 is a finite order character of A×F× (which

means that for some positive integer m, χm1 is the trivial character), the norm

| · | is the idelic norm |x| =∏v|xv|v, and λ is a purely imaginary number. Any

37

quasicharacter χ uniquely decomposes as χ = χ1 · | · |λ where λ is an arbitrary

complex number.

Note that via the inclusion F×v → A×, a 7→ (· · · , 1, a, 1, · · · ), and by continuity

of quasicharacters A× → C×, we have that any quasicharacter χ : A× → C×

breaks into local components, i.e. χ =∏vχv where χv : F×v → C× and where

almost all χv are unramified (recall that the kernel of χ must contain an open

neighborhood of 1). Define the conductor of χ to be the product of the local

conductors: n(χ) :=∏vn(χv). This product makes sense because almost all

factors are 1. One can do the same for an additive character. That is, for

ψ : A→ C×, one is able to write ψ =∏vψv and one defines the conductor of ψ

to be the product of the local conductors.

We are now in a position to define automorphic forms. We note here that

the definition of automorphic form is simplified in our context because there are

no archimedean places. Let ω be a Hecke grossencharacter of F . We will need

notions of automorphic forms both for matrix groups GL(2) and for quater-

nion groups D×. We therefore formulate the definition of automorphic forms

simultaneously for both contexts.

Definition 4.1.1. : Let ω be a Hecke character of F , and let G = GL(2) or D×,

where D is a division quaternion algebra over F . Denote by A(G(F )G(A), ω)

the space of C-valued functions Φ on G(A) that satisfy the following conditions:

(i) Φ(z · g) = ω(z)Φ(g) ∀z ∈ A,∀g ∈ G(A)

(ii) Φ is right-invariant under some open subgroup K′ of K; equivalently Φ is

right K-finite.

Condition (ii) says that Φ will be constant on the cosets gK′, hence will

be locally constant. Let G(A) act on A(G(F )G(A), ω) by right translations:

(ρ(g) · Φ)(x) := Φ(xg). Thus ρ(g) is a representation of G(A) on the infinite-

38

dimensional vector space V = A(G(F )G(A), ω). Call this space the space of

automorphic forms on G with central character ω.

When G = GL(2), it is trivial to check that the space of functions satisfy-

ing the additional condition

(iii)∫

FAΦ(

1 x

0 1

g)dx = 0 for all g ∈ G(A)

is invariant under the representation ρ. (Here, the measure on FA is any Haar

measure). The invariant subspace of such functions is denotedA0(G(F )G(A), ω)

and is called the space of cusp forms for G with central character ω. Note that

since FA is compact and the integrand is locally constant, this integral is

actually a finite sum.

An automorphic representation is any irreducible representation of G(A) which

can be realized as a subquotient of A(G(F )G(A), ω). For G = GL(2), a cus-

pidal representation is any irreducible representation of G(A) which arises as a

subquotient of A0(G(F )G(A), ω). For G = D×, by definition every automor-

phic representation is cuspidal.

4.2 Local Theory

Before going further we recall the concept of admissible representation. Let

F = Fv denote a local field, and let π be a representation of G = G(F ) on

a complex vector space V . Let K denote GL(2,Ov) for G = GL(2), and the

unique maximal order in G(F ) for G = D×. Then π is admissible if (i) the sta-

bilizer in G of each v ∈ V is open, and (ii) the subspace of V fixed by any open

subgroup of K is finite dimensional. We now wish to describe the admissible

representations when G = GL(2).

39

Let µ1, µ2 be any two quasicharacters of F× and let H (µ1, µ2) denote the

space of all locally constant compactly supported functions Φ on G such that

Φ

a ∗

0 b

g

= µ1(a)µ2(b)|a/b| 12 Φ(g)

for all a, b ∈ F×. Then G acts on H(µ1, µ2) via right translation and the re-

sulting representation of G is called a principal series representation when it

is irreducible. This representation is denoted by π(µ1, µ2).

If the above representation is not irreducible, then it must be the case that

µ1µ−12 = |x| or |x|−1. In case µ1µ

−12 (x) = |x|, the representation contains a one-

dimensional invariant subspace and the representation induced on the quotient

is irreducible. If µ1µ−12 (x) = |x|−1 then there is an irreducible invariant sub-

space of H(µ1, µ2) of codimension one. In either case, the resulting irreducible

representation is denoted by π(µ1, µ2) and is called a special representation.

Special representations and principal series representations are admissible. Any

irreducible admissible representation of G which is not principal series or special

is called supercuspidal. Although we will not work with supercuspidal represen-

tations, we still mention them in order to simplify the statements of theorems

4.6.1 and 4.6.2.

Definition 4.2.1. An irreducible representation π of G is called class 1, or

spherical, if its restriction to K contains the identity representation at least

once. That is to say, there is at least one v ∈ V fixed by all of K.

It turns out that the infinite-dimensional irreducible admissible representa-

tions π of G of class 1 are precisely the principal series representations π(µ1, µ2),

where µ1 and µ2 are unramified characters of F×. In this case, there is actually

a unique (up to scalar multiple) K-invariant vector v of π.

40

Definition/Theorem 4.2.2. : Let π be any irreducible admissible infinite-

dimensional representation of GL(2) with central character ψ. Then there is a

largest ideal c(π) of OF such that the space of vectors v such that

π

a b

c d

v = ψ(a)v for all

a b

c d

∈ Γ0(c(π))

=

a b

c d

∈ GL(2,O) : c ∼= 0 mod pc(π)

=: Kc(π)

is nonempty. In fact, this space has dimension one. We call c(π) the conductor

of π. We record for later the important

Fact 4.2.3. If π = π(µ1, µ2) is a principal series representation, then the

conductor of π is equal to the product of the conductors of µ1 and µ2. If

π = π(µ1, µ2) is a special representation, then the conductor of π is pO.

4.3 Global Theory

Now let F be a global function field, and let π be an irreducible unitary represen-

tation of G(A) on a Hilbert space H. Let σ denote any irreducible representation

of K =∏vKv and H(σ) the space of vectors in H which transform under K

according to σ. Then π is called admissible if H(σ) is finite-dimensional for

every σ.

It is well-known that every admissible irreducible unitary representation π of

G(A) is factorizable. That is, given an irreducible admissible representation π

of G(A), there exist irreducible admissible unitary representations πv of G(Fv)

such that π =⊗vπv. This last object is understood to be a representation whose

space is a certain “restricted tensor product” of the Hv with respect to Kv-fixed

vectors; see [18] chapter 4 for details. One of course defines the conductor of π

to be the product of the local conductors.

41

We now return to our cuspidal representations. It is known that these are

admissible, hence factorizable, i.e. of the form π = ⊗vπv with almost all

πv unramified. For a unitary character χ of A×F× and a cuspidal repre-

sentation (π, V ), define χ ⊗ π to be the representation on the subspace Vχ ⊆

A0(G(F )G(A), χ2ω) of all functions of the form Φχ(g) = χ(det(g))Φ(g), where

Φ ∈ V . If we decompose χ = ⊗χv, then χ⊗ π ' ⊗(χv ⊗ πv). This is the repre-

sentation obtained by twisting π by χ.

The notion of contragradient representation is important for the functional equa-

tion of the L-series associated to a representation. However, since we will not

be working heavily with the contragradient, we content ourselves to simply say

that locally, the contragradient representation of π(µ1, µ2) is π(µ−11 , µ−1

2 ), and

that the contragradient of a global (admissible irreducible unitary) represen-

tation is the tensor product of the contragradients of the local pieces. If π is

an irreducible constituent of A0(G(F )G(A), ω), then its contragradient is a

constituent of A0(G(F )G(A), ω−1), and one can show that π ' ω−1 ⊗ π.

4.4 Galois Representations

Let K be a local field, and let O, p, k, q and p denote the usual quantities. Fix

a separable closure Ksep of K with residue field k, so that K is an algebraic

closure of k. We have that Gal(k/k) ' Z because Gal(k/k) is topologically

cyclic, generated by the automorphism φ : x 7→ xq. Denote by Wk the subgroup

of Gal(k/k) generated by φ. Normalize the isomorphism Wk ' Z via φ 7→ −1.

The absolute Weil group WK is the dense subgroup of Gal(Ksep/K) consist-

ing of those elements whose image in Gal(k/k) is a power of φ. The inertia

subgroup I of Gal(Ksep/K) is the subgroup I ⊆ Gal(Ksep/K) whose image in

42

Gal(k/k) is trivial. Give I the usual profinite topology and topologize WK so

that I is open. Then we have a sequence of topological groups

1 −→ I −→WK −→Wk −→ 1

Following Deligne, any element Φ of WK or Gal(Ksep/K) whose image in

Gal(k/k) is φ−1 is called a geometric Frobenius element.

Recall that local class field theory provides us with an isomorphism of topo-

logical groups W abK ' K×. More precisely, we have a commutative diagram

with surjective vertical arrows

P −−−−→ I −−−−→ WK −−−−→ Wky y yrec y1 + p −−−−→ O× −−−−→ K×

ord−−−−→ Z

where P denotes the wild inertia subgroup of I. A representation of WK is

unramified (respectively tamely ramified) if it is trivial on I (respectively P ).

We denote by Z`(1)(Ksep) the `-adic Tate module of K×, i.e. lim←

µ`n . Then

clearly for ` 6= p the action of Gal(Ksep/K) on Z`(1)(Ksep) is unramified, since

the `n-torsion in (Ksep)× maps injectively into k× by Hensel’s lemma. Thus

for σ ∈ WK with image πn · u in K× with u ∈ O×, we see that σ acts on

Z`(1)(Ksep) as Φn where Φ is any geometric Frobenius element. More clearly,

consider the quasicharacter ω1(x) = |x| on K× (which is trivial on O×). Then

ω1(rec(Φ)) = q−1, and in general σ acts on Z`(1)(Ksep) as a 7→ aω1(rec(σ)). For

σ ∈WK we henceforth write ω1(σ) for ω1(rec(σ)).

Now let F be a global function field, and define the Weil group of F to be the

preimage WF of Wk in Gal(F sep/F ) under Gal(F sep/F ) → Gal(k/k), where k

is the field of constants of F. Then we have an exact sequence of topological

groups

43

1 −→ Gal0F −→WF −→Wk −→ 1

where Gal0F = Gal(F sep/kF ) is the set of elements of Gal(F sep/F ) which act

trivially on k. Again we topologize WF so that Wk is discrete and Gal0F is open.

For each place v of F the natural map WFv →WF is proper and injective, and

we obtain injections Wkv →Wk via φv 7→ φ[kv:k]

Global class field theory provides us with an isomorphism of topological groups

W abF ' A×/F× and the relationship between local and global class field theory

is given by the commutative diagram

W abFv−−−−→ W ab

Fy yF×v −−−−→ A×F /F×

We now discuss representations of WF and WK . Recall that for any n,

sufficiently small neighborhoods of 1 in GL(n,C) will not contain nontrivial

subgroups, so by the profinite nature of I (and G0F ), the image of inertia (and

G0F ) under any finite-dimensional complex representation will be finite. The

`-adic representations do not have this problem.

Definition 4.4.1. : A Weil-Deligne representation σ′ is a pair (σ,N) where

σ : WK → GL(n,C) is a complex n-dimensional representation and N is a

nilpotent matrix in M(n,C) such that σ(g)Nσ(g)−1 = ω1(g)N for all g ∈ WK .

σ′ is called admissible, or Φ-semisimple, if σ is semisimple. Equivalently, σ′ is

admissible if σ(Φ) is semisimple for some geometric Frobenius Φ.

We will be interested in 2-dimensional Weil-Deligne representations. The

following examples are of fundamental importance.

Example 4.4.2. Let χ be a character of K×. Then by the local class field

theory isomorphism WK ' K×, we see that χ can be regarded as a character

of WK . Then the representation

44

σ(g) =

χ(g)|g| 12 0

0 χ(g)|g|−12

, N = 0,

is a Weil-Deligne representation.

Example 4.4.3. Let χ1 and χ2 be two characters of K×, again regarded as

characters of WK . Then the representation

σ(g) =

χ1(g) 0

0 χ2(g)

, N = 0,

is a Weil-Deligne representation.

Example 4.4.4. (The special representation) Let χ be a character of K×

regarded as a character of WK . The special representation sp(2) of WK is the

representation

sp(2)(g) =

1 0

0 ω1(g)

, N =

0 0

1 0

,

so that sp(2)(Φ) =

1 0

0 q−1

. It is trivial to check that this representation

satisfies the Weil-Deligne property, and this representation is clearly semisimple,

hence admissible. The twist χ sp(2) is defined by

sp(2)(g) =

χ(g) 0

0 χ(g)ω1(g)

, N =

0 0

1 0

,

We now state the local Langlands correspondence for n = 2.

Theorem 4.4.5. There is a one-to-one correspondence σ′ = (σ,N) ↔ π(σ′)

between the set of equivalence classes of all two-dimensional representations

σ′ = (σ,N) of WK and the set of equivalence classes of all irreducible admissible

representations of GL(2,K), with L− and ε− factors, as well as conductors n,

preserved:

45

L(s, χ⊗ π(σ′)) = L(s, χ⊗ σ′)

L(s, χ⊗ π(σ′)) = L(s, χ⊗ σ′)

ε(χ⊗ π(σ′), ψ, s) = ε(χ⊗ σ′, ψ, s)

n(π(σ′)) = n(σ′)

This correspondence sends the representation from example 4.4.3 above to

the principal series representation π(χ1, χ2), and sends the special representa-

tion of example 4.4.4 above to the special representation π(χ, χ| · |). As men-

tioned before, we do not need to think about the supercuspidal representations.

We must also discuss `-adic representations before turning our attention to

elliptic curves. Let K be a nonarchimedean local field of residue characteristic

p, and let ` be a prime different from p. Let

σ` : Gal(Ksep/K) −→ GL(V`)

be a continuous homomorphism, where V` is a finite-dimensional Q`-vector

space. Such a representation is called an `-adic representation of Gal(Ksep/K).

