Kuznetsov Trace Formula and the Distribution of Fourier Coefficients

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

Jo˜ao Leit˜aoGuerreiro

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

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

Jo˜ao Leit˜aoGuerreiro

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 Poincar´eSeries ...... 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 tambémde agradecer a toda a minha fam´ılia. Aos meus pais Isabel e Lu´ıspelo seu apoio constante e por me terem dado todas as oportunidades que me levaram atéaqui. Ao meu irmãoPedro pelos seus conselhos sensatos e por me aturar hátanto tempo. And most of all, I thank my wonderful wife Jennifer, for her love and unconditional support.

iii A` minha fam´ılia

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 CHAPTER 1. INTRODUCTION

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 1 2 [ ] eigenvalue λj = 4 + xj , 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,

∞ X aj(n)aj(m) h(x ) cosh πx j j=1 j and on the other hand, a sum over Kloosterman sums S(n, m; c), weighted by values of Φh, a Bessel transform of h,

∞ √ X 1 4π nm S(n, m; c)Φ c h c c=1 . 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 (j) ν and Fourier coeﬃcients Aj(n1, n2). Let hT,R,κ be a family of smooth functions essentially

2 CHAPTER 1. INTRODUCTION supported on the spectral parameters of the Maass forms in F. Then,

(j) 4+3R X hT,R,κ(ν ) T = c + O T 3+3R+ , (T → ∞). R,κ 1/κ Lj (log T ) j≥1 Moreover,

 (j) |hT,R,κ(ν )|  P + O T 3R+3+ , if m1=n1, (j)  Lj m2=n2 X hT,R,κ(ν ) j≥1 Aj(m1, m2)Aj(n1, n2) = Lj  3R+3+ j≥1 O T , 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,   X φ(q), if n ≡ m mod q, χ(n)χ(m) = (1.1) χ mod q 0, otherwise. 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 5 on the space SL3(Z)\SL3(R)/SO3(R) says that there are T Maass forms with eigenvalue up to

3 CHAPTER 1. INTRODUCTION

T 2, by work of Miller [Miller, 2001]. Miller’s result has since been obtained in more general settings in [Lapid and Müller, 2009], [Lindenstrauss and Venkatesh, 2007], [Müller,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 Z ] × h (α) := h(t)Wα(t)d t,

T (Zp)\T (Qp) 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]).

n Theorem 1.0.2. Let H : S → C holomorphic and permutation invariant, where S is an open annulus containing the unit circle. Let n 1 Z Y H[(t) = H(β)W (t) (β − β ) dβ?. n!(2πi)n−1 1/β i j i,j=1 T i6=j 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 CHAPTER 1. INTRODUCTION 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 results 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   y x z = x + iy 7→   . 0 1

6 CHAPTER 2. PRELIMINARIES

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

  a b az + b   · z = , c d cz + d   a b 2 where   ∈ GL(2, R) and z ∈ h . c d 2 The space of complex-valued functions deﬁned on SL(2, Z)\h (with the above action) can be equipped with the following inner product.

0 2 Definition 2.1.3 (Petersson Inner Product). Let φ, φ : SL(2, Z)\h → C. We define their inner product Z dx dy φ, φ0 := φ(z)φ0(z) . y2 SL(2,Z)\h2 2 2 2 2 2 Furthermore, define the L -norm as kφk2 := hφ, φi and the L -space as L (SL(2, Z)\h ) := 2 φ : SL(2, Z)\h → C : kφk2 < ∞ .

Deﬁnition 2.1.4 (Maass form for SL(2, Z)). A Maass form of type ν ∈ C for SL(2, Z) is a 2 2 non-zero smooth function φ ∈ L (SL(2, Z)\h ) satisfying

2 ∂2 ∂2 1 2 • φ is an eigenfunction of the Laplace operator ∆ := −y ∂2x + ∂2y with eigenvalue 4 − ν ;

1 • φ is cuspidal, i.e., R φ(z) dx = 0. 0

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

az + b z 7→ , cz + d   a b with   ∈ GL(2, Z). c d

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

1 X az + b T φ(z) = √ φ , n n d ad=n 0≤b

2 2 where φ ∈ L (SL(2, Z)\h ).

7 CHAPTER 2. PRELIMINARIES

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).

3 Deﬁnition 2.1.8 (L-function). Let s ∈ C, R(s) > 2 , 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

∞ X λn L (s) := . φ ns n=1

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 ∞ 2 X |λn| L (s) := ζ(2s) , φ×φ˜ ns n=1 ∞ P 1 where ζ(s) := ns is the Riemann zeta function. n=1

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 CHAPTER 2. PRELIMINARIES

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     1 x2 x3 y1y2 0 0         x = 0 1 x1 , y =  0 y1 0 ,     0 0 1 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

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

3 From the description of h in the remark we can see that there is a group action of SL(3, Z) 3 3 on h by left multiplication. This action gives rise to the quotient space SL(3, Z)\h . We can also deﬁne an inner product for complex-valued functions on this space.

3 0 3 Deﬁnition 2.2.3 (Petersson Inner Product on h ). Let φ, φ : SL(3, Z)\h → C. We deﬁne their inner product Z φ, φ0 := φ(z)φ0(z) d∗z,

SL(3,Z)\h3

∗ dx1 dx2 dx3 dy1 dy2 where d z := 3 3 . y1 y2 2 2 2 Furthermore, deﬁne the L -norm as kφk2 := hφ, φi, and the L -space as

2 3 3 L (SL(3, Z)\h ) := φ : SL(3, Z)\h → C : kφk2 < ∞ .

∗ dx1 dx2 dx3 dy1 dy2 3 Remark 2.2.4. The measure d z = 3 3 is the left-invariant Haar measure on h (see y1 y2 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) := φ(g · exp(tα)) , ∂t t=0 with g ∈ GL(3, R).

3 The diﬀerential operators Dα with α ∈ gl(3, R) generate an associative algebra D deﬁned over 3 3 R, where multiplication of two operators is given by their composition. Let D be the center of D .

9 CHAPTER 2. PRELIMINARIES

3 3 Lemma 2.2.6. Every D ∈ D is well-deﬁned on smooth function on SL(3, Z)\h . 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

3 3 3 3 3 X X X X X ∆ := D ◦ D , ∆ := D ◦ D ◦ D . 2 Ei1,i2 Ei2,i1 3 Ei1,i2 Ei2,i3 Ei3,i1 i1=1 i2=1 i1=1 i2=1 i1=1

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

3 D = R[∆2, ∆3].

Furthermore,

2 2 2 2 2 2 2 2 ∂ 2 ∂ ∂ 2 2 2 ∂ 2 ∂ 2 ∂ 2 ∂ ∆2 = y1 2 + y2 2 − y1y2 + y1(x2 + y2) 2 + y1 2 + y2 2 + 2y1x2 ∂y1 ∂y2 ∂y1∂y2 ∂x3 ∂x1 ∂x2 ∂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.

2 3 Definition 2.2.8. For ν = (ν1, ν2) ∈ C and z = x · y ∈ h (with x and y as in Definition 2.2.1), 3 we define the function Is : h → C, by the condition:

1+ν1+2ν2 1+2ν1+ν2 Iν (z) := y1 y2 .

2 3 Lemma 2.2.9. For every ν ∈ C and D ∈ D , the function Iν is an eigenfunction of D. We denote by λν(D) the corresponding eigenvalue.

10 CHAPTER 2. PRELIMINARIES

[ ] 1 Proof. See Proposition 2.4.3 Goldfeld, 2006 . Our deﬁnition of Iν diﬀers by the map ν1 7→ ν1 − 3 , 1 ν2 7→ ν2 − 3 .

2 2 Remark 2.2.10. The Laplace eigenvalue λν(−∆2) is given by 1 − 3(ν1 + ν1ν2 + ν2 ). 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 2 2 2 2 ∂ Iν 2 ∂ Iν ∂ Iν −∆2Iν = −y1 2 − y2 2 + y1y2 ∂y1 ∂y2 ∂y1∂y2

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

2 2 = 1 − 3(ν1 + ν1ν2 + ν2 ) Iν.

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

2 Deﬁnition 2.2.11 (Maass Form for SL(3, Z)). A Maass form of type ν = (ν1, ν2) ∈ C for 2 3 SL(3, Z) is a non-zero smooth function φ ∈ L (SL(3, Z)\h ) satisfying

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

R • φ is cuspidal, i.e., φ(uz) du = 0 for U = U1,2,U2,1, where (SL(3,Z)∩U)\U      1 ∗ ∗   1 0 ∗          U1,2 = 0 1 0 ,U2,1 = 0 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).

3 Deﬁnition 2.3.1 (Siegel set). Let a, b ≥ 0. We deﬁne the Siegel set Σa,b ⊂ h 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 CHAPTER 2. PRELIMINARIES

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

Let U3(R) be the set of matrices u of the form   1 u2 u3     u = 0 1 u1 , (2.1)   0 0 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

2πi(m1u1+m2u2) ψm(u) := e . (2.2)

Deﬁnition 2.3.2 (Whittaker Function). An SL(3, Z)-Whittaker function of type ν = (ν1, ν2) ∈ 2 3 C , associated to a character ψ of U3(R), is a smooth function Ψ: h → C satisfying the following conditions:

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

3 3 • DΨ(z) = λν(D)Ψ(z), for all D ∈ D , z ∈ h ;

• R |Ψ(z)|2 d∗z < ∞. √ Σ 3 1 2 , 2

Deﬁnition 2.3.3 (Jacquet’s Whittaker Function). Fix a character ψm of U3(R) (as in 2.1, 2 2.2), with non-zero m1, m2, and ν = (ν1, ν2) ∈ C . Deﬁne Jacquet’s Whittaker function to be Z ∗ Wν(z; m) := Iν(w · u · z)ψm(u) d u,

U3R   1   3 ∗   where z ∈ h , d u = du1 du2 du3, and w =  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 2 any ν ∈ C analytically.

