Moments of Automorphic L-Functions at Special Points

Moments of automorphic L-functions at special points

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Alexander Lu Beckwith, B.A., M.S. Graduate Program in Mathematics

The Ohio State University 2020

Dissertation committee

Dr. Wenzhi Luo, Advisor Dr. James Cogdell Dr. Roman Holowinsky ⃝c Copyright by Alexander Lu Beckwith 2020 Abstract

We study the behavior of families of L-functions at exhibiting conductor-dropping behavior. We derive asymptotic expansions of the short interval ﬁrst and second moments of GL(2) × GL(2) L-functions at special points with power-saving error terms. As a consequence, we obtain an essentially optimal lower bound on the number of cusp forms for the Hecke congruence surfaces Γ0(p) \ H of prime level are destroyed under quasiconformal deformation of these surfaces. Additionally, we show that a large number of cusp forms for the Hecke congruence surfaces Γ0(p) \ H of prime level are simultaneously destroyed in two directions of the associated Teichmüller space. We also establish upper bounds for the second moment of GL(2) × GL(3) L-functions and the sixth moment of GL(2) L-functions at special points as the spectral parameter varies in a short interval. An auxiliary twisted spectral large sieve inequality for short intervals is derived for the latter two results.

ii Dedicated to my grandparents

iii Acknowledgments

During my time as a student I have been very fortunate to have had the guidance and support of a number of wise and thoughtful people. There is a very long list of individuals whom I count myself lucky to have met and to whom I wish to extend my thanks in the following—I regret that I have surely neglected some from what appears below. First and foremost, to my advisor, Wenzhi Luo, for his thoughtful guidance, constant and enduring support, for his many kind words of encouragement, and for his boundless patience: Thank you, a thousand times, thank you. To Roman Holowinsky: for all his work helping graduate students navigate the nonacademic post- academic world, from which I have personally beneﬁted so much. To Jim Cogdell: for organizing the automorphic forms seminar that I and many other students have beneﬁted from greatly over the years. To Ghaith Hiary: for serving on my candidacy committee, and for teaching a topics course on computational number theory that I found to be a formative experience. To my undergraduate professors, Judy Holdener, Bob Milnikel, and Marie Snipes: for encouraging me to study math a decade ago now. To my mentor and friend Jessen Book: for his wisdom and guidance when I needed a push in the right direction many years ago. And most of all, to my family, for all their love and support they have given me, no matter how much I stumbled.

Thank you.

iv Vita

1992 ...... Born in Midland, Michigan

2014 ...... B.A. in Mathematics, Concentration in Humane Studies, Kenyon College Gambier, Ohio

2017 ...... M.S. in Mathematics, The Ohio State University Columbus, Ohio

2014-2015 ...... University Fellow The Ohio State University Columbus, Ohio

2015-Present ...... Graduate Teaching Associate, Graduate Research Associate The Ohio State University Columbus, Ohio Fields of Study

Major ﬁeld: Mathematics

Specialization: Analytic number theory, automorphic forms

v Contents

Abstract ...... ii Acknowledgments ...... iv Vita ...... v Notation and conventions ...... ix

1 Introduction 1

I Preliminaries 7

2 Preliminaries on automorphic forms 8 2.1 Automorphic forms for GL(2, R) ...... 9 2.2 Automorphic forms for GL(3, R) ...... 33

3 Preliminaries on automorphic L-functions 42 3.1 Automorphic L-functions ...... 42 3.2 On conductor-dropping behavior of L-functions in the spectral aspect ...... 48

4 Preliminaries on the spectral theory of hyperbolic Riemann surfaces and Teichmüller theory 51 4.1 Riemann surfaces ...... 51 4.2 Some remarks on Teichmüller theory and hyperbolic 2-surfaces ...... 52 4.3 The Phillips-Sarnak deformation theory of discrete groups ...... 55

II Upper bounds for moments of L-functions at special points 58

5 A short interval large sieve inequality with spectral twists 59 5.1 Background and survey of existing literature ...... 60 5.2 The short interval twisted spectral large sieve inequality ...... 67 5.3 Proof of Theorem 5.2.1 ...... 73 5.4 Proof of Lemmas 5.2.4 and 5.2.5 ...... 75

6 The second moment of GL(2) × GL(3) L-functions at special points over a shortened spectral interval 83 6.1 Background and survey of existing literature ...... 84 6.2 Preliminaries on GL(2) × GL(3) Rankin-Selberg L-functions ...... 86 6.3 The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points 88 6.4 The shortened spectral interval sixth moment of GL(2) L-functions at the special point ... 93

vi Contents III Asymptotic expansions of moments of L-functions at special points 97

7 The nonvanishing of GL(2) × GL(2) L-functions at special points over shortened spectral intervals 100 7.1 Properties of the Rankin-Selberg L-function ...... 102 7.2 An asymptotic expansion of the ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals ...... 103

8 The simultaneous nonvanishing of GL(2) × GL(2) L-functions at special points 117 8.1 Proof of Theorem 8.0.1: an asymptotic expansion of the second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals ...... 119

IV Future directions 138

9 Future directions 139 9.1 Direct analogues ...... 140 9.2 An asymptotic expansion of the fourth moment of GL(2) L-functions at special points .... 140 9.3 The ﬁrst moment of GL(2) × GL(3) L-functions at special points ...... 141 9.4 The prime geodesic theorem ...... 142 9.5 Special point subconvexity problems ...... 144 9.6 Higher-order Fermi golden rules ...... 147 9.7 The newform Weyl law ...... 148

References 148

V Appendices 157

A Compendium of special functions and their properties 158 A.1 Bessel functions ...... 158

B Asymptotic behavoir of oscillatory integral 164

C Miscellaneous bounds for character sums 172 C.1 Compendium of properties of Kloosterman sums and Ramanujan sums and special character sums ...... 172

vii List of Figures

Chapter 2

2.1 Fundamental domain F0,1 for the modular group SL(2, Z) ...... 13 2.2 Some examples of fundamental domains for the Hecke congruence subgroups Γ0(N) ..... 14 2.3 Some examples of quotient surfaces for the Hecke congruence subgroups Γ0(N) ...... 15 2.4 Plot of cuspidal and residual spectrum of ∆ and poles of Eisenstein series for SL(2, Z) .... 22

viii List of Tables

Chapter 3

3.1 Some eﬀects of conductor-dropping for L-functions in the spectral aspect ...... 49

Chapter 6

6.1 Status of upper bounds for families of L-functions at special points ...... 85

ix Notation and conventions

We standardize the common notation and conventions used throughout this document.

• N: the set of natural numbers 1, 2, 3,...

• H: the upper half of the complex plane, H = {x + iy : x, y ∈ R, y > 0}

• SL(2, R), SL(2, Z): the special linear group over R, Z, respectively

• GL(n, R): the general linear group of degree n over R

• Γ0(N): the Hecke congruence subgroup of level N

• Γa: the stability group of a cusp a of Γ

• νf (n): the normalized Fourier coeﬃcients at ∞ of an automorphic form over GL(2, R)

• λf (n): the Hecke eigenvalues of an automorphic form f

• λf : the Laplace eigenvalue of an automorphic form f

• ΦΓ(s), φΓ(s): the scattering matrix for Γ and the determinant of ΦΓ(s)

• S(n, m; c): the Kloosterman sums for SL(2, Z), X∗ an + an S(n, m; c) = S∞∞(n, m; c) = e . c a(c)

Sab(m, n; c) denotes the Kloosterman sums of a discrete subgroup Γ ⊆ SL(2, R) attached to cusps a and b. As we will deal primarily with Kloosterman sums for the full modular group, we will not use this notation much.

• δ(condition): the delta symbol

• δ: the reﬂection operator, unless it appears in an exponent, in which case it is a positive constant.

• h·, ·i: the Petersson inner product

• k · k: the Petersson norm, unless looking at a sequence, in which case k · k denotes the (possibly truncated) ℓ2-norm

x List of Tables

• φe: the Mellin transform of a function φ (satisfying the appropriate conditions) deﬁned by Z ∞ dx φe(s) = φ(x)xs 0 x

• φb: the Fourier transforms of a function φ (satisfying the appropriate conditions) deﬁned by Z φb(ξ) = φ(x)e(−ξx)dx R

• τ2, τ3: the binary and ternary divisor functions deﬁned by X X X τ2(n) = 1 and τ3(n) = 1 = τ2(n1).

n1n2=n n1n2n3=n n1n2=n

Note that these are distinct from the divisor functions τit given by X d it τ (n) = . it k dk=n

When the order of the divisor function is a natural number (i.e., τk with k ∈ N), we mean the k-ary th divisor function. We reserve σk for the k -power-of-divisors function.

− t−T 2 • à˚u¯sfi¯sfi˚ià‹nffl(t): the Gaussian function centered at T of width M, à˚u¯sfi¯sfi˚ià‹nffl(t) = e ( M ) GT,M GT,M • e(x): the additive character e(x) = e2πix

• hT,M (t): usually the (even) twisted Gaussian centered at T of width M

it −it à˚u¯sfi¯sfi˚ià‹nffl m à˚u¯sfi¯sfi˚ià‹nffl m hT,M (t, m, n) = (t) + (−t) GT,M n GT,M n

• ρ, ψ denote the weight function and phase function, respectively, in an oscillatory integral, except when possibly ψ denotes an SL(3, Z) Maass cusp form. This will hopefully be clear from context.

• Q, Q1, and Q2 denote holomorphic modular forms of some weight and level; we will always take these to be newforms and to have trivial nebentypus. P • ∗ a(c) : denotes a sum over congruence classes a (mod c) such that (a, c) = 1. When (a, c) = 1, the multiplicative inverse of a modulo c is denoted by a.

• Tèˇi`c‚hffl(R), Tèˇi`c‚hffl(S): the Teichmüller space of a Riemann surface R, or of the underlying surface S that marks a Riemann surface R.

• When not otherwise mentioned, sj will always refer to the special point, also known as the conductor-

dropping point or spectral point, for a Hecke-Maass cusp form uj for Γ ⊆ SL(2, R) a Fuchsian 1 group of the first kind. In this case sj = 2 +itj; for other sorts of automorphic forms, the corresponding special point may be defined differently.

xi List of Tables

Remark 0.0.1. In general, we observe a notational “reset” between chapters. This is done in order to avoid having too many symbols, especially when they may serve the same purpose for corresponding objects in each chapter. Throughout any ε appearing in an exponent will usually denote a small number that may actually change from line to line in a series of equations. When δ appears in an exponent, it is an absolute constant.

Remark 0.0.2. Σi, where i = 0, 1, 2,... , is shorthand for a spectral sum that appears as a result of applying an approximate functional equation to the L-functions in question. “Mathcal” script is used to denote objects that arise from application of the Kuznetsov trace formula. These will tend to be indexed according to whether they come from the diagonal term, the terms involving Kloosterman sums, etc.

Remark 0.0.3. A number of special functions appear throughout this document, primarily Bessel functions, but also others. A brief overview of the relevant theory of these appears in Appendix A, as well as a number of useful formulas and expansions. In particular, the various integral transforms appearing in the Kuznetsov trace formula are gathered in Appendix A; we will refer to these frequently.

In an eﬀort to keep things as self-contained as possible, and to put the results presented here in greater context, throughout we will also include full statements of results attributed to others that we reference frequently. These theorems and propositions appear in blue-colored boxes. The original results in this document appear in red-colored boxes. Lemmas and other propositions appear in gray-colored boxes.

xii Chapter 1

Introduction

Let S be a compact smooth manifold of real dimension n—an “n-drum”—and let ∆S denote its associated

Laplace-Beltrami operator. The operator ∆S is a diﬀerential operator that acts on the space of square- integrable functions, L2(S), and it has the unique property that it is invariant under isometric diﬀeomor- phisms of S, which makes it an important invariant of the manifold S. It is formally self-adjoint, nonnegative, and it comes with a distinguished sequence of numbers—its spectrum or eigenvalues {λj}j≥1—that in the real world setting correspond to the resonant frequencies or tones that occur when one strikes S. Mark Kac famously once asked, “Can one hear the shape of a drum?” Put another way, what information about the geometry of S can be obtained from the λj, its resonant frequencies or spectrum? For example,

Weyl’s law for a general compact surface S expresses the volume of S in terms of the distribution of the λj: one has

p ω · (S) #{ λ ≤ T : λ is an eigenvalue of ∆ } = n ”vˆo˝l T n + O T n−1 , j j S (2π)n

n where ωn is the volume of the unit ball in R ; so one can hear the “volume” of the drum. In the time since Kac’s original question, a great deal of work has been done in studying the relationship between geometry and spectral theory. For particular classes of surfaces, one can often say a great deal more, and one can explore various reﬁnements of Kac’s question. The latter is what we will do in this dissertation.

Let Γ ⊆ SL(2, R) be a Fuchsian group of the first kind that is not cocompact. Then the quotient surface Γ \ H is a noncompact hyperbolic Riemann surface of finite volume with finitely many punctures or cusps

{a}. In this setting, the study of the spectrum of the Laplacian ∆Γ\H is the study of classical GL(2, R) automorphic forms, and with certain weakenings of the invariance conditions, classical modular forms.