It is well-known that such representations are actually defined over some finite

extension of Q`. To an `-adic representation one can associate L− and ε− fac-

tors, as well as a conductor.

There is a construction due to Grothendieck which associates a complex repre-

sentation σ′` = (σ,N) to an `-adic representation σ` and vice-versa, in such a

way that L−, ε−, and n are preserved. We will not recall the details; in the

sequel will refer to this as Grothendieck’s construction for brevity.

4.5 Representations Associated to Elliptic Curves

Let E be an elliptic curve over a local nonarchimedean field K and let ` 6=

p = char(k). Let T`(E) denote the `-adic Tate module of E; this is a free

46

Z`-module of rank 2 with a natural continuous action of Gal(Ksep/K). Let

V`(E) = T`(E)⊗Z` Q`, and consider the natural representation

σ′E,` : Gal(Ksep/K) −→ GL(V`(E)∨).

Let (σE,`, NE,`) = σ be the Weil-Deligne representation associated to σ′E,` by

Grothendieck’s construction. One can show that the isomorphism class of σ

does not depend on the choice of prime `. We briefly describe σ in terms of the

redution type of E.

First, if E has good reduction, then NE,` = 0 and σE,` is a Weil-Deligne rep-

resentation associated to a principal series representation. Now suppose E has

split multiplicative reduction. Then E is a Tate curve, so by the Tate uni-

formization E ' K×/tZ, one has that

σ′E,`(g) =

ω−11 (g) 0

∗ 1

and the image of inertia σ′E,`(I) is infinite. Tracing the details in Grothendieck’s

construction, one has that

σ′E,`(g) = σ′E,`(Φ) =

q 0

∗ 1

and that NE,` 6= 0. By work of Deligne, σ′E,` is isomorphic to some χ ⊗ sp(2)

where χ is a character of WK . Looking at the trace of this character, we see

that

χ(g)(1 + ω1(g)) = Tr(χ⊗ sp(2)(g)) = Tr(σE,`(g)) = ω−11 (g) + 1

so that χ = ω−11 . Hence σ′E,` ' ω

−11 ⊗ sp(2).

If E has nonsplit multiplicative reduction, then E acquires split multiplicative

reduction over a quadratic separable extensions ofK. Then σ′E,` ' χω−11 ⊗sp(2),

47

where χ is a quadratic character of K× which is nontrivial and unramified. Fi-

nally, we mention in passing that when E has additive (but not potentially

good) reduction, σ′E,` ' χω−11 ⊗ sp(2), where χ is now a ramified character of

K×.

Now let E be a nonisotrivial elliptic curve over F (nonisotrivial means that

jE /∈ Fq, equivalently E does not become a constant curve after some field

extension). Let ` 6= char(F ) and let ρ(E) be the `-adic representation

ρ(E) : Gal(F sep/F ) −→ GL(V`(E)∨)

Restricting ρ = ρ(E) to WFv →WF , we obtain `-adic representations ρv,`(E) of

local Weil groups. This is the same representation as that obtained by consid-

ering WFv → GL(V`(E/Fv)∨). The Weil-Deligne representation associated to

ρv,`(E) is a semisimple complex representation which up to isomorphism does

not depend on `. So, ρ(E) is a member of a family of compatible representations

in the sense of Deligne.

To each ρv(E) we can associate an irreducible admissible representation πv(E)

of GL(2, Fv), and since E has good reduction almost everywhere, almost all

πv(E) are principal series, hence unramified. Putting these local representations

together, we obtain an irreducible unitary representation π(E) =⊗vπv(E) of

GL(2,A). One has that L(π(E)⊗χ, s) = L(ρ(E)⊗χ, s) for any Hecke character

χ whose ramification locus is disjoint from the places where ρ(E), hence also

π(E), are ramified. A theorem of Grothendieck then implies that L(π(E)⊗χ, s)

is a polynomial in q−s (in particular, it is entire and bounded in vertical strips),

and satisfies a functional equation. By a theorem of Deligne, one has the lo-

cal decomposition of ε-factors ε(π(E) ⊗ χ, s) =∏vε(πv(E) ⊗ χv, ψv, s), where

ψ = ⊗ψv is a nontrivial global additive character of A/F .

48

Thus by the converse theorem of Jacquet-Langlands, we actually see that π(E)

occurs as a direct factor of A0(GL(2, F ) GL(2,A), ω) for some central char-

acter ω. It is important to remember here that the conductors are equal:

n(E) = n(π(E)).

4.6 Jacquet-Langlands Correspondence

We now briefly recall the Jacquet-Langlands correspondence, which relates cusp

forms for a quaternion algebra to certain cusp forms for GL(2). Again, the goal

here is not to give detailed proofs, but rather to simply indicate certain relations.

Let K be a local field, G = GL(2,K), and D the unique up to isomorphism

division quaternion algebra over K. Let q(x) = xxσ denote the norm form on

D, and let S(D) denote the Schwartz-Bruhat space of D, i.e. the space of locally

constant compactly supported functions Φ on D× = D×(K). Let ω denote a

nontrivial character of K×. Then by work of Weil, Shalika, and Tanaka, there

is a representation r(s) of SL(2,K) on S(D), a priori dependent on a choice of

additive character τ of K, characterized by the properties:

(i) r

1 u

0 1

Φ(x) = τ(uq(x))Φ(x)

(i) r

a 0

0 a−1

Φ(x) = ω(a)|a|12v Φ(ax)

(iii) r

0 1

−1 0

Φ(x) = −Φ(xσ)

One extends r(s) to a unitary representation r(g) of GL(2,K) which is in-

dependent of the choice of additive character τ . Finally, let π′ be an irreducible

finite-dimensional representation of D× on a C-vector space H. Then tensoring

r(s) with the trivial representation of SL(2,K) on H, we obtain a representa-

49

tion on S(D) ⊗C H, again denoted by r. Elements in S(D) ⊗ H are regarded

now as functions on D taking values in H. It turns out that the subspace of

functions in S(D) ⊗ H satisfying Φ(xh) = π′(h−1)Φ(x) for all h ∈ D× with

norm one is invariant for r(s). (Note the similarity of this construction with

the induced representations) Denote this subrepresentation by rπ′(s). Denote

by rπ′(g) also the representation of GL(2,K) induced from rπ′ . We have the

following result:

Theorem 4.6.1. Let d denote the dimension of π′, our original (finite-dimensional)

representation of D×, and r′π(g) the representation of GL(2,K) described above.

(1) rπ′(g) decomposes as a direct sum of d mutually equivalent irreducible rep-

resentations π(π′) of GL(2,K).

(2) Each π(π′) is supercuspidal if d > 1 and special if d=1.

(3) All supercuspidal and special representations of GL(2,K) are obtained in

this manner.

Of importance to us is the second statement in the theorem. Namely, to an

irreducible 1-dimensional representation (i.e. quasicharacter) of D× we can as-

sociate an infinite-dimensional irreducible (special) representation of GL(2,K).

We can specify this a bit further:

It is well known that every quasicharacter of D× is given by composition of

the norm map with a quasicharacter of K×, i.e. ψ : D× → C× factors as

ψ = χ Nrd. Then in part 2 of the above theorem, Nrd corresponds to Sp(2)

and χ Nrd corresponds to χ ⊗ Sp(2). We are now in a position to state the

Jacquet-Langlands correspondence. Let F again denote our function field and

A the adeles of F .

50

Theorem 4.6.2. There is a correspondence which associates to each irreducible

unitary representation π′ of D×(A) an irreducible unitary representation π of

GL(2,A) so that if π′ is a cusp form for D×(A) then π is a cusp form for

GL(2,A). Moreover, this correspondence π′ 7→ π = ⊗πv is one-to-one onto the

collection of cusp forms on GL(2,A) such that πv is special or supercuspidal for

all ramified places v of D. Moreover, π′ and π have the same Hecke eigenvalues

for all v.

The basic idea is as follows: For almost all places v of F , D⊗Fv is a matrix

algebra and π′v is already a representation of GL(2, Fv). For such v let πv = π′v.

For ramified v, π′v is necessarily finite-dimensional because D×v /F×v is compact.

Therefore we can construct an irreducible representation π(π′v) of GL(2, Fv).

Let πv = π(π′v) for such v. One may now form the representation π = ⊗πv and

hope that it is irreducible, unitary, and satisfies the properties in the theorem.

The proof is nontrivial and involves an application of the trace formula. We

simply mention here that clearly almost all πv are spherical, so π makes sense.

We refer the reader to [24] for a proof.

4.7 Quaternionic Uniformization of E

Let F be a global function field and E an elliptic curve over F with split multi-

plicative reduction at∞ and conductor n ·∞, where n is the squarefree product

of an even number of places of F . Let D be the unique up to isomorphism

quaternion algebra over F ramified at n. Let Γ ⊆ D be the unit group of a max-

imal order of D, XΓ the corresponding curve, and J = JD the Jacobian of XΓ.

To E we associate the `-adic representation ρE : Gal(F sep/F ) −→ Aut(V`(E)∨).

By the work from section 4.5, we have that ρE can be associated to a cusp

form πE on GL(2) with the same L, ε, n. Note now that since the conductor

of E is squarefree, E has multiplicative reduction at all places dividing n∞, so

51

that πE,v is actually a special representation of GL(2) at v|n∞. Now by the

Jacquet-Langlands correspondence, we can find a cusp form π′E for D× which is

1-dimensional at all ramified places v of D. Let K ⊆ D×(A∞) be the subgroup

generated by all local maximal compacts, i.e.

K =∏v∈RD×v ×

∏v∈|f |−R−∞

GL(2,Ov).

Then work of Laumon-Rapoport-Stuhler ([28, Theorems 14.9, 14.12]) gives us

an isomorphism of Gal(F sep/F )-modules

V`(JD)∨ ⊗Q` Q` = H1et(X ⊗F F sep, Q`) =

⊕Π∈AD,Π∞'sp∞

(Π∞)K ⊗ σ(Π) (*)

Our π′E has the property that π′E,v = ψv Nrd (for v ∈ R) for some unramified

characers ψv of F×v because E has multiplicative reduction at those places. So,

π′E appears in the RHS of (*). Therefore V`(E) is isomorphic to a quotient of

V`(J) as Gal(F sep/F )-modules. A theorem of Zarhin ([67, 68]) in characteristic

≥ 3 and Mori ([32]) in characteristic 2 tells us that

HomF (J,E)⊗Q` ' HomGal(F sep/F )(V`(J), V`(E))

Since we’ve shown that the right-hand-side is nonzero, we must have a non-

trivial (surjective) homomorphism J → E.

52

5 Analytic Uniformization of E

5.1 Harmonic Cochains for Γ

As we see from section 4.7, the automorphic representations we are interested in

are special at infinity. Thus, given a nonisotrivial elliptic curve E defined over

F with conductor n · ∞ and with split multiplicative reduction at ∞, if πE is

the automorphic form on D× corresponding to E, we have that πE is invariant

under the group

K :=∏v 6=∞

Kv × I∞.

Here I∞ is the Iwahori subgroup at∞, and Kv is GL(2,Ov) at split places and

the unit group in the unique maximal order of D ⊗F Fv for ramified places v.

That is,

πE : D×(F )D×(A)K · Z(F∞) −→ C.

Let Y (K) = D×(F )D×(A)K · Z(F∞). Let Γ0 := D×(F ) ∩ Kf , where

Kf :=∏v 6=∞

Kv. Then Γ0 is precisely Γ = O×, where O is the maximal order

of D×. By the strong approximation theorem, each adele a = (av)v ∈ D×(A)

can be written as a = γκg∞, where γ ∈ D×(F ), κ ∈ Kf , and g∞ ∈ GL(2, F∞).

Another application of the approximation theorem yields the following result.

Lemma 5.1.1. The map on Y (K) which to the double coset class of (av)v

associates the double class of g∞ in ΓGL(2, F∞)I∞ ·Z(F∞) is a well-defined

bijection.

It is well-known that GL(2, F∞)I∞ · Z(F∞) ' Y (T ), the edge set of the

Bruhat-Tits tree. We have thus identified the special automorphic forms with

certain functions on the Bruhat-Tits tree associated to F∞. We now introduce

this class of functions, the harmonic cochains for Γ.

53

Let T be the Bruhat-Tits tree associated to K∞. T is thus a q∞ + 1-regular

tree with vertex set X (T ) and oriented edge set Y (T ).

Definition 5.1.2. : Let M be an abelian group. (We will typically take M to

be the integers, complex numbers, F∞ or C∞). A harmonic cochain on T with

values in M is a map φ : Y (T )→M which satisfies:

φ (e) + φ (e) = 0 for all e ∈ Y (T ) and

∑e∈Y (T ),t(e)=v

φ (e) = 0 for all v ∈ X (T ).

Denote by Har (T ,M) the set of M -valued harmonic cochains. Har (T ,M)

is a group under pointwise addition; if M is a vector space, ring, algebra, etc,

then Har (T ,M) inherits the same structure. Recall that T can be obtained

from Ω by rigid-analytic means. It should not come as a surprise, then, that

(nice) harmonic cochains come from rigid-analytic functions on Ω.

Theorem 5.1.3. There is a canonical GL (2,K∞)-equivariant sequence

0→ C×∞ → OΩ (Ω)× → Har (T ,Z)→ 0

The first map is the obvious injection, and the second map is r from section 3.4.

Γ acts on Ω and on T compatibly. Γ therefore also acts on the harmonic

cochains via (γ · φ) (e) = φ(γ−1e

).

Definition 5.1.4. Denote by Har (T ,M)Γ

the subgroup of Har (T ,M) con-

sisting of φ which satisfy φ (γe) = φ (e) for all γ ∈ Γ and all e ∈ Y (T ).

We will investigate the structure of Har(T ,M)Γ in section 5.2. In section 5.6

we will define Hecke operators acting on the spaces of automorphic forms and

harmonic cochains; it will be important to relate these actions. We conclude

this section with the following theorem from [28].

54

Theorem 5.1.5. Under the identification in Theorem 5.1.1, the automorphic

forms for K correspond to the harmonic cochains for Γ. This identification is

compatible with the Hecke action.

