the analysis of automorphic forms

8/3/2019 The Analysis of Automorphic Forms

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The Analysis of Automorphic Forms

Sean P Gomes

An essay submitted in partial fulfillment ofthe requirements for the degree of

B.Sc. (Honours)

Pure MathematicsUniversity of Sydney

November 2011


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CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Chapter 1. Geometry of the Poincar Half-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Length, Angles and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. The action ofPSL2(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3. Classification ofPSL2(R) t r a n s f o r m a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 11.4. H as a homogeneous space and decompositions ofPSL2(R) . . . . . . . . . . . . . 1 2

Chapter 2. Harmonic Analysis on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1. Invariant Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2. Eigenfunctions of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3. Key Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4. The Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5. Invariant Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 3. Fuchsian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2. PSL2(R) as a topological group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3. Fuchsian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4. Fundamental Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5. Kloosterman Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 4. Automorphic Forms and their Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1. Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2. Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3. Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4. The Automorphic Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5. Small Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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Preface

The field of analytic number theory traditionally relied primarily on the methods ofcomplex analysis. The theory of analytic functions of one complex variable yields resultssuch as the celebrated Prime Number Theorem. The developments of abelian harmonicanalysis then took centre stage. In particular Fourier analysis on the Euclidean planeprovided a new platform from which to attack the mysteries of the primes. Prominentresults include the recent GreenTao Theorem on arithmetic progressions in primes. En-ter the hyperbolic plane. With its highly symmetric structure, the theory of functionswhich are invariant under lattice actions is key. In particular, modular forms have longbeen important to the algebraic aspects of number theory, providing for example, a shortproof of Lagranges Theorem of sums of four squares and a considerably longer proof ofFermats Last Theorem.

The study of the spectral theory of automorphic forms has been an exciting offshootfrom the study of modular forms. The field is the study of the spectral decomposition ofautomorphic functions with respect to the Laplacian, and is an exotic blend of functionalanalysis, harmonic analysis, Riemannian geometry and number theory.

Much of the progress in this field came from the 1940s onwards, with a major earlyresult being Selbergs trace formula, an analogue to the Poisson summation formula

from Fourier analysis. The spectral methods of automorphic forms maintain a dominantposition in analytic number theory today.For a more detailed overview of the historic developments of analytic number theory,

one may consult the wonderful book IwaniecKowalski [10], from which this account isdrawn.

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Introduction

In this essay, we shall develop the basic aspects of automorphic function theory, witha strong emphasis on the use of the machinery of harmonic analysis to establish spectraltheoretic results.

In Chapter 1, we establish basic properties of hyperbolic geometry through the useof the Poincare upper half-plane model. Of particular importance are the group oforientation-preserving isometries, the fractional linear transformations.

In Chapter 2, we shall discuss harmonic analysis onH

in terms of the study of theLaplaceBeltrami differential operator. In this context, we introduce the rudimentarytechniques of harmonic analysis on the hyperbolic plane and state the major results.Particularly important is Theorem 2.3.3, which establishes the Fourier series expansionof suitable functions. These Fourier series will play a key role in our subsequent analysis.

In Chapter 3, we study the Fuchsian groups. Fuchsian groups provide a rich arith-metic structure in hyperbolic space which is analogous to the structure of lattices in C.We construct fundamental domains for Fuchsian groups in order to provide some geo-metric intuition. We shall conclude this chapter with a construction of the Kloostermansum, an exponential sum that is a key ingredient in our later results.

In Chapter 4, we begin our foray into the spectral theory of automorphic forms.

We begin by gathering some general spectral theoretic results from functional analysis.We shall use these in conjunction with the machinery of previous chapters to determinethe discrete spectrum of the LaplaceBeltrami operator. To conclude, we shall discussan important conjecture due to Selberg about the spectral gap of the LaplaceBeltramioperator on the space of all automorphic functions.

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Acknowledgements

First and foremost I offer my sincerest gratitude to my supervisor, Dr Anne Thomas,

who has supported me throughout my thesis with her time and knowledge. Without her,

this thesis would not have been possible.

I would like to think all of my lecturers, from both my honours year and the entirety of

my degree. In particular I would like to thank Dr Daniel Daners and Dr James Parkin-

son for playing major roles in cultivating my interest in mathematical analysis.

I would also like to thank my fellow honours students. They have all been a pleasure

to work with, and I wish them all the best in their every endeavour.

Thanks to my family and friends, for their constant stream of love and support this

year and every year.

Special thanks to Rachel, for her love and support (and for keeping me well fed

during the more stressful moments this year!)

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CHAPTER 1

Geometry of the Poincar Half-Plane

In this chapter, we shall first in Section 1.1 define the Poincar Half-Plane and studythe basic aspects of its geometry. The rest of the chapter shall then be devoted to thedefinition and study of an action ofPSL2(R) on the Poincar Half-Plane which preservesits geometric structure. Our treatment is based on that of Katok [8, Chapter 1] andIwaniec [7, Chapter 1].

1.1. Length, Angles and AreaWe begin by establishing the basic geometry of the Poincar Half-Plane model of

the hyperbolic plane. In particular, we make the notions of length, and area precisethrough the definition of a metric and a measure on the Poincar Half-Plane model re-spectively. This is one of the most commonly-used models of the hyperbolic plane, andthe one in which we will be working for the remainder of this essay.

Definition 1.1.1. The Poincar Half-Plane is defined to be the set:

H := {z C : Im(z) > 0}.We can now define a Riemannian metric on the half-plane by making the identifica-

tion of the tangent spaces TzH = C with R2

in the obvious way and using the metrictensor:

gz :=

1/ Im(z)2 0

0 1/ Im(z)2

. (1.1.2)

This makes H a Riemannian manifold. To avoid use of results from differentialgeometry without proof, we instead use this definition of gz to motivate the followingmaterial, arriving at the same geometry with little assumed knowledge.

Definition 1.1.3. A path is a continuous and piecewise continuously differentiable map: [0, 1]H. We say that is a path from (0) to (1).Definition 1.1.4 (Hyperbolic Length). Ifis a path, we define the hyperbolic length of

to be:

h() :=

10

|(t)|Im((t))

dt

By a change of variables, it is easy to see that the length of a path is independent ofa particular choice of parametrisation. For this reason we will often use the term pathto refer to a subset ofH that is the image of a path in the sense of Definition 1.2 rather

1


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2 1. GEOMETRY OF THE POINCAR HAL F-PLANE

than the piecewise continuously differentiable map itself. We will be more explicit insituations where this leads to ambiguity.

Definition 1.1.5 (Hyperbolic Distance). For z, w H

we define:

(z, w) := inf{h() : a path from z to w}.

Proposition 1.1.6. The function : HHR is a metric on H.

Proof. Let z, w H. For any path from z to w, h() has a non-negative integrand,hence (z, w) 0 for all z, w H. Moreover equality is attained if and only if(t) = 0for all t [0, 1], that is if and only if is constant and thus z = w.If we define ()(t) := (1 t) for a given path , we have h() = h() by a changeof variables and moreover, this reversal of orientation provides a bijective correspon-dence between paths from z to w and paths from w to z. Hence for all z, w

Hwe have

(z, w) = (w, z).Now let z,w,v H. For all > 0, from the definition of hyperbolic length as aninfinum, we can find paths 1 and 2 from z to w and w to v respectively such that

h(1) < (z, w) + /2,

and

h(2) < (w, v) + /2.

Since the concatenation of these paths is a path from z to v with length h(1) + h(2)

(again by changing variables), we have:(z, v) h(1) + h(2) < (z, w) + (w, v) + .

Letting 0 establishes the triangle inequality, and completes the proof that is ametric on H.

Of all the paths between two given points in the Poincar Half-Plane, we are mostinterested in those which have minimal hyperbolic length.

Definition 1.1.7. A geodesic in H is a path between two distinct points of H whosehyperbolic length is minimal amongst the set of such paths.

We now make the first step towards finding the geodesics in H.

Lemma 1.1.8. If0 < a < b then (t) = i(a + (b a)t) is the unique path (up to repa-rameterisation) from ia to ib with minimal hyperbolic length. That is, h() = (ia, ib).

Moreover, h() = log(b/a).


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1.2. THE ACTION OF PSL2(R) 3

Proof. Let (t) := x(t) + iy(t) be an arbitrary path from ia to ib, where x(t) and y(t)are real-valued functions. Now from Definition 1.1.4 above:

h() = 1

0x

(t)2 + y(t)2

y(t)

dt

1

0

|y(t)|y(t)

dt

1

0

y(t)y(t)

dt

= log(y(1)) log(y(0))= log(b/a)

=

10

|b a|a + (b a)t dt

= h().Since equality is attained if and only ifx(t) is identically zero and y(t) is non-decreasing,(t) = i(a + (b a)t) is indeed the unique path from ia to ib of minimal hyperboliclength.

The hyperbolic metric from Definition 1.1.5 above motivates the definition of thecorresponding hyperbolic area, as follows. Let m denote the Lebesgue measure on thecomplex plane. Then we can make the following definition:

Definition 1.1.9. For A H, we define the hyperbolic area (A) ofA to be:(A) :=

A

dm

Im(z)2

whenever the above Lebesgue integral exists.

1.2. The action ofPSL2(R)

We now define an action of the group PSL2(R) on H. In doing so, we introduce adistinguished class of transformations ofH, which are exactly the orientation-preservingisometries. The first of the two main results of this section is Theorem 1.2.14, whichestablishes that PSL2(R) acts isometrically on H with respect to the hyperbolic metric.The second main result is Theorem 1.2.26, which finds all geodesics on H. We shallalso, in Definition 1.2.29, extend the action ofPSL2(R) to the compactification ofH.

Definition 1.2.1. An isometry of a metric space (M, d) is a bijective transformationf: MM such that:d(x, y) = d(f(x), f(y)) for all x, y M.