The noncompactness of Γ \ H is reﬂected in the spectral decomposition of ∆Γ\H: for these surfaces the spectrum of ∆Γ\H consists of both a discrete part and a continuous part. The continuous spectrum covers 1 ∞ the interval [ 4 , ) uniformly with multiplicity equal to the number of cusps, while the discrete eigenvalues

1 1.0. Introduction

1 2 λj can be written as λj = 4 + tj , and it is conjectured that all but ﬁnitely many of them are embedded in [ 1 , ∞). Accordingly, L2(Γ \ H) decomposes into two distinguished orthogonal subspaces: the span of the 4 { } 1 Maass cusp forms uj —the discrete spectrum—and the span of the Eisenstein series Ea z, 2 + it —the continous spectrum and residual spectrum1. The Eisenstein series have a simple deﬁnition in terms of the group Γ, but the Maass cusp forms are elusive and there are only extremely rare circumstances in which their explicit construction is known. In many circumstances, it is unclear whether any exist at all. One might wonder if a version of Weyl’s law still holds for the noncompact surface Γ \ H. Measurement

of the spectrum of ∆Γ\H is less straightforward when there are both discrete and continuous parts. When studying the distribution of eigenvalues of Γ \ H, it is diﬃcult to separate the two contributions and one is forced to consider the pieces of the spectrum jointly. Denote the counting function for the discrete spectrum up to spectral height T by

NΓ(T ) = #{tj ≤ T : λj is an eigenvalue of ∆Γ\H}.

Analogously, the continuous spectrum is measured by the integral Z T −φ′ 1 + it 1 2 MΓ(T ) = 1 dt, 4π −T φ 2 + it

which yields the winding number of the determinant φΓ of the scattering matrix in the functional equation for the Eisenstein series. This counts the poles of the Eisenstein series, also known as scattering resonances

of ∆Γ\H. Weyl’s law in this setting for the counting functions considered simultaneously takes the form

(Γ \ H) N (T ) + M (T ) = ”vˆo˝l T 2 + O (T log T ) , Γ Γ 4π

as T → ∞. It is interesting to explore whether the discrete spectrum or continuous spectrum provides a larger contribution to the righthand side. There are special circumstances in which one can isolate the counting functions to address this question. In the case where Γ is a congruence subgroup, the determinant of the scattering matrix factors in terms of classical Dirichlet L-functions attached to Dirichlet characters. Using this fact, Selberg showed that the spectrum is essentially cuspidal in this case; the continuous spectrum counting function satisﬁes

MΓ(T ) Γ T log T,

and thus NΓ(T ) provides the main contribution to Weyl’s law; the implied constant in the above depends on the level of Γ. It follows that there are inﬁnitely many cusp forms for congruence subgroups. This led Selberg to conjecture that cusp forms are the dominant part of the spectrum for generic ﬁnite volume hyperbolic

surfaces, for which such a clever approach to controlling the contribution from MΓ(T ) is no longer available. That the cusp forms apparently dominate for congruence subgroups is surprising and interesting, as there are currently only very limited ways of explicitly constructing any cusp forms.

1The residual spectrum actually comprises part of the discrete spectrum, see Chapter 2.

2 1.0. Introduction

As it turns out, the situation for congruence subgroups is rather special and also rather misleading; the dominance of the discrete spectrum appears to be related to the arithmeticity of the subgroup in question, and may be false for generic discrete, ﬁnite-volume subgroups of SL(2, R). In order to study the spectrum

of ∆Γ\H for generic discrete subgroups Γ ⊆ SL(2, R), Phillips-Sarnak restricted themselves to considering

those Γε, ε > 0, arising from quasiconformal deformations of congruence surfaces Γ \ H with Γ0 = Γ. Such \ H \ H \ H surfaces Γε lie in T`eˇi`c‚hﬄ(Γ ), the Teichmüller space of Γ . They showed that the presence of cusp forms on Γε \ H is implied by whether certain special values of a distinguished family of Rankin-Selberg L-functions are zero or not. Drawing on the work of Lax, Phillips, and Colin de Verdière, they consider perturbations of the Laplacian on the deformed surface and relate the destruction of the Maass cusp form to the nonvanishing of the inner product of a Maass cusp form and an Eisenstein series at the same spectral height. Namely, the condition is that if 1 L u ,E ·, + it =6 0, Q j 2 j

where LQ is a certain operator deﬁned in terms of a weight 4 holomorphic modular form Q for Γ, then the \ H cusp form uj is destroyed under quasiconformal deformations Γε beginning in the direction generated by h · 1 i 1 Q. Amazingly, the inner product LQuj,E , 2 + itj can be expressed in terms of the value at sj = 2 + itj of the Rankin-Selberg L-function L (s, uj ⊗ Q), and thus the techniques used to study L-functions can be used to address the question of the destruction of Maass cusp forms. It is diﬃcult to determine whether a particular L-function is nonzero on the critical line. Rather than focusing on an individual L-function, Luo studied the behavior of these special values on average. He showed

that in fact a positive proportion of the special values L (sj, uj ⊗ Q) are nonzero. Namely, he showed that

2 #{tj ≤ T : L(sj, uj ⊗ Q) =6 0} T ,

as T → ∞. His method proceeds by reducing the question of nonvanishing of the L-functions under study to obtaining an asymptotic expansion of the ﬁrst moment of those L-functions and an upper bound for their second moment. Under the assumption that the multiplicities of the discrete spectrum of congruence surfaces are essentially bounded, it follows that a positive proportion of the discrete spectrum is destroyed under quasiconformal deformation of Γ\H, and therefore that the continuous spectrum dominates for Γε \H.

In this thesis we study various reﬁnements of Luo’s nonvanishing results for the special values L (sj, uj ⊗ Q).

In short spectral intervals, many cusp forms for the surfaces Γ0(p) \ H are destroyed, where Γ0(p) denotes the Hecke congruence subgroup of prime level p ≥ 5.

Theorem 1.0.1. Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) and let Q be a 1 +ε weight-2k cuspidal holomorphic Hecke newform for Γ0(p), where p ≥ 5 is prime. Then for M T 2 , as

3 1.0. Introduction

T → ∞ one has 1−ε #{T − M ≤ tj ≤ T + M : L(sj, uj ⊗ Q) =6 0} T M.

The lower bound T 1−εM is essentially the optimal bound predicted by Phillips-Sarnak and the Weyl law. We also study the question of whether a cusp form can be destroyed in only one basis type of quasicon-

formal deformation of the surfaces Γ0(p) \ H. Over the long spectral interval, many cusp forms are destroyed in at least two basis types of deformations for p ≥ 11.

Theorem 1.0.2. Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) and let Q1 =6 Q2 be two weight-2k cuspidal holomorphic Hecke newforms for Γ0(p), where p ≥ 11 is prime. Then as T → ∞, one has 3 −ε #{tj ≤ T : L(sj, uj ⊗ Q1) =6 0 and L(sj, uj ⊗ Q2) =6 0} T 2 .

3 −ε 2−ε Unfortunately, the lower bound of T 2 in Theorem 1.0.2 is far from the optimal lower bound of T that is predicted by the Phillips-Sarnak theory. In Chapter 8 we address how this may possibly be improved by an forthcoming result of Chandee-Li on the second moment of GL(2) × GL(4) L-functions at special points. The behavior of the families of L-functions in question is actually quite interesting on its own, as at the 1 special value sj = 2 + itj the conductor of the L-function drops and the L-function behaves as though it has a lower degree. We will study this conductor-dropping behavior for additional classes of L-functions for which there is no immediate application in mind. The technical results in this dissertation are comprised of roughly of two general types: the ﬁrst is a series of upper bounds for moments of families of L-functions exhibiting conductor-dropping behavior. The second type is a series of asymptotic expansions of the same sorts of moments of L-functions. For the nonvanishing results, we are interested in combining the two types to establish a nonvanishing result for the L-functions in question. A general theme in both settings is that often one can replace a certain hypothesis of square-root cancellation with a weaker Voronoi-type summation, which apply to a wider class of arithmetic functions. The upper bound type results are the following, and their proofs are found in Chapter 6. The ﬁrst is an upper bound for the short interval second moment of GL(2) × GL(3) L-functions at the special point 1 sj = 2 + itj.

Theorem 1.0.3. Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) and let ψ be a 21 +ε GL(3, R) automorphic form. Then for T 22 M T , as T → ∞ one has the upper bound X 2 1+ε |L(sj, uj ⊗ ψ)| T M.

T −M≤tj ≤T +M

We will also obtain the following upper bound for the sixth moment of GL(2) L-functions at the special point that is consistent with the Lindelöf hypothesis.

4 1.0. Introduction

Theorem 1.0.4. Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z). Then for 21 +ε T 22 M T , as T → ∞ one has X 6 1+ε |L(sj, uj)| T M.

T −M≤tj ≤T +M

In the course of proving Theorem 1.0.3 and Theorem 1.0.4, we will require a twisted large sieve inequality for short spectral intervals. The following is proved in Chapter 5.

Theorem 1.0.5. Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) with normalized

Fourier coeﬃcients νj(n). Let M,N, and T be parameters with 1 M T and N ≥ 1. Then for any N sequence A = (an) ∈ R , one has

2 ! 7 X X 4 3 1 N itj ε 2 anνj(n)n TM + N + M 2 N 2 + T (NT ) kAk . M T −M≤tj ≤T +M n≤N

The asymptotic expansions used to obtain the nonvanishing results are the following. Their proofs are contained in Chapter 7 and Chapter 8, respectively.

Theorem 1.0.6. Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) and let Q be a 1 +ε weight 2k cuspidal holomorphic Hecke newform for Γ0(p), where p ≥ 5 is prime. Then for M T 2 , as T → ∞ one has the asymptotic expansion

( ) X t −T 2 − j 2 3 M | |2 ⊗ 2 +ε e νj(1) L(sj, uj Q) = 3 TM + O T . π 2 tj

Theorem 1.0.7. Let {uj}j be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) and let Q1 =6

Q2 be two weight 2k cuspidal holomorphic Hecke newforms for Γ0(p), where p ≥ 11 is prime. Then for 5 +ε T 6 M T , as T → ∞ one has the asymptotic expansion

( ) X t −T 2 − j 11 M | |2 ⊗ ⊗ 6 +ε e νj(1) L(sj, uj Q1)L(sj, uj Q2) = c0,Q1,Q2 TM + O T , tj

for a certain constant c0,Q1,Q2 deﬁned in terms of Q1 and Q2. Furthermore, if instead one has Q1 = Q2 = Q

5 1.0. Introduction

and p ≥ 5, then

( ) X t −T 2 − j 11 M 2 2 +ε e |νj(1)| |L(sj, uj ⊗ Q)| = c0,QTM log(T ) + c1,QTM + O T 6 ,

for certain constants c0,Q and c1,Q that are deﬁned in terms of Q.

Some remarks on organization

Part I provides an extended summary of the background material on automorphic forms for GL(2, R) and GL(3, R), automorphic L-functions, and Teichmüller theory and Phillips-Sarnak deformation theory of discrete groups. This part is quite lengthy and the familiar reader may wish to skip directly to Parts II and III, where the aforementioned results described in the introduction appear. At the head of the relevant chapters in these sections, we provide a more focused review of the relevant literature. Part IV contains some extended remarks on possible directions for future research. Integral transforms and properties of special functions that appear throughout this document can be found in Appendix A. Auxiliary tools used for evaluating integrals that appear in Parts II and III are contained in Appendix B.

6 Part I

Preliminaries

In Chapter 2 we review the classical spectral theory of automorphic forms on GL(2, R) and GL(3, R), and collect the general properties and summation formulas (Petersson, versions of Kuznetsov, Voronoi) that will be used in later chapters. We do not include the Selberg trace formula. We only discuss automorphic forms for trivial nebentypus. In Chapter 3, we collect some very basic theory of L-functions—for the sake of ease of reading, we have opted to include the more speciﬁc descriptions of various types of L-function in the relevant chapters, though the tools and formulas developed there may apply to a wider class. We also survey the literature on the behavior of L-functions exhibiting conductor-dropping in the spectral aspect. In Section 3.2, we provide a brief overview of the conductor-dropping phenomenon in the spectral aspect. In Chapter 4 we provide an outline of Phillips-Sarnak’s deformation theory of discrete groups and a brief review of the relevant material from Teichmüller theory.

7 Chapter 2

Preliminaries on automorphic forms

Contents 2.1 Automorphic forms for GL(2, R) ...... 9 2.1.1 The classical setting ...... 9 2.1.2 Fuchsian groups of the ﬁrst kind ...... 11

2.1.3 The modular group and the Hecke congruence subgroups Γ0(N) ...... 12 2.1.4 The spectral theory of ∆ on Γ \ H ...... 16 2.1.4.1 The space of incomplete Eisenstein series ...... 17 2.1.4.2 The space of cuspidal automorphic functions ...... 19 2.1.4.2.1 The space of newforms and oldforms ...... 19 2.1.4.3 The spectral theorem ...... 20 2.1.5 The Hecke operators ...... 21 2.1.6 The Weyl law ...... 21

2.1.6.1 The Weyl law for Γ0(N) \ H ...... 22 2.1.7 Holomorphic modular forms ...... 23

2.1.7.1 Holomorphic modular forms in terms of the spectral theory of ∆k .... 25 2.1.7.2 The Petersson trace formula ...... 26 2.1.8 The Kuznetsov trace formula ...... 27 2.1.9 Bounds for Fourier coefficients and twists of Fourier coefficients ...... 29 2.1.9.1 The GL(2) Voronoi summation formula ...... 30 2.1.9.2 The Hecke bound ...... 32 2.1.10 An aside: the homogenous space setting ...... 33 2.2 Automorphic forms for GL(3, R) ...... 33 2.2.1 The Maass cusp forms for SL(3, Z) ...... 35 2.2.2 The Hecke operators and Hecke-Maass cusp forms for SL(3, Z) ...... 36 2.2.3 Bounds for Fourier coefficients and twists of Fourier coefficients ...... 37 2.2.3.1 The GL(3) Voronoi summation formula ...... 37 2.2.3.2 The Miller bound for additive twists of Fourier coefficients ...... 38 2.2.4 The minimal parabolic Eisenstein series for SL(3, Z) and Voronoi summation for the ternary divisor function ...... 39

8 2.1. Automorphic forms for GL(2, R)

2.1 Automorphic forms for GL(2, R)

In an eﬀort to make this document as comprehensive as we can, in this section we provide an summary exposition of the relevant theory of GL(2, R) automorphic forms and geometry of the surfaces mentioned in the introduction. Our presentation largely follows Iwaniec’s book [Iwa02], and also draws on [Kat92], [Gol06], [Hej76], and [Mul10], to which we refer the reader for details. If the reader is already familiar with this background material we suggest they skip this section altogether.