12 CHAPTER 2. PRELIMINARIES

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

1 + 3ν 1 + 3ν 1 + 3ν + 3ν W ∗(z; m) := π−3ν1−3ν2 Γ 1 Γ 2 Γ 1 2 W (z; m). ν 2 2 2 ν

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- 2 tion of type ν = (ν1, ν2) ∈ C associated to a character ψ of U3(R). Furthermore, if Ψ is an 2 SL(3, Z)-Whittaker function of type ν = (ν1, ν2) ∈ C associated to a character ψ of U3(R), with suﬃcient decay in y1, y2, so that ∞ ∞ Z Z σ1 σ2 y1 y2 |Ψ(y)| dy1 dy2 0 0 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    |m m | ∞ 1 2   X X X A(m1, m2) ∗   γ  φ(z) = · Wν  m1    z; (1, sgn(m2)) , |m1m2|    γ∈U ( )\SL(2, ) m1=1 m26=0   1  2 Z Z 1 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:    |m1m2|y1y2 A(m , m )    1 2 ∗    · Wν  m1y1  ; (1, sgn(m2)) |m1m2|    1 1 1 1 Z Z Z −2πi(m1x1+m2x2) = φ(z)e dx1 dx2 dx3 0 0 0

13 CHAPTER 2. PRELIMINARIES

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 Wν (z) := Wν(z; (1, 1)) dropping the dependence on the character. If z = y is a diagonal ∗ ∗ ∗ matrix then we have Wν (y) = Wν (y; (1, 1)) = Wν (y; (1, −1)). We shall also denote   y1y2 0 0   ∗ : ∗   Wν (y1, y2) = Wν  0 y1 0 .   0 0 1

Deﬁne Wν(z) and Wν(y1, y2) similarly.

∗ Proposition 2.3.9 (Double Mellin Inversion). The Whittaker function Wν has an integral representation as a double Mellin inversion Q3 s1−αj s2+αj 3/2 Z Z j=1 Γ 2 Γ 2 ∗ y1y2π −s1 −s2 W (y) = y y ds1 ds2, ν 2 s1+s2 s1+s2 1 2 (2πi) 4π Γ 2 (C2) (C1)

2 where y1, y2 > 0, ν ∈ C ,

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 Wν (y) is invariant under permutations of (α1, α2, α3), and

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

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 y y y −c1 y −c2 W ∗(y) 1 2 1 2 . ν 1/2−ε (1 + |ν1| + |ν2|) 1 + |ν1| + |ν2| 1 + |ν1| + |ν2|

14 CHAPTER 2. PRELIMINARIES

Proof. See Proposition 1 [Blomer, 2013].

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

∞ ∞ Z Z 3(1−s) ∗ ∗ 2 s dy1 dy2 π Y s + αj + βk Wν (y)Wµ (y)(y1y2) 3 = 3s Γ , (y1y2) Γ 2 2 0 0 1≤j,k≤3 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.

2 3 Deﬁnition 2.4.1. For n = 1, 2,..., deﬁne the Hecke operator Tn acting on L (SL(3, Z)\h ) by    a b1 c1 1 X    Tnφ(z) = φ 0 b c  z , n  2  abc=n    0≤c1,c2

2 3 where φ ∈ L (SL(3, Z)\h ).

3 The operators Tn commute with the diﬀerential operators in D . Therefore, we may simulta- 2 3 neously diagonalize the space L (SL(3, Z)\h ) 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 CHAPTER 2. PRELIMINARIES

Tnφ = A(n, 1)φ.

Furthermore, we have the following relations

0 0 0 0 0 0 A(m1m1, m2m2) = A(m1, m2)A(m1, m2), if (m1m2, m1m2) = 1; X m1c m2a A(n, 1)A(m , m ) = A , ; 1 2 a b abc=n a|m1 b|m2 X m1b m2c A(1, n)A(m , m ) = A , ; 1 2 a b abc=n a|m1 b|m2 X m1 m2 A(m , m ) = µ(d)A , 1 A 1, , 1 2 d d d|(m1,m2) where µ is the M¨obiusfunction.

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

X n m2 A(n, 1)A(1, m ) = A , 1 A 1, . 2 d d d|(n,m2)

By M¨obiusinversion it follows that

X m1 m2 X X m1 m2 µ(d)A , 1 A 1, = µ(d) A , d d de de d|(m1,m2) d|(m1,m2) ed|(m1,m2) X m1 m2 X = A , µ(d) e0 e0 0 0 e |(m1,m2) d|e

= 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

∞ X A(1, n) L (s) := . φ ns n=1

16 CHAPTER 2. PRELIMINARIES

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 Y Lφ,p(s), p

∞ k : P A(1,p ) where Lφ,p(s) = pks . By the second and third relations in Theorem 2.4.3 one obtains k=1 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,

s s −s 2s 2s s A(p, 1)Lφ,p(s) − A(1, p)(p Lφ,p(s) − p ) = p Lφ,p(s) − p Lφ,p(s) − p − p A(1, p) .

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

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

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.

2 Theorem 2.4.8 (Functional Equation). Let φ be a Hecke-Maass form of type ν = (ν1, ν2) ∈ C ∞ ˜ P −s for SL(3, Z), with associated L-function Lφ(s). Let Lφ(s) := A(n, 1)n be the dual L-function. n=1 ˜ Then Lφ(s), Lφ(s), have holomorphic continuation for s ∈ C and satisfy the function equation

Fν(s)Lφ(s) = F˜ν(1 − s)L˜φ(1 − s), where 3 Y s − αj F (s) = π−3s/2 Γ , ν 2 j=1 3 Y s + αj F˜ (s) = π−3s/2 Γ . ν 2 j=1

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

[ ] 1 Proof. See Theorem 6.5.15 Goldfeld, 2006 . Recall that we have normalized ν1 7→ ν1 + 3 and 1 ν2 7→ ν2 + 3 .

17 CHAPTER 2. PRELIMINARIES

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 ∗

3 2 For z ∈ h and ν = (ν1, ν2) ∈ C , with R(ν1), R(ν2) suﬃciently large, deﬁne the minimal parabolic

Eisenstein series Emin(z, ν) by the series X Emin(z, ν) := Iν(γz). γ∈(P1,1,1∩SL(3,Z))\SL(3,Z)

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 1 1 1 A (m , m ) W (m y , |m |y ) Z Z Z ν 1 2 ν 1 1 2 2 −2πi(m1x1+m2x2) = Emin(z, ν)e dx1 dx2 dx3, |m1m2| ζ(1 + 3ν1)ζ(1 + 3ν2)ζ(1 + 3ν1 + 3ν2) 0 0 0 where Aν(m1, m2) satisfy the Hecke relations 2.4.3, and

X α1 α2 α3 Aν(m, 1) = d1 d2 d3 . d1d2d3=m 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 ∗

3 be a maximal standard parabolic subgroup for GL(3, R). Let z ∈ h , 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 kuk2 = 1.

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

X 1/2+s Emax(z, ν, u) := det(γz) u(τ(γz)), γ∈(P2,1∩SL(3,Z))\SL(3,Z)

18 CHAPTER 2. PRELIMINARIES where   y2 x2 τ(z) =   . 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

1 1 1 W 2µ µ (m1y1, |m2|y2) Z Z Z Bs,u(m1, m2) ( ,s− ) c 3 3 = E (z, ν, u)e−2πi(m1x1+m2x2) dx dx dx , 1/2 max 1 2 3 |m1m2| Lu(1 + 3s)LAd u(1) 0 0 0 where Bs,u(m1, m2) satisfy the Hecke relations 2.4.3, and

X s −2s Bs,u(m, 1) = λu(d1)d1d2 . d1d2=m

Here LAdu is the adjoint L-function deﬁned given by

L (s) L (s) = u×u˜ , Ad u ζ(s) and c is a nonzero absolute constant.

Proof. See page 18 [Blomer, 2013].

2 Proposition 2.5.5. Let ν ∈ C 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

ε 1/2+ε Aν(m1, m2) = O((m1m2) ),Bs,u(m1, m2) = O((m1m2) ), for every ε > 0.

Proof. By Proposition 2.5.2 we have

X α1 α2 α3 Aν(m, 1) = d1 d2 d3 d1d2d3=m ≤ σ(m)3

= O(mε).

19 CHAPTER 2. PRELIMINARIES

By Proposition 2.5.4 we have

X s −2s Bs,u(m, 1) = λu(d1)d1d2 d1d2=m   X p = O  d1 d1|m = 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 3 a constant function. For z ∈ h , s ∈ C, with R(s) suﬃciently large, deﬁne

X 1/2+s Emax(z, s, 1) := det(γz) . γ∈(P2,1∩SL(3,Z))\SL(3,Z)

This Eisenstein series only has degenerate terms (corresponding to m1m2 = 0) in its Fourier expansion (see Theorem 2.4 [Friedberg, 1987]). Deﬁne the notation R to denote the integral along the complex vertical line whose points have (C) real part equal to C.