5.2 The j Map

In this section we provide a proof for the conjecture that for an arbitrary di-

vision quaternion algebra D and finite index Γ ⊆ O×D, one has an isomorphism

of groups Γ−→Har(Γ\T ,Z)Γ. Here Γ is the group Γab modulo torsion. This is

conjectured to be true for congruence subgroups of GL(2) but so far has only

been proved in certain cases. The main reason the quaternionic case is easier is

that one has complete knowledge of the possible torsion subgroups of Γ.

It is well-known (see for example [36, Lemma 5.1]) that the only possible ver-

tex stablizers in Γ are isomorphic to F×q or F×q2 , hence are of order q − 1 or

q2 − 1. For any e ∈ Y (T ), one has that StabΓ(e) = F×q . Let v ∈ X(T ).

If #StabΓ(v) = q − 1, then none of the edges in Y (T ) incident to v can be Γ-

equivalent. On the other hand, if #StabΓ(v) = q2−1, then StabΓ(v) acts on the

edges incident to v and by the orbit-stabilizer theorem we see that all edges in-

cident to v are Γ-equivalent. Thus the graph Γ\T has only two types of vertices:

those which have valency q+1 (type 1), and those which have valency 1 (type 2).

Recall that Har(T , B) is the group of functions φ : Y (T )→ B which satisfy the

harmonic cochain conditions. For Γ ⊆ O×D, let Har(T , B)Γ denote the subgroup

of Γ-invariant functions, i.e. φ such that φ(γe) = φ(e) ∀γ ∈ Γ, e ∈ Y (T ). We

may view elements of Har(T , B)Γ as functions on the quotient graph as follows.

For e ∈ Γ\T , let m(e) be 1 if e is of type 1, and let m(e) be q+1 if e is of type

2. Then the harmonicity condition for elements of Har(T , B) amounts to saying

that

55

∑e∈Y (Γ\T )

t(e)=v

m(e)φ(e) = 0 .

Here, φ(e) is understood to be φ(e) for any ∈ T which maps to e. It is immedi-

ately clear that for any v ∈ X(Γ\T ),∑

t(e)=v

m(e) = q + 1. From this we imme-

diately see that if e is of type 2, then φ(e) must equal 0 as long as our abelian

group B has no (q + 1)-torsion. In particular, this holds for B = Z,C, C∞,

which are the cases of interest to us. Now the relationship between Γ-invariant

cochains Har(T , B)Γ and functions on Γ\T becomes even more apparent for

B = Z,C, C∞:

To φ ∈ Har(T , B)Γ we associate φ : Γ\T → B via φ(e) = φ(e). Thus the

cochain conditions hold for φ, and we have the following result.

Proposition 5.2.1. : Let Γ ⊆ O×D be an arithmetic group coming from a

division quaternion algebra, and let B be any abelian group without (q + 1)-

torsion. Then one has a canonical isomorphism of groups

Har(T , B)Γ−→Har(Γ\T , B)

We trivially have that Har(Γ\T ,Z) ' H1(Γ\T ,Z), the first (simplicial) ho-

mology group. Indeed, if φ : Γ\T → Z is a cycle, then φ satisfies the cochain

conditions and vice versa. Let Γf be the subgroup of Γ generated by torsion.

The theory developed in [49, Chapter I] identifies Γ∗ := Γ/Γf with the funda-

mental group of Γ\T , so taking abelianizations we obtain an isomorphism

Γ−→H1(Γ\T ,Z)

Putting this all together, we have proved that

Γ ' Har(T ,Z)Γ

as desired. For our applications, we will need to make this isomorphism explicit.

For two vertices v, w ∈ X(T ), let c(v, w) denote the unique geodesic from v to

w in T . Fix for now a base vertex v, and for e ∈ Y (T ), α, γ ∈ Γ, define

56

i(e, α, γ, v) := 1,−1, 0

according to whether γe belongs to c(v, αv), c(αv, v), or neither, respectively.

Since Γ acts discontinuously on T , this function is a compactly supported func-

tion of γ when α, e, v are fixed. Since Γ acts on T through the quotient z(Γ),

the function

φα,v(e) := z(Γ)−1∑γ∈Γ

i(e, α, γ, v)

on Y (T ) is well-defined and Z-valued. It is proved in [17] that φα,v has the

following properties.

Lemma 5.2.2. ([17, Lemma 3.3.3]): Let φα,v be as above.

(i) φα,v ∈ Har(T ,Z)Γ

(ii) φα := φα,v is independent of the choice of v

(iii) φαβ = φα + φβ, so that α 7→ φα induces a homomorphism j : Γ →

Har(T ,Z)Γ

(iv) j is injective with finite cokernel.

The proofs of parts (i)-(iii) are not difficult so we refer the reader to [17]

for details. We focus our attention on part (iv). Since Γ and Har(T ,Z)Γ are

both free abelian of the same rank, injectivity will follow from surjectivity. Let

T be a maximal subtree of Γ\T , and let e1, ..., eg be a set of representatives

modulo orientation of Y (Γ\T ) − Y (T ). For i = 1, 2, ..., g let vi = o(ei) and

wi = t(ei). There exists a unique geodesic c′i in T that connects wi to vi. Let ci

be the closed path around vi obtained by concatenating ei with ci. We define

φi : Y (Γ\T )→ Z as follows: φi(e) = 1,−1, or 0 according to whether e appears

in ci, ¯ci, or neither, respectively.

Then φi ∈ Har(T ,Z)Γ. In fact, φi is uniquely determined by the fact that it is

a harmonic cochain supported only on ci with value 1 at ei. (That φi(ei) = 1

follows from the definition of φα together with the fact that the stabilizer of

57

ei must be F×q , because type 2 vertices must already belong to the maximal

subtree T ). Now for any φ ∈ Har(T ,Z)Γ, one has that

φ−g∑i=1

φ(ei)φi

vanishes on all ei (i.e. vanishes outside the maximal tree T ), which implies that

it vanishes identically. Thus φ is in the image of j, and we have proved:

Theorem 5.2.3. : The map j : Γ → Hom(T ,Z)Γ = Har(Γ\T ,Z) is an

isomorphism.

5.3 Theta Functions

Holomorphic theta functions play a fundamental role in the uniformization of

Jacobians of modular curves. Indeed, Jac(XΓ) is known to be abstractly iso-

morphic to a torus modulo a lattice: Jac(XΓ) ' (C×∞)g/Λ, where the Z-rank of

Λ is g = genus(XΓ) (see also theorem 3.5.2). The meromorphic theta functions

for Γ provide us with both the torus and the lattice. Moreover, there is a way

to relate theta functions to harmonic cochains, which opens the way for explicit

calculations.

Definition 5.3.1. : A holomorphic theta function for Γ is a function u : Ω →

C∞ which for each α ∈ Γ satisfies a functional equation u (α · z) = cu (α) ·u (z),

where cu (α) ∈ C×∞ is independent of z ∈ Ω, and which is holomorphic without

zeros on Ω.

Set Θh (Γ) → Θm (Γ) for the multiplicative groups of holomorphic and mero-

morphic theta functions, respectively. Clearly, cu : Γ → C×∞, α 7→ cu (α), is a

group homomorphism. So, we obtain

c : Θm (Γ)→ Hom (Γ, C×∞) = Hom(Γab, C×∞

), u 7→ cu.

Remark 5.3.2. Let u (z) be a holomorphic theta function for Γ. Then u′(z)u(z) is

a modular form of weight 2 for Γ. This follows straightforwardly from the chain

rule.

58

The theory of theta functions in our context has been developed by Manin and

Drinfeld, see [29]. We recall the main results here.

Definition 5.3.3. : Fix a Schottky group Γ coming from a quaternion algebra.

A K∞-divisor on Ω is a function

Ω(K∞

)→ Z, z 7→ nz, such that

(1) nz1 = nz2 if z1 and z2 are conjugate over K∞.

(2) There is a finite extension L of K∞ such that nz 6= 0 =⇒ z ∈ Ω (L).

(3) The set z : nz 6= 0 (the support of the divisor) has no limit points in Ω.

Denote such a divisor by∑nz ·z = d. A divisor d is finite if supp (d) is finite.

Γ acts on Ω, as well as on Ω (L), and so acts on the group Div of K∞-divisors.

Lemma 5.3.4. (See, for example, [12] and [29]) (1) The group of Γ-invariant

K∞-divisors consists exactly of the divisors of the form∑g∈Γ

g (d), where d is a

finite K∞-divisor on Ω.

(2)∑g∈Γ

g (d) = 0 if and only if there exist gi ∈ Γ,i = 1, ..., k, and finite K∞-

divisors di, such that d =k∑i=1

(1− gi) di.

Proof. Note first that for (a) and (b), one direction is obvious, i.e. divisors of the

form∑g (d) are obviously Γ-invariant, and

∑g (d) = 0 for d =

∑(1− gi) di.

Recall that the quotient graph ΓT is finite. Each edge corresponds to an

affinoid Ui ⊂ Ω, and so taking a finite union S =⊔giUi of representatives we

obtain a domain S ⊂ Ω such that Ω =⋃g∈Γ

gS.

For such S, and for any finite extension L of K∞, we note that S(L) is com-

pact. Indeed, U0(L) is compact: It is a closed subset of the compact unit disc

in L. (Note that strict inequalities can be replaced by weak inequalities in the

definition of U0(L) because L is discretely valued). S(L) is a finite union of

59

GL (2,K∞)-translates of U0(L), so is compact as well.

(a) Let d0 be a Γ-invariant K∞-divisor, with support in Ω (L), and let S be

a domain as above. Then the support of d0 in S is finite, since otherwise (by

compactness) we obtain a limit point of z : nz 6= 0 in Ω (indeed, in S). Say,

supp (d0) ∩ S = z1, ..., zk and d0 =∑αnzα · zα.

It may happen that for g ∈ Γ, g ·S∩S 6= ∅, and so we may have some zi ≡ zj in

supp (d0)∩S. Let d1 =∑i

nzi · zi be the sum taken over distinct representatives

modulo Γ. Then d0 − Γ · d1 is a Γ-invariant divisor whose support on S is 0.

But Γ · S = Ω. So, d0 − Γ · d1 = 0.

(b) Suppose∑g∈Γ

g (d) = 0. By compactness, the support of d is finite. In-

deed, if it were not finite, we could easily show that the support of Γ · d in S

has limit points. Write d =∑nizi. Then

∑ni = 0. Since Γ · d = 0, we have

that z1 is Γ-equivalent to some zj ; say z2 = g1z1. Then the divisor

d0 =∑nizi − n1 (1− g1) z1

again satisfies Γ ·d0 = 0. But this divisor has smaller support, so we may obtain

the result by induction.

Corollary 5.3.5. There exists a degree homomorphism

DivΓ deg−→ Z ,∑g

(k∑i=1

nizi

)7→

k∑i=1

ni

Now let∑g∈Γ

g (d) be a Γ-invariant divisor of degree 0. In [29] the authors con-

struct a rigid meromorphic function on Ω having this divisor, as follows. First,

since d is a finite divisor of degree 0, there is a K∞-rational function on P1, ωd,

with divisor d. Fix z0 ∈ Ω−⋃g∈Γ

g (supp d) and consider the formal product

60

Θd,z0 (z) =∏h∈Γ

ωd(hz)ωd(hz0)

Lemma 5.3.6. For any z ∈ Ω, Θd,z0 (z) is the product of a finite number

of terms having zeros or poles at z, and a convergent infinite factor. This

convergence is locally uniform away from Γ (supp d ∪K∞ ∪ ∞) ⊂ P1 (C∞).

A proof is given in [29]. However, for our purposes we will need explicit bounds

on the rate of convergence. For now we only mention that the denominator

terms are necessary to ensure convergence. We will return to this question in

Chapter 6.

Example 5.3.7. Let d = z0 − α · z0, α ∈ Γ. Then ωd (z) = z−z0z−αz0 , and

Θd,z0 (z) =∏h∈Γ

(hz−z0hz−αz0 ·

hz0−αz0hz0−z0

)We now introduce the important automorphy factors associated to a Θ-

function.

Proposition 5.3.8. ([29]) For fixed g ∈ Γ, we have that

Θd,z0 (gz) = µd (g) Θd,z0 (z)

where µd (g) ∈ K×∞ is multiplicative in each of d, g, and is independent of z0.

Proof. We have that

Θd,z0 (gz) =∏h∈Γ

ωd(hgz)ωd(hz0) =

∏h∈Γ

ωd(hz)ωd(hg−1z0)

because g ∈ Γ. Hence, µd(g) = Θ(gz)Θ(z) =

∏h∈Γ

ωd(hz0)ωd(hg−1z0) is a constant, because it

has no zeros or poles. For z1 6= z0, we have that

Θd,z0 (z) =∏ ωd(hz)

ωd(hz1) =∏ ωd(hz)

ωd(hz0) ·∏ ωd(hz0)

ωd(hz1) .

So, Θd,z1 = const ·Θd,z0 , which implies that µd (g) does not depend on z0. Mul-

tiplicativity in d is clear, because Θd1+d2,z0 = Θd1,z0 · Θd2,z0 . Multiplicativity

in g is equally clear:

61

µd (g1g2) Θd,z0 (z) = Θd,z0 (g1g2z)

= µd (g1) Θd,z0 (g2z) = µd (g1)µd (g2) Θd,z0 (z) .

Finally, µd (g) ∈ K×∞ as follows: If z0 ∈ L for some finite extension L ofK∞, then

µd (g) =∏h∈Γ

ωd(hz0)ωd(hg−1z0) . By condition (1) in the definition of K∞-divisor, we see

that ωd is defined over K∞. Completeness of L then implies that µd (g) ∈ L.

But µd (g) is independent of z0. So, µd (g) ∈⋂L

L = K∞. Obviously then

µd (g) ∈ K×∞.

The Θ-functions, together with their automorphy factors µd (g), provide us

with a useful positive-definite symmetric bilinear form on Γ× Γ. To see this, we

apply the previous result in the special case∑g (d) = 0. In this case, Θd,z0 (z) is

a holomorphic function on Ω, because its zeros and poles cancel. The following

theorem is proved in [29].

