A GL(3) Kuznetsov Trace Formula and the Distribution of...

A GL(3) Kuznetsov Trace Formula and theDistribution of Fourier Coefficients of Maass Forms

Joao Leitao Guerreiro

Submitted in partial fulfillment of the

requirements for the degree

of Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2016

ABSTRACT

A GL(3) Kuznetsov Trace Formula and theDistribution of Fourier Coefficients of Maass Forms


We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s

law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation

for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula

on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass

forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual

forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass

forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula

for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.

Table of Contents

1 Introduction 1

2 Preliminaries 6

2.1 Maass Forms for SL(2,Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Maass Forms for SL(3,Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Fourier-Whittaker Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Hecke-Maass Forms and L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Eisenstein Series and Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Kloosterman Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Poincare Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Kuznetsov Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Inverse Lebedev-Whittaker Transform 34

3.1 Archimedean Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Nonarchimedean Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.3 Proof of Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.4 Local L-function of a Symmetric Square Lift . . . . . . . . . . . . . . . . . . 46

4 Orthogonality Relation 48

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Bound for the Inverse Lebedev-Whittaker Transform . . . . . . . . . . . . . . . . . . 51

i

4.3 Kloosterman Terms’ Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Eisenstein Terms’ Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Main Geometric Term Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 62

ii

Acknowledgments

I would like to thank my advisor Dorian Goldfeld for his guidance, and for sharing his knowledge

with me during the past years. Our conversations often made my mathematical thoughts much

clearer and his suggestions made most hurdles much easier to tackle. His help and support were

invaluable in producing this thesis.

Next I would like to thank my fellow graduate students. I would especially like to thank Rahul

Krishna, Vivek Pal and Vladislav Petkov for their mathemtical help. In addition, I would like to

thank Rob Castellano, Karsten Gimre, Andrea Heyman, Jordan Keller, Connor Mooney, Thomas

Nyberg, Natasha Potashnik and Andrey Smirnov for making my time at Columbia so much more

fun and enjoyable.

I would also like to thank Terrance Cope for helping with all the nonmathematical parts of

being a graduate student.

Gostaria tambem de agradecer a toda a minha famılia. Aos meus pais Isabel e Luıs pelo seu

apoio constante e por me terem dado todas as oportunidades que me levaram ate aqui. Ao meu

irmao Pedro pelos seus conselhos sensatos e por me aturar ha tanto tempo.

And most of all, I thank my wonderful wife Jennifer, for her love and unconditional support.

iii

A minha famılia

iv

CHAPTER 1. INTRODUCTION

Chapter 1

Introduction

Understanding the spectral theory of the Laplace operator ∆ has interested mathematicians and

physicists for a long time. This interest derives from the fact that the Laplacian spectrum is strongly

connected to the geometry of the underlying space. From an arithmetic perspective this problem is

particularly appealing when the underlying space is a quotient of the hyperbolic upper half plane

H by a discrete subgroup Γ of SL2(R), where Γ acts on H by fractional linear transformations.

When the discrete group Γ is equal to SL2(Z) (or a congruence subgroup), the quotient Γ\H has

finite volume and is noncompact, as it has a finite number of cusps. These properties are reflected

in the spectral decomposition of the Laplace operator, i.e. the spectrum of ∆ has both a discrete

part and a continuous part. The eigenfunctions of ∆ that form the discrete part of the spectrum

are the so-called Maass cusp forms. Moreover, an important feature of the Laplacian spectrum

in this case is that it can be used to decompose the space of square-integrable functions on the

quotient Γ\H.

The first result on the distribution of the discrete eigenvalues of ∆ on Γ\H is due to Selberg

[Selberg, 1956]. He developed the Selberg trace formula which relates the eigenvalues of Maass

cusp forms with the conjugacy classes of Γ (up to some contribution of the continuous spectrum).

Using this formula, Selberg was able to count the number of eigenvalues with a value less than a

given parameter X > 0, in analogy to the results already known for compact Riemannian surfaces.

Since the work of Selberg, trace formulas have been a thoroughly researched subject in number

theory. In general, these formulas relate the spectral decomposition of a space of functions (spectral

side) to information about the structure of an algebraic group (geometric side). The work in this

1


thesis is centered around a trace formula of the type first developed by Kuznetsov. For {φj}

an orthonormal basis for eigenfunctions for the discrete spectrum of ∆, indexed by increasing

eigenvalue λj = 14 + x2

j , Kuznetsov [Kuznetsov, 1980] showed a formula relating, on one hand,

a sum over the spectral information associated to the functions φj , weighted by their Fourier

coefficients aj(n) and a test function h,

∞∑j=1

aj(n)aj(m)

coshπxjh(xj)

and on the other hand, a sum over Kloosterman sums S(n,m; c), weighted by values of Φh, a Bessel

transform of h,

∞∑c=1

1

cS(n,m; c)Φh

(4π√nm

c

).

Kuznetsov’s formula is a powerful tool in analytic number theory with many applications, for

example to the Linnik’s conjecture [Deshouillers and Iwaniec, 198283] and to problems on the

distribution of quadratic roots mod p [Duke et al., 1995].

Versions of the Kuznetsov trace formula have been proved for SL3(Z)\ (SL3(R)/SO3(Z)) by

Blomer [Blomer, 2013], Buttcane [Buttcane, 2012], Goldfeld and Kontorovich [Goldfeld and Kon-

torovich, 2013] and Li [Li, 2010] and have been applied to a wide range of topics: moments of

L-functions, exceptional eigenvalues, and statistics of low-lying zeros of L-functions. As in the

GL(2) Kuznetsov trace formula, the GL(3) formulas give equalities between spectral information

about Maass forms on one side and Kloosterman sums on the other side.

In this thesis, we prove two main results; one is an application of a GL(3) Kuznetsov trace

formula and the other one deals with the problem of inverting the Lebedev-Whittaker transform,

which plays an important role in the study of this type of trace formula.

The first of those results establishes a Weyl’s law type theorem for the spectral parameters of a

“thin” family F of Maass forms, whose spectral parameters are approximately like those of self-dual

forms, and an orthogonality relation for the Fourier coefficients Aj(m1,m2) of that same family.

Theorem 1.0.1. Let φj be a set of orthogonal GL(3) Hecke-Maass forms with spectral parameters

ν(j) and Fourier coefficients Aj(n1, n2). Let hT,R,κ be a family of smooth functions essentially

2


supported on the spectral parameters of the Maass forms in F . Then,

∑j≥1

∣∣hT,R,κ(ν(j))∣∣

Lj= cR,κ

T 4+3R

(log T )1/κ+O

(T 3+3R+ε

), (T →∞).

Moreover,

∑j≥1

Aj(m1,m2)Aj(n1, n2)

∣∣hT,R,κ(ν(j))∣∣

Lj=

∑j≥1

|hT,R,κ(ν(j))|Lj +O

(T 3R+3+ε

), if m1=n1

m2=n2,

O(T 3R+3+ε

), otherwise.

A more precise version of this theorem is stated in Theorem 4.1.1. It is useful to note that the

second statement in Theorem 1.0.1 can be thought of as a version of the well known orthogonality

relation of Dirichlet characters,

∑χ mod q

χ(n)χ(m) =

φ(q), if n ≡ m mod q,

0, otherwise.

(1.1)

for integers m,n coprime to q, where the sum on the left is over all characters (mod q). Since

Dirichlet characters can be viewed as automorphic representations of GL(1,AQ), this result can

be interpreted as the simplest case of the orthogonality relation conjectured by Zhou [Zhou, 2014]

concerning Fourier-Whittaker coefficients of Maass forms on the space SLn(Z)\SLn(R)/SOn(R),

n ≥ 2. This orthogonality relation conjectured by Zhou was proved by Bruggeman [Bruggeman,

1978] in the case n = 2 and by Goldfeld-Kontorovich [Goldfeld and Kontorovich, 2013] and Blomer

[Blomer, 2013] in the case n = 3. Versions of this result have applications to the Sato-Tate

problem for Hecke operators, both in the holomorphic [Conrey et al., 1997], [Serre, 1997] and

non-holomorphic setting [Sarnak, 1987], [Zhou, 2014], as well as to the problem of determining

symmetry types of families of L-functions [Goldfeld and Kontorovich, 2013] as introduced in the

work of Katz-Sarnak [Katz and Sarnak, 1999].

The version of Weyl’s law presented above tells us that the family F of Maass forms being

studied has zero density in the space of all Maass forms, indexed by increasing spectral parameters.

This is the first result of this kind obtained using a GL(3) trace formula. After normalizing by

the weight factor T 3R (which is needed for technical reasons), Theorem 1.0.1 tells us that F has

roughly T 4 Maass forms with eigenvalue up to T 2. For comparison, Weyl’s law for all Maass forms

on the space SL3(Z)\SL3(R)/SO3(R) says that there are T 5 Maass forms with eigenvalue up to

3


T 2, by work of Miller [Miller, 2001]. Miller’s result has since been obtained in more general settings

in [Lapid and Muller, 2009], [Lindenstrauss and Venkatesh, 2007], [Muller, 2007]. Furthermore, F

contains all self-dual Maass forms, which is composed exactly of the Maass forms that are obtained

by the Gelbart-Jacquet lift [Gelbart and Jacquet, 1978], i.e. the ones that arise as symmetric square

lifts of GL(2) Maass forms.

The approach used to prove Theorem 1.0.1 is to apply the Kuznetsov trace formula for GL(3)

developed by Blomer [Blomer, 2013] and Goldfeld-Kontorovich [Goldfeld and Kontorovich, 2013],

which we rederive carefully in Section 2.8, to a suitable family of smooth test functions related to

hT,R,κ.

The other main result is an inversion formula for the Lebedev-Whittaker transform in the

nonarchimedean case, which can be described as follows. Let h : Z(Qp)\T (Qp) → C be a smooth

function. The p-adic Whittaker transform of h is defined to be

h](α) :=

∫T (Zp)\T (Qp)

h(t)Wα(t)d×t,

where Wα is a Whittaker function on GLn(Qp).

Whittaker functions on local nonarchimedean fields (such as Qp) have been studied by several

authors since they were introduced in 1967 by Jacquet [Jacquet, 1967]. Shintani [Shintani, 1976]

obtained an explicit formula for these Whittaker functions, which was then generalized in different

directions in other works (see [Casselman and Shalika, 1980], [Miyauchi, 2014]).

Theorem 1.0.2. Let H : Sn → C holomorphic and permutation invariant, where S is an open

annulus containing the unit circle. Let

H[(t) =1

n!(2πi)n−1

∫T

H(β)W1/β(t)n∏

i,j=1i 6=j

(βi − βj) dβ?.

Under certain convergence conditions,

(H[)](α) = H(α)

for α1 · · ·αn = 1.

At the end of Chapter 3 we use this inversion formula to give an integral representation for a

local L-function associated to a symmetric square lift. An example of another possible application

4


would be on the choice of test functions for an adelic Kuznetsov trace formula, similarly to what has

been achieved in [Goldfeld and Kontorovich, 2013]. We should remark that the inversion formula

was obtained in the setting of p-adic reductive groups by Delorme [Delorme, 2013]. However, the

approach presented here provides a more explicit presentation of the result in this particular setting

and its simple derivation solely relies on complex analysis.

The thesis is organized as follows. In Chapter 2, we do a quick introduction to the theory of

Hecke-Maass forms for SL(2,Z) and for SL(3,Z), Eisenstein series and Kloostermann sums. We

also rederive a Kuznetsov trace formula due to Blomer [Blomer, 2013], including many of the details

and calculations. Chapter 3 covers the Lebedev-Whittaker transform and its inversion formula, in

both the archimdean and nonarchimedean cases. The inversion formula is derived for GL(3) in the

archimedean case and for GL(n) in the nonarchimdean case (Theorem 1.0.2). Finally, Chapter 4

establishes Theorem 1.0.1, as an application of the Kuznetsov trace formula. We do a careful

analysis of all terms that arise from the trace formula, in both the spectral and geometric side,

obtaining bounds and asymptotic expressions for those terms.

5

CHAPTER 2. PRELIMINARIES

Chapter 2

Preliminaries

In this chapter we lay the foundations for the rest of the thesis, by compiling definitions and re-

sults on Maass forms for SL(3,Z), Eisenstein series and Kloosterman sums. It culminates with the

statement of a GL(3) Kuznetsov trace formula following [Blomer, 2013] and [Goldfeld and Kon-

torovich, 2013]. The GL(3) Kuznetsov trace formula is an equality which relates Fourier coefficients

of Hecke-Maass forms and Eisenstein series on one side, to Kloosterman sums on the other side.

Sections 2.2 and 2.4 introduce the theory of Hecke-Maass forms, Section 2.5 covers the theory of

Eisenstein series, and Section 2.6 that of Kloosterman sums. In the final section of this chapter

(Section 2.8) the Kuznetsov trace formula is stated and a sketch of its proof is given.

2.1 Maass Forms for SL(2,Z)

In this section we state the definition of Maass form for SL(2,Z) and establish some of its properties.

This should provide some insight into the theory of Maass forms for SL(3,Z) in the sections to follow.

Definition 2.1.1 (Upper Half Plane). Let h2 be the upper half plane

{z = x+ iy : x, y ∈ R, y > 0} .

Remark 2.1.2. Alternatively, the upper half plane can be realized as GL(2,R)/(O(2,R) · R×) via

the map

z = x+ iy 7→

y x

0 1

.

6


There is an action of GL(2,R) on h2 given by

a b

c d

· z =az + b

cz + d,

where

a b

c d

∈ GL(2,R) and z ∈ h2.

The space of complex-valued functions defined on SL(2,Z)\h2 (with the above action) can be

equipped with the following inner product.

Definition 2.1.3 (Petersson Inner Product). Let φ, φ′ : SL(2,Z)\h2 → C. We define their

inner product ⟨φ, φ′

⟩:=

∫SL(2,Z)\h2

φ(z)φ′(z)dx dy

y2.

Furthermore, define the L2-norm as ‖φ‖22 := 〈φ, φ〉 and the L2-space as L2(SL(2,Z)\h2) :={φ : SL(2,Z)\h2 → C : ‖φ‖2 <∞

}.

Definition 2.1.4 (Maass form for SL(2,Z)). A Maass form of type ν ∈ C for SL(2,Z) is a

non-zero smooth function φ ∈ L2(SL(2,Z)\h2) satisfying

• φ is an eigenfunction of the Laplace operator ∆ := −y2(∂2

∂2x+ ∂2

∂2y

)with eigenvalue 1

4 − ν2;

• φ is cuspidal, i.e.,1∫0

φ(z) dx = 0.

Remark 2.1.5. The Laplace operator is invariant under the group actions

z 7→ az + b

cz + d,

with

a b

c d

∈ GL(2,Z).

Definition 2.1.6. For n = 1, 2, . . . , define the Hecke operator Tn acting on L2(SL(2,Z)\h2) by

Tnφ(z) =1√n

∑ad=n

0≤b<d

φ

(az + b

d

),

where φ ∈ L2(SL(2,Z)\h2).