It is easy to see that the set of isometries of a given space form a group when equippedwith the binary operation of composition. We denote this group for the Poincar Half-Plane by Isom(H).


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Definition 1.2.2. Given a group G and a nonempty set X, we define a group action tobe a map : G XX denoted (g, x) g x such that:

1 x = x for all x X; and g (h x) = (gh) x

for allg, h G, x X

.

Henceforth we will often omit the . Now we recall the definition of the SpecialLinear Group of2 2 real matrices:Definition 1.2.3 (Special Linear Group). We define

SL2(R) := {A R22 : det A = 1}.If we factor out the normal subgroup {I} we obtain the Projective Special Linear

Group over R. This is the main group of interest to us in this chapter.

Definition 1.2.4 (Projective Special Linear Group). We define

G := PSL2(R) := SL2(R)/

{I

}.

For the remainder of this chapter, G shall denote PSL2(R) unless otherwise specified.

Proposition 1.2.5. The group SL2(R) acts on H via the map:a bc d

z = az + b

cz + d.

Moreover, the only elements ofSL2(R) that fix every point ofH are the elements I.

Proof. Let =

a bc d

and =

a b

c d

be elements of SL2(R). Now by direct

calculation:

Im( z) = Imaz + bcz + d= Im

(az + b)(cz + d)

|cz + d|2

=ad bc|cz + d|2 Im(z)

= |cz + d|2 Im(z) for all z H. (1.2.6)so z H for all z H. We now verify that the map (, z) z is a group action:

(

z) =

aaz+b

cz+d+ b

caz+b

cz+d + d

=a(az + b) + b(cz + d)c(az + b) + d(cz + d)

=(aa + bc)z + (ab + bd)(ca + dc)z + (cb + dd)

= () z.


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Moreover, we have:

1 z =

1 00 1

z = z + 0

0 + 1= z.

Whence is indeed a group action. Finally suppose fixes every point in H. Thenaz + b = cz2 + dz for all z H.

Since the zero polynomial is the only polynomial that is identically zero on the upperhalf plane, we can conclude that a = d and b = c = 0. Thus from the determinantcondition ad bc = 1 we have:

= Ias required.

A group action with the property that nontrivial group elements do not fix everyelement of the set that they act upon is said to be faithful. To obtain a faithful action on

H, we take the quotient of

SL2(R

)by the normal subgroup

{I} SL2(R

)of elements

which fix every point in H.Proposition 1.2.7. The quotient group G = PSL2(R) = SL2(R)/{I} acts faithfullyon H via the induced map:

a bc d

{I}

z = az + b

cz + d.

It is important to note that the elements ofG = PSL2(R) are cosets ofSL2(R), eachcoset g{I} containing the two elements g and g of SL2(R). By abuse of notationhowever, we will usually denote a coset in G by one of its representatives in SL2(R). Onthe rare occasion that distinction between these two objects is required, we will be more

explicit in our notation for cosets.

Remark 1.2.8. We will frequently identify group elements g G with the correspond-ing transformation z gz.

Before proving that G acts isometrically on H, we discuss three important examplesof transformations ofH.Example 1.2.9. The group element

g1 :=

2 00 1/2

G

acts upon H by dilating points by a factor of4 since for all z H, we have2 00 1/2

z = 2z

1/2= 4z.

This transformation is not a Euclidean isometry, since it scales all Euclidean lengths by

a factor of4. Moreover, g1 has no fixed points in H. The transformation g1 is an exampleof a hyperbolic transformation.


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Example 1.2.10. The group element

g2 :=

1 10 1

G

acts upon H by translating points 1 unit to the right since for all z H, we have1 10 1

z = z + 1

1= z + 1.

This transformation is a Euclidean translation and hence a Euclidean isometry, preserv-

ing Euclidean lengths, angles and areas. Moreover, g2 has no fixed points in H. Thetransformation g2 is an example of a parabolic transformation.

Example 1.2.11. The group element

g3 :=

0 11 0

G

acts upon H by reflecting points through the imaginary axis and inverting their modulus.To see this observe that

0 11 0

z = 1

zand hence

|g3(z)| = 1|z| .Hence g3 has the effect of inverting the modulus of z. Since a reflection in the real axisis given by the map z z , and a 180-rotation about the origin is given by the mapz z, we can conclude thatg3 has the effect of composing a reflection in the real axiswith a 180 rotation about the origin and an inversion of modulus. The element g3 is not

a Euclidean isometry since, for example, |2i i| = 1, whilst|g3(2i) g3(i)| =

1i 12i = 12 .

The transformation g3 has exactly one fixed point in H, atz = i. The transformation g3is an example of an elliptic transformation.

Proposition 1.2.12. For every g G, the transformation: g : z gz is conformal.That is, the action of every g G preserves Euclidean angles.Proof. See Katok [8, p.811].

Remark 1.2.13. It is an exercise in differential geometry to show that Euclidean anglesin tangent spaces ofH are in fact equal to the hyperbolic angles which arise from themetric tensor gz (defined in 1.1.2) and resulting inner product on tangent spaces. Thisis a consequence of the matrix gz being diagonal. Bearing this equality in mind we willtreat the word angles in this essay as being those from standard Euclidean geometry inC.

We now come to one of the main results of this section.


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Theorem 1.2.14. For every g G, the transformation: g : z gz is an isometry ofH.Hence we can regardG as a subgroup ofIsom(H).Proof. From Definition 1.1.5 above, it suffices to show that the hyperbolic length of

paths is invariant under the action of the group G. Now letg :=

a bc d

G

and let be a path in H. From the definition of hyperbolic length and the chain rule, wehave:

h(g ) =1

0

|(g )(t)|Im(g )(t) dt

=

10

|(g )(t)| |(t)|Im(g )(t) dt. (1.2.15)

Now using the quotient rule we obtain the following identity:

g(z) =a(cz + d) c(az + b)

(cz + d)2= (cz + d)2 for all z H. (1.2.16)

Furthermore, from Equation (1.2.6) in the proof of Proposition 1.2.5, we have:

Im(gz) = |cd + d|2 Im(z) for all z H.Hence from Equation (1.2.15), we can conclude that:

h(g ) =1

0

|(t)|Im(z)

dt = h()

as required.

Since G acts on H faithfully, we can regard G as a subgroup ofIsom(H) by identifyingevery group element g with the corresponding map z gz. This completes the proof.

From Theorem 1.2.14 above, we obtain the following corollary.

Corollary 1.2.17. The group G acts by homeomorphisms on H.Remark 1.2.18. It can be shown that the transformations in G, together with the reflec-tion r : z z, generate Isom H. In particular, the transformations in G are preciselythe orientation-preserving isometries ofH. For a proof of this see [8, p.811].

In our construction and study of Ford polygons in Chapter 3, we will need to considerthe set of points in

Hat which a particular

g Gdoes not distort the Euclidean metric.

This motivates the following two definitions.

Definition 1.2.19. Ifg =

a bc d

G, c = 0 and z H, we define:

jg(z) := |cz + d|2to be the deformation ofg at z.


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Note that the set of points z H such that jg(z) = 1 is the Euclidean circle:

{z H : |z (dc1)| = |c|1}.

Definition 1.2.20. We define the isometric circle Cg ofg to be the set of points at whichg has deformation equal to 1.

Remark 1.2.21. As shown in the proof of Theorem 1.2.14, the quantityjg(z) = |cz + d|2 from Definition 1.2.19 is essentially a measure of how badly a giveng G fails to be a Euclidean isometry. In particular, by using (1.2.6), (1.1.2) andTheorem 1.2.14, we can see that g does not distort the Euclidean metric on its isometriccircle Cg.

We are now going to establish an explicit formula for the hyperbolic metric , inCorollary 1.2.24, and characterise geodesics in

H, in Theorem 1.2.26. We first note:

Lemma 1.2.22. Any two points z and w in H either lie on the same vertical line or onsome common Euclidean semicircle centred on the real axis.

Proof. Ifz and w are points in H, then either they both lie on the same vertical Euclideanline, or the Euclidean perpendicular bisector of the Euclidean interval joining z and wmeets the real axis at some point c. In the latter case, the point c is equidistant from zand w, whence both z and w lie on some Euclidean semicircle centred at c R.

We now prove the useful Lemma 1.2.23, which will greatly simplify the computa-tions in Corollary 1.2.24 and Theorem 1.2.26 below.

Lemma 1.2.23. If z, w H , then there exists g G such that gz and gw lie on theimaginary axis ofH.

Proof. Let z, w H. From Lemma 1.2.22 above, z and w either lie on the same verticalline or on some common semicircle centred on the real axis. Ifz and w lie on the samevertical line, then Re(z) = Re(w) = k R. Hence the group element:

g :=

1 k0 1

G = PSL2(R)

satisfies Re(gz) = Re(gw) = 0, so gz and gw lie on the imaginary axis ofH as required.Now suppose z and w both lie on the semicircle centred at c R with radius r R+. Inparticular, suppose that z = c + rei for some (0, ). Now set:

g :=

12r

c+r2r

12r

cr2r


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and note that det(g) = 1, so we may consider g to be in G = PSL2(R). We have:

gz =c + rei (c + r)c + rei (c r)

= ei 1

ei + 1

=ei ei|ei + 1|2

which is purely imaginary. Since our choice ofg did not depend on , the same argumentshows that gw also lies on the imaginary axis, as required.

This yields a convenient expression for hyperbolic distance.

Corollary 1.2.24. Forz, w H, the distance (z, w) is given by:

cosh (z, w) = 1 + |z

w

|2

2Im(z)Im(w) (1.2.25)

Proof. We establish the result by first proving it for z and w on the imaginary axis andthen invoking Lemma 1.2.23.Suppose that z = ir and w = is, where s > r . From Lemma 1.1.8 we have:

cosh (z, w) = cosh log(b/a) =a/b + b/a

2=

a2 + b2

2ab= 1 +

(b a)22ab

as required.