Theorem 5.3.9. Let Γ = Γab/torsion. Let α, β ∈ Γ and α, β their respective

classes in Γ. Put

< α, β >= µ(β−1)z1 (α)

Then < α, β > depends only on α and β. The mapping < ·, · >: Γ × Γ → K×∞

is symmetric and bimultiplicative.

Theorem 5.3.10. The bilinear form Γ × Γ → Z given by (α, β) 7→ ord∞ <

α, β > is positive-definite. In other words, for any α ∈ Γ, one has the strict

inequality | < α, α > |∞ < 1.

Proof. This will follow from Theorem 5.3.13.

Recall the exact sequence of GL (2,K∞)-modules (Theorem 5.1.3)

0 −→ C×∞ −→ OΩ (Ω)× r−→ Har (T ,Z) −→ 0

Taking Γ-invariants gives us the exact sequence

(OΩ (Ω)×

)Γ r−→ Har (T ,Z)Γ −→ H1(C×∞,Γ) = Hom(Γ, C×∞)

62

We recall the definition of the connecting homomorphism for the sake of clarity.

If f ∈ Har(T ,Z)Γ, then there is some g ∈ OΩ(Ω)× such that r(g) = f . The

element g need not be Γ-invariant, but g(z) and g(γz) both map to f under r

since f is Γ-invariant. So, g(γz)g(z) ∈ Ker(r), hence is identically constant, and in

this manner we define the map δ : Γ→ C×∞.

Since C×∞ is an abelian group, every homomorphism Γ −→ C×∞ factors through

Γab. Recall from section 5.2 the construction j : Γ'−→ Har(T ,Z)Γ which asso-

ciates to the class of an element α ∈ Γ a Γ-invariant harmonic cochain φα. We

also have the map r : OΩ(Ω)× −→ Har(T ,Z) which associates to an invertible

holomorphic function f on Ω a harmonic cochain which essentially measures

the growth of f along affinoid regions of Ω. In particular, one has the harmonic

cochain r(Θαz1−z1,z0(z)). We thus have two recipes for cooking up harmonic

cochains from elements of Γ, and it is natural to ask if their outputs agree. This

is the content of the following important theorem, which is analagous to [17,

Theorem 5.6.1].

Theorem 5.3.11. r(Θαz1−z1,z0(z)) = φα.

Proof. We first prove that r(Θαz1−z1,z0(z)) and φα agree on the standard edge

e0 associated to U0. Let z1 ∈ Ω be such that v := λ (z1) ∈ X (T ). Then

Θαz1−z1,z0(z) =∏h∈Γ

z−hα·z1z−h·z1 ·

z0−h·z1z0−hα·z1

where almost all factors have absolute value 1 uniformly on U0. If h ∈ Γ is such

that h ·z1, hα ·z1⋂D0 = ∅, then by example 3.4.2, section 3.4 its contribution

to r (Θαz1−z1,z0(z)) is:

1 if hα · z1 ∈ Ω−, h · z1 ∈ Ω+

−1 if hα · z1 ∈ Ω+, h · z1 ∈ Ω1

0 Otherwise

63

(Note that the factor z0−h·z1z0−hα·z1 contributes nothing to the quotient of spectral

norms because it is a constant). The product

f =∏ z−hz1

z−hαz1

over those h ∈ Γ such that hz1 or hαz1 lies in U0 is finite and invertible on

U0, since its zeros and poles cancel. So, this product f satisfies the criteria of

example 3.4.3, section 3.4 .

Remember that z1 ∈ Ω was chosen so that ν := λ (z1) ∈ X (T ), so that no

Γ-translate of λ (z1) lies on the interior of an edge of T . That is, no Γ-translate

of z1 lies inside U00 . So, hz1, hαz1 ∩ U0

0 = ∅. We may thus apply observation

3.4.3 to the previous argument and obtain

1 if hα · z1 ∈ Ω−, h · z1 ∈ Ω+

−1 if hα · z1 ∈ Ω+, h · z1 ∈ Ω1

0 Otherwise

for the contribution of z−hz1z−hαz1 to r (Θαz1−z1,z0(z)), even in the case that hz1 or

hαz1 meets U0. Recall the notation c(v1, v2) from section 5.2. We thus have that

r (Θαz1−z1,z0(z)) (e0)

= #h ∈ Γ : hz1 ∈ Ω−, hαz1 ∈ Ω+ −#h ∈ Γ : hz1 ∈ Ω+, hαz1 ∈ Ω−

= #h ∈ Γ : e0 ∈ c (hν, hαν) −#h ∈ Γ : e0 ∈ c (hαν, hν)

64

= z (Γ)−1 ∑

h∈Γ

i(e0, α, h

−1, ν)

= φα (e0) by definition.

So, we have shown the coincidence of these two functions on the standard edge

e0. Note that we have done so for quite general Γ. Now let e ∈ Y (T ) be an

edge of T associated to some affinoid U ⊂ Ω. Then there is some σ ∈ D× such

that σU = U0, and on the tree this is reflected in σe = e0. Indeed, there is

certainly some element M of GL (2,K∞) taking U to U0. It is a straightforward

verification that any element of D× which is close enough to M does the job.

Here the “close enough” requirement depends on U .

Then we have that (in the notation of section 3.4) σR−1 (M∗) = C0 and

σR−1(M′∗)

= C1, and

r (f) (e) = log

(||f ||spR−1(M∗)||f ||

sp

R−1(M ′∗)

).

Consider f σ, i.e.

(f σ) (z) = f (σz) =∏h∈Γ

σz−hz1σz−hαz1

= const ·∏h∈Γ

z−σ−1hz1z−σ−1hαz1

= const ·∏h∈Γ

z−σ−1hσσ−1z1z−σ−1hσσ−1ασσ−1z1

= const ·∏β∈Γ

z−βηz−βα0η

where η = σ−1z1 and α0 = σ−1ασ. This is just a theta function associated to

another group σ−1Γσ, and the previous result applied to this theta function on

the standard edge implies the result for our original theta function on the edge

e.

Corollary 5.3.12. The map u : Γ −→ Θh (Γ)C×∞, α 7→ uα = Θαz1−z1,z0(z)

is an isomorphism. The restriction of r, r : Θh(Γ)/C×∞r−→ Har(T ,Z)Γ, is an

isomorphism.

65

Proof. : By the previous theorem we have a commutative triangle

Γ

Θh(Γ)/C×∞ Har(ΓT ,Z)

u j

r

Since j is an isomorphism, r must be surjective. But r is already injective

on Θh(Γ)/C×∞ because the kernel of r consists of the constant functions. So,

the bottom map is an isomorphism, which forces u to be an isomorphism as

well.

Next we note that the Petersson inner product on automorphic forms is

essentially the product (φ1, φ2) on Har(ΓT ,C). Let µ be a Haar measure on

the locally compact group D×(A)Z(F∞). Then the approximation theorem

implies thatD×(F )D×(A)Z(F∞) has finite volume with respect to µ. Recall

from section 5.1 the notation Y (K) = D×(F )D×(A)K·Z(F∞), which is the

edge set of the quotient graph ΓT . Let µ1 denote the pushforward measure

on Y (T ) induced by µ. Then Y (K) has finite measure with respect to µ1, and

one has explicitly that for e ∈ Y (K), µ1(e) = #stabΓ(v)#stabΓ(e) , where v = o(e). Thus

the scalar product

(φ1, φ2)µ :=∫

Y (K)

φ1(g)φ2(g)dµ(g)

on the space of C-valued functions on Y (K) induces the scalar product

(f1, f2)µ1:=

∑e∈ΓT

f1(e)f2(g)n(e).

This agrees with the inner product on ΓT discussed above. If f, g are el-

ements of Har(ΓT , B) for B = Z,Fq, C∞,C, then recall that f and g are

only supported on the edges e such that n(e) = 1. Therefore the Petersson

product on automorphic forms special at infinity takes the form (f1, f2)µ1:=∑

e∈ΓTf1(e)f2(g). We also have the pairing (·, ·) on Γ × Γ defined by (α, β) =

66

val(µ(β−1)z1(α)). The following theorem relates these pairings and is important

for computations.

Theorem 5.3.13. (cf. [60]) 2(α, β) = (φα, φβ)µ.

5.4 The Jacobian

Let Γ ⊆ D× be a finite index subgroup of the unit group of some division

quaternion algebra D over F which is split at ∞, and XΓ = ΓΩ the associ-

ated Shimura curve. As we explained in section 3.5, XΓ has split degenerate

stable reduction, and is therefore a Mumford curve. This implies (Theorem

3.5.2; see also [3, Section 1]) that Jac(XΓ) admits a uniformization of the form

Jac(XΓ) = TΓΛ, where TΓ is an analytic torus, and where Λ is a full lattice

in TΓ.

We have already alluded to using spaces of theta functions to construct Jac(XΓ);

in fact, this construction is a well-known result of Mumford in the case that Γ

is a Schottky group. This holds for example whenever D is ramified at at least

one place of even degree. If Γ is not a Schottky group, a modification is required

and is supplied by Gekeler and Reversat in the case of congruence subgroups

of GL(2, F∞) ([17] Theorem 7.4.1). In fact, the argument in [17] works for Γ

arising from quaternion algebras as well, and can be copied almost verbatim

to our situation after omitting some small talk about cusps. We therefore only

sketch the argument here, with emphasis on the points which will be needed

later.

Definition 5.4.1. (1) By a torus over a field L we mean an algebraic group

isomorphic to Ggm(L) for some g ≥ 1.

(2) Let T ' Ggm(L) be a torus defined over an intermediate field F∞ ⊂ L ⊂ C∞.

A subgroup Λ ⊂ T (L) is called a lattice if Λ ' Zg and the image of Λ under

log(| · |∞) : (C×∞)g → Rg is a lattice in Rg.

67

Fix a torus T of dimension g over C∞ and a lattice Λ ⊂ T (C∞). Since

Λ is discrete, TΛ is an analytic group variety which is compact in the rigid

analytic sense. The following well-known theorem is the natural analogue to

the Riemann conditions for abelian varieties over C.

Theorem 5.4.2. TΛ is algebraic (i.e. rigid isomorphic to the analytic space

associated to some abelian variety W ) if, and only if, there exists a homomor-

phism σ : Λ → Hom(T,Gm), the character group of T , such that the bilinear

map

(α, β) 7→ σ(α)(β) : Λ× Λ→ C×∞

is symmetric and positive-definite. Here, positive definite means that

log |σ(α)(α)|∞ > 0 for all α 6= 1 in Λ.

For our situation, we take as our torus TΓ the character group of Γ, TΓ :=

Hom(Γ,Gm) (' Ggm where g is the rank of Γ), which is a split torus defined over

K∞ with character group Hom(TΓ,Gm) ' Γ via the evaluation map. As for our

lattice, we take the image of Γ in TΓ = Hom(Γ,Gm) under the map c : α 7→ cα.

Then by Theorem 5.3.10, TΓ and Λ satisfy the requirements of Theorem 5.4.2.

Proposition 5.4.3. ([29]) Let Γ, Λ, TΓ be as above. Then Λ is a lattice, and

there exists an abelian variety AΓ over K∞ such that TΓΛ and the analytifi-

cation of AΓ are isomorphic as rigid analytic groups. AΓ is characterized by the

short exact sequence of analytic groups defined over K∞

1 −−−−→ Γc−−−−→ Hom(Γ, C×∞) −−−−→ AΓ(C∞) −−−−→ 0

α 7→ cα

Proof. This follows from Theorem 5.4.2, and the fact (Theorem 5.3.8) that Λ

and TΓ are defined over K∞.

Fact 5.4.4. Let X be an algebraic curve over a field k, and suppose X(k) 6= ∅.

Fix P0 ∈ X(k), and consider the map f : X −→ Jac(X), P 7→ [P − P0]. Then

f is an injective morphism of algebraic varieties, and X, Jac(X), f have the

68

following universal property: Let A be another abelian variety and g : X −→ A

a morphism which sends P0 to 0. Then there is a unique morphism of abelian

varieties φ : Jac(X) −→ A such that g = φ f .

The following theorem is analagous to [17, Theorem 7.4.1].

Theorem 5.4.5. The abelian variety AΓ is K∞-isomorphic to the Jacobian JΓ

of the Shimura curve XΓ.

Proof. (Sketch): (1) Fix a “base” point ω0 ∈ Ω and let Ψ : Ω→ Hom(Γ, C×∞)→

AΓ(C∞) be the map which associates to ω ∈ Ω the multiplier µω0−ω(·) of

Θω0−ω(z) [See Proposition 5.3.8] followed by the canonical projection map.

Then Ψ can be shown to be analytic, and to factor through the projection

pΓ : Ω → ΓΩ = XΓ(C∞). Writing Ψ = ΨΓ pΓ, one then shows that

ΨΓ : XΓ −→ AΓ is analytic as well. By the GAGA theorems (cf. [7, 27]), ΨΓ is

a morphism of algebraic varieties.

(2) We now think of ω0 as a base point in JΓ := Jac(XΓ). To this end, let

P0 := pΓ(ω0) and let κΓ : XΓ −→ JΓ be the morphism which to P ∈ XΓ(C∞)

associates the divisor class of P − P0. Then by 5.4.4, there exists a unique

morphism φΓ : JΓ −→ AΓ of C∞-varieties which makes the following diagram

commute:

X JΓ

AΓ

κΓ

ΨΓ φΓ

(3) φΓ is injective. Indeed, let ω1, ω2, ..., ωn ∈ Ω, Pi := pΓ(ωi), and [D] the

divisor class of D = P1 + P2 + · · ·Pn − nP0. Suppose that φΓ([D]) = 0. Then

by Proposition 5.4.3, there exists α ∈ Γ such that

cα = µαz9−z0 =∏

1≤i≤nµωi−ω0

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The function u−1α (z)

∏1≤i≤n

Θωi−ω0(z) is then a Γ-invariant, meromorphic func-

tion, hence gives a meromorphic function on XΓ = ΓΩ whose divisor equals

D. (Note that the u−1α (z) factor does not affect the divisor; it is there to provide

Γ-invariance). Therefore [D] = 0 in Jac(XΓ), and so φΓ is injective.