7


Definition 2.1.7 (Hecke-Maass Form). A Hecke-Maass form for SL(2,Z) is a Maass form φ

for SL(2,Z) satisfying:

Tnφ = λnφ

for every n = 1, 2, . . . . The λn are called the Hecke eigenvalues of φ.

With the above definitions we can now define an L-function associated to a Hecke-Maass form

for SL(2,Z).

Definition 2.1.8 (L-function). Let s ∈ C, R(s) > 32 , and let φ be a Hecke-Maass form for

SL(2,Z) with Hecke eigenvalues λn. Define the L-function associated to φ, Lφ, by the series

Lφ(s) :=∞∑n=1

λnns.

We shall also define the Rankin-Selberg L-function associated to a Hecke-Maass form for SL(2,Z)

as it will prove useful later on.

Definition 2.1.9 (Rankin-Selberg L-function). Let s ∈ C, R(s) sufficiently large, and let φ be

a Hecke-Maass form for SL(2,Z) with Hecke eigenvalues λn. Define the L-function associated to

φ, denoted Lφ, by the series

Lφ×φ(s) := ζ(2s)

∞∑n=1

|λn|2

ns,

where ζ(s) :=∞∑n=1

1ns is the Riemann zeta function.

2.2 Maass Forms for SL(3,Z)

The goal of this section is to establish a definition of a Maass form for SL(3,Z). Most of the

components of the definition of a Maass form for SL(2,Z) can be generalized in a straightforward

manner to SL(3,Z), with the exception of the Laplace eigenfunction condition which requires a bit

of work. The role of the Laplace operator will be taken over by a rank two polynomial algebra of

differential operators.

8


Definition 2.2.1 (Generalized Upper Half Plane). The generalized upper half plane h3 is the

set of 3× 3 matrices of the form z = x · y, where

x =

1 x2 x3

0 1 x1

0 0 1

, y =

y1y2 0 0

0 y1 0

0 0 1

,

with x1, x2, x3 ∈ R and y1, y2 > 0.

Remark 2.2.2. By the Iwasawa decomposition (see Proposition 1.2.6 [Goldfeld, 2006]) we have

GL(3,R) = h3 ·O(3,R) · R×, h3 ∼= GL(3,R)/(O(3,R) · R×).

From the description of h3 in the remark we can see that there is a group action of SL(3,Z)

on h3 by left multiplication. This action gives rise to the quotient space SL(3,Z)\h3. We can also

define an inner product for complex-valued functions on this space.

Definition 2.2.3 (Petersson Inner Product on h3). Let φ, φ′ : SL(3,Z)\h3 → C. We define

their inner product ⟨φ, φ′

⟩:=

∫SL(3,Z)\h3

φ(z)φ′(z) d∗z,

where d∗z := dx1 dx2 dx3 dy1 dy2y31y

32

.

Furthermore, define the L2-norm as ‖φ‖22 := 〈φ, φ〉, and the L2-space as

L2(SL(3,Z)\h3) :={φ : SL(3,Z)\h3 → C : ‖φ‖2 <∞

}.

Remark 2.2.4. The measure d∗z = dx1 dx2 dx3 dy1 dy2y31y

32

is the left-invariant Haar measure on h3 (see

Proposition 1.5.2 in [Goldfeld, 2006]).

Definition 2.2.5. Let α ∈ gl(3,R), the Lie algebra of GL(3,R), and let φ : GL(3,R) → C be a

smooth function, we define

Dαφ(g) :=∂

∂tφ(g · exp(tα))

∣∣∣∣t=0

,

with g ∈ GL(3,R).

The differential operators Dα with α ∈ gl(3,R) generate an associative algebra D3 defined over

R, where multiplication of two operators is given by their composition. Let D3 be the center of D3.

9


Lemma 2.2.6. Every D ∈ D3 is well-defined on smooth function on SL(3,Z)\h3. In other words,

for

φ : SL(3,Z)\GL(3,R)/(O(3,R) · R×

)→ C

we have

(Dφ)(γ · g · k · z) = Dφ(g),

where γ ∈ SL(3,Z), g ∈ GL(3,R), k ∈ O(3,R), z ∈ R× is a scalar matrix.

Proof. See Proposition 2.3.1 [Goldfeld, 2006].

The subalgebra D3 of these differential operators has a rather explicit description.

Proposition 2.2.7. Let Ei,j ∈ gl(3,R) be the matrix with the value 1 at the (i, j)-th entry and

zeros elsewhere, for 1 ≤ i, j ≤ 3. Define the differential operators

∆2 :=3∑

i1=1

3∑i2=1

DEi1,i2◦DEi2,i1

,∆3 :=3∑

i1=1

3∑i2=1

3∑i1=1

DEi1,i2◦DEi2,i3

◦DEi3,i1.

Then, every D ∈ D3 can be written as polynomial (with coefficients in R) in ∆2 and ∆3, i.e,

D3 = R[∆2,∆3].

Furthermore,

∆2 = y21

∂2

∂y21

+ y22

∂2

∂y22

− y1y2∂2

∂y1∂y2+ y2

1(x22 + y2

2)∂2

∂x23

+ y21

∂2

∂x21

+ y22

∂2

∂x22

+ 2y21x2

∂2

∂x1∂x3

and −∆2 is the Laplace operator

Proof. See Proposition 2.3.5 and Equation 6.1.1 [Goldfeld, 2006].

Now we define a family of functions on h3 that are eigenfunctions for all D ∈ D3.

Definition 2.2.8. For ν = (ν1, ν2) ∈ C2 and z = x · y ∈ h3 (with x and y as in Definition 2.2.1),

we define the function Is : h3 → C, by the condition:

Iν (z) := y1+ν1+2ν21 y1+2ν1+ν2

2 .

Lemma 2.2.9. For every ν ∈ C2 and D ∈ D3, the function Iν is an eigenfunction of D. We

denote by λν(D) the corresponding eigenvalue.

10


Proof. See Proposition 2.4.3 [Goldfeld, 2006]. Our definition of Iν differs by the map ν1 7→ ν1 − 13 ,

ν2 7→ ν2 − 13 .

Remark 2.2.10. The Laplace eigenvalue λν(−∆2) is given by 1 − 3(ν21 + ν1ν2 + ν2

2). This is a

straightforward computation using the explicit decription of −∆2 given in Proposition 2.2.7. Note

that Iν(z) does not depend on x1, x2, x3 so

−∆2Iν = −y21

∂2Iν∂y2

1

− y22

∂2Iν∂y2

2

+ y1y2∂2Iν∂y1∂y2

= (1 + ν1 + 2ν2)(1 + 2ν1 + ν2)Iν − (1 + ν1 + 2ν2)(ν1 + 2ν2)Iν − (1 + 2ν1 + ν2)(2ν1 + ν2)Iν

=(1− 3(ν2

1 + ν1ν2 + ν22))Iν .

We can now define the notion of Maass form for SL(3,Z).

Definition 2.2.11 (Maass Form for SL(3,Z)). A Maass form of type ν = (ν1, ν2) ∈ C2 for

SL(3,Z) is a non-zero smooth function φ ∈ L2(SL(3,Z)\h3) satisfying

• for every D ∈ D3, φ is an eigenfunction of D with eigenvalue λν(D);

• φ is cuspidal, i.e.,∫

(SL(3,Z)∩U)\Uφ(uz) du = 0 for U = U1,2, U2,1, where

U1,2 =

1 ∗ ∗

0 1 0

0 0 1

, U2,1 =

1 0 ∗

0 1 ∗

0 0 1

.

Remark 2.2.12. This definition can be easily generalized to define Maass forms for congruence

subgroups of SL(3,Z).

2.3 Fourier-Whittaker Expansion

In this section we define Whittaker functions, establish some of their properties and state the

Fourier-Whittaker expansion of a Maass form for SL(3,Z).

Definition 2.3.1 (Siegel set). Let a, b ≥ 0. We define the Siegel set Σa,b ⊂ h3 to be the set of all

matrices z = x · y such that

|x1|, |x2|, |x3| ≤ b, y1, y2 > a,

where x and y are defined as in 2.2.1.

11


For m = (m1,m2) ∈ Z2 and u ∈ U3(R) of the form

Let U3(R) be the set of matrices u of the form

u =

1 u2 u3

0 1 u1

0 0 1

, (2.1)

where u1, u2, u3 ∈ R.

Their characters can be indexed by a pair of integers m = (m1,m2) such that the character ψm

of U3(R) is defined on a matrix u as in 2.1 by

ψm(u) := e2πi(m1u1+m2u2). (2.2)

Definition 2.3.2 (Whittaker Function). An SL(3,Z)-Whittaker function of type ν = (ν1, ν2) ∈

C2, associated to a character ψ of U3(R), is a smooth function Ψ : h3 → C satisfying the following

conditions:

• Ψ(uz) = ψ(u)Ψ(z), for all u ∈ U3(R), z ∈ h3;

• DΨ(z) = λν(D)Ψ(z), for all D ∈ D3, z ∈ h3;

•∫

Σ√32 , 12

|Ψ(z)|2 d∗z <∞.

Definition 2.3.3 (Jacquet’s Whittaker Function). Fix a character ψm of U3(R) (as in 2.1,

2.2), with non-zero m1,m2, and ν = (ν1, ν2) ∈ C2. Define Jacquet’s Whittaker function to be

Wν(z;m) :=

∫U3R

Iν(w · u · z)ψm(u) d∗u,

where z ∈ h3, d∗u = du1 du2 du3, and w =

1

1

1

.

Remark 2.3.4. This integral representation only makes sense for R(ν1),R(ν2) � 1, where the

integral converges (absolutely and uniformly in compact sets). The function can be extended for

any ν ∈ C2 analytically.

12


Remark 2.3.5. We can also define the completed Jacquet’s Whittaker function to be

W ∗ν (z;m) := π−3ν1−3ν2Γ

(1 + 3ν1

2

)Γ

(1 + 3ν2

2

)Γ

(1 + 3ν1 + 3ν2

2

)Wν(z;m).

This normalization will be useful for some of the formulas.

Proposition 2.3.6 (Multiplicity One). The function Wν(z;m) is an SL(3,Z)-Whittaker func-

tion of type ν = (ν1, ν2) ∈ C2 associated to a character ψ of U3(R). Furthermore, if Ψ is an

SL(3,Z)-Whittaker function of type ν = (ν1, ν2) ∈ C2 associated to a character ψ of U3(R), with

sufficient decay in y1, y2, so that

∞∫0

∞∫0

yσ11 yσ22 |Ψ(y)| dy1 dy2

for sufficiently large σ1, σ2, then

Ψ(z) = c ·Wν(z;m)

for some c ∈ C.

Proof. See Proposition 5.5.2 and Theorem 6.1.6 [Goldfeld, 2006].

Theorem 2.3.7 (Fourier-Whittaker Expansion). Let φ be a Maass form of type ν for SL(3,Z)

then

φ(z) =∑

γ∈U2(Z)\SL(2,Z)

∞∑m1=1

∑m2 6=0

A(m1,m2)

|m1m2|·W ∗ν

|m1m2|

m1

1

γ

1

z; (1, sgn(m2))

,

where A(m1,m2) ∈ C is the (m1,m2)-th Fourier coefficient.

Proof. See Theorem 5.3.2 and Equation 6.2.1 [Goldfeld, 2006].

Remark 2.3.8. The Fourier coefficients A(m1,m2) can be computed the following way:

A(m1,m2)

|m1m2|·W ∗ν

|m1m2|y1y2

m1y1

1

; (1, sgn(m2))

=

1∫0

1∫0

1∫0

φ(z)e−2πi(m1x1+m2x2) dx1 dx2 dx3

13


Note that the Whittaker functions that show up in the Fourier-Whittaker expansion (Theo-

rem 2.3.7) are associated to either the character ψ(1,1) or the character ψ(1,−1).

Define W ∗ν (z) := Wν(z; (1, 1)) dropping the dependence on the character. If z = y is a diagonal

matrix then we have W ∗ν (y) = W ∗ν (y; (1, 1)) = W ∗ν (y; (1,−1)). We shall also denote

W ∗ν (y1, y2) := W ∗ν

y1y2 0 0

0 y1 0

0 0 1

.

Define Wν(z) and Wν(y1, y2) similarly.

Proposition 2.3.9 (Double Mellin Inversion). The Whittaker function W ∗ν has an integral

representation as a double Mellin inversion

W ∗ν (y) =y1y2π

3/2

(2πi)2

∫(C2)

∫(C1)

∏3j=1 Γ

(s1−αj

2

)Γ(s2+αj

2

)4πs1+s2Γ

(s1+s2

2

) y−s11 y−s22 ds1 ds2,

where y1, y2 > 0, ν ∈ C2,

α1 = 2ν1 + ν2, α2 = ν2 − ν1, α3 = −ν1 − 2ν2,

Ci > max(|α1|, |α2|, |α3|) and (Ci) denotes the integration path (oriented upward) along the vertical

line R(si) = Ci.

Proof. See Equation 10.1 [Bump, 1984]. Notice that [Bump, 1984] has the roles of ν1, ν2 inter-

changed in the definition of Iv and that our completed Whittaker function differs by a factor of

π3/2.

Remark 2.3.10. Note that W ∗ν (y) is invariant under permutations of (α1, α2, α3), and

W ∗(ν1,ν2)(y1, y2) = W ∗(ν2,ν1)(y2, y1) = W ∗(ν1,ν2)(y1, y2).

Using the double Mellin transform and some estimates on the Gamma function it is possible to

obtain some bounds on the Whittaker function.

Proposition 2.3.11. Let θ = max(|R(α1)|, |R(α2)|, |R(α3)|) and assume θ ≤ 1/2. Let θ < σ1 < σ2

and ε > 0. Then for any σ1 ≤ c1, c2 ≤ σ2 we have

W ∗ν (y)� y1y2

(1 + |ν1|+ |ν2|)1/2−ε

(y1

1 + |ν1|+ |ν2|

)−c1 ( y2

1 + |ν1|+ |ν2|

)−c2.

14


Proof. See Proposition 1 [Blomer, 2013].

Proposition 2.3.12 (Stade’s Formula). For s ∈ C, ν, µ ∈ C2,

∞∫0

∞∫0

W ∗ν (y)W ∗µ(y)(y21y2)s

dy1 dy2

(y1y2)3=π3(1−s)

Γ(

3s2

) ∏1≤j,k≤3

Γ

(s+ αj + βk

2

),

where the αi are defined as in Proposition 2.3.9 and the βi are defined analogously in terms of µ1,

µ2.

Proof. See Theorem 3.1 [Stade, 1993] observing that Stade’s definition of the Whittaker function

differs by a factor of π3/2.

2.4 Hecke-Maass Forms and L-functions

In this section we define Hecke operators and their simultaneous eigenfunctions, the Hecke-Maass

forms. We then define L-functions attached to these forms, which have an Euler product.