2 3 Theorem 2.5.6 (Langlands Spectral Decomposition). Let φ ∈ L (SL(3, Z)\h ) with suﬃcient decay such that hφ, Ei 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

∞ 1 Z Z 1 X Z φ(z) − hφ, E (∗, ν)iE (z, ν) dν dν − hφ, E (∗, s, u )iE (z, s, u ) ds (4πi)2 min min 1 2 2πi max j max j j=0 (0) (0) (0) 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 kujk2 = 1.

Proof. See Theorem 10.13.1 [Goldfeld, 2006].

20 CHAPTER 2. PRELIMINARIES

2.6 Kloosterman Sums

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

X∗ md + nd S(m, n, c) := e , c d (mod 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 :

      1 0 0 1 0 0 0 1 0             w1 = 0 1 0 , w2 = 0 0 1 , w3 = 1 0 0 ,       0 0 1 0 1 0 0 0 1       0 0 1 0 1 0 0 0 1             w4 = 1 0 0 , w5 = 0 0 1 , w6 = 0 1 0 .       0 1 0 1 0 0 1 0 0

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

[ GL(3, R) = Gw, w∈W where Gw = U3(R) w T U3(R).

Proof. See Proposition 10.3.2 [Goldfeld, 2006].

21 CHAPTER 2. PRELIMINARIES

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

0 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

0 X 0 Sw(ψ, ψ , d) := ψ(u1)ψ (u2), γ∈U3(Z)\(Γ∩Gw)/Γw γ=u1wdu2 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.

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

ψ(dwuw−1) = ψ0(u)

−1 3 for all u ∈ (w · U3(R) · w) ∩ U3(R). The characters ψ is extended to z = uy ∈ h 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   d1     d =  −d2/d1  ,   1/d2

1 u2 u3 2πi(n1u1+n2u2) with d1, d2 positive integers. Let ψ(u) = ψn1,n2 (u) = e , where u = 0 1 u1 . Let 0 0 1 0 0 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 CHAPTER 2. PRELIMINARIES

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

0 Sw1 (ψ, ψ , d) = δd1,1δd2,1,

0 0 Sw2 (ψ, ψ , d) = Sw3 (ψ, ψ , d) = 0, 0 ˜ Sw4 (ψ, ψ , d) = S(m1, n1, n2, d1, d2) if d1|d2, 0 ˜ Sw5 (ψ, ψ , d) = S(m2, n2, n1, d2, d1) if d2|d2,

0 Sw6 (ψ, ψ , d) = S(m1, m2, n1, n2, d1, d2), where δ is the Kronecker delta, X m1c1 + n1c1c2 n2c2 S˜(m1, n1, n2, d1, d2) := e e , d1 d2/d1 c1( mod d1), c2( mod d2) (c1,d1)=1 (c2,d2/d1)=1

X m1b1 + n1(y1d2 − z1b2) S(m1, m2, n1, n2, d1, d2) := e d1 b1,c1( mod d1), b2,c2( mod d2) (b1,c1,d1)=(b2,c2,d2)=1 b1b2+c1d2+c2d1≡0( mod d1d2) m b + n (y d − z b ) × e 2 2 2 2 1 2 1 , 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,

˜ 2 ε S(m1, n1, n2, d1, d2) min gcd(n2d1, d1d2), gcd(m1d2, n1d2, d1d2) (d1d2) .

The bound for the Kloosterman sum associated to the long Weyl element w6 is due to [Stevens, 1987].

23 CHAPTER 2. PRELIMINARIES

Proposition 2.6.7. For any ε > 0,

1/2 1/2+ε S(m1, m2, n1, n2, d1, d2) (m1m2n1n2) (d1d2) (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 Poincar´eSeries

Poincar´eseries for SL(3, Z) were ﬁrst introduced in [Bump et al., 1988]. These series relate the 3 spectral theory of SL(3, Z)\h to SL(3, Z) Kloosterman sums.

Definition 2.7.1 (PoincaréSeries). Let z ∈ h3, with z = xy as in Definition 2.2.1. Let F : 2 R+ → C, m1, m2 positive integers and define

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

The Poincar´eseries Pm1,m2 is given by the series

X Pm1,m2 (z) := Fm1,m2 (γz). γ∈U3(Z)\SL(3,Z) Proposition 2.7.2. Assume F is a bounded function with the following decay

2+ε |F (y1, y2)| (y1y2) ,

for an ε > 0 as y1 → 0 or y2 → 0. Then the Poincar´eseries Pm1,m2 (z) converges absolutely and 3 2 3 uniformly on compact subsets of h . Furthermore, Pm1,m2 ∈ L (SL(3, Z)\h ).

Proof. First note that Pm1,m2 is invariant under SL(3, Z), by construction so is a function on 3 SL(3, Z)\h . To show the convergence properties we follow the idea of Godement in the proof of 3 Theorem 9.1 [Borel, 1966]. For each z0 ∈ SL(3, Z)\h it suﬃces to show that Z ∗ |Pm1,m2 (z)| d z < ∞,

Cz0

24 CHAPTER 2. PRELIMINARIES

3 where Cz0 is a compact set of SL(3, Z)\h containing z0. Then, Z Z ∗ ∗ |Pm1,m2 (z)| d z ≤ |Fm1,m2 (z)| d z .

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

It follows from Proposition 1.3.2 [Goldfeld, 2006] there are only ﬁnitely many γ ∈ U3(Z)\SL(3, Z) √ √ 3 such that γz0 ∈ Σ 3 1 . This means that there exists an a ≥ 2 such that γz0 ∈/ Σa, 1 for every 2 , 2 2

γ ∈ U3(Z)\SL(3, Z). In addition, by taking Cz0 suﬃciently small, we also conclude that γz∈ / Σ 1 a, 2 for every γ ∈ U3(Z)\SL(3, Z), z ∈ Cz0 . This implies

3 3 SL(3,Z)\h U3(Z)\h 1 1 1 ∞ ∞ Z Z Z Z Z 2 dy1 dy2 ≤ |Fm1,m2 (z)| dx1 dx2 dx3 3 . (y1y2) 0 0 0 0 0

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

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 Poincaré 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 Poincaréseries 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 2 that F : R+ → C in the definition of the Poincaréseries (Definition 2.7.1) is a bounded function with the following decay 2+ε |F (y1, y2)| (y1y2) ,

25 CHAPTER 2. PRELIMINARIES

∗ 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 D E D E Emin(∗, ν),Pm1,m2 , Emax(∗, s, u),Pm1,m2 converge and satisfy

∞ ∞ Z Z D E Wν (y1, y2) F (y1, y2) dy1dy2 Emin(∗, ν),Pm1,m2 = m1m2 Aν(m1, m2) 3 , ζ(1 + 3ν1)ζ(1 + 3ν2)ζ(1 + 3ν1 + 3ν2) (y1y2) 0 0 ∞ ∞ D E Z Z W( 2µ ,s− µ ) (y1, y2) F (y1, y2) dy dy E (∗, s, u),P = cm m B (m , m ) 3 3 1 2 , max m1,m2 1 2 s,u 1 2 1/2 3 Lu(1 + 3s)LAd u(1) (y1y2) 0 0 for some absolute constant c > 0.

Proof. Replace Pm1,m2 (z) by its deﬁnition as a series to obtain Z D E X ∗ Emin(∗, ν),Pm1,m2 = Emin(z, ν) Fm1,m2 (γz) d z. γ∈U ( )\SL(3, ) SL(3,Z)\h3 3 Z Z

As Emin(z, ν) is invariant under the action of SL(3, Z) (on the variable z), one obtains Z D E ∗ Emin(∗, ν),Pm1,m2 = Emin(z, ν)Fm1,m2 (z)d z

3 U3(Z)\h ∞ ∞ 1 1 1 Z Z Z Z Z dy dy −2πi(m1x1+m2x2) 1 2 = F (m1y1, m2y2) Emin(z, ν)e dx1 dx2 dx3 3 (y1y2) 0 0 0 0 0 ∞ ∞ Z Z Aν(m1, m2) Wν (m1y1, m2y2) F (m1y1, m2y2) dy1dy2 = 3 , m1m2 ζ(1 + 3ν1)ζ(1 + 3ν2)ζ(1 + 3ν1 + 3ν2) (y1y2) 0 0 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 (y , y ) 7→ y1 , y2 . 1 2 m1 m2 For the second inner product we follow a similar strategy:

Z D E X ∗ Emax(z, s, u),Pm1,m2 = Emax(z, s, u) Fm1,m2 (γz)d z. γ∈U ( )\SL(3, ) SL(3,Z)\h3 3 Z Z

26 CHAPTER 2. PRELIMINARIES

As Emin(z, ν) is invariant under the action of SL(3, Z) (on the variable z), one obtains Z D E ∗ Emax(z, s, u),Pm1,m2 = Emin(z, ν)Fm1,m2 (z) d z

3 U3(Z)\h which can be written as

∞ ∞ 1 1 1 Z Z Z Z Z dy dy −2πi(m1x1+m2x2) 1 2 F (m1y1, m2y2) Emax(z, s, u)e dx1 dx2 dx3 3 (y1y2) 0 0 0 0 0 ∞ ∞ Z Z B (m , m ) W( 2µ ,s− µ ) (m1y1, m2y2) F (m1y1, m2y2) dy dy = c s,u 1 2 3 3 1 2 , 1/2 3 m1m2 Lu(1 + 3s)LAd u(1) (y1y2) 0 0 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 Poincar´eseries requires an explicit Fourier expansion for the Poincar´eseries as obtained in Theorem 5.1 [Bump et al., 1988].