Now ifg =

a bc d

G, then

|gz gw|2Im(gz)Im(gw)

=az+bcz+d aw+bcw+d 2

Im(gz)Im(gw)

=|z w|2

|cz + d|2|cw + d|2 Im(gz)Im(gw)=

|z w|2Im(z)Im(w)

using Equation (1.2.6). Hence the right-hand side of Equation (1.2.25) is invariant underthe action ofG. Since (1.2.25) holds on the imaginary axis, and by Lemma 1.2.23 everypair of points can be translated to the imaginary axis by an isometry, we are done.

With the help of Lemma 1.2.23, we can now completely characterise geodesics in H.Theorem 1.2.26. The geodesics in H are segments of vertical Euclidean lines and seg-ments of Euclidean semicircles orthogonal to the real axis.

Proof. Suppose z and w are in H, and let Sbe the set of all paths from z to w. Ifz andw both lie on the imaginary axis then Lemma 1.1.8 asserts the existence and uniquenessof a S of minimal length, which must be the segment of the imaginary axis joining


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z to w.Now suppose z and w do not both lie on the imaginary axis. Then there either existsan unique semicircle C in H orthogonal to the real axis and passing through z and w,or there exists a unique vertical Euclidean line passing through z and w. From Lemma1.2.23, we know that there exists a g G such that gz and gw lie on the imaginary axis.Now since by Theorem 1.2.14, the map g is an isometry, g provides a length-preservingbijection between the set of paths from z to w and the set of paths from gz to gw. Hencewe can conclude that there exists a unique geodesic S, which is the segment of thesemicircle C or line between z and w. This completes the proof.

Proposition 1.2.27. Elements ofG preserve hyperbolic area.

Proof. This is a consequence of Theorem 1.2.14, which says that elements ofG preservehyperbolic distance. For a complete proof see [8, p.811].

In the remainder of this essay, we will sometimes need to study the behaviour offunctions on the boundary ofH. For this reason we shall now extend H in the followingnatural way.

Definition 1.2.28 (Extended Poincar Half Plane). The extended hyperbolic plane His defined to be the topological closure ofH in the extended complex plane.

H := H R {}.The boundary of the extended hyperbolic plane is H = R {}.

We can extend each of the transformations ofH in G = PSL2(R) to H as follows:

Definition 1.2.29 (Extended PSL2(R)Action). Let g =

a bc d G.

Ifc = 0 then we define gz =

az+bd

for z = for z = .

Ifc = 0 then we define gz =

ac

for z = for z = dcaz+bcz+d for all other z H.

Proposition 1.2.30. The map: (g, z) gz defined in Definition 1.2.29 is a group actionon H.Definition 1.2.31. A hyperbolic polygon is a closed subset ofH bounded by finitelymany hyperbolic geodesic segments.

Theorem 1.2.32 (GaussBonnet). If a hyperbolic triangle A has angles ,, then(A) = ( + + ).Proof. See [8, p.1314].


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1.3. CLASSIFICATION OF PSL2(R)TRANSFORMATIONS 11

1.3. Classification ofPSL2(R)transformations

We now classify PSL2(R)transformations based on the nature of their fixed pointsin

H. To motivate our definitions, we consider the solutions in

Hto the equation:

a bc d

z = z (1.3.1)

where

g :=

a bc d

G is nontrivial and c = 0.

From Definition 1.2.7, Equation (1.3.1) simplifies to the quadratic equation:

cz2 + (d a)z b = 0.Since c is assumed to be nonzero, the number of solutions is determined by the discrim-inant:

(d a)2 + 4bc = a2 + d2 2ad + 4bc = (a + d)2 4 = tr(g)2 4where tr(g) := a + d is the trace of the chosen representative matrix.Moreover, we have the equality g{I} = (g){I} in G, so the quantity Tr(g) :=| tr(g)| is well defined and is a convenient quantity to use in the classification of trans-formations.

Definition 1.3.2. Let g G be a nontrivial transformation. IfTr(g) > 2, the transformation g is hyperbolic. IfTr(g) = 2, the transformation g is parabolic. IfTr(g) < 2, the transformation g is elliptic.

Trace is invariant under conjugacy, so the above classes of transformations are dis-joint and closed under conjugation. Bearing this in mind, we can use Jordan CanonicalForm to classify PSL2(R)transformations up to conjugacy in G, as follows.

Proposition 1.3.3. Every hyperbolic transformation is conjugate to a transformation in

the subgroup:

A :=

a 00 a1

G : a R+

G.

Define

A(a) :=

a 00 a1

A.

Proposition 1.3.4. Every parabolic transformation is conjugate to a transformation in

the subgroup:

N :=

1 t0 1

G : t R

G.

Define

N(t) :=

1 t0 1

N.


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Proposition 1.3.5. Every elliptic transformation is conjugate to a transformation in the

subgroup:

K := cos sin sin cos

G :

R

G.

Define

K() :=

cos sin sin cos

K.

The three transformations from Examples 1.2.9, 1.2.10 and 1.2.11above are in their respective normal forms.

Now an element g G fixes some x H if and only ifh1gh fixes h1x, for allh G. Hence conjugate elements ofG fix the same number of points in H. FromPropositions 1.3.3, 1.3.4 and 1.3.5 above, we can deduce the following general resultsabout the fixed points of elements ofG.

Proposition 1.3.6. Hyperbolic transformations have exactly two distinct fixed points inH. These both lie on the boundary H.Proof. This follows from the fact that A(a) fixes exactly 0 and in H. Proposition 1.3.7. Parabolic transformations have exactly one fixed point in H. This

fixed point lies on the boundary H.Proof. This follows from the fact that N(t) fixes exactly in H. Proposition 1.3.8. Elliptic transformations have exactly one fixed point in H. This fixed

point lies in the interior of

H.

Proof. This follows from the fact that K() fixes exactly i in H.

1.4. H as a homogeneous space and decompositions of PSL2(R)Up to this point we have worked with H solely using Cartesian coordinates. We can

also view H as a homogeneous space for the transformation group G = PSL2(R), andfor various aspects of harmonic analysis on H this is a convenient viewpoint. In thissection, we shall also develop several decompositions ofPSL2(R).

To begin, we show that the action ofG is transitive on H, and compute the stabiliserofi H.

Lemma 1.4.1. The group G acts transitively on H , and the stabiliser ofi H is thesubgroup K from Proposition 1.3.5.Proof. Let z and w be points in H and define r := Im(w)/ Im(z). Then w rz Rand the group element:

g :=

r1/2 r1/2(w rz)

0 r1/2

G


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1.4. H AS A HOMOGENEOUS SPACE AND DECOMPOSITIONS OFPSL2(R) 13

maps z to w. Hence G acts transitively on H.If

g = a bc d G

fixes i H, we have:ai + b = c + di.

By equating coefficients, we can deduce that:

g =

a bb a

for some a, b R. Whence by the determinant condition, a2 + b2 = 1 and we havea = cos() and b = sin() for some R, and thus g K.

As a consequence of Lemma 1.4.1 and the Orbit-Stabiliser Theorem, we can nowidentify H with G/K. That is, we identify the point z H with the coset gK of ele-ments ofG which map i to z.In the remainder of this section, we develop three ways of factorising elements of thetransformation group G. These are the Iwasawa Decomposition, the Bruhat Decomposi-tion and the Cartan Decomposition.The first of these, the Iwasawa Decomposition, provides a connection between the Carte-sian coordinates of points of H and the corresponding cosets in G/K. The Iwasawadecomposition is necesssary for developing Fourier analysis on H.Proposition 1.4.2 (Iwasawa Decomposition). Every nontrivial g G can be written inthe form:

g = N(t0)A(a0)K(0)

for some unique t0R, a0

R+ and0

[0, ).

Proof. Let

g =

a bc d

G = PSL2(R).

Given a R and [0, 2), by matrix multiplication we have:

A(a)K() =

a1 sin() a1 cos

where the denotes an arbitrary entry. By the polar coordinate representation of pointsin R2, we can find unique elements a0 R+ and 0 [0, ) such that the bottom row ofA(a0)K(0) agrees with the bottom row ofg. Now let:

A(a0)K(0) =

a bc d

.

From the determinant condition, c and d cannot both be zero. Suppose c = 0. Then bymatrix multiplication:

N

a a

c

a b

c d

=

a + caa

c

c d

=

a c d

. (1.4.3)


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Since c is nonzero, the determinant equation:

ad cx = 1has exactly one solution x

R. Hence we in fact have:

N

a a

c

a b

c d

=

a bc d

.

Whence g = N(t0)A(a0)K(0), where t0 := (a a)/c. Since c = 0 by assumption,the chosen t0 is the unique real number satisfying (1.4.3). Ifc = 0 and d = 0, we canproceed similarly. As a0 and 0 were already shown to be unique, this completes theproof.

The next decomposition, the Bruhat Decomposition, is related to the study of Kloost-erman sums, which we shall encounter in Section 3.5.

Proposition 1.4.4 (Bruhat Decomposition). Every g

G can be written in one of thetwo forms:

g = N(t1)A(a)N(t2) (1.4.5)

or

g = N(t1)A(a)N(t2) (1.4.6)

where t1, t2 R, a R+ and = K(/2) =

0 11 0

. Here, (1.4.5) factorises all

upper triangularg and(1.4.6) factorises the remaining group elements.

Proof. If g =

p q0 r

, then the determinant condition forces r = p1. Hence by the

computation: 1 t10 1

p 00 p1

1 t20 1

=

p pt2 + p1t10 p1

we can find suitable t1 and t2 to represent any upper-triangular g in the required form(1.4.6). Such a representation is not unique.

Ifg =

p qr s

is notupper-triangular, then the determinant condition forces

q = (ps 1)/r. (1.4.7)Hence by the computation:

1 p/r0 1

r 00 r1

1 s/r0 1

=

p r s

together with (1.4.7), we can conclude that any g that is not upper-triangular can beexpressed in the required form (1.4.6).