(4) JΓ and AΓ are both abelian varieties of the same dimension g. There-

fore φΓ is equal to the composition of an isogeny with a purely inseparable map.

So, if we show that φΓ is separable, we’re done, as isogenies are surjective. Let

Ω1XΓ,Ω1

AΓ,Ω1

JΓdenote the sheaves of 1-differentials on XΓ, AΓ, JΓ, respectively.

Let H0(·,Ω1· ) denote the group of holomorphic differential forms on ·. Then we

have the corresponding pullback diagram

H0(X,Ω1) H0(JΓ,Ω1)

H0(AΓ,Ω1)

κ∗Γ

Ψ∗Γ φ∗Γ

and φΓ is separable if and only if φ∗Γ is bijective. But it is well-known that κ∗Γ

is bijective. So, we are reduced to showing that

Ψ∗Γ : H0(AΓ,Ω1AΓ

) −→ H0(XΓ,Ω1XΓ

)

is bijective. We refer to [17, Prop. 2.10.2 and Thm. 6.5.4] for the proof that

this map is bijective (the proof relies on an understanding of the modular forms

for Γ, which would take us too far afield to recall for our purposes).

Corollary 5.4.6. Let χ : Γ → C×∞ be a homomorphism. There exist ω1, ω2,

..., ωn, ν1, ν2, ..., νn ∈ Ω such that χ equals the multiplier cu of u(z) =∏1≤i≤n

Θωi−νi(z).

Proof. By theorem 5.4.5, one has that

Hom(Γ, C×∞)/Γ ' JΓ(C∞).

70

So, if χ : Γ → C×∞ is a group homomorphism, then the image of χ in AΓ(C×∞)

gives a point∑Pi − P0 in JΓ. One verifies that χ is the multiplier of the theta

function ∏θωi−ω0

(z),

where ωi is any pre-image in Ω of Pi under the projection map.

5.5 Hecke Operators

We recall here the basic definitions and facts regarding Hecke operators in our

context. Proofs can be found, for example, in [31]. As usual, let Γ = O×/F×qbe our Fuchsian group coming from a quaternion algebra D over F , and acting

of course on Ω. Denote from now on ONrd6=0 the submonoid of O consisting of

elements which are invertible in D. To certain elements α ∈ D (F ) →M(2, F∞)

we associate operators Tα which act on the space of automorphic forms, as well

as on the Jacobian JΓ and on Γ itself.

Let α be a nonzero element of D (F ), viewed as a matrix in M(2, F∞). (Note,

if D is not the matrix algebra then we can not expect α to be a 2x2 matrix

over F in general. However, α can be written as a 2x2 matrix over a quadratic

extension of F contained in F∞). Then one has ([31, 2.7.1])

ΓαΓ =⊔i

Γαγi =⊔j

δjαΓ

where the disjoint unions above are finite.

Denote by RZ (Γ;ONrd6=0) the free Z-module generated by the double cosets

ΓαΓ, α ∈ ONrd6=0. Let now M be a Z-module on which D× acts, with the

action written in exponential notation: γ ·m 7→ mγ for m ∈ M,γ ∈ D×. Then

we define an action of RZ (Γ;ONrd6=0) on MΓ as follows:

71

Write ΓαΓ =⊔i

Γαi. Then we define

m|ΓαΓ =∑i

mαi

We extend this action by Z-linearity to an action ofRZ (Γ;ONrd6=0) on MΓ. This

action is well-defined, i.e. independent of the choice of coset representatives αi,

because any two such representatives differ by multiplication on the right by an

element of Γ, and because m ∈MΓ.

In fact, for any fixed ν =∑αaαΓαΓ ∈ RZ (Γ;ONrd 6=0), we have that the mapping

Tν : m 7→ m|ν

is a homomorphism of abelian groups from MΓ to itself.

Multiplication

Let α, β ∈ ONrd6=0, with ΓαΓ =⊔i

Γαi and ΓβΓ =⊔j

Γβj . We define the

product via

(ΓαΓ) · (ΓβΓ) =∑γcγΓγΓ, where

cγ = #(i, j) : ΓαiβjΓ = ΓγΓ.

That is, multiplication is defined by basically multiplying all the αi and all

the βj , and grouping together those pairs Γαiβj ,Γαi′βj′ which give the same

double coset ΓγΓ. Note that the above sum is finite because there are only

finitely many αi, βj . Extend this multiplication linearly to RZ (Γ;ONrd6=0) in

the obvious way.

Lemma 5.5.1. (1) This multiplication is independent of the choices of repre-

sentatives αi, βj , γ.

(2) This multiplication is associative, so that RZ (Γ;ONrd6=0) is a Z-algebra with

72

multiplicative identity element Γ1Γ.

(3) (m|ν1) |ν2 = m| (ν1ν2) for m ∈ MΓ and ν1, ν2 ∈ RZ (Γ;ONrd6=0). So, we

have an action of the algebra RZ (Γ;ONrd6=0) on MΓ.

Proof. (a) This is a special case of [31, Lemma 2.7.4]; we sketch the proof for the

convenience of the reader. For the remainder of this proof denote ∆ := ONrd 6=0.

Let Z[Γ∆] be the free Z-module generated by the left Γ-cosets Γα, α ∈ ∆.

Then Z[Γ∆] is a right ∆-module by right multiplication, and one sees that

the map

ΓαΓ 7→∑i

Γαi

induces an injective Z-homomorphism

RZ(Γ; ∆) → Z[Γ∆].

In fact, considering the right action of ∆, one easily sees that (via this embed-

ding)

RZ(Γ; ∆) = Z[Γ∆]Γ,

e.g. the Γ-invariant formal sums∑i

aiΓαi are precisely the elements ofRZ(Γ; ∆).

(The ⊆ direction is trivial, and the ⊇ direction can be seen either directly or by

an easy induction on∑i

|ai|.) Now let ΓαΓ =⊔i

Γαi and ΓβΓ =⊔j

Γβj . Then

by the definition of the action of RZ(Γ; ∆) on Z[Γ∆]Γ, we see that(∑i

Γαi

)| ΓβΓ =

∑i

∑j

Γαiβj .

That is, the action of RZ(Γ; ∆) on RZ(Γ; ∆) is just multiplication as defined

above. But then this multiplication must be independent of the choice of coset

representatives, because RZ(Γ; ∆) = Z[Γ∆]Γ.

(b) This follows from the preceeding discussion. Note that sinceRZ(Γ; ∆) = MΓ

for M = Z[Γ∆], we have that RZ(Γ; ∆) acts on itself by the multiplication

defined above, which implies that this multiplication is associative.

73

(c) This is immediate from (a) and (b).

The following theorem is important for our applications; we state it (without

proof) and subsequently verify that the conditions of the theorem are satisfied in

our situation. This verification is local-global in nature; it would be interesting

to find a purely global proof.

Theorem 5.5.2. ([31, Theorem 2.7.8]) Assume that there exists a mapping η

of ONrd 6=0 to itself satisfying

(i) (αβ)η

= βηαη and (αη)η

= α for α, β ∈ ONrd6=0.

(ii) Γη = Γ

(iii) ΓαηΓ = ΓαΓ.

Then we have:

(1) For any α ∈ ONrd 6=0, ΓΓαΓ and ΓαΓΓ have a common set of repre-

sentatives. That is, there exist δi ∈ ONrd6=0 such that

ΓαΓ =d⊔i=1

Γδi =d⊔i=1

δiΓ

(2) The Hecke algebra RZ (Γ;ONrd6=0) is commutative.

Proposition 5.5.3. The conditions of the previous theorem are satisfied by

RZ (Γ;ONrd 6=0). In particular, RZ (Γ;ONrd6=0) is commutative.

Proof. We take the map η to be the standard involution on D×. Clearly, then,

condition (i) is satisfied. Recall that Γ = O× where O is a maximal A-order in

D. We may choose embeddings D → M(2, Fv) for unramified places v 6=∞ of

D in such a way that Ov ( = closure of O in M(2, Fv)) equals M(2, Ov). At the

ramified places v, there is a unique maximal order Ov of Dv. Then we have

74

O = D (F )⋂( ∏

v not ramified

M(2, Ov)×∏

v ramified

Ov)

Now, η is given at the matrix places bya b

c d

7→ d −b

−c a

So, M(2, Ov)

η = M(2, Ov). At the ramified places v, we have that Oηv = Ov,

because Ov is the valuation ring of Dv, and because η preserves valuation. (η

acts as Galois conjugation on quadratic subfields, hence preserves characteristic

polynomials). So, we have that

Oη = D (F )η⋂( ∏

v not ramified

M(2, Ov)η ×

∏v ramified

Oηv)

= D (F )⋂( ∏

v not ramified

M(2, Ov)×∏

v ramified

Ov)

= O.

Thus Γη = Γ, so we have (ii).

Finally, we want to show that ΓαηΓ = ΓαΓ for α ∈ ONrd 6=0. At the rami-

fied places this is clear, because v (αη) = v (α). At the unramified places, α is

in O, so lands in Dv as a matrix

a b

c d

with entries in Ov. Multiplying on

the left and on the right by suitable elements of Γv, we can make α diagonal: say,

ΓvαΓv = Γv

a′ 0

0 d′

Γv. Then multiplying on the left and right by the matrix

0 1

1 0

∈ Γv, we obtain the following sequence of equalities:

75

ΓvαΓv = Γv

a′ 0

0 d′

Γv = Γv

d′ 0

0 a′

Γv

= Γv

a′ 0

0 d′

η

Γv = ΓvαηΓv

Here the last equality is obtained from the first equality by applying η to the

product of matrices making

a b

c d

diagonal and using the fact that Γ = Γη.

Summarizing, we have that ΓvαηΓv = ΓvαΓv for every place v 6= ∞. So,

ΓαΓ = ΓαηΓ, and the proof is complete.

5.6 Hecke Equivariance of the Jacobian Sequence

By Theorems 5.4.5 and 5.2.3 we have an exact diagram

1 −−−−→ Γc−−−−→ Hom

(Γ, C×∞

)−−−−→ JacΓ (C∞) −−−−→ 0yr

Har(T ,Z)Γ

Each object considered here comes equipped with an action of the Hecke algebra

for Γ. In this section we prove that the above maps commute with the Hecke

actions. The arguments used here are slightly less concrete than in the case of

matrix algebras because for division algebras there is no known nice set of coset

representatives for ΓΓαΓ. It’s worth noting that the arguments presented

here work equally well for matrix algebras.

Let’s first fix some notation. Let α ∈ ONrd 6=0. Instead of dealing only with

ΓΓαΓ, it will be convenient at times to use different descriptions of the Hecke

operator Tα. Fix the following objects:

76

∆+ = Γ⋂α−1Γα , Γ =

⊔∆+βi,

∆− = Γ⋂αΓα−1 , Γ =

⊔∆−γj , where βi, γj ∈ Γ.

Note that we have one-to-one correspondences ∆−Γ ←→ ΓΓα−1Γ and

∆+Γ←→ ΓΓαΓ via

∆−Γ←→ ΓΓα−1Γ ∆+Γ←→ ΓΓαΓ (**)

∆− · γj ↔ Γ · α−1γj ∆+ · βi ↔ Γ · αβi

The proof of proposition 5.5.3 allows us to compare the βi and the γj .

Lemma 5.6.1. In the above coset decompositions, we may take γj = α2

Nrd(α)βj.

In other words, after possibly permuting the indices, we have that Γαβi =

Nrd(α)Γα−1γi for each i.

Proof. In the proof of Proposition 5.5.3 we showed that ΓαΓ = ΓαΓ. Since

α = Nrd(α)α , we see that ΓαΓ = Nrd(α)Γα−1Γ, and the result follows straight-

forwardly.

Note 5.6.2. By Proposition 5.5.3, we can (hence do) choose our βi to be si-

multaneous left- and right- coset representatives for ∆+ in Γ. In the proof

of Proposition 5.6.5 we will need the easy fact that for such a set of βi, the

collection γ′i := β−1i forms a set of representatives for ∆− in Γ.

We now collect the definitions of the Hecke actions of α on the various pieces

of our Jacobian exact sequence. We will denote these actions all by the symbol

Tα. For a group G and a subgroup H ≤ G of finite index, recall the transfer (or

corestriction) map V : G −→ H, as well as the map I : H −→ G induced by

the inclusion.

77

• Tα on Γ: Let V : Γ −→ ∆− denote the transfer map, conj : ∆− −→ ∆+

the map induced by conjugation γ 7→ α−1γα, and I : ∆+ −→ Γ the

map induced by inclusion. Then we define, for α prime and coprime to

the discriminant of D, the Hecke operator Tα := I conj V . Explicitly,

Tα (δ) = α−1

( ∏βi∈∆−Γ

βiδβ−1σ(i)

)α, where σ is the permutation such that

βiδβ−1σ(i) ∈ ∆−.

• Tα on H! (T ,Z)Γ: Recall briefly that Y (T ) = GL (2,K∞)I∞ · Z (K∞),

where I∞ is the Iwahori subgroup. So, any φ ∈ H! (T ,Z)Γ

is a Γ-invariant

function on GL (2,K∞) which is, in particular, Z (K∞)-invariant. We de-

fine for such φ (Tαφ) (g) =∑

βi∈∆−Γ

φ(α−1βig

).

• Tα on Hom(Γ, C×∞

): Let φ : Γ→ C×∞ be a homomorphism. Define Tα on

φ by having Tα act on the input, i.e. (Tαφ)(δ) = φ(Tαδ).

• Tα on JacΓ (C∞): We have two maps

∆+Ω

ΓΩ ΓΩ

1 α

The left-hand map sends ∆+ · z to Γ · z, and the right-hand map sends ∆+ · z

78

to Γ · αz. One easily checks that the right-hand map is well-defined. We thus

obtain a correspondence on XΓ = ΓΩ, which extends uniquely to an endomor-

phism Tα of JacΓ(C∞). For our purposes, it will be useful to make this more

explicit. Let π : Ω → ΓΩ be the canonical mapping, let P ∈ ΓΩ be some

point, and let z ∈ Ω be such that π(z) = P . Then Tα(P ) =∑

βi∈∆+Γ

π (αβiz).