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

1 1 1 Z Z Z −2πi(m1x1+m2x2) Pn1,n2 (z)e dx1 dx2 dx3 = S1 + S2a + S2b + S3, 0 0 0 where

S1 = δm1,n1 δm2,n2 F (n1y1, n2y2), X X S2a = S˜(εm1, n1, n2, d1, d2)J˜F (y1, y2, εm1, n1, n2, d1, d2),

ε=±1 d1|d2 2 m2d1=n1d2 X X S2b = S˜(εm1, n1, n2, d1, d2)J˜F ∗ (y1, y2, εm2, n2, n1, d2, d1),

ε=±1 d2|d1 2 m1d2=n2d1 ∞ X X S3 = S(ε1m1, ε2m2, n1, n2, d1, d2)JF (y1, y2, ε1m1, ε2m2, n1, n2, d1, d2).

ε1,ε2=±1 d1,d2=1

27 CHAPTER 2. PRELIMINARIES

The Kloosterman sums S and S˜ have been deﬁned in Section 2.6 and the weight functions J and J˜ are given by

∞ ∞ Z Z ˜ 2 n1y2d2 x1x2 n2d1 x2 JF (y1, y2, εm1, n1, n2, d1, d2) = y1y2 e 2 · 2 e 2 · 2 2 d1 x1 + 1 y1y2d2 x1 + x2 + 1 −∞ −∞ p 2 2 p 2 ! n1y2d2 x1 + x2 + 1 n2d1 x1 + 1 × e (−m1x1y1) F 2 · 2 , 2 · 2 2 dx1 dx2, d1 x1 + 1 y1y2d2 x1 + x2 + 1

∞ ∞ ∞ Z Z Z 2 JF (y1, y2, m1, m2, n1, n2, d1, d2) = (y1y2) e (−m1x1y1 − m2x2y2) −∞ −∞ −∞ n1d2 x1x3 + x2 n2d1 x2(x1x2 − x3) + x1 × e − 2 · 2 2 e − 2 · 2 2 y2d1 x2 + x3 + 1 y2d2 (x1x2 − x3) + x1 + 1 p 2 2 p 2 2 ! n1d2 (x1x2 − x3) + x1 + 1 n2d1 x2 + x3 + 1 × F 2 · 2 2 , 2 · 2 2 dx1 dx2 dx3 . y2d1 x2 + x3 + 1 y2d2 (x1x2 − x3) + x1 + 1

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.

2 Deﬁnition 2.8.4 (Lebedev-Whittaker Transform). Let F : R+ → C be a function satisfying # 2.8. Deﬁne its Lebedev-Whittaker transform F : D × D → C as

∞ ∞ Z Z # ∗ dy1dy2 F (ν) = Wν (y)F (y) 3 , (2.3) (y1y2) 0 0 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 : 2 ∞ R+ → 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 CHAPTER 2. PRELIMINARIES

(j) (j) (j) coeﬃcient Aj(1, 1) is equal to 1. Denote by ν = ν1 , ν2 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 2 ∞ # (j) (j) X F ν1 , ν2 C = Aj(m1, m2)Aj(n1, n2) , (2.5) 3 (j) (j) 1+3ν 1−3ν j=0 Q k k 6LAdφj (1) Γ 2 Γ 2 k=1 (j) (j) (j) with ν3 := ν1 + ν2 ;

i∞ i∞ # 2 1 Z Z Aν(m1, m2)Aν(n1, n2) F (ν1, ν2) Emin = 2 dν1 dν2, (2.6) (4πi) 3 2 Q 1+3νk −i∞ −i∞ ζ (1 + 3νk)Γ 2 k=1 with ν3 := ν1 + ν2;

i∞ 2µ µ 2 ∞ Z B (m , m )B (n , n ) F # j , s − j c X (s,uj ) 1 2 (s,uj ) 1 2 3 3 E = ds, (2.7) max 3 2πi 2 j=0 Q 1+δk−δl −i∞ LAd uj (1) Luj (1 + 3s) Γ 2 k,l=1 k6=l with σ1 = µj + s, σ2 = s − µj, σ3 = −2s;

∞ ∞ Z Z 2 dy1dy2 D E m =n m =n Σ1 = 1 1 1 |F (y1, y2)| 3 = 1 1 1 F,F , (2.8) {m2=n2} (y1y2) {m2=n2} 0 0 ˜ r X X S(εm1, n1, n2, d1, d2) m1n1n2 Σ2a = J˜ε,F , (2.9) d1d2 d1d2 ε=±1 d1|d2 2 m2d1=n1d2 ˜ r X X S(εm2, n2, n1, d2, d1) m2n1n2 Σ2b = J˜ε,F ∗ , (2.10) d1d2 d1d2 ε=±1 d2|d1 2 m1d2=n2d1 ∞ √ √ X X S(ε1m1, ε2m2, n1, n2, d1, d2) m1n2d1 m2n1d2 Σ3 = Jε1,ε2,F , . (2.11) d1d2 d2 d1 ε1,ε2=±1 d1,d2=1 Furthermore, the weight functions J , J˜ given by

29 CHAPTER 2. PRELIMINARIES

∞ ∞ ∞ ∞ Z Z Z Z p 2 2 p 2 ! ˜ −2 x1 + x2 + 1 A x1 + 1 Jε,F (A) = A F (Ay1, y2)e(−εAx1y1)F y2 2 , 2 2 x1 + 1 y1y2 x1 + x2 + 1 0 0 −∞ −∞ x1x2 A x2 dx1dx2dy1dy2 × e y2 2 + 2 2 2 , (2.12) x1 + 1 y1y2 x1 + x2 + 1 y1y2

∞ ∞ ∞ ∞ ∞ Z Z Z Z Z −2 Jε1,ε2,F (A1,A2) = (A1A2) F (A1y1,A2y2)e(−ε1A1x1y1 − ε2A2x2y2) 0 0 −∞ −∞ −∞ p 2 2 p 2 2 ! A2 (x1x2 − x3) + x1 + 1 A1 x2 + x3 + 1 × F 2 2 , 2 2 y2 x2 + x3 + 1 y1 (x1x2 − x3) + x1 + 1 A2 x1x3 + x2 A1 x2(x1x2 − x3) + x1 dx1dx2dx3dy1dy2 × e − 2 2 − 2 2 . (2.13) y2 x2 + x3 + 1 y1 (x1x2 − x3) + x1 + 1 y1y2

P ,P Proof. We shall compute n1,n2 m1,m2 in two different ways to obtain the two sides of the equality. n1n2m1m2 To obtain the left hand side of 2.4 apply the spectral decomposition (Theorem 2.5.6) to both Poincaréseries 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 Poincaréseries satisfy the conditions of Theorem 2.5.6 according to Lemma 2.8.1 and Remark 2.8.2. It follows that

D E D ED E P ,P ∞ φ ,P φ ,P n1,n2 m1,m2 X j m1,m2 j n1,n2 = n n m m kφ k(n n m m ) 1 2 1 2 j=0 j 1 2 1 2 i∞ i∞ D ED E 1 Z Z Emin(∗, ν),Pm1,m2 Emin(∗, ν),Pn1,n2 + 2 dν1 dν2 (4πi) n1n2m1m2 −i∞ −i∞ D ED E ∞ i∞ E (∗, s, u ),P E (∗, s, u ),P 1 X Z max j m1,m2 max j n1,n2 + ds . 2πi n1n2m1m2 j=0 −i∞

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 Wν and Wν.

30 CHAPTER 2. PRELIMINARIES

To obtain the term associated to the Hecke-Maass forms for SL(3, Z) it is necessary to compute D E the inner products φj,Pm1,m2 . Start by expanding Pm1,m2 using its deﬁnition as a series to obtain

Z D E X ∗ φj,Pm1,m2 = φj(z) Fm1,m2 (γz)d z. γ∈U ( )\SL(3, ) SL(3,Z)\h3 3 Z Z

As φj is invariant under the action of SL(3, Z), one obtains Z D E ∗ φj,Pm1,m2 = φj(z)Fm1,m2 (z)d z

3 U3(Z)\h ∞ ∞ 1 1 1 Z Z Z Z Z dy dy −2πi(m1x1+m2x2) 1 2 = F (m1y1, m2y2) φj(z)e dx1 dx2 dx3 3 (y1y2) 0 0 0 0 0 ∞ ∞ Z Z Aν(m1, m2) dy1dy2 = Wν (m1y1, m2y2) F (m1y1, m2y2) 3 , m1m2 (y1y2) 0 0 where the last equality follows from Remark 2.3.8. To obtain the cuspidal term C do a change of variables (y , y ) 7→ y1 , y2 and refer to Equation 2.12 [Goldfeld and Kontorovich, 2013] for an 1 2 m1 m2 2 explicit expression for the L -norm of φj,