The third and final decomposition is the most convenient way of studying the GreenFunction, introduced in Chapter 2.


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1.4. H AS A HOMOGENEOUS SPACE AND DECOMPOSITIONS OFPSL2(R) 15

Proposition 1.4.8 (Cartan Decomposition). Every g G can be written in the form:g = K(0)A(e

r/2)K(0)

for some 0

[0, ), r

0 and0

[0, ). Moreover, ifg /

K, then this decomposition

is unique.

Proof. Ifg = K(0) K, then we have the representation:g = K(0)A(e

0)K(0).

Suppose now that g is not in K. Let g :=

a bc d

G = PSL2(R). Suppose first that

g is nota symmetric matrix. Define:

1 := cot1

a + d

c b

.

Thencos(1)sin(1)

= cot(1) =a + dc b

and by rearrangement:

b cos(1) + d sin(1) = a sin(1) + c cos(1).Hence:

K(1)g =

b cos(1) + d sin(1)a sin(1) + c cos(1)

is symmetric.

Since we have that either g = K(0)g is symmetric or K(1)g is symmetric, the

spectral theorem for symmetric matrices asserts that we can conjugate by orthogonalmatrices in order to obtain a diagonal matrix. Since the orthogonal matrices in PSL2(R)are the elements ofK and the diagonal matrices are the elements ofA, we can concludethat there exist 0, 0 [0, ) and r R such that

K(0)1gK(0)1 = A(er/2).

Using the identity:K(/2)A(a)K(/2) = A(a1)

we can ensure that r > 0. By rearrangement, this yields the desired result.

From the uniqueness of the Cartan Decomposition for g / K, and the fact that pointsof

Hcorrespond bijectively to cosets gK

G/K, we obtain an alternate coordinate

system for H, called geodesic coordinates.Definition 1.4.9 (Geodesic Polar Coordinates). Ifz H is such that z = i for somenontrivial PSL2(R), and has Cartan Decomposition:

= K(0)A(er/2)K(0),

we say that z has geodesic polar coordinates (r, 0).


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The geodesic polar coordinates defined above will be convenient when analysing theGreen Function in Section 2.4 below.

Proposition 1.4.10. Ifz H has geodesic polar coordinates (r, ), then (z, i) = r, andthe angle made between the geodesic joining z andi with the segment of the imaginaryaxis between 0 andi is .

The coordinate system introduced in Proposition 1.4.10 above is analogous to thepolar coordinates on R2.

Having established the basic properties of the Poincar Half-Plane and the action ofPSL2(R) we now turn our attention towards harmonic analysis on H.


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CHAPTER 2

Harmonic Analysis on H

In this chapter we outline harmonic analysis on H, which roughly mirrors the theoryof classical Fourier series and the Fourier transform on RN. In this setting, trigono-metric polynomials are replaced by the Whittaker functions as the functions of centralimportance. We shall first in Section 2.1 define invariant linear operators and introducethe important LaplaceBeltrami operator. We shall then find expressions for the eigen-functions of the LaplaceBeltrami operator, and state without proof the major results ofharmonic analysis on

H. Notably we will state analogues to the Fourier Inversion Theo-

rem and the Convergence Theorem for the classical Fourier series of periodic functionsin Section 2.3. The theory of Fourier series in particular will be an indispensable toolin our study of automorphic functions in Chapter 4. We will then conclude the chapterwith a study of the Green Function on H in Section 2.4. The results from this sectionwill be important for our study of the automorphic Green Function in Section 4.4. Ourtreatment will be similar to that in [6] and [7].

2.1. Invariant Operators

It will be important later to consider PSL2(R)invariant functions, so in this sectionwe develop the related notion of an invariant operator on the space of functions f:

HC.Let G = PSL2(R) act isometrically on H, as in Chapter 1.Definition 2.1.1. For all functions f: HC, define the operator Tg as follows:

(Tgf)(z) = f(gz) for all g G, z H.Definition 2.1.2. We define an invariant linear operator T to be a linear operator onthe space of all functions f: H C, that commutes with Tg for every g G. That isfor all functions f: HC, and for all g G, we have:

(T(Tgf))(z) = (T f)(gz)

for all z H

.

Important for the spectral theory of linear operators is the resolvent.

Definition 2.1.3. We define the resolvent Rs of a linear operator T to be the functionRs() = (I T)1 = (s(1 s)I T)1 for C, with = s(1 s) and s C.

The analytic structure of the resolvent encodes the spectral theory of the correspond-ing operator. We will encounter the resolvent again in Chapter 4.

17


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18 2. HARMONIC ANALYSIS O NH

In practice we shall often restrict our attention to subspaces of the space of complex-valued functions on H by imposing conditions on smoothness and growth. We shall nowdefine the analogue of the Laplacian =

2

x2+

2

y2on R2.

Remark 2.1.4. We work in rectangular coordinates here, so by a slight abuse of notation,for functions f : H C we define the following:

f

x:=

f

x

where f:R2 C is given by:f(x, y) = f(x + iy).

Definition 2.1.5 (LaplaceBeltrami Operator). We define the Laplace-Beltrami opera-tor on H to be:

:= y2

2

x2 + 2

y 2

.

Remark 2.1.6. On every complete Riemannian manifold, aLaplaceBeltrami operator can be defined using the metric tensor. See Buser [4, p.184].

The LaplaceBeltrami operator on H can also be written in terms of partial complexderivatives:

= (z z)2 z

z. (2.1.7)

Proposition 2.1.8. The Laplace-Beltrami operator is an invariant differential operator.

Proof. Let g =

a bc d

PSL2(R). Observe first that the map on H given by z gz

is a rational function with no poles in H, and is hence holomorphic. If we decomposethe map g into its real and imaginary components as g(z) = u(z) + iv(z), we have bythe CauchyRiemann equations:

u

x=

v

yand

u

y= v

x. (2.1.9)

In particular, this implies that:

2u

x2+

2u

y2=

2v

x2+

2v

y2= 0 (2.1.10)

by the equality of second-order mixed partial derivatives. Now let f: HC be a func-tion that has second-order partial derivatives and continuous first-order partial deriva-tives. We use the chain rule together with Equation (2.1.9) above to obtain an expressionfor (f g). First we have by the chain rule

f

x=

f

u

u

x+

f

v

v

x.


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2.1. INVARIANT OPERATORS 19

Thus2f

x2=

2f

u2u

x+

2f

uv

u

x+

2f

vu

u

x+

2f

v 2v

x

v

x+

f

u

2u

x2+

f

v

2v

x2.

and similarly2f

y2=

2f

u2u

y+

2f

uv

u

y+

2f

vu

u

y+

2f

v2v

y

v

y+

f

u

2u

y 2+

f

v

2v

y 2.

Since f(u + iv) has continuous first-order partial derivatives, its mixed second-orderpartial derivatives are equal. Hence by using Equations (2.1.9) and (2.1.10) to cancelterms we have:

2f

x2+

2f

y2=

2f

u2+

2f

v2

v

x

2+

v

y

2.

Computing the partial derivatives ofv directly from Equation (1.2.6) yields:

vx

=2cy(cx + d)

|cz + d|4 andvy

=|cz + d|2 2c2y2

|cz + d|4 .Hence:

v

x

2+

v

y

2=

(|cz + d|4 + 4c2y2((cx + d)2 + c2y2 |cz + d|2))|cz + d|8

= |cz + d|4.Hence, again using Equation (1.2.6) and the definition of the LaplaceBeltrami operator,we can conclude that:

(Tg)(f) = y

2

|cz + d|42f

u2 +

2f

v2

= v2

2f

u2+

2f

v 2

= (Tg)(f)

as required.

Part of the importance of the LaplaceBeltrami operator stems from the followingresults.

Proposition 2.1.11. Every invariant differential operator can be written as a polynomial

in the LaplaceBeltrami operator with constant complex coefficients.

Proof. See [6, p.288].

Corollary 2.1.12. If the function f: HC is an eigenfunction of the LaplaceBeltramioperator, then f is an eigenfunction of every invariant differential operator.

Having established their importance, in the next section we compute several eigen-functions of the LaplaceBeltrami operator.


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2.2. Eigenfunctions of

For now we shall work exclusively in the rectangular coordinatesz = x+iy. Foreshadowing our generalisation of periodic functions inRN to automorphic

functions in H in Chapter 4, we search for eigenfunctions of which possess desirablesymmetries. In particular, we shall restrict our attention to the functions f(x + iy) whichare either constant or of the form f(x + iy) = F(2y)e2ix. This method is calledseparation of variables.

We will make frequent use of the function e2iz in the remainder of this essay, so weintroduce some convenient notation here.

Definition 2.2.1. Define the function e(z) as follows:

e(z) := e2iz .

We also make precise the notion of asymptotic equivalence, which we use to describethe behaviour of functions for large imaginary part.

Definition 2.2.2. We say that two functions f, g : H C are asymptotically equiva-lent if:

limy

|f(x + iy)||g(x + iy)| = 1

Iff is asymptotically equivalent to g, we write: f g.The eigenfunctions f: H C of the LaplaceBeltrami operator by definition are

the non-zero solutions to the following equation:

( + )f = 0 for some C. (2.2.3)Proposition 2.2.4. Every solution f: HC of (2.2.3) with = 1/4, which is constantin x, is a linear combination of:

ys and y1s (2.2.5)

where s(1 s) = . In the special case of = 1/4, f is a linear combination of:y1/2 and y1/2 log(y). (2.2.6)

Proof. This is a CauchyEuler equation of order 2.

In order to state the solutions of Equation (2.2.3) which are of the form f(x + iy) =F(2y)e(x), we will need to define the Bessel functions and state their basic properties.We shall do this now.