2.1.1 The classical setting

There are many possible starting points for describing automorphic forms on GL(2, R). We begin with the classical setting, which is most natural for us in describing the Riemann surfaces mentioned in the introduction. Classically, one uses the upper half plane model of the hyperbolic plane,

H = {z = x + iy : x, y ∈ R, y > 0} .

The upper half plane is a Riemannian manifold, with line element

dx2 + dy2 ds2 = . y2

This equips H with a distance function, via the usual length formula Z Z 1 p dt ρ(z, w) = inf ds = inf x′(t)2 + y′(t)2 , p p p 0 y(t)

where the inﬁmum is taken over smooth paths p = (x, y) : [0, 1] → H starting at z and ending at w. Explicitly, the distance function (the Poincaré distance function) is given by |z − w| + |z − w| ρ(z, w) = log , for z, q ∈ H, |z − w| − |z − w|

with geodesics given by half-circles intersecting the real line Rb. The special linear group consisting of 2 × 2 matrices with determinant 1, ( ! ) a b SL(2, R) = γ = : det(γ) = ad − bc = 1 , c d ! a b acts transitively on H via linear fractional transformations: for a matrix γ = ∈ SL(2, R), γ acts on c d H via az + b γz = ∈ H. cz + d This action is isometric in the sense that the metric derived from the line element ds2 is SL(2, R)-invariant. Note that both γ and −γ determine the same action on H, so one can instead consider the action of

9 2.1. Automorphic forms for GL(2, R)

PSL(2, R) on H. We will regularly abuse notation in this way. The whole group of isometries on H is generated by PSL(2, R) and the reﬂection operator δ : z 7→ −z. The associated area (or volume) element is the SL(2, R)-invariant Haar measure dxdy dµ(z) = , y2 also known as the Poincaré measure. ! a b The nontrivial motions given by γ = ∈ SL(2, R) on H take one of three types. Up to conjugacy c d class [γ] of γ, these are:

1. parabolic: [γ] ﬁxes ∞: action by translation ⇔ |a + d| = 2,

2. hyperbolic: [γ] ﬁxes 0 and ∞: action by dilation ⇔ |a + d| > 2,

3. elliptic: [γ] ﬁxes i: action by rotation ⇔ |a + d| < 2.

We refer to the points ﬁxed by parabolic and elliptic motions as parabolic or elliptic ﬁxed points, respectively. Later we will outline how the geometry of quotient surfaces of H is described by the above types of motions, and how the geometry of the quotient surface is related to the spectrum of the Laplace-Beltrami on the surface. As a Riemannian manifold, H comes with an associated Laplace-Beltrami operator (or the Lapla-

cian) ∆H = div◦grad. In rectangular coordinates, the Laplacian on H (we will often suppress the underlying surface and simply denote the Laplacian by ∆) is given by 2 2 2 ∂ ∂ 2 ∂ ∂ ∆H = ∆ = y + = − (z − z) , ∂x2 ∂y2 ∂z ∂z

where ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = − i and = + i ∂z 2 ∂x ∂y ∂z 2 ∂x ∂y are the Wirtinger derivatives. It was mentioned in the introduction that the Laplace-Beltrami operator is an important invariant of a Riemannian manifold—in general, the Laplace-Beltrami operator associated with a Riemannian manifold determines the diﬀeomorphic isometries of the manifold: a diﬀeomorphism of the manifold is an isometry if and only if it leaves the Laplace-Beltrami operator invariant. Indeed, it is straightforward to check that ∆ is invariant under the action of SL(2, R) on H and δ. It is interesting to consider the eigenfunctions of ∆ on H; that is, those f : H → C satisfying

(∆ + λ)f = 0

with eigenvalue λ ∈ C and the appropriate diﬀerentiation conditions. The simplest eigenfunctions of ∆ on the full space H with nonzero eigenvalue1 are the simple power functions

1 1 I (z) = ys + y1−s and I (z) = ys − y1−s , 1,s 2 2,s 2s − 1

1The eigenfunctions of ∆ with eigenvalue λ = 0 are numerous: they are the harmonic functions on H.

10 2.1. Automorphic forms for GL(2, R)

− 6 1 both having eigenvalue λs = λs(∆) = s(1 s), when s = 2 . Generally, one makes the problem more interesting by restricting to subspaces of functions satisfying additional conditions. For example, by periodizing, one can show that the Whittaker functions

1 Ws(z) = 2y 2 K − 1 (2πy)e(x) s 2 ! 1 1 are eigenfunctions of ∆ that are invariant under the shifts z = z+1, and that the family of Whittaker 1 functions {W 1 (rz)}r,t∈R forms a complete eigenpacket on H. 2 +it

2.1.2 Fuchsian groups of the ﬁrst kind

The action of the various kinds of subgroups of SL(2, R) are of interest. The subgroups that we are interested in are the ones yielding ﬁnite-volume Riemann surfaces. These are the Fuchsian groups of the ﬁrst kind.

Deﬁnition 2.1.1. A subgroup Γ ⊆ SL(2, R) is said to be a Fuchsian group (or discrete) if Γ/{I} ⊆ PSL(2, R) acts discontinuously on H.

The action of a Fuchsian group Γ is completely determined by its action on a fundamental domain for Γ; a fundamental domain F ⊆ H for Γ is a domain in H such that ΓF ∩ F = ∅ and for all z ∈ H, Γz ∩ F =6 ∅.

Remark 2.1.1. Often we abuse notation and denote a fundamental domain for Γ by Γ \ H, although this is not well-deﬁned since there are many fundamental domains for a given Fuchsian group; however, in the contexts where we use Γ \ H in this way the behavior of interest will be independent of our choice of a corresponding subset of H.

Definition 2.1.2. A Fuchsian group Γ is a Fuchsian group of the first kind if it is finitely generated

and has a fundamental domain FΓ = Γ \ H with ﬁnite volume, which is given by Z \ H ”vˆo˝l(Γ ) = dµ(z). FΓ

We say that Γ is co-compact if it contains no parabolic elements; in this case the fundamental domain is a compact polygon in H with an even number of sides.

With the appropriate choice of a complex atlas on the quotient space Γ \ H, one obtains a Riemann surface of constant negative curvature. A finite-volume group Γ may have finitely many cusps or punctures—these are points on the boundary of the fundamental domain lying on Rb, so they are the fixed point of a parabolic element of Γ. For any cusp

a of Γ, there is a scaling matrix σa ∈ SL(2, R) such that σa∞ = a and !

−1 1 1 σ γaσa = , a 1

where γa is the generator of the stability group Γa = {γ ∈ Γ: γa = a} = hγai of a.

11 2.1. Automorphic forms for GL(2, R)

The boundary of the fundamental domain of Γ may also contain elliptic ﬁxed points. These correspond to branch points in the covering H → Γ \ H.

Deﬁnition 2.1.3. Let Γ be a Fuchsian group of the ﬁrst kind such that Γ\H has genus g, inequivalent elliptic

ﬁxed points ν1, . . . , νℓ of orders m1, . . . , mℓ, and h inequivalent cusps. Then we say that the signature of Γ

(or Γ \ H) is (g; m1, . . . , mℓ; h).

The signature of Γ \ H is an important invariant of Γ. For example, the volume of Γ \ H is completely determined by the signature of Γ \ H; one has the Gauss-Bonnet formula   Xℓ 1 (Γ \ H) = 2π 2g − 2 + 1 − + h . ”vˆo˝l m j=1 j

In Chapter 4 we will see that the signature of Γ \ H characterizes the Riemann surface up to quasiconformal equivalence.

Remark 2.1.2. From this point forward, all Fuchsian groups of the ﬁrst kind that we deal with will be assumed to be noncocompact unless otherwise speciﬁed, and will be denoted by Γ. Furthermore we will assume that ∞ is a cusp.

When Γ \ H has cusps, a great deal of information about the spectrum of ∆Γ\H is captured by the

Kloosterman sums associated with Γ \ H. Suppose that a and b are cusps of Γ, and c ∈ Cab, where ( ! ) ∗ ∗ −1 Cab = c > 0 : ∈ σ Γσb . c ∗ a

Then the Kloosterman sum of modulus c attached to a and b is deﬁned by X md na Sab(n, m; c) = e + , (2.1)   c c ∗ a  −1  ∈Γ∞\σa Γσb/Γ∞ c d for m, n ∈ Z.

2.1.3 The modular group and the Hecke congruence subgroups Γ0(N)

Without imposing any restrictions on the eigenfunctions of ∆, the spectral theory on H is not particularly interesting. The natural restrictions to impose are invariance under the action of subgroups Γ ⊆ SL(2, R), which amounts to studying the spectrum of ∆ on the quotient surface Γ \ H. We shall deal with two distinguished types of Fuchsian groups of the ﬁrst kind: the Hecke congruence subgroups of level N (which we will assume later to be prime), ( ! ) a b Γ0(N) = γ = ∈ SL(2, Z): c ≡ 0 (mod N) , c d

12 2.1. Automorphic forms for GL(2, R)

and those subgroups that arise from quasiconformal deformations of Γ0(N) \ H, described in Chapter 4. The

full modular group, SL(2, Z) = Γ0(1), has fundamental domain 1 F = z = x + iy : |x| < , |z| > 1 0,1 2

π Z \ H with volume 3 . The quotient surface SL(2, ) is a noncompact, finite-volume Riemann√ surface with one ∞ 1 i 3 cusp at , one elliptic fixed point i of order two, and one elliptic fixed point 2 + 2 of order three.

SL(2, Z) \ H

i √ i 1 i 3 2 + 2

(b) The Riemann surface Γ0(1) \ H

1 1 − 2 2

(a) Fundamental domain for Γ0(1) \ H

Figure 2.1: Fundamental domain F0,1 for the modular group SL(2, Z) and the resulting Riemann surface.

For higher level, the surfaces Γ0(N) \ H are noncompact, finite-volume Riemann surfaces whose volume is given by \ H \ H · ”vô˝l (Γ0(N) ) = ”vô˝l (Γ0(1) ) [Γ0(1) : Γ0(N)] ,

with Y Y 1 µ (N) := [Γ (1) : Γ (N)] = pα + pα−1 = N 1 + . 0 0 0 p pα∥N p|N

The fundamental domain of Γ0(N) is tessellated by µ0(N) copies of the fundamental domain of Γ0(1). Below are several examples.

13 2.1. Automorphic forms for GL(2, R)

i Γ0(11) \ H i Γ0(14) \ H i Γ0(37) \ H

√ 21+i 3 74 74 √ 53+i 3 74 74

6 + i 37 37

31+ i 37 37

1 4 1 1 2 1 12 1 1 2 53 1 15 1 1 2 1 3 2 3 55 1 3 2 3 47 3 2 3 74 6 4 3 5 2 5 3 4 6

(a) Γ0(11) \ H (b) Γ0(14) \ H (c) Γ0(37) \ H

Figure 2.2: Some examples of fundamental domains for Hecke congruence subgroups Γ0(N), truncated to finite height inside the standard polygon; the pairings of sides of the hyperbolic polygon are indicated by color. The Riemann surface corresponding to Γ0(11) \ H is a torus with two inequivalent cusps at 0 and ∞, \ H 1 1 ∞ \ H while Γ0(14) is a torus with four inequivalent cusps at 0, 7 , 2 and . As a Riemann surface, Γ0(37) ∞ has genus 2 and two inequivalent cusps at 0 and , as well as two inequivalent elliptic√ fixed points√ of order 6 i 31 i 21 i 3 53 i 3 2 at 37 + 37 and 37 + 37 , and inequivalent elliptic fixed points of order 3 at 74 + 74 and 74 + 74 .

The number of inequivalent cusps of Γ0(N) \ H is X c0(N) = φ ((n, m)) . nm=N

In the case where N is square-free, then c0(N) = τ2(N). The number of inequivalent elliptic fixed points of order 2 is   0 if 4|N, Y ν2(N) = −1  1 + otherwise,  p p|N and the number of inequivalent elliptic fixed points of order 3 is   0 if 9|N, Y ν3(N) = −3  1 + otherwise,  p p|N · where · is the Jacobi symbol and φ is the Euler totient function. One shows that Γ0(N) \ H has genus given by µ (N) ν (N) ν (N) c (N) (Γ (N) \ H) = 1 + 0 − 2 − 3 − 0 . `gé›n˚u¯s 0 12 4 3 2

14 2.1. Automorphic forms for GL(2, R)

∞ ←− −→0

(a) Γ0(11) \ H

1 2 −→

←−∞ −→0

1 7 ←−

(b) Γ0(14) \ H √ i 11 µ = 31 + i µ = 26 + 2,0 37 37 3,0 37 37 ∞ ←− −→0

√ 6 i µ = + 10 3i 3 2,1 37 37 µ = + 3,1 37 37

Figure 2.3: Some examples of quotient surfaces for Hecke congruence subgroups Γ0(N).

15 2.1. Automorphic forms for GL(2, R)

With the additional symmetries imposed by requiring a function f : H → C to be invariant under the action

of Γ0(N), the spectral theory of ∆ on Γ0(N) \ H becomes much more interesting.