3 (j) ! (j) ! Y 1 + 3ν 1 − 3ν kφ k = 6L (1) Γ k Γ k . j Adφj 2 2 k=1 Collecting the results obtained so far we get D E Pn1,n2 ,Pm1,m2 = C + Emin + Emax. n1n2m1m2 D E We will now compute the inner product Pn1,n2 ,Pm1,m2 directly using the Fourier expansion of Poincar´eseries as in Theorem 2.8.3. Z D E ∗ Pn1,n2 ,Pm1,m2 = Pn1,n2 (z)Fm1,m2 (z)d z

3 U3(Z)\h ∞ ∞ 1 1 1 Z Z Z Z Z dy dy −2πi(m1x1+m2x2) 1 2 = F (m1y1, m2y2) Pn1,n2 (z)e dx1 dx2 dx3 3 (y1y2) 0 0 0 0 0 ∞ ∞ Z Z dy1dy2 = F (m1y1, m2y2)(S1 + S2a + S2b + S3) 3 , (y1y2) 0 0

31 CHAPTER 2. PRELIMINARIES

where S1, S2a, S2b, S3 are as in Theorem 2.8.3. We then proceed to compute the integrals ∞ ∞ R R dy1dy2 F (m1y1, m2y2)Sτ 3 for τ ∈ {1, 2a, 2b, 3}. Recall the deﬁnitions of Σ1, Σ2a, Σ2b and Σ3 (y1y2) 0 0 as given by equations 2.8, 2.9, 2.10 and 2.11, respectively. We get

∞ ∞ Z Z dy1dy2 F (m1y1, m2y2)S1 3 = n1n2m1m2Σ1, (y1y2) 0 0 after changing variables (y , y ) 7→ y1 , y2 . Furthermore, 1 2 m1 m2

∞ ∞ Z Z dy1dy2 X X ˜ F (m1y1, m2y2)S2a 3 = S(εm1, n1, n2, d1, d2) (y1y2) 0 0 ε=±1 d1|d2 2 m2d1=n1d2 ∞ ∞ Z Z ˜ dy1dy2 × F (m1y1, m2y2)JF (y1, y2, εm1, n1, n2, d1, d2) 3 (y1y2) 0 0 r X X n1n2m1m2 m1n1n2 = S˜(εm1, n1, n2, d1, d2) J˜ε d1d2 d1d2 ε=±1 d1|d2 2 m2d1=n1d2

= n1n2m1m2Σ2a, where the penultimate equality follows from the change of variables

r 2 n1n2 d1 (y1, y2) 7→ y1, y2 . m1d1d2 n1d2 A similar calculation gives

∞ ∞ Z Z dy1dy2 F (m1y1, m2y2)S2b 3 = n1n2m1m2Σ2b. (y1y2) 0 0

32 CHAPTER 2. PRELIMINARIES

Finally,

∞ ∞ Z Z ∞ dy1dy2 X X F (m1y1, m2y2)S3 3 = S(ε1m1, ε2m2, n1, n2, d1, d2) (y1y2) 0 0 ε1,ε2=±1 d1,d2=1 ∞ ∞ Z Z dy1dy2 × F (m1y1, m2y2)JF (y1, y2, ε1m1, ε2m2, n1, n2, d1, d2) 3 (y1y2) 0 0 ∞ X X n1n2m1m2 = S(ε1m1, ε2m2, n1, n2, d1, d2) d1d2 ε1,ε2=±1 d1,d2=1 √ √ m1n2d1 m2n1d2 × Jε1,ε2 , d2 d1

= n1n2m1m2Σ3,

q n2d1 n1d2 where the penultimate equality follows from the change of variables (y1, y2) 7→ 2 y1, 2 y2 . m1d2 m2d1 It follows from the above computations that

hPn1,n2 ,Pm1,m2 i C + Emin + Emax = = Σ1 + Σ2a + Σ2b + Σ3, n1n2m1m2 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.

2 Deﬁnition 3.1.1 (Lebedev-Whittaker Inverse). Let g : iR → C. Deﬁne its Lebedev-Whittaker inverse g[ by i∞ i∞ 1 Z Z dν ν g[(y) := g(ν)W ∗(y) 1 2 , (πi)2 ν 3 Q 3νj 3νj −i∞ −i∞ Γ 2 Γ − 2 j=1 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 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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

 3  3 3π X Y |g(ν)| exp − |ν | (1 + |ν |)−10 (3.1)  4 j  j j=1 j=1

See Section 2.2 [Goldfeld and Kontorovich, 2013].

2 Theorem 3.1.3 (Lebedev-Whittaker Inversion). Let δ > 0 and g : D → 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,

i∞ i∞ # 2 D E 1 Z Z g (ν) dν1dν2 g#, g# := = hg, gi . (3.2) (πi)2 3 Q 3νj −3νj −i∞ −i∞ Γ 2 Γ 2 j=1

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 Wν = Wν . The proof ] [ 2 2 in Theorem 3.4 shows the equality of (g ) and g on (iR) and this equality can be extended to D 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 0 character on U induced from a character ψ on Qp via

n−1 ! 0 X ψ(u) = ψ ui,i+1 , (where u = (ui,j) ∈ U) . i=1 Deﬁnition 3.2.1 (Spherical Hecke Algebra). Deﬁne the spherical Hecke algebra K HK to be the set of locally constant, compactly supported function f : GLn(Qp) → C satisfying

f(k1gk2) = f(g)

35 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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 Z W (gx)φ(x) dx = λ(φ)W (g)

K K K K for each algebra homomorphism λ : H → C, φ ∈ H 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 h(t)W (t)d×t,

Z(Qp)\T (Qp) 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 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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

  t1 ··· tn−1    ..   .  g = zu   k,    t1    1 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 n Y −s−1 L(s, π) = 1 − αip , i=1

37 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

where αi ∈ C are all nonzero and α1 ··· αn = 1. Then, for   t1 ··· tn−1    ..   .  t =      t1    1 we have   1/2 δ (t)s-log |t|p (α), if ti ∈ Zp for i = 1, . . . , n − 1, Wπ(t) = 0, otherwise; where n−1 Y k(n−k) δ(t) = |tk|p , k=1

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

αn−1+λ1+λ2+···+λn−1 αn−1+λ1+λ2+···+λn−1 ··· αn−1+λ1+λ2+···+λn−1 1 2 n

n−2+λ1+λ2+···+λn−2 n−2+λ1+λ2+···+λn−2 n−2+λ1+λ2+···+λn−2 α1 α2 ··· αn

......

1+λ1 1+λ1 1+λ1 α α ··· αn 1 2

1 1 ··· 1 sλ(α) = αn−1 αn−1 ··· αn−1 1 2 n

n−2 n−2 n−2 α1 α2 ··· αn

......

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].

n Remark 3.2.6. We will also use the notation Wα, for α ∈ (C\{0}) , to denote the function   1/2 δ (t)s-log |t|p (α), if ti ∈ Zp for i = 1, . . . , n − 1, Wα(t) = 0, otherwise.

38 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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 ] n Whittaker transform h : C → C by Z ] × h (α) = h(t)Wα(t)d t,

T (Zp)\T (Qp) provided the integral converges, where

n−1 × Y −k(n−k) dtk d t = |tk|p . |tk|p k=1 Remark 3.2.8. A suﬃcient condition for the convergence of the above integral is that h satisﬁes the decay condition 1/2 τ+ε |h(t)|p δ (t)|t1 ··· tn−1|p for any ε > 0 and τ = max − log|αi|p. 1≤i≤n−1

n Deﬁnition 3.2.9 (Inverse Whittaker Transform). Let H : S → C holomorphic, where S is an open annulus containing the unit circle. Further, assume that H is invariant under the trans- n formations (α1, . . . , αn) 7→ (ασ(1), . . . , ασ(n)), for all (α1, . . . , αn) ∈ S and for every permutation

σ ∈ Sn. Let   1/2 δ (t)s-log |t|p (α), if ti ∈ Zp for i = 1, . . . , n − 1, Wα(t) = 0, otherwise;

[ where α = (α1, . . . , αn). Deﬁne the inverse Whittaker transform of H, H : Z(Qp)\T (Qp) → C by Z n 1 Y dβ1 ··· dβn−1 H[(t) = H(β)W (t) (β − β ) , n!(2πi)n−1 1/β i j β ··· β i,j=1 1 n−1 T i6=j where = (β , . . . , β ) ∈ n−1 : |β | = 1 , 1/β = (1/β ,..., 1/β ) and β = 1 . T 1 n−1 C i C 1 n n β1···βn−1

The inverse transform is given in the following theorem.

39 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

n Theorem 3.2.10 (Inversion Formula). Let H : S → 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) ∈ n n C : |αi|C = 1 ⊂ D ⊂ S . Then, (H[)](α) = H(α) for α1 ··· αn = 1.

Corollary 3.2.11. Let h : Z(Qp)\T (Qp) → C. For   t1 ··· tn−1    ..   .  t =      t1    1 assume h(t) is compactly supported on the region {ti ∈ Zp\{0} : i = 1, . . . , n − 1}, and that n−1 h(t) = f(−|t1|p,..., −|tn−1|p) for some function f : Z≥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,

X 1 − α1 ··· αnβ1 ··· βn sm (α) sm (β) = n , Q m1,...,mn−1≥0 (1 − αiβj) i,j=1 where m = (m1, . . . , mn−1).