Definition 2.4.1. For n = 1, 2, . . . , define the Hecke operator Tn acting on L2(SL(3,Z)\h3) by

Tnφ(z) =1

n

∑abc=n

0≤c1,c2<c0≤b1<b

φ

a b1 c1

0 b c2

0 0 c

z

,

where φ ∈ L2(SL(3,Z)\h3).

The operators Tn commute with the differential operators in D3. Therefore, we may simulta-

neously diagonalize the space L2(SL(3,Z)\h3) by all these operators.

Definition 2.4.2 (Hecke-Maass form). A Hecke-Maass form for SL(3,Z) is a Maass form φ

for SL(3,Z) satisfying:

Tnφ = λnφ

for every n = 1, 2, . . . . The λn are called the Hecke eigenvalues of φ.

Theorem 2.4.3 (Hecke Relations). Let φ be a Hecke-Maass form for SL(3,Z) with Fourier

coefficients A(m1,m2), normalized such that A(1, 1) = 1. Then,

15


Tnφ = A(n, 1)φ.

Furthermore, we have the following relations

A(m1m′1,m2m

′2) = A(m1,m2)A(m′1,m

′2), if (m1m2,m

′1m′2) = 1;

A(n, 1)A(m1,m2) =∑abc=na|m1

b|m2

A(m1c

a,m2a

b

);

A(1, n)A(m1,m2) =∑abc=na|m1

b|m2

A

(m1b

a,m2c

b

);

A(m1,m2) =∑

d|(m1,m2)

µ(d)A(m1

d, 1)A(

1,m2

d

),

where µ is the Mobius function.

Proof. See Theorem 6.4.11 [Goldfeld, 2006] for the first statement and the first three relations

between the Fourier coefficients. To obtain the last relation, we start by choosing m1 = 1 in the

second relation to obtain

A(n, 1)A(1,m2) =∑

d|(n,m2)

A(nd, 1)A(

1,m2

d

).

By Mobius inversion it follows that

∑d|(m1,m2)

µ(d)A(m1

d, 1)A(

1,m2

d

)=

∑d|(m1,m2)

µ(d)∑

ed|(m1,m2)

A(m1

de,m2

de

)=

∑e′|(m1,m2)

A(m1

e′,m2

e′

)∑d|e′

µ(d)

= A(m1,m2).

Definition 2.4.4 (Standard L-function). Let s ∈ C, R(s) > 2, and let φ be a Hecke-Maass

form for SL(3,Z). Define the standard L-function associated to φ, denoted Lφ, by the series

Lφ(s) :=

∞∑n=1

A(1, n)

ns.

16


Remark 2.4.5. The series that defines Lφ(s) is absolutely convergent in the region R(s) > 2 as

A(1, n) = O(n) (see Lemma 6.2.2 [Goldfeld, 2006]).

Remark 2.4.6. By the first relation in Theorem 2.4.3 one can factor Lφ(s) as∏p

Lφ,p(s),

where Lφ,p(s) :=∞∑k=1

A(1,pk)pks

. By the second and third relations in Theorem 2.4.3 one obtains

A(p, 1)A(1, pk)−A(1, p)A(1, pk+1) = A(1, pk−1)−A(1, pk+2) for k ≥ 1,

A(p, 1)−A(1, p)A(1, p) = −A(1, p2).

Therefore,

A(p, 1)Lφ,p(s)−A(1, p) (psLφ,p(s)− ps) = p−sLφ,p(s)−(p2sLφ,p(s)− p2s − psA(1, p)

).

Solving the above equation for Lφ,p(s) yields

Lφ,p(s) =(1− p−sA(1, p) + p−2sA(p, 1)− p−3s

)−1.

Definition 2.4.7 (Satake Parameters at p). The roots α1,p, α2,p, α3,p ∈ C of the polynomial

1− p−sA(1, p) + p−2sA(p, 1)− p−3s

are called the Satake parameters of φ at p.

Theorem 2.4.8 (Functional Equation). Let φ be a Hecke-Maass form of type ν = (ν1, ν2) ∈ C2

for SL(3,Z), with associated L-function Lφ(s). Let Lφ(s) :=∞∑n=1

A(n, 1)n−s be the dual L-function.

Then Lφ(s), Lφ(s), have holomorphic continuation for s ∈ C and satisfy the function equation

Fν(s)Lφ(s) = Fν(1− s)Lφ(1− s),

where

Fν(s) = π−3s/23∏j=1

Γ

(s− αj

2

),

Fν(s) = π−3s/23∏j=1

Γ

(s+ αj

2

).

Here α1 = 2ν1 + ν2, α2 = ν2 − ν1, α3 = −ν1 − 2ν2 are the Langlands parameters of φ.

Proof. See Theorem 6.5.15 [Goldfeld, 2006]. Recall that we have normalized ν1 7→ ν1 + 13 and

ν2 7→ ν2 + 13 .

17


2.5 Eisenstein Series and Spectral Decomposition

Definition 2.5.1 (Minimal Parabolic Eisenstein Series). Define the minimal standard parabolic

subgroup for GL(3,R) as

P1,1,1 :=

∗ ∗ ∗

0 ∗ ∗

0 0 ∗

.

For z ∈ h3 and ν = (ν1, ν2) ∈ C2, with R(ν1), R(ν2) sufficiently large, define the minimal parabolic

Eisenstein series Emin(z, ν) by the series

Emin(z, ν) :=∑

γ∈(P1,1,1∩SL(3,Z))\SL(3,Z)

Iν(γz).

This function has meromorphic continuation to all ν1, ν2 ∈ C.

Proposition 2.5.2. The minimal parabolic Eisenstein series Emin(z, ν) has a Fourier-Whittaker

expansion (as in 2.3.7) and its Fourier coefficients (for nonzero m1, m2) are given by

Aν(m1,m2)

|m1m2|Wν (m1y1, |m2|y2)

ζ(1 + 3ν1)ζ(1 + 3ν2)ζ(1 + 3ν1 + 3ν2)=

1∫0

1∫0

1∫0

Emin(z, ν)e−2πi(m1x1+m2x2) dx1 dx2 dx3,

where Aν(m1,m2) satisfy the Hecke relations 2.4.3, and

Aν(m, 1) =∑

d1d2d3=m

dα11 dα2

2 dα33 .

Proof. See Theorem 10.8.1 [Goldfeld, 2006]. Due to the normalization in that statement the zeta

factors do not show up. See Theorem 7.2 [Bump, 1984] for the computation of those factors.

Definition 2.5.3 (Maximal Parabolic Eisenstein Series). Let

P2,1 :=

∗ ∗ ∗

∗ ∗ ∗

0 0 ∗

be a maximal standard parabolic subgroup for GL(3,R). Let z ∈ h3, s ∈ C, with R(s) sufficiently

large, and let u be a Hecke-Maass form of type µ for SL(2,Z), normalized such that ‖u‖2 = 1.

Define the maximal parabolic Eisenstein series associated to u, denoted Emax(z, s, u), by

Emax(z, ν, u) :=∑

γ∈(P2,1∩SL(3,Z))\SL(3,Z)

det(γz)1/2+su(τ(γz)),

18


where

τ(z) =

y2 x2

0 1

.

Proposition 2.5.4. The maximal parabolic Eisenstein series Emax(z, s, u) has a Fourier-Whittaker

expansion (as in 2.3.7) and its Fourier coefficients (for nonzero m1, m2) are given by

cBs,u(m1,m2)

|m1m2|

W( 2µ3,s−µ

3 ) (m1y1, |m2|y2)

Lu(1 + 3s)LAd u(1)1/2=

1∫0

1∫0

1∫0

Emax(z, ν, u)e−2πi(m1x1+m2x2) dx1 dx2 dx3,

where Bs,u(m1,m2) satisfy the Hecke relations 2.4.3, and

Bs,u(m, 1) =∑

d1d2=m

λu(d1)ds1d−2s2 .

Here LAdu is the adjoint L-function defined given by

LAd u(s) =Lu×u(s)

ζ(s),

and c is a nonzero absolute constant.

Proof. See page 18 [Blomer, 2013].

Proposition 2.5.5. Let ν ∈ C2 such that R(ν1) = R(ν2) = 0, and s ∈ C, such that R(s) = 0.

Then, the Fourier coefficients Aν(m1,m2) and Bs,u(m1,m2) satisfy

Aν(m1,m2) = O((m1m2)ε), Bs,u(m1,m2) = O((m1m2)1/2+ε),

for every ε > 0.

Proof. By Proposition 2.5.2 we have

Aν(m, 1) =∑

d1d2d3=m

dα11 dα2

2 dα33

≤ σ(m)3

= O(mε).

19


By Proposition 2.5.4 we have

Bs,u(m, 1) =∑

d1d2=m

λu(d1)ds1d−2s2

= O

∑d1|m

√d1

= O(m1/2+ε),

where the first inequality follows from λu(n) = O(√n) (see Proposition 3.6.3 [Goldfeld, 2006]).

Analogously one can obtain the same bounds for Aν(1,m), Bs,u(1,m) (by considering the dual

objects). The result then follows from the Hecke relations for Aν(m1,m2), Bs,u(m1,m2).

A degenerate case of a maximal parabolic Eisenstein series is given by Emax(z, s, u), where u is

a constant function. For z ∈ h3, s ∈ C, with R(s) sufficiently large, define

Emax(z, s,1) :=∑

γ∈(P2,1∩SL(3,Z))\SL(3,Z)

det(γz)1/2+s.

This Eisenstein series only has degenerate terms (corresponding to m1m2 = 0) in its Fourier

expansion (see Theorem 2.4 [Friedberg, 1987]).

Define the notation∫

(C)

to denote the integral along the complex vertical line whose points have

real part equal to C.

Theorem 2.5.6 (Langlands Spectral Decomposition). Let φ ∈ L2(SL(3,Z)\h3) with sufficient

decay such that 〈φ,E〉 converges for all Eisenstein series defined in this section. Assume that φ is

orthogonal to all residues of all such Eisenstein series. Then the function

φ(z)− 1

(4πi)2

∫(0)

∫(0)

〈φ,Emin(∗, ν)〉Emin(z, ν) dν1 dν2−1

2πi

∞∑j=0

∫(0)

〈φ,Emax(∗, s, uj)〉Emax(z, s, uj) ds

is a Maass form for SL(3,Z), where {uj} for j = 1, 2, . . . is a basis of Maass forms for SL(2,Z)

normalized such that ‖uj‖2 = 1.

Proof. See Theorem 10.13.1 [Goldfeld, 2006].

20


2.6 Kloosterman Sums

The standard Kloosterman sum S(m,n, c) is defined by

S(m,n, c) :=∑∗

d (mod c)

e

(md+ nd

c

),

where d denote the inverse of d (mod c) and e(x) := e2πix. Here ∗ means that the sum is over

invertible classes (mod c).

Using geometric arguments Weil [Weil, 1948] showed the following estimate for these sums

|S(m,n, c)| � c1/2+ε(m,n, c)ε.

In order to define Kloosterman sums for SL(3,Z) we will need a few facts about the structure

of GL(3,R).

Definition 2.6.1 (Weyl Group). Let T be the set of diagonal matrices of GL(3,R). Define the

Weyl group W of GL(3,R) to be the quotient N/T , where N is the normalizer of T in GL(3,R),

i.e, every element n ∈ N commutes with every element of T .

We can choose the following representatives for W :

w1 =

1 0 0

0 1 0

0 0 1

, w2 =

1 0 0

0 0 1

0 1 0

, w3 =

0 1 0

1 0 0

0 0 1

,

w4 =

0 0 1

1 0 0

0 1 0

, w5 =

0 1 0

0 0 1

1 0 0

, w6 =

0 0 1

0 1 0

1 0 0

.

Proposition 2.6.2 (Bruhat Decomposition). The group GL(3,R) decomposes as

GL(3,R) =⋃w∈W

Gw,

where Gw = U3(R)wT U3(R).

Proof. See Proposition 10.3.2 [Goldfeld, 2006].

21


Let Γ := SL(3,Z) and Γw := (w−1 · U3(Z)t · w) ∩ U3(Z), where t denotes matrix transposition.

Definition 2.6.3 (SL(3,Z) Kloosterman Sum). Let w ∈W , d ∈ T , and ψ, ψ′ be characters of

U3. Define the SL(3,Z) Kloosterman sum Sw by

Sw(ψ,ψ′, d) :=∑

γ∈U3(Z)\(Γ∩Gw)/Γwγ=u1wdu2

ψ(u1)ψ′(u2),

provided the sum is well-defined, i.e., it does not depend on the Bruhat decomposition of γ. Other-

wise, define the Kloosterman sum as zero.

A necessary and sufficient condition for these Kloosterman sums to be well-defined is given in

the following lemma.

Lemma 2.6.4. The Kloosterman sum Sw(ψ,ψ′, d) is well-defined if and only if

ψ(dwuw−1) = ψ′(u)

for all u ∈ (w−1 ·U3(R) ·w)∩U3(R). The characters ψ is extended to z = uy ∈ h3 by ψ(uy) = ψ(u)

with the matrix decomposition as in Definition 2.2.1.

Proof. See Lemma 10.6.3 and Proposition 11.2.10 [Goldfeld, 2006].

Let

d =

d1

−d2/d1

1/d2

,

with d1, d2 positive integers. Let ψ(u) = ψn1,n2(u) = e2πi(n1u1+n2u2), where u =(

1 u2 u30 1 u10 0 1

). Let

ψ′ = ψm2,m1 . Assume both ψ,ψ′ are non-degenerate, i.e, n1n2 6= 0, m1m2 6= 0.

These Kloosterman sums have been computed in an explicit form by Bump, Friendberg and

Golfeld [Bump et al., 1988]. We collect this information in the following proposition.

22


Proposition 2.6.5. Let d ∈ T and ψ,ψ′ be non-degenerate characters of U3(R), as above. Then,

Sw1(ψ,ψ′, d) = δd1,1δd2,1,

Sw2(ψ,ψ′, d) = Sw3(ψ,ψ′, d) = 0,

Sw4(ψ,ψ′, d) = S(m1, n1, n2, d1, d2) if d1|d2,

Sw5(ψ,ψ′, d) = S(m2, n2, n1, d2, d1) if d2|d2,

Sw6(ψ,ψ′, d) = S(m1,m2, n1, n2, d1, d2),

where δ is the Kronecker delta,

S(m1, n1, n2, d1, d2) :=∑

c1( mod d1), c2( mod d2)(c1,d1)=1

(c2,d2/d1)=1

e

(m1c1 + n1c1c2

d1

)e

(n2c2

d2/d1

),

S(m1,m2, n1, n2, d1, d2) :=∑

b1,c1( mod d1), b2,c2( mod d2)(b1,c1,d1)=(b2,c2,d2)=1

b1b2+c1d2+c2d1≡0( mod d1d2)

e

(m1b1 + n1(y1d2 − z1b2)

d1

)

× e(m2b2 + n2(y2d1 − z2b1)

d2

),

and y1, y2, z1, z2 are such that

y1b1 + z1c1 ≡ 1 (mod d1) and y2b2 + z2x2 ≡ 1 (mod d2).