2.1.4 The spectral theory of ∆ on Γ \ H

Consider the automorphic functions deﬁned on Γ \ H having moderate growth in cusps. These functions f : H → C are automorphic with respect to Γ; namely, those satisfying

f(γz) = f(z) for all γ ∈ Γ;

if f is additionally an eigenfunction of ∆, then we say that f is an automorphic form for Γ. An automorphic

form f with Laplace eigenvalue λf = s(1 − s) for some s ∈ C has Fourier expansions at each cusp a of Γ given by X b f(σaz) = Fa(y) + fa(n)Ws(nz), n=0̸

provided again that f does not grow too quickly in the cusps. We say that f is cuspidal if fa(y) ≡ 0 for all cusps a. Here the coeﬃcients are denoted by Z 1 b fa(n) = f (σax) e (−nx) dx. 0 ≥ < 1 = = ∈ 1 Since ∆ is nonnegative and symmetric, we have λf 0, so that (s) = 2 and (s) = 0 or (s) 0, 2 . The space of square-integrable functions L2(Γ\H) is an inner product space with respect to the Petersson inner product, deﬁned by Z hf, gi = f(z)g(z)dµ(z) Γ\H for f, g ∈ L2(Γ \ H). We denote the corresponding Petersson norm by kfk2 = hf, fi. The Laplacian acts on the space {f ∈ L2(Γ \ H): f and ∆f are smooth and ∆f ∈ L2(Γ \ H)}.

This is a dense subspace of L2(Γ \ H), from which ∆ has a self-adjoint extension to L2(Γ \ H) that is nonnegative. Then L2(Γ \ H) has the orthogonal decomposition

L2(Γ \ H) = C(Γ \ H) ⊕ E(Γ \ H), where C(Γ \ H) denotes the space of cuspidal automorphic functions, and E(Γ \ H) denotes the space of incomplete Eisenstein series; we will describe how these spaces decompose with respect to ∆ concretely in the two sections that follow.

16 2.1. Automorphic forms for GL(2, R)

2.1.4.1 The space of incomplete Eisenstein series

The incomplete Eisenstein series attached to cusps are deﬁned by X | = −1 ∈ ∞ ∞ Ea(z ψ) = ψ (σa γz) , for ψ C0 (0, ).

γ∈Γa\Γ

Taking ψ(z) = zs yields a concrete class of automorphic forms: the Eisenstein series attached to the cusps {a} of Γ. These are deﬁned by X = −1 s < Ea(z, s) = σa γz , for (s) > 1; (2.2)

γ∈Γa\Γ

However, the Eisenstein series are not in L2(Γ \ H), and the above series only converges absolutely for <(s) > 1. It turns out that the Eisenstein series span the space of incomplete Eisenstein series—we make this explicit shortly. The Eisenstein series have the Fourier expansions at cusps given by X s 1−s Ea(σbz, s) = δ(a = b)y + φab(s)y + φab(n, s)Ws(nz), n=0̸

where the functions deﬁning the zeroth coeﬃcients are given by the Dirichlet series √ Γ s − 1 X S (0, 0; c) φ (s) = π 2 ab ab Γ(s) c2s c≥1

and π2|n|s−1 X S (n, 0; c) φ (n, s) = ab . ab Γ(s) c2s c≥1

In the above the Sab(n, m; c) are the Kloosterman sums deﬁned in (2.1).

Example 2.1.3. For the modular group SL(2, Z), there is only the classical Kloosterman sum S∞∞(n, m; c) = S(n, m; c), and we have S(0, 0; c) = φ(c), where φ is the Euler totient function. The Fourier coeﬃcients of the single Eisenstein series E∞(z, s) = E(z, s) have pleasant closed forms, owing to the special formulas √ − 1 X √ − 1 − Γ s 2 φ(c) Γ s 2 ζ(2s 1) φ∞∞(s) = φ Z (s) = π = π SL(2, ) Γ(s) c2s Γ(s) ζ(2s) c≥1

and 2 s−1 X 2 s−1 π |n| S(n, 0; c) π |n| σ1−2s(n) φ∞∞(n, s) = = . Γ(s) c2s Γ(s) ζ(2s) c≥1 In this case the Fourier expansion takes the form √ X s 1−s E(z, s) = y + φSL(2,Z)(s)y + 4 y τ − 1 (n)K − 1 (2πny) cos(2πnx). s 2 s 2 n≥1

17 2.1. Automorphic forms for GL(2, R)

Since the Eisenstein series are only deﬁned by the series (2.2) for <(s) > 1, one uses their functional equation to extend them meromorphically to C. Deﬁne the column vector E(z, s) of the Eisenstein series attached to the inequivalent cusps a1,..., ar of Γ by   E (z, s)  a1   .  E(z, s) =  .  .

Ear (z, s) and deﬁne the scattering matrix of Γ by   φ (s) ··· φ (s)  a1a1 a1ar   . . .  ΦΓ(s) =  . .. .  . ··· φar a1 (s) φar ar (s)

Then E(z, s) satisﬁes the functional equation

E(z, s) = ΦΓ(s)E(z, 1 − s), (2.3) and this provides the analytic continuation of the Eisenstein series to <(s) ≤ 1. The Eisenstein series Ea(z, s) < ≥ 1 1 has at most ﬁnitely many poles in (s) 2 ; these are all simple, lie in ( 2 , 1], and they are also poles for 2 the diagonal entries φaa(s). The residues of these poles lie in L (Γ \ H) and are orthogonal to cusp forms, but form part of the discrete spectrum of ∆; we refer to these as the residual spectrum. When we have a particular cusp a in mind, we denote the Maass forms in the residual spectrum by uaj(z). However, there can be linear dependences between the uaj(z) for inequivalent cusps of Γ, so without reference to any cusps we denote a general basis element for the span of the residues by uj(z). We also normalize the residues to

have kuajk = 1. ≤ < 1 One also expects there to be many poles of the Eisenstein series in the segment in 0 (s) < 2 ; these are referred to as resonances.

Example 2.1.4. In the case of Hecke congruence subgroups Γ0(N), when N is odd and square-free the \ H ≡ 1 ≡ 1 entries of ΦΓ0(N)(s) are given as follows. For cusps a and b of Γ0(N) , say a v and b v′ with vw = v′w′ = N, we have Y Y ′ ′ 2s −1 s 1−s φab(s) = φSL(2,Z)(s) · φ ((w, w ) · (v, v )) p − 1 p − p . p|N p|(w,w′)·(v,v′)

< ≥ 1 These are all holomorphic in (s) 2 , except for a simple pole at s = 1, which correspond to the constant functions.

The space E(Γ \ H) of incomplete Eisenstein series Ea(z|ψ) is spanned by the Eisenstein series in the sense that each incomplete Eisenstein series has the expansion Z X 1 X 1 1 Ea(z|ψ) = hEa(·|ψ), uajiuaj(z) + Ea(·|ψ),Eb ·, + it Eb z, + it dt. 4π R 2 2 1 ≤ b 2

18 2.1. Automorphic forms for GL(2, R) 1 For the Eisenstein series Eab z, 2 + it , the normalized coeﬃcients are deﬁned to be

1 4π|n| 2 1 η (n, t) = φ n, + it . ab cosh(πt) ab 2

2.1.4.2 The space of cuspidal automorphic functions

The cuspidal automorphic forms are called cusp forms or Maass cusp forms. They are those automorphic forms whose zeroth eigenvalues are zero at all cusps.

Deﬁnition 2.1.4. A cusp form or Maass cusp form for Γ is a function u ∈ L2(Γ \ H) satisfying the following:

− 1 2 1. ∆u = λuu, where λu = su(1 su) = 4 + tu is the Laplace eigenvalue of u,

2. For each cusp a of Γ, Z 1 fa(y) = f (σaz) dx ≡ 0. 0

By the abstract spectral theorem, C(Γ \ H) is spanned by the Maass cusp forms, and we denote an orthonormal basis for it by {uj}. The eigenspaces are all ﬁnite dimensional, and for any f ∈ C(Γ \ H) such that ∆f is smooth and k∆fk < ∞, one has X f(z) = hf, ujiuj(z), j

with the eigenvalues being counted with multiplicity. Given an orthonormal basis {uj} for C(Γ \ H), we

denote the corresponding Laplace eigenvalue of each uj by

1 1 λ = + t2 = s (1 − s ), with s = + it . j 4 j j j j 2 j

The Fourier expansion at a cusp a is given by X

uj(σaz) = ρj,a(n)Wsj (nz), n=0̸

and we deﬁne the normalized Fourier coeﬃcients to be

1 4π|n| 2 νj,a(n) = ρj,a(n) cosh(πtj)

The distinguished normalized Fourier coeﬃcients in the cusp ∞ are denoted by νj(n).

2.1.4.2.1 The space of newforms and oldforms

For the Hecke congruence subgroups, C(Γ \ H) decomposes further depending on whether the ﬁrst Fourier

coeﬃcient is zero or not. For M|N and M < N, one has Γ0(N) ⊆ Γ0(M), and if v(z) is a Maass cusp form for

19 2.1. Automorphic forms for GL(2, R)

| Γ0(M), then for each D such that DM N, vD(z) = v(Dz) is a cusp form for Γ0(N). Furthermore, νvD (n) = 0

for n 6≡ 0 (mod D). The span of such cusp forms is the space of oldforms, denoted by C`o˝l´dﬄ(Γ0(N) \ H).

The orthogonal complement C`o˝l´dﬄ(Γ0(N) \ H) comprises the newforms of level N, and we denote the space

of these by C”nè›w(Γ \ H). A newform of level M|N with M < N residing in Cò˝l´dffl(Γ0(N) \ H) of the form

u(z) = vD(z) for some DM|N is called an oldform.

2.1.4.3 The spectral theorem

Combining the expansions in the previous two sections yields the spectral theorem, or the full spectral decomposition of L2(Γ \ H) with respect to ∆.

Theorem 2.1.1. [The Spectral Theorem, [Iwa02]] Let Γ ⊆ SL(2, R) be a Fuchsian group of the ﬁrst kind. The cuspidal space C(Γ \ H) is spanned by countably many eigenfunctions of ∆, the Maass cusp forms. The eigenspaces of ∆ on C(Γ \ H) each have ﬁnite dimension. The space E(Γ \ H) of incomplete Eisenstein series has the orthogonal decomposition M E(Γ \ H) = R(Γ \ H) ⊕ Ea(Γ \ H), a

R \ H ∈ 1 where (Γ ) is the span of the Eisenstein series with residual eigenvalues having sj ( 2 , 1], M R \ H R \ H (Γ ) = sj (Γ ), 1 ≤ 2

R \ H where sj (Γ ) is the space spanned by the residue of all Eisenstein series regardless of cusps at sj. The multiplicity of each residual eigenvalue is bounded above by the number of cusps of Γ. For each cusp, the E \ H 1 ∞ ∈ 2 \ H spectrum of ∆ in a(Γ ) covers [ 4 , ) uniformly with multiplicity 1. Thus any f L (Γ ) has the expansion Z X X 1 ∞ 1 1 f(z) = hf, ujiuj(z) + f, Ea ·, + it Ea z, + it dt. 4π −∞ 2 2 j a

Remark 2.1.5. In the case that Γ = SL2(Z), there is only one residual eigenvalue at s = 1, which corre- ≡ 3 sponds to the eigenvalue λ0 = 0. The corresponding residue of the Eisenstein series is u0(z) π , so we have R(SL(2, Z) \ H) = C. In general, there is always a residual eigenvalue at s = 1 for which the corresponding ≡ \ H −1 residue of the Eisenstein series is u0(z) (”vˆo˝l (Γ )) , which spans the constant functions.

20 2.1. Automorphic forms for GL(2, R)

2.1.5 The Hecke operators

2 The Hecke congruence subgroups Γ0(N) come with a special collection of operators, the Hecke operators 2 2 Tn : L (Γ0(N) \ H) → L (Γ0(N) \ H), that commute with ∆. These are deﬁned by 1 X 1 X X az + d (T f)(z) = √ f(γz) = √ f , n n n d γ∈SL(2,Z)\Γn ad=n b(d)

where Γn = {γ ∈ GL(2, Z) : det γ = n}. The Tn with (n, N) = 1 form a family of self-adjoint operators that 2 2 commute with the action of ∆ on L (Γ0(N) \ H), so we can choose a basis for L (Γ0(N) \ H) of simultaneous

eigenfunctions for ∆, δ, and {Tn}(n,N)=1. These are the Hecke-Maass forms, and for cusp forms we denote

the Hecke eigenvalues of a basis element uj by λj(n). For the Fourier coeﬃcients in the cusp a = ∞, one has

νj(n) = νj(1)λj(n) for (n, N) = 1,

and the Hecke eigenvalues satisfy the multiplicativity relation X mn ν (m)λ (n) = ν , for (n, N) = 1. (2.4) j j j d2 d|(m,n)

The Hecke operators preserve Cò˝l´dffl(Γ0(N) \ H) and C”nè›w(Γ0(N) \ H), and we can choose a basis for these

consisting of simultaneous eigenfunctions. When uj is a newform of level N, then νj(1) =6 0, and in this case one further has X mn λ (m)λ (n) = λ , for (n, N) = 1, j j j d2 d|(m,n) and

λj(m)λj(p) = λj(mp), for p|N prime. 1 ∞ For Eisenstein series E∞ z, 2 + it , the eigenvalues at the cusp are τit(n), and these also satisfy the above multiplicativity relation.

If δuj = uj, we say that uj is even, and if δuj = −uj we say it is odd. The Fourier coeﬃcients of even

Hecke-Maass cusp forms satisfy νj(−n) = νj(n) for all n, while the odd cusp forms satisfy νj(−n) = −νj(n).

2.1.6 The Weyl law

As we described in the introduction, the distribution of the discrete eigenvalues of ∆ is of great interest and is related to the geometry of the surface in question. Recall that the counting function for the continuous spectrum is deﬁned by Z T −φ′ 1 + it 1 Γ 2 MΓ(T ) = 1 dt, 4π −T φΓ 2 + it 2Hecke operators are also deﬁned in more general settings; see [Ven90].