40 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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 Q Q 1−α1β1 1−αnβ1 (αi − αj) (βi − βj)

. .. . 1≤i

Expanding each term 1 as 1 + α β + α2β2 + ··· , we obtain 1−αiβj i j i j

1 ··· 1 1−α1β1 1−αnβ1 . . X X . .. . = sgn(σ)αl1 ··· αln βl1 ··· βln . . . . σ(1) σ(n) 1 n l1,...,ln≥0 σ∈Sn 1 ··· 1 1−α1βn 1−αnβn

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

1 ··· 1 1−α1β1 1−αnβ1 . . X . .. . = sgn(σ)sgn(τ)αl1 ··· αln βl1 ··· βln . . . σ(1) σ(n) τ(1) τ(n) l >...>l ≥0 1 1 1 n ··· σ,τ∈Sn 1−α1βn 1−αnβn

l1 l1 l1 l1 α1 ··· αn β1 ··· βn

X ...... = ...... l1>...>ln≥0 ln ln ln ln α1 ··· αn β1 ··· βn

By making the substitution (l1, . . . , ln) = (m0+···+mn−1+(n−1), m0+···+mn−2+(n−2), . . . , m0) Q Q and dividing by (αi − αj) (βi − βj) we obtain 1≤i

1 X m0 n = sm (α) sm (β)(α1 ··· αnβ1 ··· βn) , Q (1 − αiβj) m0,m1,...,mn−1≥0 i,j=1 where right hand side simpliﬁes to

1 X sm (α) sm (β) . 1 − α1 ··· αnβ1 ··· βn m1,...,mn−1≥0

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 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

Proposition 3.2.13. Let α1, . . . , αn, β1, . . . , βn ∈ C and > 0. Let Wα, Wβ be Whittaker functions as deﬁned in Deﬁnition 3.2.9. Then Z n−1 α1···αnβ1···βn Y 1 − pn W (t)W (t) |t |(n−k)d×t = . α β k p n Q αiβj k=1 1 − Z(Qp)\T (Qp) p i,j=1 Proof. We start by applying Theorem 3.2.5 to write down the Whittaker functions explicitly: Z n−1 Y (n−k) × Wα(t)Wβ(t) |tk|p d t k=1 Z(Qp)\T (Qp) Z n−1 Y −(k−)(n−k) dtk = δ(t)s-log |t|p (α)s-log |t|p (β) |tk|p |tk|p n−1 k=1 Zp Z n−1 Y (n−k) dtk = s-log |t|p (α)s-log |t|p (β) |tk|p . |tk|p n−1 k=1 Zp Breaking up the region of integration by absolute value we get Z n−1 Y (n−k) × Wα(t)Wβ(t) |tk|p d t k=1 Z(Qp)\T (Qp) n−1 times Zz }| Z{ n−1 X Y (n−k) dtk = ··· s-log |t|p (α)s-log |t|p (β) |tk|p |tk|p m1,...,mn−1≥0 m × k=1 p k Zp

X −(n−1)m1−(n−2)m2−···−mn−1 = sm(α)sm (β) p

m1,...,mn−1≥0 X α = s s (β) m p m m1,...,mn−1≥0

α1···αnβ1···βn 1 − n = p , n Q αiβj 1 − p i,j=1 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 Z [ × H(α) = H (t)Wα(t)d t.

Z(Qp)\T (Qp)

42 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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

Z n−1 [ Y (n−k) × H(α) = H (t)Wα(t) |tk|p d t. k=1 Z(Qp)\T (Qp) Then Z [ × lim H(α) = H (t)Wα(t)d t. →0 Z(Qp)\T (Qp)

It suﬃces to show that lim H(α) = H(α) as well. →0 Replacing H[(t) by its explicit formula and swapping the order of integration we get   n−1 1 Z Z Y H (α) =  W (t)W (t) |t |(n−k)d×t n!(2πi)n−1  α 1/β k p  k=1 T Z(Qp)\T (Qp) n Y dβ1 ··· dβn−1 × H(β) (β − β ) . i j β ··· β i,j=1 1 n−1 i6=j

We use Proposition 3.2.13 to compute the innermost integrals and get 1 n Z H(β) 1 − n 1 p Y dβ1 ··· dβn−1 H(α) = n−1 n (βi − βj) . n!(2πi) α β ··· β Q β − i i,j=1 1 n−1 T j p i,j=1 i6=j

Now shift each variable β (i = 1, . . . , n − 1) to the contour given by the equation |z| = 1 , in i C p+0 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

α α β = 1 , . . . , β = n . 1 p 1 p

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 1 X σ H (α) = R + I 0 , n! , σ∈Sn

43 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM where n−1 ασ(1) ασ(n) 1 Q ασ(i) (n−1) H p ,..., p 1 − pn p − p ασ(n) Rσ = i=1 n−1 n−1 (n−1) ασ(n) Q ασ(i) ασ(n) Q ασ(i) p ασ(n) − p p − p p i=1 i=1 and I,0 is a sum of residue integrals with each integrand bounded (in absolute value) by

1 (n3+n) 1 − pn p CH 1 − 1 p0 for some constant CH > 0 depending only on the function H. Therefore, n−1 α α Q ασ(i) (n−1) σ(1) σ(n) 1 − p α H ,..., 1 − n p σ(n) σ p p p i=1 lim R = lim →0 →0 n α 1 n−1 Q σ(i) p(n−1) − ασ(i) ασ(n) p Q p p − p i=1 i=1 n−1 Q α − α H α , . . . , α σ(i) σ(n) = σ(1) σ(n) i=1 n n−1 Q Q ασ(i) ασ(i) − ασ(n) i=1 i=1 = H ασ(1), . . . , ασ(n) = H(α) and 1 (n3+n) 1 − pn p lim |I,0 | H lim = 0. →0 →0 1 − 1 p0 In conclusion, we obtain 1 X lim H(α) = H(α) = H(α), →0 n! σ∈Sn 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 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

 1, if log |ti|p = −λi for i = 1, . . . , n − 1, fλ1,...,λn−1 (t) = 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 λ [ Wβ (p ) H 7→ H . Let H(β) = δ(pλ) , where   pλ1 ··· pλn−1  .   ..  λ   p =   .  λ1   p    1

Then

H[(t) Z n 1 Y dβ1 ··· dβn−1 = W (pλ)W (t) (β − β ) δ(pλ)n!(2πi)n−1 β 1/β i j β ··· β i,j=1 1 n−1 T i6=j 1/2 Z n δ (t) Y dβ1 ··· dβn−1 = s (β)s (1/β) (β − β ) δ1/2(pλ)n!(2πi)n−1 λ -log |t|p i j β ··· β i,j=1 1 n−1 T i6=j 1/2 Z δ (t) X (n−1)+λ +···+λ = β 1 n−1 ··· β1+λ1 δ1/2(pλ)n!(2πi)n−1 σ(1) σ(n−1) σ,τ∈S T n

−(n−1)+log |t1|p+···+log |tn−1|p −1+log |t1|p dβ1 ··· dβn−1 × βτ(1) ··· βτ(n−1) β1 ··· βn−1

δ1/2(t) = δ1/2(pλ)n!(2πi)n−1 Z X λ1+···+λn−1+log |t1|p+···+log |tn−1|p λ1+log |t1|p dβ1 ··· dβn−1 × βσ(1) ··· βσ(n−1) β1 ··· βn−1 σ∈S T n δ1/2(t) = f (t) δ1/2(pλ) λ1,...,λn−1

= fλ1,...,λn−1 (t), as desired.

45 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

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

2 −s−1 −s−1 2 −s−1 Lp(s, π) = 1 − α1p 1 − α1α2p 1 − α2p = 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

Z [ × Lp(s, π) = (hs,p) (t)Wα(t)d t,

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

Z [ 1 −1 −1 dβ1 (hs,p) (t) = hs,p(β)W1/β(t)(β1 − β1 )(β1 − β1) 4πi β1 |β1|=1 ! |t |1/2 1 = 1 p 1 − p−s 4πi Z 1 1 dβ × W (t) βλ+1 − β−(λ+1) β−1 − β 1 , 2 −s −2 −s 1/β 1 1 1 1 1 − β1 p 1 − β1 p β1 |β1|C=1   t 0 1 −λ with t =   and |t1|p = p . 0 1

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,

1/2 [ |t1|p (hs,p) (t) = Resβ =0 i(β1) + Res −s/2 i(β1) + Res −s/2 i(β1) , 2(1 − p−s) 1 β1=p β1=−p

where the residues are equal to

46 CHAPTER 3. INVERSE LEBEDEV-WHITTAKER TRANSFORM

λ+1 −λ−1  ps/2 (ps/2) −(ps/2)   (1+p−s) , if λ even, Resβ1=0 i(β1) = 0, if λ odd;

λ+1 −λ−1 ps/2 ps/2 − ps/2 Res −s/2 i(β1) = ; β1=p 2 (1 + p−s) λ+1 −λ−1 ps/2 −ps/2 − −ps/2 Res −s/2 i(β1) = − . β1=−p 2 (1 + p−s)

Combining all the above terms we obtain

 s −(s−1)/2  p |t1|p , if λ even, [  1−p−2s (hs,p) (t) = 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]. (j) Let φj (j = 1, 2,... ) be a set of orthogonal Hecke-Maass forms for SL(3, Z) of type ν = (j) (j) (j) (j) (j) ν1 , ν2 , and deﬁne ν3 = −ν1 − ν2 . 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

2 2 2 2 2 2 λφ = 1 − 3(ν1 + ν2 + ν3 ) = 1 − (α1 + α2 + α3).