Proof. See Table 5.4 [Bump et al., 1988].

Furthermore, we have sharp bounds for these Kloosterman sums. The following bound is due

to Larsen in the Appendix [Bump et al., 1988].

Proposition 2.6.6. For any ε > 0,

S(m1, n1, n2, d1, d2)� min(gcd(n2d

21, d1d2), gcd(m1d2, n1d2, d1d2)

)(d1d2)ε.

The bound for the Kloosterman sum associated to the long Weyl element w6 is due to [Stevens,

1987].

23


Proposition 2.6.7. For any ε > 0,

S(m1,m2, n1, n2, d1, d2)� (m1m2n1n2)1/2(d1d2)1/2+ε(d1, d2).

Proof. See Theorem 5.1 [Stevens, 1987]. Note that the theorem does not make the dependence on

m1, m2, n1, n2 as these are assumed to be fixed. This dependence can be found by looking into

the proof of the theorem, in equation 5.9.

2.7 Poincare Series

Poincare series for SL(3,Z) were first introduced in [Bump et al., 1988]. These series relate the

spectral theory of SL(3,Z)\h3 to SL(3,Z) Kloosterman sums.

Definition 2.7.1 (Poincare Series). Let z ∈ h3, with z = xy as in Definition 2.2.1. Let F :

R2+ → C, m1,m2 positive integers and define

Fm1,m2(z) := e(m1x1 +m2x2)F (m1y1,m2y2).

The Poincare series Pm1,m2 is given by the series

Pm1,m2(z) :=∑

γ∈U3(Z)\SL(3,Z)

Fm1,m2(γz).

Proposition 2.7.2. Assume F is a bounded function with the following decay

|F (y1, y2)| � (y1y2)2+ε,

for an ε > 0 as y1 → 0 or y2 → 0. Then the Poincare series Pm1,m2(z) converges absolutely and

uniformly on compact subsets of h3. Furthermore, Pm1,m2 ∈ L2(SL(3,Z)\h3).

Proof. First note that Pm1,m2 is invariant under SL(3,Z), by construction so is a function on

SL(3,Z)\h3. To show the convergence properties we follow the idea of Godement in the proof of

Theorem 9.1 [Borel, 1966]. For each z0 ∈ SL(3,Z)\h3 it suffices to show that∫Cz0

|Pm1,m2(z)| d∗z <∞,

24


where Cz0 is a compact set of SL(3,Z)\h3 containing z0. Then,∫Cz0

|Pm1,m2(z)| d∗z ≤∫

(U3(Z)\SL(3,Z))Cz0

|Fm1,m2(z)| d∗z .

It follows from Proposition 1.3.2 [Goldfeld, 2006] there are only finitely many γ ∈ U3(Z)\SL(3,Z)

such that γz0 ∈ Σ√32, 12

. This means that there exists an a ≥√

32 such that γz0 /∈ Σa, 1

2for every

γ ∈ U3(Z)\SL(3,Z). In addition, by taking Cz0 sufficiently small, we also conclude that γz /∈ Σa, 12

for every γ ∈ U3(Z)\SL(3,Z), z ∈ Cz0 . This implies

∫(U3(Z)\SL(3,Z))Cz0

|Fm1,m2(z)| d∗z ≤1∫

0

1∫0

1∫0

a∫0

a∫0

|F (m1y1,m2y2)| dx1 dx2 dx3dy1 dy2

(y1y2)3,

and the latter integral converges for F under the decay conditions in the statement. To show the

second part of the proposition note that∫SL(3,Z)\h3

|Pm1,m2(z)|2 d∗z ≤∫

U3(Z)\h3

|Fm1,m2(z)|2 d∗z

≤1∫

0

1∫0

1∫0

∞∫0

∞∫0

|Fm1,m2(z)|2 dx1 dx2 dx3dy1 dy2

(y1y2)3.

The last integral converges for bounded F satisfying |F (y1, y2)| � (y1y2)2+ε.

2.8 Kuznetsov Trace Formula

In this section we present a GL(3) Kuznetsov trace formula following [Blomer, 2013] and [Goldfeld

and Kontorovich, 2013]. It is obtained by computing the Petersson inner product of two Poincare

series in different ways. The first way is to use the Langlands spectral decomposition (Theo-

rem 2.5.6). The second way is to use the geometric structure GL(3,R), as given by the Bruhat

decomposition.

To apply the spectral decomposition to a Poincare series Pm1,m2 we first need to make sure that

such a function satisfies the conditions of Theorem 2.5.6. For the remainder of the section, assume

that F : R2+ → C in the definition of the Poincare series (Definition 2.7.1) is a bounded function

with the following decay

|F (y1, y2)| � (y1y2)2+ε,

25


for an ε > 0 as y1 → 0 or y2 → 0. In addition, define F ∗(y1, y2) := F (y2, y1) which satisfies the

same decay condition.

Lemma 2.8.1. Let Emin(z, ν) be the minimal parabolic Eisenstein series and assume R(ν1) =

R(ν2) = 0. For u be a Hecke-Maass form of type µ, let Emax(z, s, u) be the maximal parabolic

Eisenstein series associated to u. Assume R(s) = R(µ) = 0. Then the Petersson inner products⟨Emin(∗, ν), Pm1,m2

⟩,

⟨Emax(∗, s, u), Pm1,m2

⟩converge and satisfy

⟨Emin(∗, ν), Pm1,m2

⟩= m1m2

∞∫0

∞∫0

Aν(m1,m2)Wν (y1, y2)F (y1, y2)

ζ(1 + 3ν1)ζ(1 + 3ν2)ζ(1 + 3ν1 + 3ν2)

dy1dy2

(y1y2)3,

⟨Emax(∗, s, u), Pm1,m2

⟩= cm1m2

∞∫0

∞∫0

Bs,u(m1,m2)W( 2µ

3,s−µ

3 ) (y1, y2)F (y1, y2)

Lu(1 + 3s)LAd u(1)1/2

dy1dy2

(y1y2)3,

for some absolute constant c > 0.

Proof. Replace Pm1,m2(z) by its definition as a series to obtain⟨Emin(∗, ν), Pm1,m2

⟩=

∫SL(3,Z)\h3

Emin(z, ν)∑

γ∈U3(Z)\SL(3,Z)

Fm1,m2(γz) d∗z.

As Emin(z, ν) is invariant under the action of SL(3,Z) (on the variable z), one obtains⟨Emin(∗, ν), Pm1,m2

⟩=

∫U3(Z)\h3

Emin(z, ν)Fm1,m2(z)d∗z

=

∞∫0

∞∫0

F (m1y1,m2y2)

1∫0

1∫0

1∫0

Emin(z, ν)e−2πi(m1x1+m2x2) dx1 dx2 dx3dy1dy2

(y1y2)3

=

∞∫0

∞∫0

Aν(m1,m2)

m1m2

Wν (m1y1,m2y2)F (m1y1,m2y2)

ζ(1 + 3ν1)ζ(1 + 3ν2)ζ(1 + 3ν1 + 3ν2)

dy1dy2

(y1y2)3,

where the last equality follows from Proposition 2.5.2. The latter integral converges absolutely due

to the decay properties of F (2.8) and the bounds on the Whittaker functions (2.3.11). To finish

the proof do the change of variables (y1, y2) 7→(y1m1, y2m2

).

For the second inner product we follow a similar strategy:

⟨Emax(z, s, u), Pm1,m2

⟩=

∫SL(3,Z)\h3

Emax(z, s, u)∑

γ∈U3(Z)\SL(3,Z)

Fm1,m2(γz)d∗z.

26


As Emin(z, ν) is invariant under the action of SL(3,Z) (on the variable z), one obtains⟨Emax(z, s, u), Pm1,m2

⟩=

∫U3(Z)\h3

Emin(z, ν)Fm1,m2(z) d∗z

which can be written as

∞∫0

∞∫0

F (m1y1,m2y2)

1∫0

1∫0

1∫0

Emax(z, s, u)e−2πi(m1x1+m2x2) dx1 dx2 dx3dy1dy2

(y1y2)3

= c

∞∫0

∞∫0

Bs,u(m1,m2)

m1m2

W( 2µ3,s−µ

3 ) (m1y1,m2y2)F (m1y1,m2y2)

Lu(1 + 3s)LAd u(1)1/2

dy1dy2

(y1y2)3,

where the last equality follows from Proposition 2.5.4. The latter integral again converges absolutely

due to the decay properties of F (2.8) and the bounds on the Whittaker functions (2.3.11). Perform

a change of variables as before to obtain the equality in the theorem.

Remark 2.8.2. Note that Pm1,m2 is orthogonal to the degenerate Eisenstein series Emax(z, s, u)

with u a constant function. This follows from Emax(z, s, u) only having degenerate terms in its

Fourier-Whittaker expansion. The same is true for all residues of Eisenstein series.

The second way to compute the Petersson inner product of two Poincare series requires an

explicit Fourier expansion for the Poincare series as obtained in Theorem 5.1 [Bump et al., 1988].

Theorem 2.8.3. Let m1,m2, n1, n2 be positive integers. Then

1∫0

1∫0

1∫0

Pn1,n2(z)e−2πi(m1x1+m2x2) dx1 dx2 dx3 = S1 + S2a + S2b + S3,

where

S1 = δm1,n1δm2,n2F (n1y1, n2y2),

S2a =∑ε=±1

∑d1|d2

m2d21=n1d2

S(εm1, n1, n2, d1, d2)JF (y1, y2, εm1, n1, n2, d1, d2),

S2b =∑ε=±1

∑d2|d1

m1d22=n2d1

S(εm1, n1, n2, d1, d2)JF ∗(y1, y2, εm2, n2, n1, d2, d1),

S3 =∑

ε1,ε2=±1

∞∑d1,d2=1

S(ε1m1, ε2m2, n1, n2, d1, d2)JF (y1, y2, ε1m1, ε2m2, n1, n2, d1, d2).

27


The Kloosterman sums S and S have been defined in Section 2.6 and the weight functions J and

J are given by

JF (y1, y2, εm1, n1, n2, d1, d2) = y21y2

∞∫−∞

∞∫−∞

e

(n1y2d2

d21

· x1x2

x21 + 1

)e

(n2d1

y1y2d22

· x2

x21 + x2

2 + 1

)

× e (−m1x1y1)F

(n1y2d2

d21

·√x2

1 + x22 + 1

x21 + 1

,n2d1

y1y2d22

·√x2

1 + 1

x21 + x2

2 + 1

)dx1 dx2,

JF (y1, y2,m1,m2, n1, n2, d1, d2) = (y1y2)2

∞∫−∞

∞∫−∞

∞∫−∞

e (−m1x1y1 −m2x2y2)

× e(−n1d2

y2d21

· x1x3 + x2

x22 + x2

3 + 1

)e

(−n2d1

y2d22

· x2(x1x2 − x3) + x1

(x1x2 − x3)2 + x21 + 1

)× F

(n1d2

y2d21

·√

(x1x2 − x3)2 + x21 + 1

x22 + x2

3 + 1,n2d1

y2d22

·√x2

2 + x23 + 1

(x1x2 − x3)2 + x21 + 1

)dx1 dx2 dx3 .

Proof. Apply Theorem 5.1 [Bump et al., 1988] replacing Iν1,ν2(τ)En1,n2(τ) by Fn1,n2 and setting

x1, x2 equal to zero. Note that the conditions R(ν1),R(ν2) > 2/3 is equivalent to the decay

condition 2.8. To obtain the weight functions J and J do the change of variables that maps(ζ1, ζ2, ζ3, ζ4

)to(y1ζ1, y2ζ2, y1y2ζ3, y1y2ζ4

)on the integrals in Table 5.4 [Bump et al., 1988].

As the Fourier coefficients of Maass forms and Eisenstein series are Petersson inner products

involving a Whittaker function we make the following definition.

Definition 2.8.4 (Lebedev-Whittaker Transform). Let F : R2+ → C be a function satisfying

2.8. Define its Lebedev-Whittaker transform F# : D ×D → C as

F#(ν) =

∞∫0

∞∫0

W ∗ν (y)F (y)dy1dy2

(y1y2)3, (2.3)

where Wν(y) is the Whittaker function and D ⊂ C is of the form (−δ, δ)× iR for some δ > 0.

We can finally state the GL(3) Kuznetsov trace formula.

Theorem 2.8.5 (Kuznetsov Trace Formula). Let m1,m2, n1, n2 be positive integers. Let F :

R2+ → C be a function satisfying 2.8. Let {φj}∞j=0 be an orthonormal basis of Hecke-Maass forms

for SL(3,Z), ordered by Laplacian eigenvalue and normalized such that the first Fourier-Whittaker

28


coefficient Aj(1, 1) is equal to 1. Denote by ν(j) =(ν

(j)1 , ν

(j)2

)the type of the Maass form φj. Let

{uj} be a basis of Hecke-Maass form for SL(2,Z), ordered by Laplacian eigenvalue and normalized

to have norm 1. Denote by µj the type of of the Maass form uj. Then for some absolute constant

c > 0 the following equality holds:

C + Emin + Emax = Σ1 + Σ2a + Σ2b + Σ3, (2.4)

where

C =

∞∑j=0

Aj(m1,m2)Aj(n1, n2)

∣∣∣F#(ν

(j)1 , ν

(j)2

)∣∣∣26LAdφj (1)

3∏k=1

Γ

(1+3ν

(j)k

2

)Γ

(1−3ν

(j)k

2

) , (2.5)

with ν(j)3 := ν

(j)1 + ν

(j)2 ;

Emin =1

(4πi)2

i∞∫−i∞

i∞∫−i∞

Aν(m1,m2)Aν(n1, n2)∣∣F# (ν1, ν2)

∣∣23∏

k=1

∣∣ζ (1 + 3νk) Γ(

1+3νk2

)∣∣2 dν1 dν2, (2.6)

with ν3 := ν1 + ν2;

Emax =c

2πi

∞∑j=0

i∞∫−i∞

B(s,uj)(m1,m2)B(s,uj)(n1, n2)∣∣∣F#

(2µj3 , s− µj

3

)∣∣∣2LAduj (1)

∣∣Luj (1 + 3s)∣∣2 3∏k,l=1k 6=l

Γ(

1+δk−δl2

) ds, (2.7)

with σ1 = µj + s, σ2 = s− µj, σ3 = −2s;

Σ1 = 1{m1=n1m2=n2}

∞∫0

∞∫0

|F (y1, y2)|2 dy1dy2

(y1y2)3= 1{m1=n1

m2=n2}⟨F, F

⟩, (2.8)

Σ2a =∑ε=±1

∑d1|d2

m2d21=n1d2

S(εm1, n1, n2, d1, d2)

d1d2Jε,F

(√m1n1n2

d1d2

), (2.9)

Σ2b =∑ε=±1

∑d2|d1

m1d22=n2d1

S(εm2, n2, n1, d2, d1)

d1d2Jε,F ∗

(√m2n1n2

d1d2

), (2.10)

Σ3 =∑

ε1,ε2=±1

∞∑d1,d2=1

S(ε1m1, ε2m2, n1, n2, d1, d2)

d1d2Jε1,ε2,F

(√m1n2d1

d2,

√m2n1d2

d1

). (2.11)

Furthermore, the weight functions J , J given by

29


Jε,F (A) = A−2

∞∫0

∞∫0

∞∫−∞

∞∫−∞

F (Ay1, y2)e(−εAx1y1)F

(y2

√x2

1 + x22 + 1

x21 + 1

,A

y1y2

√x2

1 + 1

x21 + x2

2 + 1

)

× e(y2

x1x2

x21 + 1

+A

y1y2

x2

x21 + x2

2 + 1

)dx1dx2dy1dy2

y1y22

, (2.12)

Jε1,ε2,F (A1, A2) = (A1A2)−2

∞∫0

∞∫0

∞∫−∞

∞∫−∞

∞∫−∞

F (A1y1, A2y2)e(−ε1A1x1y1 − ε2A2x2y2)

× F

(A2

y2

√(x1x2 − x3)2 + x2

1 + 1

x22 + x2

3 + 1,A1

y1

√x2

2 + x23 + 1

(x1x2 − x3)2 + x21 + 1

)

× e(−A2

y2

x1x3 + x2

x22 + x2

3 + 1− A1

y1

x2(x1x2 − x3) + x1

(x1x2 − x3)2 + x21 + 1

)dx1dx2dx3dy1dy2

y1y2. (2.13)

Proof. We shall compute

⟨Pn1,n2 ,Pm1,m2

⟩n1n2m1m2

in two different ways to obtain the two sides of the equality.