Consider the modified Bessels equation:

x2y

x2+ x

y

x (x2 + 2)y = 0 where C. (2.2.7)

This equation has two linearly independent solutions I and K, known as modifiedBessel functions of the first and second kind respectively. A more detailed account ofBessel functions can be found in [2, p.222223]. For our analysis, the following proper-ties of the modified Bessel functions are sufficient.


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2.2. EIGENFUNCTIONS OF 21

Proposition 2.2.8. The general solution to Equation (2.2.7) is given by linear combina-tions of the two functions:

I(x) :=x

2

n=0

(x2 )2n

( + n + 1)n! ,

where is the Gamma Function and

K(x) :=

2sin()(I(x) I(x)).

Moreover, as x , we have the asymptotic equalities:I(x) (2x)1/2ex

and

K(x)

2y

1/2

ex.

Proposition 2.2.9. Every solution f: HC of (2.2.3) which is of the form f(x+iy) =e(x)F(2y) is a linear combination of:

(21y)1/2Ks1/2(y)e(x) and (2y)1/2Is1/2(y)e(x). (2.2.10)

Proof. This result follows from the definition of the Bessel functions and the direct com-putation of partial derivatives.

Since the two linearly independent solutions to (2.2.3) found in Proposition 2.2.9above have very different asymptotic behaviour, we can express solutions to (2.2.3)which are of the form f(x + iy) = e(x)F(2y) and satisfy suitable growth conditionsentirely in terms of the left hand solution in 2.2.10. This motivates the following defini-tion.

Definition 2.2.11. The Whittaker function is given by:

Ws(z) := 2y1/2Ks1/2(2y)e(x)

in the upper half-plane. Define:

Ws(z) := Ws(z)

for the lower half-plane.

We now introduce further growth constraints into our study of harmonic analysis.Definition 2.2.12 (Little-O Notation). Iff, g : HC are functions such that:

limy

|f(x + iy)||g(x + iy)| = 0

for all x R, we write f(z) = o(g(z)).


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Definition 2.2.13 (Big-O Notation). IfM > 0 and f, g : HC are functions such that:|f(x + iy)||g(x + iy)| M

for sufficiently large y > 0 and all x R, we write f(z) = O(g(z)). We call the constantM an implied constant in our usage of such notation.

Remark 2.2.14. Note that f(z) = o(g(z)) is a stronger condition than f(z) = O(g(z)).

From Propositions 2.2.8 and 2.2.9, we have the following result.

Corollary 2.2.15. Every solution f: H C of (2.2.3) which is of the form f(z) =e(x)F(2y) and obeys the growth condition:

f(z) = o(e2y) (2.2.16)

is a constant multiple of the Whittaker function Ws(z), where s(1 s) = .

2.3. Key Theorems

We shall now state the analogues on H of two of the key theorems of classical Fourieranalysis.Consider the space of all smooth functions on H that decay to zero towards its boundary:Definition 2.3.1. Define the set C0 (H) to be the collection of all functions f: HCsuch that:

f has continuous derivatives of all orders. lim|z|

f(z) = 0.

limIm z

0

f(z) = 0.

Theorem 2.3.2 (Analogue to Fourier Inversion Theorem). Letf C0 (H).Letfs(r) :=

H f(z)Ws(rz) dz. Then:

f(z) =1

2i

C

R

Ws(rz)fs(r)s(r) drds

where C is the line in C given by Re(z) = 1/2.

Proof. [7, p.30]

Theorem 2.3.3 (Analogue to Fourier Series Expansion). Letf(z) be an eigenfunction of with eigenvalue = s(1 s) which satisfies:

f(z + 1) = f(z) for all z Hand:

f(z) = o(e2y).

Then we have the following series expansion:

f(z) = f0(y) +

nZ\{0}anWs(nz) (2.3.4)


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2.4. THE GREEN FUNCTION 23

where the zero-th term f0(y) is the linear combination of functions from Proposition2.2.4. This series converges absolutely on H and uniformly on compact subsets ofH.Definition 2.3.5. We say that the series in Equation (2.3.4) is the Fourier expansion for

f.Corollary 2.3.6. In the Fourier expansion forf: HC, we have the following asymp-totic bound on coefficients for all > 0:

an = O(e|n|). (2.3.7)

where the implied constant depends only on our choice of andf.

2.4. The Green Function

In this section we will discuss the important Green Function on H. To do this, weshall need to study the invariant integral operators. We proceed using the coordinate

system introduced in Definition 1.4.9, the geodesic polar coordinates.We first note:

Proposition 2.4.1. The LaplaceBeltrami operator on H is given in geodesic polar co-ordinates by:

=2

r2+

1

tanh(r)

r+

1

(2 sinh r)22

2.

If we replace the coordinate r by the coordinate u := (cosh(r) 1)/2 (see Equation(1.2.25)), we obtain:

= u(u + 1)2

u2+ (2u + 1)

u+

1

16u(u + 1)

2

2. (2.4.2)

We now repeat the analysis of Section 2.2 in geodesic polar coordinates.

Proposition 2.4.3. Every solution to Equation (2.2.3) that is of the form: f(z) = F(u)e2im

in geodesic polar coordinates is a linear combination of the two functions:

Fs(u)e2im and Gs(u)e

2im

where:

Fs(u) = F(s, 1 s; 1, u)and

Gs(u) =(s)2

4(2s)usF(s, s; 2s, u1) =

1

4

10

((1 ))s1( + u)s d

in terms of the hypergeometric functions F defined in [1, p.315]. We call the function Gsthe Green Function.

We now establish an important asymptotic bound for the function Gs.


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Lemma 2.4.4. Suppose s = + it with > 0 andx ranges overR+. Then we have:

Gs(x) = O(x).

Proof. For > 0, the integral formula for Gs converges absolutely and we have:

Gs(u) |us|

4

10

((1 ))s1 d = u

4

10

((1 ))s1 d = O(x).

The most important property of the Green Function is connected to the theory ofinvariant integral operators, which we now introduce.

2.5. Invariant Integral Operators

Using the linearity of the Lebesgue integral, we can define a class of linear operatorson spaces of complex-valued functions ofH.Definition 2.5.1. Given a space of complex-valued functions f: HC, we define theintegral operator L by:

(Lf)(z) :=

H

k(z, w)f(w) d(w)

where k : H H C is such that absolute convergence is ensured. The functionk : HHC is called the kernel of the integral operator L.Definition 2.5.2. The kernel k of an integral operator is said to be point-pair invariantif:

k(gz,gw) = k(z, w) for all g G,z,w H.

Proposition 2.5.3. An integral operator is invariant if and only if its kernel is point-pairinvariant.

In light of Theorem 1.2.14 and Equation 1.2.25, we will usually write point-pairinvariant kernels as k(u(z, w)), in terms of

u(z, w) :=|z w|2

4Im(z)Im(w).

The integral operators will be important in our analysis of the spectral theory ofautomorphic forms in Chapter 4. The main result is that the resolvent Rs of is aninvariant integral operator with the Green function as its kernel. Explicitly:

(Rsf)(z) = H

Gs(u(z, w))f(w) d(w).

The proof of this fact is not easy. A Lie algebraic approach can be found in [7,p.3538].

Our next chapter will provide an introduction to the theory of Fuchsian groups.


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CHAPTER 3

Fuchsian Groups

In this chapter, we study the Fuchsian groups, which as we explain in Section 3.1 areroughly analogous to the lattices in the complex plane.

In Section 3.2 we will collect some relevant results about topological groups, and inparticular about PSL2(R). Most importantly, we will establish a characterisation of dis-crete subgroups ofPSL2(R) in terms of their action on H, in Theorem 3.2.12. In Section3.3 we will define and provide several examples of Fuchsian groups. We will then inSection 3.4 define a fundamental domain for a Fuchsian group, and provide several con-structions thereof. Our final section will be more arithmetic in nature. We will constructa special exponential sum over double-cosets of a Fuchsian group, and establish somebasic bounds thereof. This Kloosterman sum will prove to be a key tool in the spectraltheory that is to ensue in Chapter 4. Our work on Fuchsian groups themselves is looselybased on that of Katok in [8, Chapter. 2], and the section on Kloosterman sums is basedon the treatment of Iwaniec in [7, Chapter. 2].

First, we shall provide some motivation for the Fuchsian groups using our well-established intuition in C.

3.1. Motivation

Having defined a group action of PSL2(R) upon H, we shall now consider the in-duced action of certain subgroups PSL2(R) on H. Of particular interest are thecomplex functions which are invariant under the action of on H.

Recall the doubly periodic functions on C.

Definition 3.1.1. A function f:CC is doubly-periodic with periods 1, 2 C iff(z + mw1 + nw2) = f(z) for all z C,m,n Z.

An equivalent formulation of the Definition 3.1.1 is the statement that f is invariantwith respect to an action ofZ2 on C. Also note that Z2 can be regarded as an additivesubgroup ofC that is in some sense spread out.We will now consider the analogous class of functions on

H.

Definition 3.1.2. Suppose that PSL2(R). We say that f: H C is an automor-phic function with respect to , if

f(z) = f(z) for all .Much more will be said about automorphic functions in Chapter 4.

25


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26 3. FUCHSIAN GROUPS

For now it is natural to ask the following question: For which subgroups ofPSL2(R) is the theory ofautomorphic functions sufficiently interesting?Of course if we take = PSL2(R) itself, then the space of automorphic functionsconsists only of constant functions, since PSL2(R) acts transitively on

H. At the other

extreme, if we take = {I}, the trivial subgroup, then the space of automorphicfunctions is the space of complex-valued functions on H with no restrictions whatso-ever! The subgroups ofPSL2(R) that we shall study for the remainder of this essay arethe Fuchsian groups, whose automorphic functions possess a rich theory. In order todefine the Fuchsian groups, we shall first need to topologise PSL2(R).