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

48 CHAPTER 4. ORTHOGONALITY RELATION to 2 (mod 4), we deﬁne

κ κ κ 2 2 2 2 2 (α1/R) (α2/R) (α3/R) 2(α +α +α )/T hT,R,κ(ν) = T + T + T e 1 2 3 !2 3 Q 2+R+3νj 2+R−3νj Γ 4 Γ 4 (4.1) j=1 × . 3 Q 1+3νj 1−3νj Γ 2 Γ 2 j=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

" 3 #R " 3 #R (1) Y (2) Y cR (1 + |νi|) ≤ hT,R,κ(ν) ≤ cR (1 + |νi|) , i=1 i=1 (2) (1) for some cR ≥ cR > 0. If one of the |αi| is smaller than R, then

[(1 + |ν |)(1 + |ν |)(1 + |ν |)]R h (ν) 1 2 3 . T,R,κ R T 2 1+ If |α1|, |α2|, |α3| > R , then

1 h (ν) . T,R,κ T Rκ

Theorem 4.1.1. For j = 1, 2,..., let φj be a set of orthogonal Hecke-Maass forms for SL(3, Z) (j) (j) (j) with spectral parameters ν = ν1 , ν2 . Let hT,R,κ be given as in (4.1). We have that for ﬁxed

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

(j) 4+3R X hT,R,κ(ν ) T = c + O (T 3+3R+), (T → ∞). R,κ 1/κ R, Lj (log T ) j≥1

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

(j) X hT,R,κ(ν ) Aj(m1, m2)Aj(n1, n2) Lj j≥1

 (j) |hT,R,κ(ν )|  P + O (m m n n )6 T 3R+3+ , if m1=n1,  Lj R, 1 2 1 2 m2=n2 j≥1 =  6 3R+3+ OR, (m1m2n1n2) T , otherwise.

49 CHAPTER 4. ORTHOGONALITY RELATION

Here the weights Lj are the L-values LAd φj (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 5+3R T . 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 ; the second expo- 1+ nential term contributes with exponential decay when one of |νi| > T ; 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

!−1 ! 3 3 Y (j) Y (j) 1 + νi Lj 1 + νi (4.2) i=1 i=1 and, conjecturally, the lower bound is expected to be

!− 3 Y (j) 1 + νi Lj. (4.3) i=1

50 CHAPTER 4. ORTHOGONALITY RELATION

4.2 Bound for the Inverse Lebedev-Whittaker Transform

Let

κ κ κ 2 2 2 2 # (α1/R) (α2/R) (α3/R) (α1+α2+α3)/T FT,R,κ(ν1, ν2) = T + T + T e   3 Y 2 + R + 3νj 2 + R − 3νj × Γ Γ .  4 4  j=1

# As FT,R has enough exponential decay on a strip |<(ν1)|, |<(ν2)| < (see 3.1) then by Lebedev- Whittaker inversion as in Theorem 3.1.3,

i∞ i∞ 1 Z Z dν dν F (y) = F # (ν)W ∗(y) 1 2 . T,R,κ (πi)2 T,R,κ ν 3 Q 3νj 3νj −i∞ −i∞ Γ 2 Γ − 2 j=1 We also have the Parseval-type identity as in the second part of Theorem 3.1.3

2 i∞ i∞ # 1 Z Z FT,R,κ(ν) dν1dν2 D E hF ,F i = = F # ,F # . (4.4) T,R,κ T,R,κ (πi)2 3 T,R,κ T,R,κ Q 3νj −3νj −i∞ −i∞ Γ 2 Γ 2 j=1

Proposition 4.2.1. Fix C1,C2 > 0, R > 3 max(C1,C2) + 6 and > 0. For any y1, y2 > 0, T 1, we have y C1 y C2 |F (y)| y y T 3R/2+11/2+C1/2+C2/2+ 1 2 . (4.5) T,R,κ C1,C2,R, 1 2 T T 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 Wν as an inverse Mellin transform as in Proposition 2.3.9,

Q3 s1−αj s2+αj 3/2 Z Z j=1 Γ 2 Γ 2 ∗ y1y2π −s1 −s2 W (y) = y y ds1 ds2, ν 2 s1+s2 s1+s2 1 2 (2πi) 4π Γ 2 (C2) (C1) for C1,C2 > 0. Combining this with the Lebedev-Whittaker inverse transform of FT,R,κ, we observe that

51 CHAPTER 4. ORTHOGONALITY RELATION

# Z Z Z Z F (ν1, ν2) F (y) = T,R,κ T,R,κ 3 Q 3νj 3νj (0) (0) (C2) (C1) Γ 2 Γ − 2 j=1 3 Q s1−αj s2+αj Γ 2 Γ 2 j=1 1−s1 1−s2 × y y ds1ds2dν1dν2. s1+s2+5/2 s1+s2 1 2 16π Γ 2

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

k 3 X X FT,R,κ(y) = M + Rm,l, (4.6) m=0 l=1 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. 0 0 0 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,

k 3 k,k0 3 ˜ X X X X X FT,R,κ(y) = M + Mm0,l0 + Rm,l,m0,l0 , (4.7) m0=0 l0=1 m,m0=0 l=1 l06=l where M˜ is the main term, given by the same expression as M with the contour (C2) replaced by the contour (−C2), and Mm0,l0 are the residues corresponding to the poles of the integrand of M 0 0 0 at s2 = −αl0 − 2m for m = 0, . . . , k . The terms Rm,l,m0,l0 are the residues corresponding to the 0 0 0 0 poles of the integrand of Rm,l at s2 = −αl0 − 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 FT,R,κ is bounded (independent of κ) and the second one has exponential decay for 1+δ |tj| > T . Using Stirling’s formula to estimate the Gamma factors we get

ZZ ZZ ˜ 1+C1 1+C2 |M| y1 y2 P· exp (E) du1du2dt1dt2, 1+δ 2 |t1|,|t2|

52 CHAPTER 4. ORTHOGONALITY RELATION where the exponential factor is given by

3 3 3 4E X X X = 3 |t | + |u + u | − |iu − α | − |iu + α |, π j 1 2 1 j 2 j j=1 j=1 j=1 and the polynomial term is given by

(R+2)/2  3  Y (1+C1+C2)/2 P =  (1 + |tj|) (1 + |u1 + u2|) j=1 (−C −1)/2 (−C −1)/2  3  1  3  2 Y Y ×  (1 + |iu1 − αj|)  (1 + |iu2 + αj|) . j=1 j=1 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

3 4E X ≤ 3 |t | − |iu − α | − |iu − α | − |iu + α | − |iu + α | π j 1 1 1 3 2 1 2 3 j=1

≤ 6|t1| + 6|t2| − 2(|α1| + |α3|)

= 0.

1+δ 1+δ 1+δ For either |u1| > 5T or |u2| > 5T , the exponential factor is bounded above by −T giving exponential decay to the integral. Integrating ﬁrst over u1, u2 we get

ZZZZ 1+C1 1+C2 1+C1 1+C2 (3R+11−C1−C2)/2+ |M| y1 y2 Pdu1du2dt1dt2 y1 y2 T 1+δ |t1|,|t2|

1+B0 +2m0 ZZ Z 1+C1 l0 |Mm0,l0 | y1 y2 T P· exp (E) du1dt1dt2, 1+δ |t1|,|t2|

53 CHAPTER 4. ORTHOGONALITY RELATION where the exponential factor is given by

3 3 4E X X X = 3 |t | + |u − =(α 0 )| − |u − =(α )| − |=(α − α 0 )|, π j 1 l 1 j j l j=1 j=1 j6=l0 and the polynomial term is given by

(R+2)/2 (−C −1)/2  3   3  1 0 0 Y (1+C1+B 0 +2m )/2 Y P =  (1 + |tj|) (1 + |u1 − =(αl0 )|) l  (1 + |u1 − =(αj)|) j=1 j=1

0 0 0 Y (B −B 0 −2m −1)/2 × (1 + |=(−αl0 + αj)|) j l . j6=l0

The exponential factor is again non-positive as

3 4E X X X ≤ 3 |t | − |u − =(α )| − |=(α − α 0 )| ≤ 0, π j 1 j j l j=1 j6=l0 j6=l0 by using the triangle inequality on the second sum to get a diﬀerence of =(αj). We now pick 0 0 0 0 Bl0 = C2 − 2m and Bj < 0 for j 6= l to get

ZZZ 1+C1 1+C2 |Mm0,l0 | y1 y2 T Pdu1dt1dt2 1+δ |t1|,|t2|

1+C1 1+C2 (3R+11−C1−C2)/2+ y1 y2 T .