One extends this definition to all divisors on XΓ, in particular to the degree 0

divisors, to obtain the action of Tα on JacΓ(C∞). One easily checks that this

Tα sends principal divisors to principal divisors. Indeed, if f : XΓ → P1 has

divisor D, then g(z) :=∏βi

f(αβi · z) has divisor Tα(D).

We now prove Hecke equivariance of our maps. We begin with r : Γ →

Har(T ,Z)Γ.

Proposition 5.6.3. Fix α ∈ ONrd6=0 and δ ∈ Γ. Write Φδ = r(δ) ∈ Har(T ,Z)Γ.

Then we have ΦTα(δ) = Tα(Φδ).

Proof. Exactly as was shown in [17], we have the equalities

(Tαφδ) (g) =∑

γi∈∆−Γ

φδ(α−1γig

)and

φTα(δ) (g) =∑

βi∈∆+Γ

φδ (αβig).

So, we are reduced to showing the equality of these two expressions.

But by Lemma 5.6.1, we see that∑βi∈∆+Γ

φδ (αβig) =∑

γi∈∆−Γ

φδ(Nrd(α)α−1γig

)=

∑γi∈∆−Γ

φδ(α−1γig

)because φδ is Z(K∞)-invariant.

79

Proposition 5.6.4. The map c : Γ → Hom(Γ, C×∞) is Hecke equivariant; i.e.

for δ ∈ Γ one has c(Tαδ) = Tα(cδ).

Proof. The proof of [17, Prop 9.3.3] goes through verbatim.

Finally, we consider the map d : Hom(Γ, C×∞) → JacΓ(C∞). Recall that this

map proceeds via the theta functions θ(ω0, ω, ·), and can be traced through the

following summarizing diagram:

Ω Hom(Γ, C×∞) AΓ(C∞)

XΓ = ΓΩ

JacΓ(C∞)

c(ω0 − ω, ·)

pΓ

κΓ

ΨΓ

φΓ

By 5.4.6, every homomorphism ψ : Γ → C×∞ is of the form cu for some mero-

morphic theta function u. In particular, let v, w ∈ Ω be two points, representing

P1 and P2 in ΓΩ, respectively. Let z0 /∈ limit points of Γ, and consider the

theta function θ(v − w, z0, z) =∏h∈Γ

z−hvz−hw

z0−hwz0−hv . This theta function has

zeros at Γ · v and poles at Γ · w. If ψ(g) = θ(v − w, z0, gz)/θ(v − w, z0, z), then

d(ψ) = P1 − P2.

Proposition 5.6.5. The map d : Hom(Γ, C×∞)→ JacΓ(C∞) is Hecke equivari-

ant, i.e. Tα(d(ψ)) = d(Tαψ).

Proof. It suffices to show this for ψ = cu where u = θ(v −w, z0, z). Write P1 =

π(v) and P2 = π(w) so that d(ψ) = P1 − P2. We have that Tα(ψ)(δ) = ψ(Tαδ)

80

= θ (v − w, z0, Tα(δ)z0) by definition

=∏

γi∈∆−Γ

θ(v − w, z0, α

−1γiδγ−1σ(i)α · z0

)

=∏γi

θ(γ−1i αv − γ−1

i αw, γ−1i αz0, δγ

−1σ(i)αz0

)by the invariance of the

cross-ratio

=∏γi

θ(γ−1i αv − γ−1

i αw, γ−1i γσ(i)z1, δz1

)changing variables, z1 =

γ−1σ(i)αz0

=∏γi

θ(γ−1i αv − γ−1

i αw, z1, δz1

)because our multipliers do

not depend on the choice

of base point.

This quantity is just the multiplier cf (δ), where

f(z) :=∏

γi∈∆−Γ

∏h∈Γ

z−hγ−1i αv

z−hγ−1i αw

z0−hγ−1i w

z0−hγ−1i v

.

Hence, d(Tα(ψ)) is equal to the divisor of f(z). Since the theta functions con-

verge locally uniformly, this product’s divisor can be found by simply finding

all z ∈ Ω which make some numerator or denominator term vanish. So, the

zeros of f(z) are those z ∈ Ω such that z = hγ−1i αv for some h ∈ Γ. The

poles are found in an identical manner. By Note 5.6.2, the divisor of f(z) is∑βi∈∆+Γ

Γ · βiαv − Γ · βiαw. On ΓΩ this divisor is simply∑βi

βiα(P1 − P2).

But this is precisely Tα(P1 − P2), so we are done.

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5.7 Uniformization of Elliptic Curves

Fix an elliptic curve E over F with conductor n · ∞, where n is the squarefree

product of an even number of finite places. In section 4.7 we saw that we can

associate to E an automorphic form f on D×, which we call the newform asso-

ciated to E. In sections 5.1 and 5.2 we saw that we can regard f as an element

of Γ. We saw in 4.7 that there exists a surjective morphism Jac(XΓ) −→ E; our

goal in this section is to give an analytic proof of this result. As we have seen

in section 5.4, the set cα(β)|α, β ∈ Γ gives a full lattice L in (C×∞)g which

uniformizes JΓ(C∞):

1 −−−−→ Γc−−−−→ Hom

(Γ, C×∞

)−−−−→ JacΓ (C∞) −−−−→ 0

Since sublattices L1 ⊂ L correspond to analytic quotients of JΓ(C∞), it

seems natural (by the GAGA theorems) to look for a 1-dimensional sublattice

of L corresponding to E. Let Λ = cα(f)|α ∈ Γ. Then we will see that a finite

index subgroup of Λ does the job. We now fix a few conventions.

Notation 5.7.1. As before, f is a simultaneous eigenform for all Hecke opera-

tors Tv where v is a split place of D. We may regard f both as an element of Γ

and as an element of Har(T ,Z)Γ, depending on our situation. As an element

of Har(T ,Z)Γ we assume that f is primitive, i.e. f /∈ n ·Har(T ,Z)Γ for any

n > 1. Finally, we note that the Hecke eigenvalues of f must be rational, hence

integral.

Let T0 denote the algebra generated by the operators Tv for v coprime

to the ramification locus of D. Then T0 ⊗ Q acts on Har(ΓT ,Q) as a

semisimple, maximal commutative subalgebra of End(Har(ΓT ,Q)); i.e. with

an appropriate choice of Q-basis for Har(ΓT ,Q), the image of T0 ⊗ Q in

End(Har(ΓT ,Q)) is the collection of all diagonal matrices. This abstractly

implies that

82

Fact 5.7.2. Har(ΓT ,Q) is a cyclic T0 ⊗Q-module.

In other words, there exists some h ∈ Har(ΓT ,Q) such that

(T0 ⊗Q)(g) = Har(ΓT ,Q).

For the sake of concreteness, we mention that one may take h =∑fi, where

the fi form a simultaneous eigenbasis of Har(ΓT ,Q).

Proposition 5.7.3. (cf. [17, Prop. 9.5.1]) Let f ∈ j(Γ) → Har(T ,Q)Γ be as

in 5.7.1, regarded as an element, also labelled f , of Γ. Put Λ for the subgroup

cα(f)|α ∈ Γ ⊂ C×∞. Then there exists t ∈ C×∞ such that |t| < 1 and tZ has

finite index in Λ.

Proof. By 5.7.2 we have that Har(T ,Q)Γ is a cyclic T0 ⊗Q-module. Therefore

there exists β ∈ Γ such that

Λβ := T0(β) −→ Γ

is a finite index sublattice of Γ. Let α ∈ Λβ be written additively, α =∑nvTv(β), the sum taken over a finite number of v coprime to disc(D). Then

since Tv is self-adjoint with respect to the bilinear pairing ca(b), we get

cα(f) = cf (α) = cf (∑nvTv(β))

= cβ(∑nvTv(f)) = cβ(n · f) = cβ(f)n for some n ∈ Z

since f has integral eigenvalues under Tv. Therefore cα(f)|α ∈ Λβ = tZ with

t = cβ(f). Since |cf (f) > 1, |t| 6= 1, so we may choose |t| < 1, which completes

the proof.

Since ca(b) ∈ K×∞, Λ must be a subgroup of K×∞. The only torsion in K×∞ is

F×q . Therefore taking |t| maximal, we obtain the

Corollary 5.7.4. Let f be as in 5.7.1 and Λ ⊂ K×∞ be the group cα(f)|α ∈ Γ.

Then there exists a divisor d of q − 1 and t ∈ K×∞ with |t| < 1 such that

Λ = µd × tZ.

83

If we eliminate the torsion, then the resulting lattice gives rise to a Tate

curve. That is, the map

C×∞Λ'−→ C×∞tdZ

z 7−→ zd

associates to our eigenform f ∈ Har(T ,Q)Γ the Tate curve Tate(td) over K∞

with period td. (The map z 7→ zd is easily seen to be injective, and is surjective

because p = char(C∞) does not divide d). Denote this curve by Ef . Then we

have a diagram

1 −−−−→ Γc−−−−→ Hom

(Γ, C×∞

)−−−−→ JacΓ (C∞) −−−−→ 0ycf (·)

yev yprf (C∞)

1 −−−−→ Λ −−−−→ C×∞ −−−−→ Ef (C∞) −−−−→ 0.

Here the ev map is φ 7→ f(φ) and prf (C∞) is the natural projection map.

Let T ∈ T0 be a Hecke operator, and let λ be the eigenvalue of T acting on f

(seen as an element of Γ, say). Then by Hecke equivariance and a simple diagram

chase, we actually see that T acts on Ef as multiplication by λ. Therefore the

isogeny class of Ef is the isogeny class of the curve E giving rise to f in the

first place. We summarize our results in the following

Theorem 5.7.5. Let E be an elliptic curve over F with conductor n ·∞, where

n is the squarefree product of an even number of finite places, and assume that

E has split multiplicative reduction at ∞. Let D be the quaternion algebra over

F with conductor n, and let Γ and XΓ denote the usual objects. Let α denote the

eigen-element in Γ associated to E. Then the uniformization Jac(XD) −→ E

is given explicitly via the evaluation map at α; that is, on the level of lattices,

the uniformization is given by

Hom(Γ, C×∞)/Γ −→ C×∞/Λ

φ 7→ φ(α)

84

Now, E = Ef is maximal with respect to the diagram

JacΓ(C∞) Ef

E′

i.e. every morphism from JacΓ(C∞) to another elliptic curve E′ in the

isogeny class of E sending 0 to 0 factors through Ef . This can be seen in

two steps: First, it is a fact that such E always exists. Second, if E 6= Ef ,

then we have a diagram JΓ −→ E −→ Ef whose composition is prEf . But then

the lattice for E is intermediate between Λ and Γ and of rank 1, a contradiction.

Since Ef is maximal in the above sense, Ef actually embeds into Jac(XΓ) as a

subvariety; that is, JΓ is the extension of an abelian variety A by Ef . We will

prove and use this fact in section 5.8.

We end this section by highlighting the relationship between the modular form

associated to E and its invariant differential. Let ω0 ∈ Ω have class P0 ∈ ΓΩ

and let κΓ : XΓ → JΓ be the map P 7→ [P − P0] as in the proof of Theorem

5.4.5. Put pf for the composite map prf κΓ. Let uf be the theta function

associated to f . Then we have a diagram

Ωuf−−−−→ C×∞y y

ΓΩ C×∞Λy=

y=

XΓ(C∞)pf−−−−→ Ef (C∞)

Let dww be the invariant differential on Ef coming from a coordinate w on C×∞.

Then the above diagram says that the pullback differential is given by

85

p∗f (dww ) =u′f (z)

uf (z)dz

which by the work of Gekeler can be thought of as the reduction of f ∈

Har(T ,Z)Γ modulo p = char(Fq).

5.8 The Degree of the Parametrization

In this section we derive a formula for the degree of the strong Weil uniformiza-

tion XΓ −→ E given in section 5.7. We largely follow section 3 of the pa-

per [15]. Write (α, β) for logq |cα(β)|q, H for Har(T ,Z)Γ = Γ, and φ⊥ for

α ∈ Γ = H|(φ, α) = 0. (α, β) is positive-definite and Z-valued by The-

orem 5.3.10. The reader should take note that φ⊥ may include elements α

such that cφ(α) 6= 1 - namely, cφ(α) may lie in µd, where Λ = tZ×µd . Let

m := min(φ, α) > 0 | α ∈ Γ.

Lemma 5.8.1. r := [H : Zφ⊕ φ⊥] = (φ,φ)m ( = congruence number of φ).

Proof. Write V = Γ⊗Z Q.

(1) Γφ⊥ ' Z. Indeed, φ⊥ ⊗ Q is a codimension one subspace of V , so we

only need to check that Γφ⊥ is torsion-free. Let α ∈ Γ with αk ∈ φ⊥. Then

k(α, φ) = (αk, φ) = 0, so α ∈ φ⊥ as well.

(2) Fix a Z-basis γ2, ..., γg of φ⊥. Let µ ∈ Γ be such that (µ, φ) = m. Then

SpanZ(µ, γ2, ..., γg) = Γ. To see this, note that the set (α, φ)|α ∈ Γ is the

cyclic subgroup of Z generated by m. Now for α ∈ Γ, one has that (α,φ)(µ,φ) is an

integer, and α− (α,φ)(µ,φ)µ is in φ⊥.

(3) Write φ = a1µ + a2γ2 + · · · + agγg. Then [Γ : φ⊥ ⊕ Zφ] = a1. On the

other hand, (φ, φ) = (a1µ + a2γ2 + · · · + agγg, φ) = a1(µ, φ) = a1 · m. So,

a1 = (φ,φ)m .

86

Let pφ : XΓ(C∞) −→ Eφ(C∞) be the strong Weil uniformization, and let

p∗φ : Eφ −→ JacΓ(C∞) be the map on Jacobians induced by Picard functoriality.

Then it is well-known that p∗φ is injective. Indeed, since pφ is a strong Weil

uniformization, we obtain an exact sequence of abelian varieties

0 −→ A −→ JacΓ −→ Eφ −→ 0.