To obtain the left hand side of 2.4 apply the spectral decomposition (Theorem 2.5.6) to both

Poincare series and also decompose the space of Maass form via a basis of Hecke-Maass form as in

the statement of the theorem. Note that these Poincare series satisfy the conditions of Theorem 2.5.6

according to Lemma 2.8.1 and Remark 2.8.2. It follows that

⟨Pn1,n2 , Pm1,m2

⟩n1n2m1m2

=

∞∑j=0

⟨φj , Pm1,m2

⟩⟨φj , Pn1,n2

⟩‖φj‖(n1n2m1m2)

+1

(4πi)2

i∞∫−i∞

i∞∫−i∞

⟨Emin(∗, ν), Pm1,m2

⟩⟨Emin(∗, ν), Pn1,n2

⟩n1n2m1m2

dν1 dν2

+1

2πi

∞∑j=0

i∞∫−i∞

⟨Emax(∗, s, uj), Pm1,m2

⟩⟨Emax(∗, s, uj), Pn1,n2

⟩n1n2m1m2

ds .

Applying Lemma 2.8.1 to compute the inner products involving the Eisenstein series one obtains

the terms Emin and Emax. Note that the Hecke-Maass forms for SL(2,Z) satisfy the Ramanujan

conjecture at infinity so their type is purely imaginary. The Gamma factors in the integrands of

Emin and Emax appear as the quotient between W ∗ν and Wν .

30


To obtain the term associated to the Hecke-Maass forms for SL(3,Z) it is necessary to compute

the inner products⟨φj , Pm1,m2

⟩. Start by expanding Pm1,m2 using its definition as a series to

obtain

⟨φj , Pm1,m2

⟩=

∫SL(3,Z)\h3

φj(z)∑

γ∈U3(Z)\SL(3,Z)

Fm1,m2(γz)d∗z.

As φj is invariant under the action of SL(3,Z), one obtains⟨φj , Pm1,m2

⟩=

∫U3(Z)\h3

φj(z)Fm1,m2(z)d∗z

=

∞∫0

∞∫0

F (m1y1,m2y2)

1∫0

1∫0

1∫0

φj(z)e−2πi(m1x1+m2x2) dx1 dx2 dx3

dy1dy2

(y1y2)3

=

∞∫0

∞∫0

Aν(m1,m2)

m1m2Wν (m1y1,m2y2)F (m1y1,m2y2)

dy1dy2

(y1y2)3,

where the last equality follows from Remark 2.3.8. To obtain the cuspidal term C do a change of

variables (y1, y2) 7→(y1m1, y2m2

)and refer to Equation 2.12 [Goldfeld and Kontorovich, 2013] for an

explicit expression for the L2-norm of φj ,

‖φj‖ = 6LAdφj (1)

3∏k=1

Γ

(1 + 3ν

(j)k

2

)Γ

(1− 3ν

(j)k

2

).

Collecting the results obtained so far we get⟨Pn1,n2 , Pm1,m2

⟩n1n2m1m2

= C + Emin + Emax.

We will now compute the inner product⟨Pn1,n2 , Pm1,m2

⟩directly using the Fourier expansion of

Poincare series as in Theorem 2.8.3.⟨Pn1,n2 , Pm1,m2

⟩=

∫U3(Z)\h3

Pn1,n2(z)Fm1,m2(z)d∗z

=

∞∫0

∞∫0

F (m1y1,m2y2)

1∫0

1∫0

1∫0

Pn1,n2(z)e−2πi(m1x1+m2x2) dx1 dx2 dx3dy1dy2

(y1y2)3

=

∞∫0

∞∫0

F (m1y1,m2y2) (S1 + S2a + S2b + S3)dy1dy2

(y1y2)3,

31


where S1, S2a, S2b, S3 are as in Theorem 2.8.3. We then proceed to compute the integrals∞∫0

∞∫0

F (m1y1,m2y2)Sτdy1dy2(y1y2)3

for τ ∈ {1, 2a, 2b, 3}. Recall the definitions of Σ1,Σ2a,Σ2b and Σ3

as given by equations 2.8, 2.9, 2.10 and 2.11, respectively. We get

∞∫0

∞∫0

F (m1y1,m2y2)S1dy1dy2

(y1y2)3= n1n2m1m2Σ1,

after changing variables (y1, y2) 7→(y1m1, y2m2

). Furthermore,

∞∫0

∞∫0

F (m1y1,m2y2)S2ady1dy2

(y1y2)3=∑ε=±1

∑d1|d2

m2d21=n1d2

S(εm1, n1, n2, d1, d2)

×∞∫

0

∞∫0

F (m1y1,m2y2)JF (y1, y2, εm1, n1, n2, d1, d2)dy1dy2

(y1y2)3

=∑ε=±1

∑d1|d2

m2d21=n1d2

S(εm1, n1, n2, d1, d2)n1n2m1m2

d1d2Jε(√

m1n1n2

d1d2

)

= n1n2m1m2Σ2a,

where the penultimate equality follows from the change of variables

(y1, y2) 7→(√

n1n2

m1d1d2y1,

d21

n1d2y2

).

A similar calculation gives

∞∫0

∞∫0

F (m1y1,m2y2)S2bdy1dy2

(y1y2)3= n1n2m1m2Σ2b.

32


Finally,

∞∫0

∞∫0

F (m1y1,m2y2)S3dy1dy2

(y1y2)3=

∑ε1,ε2=±1

∞∑d1,d2=1

S(ε1m1, ε2m2, n1, n2, d1, d2)

×∞∫

0

∞∫0

F (m1y1,m2y2)JF (y1, y2, ε1m1, ε2m2, n1, n2, d1, d2)dy1dy2

(y1y2)3

=∑

ε1,ε2=±1

∞∑d1,d2=1

S(ε1m1, ε2m2, n1, n2, d1, d2)n1n2m1m2

d1d2

× Jε1,ε2(√

m1n2d1

d2,

√m2n1d2

d1

)= n1n2m1m2Σ3,

where the penultimate equality follows from the change of variables (y1, y2) 7→(√

n2d1m2

1d2y1,

n1d2m2

2d1y2

).

It follows from the above computations that

C + Emin + Emax =〈Pn1,n2 , Pm1,m2〉n1n2m1m2

= Σ1 + Σ2a + Σ2b + Σ3,

as desired.

33

CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

Chapter 3

Inverse Lebedev-Whittaker Transform

This chapter is devoted to the Lebedev-Whittaker transform on GL(n), in particular to its inverse

transform. We will analyse this transform in both the archimedean (for the real group GL(3,R))

and nonarchimedean (for p-adic groups GL(n,Qp)) cases. The inverse transform for GL(3,R) will

be crucial to the application of GL(3) Kuznetsov trace formula that shall be described in the

following chapter.

3.1 Archimedean Case

On archimedean local fields Whittaker functions have been thoroughly studied and an inversion

formula for the Lebedev-Whittaker transform in known due to the works of Wallach [Wallach,

1992] and Goldfeld-Kontorovich [Goldfeld and Kontorovich, 2012]. We will follow in this section

the presentation of this inversion formula in the latter. We shall restrict ourselves to the inverse

transform for GL(n,R) for n = 3 as it will be the only case we will need to use.

Definition 3.1.1 (Lebedev-Whittaker Inverse). Let g : iR2 → C. Define its Lebedev-Whittaker

inverse g[ by

g[(y) :=1

(πi)2

i∞∫−i∞

i∞∫−i∞

g(ν)W ∗ν (y)dν1ν2

3∏j=1

Γ(

3νj2

)Γ(−3νj

2

) ,where ν3 = ν1 + ν2, provided the integral converges.

Remark 3.1.2. A sufficient condition for the convergence of the integral is that g extends holo-

34


morphically to a function defined on D2, with D = (−δ, δ)× iR, and satisfying

|g(ν)| � exp

−3π

4

3∑j=1

|νj |

3∏j=1

(1 + |νj |)−10 (3.1)

See Section 2.2 [Goldfeld and Kontorovich, 2013].

Theorem 3.1.3 (Lebedev-Whittaker Inversion). Let δ > 0 and g : D2 → C holomorphic, with

D = (−δ, δ)× iR. Assume that g(ν1, ν2) is invariant under permutations of (2ν1 +ν2, ν2−ν1, −ν1−

2ν2), and satisfies 3.1. Then

(g])[(ν) = g(ν)

for ν ∈ D2. Furthermore,

⟨g#, g#

⟩:=

1

(πi)2

i∞∫−i∞

i∞∫−i∞

∣∣g#(ν)∣∣2 dν1dν2

3∏j=1

Γ(

3νj2

)Γ(−3νj

2

) = 〈g, g〉 . (3.2)

Proof. See Theorem 1.6, Corollary 1.9 and Theorem 3.4 [Goldfeld and Kontorovich, 2012]. Note

that our definitions of f 7→ f ] and g 7→ g[ differ from the ones in [Goldfeld and Kontorovich, 2012]

by conjugation of the Whittaker functions. This does not affect the result as W ∗ν = W ∗ν . The proof

in Theorem 3.4 shows the equality of (g])[ and g on (iR)2 and this equality can be extended to D2

as both functions are holomorphic.

3.2 Nonarchimedean Case

Let G = GLn(Qp) for which we have the Iwasawa decomposition G = UTK, where U = U(Qp) is

the unipotent radical of the standard Borel subgroup, T = T (Qp) is the torus of diagonal matrices

and K = GLn(Zp) is the maximal compact subgroup. Let Z be the center of G. Let ψ be a

character on U induced from a character ψ′ on Qp via

ψ(u) = ψ′

(n−1∑i=1

ui,i+1

), (where u = (ui,j) ∈ U) .

Definition 3.2.1 (Spherical Hecke Algebra). Define the spherical Hecke algebra KHK to be

the set of locally constant, compactly supported function f : GLn(Qp)→ C satisfying

f(k1gk2) = f(g)

35


for k1, k2 ∈ K, g ∈ GLn(Qp).

One can now define Whittaker functions on GLn(Qp).

Definition 3.2.2 (Whittaker Function on GLn(Qp)). A Whittaker function on GLn(Qp) is a

K-finite smooth function of moderate growth W : GLn(Qp)→ C that satisfies

W (ug) = ψ(u)W (g)

for some fixed character ψ as defined above and any u ∈ U , and∫G

W (gx)φ(x) dx = λ(φ)W (g)

for each algebra homomorphism λ : KHK → C, φ ∈ KHK and g ∈ GLn(Qp).

Let h : Z(Qp)\T (Qp) → C be a smooth function. The p-adic Whittaker transform of h is

defined to be ∫Z(Qp)\T (Qp)

h(t)W (t)d×t,

where W is a Whittaker function on GLn(Qp).

The main goal of the present section is to obtain an inversion formula for the p-adic Whittaker

transform. The precise inversion formula for the Whittaker transform is given in Theorem 3.2.10.

The work in this section is inspired by the one of Goldfeld-Kontorovich [Goldfeld and Kontorovich,

2012] for the archimedean case. A crucial step in our approach is the computation of an integral of

a product of two p-adic Whittaker functions. One can view this computation as a nonarchimedean

analogue of Stade’s formula [Stade, 2001].

3.2.1 Inversion Formula

The goal of this subsection is to provide all the necessary definitions and results that allow us

to state the main theorem of this section 3.2.10, which gives an inverse formula for the p-adic

Whittaker transform.

We start by noting Whittaker functions on GLn(Qp) arise naturally from certain irreducible

representation of this group.

36


Definition 3.2.3 (Whittaker Function associated to a Representation π). Let V be a

complex vector space and let (π, V ) be an irreducible generic representation of G. Then the space

HomU (π, ψ) = {f : V → C linear : f(π(u)v) = ψ(u)f(v), ∀v ∈ V, u ∈ U}

is one-dimensional generated by an element λ. For v ∈ V a newform, define a Whittaker function

W associated to π as

W (g) = λ(π(g)v).

Remark 3.2.4. As the space of newforms in V is one-dimensional then the Whittaker function

associated to a representation π is well-defined up to a constant. Furthermore, a Whittaker function

associated to an irreducible generic representation π of GLn(Qp) is a nonzero Whittaker function

on GLn(Qp). For proofs of these statements see [Jacquet et al., 1981] and [Gelfand and Kazhdan,

1975].

In order to explicitly describe Whittaker functions associated to certain representations of

GLn(Qp) we proceed to quote a theorem of Shintani [Shintani, 1976]. For g = zutk ∈ GLn(Qp)

define the Iwasawa coordinates

g = zu

t1 · · · tn−1

. . .

t1

1

k,

where z ∈ Z, u ∈ U , t ∈ T and k ∈ K.