21 2.1. Automorphic forms for GL(2, R)

where φΓ is determinant of the scattering matrix for Γ. By the appropriate choice of a test function in the Selberg trace formula, one obtains the Weyl law for noncompact Γ \ H: (Γ \ H) h T N (T ) + M (T ) = ”vˆo˝l T 2 − Γ T log T + c T + O , Γ Γ 4π π Γ log T

\ H hΓ − where hΓ is the number of inequivalent cusps of Γ and cΓ = π (1 log 2); see [Sel91].

50i

40i

30i

- cuspidal spectrum 20i < ≤ 1 - poles of φΓ0(1) in (s) 2 - residual spectrum

10i

1 1 1 4 2

Figure 2.4: Plot of small eigenvalues of ∆, the residual spectrum, and poles of Eisenstein series for SL(2, Z). The cusp forms dominate.

2.1.6.1 The Weyl law for Γ0(N) \ H

In the case where Γ = Γ0(N) is a Hecke congruence subgroup, a great deal more can be said about the continuous spectrum. In the case where N = p is prime, the scattering matrix is given by ! 1 − 1 √ − − − − s − 1 s √ − − Γ s 2 ζ (2 2s) 2s 1 p 1 p p Γ s 2 ζ (2 2s) ΦΓ (p)(s) = π p − 1 = π Np(s). 0 Γ(s) ζ (2s) ps − p1−s p − 1 Γ(s) ζ (2s)

22 2.1. Automorphic forms for GL(2, R)

More generally, for N = p1p2 ··· pr square-free, the scattering matrix for Γ0(N) is given by √ Γ s − 1 ζ (2 − 2s) O Φ (s) = π 2 N (s); Γ0(N) Γ(s) ζ (2s) p p|N

the computation can be found in [Hej76]. Explicitly, then the determinant of the scattering matrix φΓ0(N)(s) has the factorization

− − ⊮ 2 − 2r 1 Λ (2 2s, ) 1−2s Λ (2 2s, χN ) φΓ0(N)(s) = det ΦΓ0(N)(s) = N , Λ (2s, ⊮) Λ (2s, χN )

where χN is the principal Dirichlet modulo N, and Λ(s, ⊮) and Λ(s, χN ) are the completed L-functions for

ζ(s) and L(s, χN ), respectively. In these cases, MΓ0(N)(T ) essentially counts the nontrivial zeros of those L-functions in the critical strip, so that by known standard techniques one has

MΓ0(N)(T ) T log T, so that one can see that the discrete spectrum dominates. Namely, one has (Γ (N) \ H) T N (T ) = ”vˆo˝l 0 T 2 − c T log T − c T + O . Γ0(N) 4π 1,Γ0(N) 2,Γ0(N) log T

2.1.7 Holomorphic modular forms

There is an extensive theory of holomorphic modular forms as well. We shall only describe a small portion of it; for a basic introduction, see Iwaniec-Kowalski’s book [IK04].

Deﬁnition 2.1.5. Let k ≥ 0 be an even integer. A modular form of weight k for Γ is a holomorphic function on H that respects the action of Γ as follows: ! a b Q(γz) = (cz + d)kQ(z) for all γ = ∈ Γ. c d

We also require Q to be holomorphic at all cusps of Γ; that is, in local coordinates near each cusp a of Γ \ H, f is complex-diﬀerentiable.

Remark 2.1.6.! The j-factor cz + d in the modularity condition is often denoted by j(γ, z) = cz + d, for a b γ = ∈ Γ. The j-factor satisﬁes j(γ1γ2, z) = j(γ1, γ2z) · j(γ2, z). c d

Remark 2.1.7. The more convenient deﬁnition of holomorphicity at a cusp a of Γ \ H is phrased in terms −k of the Fourier series for Q. Note that by the modularity condition, the function Qa(z) = Q(σaz)j(σa, z)

23 2.1. Automorphic forms for GL(2, R) ! 1 1 is invariant under the action of , since 1

! ! ! !−k 1 1 1 1 Qa(z + 1) = Q σa z · j σa , z 1 1

−k = Q(γaσaz) · j(γaσa, z) = Qa(z).

Thus Q has Fourier expansion at a given by X X c k−1 Qa(z) = Qa(n)e(nz) = νQ,a(n)n 2 e(nz), (2.5) n∈Z n∈Z

c and Q is holomorphic at a if Qa(n) = 0 for n < 0. We refer to the νQ,a(n) as the normalized Fourier

coefficients of Q at a. If νQ,a(0) = 0 for all cusps, then we say that Q is cuspidal or a cusp form; note that in this case Q has exponential decay as z → a. If not, the growth is moderate. The Fourier expansion (2.5) coincides with the usual definition of holomorphicity in terms of the Riemann surface structure of Γ \ H in the following sense: taking local coordinates q = e(z) of Γ \ H near ∞, the condition is that the Laurent series expansion in q of Q at a has negative index coefficients all zero; hence, Q is holomorphic at a.

Remark 2.1.8. The Fourier expansion at ∞ is special and we denote the corresponding Fourier coeﬃcients d by Q∞(n) = ρQ(n) and νQ,∞(n) = νQ(n). By the work of Deligne ([Del74]) these are known to satisfy the

Ramanujan conjecture: one has νQ(n) τ(n) for all n.

The spaces Mk(Γ \ H) and Sk(Γ \ H) of weight-k holomorphic modular forms for Γ that are noncuspidal and cuspidal, respectively, are both ﬁnite dimensional. The space of weight-k cusp forms is an inner product space, with the Petersson inner product deﬁned by Z k hQ1,Q2ik = y Q1(z)Q2(z)dµ(z), for Q1,Q2 ∈ Sk(Γ \ H). Γ\H

By interpreting the (even) weight-k modular forms as k-fold diﬀerential forms on Γ \ H, using the Riemann-

Roch theorem one ﬁnds that for Γ with signature (g; m1, . . . , mℓ; h), still assuming h > 0, the dimension of

Mk(Γ \ H) is given by   Xℓ  k 1 k (k − 1)(g − 1) + 1 − + − 1 h if k ≥ 4,  2 mj 2 M \ H j=1 `d˚i‹mﬄ ( k(Γ )) = g + h − 1 if k = 2,   1 if k = 0,

24 2.1. Automorphic forms for GL(2, R)

and the dimension of Sk(Γ \ H) is given by   M \ H − ≥ `d˚i‹mﬄ ( k(Γ )) h if k 4,

(Sk(Γ \ H)) = g if k = 2, `d˚i‹mﬄ   0 if k = 0.

For the Hecke congruence subgroups specifically, the dimension of the space of cusp forms is given by k k k (S (Γ (N) \ H)) = (k − 1) · ( (Γ (N) \ H) − 1) + − 1 · c (N) + ν (N) · + ν (N) · , `d˚i‹mffl k 0 `gé›n˚u¯s 0 2 0 2 4 3 3

\ H − for k > 2, and `d˚i‹mffl (S2(Γ0(N) )) = `gé›n˚u¯s(Γ0(N))+c0(N) 1, where c0(N), ν2(N), and ν3(N) are as defined earlier.

Remark 2.1.9. Note that modular forms are not actually well-deﬁned on the surface Γ \ H, so we are

abusing notation when we write Mk(Γ \ H) and Sk(Γ \ H); the standard notation is Mk(Γ) and Sk(Γ). By

the uniformization theorem, however, it makes sense to write Sk(Γ\H). The interpretation of Q ∈ Sk(Γ\H)

in terms of k-fold diﬀerential forms on Γ \ H is natural: there is a bijection between Sk(Γ \ H) and sections of the line bundle ω⊗k for the universal elliptic curve π : E → Γ \ H over Γ \ H. This connection allows one to fruitfully carry over homological tools from algebraic geometry.

For congruence subgroups, the Hecke operators also act on Sk(Γ0(N) \ H), so we can choose a basis for

it consisting of simultaneous eigenfunctions. We will occasionally denote such a basis for Sk(Γ0(N) \ H) by

Bk(Γ0(N) \ H). We denote the normalized Hecke eigenvalues by λQ(n), with

k−1 TnQ = λQ(n)n 2 f, for (n, N) = 1

so that these satisfy the Hecke relations X mn λ (m)λ (n) = λ , for (n, N) = 1. (2.6) Q Q Q d2 d|(m,n)

One ﬁnds that νQ(n) = λQ(n)νQ(1) for (n, N) = 1. As before, one needs to restrict attention to the space of newforms in order for the Fourier coeﬃcients to be multiplicative; in this case one has λQ(n)νQ(1) = νQ(n) for all n, and so in addition to satisfying (2.6), these satisfy

λQ(m)λQ(p) = λ(mp), for p|N.

Throughout we will take the basis newforms to be normalized so that λQ(1) = 1.

2.1.7.1 Holomorphic modular forms in terms of the spectral theory of ∆k

In the previous discussion of the spectral theory of Γ \ H, one could also have viewed the action of SL(2, R) on H as an action on the space of smooth functions C∞(H) instead. Similarly, one can view the weight-k holomorphic modular forms for Γ in terms of a modiﬁed action of SL(2, R) on C∞(H): one deﬁnes the

25 2.1. Automorphic forms for GL(2, R)

∞ weight-k action SL(2, R) ⟳ C (H) sending γ : Q 7→ Q|kγ by

− cz + d k (Q| γ)(z) = Q(γz). (2.7) k |cz + d|

The weight-k automorphic functions are those f ∈ C∞(H) that are invariant under (2.7):

∞ ∞ C (Γ \ H, k) := {f ∈ C (H): f|kγ = Q} .

The weight-k Laplacian is deﬁned by ∂2 ∂2 ∂ ∆ = y2 + − iky , k ∂x2 ∂y2 ∂x

∞ and the weight-k Maass forms are those Q ∈ C (Γ \ H, k) that are eigenfunctions of ∆k and that have moderate growth in cusps. As in the weight-0 case, one has the spectral decomposition

L2(Γ \ H, k) = C(Γ \ H, k) ⊕ E(Γ \ H, k), where the space of cuspidal weight-k automorphic functions and the space of incomplete Poincaré series are deﬁned analogously. The spectral resolution of ∆k in this case decomposes C(Γ \ H, k) in terms of a basis of k ∈ \ H 2 weight-k Maass cusp forms. Furthermore, if Q Sk(Γ )is a cusp form, then y Q(z) is a weight-k Maass k k − cusp form for Γ with weight-k Laplace eigenvalue λQ = 2 2 1 . The automorphic functions of various weights for Γ form a graded ring: one has M M 2 + 2 + ∼ 2 L (Γ \ GL(2, R) ) = L (Γ \ GL(2, R) , k) = L (Γ \ H, k), k k and the mass-raising and mass-lowering operators

∂ ∂ k ∂ k R = iy + y + = (z − z) + k ∂x ∂y 2 ∂z 2

and ∂ ∂ k ∂ k L = −iy + y − = −(z − z) − k ∂x ∂y 2 ∂z 2

2 2 2 2 such that Rk : L (Γ \ H, k) → L (Γ \ H, k + 2) and Lk : L (Γ \ H, k) → L (Γ \ H, k − 2).

2.1.7.2 The Petersson trace formula

The Petersson trace formula relates the Fourier coeﬃcients of weight-k holomorphic modular forms to the Kloosterman sums attached to Γ. The deﬁnitions of the various integral transforms in this section can be found in Appendix A.

26 2.1. Automorphic forms for GL(2, R)

Theorem 2.1.2. Let k, m, n ∈ N and let k be even. Let a and b be two cusps of Γ \ H, and let Bk(Γ \ H) denote an orthonormal basis of weight-k holomorphic modular forms for Γ. Then √ − X X (k 2)! k Sab(m, n; c) 4π mn ν (m)ν (n) = δ(a = b)δ(m = n) + 2πi J − . (4π)k−1 f,a f,b c k 1 c f∈Bk(Γ\H) c∈Cab

Summing over all weights k and applying the Sears-Titchmarsh inversion formulas (see Appendix A) one arrives at the following general form of the Petersson trace formula. This comprises an important piece of the Kuznetsov trace formula, and corresponds to the contribution holomorphic discrete series representations of Γ in the representation theoretic perspective; see [CPS90].