Taking Hom(·,Gm) gives a long exact sequence in cohomology,

0 −→ Hom(E,Gm) −→ Hom(J,Gm) −→ Hom(A,Gm) −→

Ext1(E,Gm) −→ Ext1(J,Gm) −→ · · ·

There are no nontrivial morphisms from an abelian variety to a torus, so

Hom(A,Gm) = 0. Finally, Ext1(E,Gm) = E∗ and Ext1(JacΓ,Gm) = Jac∗Γ,

where V ∗ is the dual abelian variety to V . Finally, it was proved by Weil that

when V is a Jacobian variety, V ' V ∗. So, Eφ embeds into JacΓ(C∞) as a

subvariety. Now let n denote the degree of the uniformization pφ. We have the

following

Fact 5.8.2. prφ p∗φ : Eφ −→ Eφ is multiplication by n.

Proof. (Sketch) p∗φ([P ] − [P0]) = [p−1φ (P ) − p−1

φ (P0)], where p−1φ (Q) typically

consists of n distinct points of XΓ. Now, prφ simply pushes these points back to

E: pr(∑aiQi) =

∑aipφ(Qi). So the composition sends the divisor [P ] − [P0]

to [nP ]− [nP0].

Theorem 5.8.3. (cf. [15, Prop. 3.8]) Let d, n, r be as above. Then n = d · r.

Proof. We will work with the C∞-valued points on our varieties, so when it’s

convenient we will omit the field. Let Eφ = im(p∗φ) ⊂ JacΓ. Eφ is the subva-

riety corresponding to the subtorus Hom(Γφ⊥, C×∞) of Hom(Γ, C×∞). There-

fore Eφ(C×∞ = Hom(Γφ⊥, C×∞)L, where L = cα| cα|φ⊥ = 1. The map

prφ|Eφ : Eφ −→ Eφ is given by the commutative diagram

87

0 −−−−→ L −−−−→ Hom(Γφ⊥, C×∞) −−−−→ Eφ(C∞) −−−−→ 0ycφ(·)yev yprφ|Eφ

0 −−−−→ Λ −−−−→ C×∞ −−−−→ Eφ(C∞) −−−−→ 0.

Here ev(ψ) = ψ(φ). Note that ev|L is injective. Indeed, if cα is in the ker-

nel, then cα equals 1 on φ⊥ and at φ, i.e. cα = 1 on φ⊥ ⊕ Zφ. Therefore cα

has values in the rth roots of unity. Thus the theta function uα is entire and

bounded, hence constant, so cα is identically 1.

We have that ev(L) = cα(φ)| cα|φ⊥ = 1 = cφ(φ)dZ = trdZ has index d2 · r

in Λ = µd · tZ (see the following lemma). For any natural number i, let i∗

denote the p-free part of i. Then r∗ = #ker(ev) = #Hom(Γ(Zφ⊕ φ⊥), C×∞).

Indeed, if G is a finite abelian group which decomposes as G = G1×G2, G2 the

Sylow-p subgroup of G, then Hom(G,C×∞) = Hom(G1, C×∞)×Hom(G2, C

×∞) =

Hom(G1, C×∞) = G∗1. This is because C×∞ has k-torsion exactly when (k, p) = 1.

We also obtain n · n∗ = #ker(prφ|Eφ) = #ker(mult by n). Indeed, Eφ is a

Tate curve, so its n-torsion is given by µn× < t1n >. Since C×∞ has no p-torsion,

#µn = n∗. An application of the snake lemma gives us

0 −→ ker(Hom(Γφ⊥, C×∞)→ C×∞) −→ ker(Eφ → Eφ)

−→ coker(L→ Λ) −→ 0

from which we get n ·n∗ = d2 · r · r∗, hence n = d · r. Therefore the next lemma

completes the proof.

Lemma 5.8.4. cα(φ)| cα|φ⊥ = 1 = cφ(φ)dZ

Proof. Let A = α ∈ Γ| cα|φ⊥ = 1. A is clearly a group, and φd ∈ A,

which proves ⊇. Let β ∈ Γ be such that cφ(β) generates µd. If α ∈ A then

α = φjβkη, where j, k ∈ Z and cφ(η) = 1. Then αd ∈ A as well. From this and

88

αd = φjdβkdηd we see that βkdηd ∈ A. Therefore cβkdηd(βkdηd) = 1. But c(·, ·)

is nondegenerate, so 1 = βkdηd = (βkη)d. But Γ is torsion-free, so βkη = 1.

Thus α = φj . Finally, d divides j because 1 = cα(β) = cφ(β)j .

We summarize the preceeding discussion in the

Theorem 5.8.5. Let pφ : XΓ −→ E denote the strong Weil uniformization,

let φ ∈ Γ be a simultaneous Hecke eigenelement corresponding to E, let m :=

min(φ, α) > 0|α ∈ Γ, and let d ∈ Z≥1 be as in corollary 5.7.4. Then

deg(pφ) = d·(φ,φ)m = d · [H : Zφ⊕ φ⊥].

This is analagous to the corresponding formula over C. Let E be an elliptic curve

over C of conductor n, and consider the modular curve X := XΓ0(n) arising from

the congruence subgroup Γ0(n). It has been shown by Wiles, Breuil, Conrad,

Diamond, and Taylor that there exists a morphism of curves

φ : XΓ0(n) −→ E.

Take a minimal such φ - that is, suppose φ does not factor as a map of smaller

degree composed with an isogeny. Let ωE = dx2y+a1x+a3

be a Neron differential

on E, and let ω∗E := φ∗(ωE) be the pullback to X. Then ω∗E is a holomorphic

differential on X, from which it is clear that ω∗E = fE(z)dz where fE is a cusp

form of weight 2 for Γ0(n). By a theorem of Shimura ([52]) one knows that fE

is a simultaneous eigenform for the Hecke algebra; therefore fE = cEfE , where

fE is a normalized new form of weight 2 and where cE is some constant. It is

known that cE ∈ Z.

The Faltings height of E is the quantity

− 12 log

(1

2π

∫E(C)

|ωE ∧ ωE |

),

which is by standard arguments equal to the quantity

89

− 12 log

(c2E

2πdeg(φ)

∫X

|fE |2dz)

.

The quantity∫X

|fE |2dz is denoted by < fE , fE >; it is the Petersson product

in the classical setting. Putting the above information together, we obtain the

formula

deg(φ) =c2E<fE ,fE>∫E(C)

|ωE∧ωE | .

We now return to our function field setting; let pφ : XΓ −→ E denote our

optimal morphism, and let t denote the Tate period of E. The quantity

∫E(F∞)

|ωE ∧ ωE |

can be described as the volume of a fundamental domain for F×∞/tZ, where the

measure of the annulus U := z ∈ F×∞ | q−1 ≤ |z| < 1 is 1. The fundamental

domain for F×∞/tZ is the union of val(t) multiplicative translates of U ; therefore

we see that

∫E(F∞)

|ωE ∧ ωE | = val(t) = #ΦE ,

where ΦE is the component group of the Neron model of E; see [54]. Finally,

Gekeler has shown [15, Corollary 3.19] that at least in certain cases, one has the

equality m = #ΦE . Now the analogy with the classical case is clear.

90

6 Computer Applications

We now turn our attention to explicit computations. Fix once and for all the

objects F = Fq(T ), ∞ = 1T , D a division quaternion algebra over F split at

∞, O a maximal Fq[T ]-order in D, Γ the image of O× in PGL(2, F∞), and E

an elliptic curve over F with conductor cond(D) · ∞ with split multiplicative

reduction at ∞. Recall that the map Jac(XΓ) −→ E is given on the level

of lattices by holomorphic theta functions. More concretely, let α ∈ Γ be a

simultaneous eigenelement associated to E as in section 4.7, and let β ∈ Γ be

such that |µα(β)| < 1 is as large as possible. Denote t := µα(β). Then there

exists a divisor d of q − 1 such that E ' Tate(td). Thus at the very least one

would desire methods for

• computing a basis of simultaneous eigenelements αi ⊂ Γ;

• computing µα(β) to arbitrary precision, or at least computing |µα(β)|;

• finding β ∈ Γ which maximizes |µα(β)| < 1;

• computing the constant d.

The above probems may be broken down further until it becomes evident that

we are concerned with (i) a general procedure for finding coset representatives

ΓαΓ =⊔i

Γαi, and (ii) bounds on the rate of convergence for the theta functions

defined in section 5.3. We will address (i) by considering fundamental domains

in T for certain group actions. As for (ii), we will use the Goldman-Iwahori

identification of the Bruhat-Tits tree T with the reduction of Ω, together with

a case-by-case analysis of elements of Γ, to determine which elements of Γ one

needs in order to compute µα(β) to any desired precision. Our computations

depend crucially on an algorithm due to Butenuth and Bockle ([2]), which com-

putes for two vertices v1, v2 ∈ X(T ) an element γ ∈ Γ which sends v1 to v2 if

such an element exists, and which returns 0 if such an element does not exist. In

91

fact, their algorithm also computes the quotient graph ΓT . For convenience

we will refer to this algorithm as Algorithm I. We use many facts about groups

acting on trees; the book [49] by Serre is an excellent reference. We will assume

for simplicity that the only torsion in Γ is F×q ; this assumption may be dropped

at the expense of lengthier arguments and looser bounds on convergence.

6.1 Explicit Hecke Operators

In order to describe elements of the Hecke algebra explicitly in terms of coset

representatives, it will be convenient to fix a Z-basis of Γ. This may be ac-

complished as follows: let T ⊂ T be a fundamental domain (which we assume

contains the standard edge e0) for the action of Γ; then T is a finite subgraph

of T and ΓT is obtained by identifying some vertices on the boundary of T .

Then Γ is generated by those elements γ which send some boundary vertex v1

to some other boundary vertex v2. We may compute these elements by applying

algorithm I; therefore fix once and for all a generating set Γ =< γ1, ..., γg >.

Note that the Γ-translates of the fundamental domain T induce a tiling⋃γ∈Γ

γ ·T

of the Bruhat-Tits tree T . In fact, if γi is in our generating set, then γi · T and

T intersect in a unique vertex, and conversely (by construction) if T ∩ γT 6= ∅

then γ = γi for some i. More generally, if diam denotes the diameter of T , then

all vertices of distance ≤ n ·diam from a vertex in T may be obtained by acting

on an element of T by a word of length ≤ n in < γ1, ..., γg >. One may prove

this (for example) by induction.

Now let α ∈ O, and assume that Nrd(α) is a prime in Fq[T ] of degree d (and

coprime to disc(D)). We are interested in computing a set of coset representa-

tives

ΓαΓ =⊔

Γαβi.

92

By the discussion preceeding Lemma 5.6.1, this is the same as finding coset

representatives for

∆+Γ, ∆+ = Γ ∩ α−1Γα.

It is well-known that when Nrd(α) is prime, the set ∆+Γ is in bijection

with the projective line P1(Fqd). This in turn implies that the natural map

∆+T −→ ΓT is a (qd+ 1)-fold cover. Let T∆ denote a fundamental domain

in T for the action of ∆+ which contains T . If ζj is a collection of elements

of Γ with the property that⊔ζjT covers T∆, then we claim that ζj contains

a set of representatives for ∆+Γ. Indeed, if γ ∈ Γ is any element, then γ · e0

is ∆+-equivalent to some edge e1 ∈ Y (T∆); say δγ · e0 = e1, where δ ∈ ∆+. By

the construction of our set ζi and the freeness of the action of Γ on T , we see

that δγ = ζj for some unique j. This proves the claim.

Since the map ∆T −→ ΓT is a (qd + 1)-fold cover, we see that it takes

qd + 1 translates of T to cover T∆. Therefore we may take ζj to be the set of

words in γ1, ..., γg of length less than or equal to qd + 1. In general one expects

that significantly fewer words will need to be considered: if the fundamental

domain for T∆ is “ball-like” then it suffices to consider words of length ≤ d.

If T∆ is “line-like” then one actually will need to consider one word of each

intermediate length between 1 and qd + 1. It would be an interesting problem

to investigate the general structure of fundamental domains of the form T∆. In

any case, we have proved the

Theorem 6.1.1. Let γ1, ..., γg be generators of Γ as defined above, and let α ∈ O

be such that Nrd(α) is an irreducible polynomial of degree d. Then the set ζj

of words of length ≤ qd + 1 in γ1, ..., γg contains a set of representatives for the

coset space ∆+Γ.

From such a set S of words, it is an easy matter to pick a subset containing

exactly one representative for each coset. Indeed, one may verify immediately

93

that ∆+ζ1 = ∆+ζ2 if, and only if, ζ1ζ−12 ∈ Γ ∩ α−1Γα. This condition is very

quickly checked on a computer; for example, if one writes ζ1ζ−12 ∈ D× in terms

of a fixed Fq[T ]-basis for O, then one only needs to check if the coefficients

involved belong to Fq[T ]. Therefore, for α prime of degree d fix a set of coset

representatives ζj for ∆+Γ.

6.2 Computing Eigenforms

For a fixed α ∈ O and set of representatives ζj as in section 6.1, we wish

to find eigen-elements α ∈ Γ for the Hecke operator Tα. Doing so directly in-

volves writing the class of Tα(γ) ∈ Γ as a sum of basis elements γ1, ..., γg, which

seems to be a computationally expensive task. Therefore, we instead consider

the equivalent problem of finding an eigen-cochain for Tα, which turns out to

be a relatively quick computation. We note here that we only need to compute

Tα(γi) for a basis γi of Γ, since Tα acts linearly.

Recall that to γ ∈ Γ we may associate a harmonic cochain φγ by considering

the path in T from any given vertex v0 to γv0, and then projecting this path

onto ΓT (see section 5.2). Let φi := φgi for i = 1, ..., g. Then by theorem

5.2.3, the φi form a basis for Har(ΓT ,Z). One straightforwardly computes

Tα(φi) via the formula given in section 5.6; therefore we have integers ci,j ∈ Z

such that Tα(φγi) =∑j

ci,jφγj . Thus the computation of Hecke eigen-cochains

is reduced to a straightforward linear algebra problem, which may be solved

quickly. Given a Hecke eigen-cochain φ =∑j

cjφgj , denote by ρφ the element of

Γ given by ρφ =∏j

gcjj . Then the class of ρφ in Γ will be a Hecke eigen-element

of Γ with respect to the operator Tα.