Theorem 3.2.5 (Shintani). Let π be an irreducible unramified generic representation of GLn(Qp)

and Wπ the Whittaker function associated to a representation π (normalized such that Wπ(id) = 1,

where id denotes the n × n identity matrix) as in Definition 3.2.3. We can write the L-function

associated to π as

L(s, π) =

n∏i=1

(1− αip−s

)−1,

37


where αi ∈ C are all nonzero and α1 · · ·αn = 1. Then, for

t =

t1 · · · tn−1

. . .

t1

1

we have

Wπ(t) =

δ1/2(t)s-log |t|p(α), if ti ∈ Zp for i = 1, . . . , n− 1,

0, otherwise;

where

δ(t) =n−1∏k=1

|tk|k(n−k)p ,

log |t|p = (log |t1|p, . . . , log |tn−1|p)

and

sλ(α) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

αn−1+λ1+λ2+···+λn−1

1 αn−1+λ1+λ2+···+λn−1

2 · · · αn−1+λ1+λ2+···+λn−1n

αn−2+λ1+λ2+···+λn−2

1 αn−2+λ1+λ2+···+λn−2

2 · · · αn−2+λ1+λ2+···+λn−2n

......

. . ....

α1+λ11 α1+λ1

2 · · · α1+λ1n

1 1 · · · 1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

αn−11 αn−1

2 · · · αn−1n

αn−21 αn−2

2 · · · αn−2n

......

. . ....

α1 α2 · · · αn

1 1 · · · 1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣is the Schur polynomial, where λ = (λ1, . . . , λn) and λi ∈ Z.

Proof. See [Shintani, 1976]. For the relation between the L-function L(s, π) and the Whittaker

function Wπ see Subsection 3.1.3 [Cogdell, 2008].

Remark 3.2.6. We will also use the notation Wα, for α ∈ (C\{0})n, to denote the function

Wα(t) =


0, otherwise.

38


This coincides with the definition of Wπ when α = (α1, . . . , αn) are the Langlands parameters

associated to π.

We are now able to define the p-adic Whittaker transform precisely.

Definition 3.2.7 (p-adic Whittaker Transform). Let h : Z(Qp)\T (Qp) → C. Define the

Whittaker transform h] : Cn → C by

h](α) =

∫T (Zp)\T (Qp)

h(t)Wα(t)d×t,

provided the integral converges, where

d×t =n−1∏k=1

|tk|−k(n−k)p

dtk|tk|p

.

Remark 3.2.8. A sufficient condition for the convergence of the above integral is that h satisfies

the decay condition

|h(t)|p � δ1/2(t)|t1 · · · tn−1|τ+εp

for any ε > 0 and τ = max1≤i≤n−1

− log|αi|p.

Definition 3.2.9 (Inverse Whittaker Transform). Let H : Sn → C holomorphic, where S is

an open annulus containing the unit circle. Further, assume that H is invariant under the trans-

formations (α1, . . . , αn) 7→ (ασ(1), . . . , ασ(n)), for all (α1, . . . , αn) ∈ Sn and for every permutation

σ ∈ Sn. Let

Wα(t) =


0, otherwise;

where α = (α1, . . . , αn). Define the inverse Whittaker transform of H, H[ : Z(Qp)\T (Qp)→ C by

H[(t) =1

n!(2πi)n−1

∫T

H(β)W1/β(t)n∏

i,j=1i 6=j

(βi − βj)dβ1 · · · dβn−1

β1 · · ·βn−1,

where T ={

(β1, . . . , βn−1) ∈ Cn−1 : |βi|C = 1}

, 1/β = (1/β1, . . . , 1/βn) and βn = 1β1···βn−1

.

The inverse transform is given in the following theorem.

39


Theorem 3.2.10 (Inversion Formula). Let H : Sn → C holomorphic, where S is an open annu-

lus containing the unit circle. Let D be a domain where (H[)] converges, and assume{

(α1, . . . , αn) ∈

Cn : |αi|C = 1}⊂ D ⊂ Sn. Then,

(H[)](α) = H(α)

for α1 · · ·αn = 1.

Corollary 3.2.11. Let h : Z(Qp)\T (Qp)→ C. For

t =

t1 · · · tn−1

. . .

t1

1

assume h(t) is compactly supported on the region {ti ∈ Zp\{0} : i = 1, . . . , n − 1}, and that

h(t) = f(−|t1|p, . . . ,−|tn−1|p) for some function f : Zn−1≥0 → C. Then

(h])[ = h.

We postpone the proofs of the results stated above to the last subsection.

3.2.2 Background

Here we show two of the ingredients needed to prove the inverse transform theorem. Both results

can also be found in Section 4 of [Bump, 2005]. They are included here as their proofs are concise

and improve the exposition. The first of these is a version of Cauchy’s identity.

Lemma 3.2.12 (Cauchy’s Identity). Let sλ be the Schur polynomial defined in the previous

section. Let α1, . . . , αn, β1, . . . , βn ∈ C such that |αiβj | < 1 for every 1 ≤ i, j ≤ n. Then,

∑m1,...,mn−1≥0

sm (α) sm (β) =1− α1 · · ·αnβ1 · · ·βn

n∏i,j=1

(1− αiβj),

where m = (m1, . . . ,mn−1).

40


Proof. This proof follows [Goldfeld, 2006] and [Macdonald, 1979]. We start by stating the Cauchy

determinant identity (lemma 7.4.18 in [Goldfeld, 2006])∣∣∣∣∣∣∣∣∣1

1−α1β1· · · 1

1−αnβ1...

. . ....

11−α1βn

· · · 11−αnβn

∣∣∣∣∣∣∣∣∣ =

∏1≤i<j≤n

(αi − αj)∏

1≤i<j≤n(βi − βj)

n∏i,j=1

(1− αiβj).

Expanding each term 11−αiβj as 1 + αiβj + α2

i β2j + · · · , we obtain

∣∣∣∣∣∣∣∣∣1

1−α1β1· · · 1

1−αnβ1...

. . ....

11−α1βn

· · · 11−αnβn

∣∣∣∣∣∣∣∣∣ =∑

l1,...,ln≥0

∑σ∈Sn

sgn(σ)αl1σ(1) · · ·αlnσ(n)β

l11 · · ·β

lnn .

Notice that if the li are not all distinct then the sum over Sn vanishes. Therefore∣∣∣∣∣∣∣∣∣1

1−α1β1· · · 1

1−αnβ1...

. . ....

11−α1βn

· · · 11−αnβn

∣∣∣∣∣∣∣∣∣ =∑

l1>...>ln≥0σ,τ∈Sn

sgn(σ)sgn(τ)αl1σ(1) · · ·αlnσ(n)β

l1τ(1) · · ·β

lnτ(n)

=∑

l1>...>ln≥0

∣∣∣∣∣∣∣∣∣αl11 · · · αl1n...

. . ....

αln1 · · · αlnn

∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣βl11 · · · βl1n...

. . ....

βln1 · · · βlnn

∣∣∣∣∣∣∣∣∣ .By making the substitution (l1, . . . , ln) = (m0+· · ·+mn−1+(n−1),m0+· · ·+mn−2+(n−2), . . . ,m0)

and dividing by∏

1≤i<j≤n(αi − αj)

∏1≤i<j≤n

(βi − βj) we obtain

1n∏

i,j=1(1− αiβj)

=∑

m0,m1,...,mn−1≥0

sm (α) sm (β) (α1 · · ·αnβ1 · · ·βn)m0 ,

where right hand side simplifies to

1

1− α1 · · ·αnβ1 · · ·βn

∑m1,...,mn−1≥0

sm (α) sm (β) .

Multiplying out by 1− α1 · · ·αnβ1 · · ·βn finishes the proof.

The second necessary ingredient is an identity for the integral of a product of two Whittaker

function, as in [Stade, 2001].

41


Proposition 3.2.13. Let α1, . . . , αn, β1, . . . , βn ∈ C and ε > 0. Let Wα, Wβ be Whittaker functions

as defined in Definition 3.2.9. Then∫Z(Qp)\T (Qp)

Wα(t)Wβ(t)n−1∏k=1

|tk|ε(n−k)p d×t =

1− α1···αnβ1···βnpεn

n∏i,j=1

(1− αiβj

pε

) .Proof. We start by applying Theorem 3.2.5 to write down the Whittaker functions explicitly:∫

Z(Qp)\T (Qp)

Wα(t)Wβ(t)

n−1∏k=1

|tk|ε(n−k)p d×t

=

∫Zn−1p

δ(t)s-log |t|p(α)s-log |t|p (β)

n−1∏k=1

|tk|−(k−ε)(n−k)p

dtk|tk|p

=

∫Zn−1p

s-log |t|p(α)s-log |t|p (β)n−1∏k=1

|tk|ε(n−k)p

dtk|tk|p

.

Breaking up the region of integration by absolute value we get∫Z(Qp)\T (Qp)

Wα(t)Wβ(t)

n−1∏k=1

|tk|ε(n−k)p d×t

=∑

m1,...,mn−1≥0

n−1 times︷︸︸︷∫· · ·∫

pmkZ×p

s-log |t|p(α)s-log |t|p (β)

n−1∏k=1

|tk|ε(n−k)p

dtk|tk|p

=∑

m1,...,mn−1≥0

sm(α)sm (β) p−ε(n−1)m1−ε(n−2)m2−···−εmn−1

=∑

m1,...,mn−1≥0

sm

(α

pε

)sm (β)

=1− α1···αnβ1···βn

pεn

n∏i,j=1

(1− αiβj

pε

) ,where the last equality follows from Cauchy’s identity 3.2.12.

3.2.3 Proof of Inversion Formula

We are now ready to proceed to the proof of the main theorem in this section. Restate the theorem

as

H(α) =

∫Z(Qp)\T (Qp)

H[(t)Wα(t)d×t.

42


Recall that α ∈ D and α1 · · ·αn = 1. Further assume that |α1|C = · · · = |αn|C = 1. We can later

remove this assumption as both sides of the equality are holomorphic. Note that H is invariant

under permutations of (α1, . . . , αn) as the Whittaker function also has those symmetries. Define

Hε(α) =

∫Z(Qp)\T (Qp)

H[(t)Wα(t)n−1∏k=1

|tk|ε(n−k)p d×t.

Then

limε→0

Hε(α) =

∫Z(Qp)\T (Qp)

H[(t)Wα(t)d×t.

It suffices to show that limε→0

Hε(α) = H(α) as well.

Replacing H[(t) by its explicit formula and swapping the order of integration we get

Hε(α) =1

n!(2πi)n−1

∫T

∫Z(Qp)\T (Qp)

Wα(t)W1/β(t)n−1∏k=1

|tk|ε(n−k)p d×t

×H(β)

n∏i,j=1i 6=j

(βi − βj)dβ1 · · · dβn−1

β1 · · ·βn−1.

We use Proposition 3.2.13 to compute the innermost integrals and get

Hε(α) =1

n!(2πi)n−1

∫T

H(β)(

1− 1pεn

)n∏

i,j=1

(βj − αi

pε

) n∏i,j=1i 6=j

(βi − βj)dβ1 · · · dβn−1

β1 · · ·βn−1.

Now shift each variable βi (i = 1, . . . , n − 1) to the contour given by the equation |z|C = 1pε+ε′

, in

order. For the sake of clearness, we shall assume that all the αi are distinct. When the contour of

integration for β1 is shifted one picks up several residues at

β1 =α1

pε, . . . , β1 =

αnpε.

For which residue integral obtained in this manner one can shift the contour of integration for β2

picking up n− 1 residues in the process. Repeating this process for which βi, i = 3, . . . , n− 1 one

obtains

Hε(α) =1

n!

∑σ∈Sn

Rσε + Iε,ε′ ,

43


where

Rσε =

H(ασ(1)pε , . . . ,

ασ(n)pε

)(1− 1

pεn

) n−1∏i=1

(ασ(i)pε − p

(n−1)εασ(n)

)(p(n−1)εασ(n) −

ασ(n)pε

) n−1∏i=1

(ασ(i)pε −

ασ(n)pε

) n−1∏i=1

ασ(i)pε

and Iε,ε′ is a sum of residue integrals with each integrand bounded (in absolute value) by

CH

(1− 1

pnε

)pε(n

3+n)

1− 1pε′

for some constant CH > 0 depending only on the function H. Therefore,

limε→0Rσε = lim

ε→0

H(ασ(1)pε , . . . ,

ασ(n)pε

)n∏i=1

ασ(i)pε

(1− 1

pεn

)(p(n−1)ε − 1

pε

)n−1∏i=1

(ασ(i)pε − p

(n−1)εασ(n)

)n−1∏i=1

(ασ(i)pε −

ασ(n)pε

)

=H(ασ(1), . . . , ασ(n)

)n∏i=1

ασ(i)

n−1∏i=1

(ασ(i) − ασ(n)

)n−1∏i=1

(ασ(i) − ασ(n)

)= H

(ασ(1), . . . , ασ(n)

)= H(α)

and

limε→0|Iε,ε′ | �H lim

ε→0

(1− 1

pnε

)pε(n

3+n)

1− 1pε′

= 0.

In conclusion, we obtain

limε→0

Hε(α) =1

n!

∑σ∈Sn

H(α) = H(α),

as desired. If the αi are not all distinct then the original integrand would have higher order poles.

This means that the residue sum would have a small number of residue but some residues would

contribute to the sum with a multiple of H(α).

To prove the corollary one simply needs to compute the image of the inverse transform H 7→ H[

as any function h in the image of this mapping will satisfy (h])[ = h. Simply choose H such that

h = H[, and apply the inversion formula 3.2.10 to H, to obtain (h])[ = ((H[)])[ = H[ = h. We

start by noting that any function h satisfying the conditions in the corollary is a finite sum of scalar

multiples of function of the form:

44


fλ1,...,λn−1(t) =

1, if log |ti|p = −λi for i = 1, . . . , n− 1,

0, otherwise;

where λ1, . . . , λn−1 are nonnegative integers. Therefore, it suffices to show that all function

fλ1,...,λn−1 for λ1, . . . , λn−1 nonnegative integers are in the image of the map of the inverse transform

H 7→ H[. Let H(β) =Wβ(pλ)

δ(pλ), where

pλ =

pλ1 · · · pλn−1

. . .

pλ1

1

.

Then

H[(t)

=1

δ(pλ)n!(2πi)n−1

∫T

Wβ(pλ)W1/β(t)

n∏i,j=1i 6=j

(βi − βj)dβ1 · · · dβn−1

β1 · · ·βn−1

=δ1/2(t)

δ1/2(pλ)n!(2πi)n−1

∫T

sλ(β)s-log |t|p(1/β)n∏

i,j=1i 6=j

(βi − βj)dβ1 · · · dβn−1

β1 · · ·βn−1

=δ1/2(t)

δ1/2(pλ)n!(2πi)n−1

∫T

∑σ,τ∈Sn

(β

(n−1)+λ1+···+λn−1

σ(1) · · ·β1+λ1σ(n−1)

)

×(β−(n−1)+log |t1|p+···+log |tn−1|pτ(1) · · ·β−1+log |t1|p

τ(n−1)

) dβ1 · · · dβn−1

β1 · · ·βn−1

=δ1/2(t)

δ1/2(pλ)n!(2πi)n−1

×∫T

∑σ∈Sn

(βλ1+···+λn−1+log |t1|p+···+log |tn−1|pσ(1) · · ·βλ1+log |t1|p

σ(n−1)

) dβ1 · · · dβn−1

β1 · · ·βn−1

=δ1/2(t)

δ1/2(pλ)fλ1,...,λn−1(t)

= fλ1,...,λn−1(t),

as desired.