3.2. PSL2(R) as a topological group

In this section we prove some results on topological groups that will be important forour analysis of Fuchsian groups.

Definition 3.2.1 (Topological Group). A group G endowed with a topology P

(G)

is said to be a topological group if: The group multiplication map (g, h) gh is continuous with respect to the

product topology on G G. The group inversion map g g1 is continuous with respect to .

As a first example, the additive group (Rn, +) is a topological group with respect tothe standard Euclidean topology. Since we can view SL2(R) as being embedded in R4

via the injective map:

i :

a bc d

(a,b,c,d)

we can topologise SL2(R

) using the subspace topology inherited from the standard Eu-clidean topology on R4. Explicitly, we declare U SL2(R) to be open in SL2(R) if andonly ifi(U) is open in R4.

Having topologised SL2(R) as above, there is a natural way to topologise PSL2(R) =SL2(R)/{I} as a quotient space. Explicitly, we declare U PSL2R to be open if andonly if the union of all cosets in U is an open subset ofSL2(R).

Proposition 3.2.2. Endowed with the topology above, the group PSL2(R) is a topologi-cal group.

Proof. The group SL2(R) is a topological group because multiplication and inversionof matrices are rational functions in the matrix entries and hence continuous with respect

to the chosen topology. The claim follows from the general result that closed subgroupsof topological groups are topological groups, and quotients by closed subgroups aretopological groups.

Henceforth we shall always assume PSL2(R) to be equipped with this topology.

Definition 3.2.3. A subset Y of a topological space (X, ) is said to be discrete ifevery subset ofY is open with respect to the subspace topology on Y. In particular, a


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3.2. PSL2(R) AS A TOPOLOGICAL GROUP 27

subgroup of a topological group is discrete if every subset thereof is open with respect tothe subspace topology.

For example, the subgroup (Zn, +) < (Rn, +) is discrete with respect to the standard

Euclidean topology. The following lemma is immediate from the definition of conver-gence.

Lemma 3.2.4. IfXis a discrete topological space and the sequence (xn) in Xconvergesto x X, then xn = x for sufficiently large n.

The primary reason that we will work with discrete subgroups of PSL2(R) is thatthey are precisely the subgroups ofPSL2(R) that act discontinuously on H, in a sensewe will now make precise.

Definition 3.2.5. A collection of subsets {M}J of a topological space X is said tobe locally finite if for every point x X, there exists a neighbourhood N ofx such thatN M = for only finitely many J.Definition 3.2.6. A group G acting on the topological space X is said to act discontin-uously if every orbit ofG is locally finite in X. In the case that X is a locally compactmetrisable space, this is equivalent to the statement that for every x X and everycompact set K X, the set:

{g G : gx K}is finite.

Proposition 3.2.7. Suppose a group G acts discontinuously on a locally compact topo-logical space X. Then for each x X, the orbit:

Gx = {gx : g G}has no limit points in H.Proof. Suppose that the orbit Gx has a limit point y H. Then by the local compactnessof H, there exists a compact neighbourhood of y containing infinitely many distinctpoints ofGx. Hence there are infinitely many g G such that gx K, contradictingour assumption that G acts discontinuously on H. Proposition 3.2.8. Suppose a group G acts discontinuously on a topological space X.Then for every x X, the stabiliser:

Gx := { G : x = x}is finite.

Proof. Suppose that the stabiliser Gx was not finite for some x H. In particular, thisimplies that there exist infinitely many g G such that gx = x Kan arbitrary compactneighbourhood ofx. This again contradicts our assumption that G acts discontinuouslyon H.


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We now show that the properties defined in Definitions 3.2.3 and 3.2.6 are equivalentin our setting. We shall need three prepatory lemmas in order to prove our main result,Theorem 3.2.12.

Lemma 3.2.9. Suppose z0 H is fixed andK H is compact. Then the set:S := {g PSL2(R) : gz0 K}

is compact.

Proof. Recall that PSL2(R) is topologised as a quotient ofSL2(R). Hence the naturalprojection map : SL2(R)PSL2(R) is continuous. Since the continuous image of acompact set is compact, it suffices to show that the set:

S :=

a bc d

SL2(R) : az0 + b

cz0 + d K

is compact as a subset ofSL2(R). By the HeineBorel theorem in R4

, we can establishthis by showing that S is closed and bounded when regarded as a subset ofR4. Nowsince the fractional linear transformations are continuous in their coefficients and S isthe pre-image of a closed set K, S is certainly closed. Since K is compact in H, it isalso compact as a subset ofC and hence bounded. So there exists a positive constant Msuch that: az0 + bcz0 + d

Mfor all

a bc d

S. Furthermore, by compactness, there exists an element g ofSwhich

minimises Im gz0, so there exists a positive constant L such that:

Im

az0 + b

cz0 + d

L

for all

a bc d

S. Using Equation (1.2.6) and rearrangement, these two bounds give

us:

|cz0 + d|

Im(z0)

L

1/2and

|az0 + b| MIm(z0)

L1/2

.Hence the coefficients a,b,c and d are bounded. This completes the proof.

The proof of the following result is from [13] due to Mariano Surez-Alvarez).

Proposition 3.2.10. Suppose G is a Hausdorff topological group. Then any discretesubgroup H ofG is closed.


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3.2. PSL2(R) AS A TOPOLOGICAL GROUP 29

Proof. For each x H, the set {x} is closed in G. We first prove that the collection ofall elements ofH is locally finite, and then show that this implies H is closed. Supposefor the sake of contradiction that there exists y G such that every neighbourhoodof y contains infinitely many elements of H. By the definition of discreteness, thereexists an open subset U G such that U H = {1}. Now by the continuity of groupmultiplication and inversion in a topological group, we can find an open set V Gsuch that 1 V and V1V U. Hence yV is an open neighbourhood ofy which byassumption must contain two distinct elements h1 = h2 ofH. Now:

h11 h2 = (y1h1)1(y1h2) V1V U.

So h1h2 U H = {1}, which is absurd. Hence no such y exists and the collection ofelements ofH is locally finite in G. Now let z be an element that is not in H. From theproperty of local finiteness, we may choose a neighbourhood N ofz that only containsfinitely many h H. Hence by taking the finite intersection ofN with the complementsin G of each of these elements h

H, we obtain a neighbourhood N of z such that

N H = . Since z is arbitrary, we can conclude that H is closed. Lemma 3.2.11. Suppose that PSL2(R) acts discontinuously on H, andz0 H is

fixed by a nontrivial elementof. Then there exists a neighbourhoodN ofz0 such thatno other point in N is fixed by any nontrivial elements ofPSL2(R).

Proof. Suppose for the sake of contradiction that every neighbourhood of z0 contains apoint other than z0 which is fixed by a nontrivial element of. In particular, this impliesthat there exists a sequence (zn) in H and a sequence (n) in PSL2(R) \ {1} such thatzn z0 and nzn = zn. Now since acts discontinuously, the set: { : z0 B(z0, 3)} is finite, where > 0 is arbitrary, and B(z, r) denotes the hyperbolic ball ofradius r centred at z. Hence for sufficiently large n:

(nz0, z0) > 3.

However, from the triangle inequality and the fact that acts by isometries, we have:

(nz0, z0) (nz0, nzn) + (nzn, z0) = (z0, zn) + (zn, z0) < 2for sufficiently large n. This yields the desired contradiction.

We now have the following theorem due to Poincar.

Theorem 3.2.12. A subgroup ofPSL2(R) is discrete if and only if it acts discontinu-ously on H.Proof. We first show that a discrete subgroup ofPSL2(R) acts discontinuously on

H.

Let K H be compact and let z H. From Proposition 3.2.10, is closed as a subsetofPSL2(R). Moreover, by Lemma 3.2.9, the set {g PSL2(R) : gz K} is compact.Hence the set:

S := {g : gz K} = {g PSL2(R) : gz K} is both discrete and compact. Since S is both discrete and compact, we can concludethat S is finite, whence acts discontinuously on H.


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Conversely, suppose that acts discontinuously on H but is not discrete as a subgroupof PSL2(R). Let z0 H be a point which is not fixed by any nontrivial element ofPSL2(R). (The existence of such points is guaranteed by Lemma 3.2.11.) Since isnot discrete, there exists a sequence of distinct elements (n) in with n

1. By

continuity, this gives us n(s) s and we can conclude the intersection s K is notfinite, for any neighbourhood of s. This contradicts the definition of a discontinuousaction, hence our assumption that was not discrete is absurd. This completes theproof.

We are now ready to define the Fuchsian groups, which will be central to the remain-der of this essay.

3.3. Fuchsian Groups

In this section we define the Fuchsian groups and establish their basic properties.

Definition 3.3.1 (Fuchsian Group). A subgroup PSL2(R) is said to be a Fuchsiangroup if is a discrete subset ofPSL2(R).

Note that Theorem 3.2.12 provides us with the equivalent and useful characterisationof Fuchsian groups as subgroups ofPSL2(R) that act discontinuously on H.

We now provide several examples of Fuchsian groups. Recall the classification ofelements ofPSL2(R) as hyperbolic, parabolic or elliptic in Propositions 1.3.3, 1.3.4 and1.3.5. Correspondingly we can form cyclic Fuchsian groups.

Proposition 3.3.2. We have the following three examples of Fuchsian groups:

Every cyclic subgroup of PSL2(R) consisting only of hyperbolic elements is aFuchsian group.

Every cyclic subgroup of PSL2(R) consisting only of parabolic elements is aFuchsian group.

Every finite cyclic subgroup ofPSL2(R) consisting only of elliptic elements is aFuchsian group.