In this manner we may obtain a basis ρ1, ..., ρg for Γ consisting of eigen-elements

for Tα. If the eigenvalues are all distinct, then this basis is actually a simulta-

94

neous eigenbasis for the entire Hecke algebra. However, if some basis elements

share an eigenvalue, one needs to consider other Hecke operators Tα′ (where α′

is also chosen to have prime norm coprime to disc(D)) in order to determine

a basis of simultaneous eigenforms. From now on, we therefore assume that

we have found a simultaneous eigenbasis B := α1, ..., αg of Γ for the Hecke

algebra.

6.3 Computing the Tate Period

We now fix an element α ∈ B - i.e. we fix a simultaneous eigen-element of Γ for

the Hecke algebra. Given any other element β ∈ Γ, one may compute (α, β) via

the formula (α, β) =∑

e∈Y (ΓT )

φα(e)φβ(e). By theorem 5.3.13, we are interested

in finding β ∈ Γ which minimizes the quantity (α, β) > 0. Let ni := (α, gi),

where the gi are the basis from section 6.1; thus the ni are rational integers. Let

N denote the gcd of the set ni, with realization N =∑i

aini. Then β :=∏i

gaii

is easily seen to have the property that (α, β) = N ; moreover, it is easy to verify

that for any β′ ∈ Γ, one has that N |(α, β′). We have therefore proved the

Proposition 6.3.1. Let α ∈ Γ be an eigen-element for the Hecke algebra cor-

responding to an elliptic curve E. Then there exists β ∈ Γ such that (α, β) =

gcd((α, gi)). One has that (α, β) is minimal positive among all (α, β′), β′ ∈ Γ.

In other words, E ×K∞ ' Tate(µα(β)d) for some divisor d of q − 1.

We remark in passing that each algorithm mentioned above is quick when

carried out on a computer; the most time-consuming step is Algorithm I, which

runs in time O(a2), where a is the distance between the two tree vertices in

question. We also note that proposition 6.3.1 is already enough to determine

v∞(j(E)) up to a divisor of q− 1; see corollary 5.7.4 and fact 2.5.1(a). We now

turn our attention to computing µα(β) to a prescribed precision, which allows

95

one in turn to compute j(E) to a prescribed precision.

By definition, µα(β) =∏h∈Γ

h·z0−z0h·z0−αz0

hβ−1z0−αz0hβ−1z0−z0 , where z0 ∈ Ω is some fixed

element. For computational convenience we take z0 =√T . For the sake of

clarity in what follows, we denote a := z0, b := αz0, c := β−1z0, and we denote

the “h-factor” of µα(β) by

Ph := h·a−ah·a−b

h·c−bh·c−a .

We are interested in knowing for which h the factor Ph is close to 1. A quick

calculation shows that

1− Ph = (a−b)(h·c−h·a)(h·a−b)(h·c−a) .

We make some imprecise motivating remarks before going on to study what

happens in general.

• If h ·c and h ·a are both small, then (h ·c−h ·a) is also small, but (h ·a−b)

and (h · c− a) are larger. In this case, 1− Ph is small.

• If h · c and h · a are both large, then (h · c − h · a) is possibly large but

bounded by the larger of the two, whereas (h · a − b) and (h · c − a) are

both large (and at least one of these terms dominates (h · c − h · a)). In

this case, we therefore also see that 1− Ph is small.

We make the following ad-hoc definition, which depends on our choices of a and

b.

Definition 6.3.2. An element x ∈ C∞ is small if |x| < min(|a|, |b|), medium−

sized if min(|a|, |b|) ≤ |x| ≤ max(|a|, |b|), and large if |x| > max(|a|, |b|).

We therefore have 9 cases, depending on the sizes of h · a and h · c. At least

on the surface there appears to be a symmetry which reduces the number of

cases to 5; however, some care must be taken because the formula for 1 − Ph

96

is not symmetric with respect to h · a and h · c. We will restrict ourselves to

finding all h ∈ Γ such that |1−Ph| ≥ 1. The reasoning for this is twofold. First,

this restriction will simplify our exposition, and it will be clear how to extend

the ideas presented here to find all h ∈ Γ such that |1 − Ph| ≥ 1qm for any m.

Second, we will only need to know the first nonzero term of µα(β) in order to

compute the constant d of corollary 5.7.4. We now revisit the two cases already

mentioned before moving on to the other cases.

Case 1: If h is such that |h · c| < min(|a|, |b|) and |h · a| < min(|a|, |b|), then

|h · c− h · a| < min(|a|, |b|) and we have∣∣∣ (a−b)(h·c−h·a)(h·a−b)(h·c−a)

∣∣∣ < max(|a|,|b|) min(|a|,|b|)|b||a| = 1

In this case, we therefore always have |1 − Ph| < 1. Note that if h · a and h · c

are both made smaller, then this entire quantity will become smaller as well,

because the terms (a− b), (h · a− b), (h · c− a) will not be affected in absolute

value, whereas (h · c− h · a) will be.

Case 2: If h is such that |h · c| > max(|a|, |b|) and |h · a| > max(|a|, |b|), then

each denominator term (h ·a− b), (h · c−a) is larger than (a− b). Furthermore,

h · c− h · a) is no larger than max(|h · a− b|, |h · c− a|). So |1− Ph| < 1 here as

well. One may show that if h is chosen so that |h · c| and |h · a| are larger, then

|1− Ph| is made accordingly smaller.

For the remaining 7 cases, let d := dist(c, a) denote the distance between c

and a in the tree T . By this we mean the minimum length of a subgraph of

T containing the images of both a and c in T via the canonical reduction map

R : Ω→ T . Then clearly for any h ∈ Γ, d is also the distance between h · c and

h · a in T . We briefly recall some relevant details on the reduction map Ω→ T .

First recall the notation D(n,x) = πnD0 + x, where

97

D0 = z ∈ C∞| 1q ≤ |z| ≤ 1, |z − c| ≥ 1, |z − cπ| ≥ 1q ∀c ∈ F

×q .

We have seen in chapter 3 that the D(n,x) correspond to the edges of T and the

pairs D(n1,x1), D(n2,x2) which intersect nontrivially correspond to the vertices

of T . Informally, as one walks along the edges of the tree T , the corresponding

sequence of affinoids looks like a combination of the following sequences:

• D(n,x) · · ·D(n+1,x) · · ·D(n+2,x) · · · · · ·D(n+k,x).

As we increase n, the quantity |z| for z ∈ D(n,x) stays at a constant

|x|. In other words, increasing n amounts to “zooming in” on a particular

region of Ω, which implies that |z| must remain constant in this process

since z must remain close to x.

• D(n,x) · · ·D(n−1,x) · · · · · ·D(n−k,x).

Decreasing n corresponds to “zooming out”. As n is decreased, eventually

one will reach a point where D(ni,x) = D(ni,0).

• D(n,0) · · ·D(n+1,0) · · · · · ·D(n+k,0).

Moving along the “standard line” in T , e.g. the line determined by the

ends 0 and ∞ ∈ P1(K∞).

In general, if one begins at an affinoid Di = D(ni,xi) and “zooms out” until the

xi is absorbed, then one arrives at some D(n′1,0). Given another affinoid D(n2,x2),

one similarly constructs a path to some D(n′2,0). Combining these paths with

the unique path from D(n′1,0) to D(n′2,0) gives a path from Di1 to Di2 in T

which becomes a geodesic once any backtracking has been deleted. From this

discussion we single out the important

98

Observation 6.3.3. |z| remains constant along any edge of T which does not

intersect the standard line⋃n∈Z

R(D(n0)).

Let P(Di1 , Di2) denote the unique smallest geodesic in T which contains

the edges corresponding to Di1 and Di2 , oriented so that it begins at a vertex

incident to R(Di1). For now assume that |h · c| > |h · a| with |h · c| = |π|k1 and

|h · a| = |π|k2 , so that k1 < k2. Then the path in T connecting h · c and h · a

looks like

P (D(n1,x1), D(k1,x1)) ∪ P (D(k1,0), D(k2,0)) ∪ P (D(k2,x2), D(n2,x2))

where we understand that D(k1,x1) = D(k1,0) and D(k2,x2) = D(k2,0).

Definition 6.3.4. Let D(n1,x1) and D(n2,x2) be two affinoids with corresponding

graph edges e1, e2 respectively. Let P be the unique minimal geodesic in T

containing e1 and e2. Let S denote the standard line in T . If P ∩ S 6= ∅, then

we call ∂(P ∩ S) the set of pivot points for D(n1,x1) and D(n2,x2).

Lemma 6.3.5. Fix integers k1, k2 with k2 > k1. Then there exist only finitely

many h ∈ Γ such that |h · c| = |π|k1 and |h · a| = |π|k2 . In fact, there are no

more than (q + 1)qd−k2+k1−1 such h ∈ Γ.

Proof. If |h · c| = |π|k1 and |h · a| = |π|k2 , then D(k1,0) and D(k2,0) must both

be pivot points for the path connecting h · c and h · a. Let r := k2 − k1, and let

n1, n2 be as above. Then we see that n1 − k1 can be (and must be) anything

between 0 and d − r. So, h · c must be of distance ≤ d − r from D(k1,0). Only

finitely many h satisfy this condition, and since Γ acts freely on T a simple edge

count gives the second part of the lemma.

It is now possible to examine the remaining cases. It turns out that the

arguments are all very similar, so we will simply cover one case to provide a

prototype and then state the general result.

99

Case 3: If h ∈ Γ is such that h·c is small and h·a is large, then the pivots for the

path from h·a to h·c must have distance at least s+2 from each other, where s =

max(v(a), v(b))−min(v(a), v(b)). Therefore Dist(h ·c,D(min(v(a),v(b)),0)) ≤ d−s.

This implies that Dist(h ·D(0,0)) ≤ d−s+min(v(a), v(b)) = d+max(v(a), v(b)).

Proposition 6.3.6. The only h ∈ Γ for which |Ph − 1| ≥ 1 are such that

Dist(h · c,D(0,0)) ≤ d+ max(|v(a)|, |v(b)|).

Finally, let w = Dist(c,D(0,0)). Then the above h have the property that

Dist(D(0,0), h ·D(0,0)) ≤ d+w+ max(|v(a)|, |v(b)|) =: d′. This set can be found

in exponential time in d′. Since µα(β) is linear in α and β, we only need to do

this computation for α, β in our fixed basis for Γ. Considering only such β, we

may bound the quantity d′ because in this case |v(b)| ≤ 2 · diam(ΓT ), and

similarly Dist(c, a) ≤ 2 · diam(ΓT ) and w ≤ 2 · diam(ΓT ). We thus see that

d′ ≤ 6 · diam(ΓT ). This bound may be combined with [2, Proposition 9.4] to

obtain the following general result.

Theorem 6.3.7. Let α, β ∈ Γ be generators of Γ as in Theorem 6.1.1. Then

the polynomial part of µα(β) is given by the product

∏h∈S0

h·z0−z0h·z0−αz0

hβ−1z0−αz0hβ−1z0−z0 ,

where S0 = h ∈ Γ : dist(h · e0, e0) ≤ N, where N = 12 · (deg(disc(D)) +

2 logq(2)+1− logq(q−1)). Taking into account the arguments from section 6.1,

S0 is the collection of words in γ1, ..., γg of length ≤ N .

6.4 Computing Torsion

Our final task is to compute the order d of the torsion subgroup of µα(Γ). First

write µα(Γ) = tZ ⊕ Z/dZ, where t = µα(β) is such that |t| < 1 is as large

as possible. For each torsion element µα(γ) ∈ µα(Γ), one sees immediately

that t′ = µα(γβ) also has the property that |t′| < 1 is minimal. Conversely, if

t1 = µα(β1) is such that |t1| > 1 is minimal, then ββ−11 is sent via µα to a root

100

of unity in K×∞. Note that for such γ = ββ−11 , one only needs to compute µα(γ)

to precision 1q to fully determine µα(γ). In other words, µα(γ) ∈ F×q , so that

µα(γ) equals the first nonzero term in its Laurent expansion.

Recall that we have fixed a basis γ1, ..., γg for Γ. Write v(µα(γi)) =: ni.

Then a product γ =∏γaii maps to a torsion element under µα if, and only if,∑

aini = 0. Let A ⊂ Zg denote the plane

A = (x1, ..., xg) ∈ Zg | n1x1 + · · ·+ ngxg = 0.

Then the torsion subgroup of µα(Γ) is clearly given by

µα(∏γxii ) | (x1, ..., xg) ∈ A.

Furthermore, since A is a submodule of Zg, we have that A is finitely generated.

In fact, a set of generators can be effectively computer given the initial tuple

(n1, ..., ng). This is easy to see: first, one considers the set S1 ⊂ A given by

S1 = (nj , 0, ..., 0,−n1, ..., 0) | j = 2, ..., g,

where the nonzero term −n1 occurs in the jth coordinate. Let ZS1 ⊂ A denote

the Z-span of this set. Then [A : ZS1] is finite, and every element of A is

equivalent modulo ZS1 to a tuple (c1, ..., cg) with all |ci| < |ni| for i = 2, ..., g

(and |c1| <g∑i=2

|ni|). Thus one only has to search the finitely many such tuples to

find a full generating set S for A. The torsion subgroup of µα(Γ) is immediately

seen to be the subgroup of F×q generated by µα(γ) | γ ∈ S.

101

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107

Vita

The author was born on March 25, 1986 in Miami, Florida. In 2004 he received

his high school diploma from Miami Palmetto Senior High in Miami, Florida.

In June 2008, the author graduated with highest honors from the University

of Florida with a B.S. in mathematics. The author then pursued his graduate

studies in number theory at the Pennsylvania State University. He completed

his thesis under the advisement of Dr. Mihran Papikian in the Spring semester

of 2013 and was awarded with his PhD in Mathematics in August 2013.