45


3.2.4 Local L-function of a Symmetric Square Lift

Let L(s, π) be the L-function associated to an irreducible automorphic representation of GL(3,AQ)

with local factors (at the unramified places p) given by

Lp(s, π) =(1− α2

1p−s)−1 (

1− α1α2p−s)−1 (

1− α22p−s)−1

= hs,p(α),

with α1α2 = 1 and R(s) large enough. These are the L-factors that occur when π is the symmetric

square lift of an irreducible automorphic representation of GL(3,AQ). We shall obtain an integral

representation for the L-factors Lp(s, π) using Theorem 3.2.10.

By Theorem 3.2.10 we can write

Lp(s, π) =

∫T (Zp)\T (Qp)

(hs,p)[(t)Wα(t)d×t,

with α = (α1, α2), assuming R(s) is large enough. We will now compute (hs,p)[ explicitly.

(hs,p)[(t) =

1

4πi

∫|β1|=1

hs,p(β)W1/β(t)(β1 − β−11 )(β−1

1 − β1)dβ1

β1

=

(|t1|1/2p

1− p−s

)1

4πi

×∫

|β1|C=1

(1

1− β21p−s

)(1

1− β−21 p−s

)W1/β(t)

(βλ+1

1 − β−(λ+1)1

) (β−1

1 − β1

) dβ1

β1,

with t =

t1 0

0 1

and |t1|p = p−λ.

Note that in the last integral the integrand i(β1) has possible poles at inside the unit circle at

β = 0,±p−s/2. By the residue theorem,

(hs,p)[(t) =

|t1|1/2p

2(1− p−s)

(Resβ1=0 i(β1) + Resβ1=p−s/2 i(β1) + Resβ1=−p−s/2 i(β1)

),

where the residues are equal to

46


Resβ1=0 i(β1) =

ps/2

((ps/2)

λ+1−(ps/2)−λ−1

)(1+p−s) , if λ even,

0, if λ odd;

Resβ1=p−s/2 i(β1) =ps/2

((ps/2

)λ+1 −(ps/2

)−λ−1)

2 (1 + p−s);

Resβ1=−p−s/2 i(β1) = −ps/2

((−ps/2

)λ+1 −(−ps/2

)−λ−1)

2 (1 + p−s).

Combining all the above terms we obtain

(hs,p)[(t) =

ps|t1|−(s−1)/2

p

1−p−2s , if λ even,

0, if λ odd.

47

CHAPTER 4. ORTHOGONALITY RELATION

Chapter 4

Orthogonality Relation

4.1 Introduction

In this chapter, we prove an orthogonality relation for the Fourier coefficients of a ”thin” family of

Hecke-Maass forms for SL(3,Z). In addition, we also obtain a Weyl’s law type result (weighted by

some residues) for the same family of Maass forms. Both results slightly generalize previous work

by the author [Guerreiro, 2015].

Let φj (j = 1, 2, . . . ) be a set of orthogonal Hecke-Maass forms for SL(3,Z) of type ν(j) =(ν

(j)1 , ν

(j)2

), and define ν

(j)3 = −ν(j)

1 − ν(j)2 . Note that for tempered forms the type is purely

imaginary. Recall that for a particular Hecke-Maass form φ of type ν = (ν1, ν2) define ν3 = −ν1−ν2,

the Langlands parameters are

α1 = 2ν1 + ν2, α2 = ν2 − ν1, α3 = −ν1 − 2ν2.

and the Laplace eigenvalue of φ is given by

λφ = 1− 3(ν21 + ν2

2 + ν23) = 1− (α2

1 + α22 + α2

3).

For T � 1, ν = (ν1, ν2) ∈ C2 (with ν3 = −ν1 − ν2), R > 0 and κ a positive number congruent

48


to 2 (mod 4), we define

hT,R,κ(ν) =(T (α1/R)κ + T (α2/R)κ + T (α3/R)κ

)2e2(α2

1+α22+α2

3)/T 2

×

(3∏j=1

Γ(

2+R+3νj4

)Γ(

2+R−3νj4

))2

3∏j=1

Γ(

1+3νj2

)Γ(

1−3νj2

) .

(4.1)

This function is essentially supported on the region where |ν1|, |ν2|, |ν3| � T . If φ is tempered

and |ν1|, |ν2|, |ν3| < T then hT,R,κ(ν) is real valued and positive. We estimate this function in three

regions. If one of the αi is equal to 0, then

c(1)R

[3∏i=1

(1 + |νi|)

]R≤ hT,R,κ(ν) ≤ c(2)

R

[3∏i=1

(1 + |νi|)

]R,

for some c(2)R ≥ c

(1)R > 0. If one of the |αi| is smaller than R, then

hT,R,κ(ν)�R[(1 + |ν1|)(1 + |ν2|)(1 + |ν3|)]R

T 2.

If |α1|, |α2|, |α3| > R1+ε, then

hT,R,κ(ν)� 1

TRεκ.

Theorem 4.1.1. For j = 1, 2, . . . , let φj be a set of orthogonal Hecke-Maass forms for SL(3,Z)

with spectral parameters ν(j) =(ν

(j)1 , ν

(j)2

). Let hT,R,κ be given as in (4.1). We have that for fixed

R > 50, ε > 0, κ ≡ 2 (mod 4) positive and some cR,κ > 0,

∑j≥1

∣∣hT,R,κ(ν(j))∣∣

Lj= cR,κ

T 4+3R

(log T )1/κ+OR,ε(T 3+3R+ε), (T →∞).

Moreover, for positive integers m1,m2, n1, n2, we have

∑j≥1

Aj(m1,m2)Aj(n1, n2)

∣∣hT,R,κ(ν(j))∣∣

Lj

=

∑j≥1

|hT,R,κ(ν(j))|Lj +OR,ε

((m1m2n1n2)6 T 3R+3+ε

), if m1=n1

m2=n2,

OR,ε(

(m1m2n1n2)6 T 3R+3+ε

), otherwise.

49


Here the weights Lj are the L-values LAdφj (1).

Remark 4.1.2. This result generalizes previous work by the author in [Guerreiro, 2015], where the

case κ = 2 was shown.

Remark 4.1.3. Note that an analogous result in [Goldfeld and Kontorovich, 2013] for all Hecke-

Maass forms for SL(3,Z) with spectral parameters |ν1|, |ν2|, |ν3| � T yields a main term of size

T 5+3R. This implies that the family of Maass forms picked out by our test function hT,R,κ is indeed

significantly thinner (by a factor of T (log T )1/κ) than the family of all Hecke-Maass forms for

SL(3,Z).

Remark 4.1.4. The test functions hT,R,κ appearing in this theorem are a product of three terms

chosen with the following objectives: the first exponential term contributes with polynomial decay

when all |α1|, |α2|, |α3| � 0 and exponential decay when all |α1|, |α2|, |α3| � T ε; the second expo-

nential term contributes with exponential decay when one of |νi| > T 1+ε; the product of Gamma

factors is already partially present in the Kuznetsov trace formula and its particular form is such

that it has polynomial growth in ν. We note that the methods presented here have also been carried

out for a different family of test functions in [Goldfeld and Kontorovich, 2013].

Remark 4.1.5. Blomer [Blomer, 2013] shows that the weights Lj are bounded by

(3∏i=1

(1 +

∣∣∣ν(j)i

∣∣∣))−1

� Lj �ε

(3∏i=1

(1 +

∣∣∣ν(j)i

∣∣∣))ε (4.2)

and, conjecturally, the lower bound is expected to be

(3∏i=1

(1 +

∣∣∣ν(j)i

∣∣∣))−ε �ε Lj . (4.3)

50


4.2 Bound for the Inverse Lebedev-Whittaker Transform

Let

F#T,R,κ(ν1, ν2) =

(T (α1/R)κ + T (α2/R)κ + T (α3/R)κ

)e(α2

1+α22+α2

3)/T 2

×

3∏j=1

Γ

(2 +R+ 3νj

4

)Γ

(2 +R− 3νj

4

) .

As F#T,R has enough exponential decay on a strip |<(ν1)|, |<(ν2)| < ε (see 3.1) then by Lebedev-

Whittaker inversion as in Theorem 3.1.3,

FT,R,κ(y) =1

(πi)2

i∞∫−i∞

i∞∫−i∞

F#T,R,κ(ν)W ∗ν (y)

dν1dν2

3∏j=1

Γ(

3νj2

)Γ(−3νj

2

) .We also have the Parseval-type identity as in the second part of Theorem 3.1.3

〈FT,R,κ, FT,R,κ〉 =1

(πi)2

i∞∫−i∞

i∞∫−i∞

∣∣∣F#T,R,κ(ν)

∣∣∣2 dν1dν2

3∏j=1

Γ(

3νj2

)Γ(−3νj

2

) =⟨F#T,R,κ, F

#T,R,κ

⟩. (4.4)

Proposition 4.2.1. Fix C1, C2 > 0, R > 3 max(C1, C2) + 6 and ε > 0. For any y1, y2 > 0, T � 1,

we have

|FT,R,κ(y)| �C1,C2,R,ε y1y2T3R/2+11/2+C1/2+C2/2+ε

(y1

T

)C1(y2

T

)C2

. (4.5)

Proof. This proof follows the same steps as the proof of Proposition 3.1 in [Guerreiro, 2015], where

it was carried out for FT,R,2. We start by writing out the representation of W ∗ν as an inverse Mellin

transform as in Proposition 2.3.9,

W ∗ν (y) =y1y2π

3/2

(2πi)2

∫(C2)

∫(C1)

∏3j=1 Γ

(s1−αj

2

)Γ(s2+αj

2

)4πs1+s2Γ

(s1+s2

2

) y−s11 y−s22 ds1 ds2,

for C1, C2 > 0. Combining this with the Lebedev-Whittaker inverse transform of FT,R,κ, we observe

that

51


FT,R,κ(y) =

∫(0)

∫(0)

∫(C2)

∫(C1)

F#T,R,κ(ν1, ν2)

3∏j=1

Γ(

3νj2

)Γ(−3νj

2

)

×

3∏j=1

Γ(s1−αj

2

)Γ(s2+αj

2

)16πs1+s2+5/2Γ

(s1+s2

2

) y1−s11 y1−s2

2 ds1ds2dν1dν2.

Assume 2k < C1 < 2k + 2 for some non-negative interger k and pull s1 from the contour (C1)

to the contour (−C1). Then

FT,R,κ(y) =M+k∑

m=0

3∑l=1

Rm,l, (4.6)

where M is the main term and is given by the same expression as FT,R,κ with the contour (C1)

replaced by (−C1). Here Rm,l are the residues corresponding to the poles of the integrand at

s1 = αl − 2m for m = 0, . . . , k and l = 1, 2, 3. Note that there will be no contribution of residues

of higher order poles, which could possibly show up when two of the αi differ by an even integer.

In this situation, at least one of ±3ν1, ±3ν2, ±3ν3 is equal to a non-positive even integer, making

the integrand identically zero in that region.

Then assume 2k′ < C2 < 2k′ + 2 for some non-negative interger k′. For each term in (4.6) shift

variable s2 from the contour (C2) to (−C2) to get,

FT,R,κ(y) = M+

k∑m′=0

3∑l′=1

Mm′,l′ +

k,k′∑m,m′=0

3∑l=1

∑l′ 6=lRm,l,m′,l′ , (4.7)

where M is the main term, given by the same expression as M with the contour (C2) replaced by

the contour (−C2), and Mm′,l′ are the residues corresponding to the poles of the integrand of M

at s2 = −αl′ − 2m′ for m′ = 0, . . . , k′. The terms Rm,l,m′,l′ are the residues corresponding to the

poles of the integrand of Rm,l at s2 = −αl′ − 2m′ for m′ = 0, . . . , k′ and l′ 6= l.

Let νj = itj and sj = Cj + iuj . Note that, for <(νj) = 0, the first exponential term in the

definition of F#T,R,κ is bounded (independent of κ) and the second one has exponential decay for

|tj | > T 1+δ. Using Stirling’s formula to estimate the Gamma factors we get

|M| � y1+C11 y1+C2

2

∫∫|t1|,|t2|<T 1+δ

∫∫R2

P · exp (E) du1du2dt1dt2,

52


where the exponential factor is given by

4Eπ

= 33∑j=1

|tj |+ |u1 + u2| −3∑j=1

|iu1 − αj | −3∑j=1

|iu2 + αj |,

and the polynomial term is given by

P =

3∏j=1

(1 + |tj |)

(R+2)/2

(1 + |u1 + u2|)(1+C1+C2)/2

×

3∏j=1

(1 + |iu1 − αj |)

(−C1−1)/2 3∏j=1

(1 + |iu2 + αj |)

(−C2−1)/2

.

We now show that the exponential factor is non-positive. As the exponential factor is invariant

under cyclic permutations of (t1, t2, t3), we may assume, without loss of generality, that t1 and t2

have the same sign. Then |α1|+ |α3| = 3|t1|+ 3|t2|. As |u1 + u2| ≤ |iu1 − α2|+ |iu2 + α2|, we get

4Eπ≤ 3

3∑j=1

|tj | − |iu1 − α1| − |iu1 − α3| − |iu2 + α1| − |iu2 + α3|

≤ 6|t1|+ 6|t2| − 2(|α1|+ |α3|)

= 0.

For either |u1| > 5T 1+δ or |u2| > 5T 1+δ, the exponential factor is bounded above by −T 1+δ

giving exponential decay to the integral. Integrating first over u1, u2 we get

|M| � y1+C11 y1+C2

2

∫∫∫∫|t1|,|t2|<T 1+δ

|u1|,|u2|<5T 1+δ

Pdu1du2dt1dt2 � y1+C11 y1+C2

2 T (3R+11−C1−C2)/2+ε

by choosing δ appropriately. To bound the residuesMm′,l′ we start by shifting variables ν1 and ν2

to contours (B1) and (B2), respectively, where |B1|, |B2| < R/3 and B′j < C1 for j 6= l, defining

B′1 = 2B1 +B2, B′2 = B2−B1, B′3 = −B1− 2B2, in a manner similar to the νj . Note that the first

exponential term is now bounded by 3T (max (B′1,B′2,B′3)/R)κ ≤ T . It follows that

|Mm′,l′ | � y1+C11 y

1+B′l′+2m′

2 T

∫∫|t1|,|t2|<T 1+δ

∫R

P · exp (E) du1dt1dt2,

53


where the exponential factor is given by

4Eπ

= 33∑j=1

|tj |+ |u1 −=(αl′)| −3∑j=1

|u1 −=(αj)| −∑j 6=l′|=(αj − αl′)|,


P =

3∏j=1

(1 + |tj |)

(R+2)/2

(1 + |u1 −=(αl′)|)(1+C1+B′l′+2m′)/2

3∏j=1

(1 + |u1 −=(αj)|)

(−C1−1)/2

×∏j 6=l′

((1 + |=(−αl′ + αj)|)(B′j−B′l′−2m′−1)/2

).