Proof. Since the map 1 is a homeomorphism from PSL2(R) to itself forfixed PSL2(R), any subgroup ofPSL2(R) that is conjugate to a discrete subgroupof PSL2(R) is also discrete. Using this fact and Propositions 1.3.3,1.3.4, and 1.3.5, itsuffices to prove the assertion for the three cyclic groups:

H :=

n0 00 n0

: n Z, 0 > 0 fixed

,

P :=

1 nt00 1

: n Z, t0 > 0 fixed

,

and


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3.3. FUCHSIAN GROUPS 31

E :=

cos(k/n) sin(k/n)sin(k/n) cos(k/n)

: k Z, n N fixed

.

To show that H is discrete, it suffices to observe that taking the norm on R4,(2n0 +

2n0 )

1/2 blows up for |n| large. Hence only finitely many representatives forelements of H lie in any given ball in R4 about the origin. Similarly, examining thenorm (2 + n2t20)

1/2 shows that P is discrete. Finally, observe that the projection map : SO2(R) K has closed kernel {I}. Hence K is Hausdorff, and any finite sub-set thereof is discrete. This shows that E is discrete and completes the proof of theproposition.

Remark 3.3.3. Note that infinite cyclic groups consisting only of elliptic elements arenot discrete. This is an easy consequence of Proposition 3.2.8 below.

A more important example for us is the modular group.

Proposition 3.3.4 (Modular Group). The modular group:

PSL2(Z) :=

a bc d

{I} PSL2(R) : a,b,c,d Z

is a Fuchsian group.

Proof. The discreteness ofPSL2(Z) in PSL2(R) follows from the discreteness ofZ4 inR4.

Of particular importance to number theory are the congruence subgroups.

Proposition 3.3.5 (Principal Congruence Subgroup). Let N be a positive integer. Thenatural projection homomorphism:

N : SL2(Z)SL2(Z/NZ)induces a surjective homomorphism:

: PSL2(Z)PSL2(Z/NZ).The principal congruence subgroupof level N:

(N) := ker(N)

is a Fuchsian group.

Proof. See [8, p.133-134].

Proposition 3.3.6 (Congruence Subgroup). Every such that

(N) PSL2(Z)is a Fuchsian group, said to be a congruence subgroupof level N.

Proposition 3.3.7. Let P be a hyperbolic triangle with angles /2, /3 and /7). Let be the subgroup ofIsom(H) generated by reflections in the sides of P. The subgroup : = PSL2(R) of orientation-preserving elements of forms a Fuchsian groupknown as the (2, 3, 7)-triangle group.


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We now prove some of the basic results about Fuchsian groups.

Proposition 3.3.8. Suppose that is a Fuchsian group. Then for all z H, the stabiliserz is finite and and cyclic.

Proof. The stabiliser z is finite by Proposition 3.2.8. By Lemma 1.4.1, we can find PSL2(R) such that i = z. Hence we have:

1z = i = K.Now K is certainly a discrete subgroup of K. Since the conjugate sugbroup ofa cyclic subgroup is cyclic, it now suffices to prove that any discrete subgroup H ofK is cyclic. By discreteness, there exists a nontrivial element = K(0) H with 0minimal amongst all elements ofH. Now let = K(1) Hbe an arbitrary element ofH. We have: 1 = m0 + for some unique non-negative integer m and 0 < 0. ButK() = K(1)K(0)

m H. Hence by minimality of0, we are forced to conclude that = 0. This shows that every element ofH lies in the cyclic subgroup ofK generated

by K(0) and completes the proof. Remark 3.3.9. From Proposition 1.2.30, the action of a Fuchsian group extends to H,not necessarily discontinuously.

We have the following result, the proof of which runs along similar lines to Proposi-tion 3.3.8.

Proposition 3.3.10. Suppose that is a Fuchsian group. Then for alls H = R {}, the stabilisers is cyclic.Remark 3.3.11. For s H, the stabiliser s is notnecessarily finite. For example, thestabiliser of the point in the Fuchsian group = PSL2(Z) is the infinite cyclicsubgroup ofPSL2(Z) generated by N(1).

In order to obtain more structure than is possible for arbitrary Fuchsian groups, wenow define a special kind of Fuchsian group. These are the Fuchsian groups that arelarge enough for us to have a reasonable chance of obtaining useful information aboutthe functions which are automorphic with respect to .

Definition 3.3.12. A Fuchsian group is said to be a Fuchsian group of the first kind ifevery point s H is is a limit point of some orbit z, where z H.

An example of a Fuchsian group that is not of the first kind is the cyclic group ofparabolic elements from Proposition 3.3.2.

We now introduce the extremely important notion of a cusp of a Fuchsian group. We

shall provide a geometric motivation for the usage of the term cusp later in Section 3.4.Definition 3.3.13 (Cusp). Ifs H is a fixed point of some parabolic transformation in, we say that the orbit s is a cusp of.

For example, the point s = is fixed by the parabolic transformation N(1) inPSL2(Z) and hence is a representative of a cusp thereof. We will often blur the distinc-tion between cusps and representatives thereof.


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3.4. FUNDAMENTAL DOMAINS 33

Restricting our attention to the Fuchsian groups of the first kind for now, we shallnow further divide Fuchsian groups according to whether or not they possess cusps.

For this, we establish another characterisation of Fuchsian groups with cusps in termsof the topological quotient space

\H.

Proposition 3.3.14. If PSL2(R), then the relation on H given by:z w w = z for some

is an equivalence relation on H. Ifz w, we say thatz andw are equivalentor thatz andw are congruent modulo.

In light of Proposition 3.3.14 above, we may consider the quotient topological space\H.Definition 3.3.15. We say that a Fuchsian group is co-compact if the topologicalspace

\His compact.

Proposition 3.3.16. A Fuchsian group is co-compact if and only if it does not have any

cusps.

Proof. [8, p.8490]

We now briefly refer back to our motivational example of the doubly periodic func-tions on C. The analysis of a doubly periodic function on C is often described using itsperiod parallelogram which tessellates C.

Such a tessellation is possible because for fixed w1, w2 C, the group of translations:{z z + mw1 + nw2 : m, n Z}

acts discontinuously on C. We will now define an analogous construction for Fuchsiangroups, the fundamental domain.

3.4. Fundamental Domains

A convenient way of analysing the action of a particular Fuchsian group on H isthrough the use of a fundamental domain.

Definition 3.4.1 (Fundamental Domain). A fundamental domain for the Fuchsian group is an open connected subset F ofH such that:

For all nontrivial and all z F, z / F. For all z H, the intersection z F is nonempty.

Since every -orbit in H contains a point in F, and the points in the interior ofF arepairwise inequivalent with respect to the relation defined in Proposition 3.3.14, we canview the family of sets:

{(F) : }as a tessellation ofH.


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Example 3.4.2. The cyclic Fuchsian group:

:= {N(m) : m Z} =

1 m0 1

: m Z

has the stripP := {z H : 0 < Re(z) < 1}

as a fundamental domain.

Remark 3.4.3. There certainly exists a fundamental domain for an arbitrary Fuchsiangroup , as we shall construct in Proposition 3.4.6. The fundamental domain for a Fuch-sian group is by no means unique though, even if we identify fundamental domains thatare equivalent modulo . For example we can construct a different fundamental domainfor the Fuchsian group in Example 3.4.2 by simply making a small indent in both sidesof the strip.

We shall now construct a fundamental domain for an arbitrary Fuchsian group intwo different ways.

Definition 3.4.4. Suppose z1, z2 H with z1 = z2. We define the set:{z H : (z, z1) = (z, z2)}

to be the perpendicular bisector of the geodesic segment joining z1 to z2.

Proposition 3.4.5. The perpendicular bisector of a geodesic segment L is itself a geo-desic segment that meets L orthogonally.

Proof. See [8, p.5354].

In particular, if

andp H

, we denote the perpendicular bisector of thegeodesic segment joining p to p by Lp(). The geodesic segment Lp() divides thehyperbolic plane into two open half-planes. We denote the half-plane containing p byHp().

Proposition 3.4.6 (Dirichlet Normal Polygon). Let be a Fuchsian group. If p H isnot fixed by all nontrivial , then the set:

D(p) := {z H : (z, p) < ((z), p) for all nontrivial }is a fundamental domain for bounded by geodesics in H.Proof. The set D(p) can be characterised as:

D(p) =

\{1}Hp()

so is certainly bounded by geodesics in H. Now let z H be arbitrary. Since the orbitz does not accumulate in H by Proposition 3.2.7, there must exist at least one point inH of minimal distance from p. That is, there exists z0 z such that:

(z0, p) (z0, p) for all .


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Hence z0 D(p) and D(p) contains a point from each orbit. It remains to showthat the points in D(p) are noncongruent modulo . Suppose z1, z2 D(p) are distinctbut congruent modulo . Then we have z2 = z1 for some nontrivial . By thedefinition ofD(p), this gives us:

(z1, p) < (z1, p) = (z2, p) < (1z2, p) = (z1, p)

which is absurd. Hence D(p) contains at most one one element from each orbit in Hand is a fundamental domain as required.

The next proposition determines the Dirichlet normal polygon for the modular group.

Proposition 3.4.7 (Normal Polygon for PSL2(Z)). A fundamental domain forPSL2(Z)is the set:

{z H : |z| > 1, | Re(z)| < 1/2}.Proof. First, we show that for k > 1, the point p = ki H has trivial stabiliser inPSL2(Z). Suppose that:

a bc d

ki = ki.

By expanding and equating real and imaginary parts, we obtain a = d and b = ck2.Hence by the determinant condition, we have:

a2 + c2k2 = 1.

Since a, c Z, this forces c = 0 and a = 1, whence ki has no nontrivial stabilisers inPSL2(

Z). Hence from Proposition 3.4.6, we have that the polygon D(ki) for PSL2(

Z)is a fundamental domain.