Theorem 2.1.3. Let m, n ∈ N. Let a and b be two cusps of Γ \ H. Let φ be a smooth function on [0, ∞) satisfying

• |φ(x)| x as x → 0,

• φ(r)(x) x−3 as x → ∞ for 0 ≤ r ≤ 3. Then √ X k X X ∞ i Γ(k)N (k − 1) S (m, n; c) 4π mn δ(a = b)δ(m = n)φ + φ ν (m)ν (n) = ab φS , (4π)k−1 f,a f,b c c k≥2 f∈Bk(Γ\H) c∈Cab 2|k

where Z ∞ Z ∞ ∞ 1 2 T φ = 2 J0(x)φ(x)dx = 2 φ(t)t tanh(πt)dt. 2π 0 π 0

2.1.8 The Kuznetsov trace formula

Similar to the Petersson trace formula, the Kuznetsov trace formula captures the deep connection between

automorphic forms for Γ ⊆ SL(2, R) and the Kloosterman sums Sab(m, n; c). In one direction, it yields an expansion of spectral features of Γ in terms of algebro-geometric features of Γ. There are many generalizations of Kuznetsov’s original formula; in this section we document the versions we will make use of later. The various integral transforms that appear in the statements below can be found in Appendix A.

|= | ≤ 1 Theorem 2.1.4. Let h be an even function that is regular in the strip (t) 2 , where it satisﬁes the bound 1 |h(t)| (1 + |t|)2+δ

27 2.1. Automorphic forms for GL(2, R)

for some δ > 0. Let a and b be cusps of Γ \ H. Then for any m, n ∈ Z with mn =6 0, we have Z X X 1 ∞ h(tj)νj,a(m)νj,b(n) + 2 h(t)ηac(m, t)ηbc(n, t)dt 4π −∞ j c p ! Z X δ(a = b)δ(n = m) Sab(m, n; c) 4π |mn| ± = 2 t tanh(πt)h(t)dt + h , (2.8) π R c c c∈Cab where = sgn(mn) and the integral transforms are deﬁned to be Z 2i tdt h+(x) = J2it (x) h(t) , π R cosh(πt)

and Z 4 − h (x) = 2 K2it (x) h(t)t sinh(πt)dt. π R

Remark 2.1.10. In the speciﬁc case of the full modular group Γ = SL(2, Z), there is only one cusp at ∞ and the normalized Fourier coeﬃcients of the Eisenstein series are 1 1 − 1 1 1 | | 2 2 +it| | 2 +it 2 2 +it 4π n π n σ−2it(n) 4π π τit(n) η∞∞(n, t) = η(n, t) = 1 = 1 . cosh(πt) Γ 2 + it ζ(1 + 2it) cosh(πt) Γ 2 + it ζ(1 + 2it)

In the contribution from the Eisenstein series, we then have

4πτ (m)τ (n) η (m, t)η (n, t) = it it , ac bc |ζ(1 + 2it)|2 π 1 1 − since cosh(πt) = Γ 2 + it Γ 2 it . Thus the Kuznetsov trace formula takes the form Z X 2 ∞ τ (n)τ (m) h(t )ν (m)ν (n) + h(t) it it dt j j j | |2 π 0 ζ(1 + 2it) j Z √ δ(n = m) X S(m, n; c) 4π mn ± = 2 t tanh(πt)h(t)dt + h . π R c c c≥1

There is a perhaps more illuminating form of Kuznetsov that fully displays the relationship between Kloosterman sums and the representation theory of GL(2, R). This form is better regarded as the full spectral decomposition of the Kloosterman zeta function

X S (m, n; c) Z (m, n) = ab , for <(s) > 1. ab,s cs c∈Cab

28 2.1. Automorphic forms for GL(2, R)

Theorem 2.1.5. Let φ be a smooth function on [0, ∞) satisfying

• |φ(x)| x as x → 0,

• φ(r)(x) x−3 as x → ∞ for 0 ≤ r ≤ 3. Then for m, n ∈ N, we have √ X X X Z ∞ Sab(m, n; c) 4π mn T 1 T φ = νj,a(m)νj,b(n) φ(tj) + 2 φ(t)ηac(m, t)ηbc(n, t)dt c c 4π −∞ c∈Cab j c X πikΓ(k)N (k − 1) X + φ ν (m)ν (n). (4π)k−1 f,a f,b k≥2 f∈Bk(Γ\H) 2|k

Without the contribution from the holomorphic modular forms, one has √ X S (m, n; c) 4π mn δ(a = b)δ(m = n)φ∞+ ab φ c c c∈Cab X X Z ∞ T 1 T = νj,a(m)νj,b(n) φ(tj) + 2 φ(t)ηac(m, t)ηbc(n, t)dt, 4π −∞ j c

and if mn < 0, then one has (by the Kontorovich-Lebedev inversion formula—see Appendix A) √ X X X Z ∞ Sab(m, n; c) 4π mn L 1 L φ = νj,a(m)νj,b(n) φ(tj) + 2 φ(t)ηac(m, t)ηbc(n, t)dt. c c 4π −∞ c∈Cab j c

Remark 2.1.11. For the full modular group, Theorem 2.1.5 takes the form √ Z X S(m, n; c) 4π mn X 2 ∞ τ (m)τ (n) φ = ν (m)ν (n)T (t ) + T (t) it it dt c c j j φ j π φ |ζ(1 + 2it)|2 c≥1 j 0 X πikΓ(k)N (k − 1) X + φ ν (m)ν (n). (4π)k−1 f f k≥2 f∈Bk(Γ0(1)\H) 2|k

2.1.9 Bounds for Fourier coeﬃcients and twists of Fourier coeﬃcients

For holomorphic modular forms, the individual Fourier coefficients satisfy λQ(n) τ(n), known as the Deligne bound for the Fourier coefficients ([Del74]). In general, the Ramanujan-Petersson conjecture ε predicts that the same bound holds for the Fourier coefficients of Maass cusp forms; namely, that |λj(n)| n 7 +ε for any ε > 0. The current best bound towards the Ramanujan-Petersson conjecture is |λj(n)| n 64 , due to Kim-Sarnak [Kim03]. On the other hand, by the Rankin-Selberg method, one shows that on average the Fourier coefficients

29 2.1. Automorphic forms for GL(2, R)

satisfy the Ramanujan conjecture: for N 1, one has X 2 1+ε |λj(n)| N , n≤N

so one can see that very few of the λj(n) can possibly deviate from their expected bound.

2.1.9.1 The GL(2) Voronoi summation formula

The additive twists of the Fourier coefficients of GL(2) automorphic forms exhibit a special type of symmetry owing to the invariance of the underlying automorphic object under the action of Γ. These are the Voronoi summation formulas. In the modern formulation, one can regard these as a consequence of the functional equation for the twisted L-functions attached to the given automorphic form. The presence of gamma factors in the functional equation results in the appearance of integral transforms on the dual side of the formula defined in terms of Bessel functions. Associated to each GL(2) automorphic form are two distinguished Bessel functions. There is a pleasant representation theoretic description of these that the reader may find in Cogdell-Piatetskii-Shapiro’s book [CPS90] in terms of the Whittaker and Kirillov models for GL(2, R). For Q a holomorphic modular form of even weight k, we have + k JQ (ξ) = 2πi Jk−1(ξ)

− + and JQ (ξ) = 0. Furthermore, JQ has a representation as an integral of Marnes-Barnes type, with s+ k−1 s+ k+1 Z −s 2 2 1 ξ Γ 2 Γ 2 J +(ξ) = ds Q 1−s+ k−1 1−s+ k+1 2πi (σ) 2 2 2 Γ 2 Γ 2

for σ > 0. For Maass cusp forms uj, the Bessel functions are − + π J (ξ) = J (ξ) − J− (ξ) uj 2itj 2itj sinh(πtj) and J − (ξ) = 4ε cosh(πt )K (ξ). uj j j 2itj For both of these, there is the integral representation of Mellin-Barnes type   − − Z −s s+itj s itj 1+s+itj 1+s itj 1 ξ Γ 2 Γ 2 Γ 2 Γ 2 J ± (ξ) =   ds, uj 1−s+it 1−s−it 2−s+it 2−s−it 2πi (σ) 2 j j j j Γ 2 Γ 2 Γ 2 Γ 2

for σ > 0.

Theorem 2.1.6 (Kowalski-Michel-VanderKam, [KMV02], Theorem A.4). Let N ∈ N and let f be either a holomorphic modular form of even weight k and level N, or a newform Maass cusp form of level N and

30 2.1. Automorphic forms for GL(2, R)

1 2 ∈ N Laplace eigenvalue 4 + t ; in either case let f have trivial nebentypus. Let a, c with (a, c) = 1 and set N ∈ ∞ ∞ N1 = (c,N) . Let φ C0 ((0, )) be a smooth weight function. Then there is a newform fN1 of level N1 and weight k such that X an η (N ) X X naN n λ (n)e φ(n) = f√ 1 λ (n)e ∓ 1 Φ± , (2.9) f fN1 f 2 c c N c c N1 n≥1 1 ± n≥1

± where the integral transforms Φf are deﬁned by

Z ∞ ± ± √ Φf (x) = Jf (4π xy) φ(y)dy. (2.10) 0

The Voronoi summation formula is named for Georgii Voronoi, who derived a similarly shaped summation formula for the binary divisor function, X τ2(n) = 1,

n1n2=n in his work on the divisor problem and Gauss circle problem. In the modern formulation, the binary divisor function appears in the Fourier coeﬃcients of the degenerate Eisenstein series

E ∂ 2(z) = [E(z, s)]s= 1 ∂s 2 √ √ X = y log y + 4 y τ2(n)K0(2πny) cos(2πnx). n≥1

Explicitly, the τ2-Voronoi summation formula is the following.

∈ ∞ ∞ Theorem 2.1.7 (Kowalski-Michel-VanderKam, [KMV02], Theorem A.4). Let φ C0 ((0, )) be a smooth weight function, let a, c ∈ N with (a, c) = 1, and let 1 γ = γ0 = lim ζ(s) − s→1 s − 1

denote the Euler-Mascheroni constant. Then we have X an X X an n τ (n)e φ(n) = ∆(c; φ) + τ (n)e ∓ Φ± , (2.11) 2 2 E2 2 c c c N1 n≥1 ± n≥1 where the integral transforms Φ± are deﬁned by E2 Z ∞ √ Φ± (ξ) = J ± (4π xy) φ(y)dy, E2 E2 0

31 2.1. Automorphic forms for GL(2, R)

− with Bessel kernels given by JE2,+(ξ) = 2πY0(ξ) and JE2,+(ξ) = 4K0(ξ), and where we have denoted Z √ ∞ y ∆(c; φ) = 2 log + γ φ(y)dy. 0 c

Remark 2.1.12. Both of the above GL(2) Voronoi-type summations has an interpretation in terms of the functional equation of the Estermann zeta function or an analogue of the Estermann zeta function. One considers the L-functions twisted by additive characters X an λf (n)e c L(s, f ⊗ ξ a ) = , for <(s) > 1, c ns n≥1 an where ξ a (n) = e for ﬁxed a, c ∈ N with (a, c) = 1. For example, when f = Q is a weight-k holomorphic c c Hecke eigencuspform for SL(2, Z), then the completed L-function ! ! − −s k 1 k+1 π s + 2 s + 2 Λ(s, Q ⊗ ξ a ) = Γ Γ L(s, Q ⊗ ξ a ) c c 2 2 c

k satisﬁes the functional equation Λ(s, Q ⊗ ξ a ) = i Λ(1 − s, Q ⊗ ξ −a ). The situation is similar but slightly c c more complicated for the other cases.

2.1.9.2 The Hecke bound

The Fourier coeﬃcients of GL(2, R) automorphic forms satisfy a square-root cancellation when twisted by an additive character. This is refered to as the Hecke bound, and it is essentially the strongest type of bound one expects for Fourier coeﬃcients.

Proposition 2.1.8 (Godber, [God13], Theorem 1.2). Let f be an automorphic form for a discrete subgroup Γ ⊆ SL(2, R). Let α ∈ R and N 1. Then X 1 +ε 1 λf (n)e (αn) N 2 Q(f, 0) 4 . n≤N

The implied constant depends only on ε.

One method of proving Proposition 2.1.8 is to use GL(2) Voronoi summation, and this is the approach taken when proving the analogous bound for additive twists of GL(3) Fourier coeﬃcients (the Miller bound, 1 which only yields 4 -cancellation)—see Section 2.2.3.2. We will also require the following form of the Hecke bound for twisted double sums of Fourier coeﬃcients It actually holds for the double sum of any two additively twisted arithmetic functions that exhibit square-root cancellation. Later we refer to this bound as the double Hecke bound.

32 2.2. Automorphic forms for GL(3, R)

Proposition 2.1.9 (Deshouillers-Iwaniec, [DI86], Lemma 11.7). Let f be a cuspidal automorphic form for

a discrete subgroup Γ ⊆ SL(2, R). Let α, β ∈ R and N 1, and suppose that 0 < λ0 < λ. Then X X 1 1 2 +ε λf (n1)λf (n2)e (αn1 + βn2) (N1N2) Q(f, 0) 4 .

n1≤N1 n2≤N2 n1 λ0< <λ n2

The implied constant depends only on ε.

2.1.10 An aside: the homogenous space setting

It is natural to seek to generalize of the classical setting for automorphic forms to higher dimensions. One way of doing this is to consider the invariant functions under actions of subgroups of GL(n, R); this is cast in the language of homogenous spaces and group actions. The generalization of automorphic forms to GL(n, R) will seem more natural if we rephrase the discussion in Section 2.1.1 in terms of GL(2, R). One can identify the upper half place H with

∼ 2 × H = h = GL(2, R)/hO(2, R) · R i.

By the Iwasawa decomposition of GL(2, R), we see that every z ∈ h2 can be written as ! ! ! y x 1 x y z = = , 1 1 1 ! y x so the above isomorphism identiﬁes z = x + iy ∈ H with the element of h2 that has Iwasawa form . 1 This makes GL(2, R) ⟳ h2 a homogenous space, and the (left) GL(2, R)-invariant measure is still dµ(z) = dxdy R 2 y2 . The center of the universal enveloping algebra of gl(2, ), D , is generated by the Laplacian ∆; that is, D2 = C(∆). Working with SL(2, Z), the minimal parabolic subgroup of SL(2, Z) consists of upper triangular matrices ( ! ) 1 m Γ∞ = : m ∈ Z . 1

We shall only be working with GL(2, R) and GL(3, R), so we won’t present the general theory of automorphic forms on GL(n, R); for that we refer the reader to Goldfeld’s book [Gol06].

2.2 Automorphic forms for GL(3, R)

In this section we record some of the basic theory of automorphic forms for the group GL(3, R). We only provide enough to reach the results in Chapter 5, where we study moments of GL(2) × GL(3) L-functions.

33 2.2. Automorphic forms for GL(3, R)

Absent is any spectral theory in the guise of the spectral theorem and the GL(3) Kuznetsov trace formula. We only describe the minimal parabolic Eisenstein series, which we require for a GL(3)-type Voronoi summation

for the ternary divisor function τ3—see Section 2.2.4. The material presented in this section follows Goldfeld’s book [Gol06]. Analogous to the description of H in Section 2.1.10 in terms of homogenous spaces, the generalized 3-dimensional upper half plane is deﬁned by

h3 = GL(3, R)/hO(3, R) · R×i.