The exponential factor is again non-positive as

4Eπ≤ 3

3∑j=1

|tj | −∑j 6=l′|u1 −=(αj)| −

∑j 6=l′|=(αj − αl′)| ≤ 0,

by using the triangle inequality on the second sum to get a difference of =(αj). We now pick

B′l′ = C2 − 2m′ and B′j < 0 for j 6= l′ to get

|Mm′,l′ | � y1+C11 y1+C2

2 T

∫∫∫|t1|,|t2|<T 1+δ

|u1|<10T 1+δ

Pdu1dt1dt2

� y1+C11 y1+C2

2 T (3R+11−C1−C2)/2+ε.

(4.8)

To bound the residues Rm,l,m′,l′ we again shift variables ν1 and ν2 to contours (B1) and (B2),

respectively, where |B1|, |B2| < R/3. To simplify notation, we will assume without loss of generality

that l = 1 and l′ = 2. We obtain

|Rm,1,m′,2| � y1−B′1+2m1 y

1+B′2+2m′

2 T

∫∫|t1|,|t2|<T 1+δ

P · exp (E) dt1dt2,

where the exponential factor E is given by

4Eπ

= 33∑j=1

|tj | −∑j 6=1

|=(αj − α1)| −∑j 6=2

|=(αj − α2)|+ |=(α1 − α2)| = 0,


54


P =

3∏j=1

(1 + |tj |)

(R+2)/2

(1 + |t1|)(3B1−1)/2(1 + |t2|)(−3B2−2m′−1)/2(1 + |t3|)(3B3−2m−1)/2,

where B3 = B1 +B2. Then pick B′1 = −C1 + 2m and B′2 = C2 − 2m′ to get

|Rm,1,m′,2| � y1+C11 y1+C2

2 T

∫∫|t1|,|t2|<T 1+δ

Pdt1dt2

� y1+C11 y1+C2

2 T (3R+9−2C1−2C2+2m+2m′)/2+ε

� y1+C11 y1+C2

2 T (3R+9−C1−C2)/2+ε.

(4.9)

Combining all the bounds we get the desired result.

Remark 4.2.2. It follows from this proposition that FT,R,κ satisfies the decay condition 3.1.

4.3 Kloosterman Terms’ Bounds

We start by getting bounds for the Kloosterman integrals Jε,F and Jε1,ε2,F . Break the integral

Jε,F = J1 +J2 +J3 +J4 depending on whether y1 and y2 are smaller or greater than 1. To estimate

J1, start by taking absolute values inside the integral

|J1| � A−2

∫ 1

0

∫ 1

0

∫∫R2

|FT,R,κ(Ay1, y2)|

×

∣∣∣∣∣FT,R,κ(y2

√1 + x2

1 + x22

1 + x21

,A

y1y2

√1 + x2

1

1 + x21 + x2

2

)∣∣∣∣∣ dx1dx2dy1dy2

y1y22

.

Use (4.5) with C1 = C2 = 6− ε/2 to bound the first instance of FT,R,κ and C1 = ε, C2 = 6− ε to

bound the second instance of FT,R,κ in order to get

|J1| � A12−3ε/2 T 3R+2+3ε/2

∫ 1

0

∫ 1

0

∫∫R2

y−1+ε/21 y

−1+3ε/22 (1 + x2

1)3−3ε/2

(1 + x21 + x2

2)6−3ε/2dx1dx2dy1dy2.

55


It is clear that the integral in y converges and the integral in x also converges for small values

of ε. After a suitable redefinition of ε we get

|J1| �R,ε A12−ε T 3R+2+ε.

For the remaining three integrals the argument is almost identical. The only necessary changes

are to pick C1 = 6 − 3ε/2 when y1 > 1 and C2 = 6 − 5ε/2 when y2 > 1, while bounding the first

instance of FT,R,κ. Putting all the bounds together, we obtain

|Jε,F (A)| �R,ε A12−ε T 3R+2+ε. (4.10)

To bound Jε1,ε2,F , we also break it into Jε1,ε2,F = J1 + J2 + J3 + J4 depending on whether yi is

smaller or greater than 1. We will first bound J1 by taking absolute values and doing the change

of variables that sends x3 to x3 + x1x2. We have

|J1| � (A1A2)−2

∫ 1

0

∫ 1

0

∫∫∫R3

|FT,R,κ(A1y1, A2y2)|

×

∣∣∣∣∣FT,R,κ(A2

y2

√x2

3 + x21 + 1

1 + x22 + x2

3

,A1

y1

√1 + x2

2 + x23

x23 + x2

1 + 1

)∣∣∣∣∣ dx1dx2dx3dy1dy2

y1y2.

Then apply (4.5) with C1 = C2 = 6− ε to bound the first instance of FT,R,κ and C1 = C2 = 6− 2ε

to bound the second instance of FT,R,κ in order to get

|J1| � (A1A2)10−3ε T 3R−1+2ε

∫ 1

0

∫ 1

0

∫∫∫R3

(y1y2)−1+εdx1dx2dx3dy1dy2((1 + x2

2 + x23)(1 + x2

1 + x23))3−ε .

As the integral in y is convergent and the x integral is also convergent for small values of ε then,

after redefining ε

|J1| �R,ε (A1A2)10−ε T 3R−1+ε.

For the remaining three integrals the same argument works by choosing C1 = 6− ε/2 when y1 > 1

and C2 = 6 − ε/2 when y2 > 1, while bounding the second instance of FT,R,κ. Combining all four

bounds, one obtains

56


|Jε1,ε2,F (A1, A2)| �R,ε (A1A2)10−ε T 3R−1+ε. (4.11)

We may now bound the Kloosterman terms Σ2a, Σ2b and Σ3. The only necessary bounds for the

Kloosterman sums will be

S(m1, n1, n2, d1, d2)�ε (d1d2)1+ε,

S(m1,m2, n1, n2, d1, d2)�ε (m1m2n1n2)1/2(d1d2)1+ε.

which follow from Proposition 2.6.6 and Proposition 2.6.7, respectively. To bound Σ2a we use (4.10)

together with the first bound for Kloosterman sums, to obtain

|Σ2a| �R,κ,ε

∑ε=±1

∑d1|d2

m2d21=n1d2

|S(εm1, n1, n2, d1, d2)|d1d2

∣∣∣∣Jε,F (√m1n1n2

d1d2

)∣∣∣∣� T 3R+2+ε

∑d1,d2

(m1n1n2)6

(d1d2)6−3ε/2

� (m1n1n2)6 T 3R+2+ε.

The bound for Σ2b is obtained in the same manner. In order to bound Σ3 we use (4.11) together

with the second bound for Kloosterman sums, to obtain

|Σ3| �∑

ε1,ε2=±1

∑d1,d2

|S(ε1m1, ε2m2, n1, n2, d1, d2)|d1d2

∣∣∣∣Jε1,ε2,F (√m1n2d1

d2,

√m2n1d2

d1

)∣∣∣∣� (m1m2n1n2)11/2T 3R−1+ε

∑ε1,ε2=±1

∑d1,d2

1

(d1d2)5−3ε/2

� (m1m2n1n2)11/2 T 3R−1+ε.

For future reference, we write down these bounds in the following proposition.

57


Proposition 4.3.1. Fix R > 50, T � 1 and ε > 0. We have the following bounds for the

Kloosterman terms:

|Σ2a| �R,ε (m1n1n2)6 T 3R+2+ε;

|Σ2b| �R,ε (m2n1n2)6 T 3R+2+ε;

|Σ3| �R,ε (m1m2n1n2)11/2 T 3R−1+ε.

4.4 Eisenstein Terms’ Bounds

We start by obtaining a bound for the contribution to the Kuznetsov trace formula of the minimal

Eisenstein series Emin. We will require the de la Vallee Poussin bound for the Riemann zeta function,

|ζ(1 + it)| � 1

log(2 + |t|).

Using the bounds for the Fourier coefficients in Proposition 2.5.5, the de la Vallee Poussin bound

and Stirling’s formula for the Gamma factors, we get

|Emin| �iT 1+ε∫−iT 1+ε

iT 1+ε∫−iT 1+ε

(m1m2n1n2)ε

3∏k=1

∣∣∣Γ(2+R+3νk4

)Γ(

2+R−3νk4

)∣∣∣23∏

k=1

∣∣ζ (1 + 3νk) Γ(

1+3νk2

)∣∣2 |dν1dν2|

� (m1m2n1n2)εiT 1+ε∫−iT 1+ε


3∏k=1

((1 + |νk|)R log(2 + |νk|)2

)|dν1dν2|

�R,ε (m1m2n1n2)ε T 3R+2+ε.

To bound the maximal Eisenstein series contribution Emax we require the following lower bounds

for L-functions

L(1,Ad uj)�ε (1 + |rj |)−ε, |L(1 + 3ν, uj)| �ε (1 + |ν|+ |rj |)−ε,

where the eigenvalue of uj is 1/4+r2j . These lower bounds can be found in [Hoffstein and Lockhart,

1994] and [Hoffstein and Ramakrishnan, 1995]. It follows from the bounds above and Proposi-

tion 2.5.5 that

58


|Emax| �∑

rj<T 1+ε


(m1m2n1n2)1/2+ε

×

∣∣∣Γ(2+R+3ν+irj4

)∣∣∣8 ∣∣∣Γ(2+R+2irj4

)∣∣∣4 (1 + |rj |)ε(1 + |ν|+ |rj |)2ε∣∣∣Γ(1+3ν−irj2

)Γ(

1+2irj2

)Γ(

1+3ν+irj2

)∣∣∣2 |dν|

� (m1m2n1n2)1/2+ε∑

rj<T 1+ε


(1 + |rj |)R+ε(1 + |ν|+ |rj |)2R+2ε|dν|

�R,ε (m1m2n1n2)1/2+ε T 3R+3+ε.

For the last inequality, we use Weyl’s law for GL(2) Maass forms which tells us that

∣∣{φj : rj < T}∣∣ ∼ cT 2

for some constant c > 0. In summary, we obtain the following proposition.

Proposition 4.4.1. Fix R > 50, T � 1 and ε > 0. We have the following bounds for the Eisenstein

terms in the Kuznetsov trace formula:

|Emin| �R,ε (m1m2n1n2)ε T 3R+2+ε; |Emax| �R,ε (m1m2n1n2)1/2+ε T 3R+3+ε.

4.5 Main Geometric Term Computation

To estimate the main term on the geometric side, Σ1 = 〈FT,R,κ, FT,R,κ〉, we use the Parseval-type

identity (4.4) which says that Σ1 =⟨F#T,R,κ, F

#T,R,κ

⟩. Hence

⟨F#T,R,κ, F

#T,R,κ

⟩=

1

(πi)2

i∞∫−i∞

i∞∫−i∞

∣∣∣F#T,R,κ(ν)

∣∣∣2 dν1dν2

3∏j=1

Γ(

3νj2

)Γ(−3νj

2

)

=27R+1π

64R

∞∫−∞

∞∫−∞

3∑j=1

T−(βj/R)κ

2

exp

−2

3∑j=1

β2j /T

2

×

3∏j=1

(|tj |R+1 +O(|tj |R + 1)

)dt1dt2,

59


where νj = itj and αj = iβj . The second equality follows from Stirling’s approximation of the

Gamma factors. Making a linear change of variables of integration from t1, t2 to β1 = t3 − t1 =

2t1 + t2, β2 = t2 − t1 and using the symmetry of the integral in the βi, one gets

9R+14π

64R

∞∫0

β2∫0

3∑j=1

T−(βj/R)κ

2

exp

−23∑j=1

β2j /T

2

×∏j 6=k

(|βj − βk|R+1 +O(|βj − βk|R + 1)

)dβ1dβ2

=9R+18π

32R

∞∫0

β2∫0

3∑j=1

T−(βj/R)κ

2

exp

−23∑j=1

β2j /T

2

×(β3R+3

2 +O(β3R+22 β1 + β3R+2

2 + 1)

)dβ1dβ2.

We now multiply out the integrand to get⟨F#T,R,κ, F

#T,R,κ

⟩= M + Error1 + Error2,

where

M =9R+18π

32R

∞∫0

β2∫0

exp

−2 log T (β1/R)κ − 2

3∑j=1

β2j /T

2

β3R+32 dβ1dβ2,

Error1 includes all the terms not involving T−(β1/R)κ and Error2 the remaining terms involving

β3R+22 β1 + β3R+2

2 + 1. We can bound the error terms by

Error1 �∞∫

0

β2∫0

T−(β2/R)κp(β1, β2)dβ1dβ2 �∞∫

0

T−(β2/R)κ p(β2)dβ2 � 1,

where p, p are polynomials, and

Error2 �∞∫

0

β2∫0

T−(β1/R)κ exp

− 3∑j=1

β22/T

2

(β3R+22 β1 + β3R+2

2 + 1)dβ1dβ2

�∞∫

0

exp(−β2

2/T2)

(β3R+32 + β2)dβ2

� T 3R+3.

60


We then write

M =9R+18π

32R

∞∫0

β2∫0

exp(−c2

1βκ1 − c2

2(β21 + β1β2 + β2

2))β3R+3

2 dβ1dβ2,

where c21 = 2 log T/Rκ, c2

2 = 4/T 2.

Note that this integral is going to have exponential decay outside the region where β1 �

c−2/κ1 , β2 � c−1

2 . Therefore,

M =9R+18π

32R

∞∫0

β2∫0

exp(−c2

1βκ1 − c2

2β22

)β3R+3

2

(1 +O

(c2

2(β21 + β1β2)

))dβ1dβ2

=9R+18π

32R

∞∫0

β2∫0

exp(−c2

1βκ1 − c2

2β22

)β3R+3

2 dβ1dβ2 +O(c−(3R+2)2 c

−6/κ1 + c

−(3R+3)2 c

−4/κ1

)

=9R+18π

32R

∞∫0

∞∫0

exp(−c2

1βκ1 − c2

2β22

)β3R+3

2 dβ1 +O(

exp(−c2

1βκ2

)β3R+3

2

)dβ2 +O

(T 3R+3

)

=9R+18π

32Rc−2/κ1 c

−(3R+4)2

∞∫0

∞∫0

exp(−βκ1 − β2

2

)dβ1dβ2 +O(T 3R+3)

=9R+1Rπ3/2Cκ

28R+2+1/κ

T 3R+4

(log T )1/κ+ +O(T 3R+3),

where Cκ =∞∫0

exp (−xκ) dx.

Combining the bounds for the Kloosterman terms in Proposition 4.3.1, the bounds for the

Eisenstein terms in Proposition 4.4.1 and the computation above we obtain Theorem 4.1.1.

61

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