We now construct this normal polygon. Since we have N(1), N(1) PSL2(Z), thepolygon D(p) is contained in the intersection of half-planes given by:

Hp(N(1)) Hp(N(1)) = {z H : | Re(z)| < 1/2}.Moreover, D(p) is contained in the half-plane Hp(K(/2)) = {z H : |z| > 1}. Hencewe have:

D(p) {z H : |z| > 1, | Re(z)| < 1/2} =: F. (3.4.8)

We now show that the containment in Equation 3.4.8 above is in fact equality. Supposefor the sake of contradiction, that F is nota normal polygon for PSL2(Z). Then we havesome z in the set F\D(p). Now since D(p) is a fundamental domain for PSL2(Z), thereexists a representative z D(p) for some nontrivial PSL2(Z). Let

=

a bc d

.


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Then since z F, we have:|cz + d|2 = c2|z|2 + 2 Re(z)cd + d2

> c2 + d2

|cd

|= (|c| |d|)2 + |cd| 0.

Moreover, equality cannot be obtained unless c = d = 0, which violates the determinantcondition. Since c and d are integral, this gives us |cz + d| > 1, whence:

Im(z) < Im(z). (3.4.9)

However, we can repeat the above calculation using the pair (z,1(z)). The onlydifference in this calculation is that z possibly lies on the boundary F. Hence wehave:

Im(z) = Im(1z) Im(z). (3.4.10)Combining the inequalities from Equations 3.4.9 and 3.4.10 yields the desired contradic-tion. Hence we can conclude that F = D(p) is a fundamental domain for the Fuchsiangroup PSL2(Z).

The fundamental domain constructed above for PSL2(Z) is an extended hyperbolictriangle with vertex at . Notice that this vertex has stabiliser = N, generated bythe parabolic element N(1), and is hence a representative of a cusp.

In order to treat all cusps of a Fuchsian group on an equal footing, we introduce thescaling matrices.

Proposition 3.4.11. Ifs H is a representative for a cusp of with stabiliser generatedby s , there exists a scaling matrixs PSL2(R) such that:

s = s

1s ss = 1 10 1.Scaling matrices are unique up to right-multiplication by elements of the subgroup N PSL2(R).

Proof. Suppose first that s = and s =

1 r0 1

, for some r R+.


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We then have by direct computation that A(

r)1sA(

r) = N(1), so A(

r) is ascaling matrix for s as required.

Ifs = , by the transitivity of the action ofPSL2(R) on H, there exists an elementg

PSL2(R) such that g

= s. Whence from the case s =

, it easily follows that the

matrix gA(a0) is a scaling matrix for s, given suitable choice ofa0. We omit for brevitythe proof of the uniqueness statement.

Using scaling matrices, we will show how to construct a fundamental domain for aFuchsian group with a cusp at . This construction will involve the use of the isometriccircles from Definition 1.2.20. We first prove two preliminary results. The first is alemma about the effect ofPSL2(R)-transformations on isometric circles.

Lemma 3.4.12. Suppose g PSL2(R) and z H lies strictly outside the isometriccircle Cg. Then the pointgz lies strictly inside the isometric circle Cg1 .

Proof. Since z lies outside Cg, we have: jg(z) < 1. Now by Equation (1.2.6) and

Definition 1.2.19, we have:Im(z) = Im(g1gz) = jg1(gz)jg(z)Im(z).

Hence we can conclude that jg1(gz) > 1 and that gz lies strictly inside the isometriccircle Cg1.

We will also need:

Lemma 3.4.13. Suppose is a Fuchsian group with cusp at, whose stability group isgenerated by the parabolic element:

= 1 10 1 .

Then for all z H, the orbitz contains a point with maximal imaginary part.Proof. We first show that the set {Im(gz) : g } is bounded for each z H. Fixz H. Suppose that (gn) is a sequence in with Im(gnz) . Then by the definitionof the hyperbolic metric we have:

(z, g1n gnz) = (gnz, gnz) 0as n . This contradicts Proposition 3.2.7. Hence we can conclude that no suchsequence exists and we have M := sup{Im(gz) : g } < . It remains to showthat M is actually attained. Suppose that Im(gz) < M for every g . Since ,every point in

His equivalent to a point in the closure of the strip P :=

{z

H:

0 < Re(z) < 1}. Hence the supposition that Im(gz) < M for every g impliesthat there are infinitely many distinct point in the orbit z that lie in the compact set{z P : M/2 Im(z) M}. Since Fuchsian groups act discontinuously on H, this isabsurd. This completes the proof.

Our second construction for the fundamental domain, the Ford Polygon, is then asfollows.


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Proposition 3.4.14 (Ford Standard Polygon). Suppose that is a Fuchsian group that isnot co-compact. Leta be a representative of a cusp for . If we set:

F := {z H : 0 < Re(z) < 1, z lies outside every isometric circle Cg, g \ {1}}then the setF := a(F) is a fundamental domain for.Proof. We first show that F is a fundamental domain for 1a a.Suppose z F. By Lemma 3.4.12, for any g 1a a, the point gz will lie insidesome isometric circle Cg1 . Hence gz / F, and we have that the points in F arepairwise inequivalent.It remains to show that every orbit 1

aaz contains a representative in F. By Lemma

3.4.13, every point z H is equivalent to a point z H of maximal imaginary part.Moreover, from the definition of the scaling matrices, we have:

:=

1 10 1

1a

a.

Hence we can assume that 0 Re(z) 1. Now since Im(z) Im(gz ) for allg 1

aa, we have from Equation (1.2.6) and Definition 1.2.19 that jg(z) 1 for all

g 1a a. This implies that z lies on or outside every isometric circle, and is hencean element ofF. Hence F is a fundamental domain for 1a a.Now suppose we have two distinct points z, w F = a(F). Then z = a(z) andw = a(w) for two distinct points z and w in F. Ifz and w are equivalent modulo then w = gz for some nontrivial g . Hence by rearrangement, w = 1

agaz and w

is equivalent to z modulo 1a a. Since z, w F which is a fundamental domain for1a a, this is absurd, whence points in F are pairwise inequivalent modulo . Now letz = a(z) H be arbitrary. Similarly, since F is a fundamental domain for 1a a,there exists g

such that 1a gaz

F

which upon rearrangement yields z

F.

Hence F is a fundamental domain for , and we are done.

The Ford Polygon will prove useful in establishing some bounds of the Kloostermansum in Section 3.5.We shall now prove some key statements about fundamental domains for a Fuchsiangroup. The first establishes a numerical invariant of a Fuchsian group.

Proposition 3.4.15. Let be a Fuchsian group. Then any two fundamental domains for have the same hyperbolic area.

Proof. Suppose F1 and F2 are two fundamental domains for , with (F1) < . Wehave:

F1 F1

(F2)

=

F1 (F2). (3.4.16)

Since F2 is a fundamental domain, the final union in Equation (3.4.16) is disjoint. Now,from Proposition 1.2.27:

(F1)

(F1 (F2)) =

(1(F1) F2) =

((F1) F2).


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Since F1 is a fundamental domain for , we have:

(F1) = H.

Hence:(F1)

((F1) F2) =

((F1) F2) = (F2).

By symmetry, we can conclude that:

(F1) = (F2)

as required.

We can now define the covolume of a Fuchsian group.

Definition 3.4.17. If is a Fuchsian group, the covolume (\H

) of is the hyperbolicarea of any fundamental domain thereof. In particular, if (\H) < , we say that isa finite-volume group.

Most of the Fuchsian groups we have discussed are finite-volume groups, an excep-tion being the cyclic parabolic group from Proposition 3.3.2. For example, we considerthe Dirichlet polygon for PSL2(Z) which we found in 3.4.7. This is a hyperbolic trianglewith angles (0, /3, /3), and hence by the GaussBonnet Theorem has area /3.

There is in fact an intimate connection between finite-volume groups and Fuchsiangroups of the first kind.

Proposition 3.4.18. A finite-volume Fuchsian group is a Fuchsian group of the first kind.

Proof. The proof of this theorem is somewhat technical. Details can be found in [8,p.103].

Theorem 3.4.19 (Siegels Theorem). A Fuchsian group of the first kind which has ahyperbolic n-gon as a fundamental domain for some n N is a finite-volume group.Proof. The proof of of this theorem may be found in [8, p.80-84].

Remark 3.4.20. As a consequence of our constructions of the Dirichlet and Ford poly-gons, the requirement that a polygonal fundamental domain for a Fuchsian group existsis satisfied by all Fuchsian groups. However the requirement that this polygon has onlyfinitely many sides is a nontrivial condition. For more details on this condition see [8,

p.8084].

Since most of the Fuchsian groups listed at the start are simply congruence subgroupsof the modular group, and so are of finite index in PSL2(Z), we will now show how touse the fundamental domain of a Fuchsian group to find fundamental domains of itsfinite-index subgroups. In particular, we have the following proposition, whose proof isfairly straightforward.


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Proposition 3.4.21. Let be a Fuchsian group with a fundamental domain F, and letbe a finite-index subgroup of . If

=n

i=1

i

is a -coset decomposition of, then the set:

F :=ni=1

i(F)

is a fundamental domain for . Moreover, if(F) < and(F) = 0, then (F) =n(F).

We now examine the geometric structure of fundamental domains in some greaterdetail. In the following section let be a fixed Fuchsian group and let F be a fixed

Dirichlet polygon thereof.Definition 3.4.22 (Elliptic Point). Suppose that is a Fuchsian group. For z H, wesay that the orbit z is an elliptic point of if the stabiliser z in is nontrivial.

As with cusps, we shall often refer to representatives of elliptic points as ellipticpoints.

From Definition 3.4.1, it is immediate that any point z F with nontrivial stabiliserz in must lie on the boundary F.

From Theorem 3.4.6, the set F is a hyperbolic polygon, bounded by geodesic seg-ments in H and possibly segments of the real axis.

Definition 3.4.23. Suppose that is a Fuchsian group with a Dirichlet polygon F. Wedefine a vertex of F to be a point s F such that s is either the point of intersec-tion of two bounding geodesics of F, or s is a

the analysis of automorphic forms

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