Owing to the Iwasawa decomposition, each z ∈ h3 can be written in Iwasawa form as     1 x x y y  2 3  1 2  z =  1 x1  y1  , 1 1

3 with x1, x2, x3 ∈ R and y1, y2 > 0. Then GL(3, R) ⟳ h , and the (left) GL(3, R)-invariant Haar measure has the explicit formula dx1dx2dx3dy1dy2 dµ(z) = 3 . (y1y2) While in the two-dimensional setting D2 is generated by a single operator, namely the hyperbolic Laplacian 2 ∂2 ∂2 R 3 ∆ = y ∂x2 + ∂y2 , the center of the universal enveloping algebra of gl(3, ), denoted by D , is generated by two invariant diﬀerential operators,

2 2 2 2 2 2 2 2 ∂ 2 ∂ − ∂ 2 2 2 ∂ 2 ∂ 2 ∂ 2 ∂ ∆1 = y1 2 + y2 2 y1y2 + y1 x2 + y2 2 + y1 2 + y2 2 + 2y1x1 ∂y1 ∂y2 ∂y1∂y2 ∂x3 ∂x2 ∂x1 ∂x2∂x3

and

3 3 3 3 − 2 ∂ 2 ∂ − 3 2 ∂ 2 ∂ ∆2 = y1y2 2 + y1y2 2 y1y2 2 + y1y2 2 ∂y1∂y2 ∂y1∂y2 ∂x3∂y1 ∂x1∂y1 3 3 3 3 − 2 ∂ 2 − 2 2 ∂ − 2 ∂ 2 2 ∂ 2y1y2x1 + y2 x1 y1y2 2 y1y2 2 + 2y1y2 ∂x2∂x3∂y2 ∂x3∂y2 ∂x2∂y2 ∂x2∂x1∂x3 3 2 2 2 2 2 2 2 2 ∂ 2 ∂ − 2 ∂ 2 ∂ 2 2 2 ∂ 2 ∂ − 2 ∂ + 2y1y2x1 2 + y1 2 y2 2 + 2y1x1 + x1 + y2 y1 2 + y1 2 y2 2 . ∂x1∂x3 ∂y1 ∂y2 ∂x2∂x3 ∂x3 ∂x2 ∂x1

The simplest eigenfunctions of D3 are the power functions deﬁned by

ν1+2ν2 2ν1+ν2 Iν (z) = I(ν1,ν2)(z) = y1 y2 ,

2 for a tuple ν = (ν1, ν2) ∈ C . We have 2 2 ∆1Iν (z) = (ν1 + 2ν2) − (ν1 + 2ν2)(2ν1 + ν1) + (2ν1 + ν1) Iν (z) = λν (∆1)Iν (z)

34 2.2. Automorphic forms for GL(3, R)

and

∆2Iν (z) = λν (∆2)Iν (z),

3 writing λν (∆2) for the corresponding polynomial in ν1, ν2. We say that Iν (z) is an eigenfunction of D of

type ν = (ν1, ν2).

2.2.1 The Maass cusp forms for SL(3, Z) Z ∈ The Maass cusp forms for SL(3, ) of type ν = (ν1, ν2) are those (non-zero) smooth functions ψ 2 3 L SL(3, Z) \ h that are eigenfunctions of ∆1 and ∆2 with eigenvalues λν (∆1) and λν (∆2), respectively, satisfying a certain decay condition.

Z Deﬁnition 2.2.1. AMaass cusp form for SL(3, ) of type ν = (ν1, ν2) is a non-zero smooth function 2 3 ψ ∈ L SL(3, Z) \ h such that ∆1ψ = λν (∆1)ψ and ∆2ψ = λν (∆2), and which satisﬁes Z ψ(uz)du = 0 (SL(3,Z)∩U)\U

for all subgroups U of SL(2, R) consisting of upper triangular matrices of the form   I ∗  r1    Ir2

Ir3

with r1 + r2 + r3 = 3.

× where Un(G) denotes the n n upper triangular matrices with 1 on the diagonal and entries in G, φε1,ε2 :

U3(R) → C denotes the character of U3(R) deﬁned by   1 u u  1 3   ∈ R φε1,ε2 1 u2 = e (ε1u1 + ε2u2) , with ε1, ε2 , 1

and WJacquet (z, ν, φε1,ε2 ) is the Jacquet-Whittaker function Z Z Z · · WJacquet (z, ν, φε1,ε2 ) Iν (w3 u z)φε1,ε2 (u)du1du2du3, R R R

35 2.2. Automorphic forms for GL(3, R)   −1   with w3 =  1 . The Fourier coeﬃcients Aψ(m1, m2) are normalized such that A(m1, m2) = 1 O(|m1m2|). In what follows the Langlands parameters, or spectral parameters, for a Maass cusp form ψ of type ν will be denoted by

α = −ν1 − 2ν2 + 1, β = −ν1 + ν2 and γ = 2ν1 + ν2 − 1.

∗ ∗ Associated to ψ is a dual Maass cusp form ψ of type (ν2, ν1). Furthermore, the Fourier coeﬃcients of ψ are given by

Aψ∗ (m1, m2) = Aψ(m2, m1),

and the Langlands parameters are

∗ ∗ ∗ α = ν2 + 2ν1 − 1, β = −ν2 + ν1, and γ = 1 − 2ν2 − ν1.

2.2.2 The Hecke operators and Hecke-Maass cusp forms for SL(3, Z) Z Z There is an analogous theory of Hecke operators for SL(3, ) as in the SL(2, ) setting. The Hecke operators 2 3 2 3 Tn : L SL(3, Z) \ h → L SL(3, Z) \ h are deﬁned by    a b1 c1 1 X    T f(z) = f  b c  z , n n 2 abc=n 0≤c1,c2

for n ∈ N. For diagonal matrices   δ δ  1 2  δ =  δ1  , 1 2 3 2 3 where δ1, δ2 = 1, we have the involutions Tδ : L SL(3, Z) \ h → L SL(3, Z) \ h given by

Tδψ(z) = ψ(δzδ).

These all commute with ∆1 and ∆2, and in fact, for Maass cusp forms, one has Tδψ = ψ for each Tδ; that is, all Maass cusp forms for SL(3, Z) are even.

The Hecke operators {Tn} comprise a family of commuting normal operators, and they also commute { } with all the Tδ and ∆1 and ∆2. It follows from the spectral theorem we may choose a basis ψj for 2 3 L SL(3, Z) \ h consisting of simultaneous eigenfunctions of all the Tn, Tδ, ∆1, and ∆2; accordingly, the

ψj are referred to as Hecke-Maass cusp forms. ∈ N We normalize these such that Aψj (1, 1) = 1, and we have Tnψj = Aψj (n, 1) for all n . Furthermore,

36 2.2. Automorphic forms for GL(3, R)

′ ′ ∈ Z the Fourier coeﬃcients satisfy the following Hecke relations: for m1, m2, m1, m2, n ,

′ ′ ′ ′ Aψj (m1m1, m2m2) = Aψj (m1, m2)Aψj (m1, m2), ′ ′ provided (m1m1, m2m2) = 1, X d0m1 d1m2 Aψj (n, 1)Aψj (m1, m2) = Aψj , . d1 d2 d0d1d2=n d1|m1 d2|m2

2.2.3 Bounds for Fourier coeﬃcients and twists of Fourier coeﬃcients

The Fourier coeﬃcients of an SL(3, Z) Maass cusp form ψ are predicted to satisfy the Ramanujan-Petersson ε conjecture as well; namely, one expects that |Aψ(m, n)| (mn) . The current best bound towards this is

5 +ε |Aψ(m, n)| (mn) 14 , due to Luo-Rudnick-Sarnak ([LRS99])3. Meanwhile, on average, one has the Rankin-Selberg bound X 2 |Aψ(m, n)| ψ N. m2n≤N

By the Cauchy-Schwarz, it follows that X X 2 |Aψ(m, n)| ψ N|m|. m≤N n≤N

2.2.3.1 The GL(3) Voronoi summation formula

As in the GL(2) setting, additive twists of the Fourier coeﬃcients of ψ an SL(3, Z) automorphic form satisfy a Voronoi summation formula owing to the invariance of ψ under the action of SL(3, Z). The proof amounts to the functional equation for a twisted L-function, so that gamma factors appear on the dual side. These give rise to an integral transform similar to the classical Hankel-type transforms deﬁned in terms of Bessel functions in the GL(2) setting.

As described in [Qi15], an analogously to the GL(2) situation, associated to each ψ of type (ν1, ν2) outlined above are two GL(3) Bessel functions. For ξ > 0 they are deﬁned by the Mellin-Barnes integrals Z s−α+δ s−β+δ s−γ+δ X δ −s Γ Γ Γ ± ( i) ξ 2 2 2 Jψ (ξ) = ds, (2.13) 4πi C 2 1−s+α+δ 1−s+β+δ 1−s+γ+δ δ=0,1 Γ 2 Γ 2 Γ 2 where C is a curved contour avoiding the poles of the gamma functions described in [Qi15]. Note that the Langlands parameters appearing in the denominator of (2.13) are the Langlands parameters of ψ∗, so the

3 2 7 +ε For symmetric square lifts ψ = sym uj , where uj is a Maass cusp forms for SL(2, Z), the bound is improved to (mn) 32 by the result of Kim-Sarnak ([Kim03]).

37 2.2. Automorphic forms for GL(3, R)

ratio of gamma functions is the ratio of gamma factors of the associated L-function L(s, ψ). In [Qi15], Qi shows that the GL(3) Bessel functions have the asymptotic expansion

− ± e (3ξ) KX1 B J ±(ξ) = ψ,k + E±(ξ) + O ξ−K , (2.14) ψ ξ ξk ψ K,ψ k=0 ∈ Z for K ≥0 and ξ 1 and where √ −3 3πξ (j) e E± (ξ) . ψ ψ,j ξ The Fourier coeﬃcients of ψ satisfy the following GL(3) Voronoi summation formula.

Theorem 2.2.1 (Li, [Li11]). Let ψ be an SL(3, Z) Maass cusp form of type with Fourier coeﬃcients denoted ∈ ∞ ∞ ∈ Z by Aψ(m, n). Let φ Cc ((0, )). Let a, c, m with (a, c) = 1. Then we have X X X X 2 an 1 n1 cm ± n2n1 ∗ Aψ(m, n)e φ(n) = Aψ (n1, n2)S am, n2; Φψ∗ 3 , (2.15) c c ± cm n1 c m n≥1 n1|cm n2≥1 where the integral transform is deﬁned by

Z ∞ ± ∓ 1 3 Φψ∗ (x) = φ(y)Jψ∗ 2π (xy) dy. (2.16) 0

2.2.3.2 The Miller bound for additive twists of Fourier coeﬃcients

While the Fourier coefficients of GL(2, R) automorphic forms satisfy the square-root cancellation of the Hecke bound, the best known bound for additive twists of Fourier coefficients of SL(3, Z) automorphic forms 1 is weaker; one only has a 4 -saving bound. The first nontrivial bound of this sort for GL(3) was achieved by Miller ([MS06]), who used GL(3) Voronoi summation. The following version is due to Xiannan Li ([Li]).

Proposition 2.2.2 (Li, [Li]). Let ψ be a (tempered) cusp form for SL(3, Z) of type (ν1, ν2). Let α ∈ R and N 1. Then X 3 +ε D Aψ(1, n)e (αn) N 4 Q(ψ, 0) , n≤N

1 5 where the exponent can be taken to be D = 4 assuming the Ramanujan conjecture and D = 12 uncondition- ally. The implied constant depends only on ε.

38 2.2. Automorphic forms for GL(3, R)

2.2.4 The minimal parabolic Eisenstein series for SL(3, Z) and Voronoi summation for the ternary divisor function

In this section we provide a statement of a GL(3)-type Voronoi summation formula for the ternary divisor

function τ3 that was obtained in [Li14]. This is best viewed as a consequence of the symmetries of a degenerate minimal parabolic Eisenstein series for SL(3, Z).

The minimal parabolic subgroup Γ∞ of SL(3, Z) consists of the upper triangular matrices     1 m2 m3    Γ∞ =  1 m  : m1, m2, m3 ∈ Z .  1  1

< < 2 ∈ 3 For (ν1), (ν2) > 3 and z h , the minimal parabolic Eisenstein series E(z, ν1, ν2) is constructed by the same method as are the GL(2) Eisenstein series; namely, by periodizing the power functions Iν (z). Explicitly, they are deﬁned by X 2ν1+ν2 ν1+2ν2 E(z, ν1, ν2) = (y1(γz)) (y2(γz)) .

γ∈Γ∞\SL(3,Z) One then has the completed Eisenstein series 2 2 Eb(z, ν , ν ) = ν − ν − ζ(3ν )ζ(3ν )E(z, ν , ν ), 1 2 1 3 2 3 1 2 1 2

which extends to an entire function of ν1, ν2 ∈ C, and which has a Fourier expansion where the Fourier coeﬃcients for the principal part4 are deﬁned by X X 1−3ν1 2−3ν1−3ν2 σ1−3ν1,2−3ν1−3ν2 (n, m) = d1 d2 . | | m d1 m d2 d d >0 1 1 d2>0 (d2,n)=1

A relationship similar to (2.11) involving the degenerate GL(2, R) Eisenstein series and the binary divisor

function τ2(n) holds for the ternary divisor function X τ3(n) = 1, n1n2n3=n n1,n2,n3∈N which forms the Fourier coeﬃcients of a certain derivative of the minimal parabolic Eisenstein series for SL(3, Z); namely, the degenerate Eisenstein series is

∂3 E3(z) = [E(z, ν1, ν2)] 1 . 2 ν1=ν2= 3 ∂ν1∂ν2

Furthermore, one can obtain a Voronoi-type summation formula for the ternary divisor function τ3(n) from

4There are several other complicated terms that arise with “Fourier coeﬃcients” deﬁned in terms of the binary divisor function. We will not reproduce them here: see [Li14].