To bound the residues Rm,l,m0,l0 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 l0 = 2. We obtain

0 0 0 ZZ 1−B1+2m 1+B2+2m |Rm,1,m0,2| y1 y2 T P· exp (E) dt1dt2, 1+δ |t1|,|t2|

3 4E X X X = 3 |t | − |=(α − α )| − |=(α − α )| + |=(α − α )| = 0, π j j 1 j 2 1 2 j=1 j6=1 j6=2 and the polynomial term is given by

54 CHAPTER 4. ORTHOGONALITY RELATION

(R+2)/2  3  0 Y (3B1−1)/2 (−3B2−2m −1)/2 (3B3−2m−1)/2 P =  (1 + |tj|) (1 + |t1|) (1 + |t2|) (1 + |t3|) , j=1

0 0 0 where B3 = B1 + B2. Then pick B1 = −C1 + 2m and B2 = C2 − 2m to get

ZZ 1+C1 1+C2 |Rm,1,m0,2| y1 y2 T Pdt1dt2 1+δ |t1|,|t2|

0 (4.9) 1+C1 1+C2 (3R+9−2C1−2C2+2m+2m )/2+ y1 y2 T

1+C1 1+C2 (3R+9−C1−C2)/2+ y1 y2 T .

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

J˜ε,F = J˜1 +J˜2 +J˜3 +J˜4 depending on whether y1 and y2 are smaller or greater than 1. To estimate

J˜1, start by taking absolute values inside the integral

Z 1 Z 1 ZZ −2 |J˜1| A |FT,R,κ(Ay1, y2)| 0 0 R2 ! p1 + x2 + x2 A p1 + x2 dx dx dy dy × F y 1 2 , 1 1 2 1 2 . T,R,κ 2 2 2 2 2 1 + x1 y1y2 1 + x1 + x2 y1y2

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

Z 1 Z 1 ZZ y−1+/2y−1+3/2(1 + x2)3−3/2 |J˜ | A12−3/2 T 3R+2+3/2 1 2 1 dx dx dy dy . 1 2 2 6−3/2 1 2 1 2 0 0 (1 + x1 + x2) R2

55 CHAPTER 4. ORTHOGONALITY RELATION

It is clear that the integral in y converges and the integral in x also converges for small values of . After a suitable redeﬁnition of we get

12− 3R+2+ |J˜1| R, A T .

For the remaining three integrals the argument is almost identical. The only necessary changes are to pick C1 = 6 − 3/2 when y1 > 1 and C2 = 6 − 5/2 when y2 > 1, while bounding the ﬁrst instance of FT,R,κ. Putting all the bounds together, we obtain

12− 3R+2+ |J˜ε,F (A)| R, A T . (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

Z 1 Z 1 ZZZ −2 |J1| (A1A2) |FT,R,κ(A1y1,A2y2)| 0 0 R3 ! A px2 + x2 + 1 A p1 + x2 + x2 dx dx dx dy dy × F 2 3 1 , 1 2 3 1 2 3 1 2 . T,R,κ 2 2 2 2 y2 1 + x2 + x3 y1 x3 + x1 + 1 y1y2

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

Z 1 Z 1 ZZZ −1+ 10−3 3R−1+2 (y1y2) dx1dx2dx3dy1dy2 |J1| (A1A2) T . 2 2 2 2 3− 0 0 (1 + x2 + x3)(1 + x1 + x3) R3 As the integral in y is convergent and the x integral is also convergent for small values of then, after redeﬁning

10− 3R−1+ |J1| R, (A1A2) T .

For the remaining three integrals the same argument works by choosing C1 = 6 − /2 when y1 > 1 and C2 = 6 − /2 when y2 > 1, while bounding the second instance of FT,R,κ. Combining all four bounds, one obtains

56 CHAPTER 4. ORTHOGONALITY RELATION

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

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

1+ S˜(m1, n1, n2, d1, d2) (d1d2) ,

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

˜ r X X |S(εm1, n1, n2, d1, d2)| ˜ m1n1n2 |Σ2a| R,κ, Jε,F d1d2 d1d2 ε=±1 d1|d2 2 m2d1=n1d2 6 X (m1n1n2) T 3R+2+ 6−3/2 (d1d2) d1,d2 6 3R+2+ (m1n1n2) T .

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

√ √ X X |S(ε1m1, ε2m2, n1, n2, d1, d2)| m1n2d1 m2n1d2 |Σ3| Jε1,ε2,F , d1d2 d2 d1 ε1,ε2=±1 d1,d2 X X 1 (m m n n )11/2T 3R−1+ 1 2 1 2 5−3/2 (d1d2) ε1,ε2=±1 d1,d2 11/2 3R−1+ (m1m2n1n2) T .

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

57 CHAPTER 4. ORTHOGONALITY RELATION

Proposition 4.3.1. Fix R > 50, T 1 and > 0. We have the following bounds for the Kloosterman terms:

6 3R+2+ |Σ2a| R, (m1n1n2) T ;

6 3R+2+ |Σ2b| R, (m2n1n2) T ;

11/2 3R−1+ |Σ3| R, (m1m2n1n2) T .

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 Vall´eePoussin bound for the Riemann zeta function,

1 |ζ(1 + it)| . log(2 + |t|) Using the bounds for the Fourier coeﬃcients in Proposition 2.5.5, the de la Vall`ee Poussin bound and Stirling’s formula for the Gamma factors, we get

To bound the maximal Eisenstein series contribution Emax we require the following lower bounds for L-functions

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

2 where the eigenvalue of uj is 1/4+rj . 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 CHAPTER 4. ORTHOGONALITY RELATION

iT 1+ Z X 1/2+ |Emax| (m1m2n1n2) 1+ rj

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

2 φj : rj < T ∼ cT for some constant c > 0. In summary, we obtain the following proposition.

Proposition 4.4.1. Fix R > 50, T 1 and > 0. We have the following bounds for the Eisenstein terms in the Kuznetsov trace formula:

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

4.5 Main Geometric Term Computation

To estimate the main term on the geometric side, Σ1 = hFT,R,κ,FT,R,κi, we use the Parseval-type D # # E identity (4.4) which says that Σ1 = FT,R,κ,FT,R,κ . Hence

2 i∞ i∞ # D E 1 Z Z FT,R,κ(ν) dν1dν2 F # ,F # = T,R,κ T,R,κ (πi)2 3 Q 3νj −3νj −i∞ −i∞ Γ 2 Γ 2 j=1 ∞ ∞  2   R+1 Z Z 3 3 27 π X κ X = T −(βj /R) exp −2 β2/T 2 64R    j  −∞ −∞ j=1 j=1 3 Y R+1 R × |tj| + O(|tj| + 1) dt1dt2, j=1

59 CHAPTER 4. ORTHOGONALITY RELATION

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

∞ β2  2   R+1 Z Z 3 3 9 4π X κ X T −(βj /R) exp −2 β2/T 2 64R    j  0 0 j=1 j=1 Y R+1 R × |βj − βk| + O(|βj − βk| + 1) dβ1dβ2 j6=k

∞ β2  2   R+1 Z Z 3 3 9 8π X κ X = T −(βj /R) exp −2 β2/T 2 32R    j  0 0 j=1 j=1 3R+3 3R+2 3R+2 × β2 + O(β2 β1 + β2 + 1) dβ1dβ2.

We now multiply out the integrand to get

D # # E FT,R,κ,FT,R,κ = M + Error1 + Error2, where ∞ β2  3  9R+18π Z Z X M = exp −2 log T (β /R)κ − 2 β2/T 2 β3R+3dβ dβ , 32R  1 j  2 1 2 0 0 j=1 −(β /R)κ Error1 includes all the terms not involving T 1 and Error2 the remaining terms involving 3R+2 3R+2 β2 β1 + β2 + 1. We can bound the error terms by

∞ β2 ∞ Z Z κ Z κ −(β2/R) −(β2/R) Error1 T p(β1, β2)dβ1dβ2 T p˜(β2)dβ2 1, 0 0 0 where p, p˜ are polynomials, and

∞ β2  3  Z Z κ −(β1/R) X 2 2 3R+2 3R+2 Error2 T exp − β2 /T  (β2 β1 + β2 + 1)dβ1dβ2 0 0 j=1 ∞ Z 2 2 3R+3 exp −β2 /T (β2 + β2)dβ2 0 T 3R+3.

60 CHAPTER 4. ORTHOGONALITY RELATION

We then write

∞ β2 9R+18π Z Z M = exp −c2βκ − c2(β2 + β β + β2) β3R+3dβ dβ , 32R 1 1 2 1 1 2 2 2 1 2 0 0 2 κ 2 2 where c1 = 2 log T/R , c2 = 4/T .

Note that this integral is going to have exponential decay outside the region where β1 −2/κ −1 c1 , β2 c2 . Therefore,

∞ β2 9R+18π Z Z M = exp −c2βκ − c2β2 β3R+3 1 + O c2(β2 + β β ) dβ dβ 32R 1 1 2 2 2 2 1 1 2 1 2 0 0

∞ β2 9R+18π Z Z = exp −c2βκ − c2β2 β3R+3dβ dβ + O c−(3R+2)c−6/κ + c−(3R+3)c−4/κ 32R 1 1 2 2 2 1 2 2 1 2 1 0 0 ∞ ∞ 9R+18π Z Z = exp −c2βκ − c2β2 β3R+3dβ + O exp −c2βκ β3R+3 dβ + O T 3R+3 32R 1 1 2 2 2 1 1 2 2 2 0 0 ∞ ∞ 9R+18π Z Z = c−2/κc−(3R+4) exp −βκ − β2 dβ dβ + O(T 3R+3) 32R 1 2 1 2 1 2 0 0 9R+1Rπ3/2C T 3R+4 = κ + +O(T 3R+3), 28R+2+1/κ (log T )1/κ

∞ R κ where Cκ = exp (−x ) dx. 0 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 BIBLIOGRAPHY

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