39 2.2. Automorphic forms for GL(3, R)

the functional equation for additive twists of E(z, ν1, ν2); see [Li14] for a full exposition. We will need this GL(3) Voronoi-type summation formula in Chapter 6, where we derive an upper bound for the sixth moment

of L(sj, uj), which is essentially L(sj, uj ⊗ E3). Speciﬁcally, one has the following.

∈ ∞ ∞ Theorem 2.2.3 (Li, [Li14]). Let φ(x) Cc (0, ) be a smooth compactly supported weight function, and let h, ℓ, c ∈ N with hℓ ≡ 1 (mod c). Denote X X σ0,0(m, n) = 1 | | n d1 n d2 d d >0 1 1 d2>0 (d2,m)=1 and let 1 γ = γ0 = lim ζ(s) − s→1 s − 1 be the Euler-Mascheroni constant and

− d − 1 γ1 = ζ(s) − ds s 1 s=1

the first Stieltjes constant5. Let Φ± denote the GL(3) Bessel function defined in (2.13) with Langlands E3 parameters α = β = γ = 0. Then τ3(n) satisfies the Voronoi-type summation X mℓ τ (m)e φ(m) 3 c m≥1 c X X X 1 X X n c mn2 = σ , m S ηm, h; Φ± 3 0,0 E3 3 2 2π ± nm n n1n2 n c η= 1 n|c m≥1 n1|n n2| n1

+ ∆0(h, c; φ) + ∆1(h, c; φ) + ∆2(h, c; φ), (2.17) where we have denoted 1 X nτ (n) n ∆ (h, c; φ) = 2 S 0, h; T (n, c)ψe(1), (2.18) 0 2 c2 c 3,0 n|c with

2 9 2 2 T3,0(n, c) = (log n) − 5(log c)(log n) + (log c) + 3γ + 7γ log n − 9γ log c − 3γ1 2 1 X 3 + log(d) (log(nc) − 5γ) − (log d)2 , τ2(n) 2 d|n

1 X nτ (n) n ∆ (h, c; φ) = 2 S 0, h; T (n, c)ψe′(1), (2.19) 1 2 c2 c 3,1 n|c

40 2.2. Automorphic forms for GL(3, R)

with 5 1 X T3,1(n, c) = log n − 3 log c + 3γ − log d, 3 3τ2(n) d|n

and 1 X nτ (n) n ∆ (h, c; φ) = 2 S 0, h; ψe′′(1). (2.20) 2 4 c2 c n|c

Remark 2.2.1. Before continuing, we remark that there is a much more general Voronoi-type summation formula for the n-ary divisor functions. The general shape of the formula is similar, but the proof is via the functional equation for an analogue of the Estermann zeta function, rather than the automorphy of an explicit GL(2, R) automorphic form. See [Ivi97] for a full treatment.

5 b The Stieltjes constants arise naturally here because of the presence of the zeta function in the deﬁnition of the E(z, ν1, ν2), since they appear in the Laurent series expansion of ζ(s) at s = 1.

41 Chapter 3

Preliminaries on automorphic L-functions

Contents 3.1 Automorphic L-functions ...... 42 3.1.1 The approximate functional equation ...... 44 3.1.2 The convexity bound ...... 47 3.2 On conductor-dropping behavior of L-functions in the spectral aspect .... 48

The main tools we will use to study automorphic forms are their associated L-functions. In general, all L-functions we study have the same basic properties: a Dirichlet series definition in a half-plane <(s) > 1, a functional equation reflecting s → 1 − s giving the L-function a meromorphic continuation to C, an Euler product, etc. There is a general formal framework of L-functions, an outline of which we provide here, but we will leave specific properties of the L-functions we study in each chapter to the preliminaries section of each. We largely follow the presentation in Iwaniec-Kowalski’s book [IK04]. Since later we study the behavior of L-functions exhibiting conductor-dropping, we will describe this phenomenon and the literature surrounding it in greater detail in Section 3.2.

3.1 Automorphic L-functions

Deﬁnition 3.1.1. We say that a function L(s, f), where f is some sort of automorphic object and s ∈ C (possibly not at s = 0 or s = 1), is an L-function if it satisﬁes the following.

N 1. There is a sequence of Dirichlet series coeﬃcients (λf (n)) ∈ C such that for <(s) > 1, one has an absolutely convergent Dirichlet series

X λ (n) L(s, f) = f . ns n≥1

42 3.1. Automorphic L-functions

2. Furthermore, we assume that for <(s) > 1 the Dirichlet series has an Euler product of degree d,

− − Y α (p) 1 α (p) 1 L(s, f) = 1 − 1 ··· 1 − d ps ps p

where the p are prime (possibly excluding some ﬁnite subset) and local parameters αi(p) ∈ C and

|αi(p) < p for all 1 ≤ i ≤ d. We say that d is the degree of L(s, f), or L(s, f) is an L-function of degree d. That L(s, f) has an Euler product means the Dirichlet series coeﬃcients are multiplicative. From the adelic point-of-view, it makes sense to occasionally denote the local L-functions by

− − α (p) 1 α (p) 1 L (s, f) = 1 − 1 ··· 1 − d . p ps ps

3. There are gamma factors associated to L(s, f), Yd − ds s + κj γ(s, f) = π 2 Γ , 2 j=1

where κj ∈ C with <(κj) > −1 are the local parameters of L(s, f) at ∞, or the archimedean param-

eters of L(s, f). For this reason, sometimes the adelic notation L∞(s, f) = γ(s, f) is used. The κj are either real or come in complex conjugate pairs.

4. The (arithmetic) conductor Q(f) ∈ Z is such that αi(p) =6 0 for p 6 |Q(f) and 1 ≤ i ≤ d, and we say such a prime p is unramiﬁed.

5. The completed L-function s Λ(s, f) = Q(f) 2 γ(s, f)L(s, f)

satisﬁes a functional equation of the form

Λ(s, f) = ε(f)Λ(1 − s, f ∗),

which extends L(s, f) to a meromorphic function on C with at most poles at s = 0 and s = 1. In the above, |ε(f)| = 1 is the root number of L(s, f). The dual object f ∗ is another automorphic object whose L-function L(s, f ∗) has the same degree, conductor, and gamma factors as L(s, f), and has ∗ Dirichlet series coeﬃcients λf ∗ (n) = λf (n). In the case where L(s, f) = L(s, f ), we say that the underlying object is self-dual.

Typically we are interested of the behavior of L-functions in a particular family. In this case it is useful to describe the behavior of the L-functions in question in terms of their analytic conductor, deﬁned to be

Yd Q(s, f) = Q(f) (|s + κj| + 3) = Q(f)Q∞(s, f). j=1

43 3.1. Automorphic L-functions

This is essentially just the bound that comes from applying Stirling’s approximation, √ it σ− 1 − πt t 1 Γ(σ + it) = 2πt 2 e 2 1 + O for t > 0, e σ t to the gamma factors. If one prefers, one can simply deﬁne the analytic conductor to be γ(s, f).

Example 3.1.1. An example of a family of L-functions we will use throughout this section are the L-

functions associated to a basis {uj} of Hecke-Maass cusp forms for SL(2, Z). We deﬁne the L-function

L(s, uj) by the Dirichlet series

X λ (n) L(s, u ) = j , for <(s) > 1, j ns n≥1

and these have the degree two Euler product Y −1 −1 Y −1 − αj(p) − βj(p) − λj(p) 1 L(s, uj) = 1 s 1 s = 1 s + 2s . p p p p p p prime prime

The gamma factors are  − Γ s itj Γ s+itj if u is even, −s 2 2 j γ(s, uj) = π  s+1−itj s+1+itj Γ 2 Γ 2 if uj is odd,

with the archimedean parameters being itj if uj is even and 1 itj if uj is odd. The completed L-function

Λ(s, uj) = γ(s, uj)L(s, uj) is entire and satisﬁes the functional equation

Λ(s, uj) = εjΛ(1 − s, uj), where εj = 1 if uj is even and εj = −1 if uj is odd; in both cases uj is self-dual. The conductor is Q(uj) = 1.

3.1.1 The approximate functional equation

Let L(s, f) be an L-function. The standard approximate functional equation allows one to express L(s, f) as the sum of two weighted Dirichlet series for s in the critical strip 0 < <(s) < 1, where the L(s, f) lacks a Dirichlet series representation.

Theorem 3.1.1 (Iwaniec-Kowalski, [IK04], Theorem 5.3). Let L(s, f) be an L-function. Let G(u) be any function which is holomorphic and bounded in the strip −4 < <(u) < 4, even, and normalized by G(0) = 1. Let X > 0. Then for s in the strip 0 ≤ σ ≤ 1 we have ! ! X X λf (n) n λf (n) nX L(s, f) = V p + ε(s, f) V − p + R (3.1) ns s n1−s 1 s f n≥1 X Q(f) n≥1 Q(f)

44 3.1. Automorphic L-functions

where Vs(y) is a smooth function deﬁned by Z 1 −u γ(s + u, f) du Vs(y) = y G(u) 2πi (3) γ(s, f) u

and − 1 −s γ(1 s, f) ε(s, f) = ε(f)Q(f) 2 . γ(s, f)

The last term is Rf = 0 if Λ(s, f) is entire, otherwise Λ(s + u, f) G(u) u Rf = Res + Res X . u=1−s u=−s Q(f)s/2γ(s, f) u

The general mantra is that the approximate functional equation splits L(s, f) into two sums of length 1 Q(s, f) 2 . One can see that this is the case by shifting the contours deﬁning Vs(y) very far to the right, where they satisfy !−A y Vs(y) 1 + p Q∞(s, f)

and !α! y Vs(y) = 1 + O p , Q∞(s, f) if <(s + κj) ≥ 3α > 0 and where A > 0 is arbitrary. The weight functions appearing in the sums are deﬁned in terms of the gamma factors of L(s, f). Often we will be interested in obtaining an asymptotic expansion of a moment of L-functions in some family, and to extract the main term of the asymptotic expansion we will need to remove the dependence of the weight function on the parameters in question. To do so, we use an unbalanced approximate functional equation that contains all the parameters of interest in only one of the weight functions. This provides us with the additional ﬂexibility needed to achieve power-saving error terms in the asymptotic expansions. The derivation is similar for the other L-functions we will study, but since the proofs are all similar, we will only provide one for the example below. 1 Example 3.1.2 (Unbalanced approximate functional equation for L 2 + itj, uj ). Let uj be an even Hecke- 1 2 1 Maass cusp form with Laplace eigenvalue λj = 4 + tj . Then at the special point sj = 2 + itj (where the analytic conductor drops), for any N > 0, L(sj, uj) can be expanded as X λ (n) n X λ (n) j 2itj j L(sj, uj) = H + π Pt (πnN) , (3.2) nsj N nsj j n≥1 n≥1

45 3.1. Automorphic L-functions

where Z s+ 1 Γ 2 1 2 −s ds H(ξ) = 1 ξ , (3.3) 2πi (σ) Γ 4 s and 1 1 − Z s+ 2 2 +s 2itj 1 Γ 2 Γ 2 ds P (ξ) = ξ−s , (3.4) tj 1 1 −s+2it 2πi (σ) Γ 2 j s 4 Γ 2

1 with σ > 2 . In this form, the weight function H(ξ) will not participate if we apply the Kuznetsov trace formula. Furthermore, we are free to choose N however we wish in order to compensate for whatever crude estimates to the integral transforms that appear after applying Kuznetsov to the dual sum, where the weight function does participate. To prove (3.2), as typical, we evaluate the following integral in two diﬀerent ways: Z s+ 1 Γ 2 1 2 ds I(σ) = 1 L(s + sj, uj) . 2πi (σ) Γ 4 s

1 For σ > 2 , we can expand the L-function as a Dirichlet series, yielding Z s+ 1 X Γ 2 λj(n) 1 2 ds I(σ) = L(s + sj, uj) , nsj 2πi Γ 1 s n≥1 (σ) 4

whence we deﬁne H(ξ) as in (3.3). Meanwhile, by the functional equation we have 1 − 1 1 − − s+ 2 s+ 2 2 s 2itj Γ 2 Γ 2 Γ 2 L(s + s , u ) = πs+2itj L(s − s, u ). 1 j j 1 1 +s+2it j j Γ Γ 2 j 4 4 Γ 2

Since L(s, uj) is entire, shifting the contour to <(s) = −σ we cross a simple pole at s = 0, so that Z s+ 1 Γ 2 1 2 ds I(σ) = L(sj, uj) + 1 L(s + sj, uj) 2πi − Γ s ( σ) 4 − 1 1 − − Z s+ 2 2 s 2itj 1 Γ 2 Γ 2 ds s+2itj − = L(sj, uj) + π 1 L(sj s, uj) 1 +s+2itj 2πi (−σ) Γ 4 Γ 2 s 2 1 1 Z s+ +s−2itj Γ 2 Γ 2 1 − 2 2 ds = L(s , u ) − π2itj π s L(s + s, u ) , j j 1 1 −s+2it j j 2πi (σ) Γ 2 j s 4 Γ 2

46 3.1. Automorphic L-functions

after changing variables s 7→ −s. Expanding the L-function appearing in the ﬁnal integral, we see that 1 1 Z s+ +s−2itj Γ 2 Γ 2 1 − 2 2 ds L(s , u ) = I(σ) + π2itj π s L(s + s, u ) j j 1 1 −s+2it j j 2πi (σ) Γ 2 j s 4 Γ 2 X λ (n) n X λ (n) j 2itj j = H + π Pt (πnN) , nsj N nsj j n≥1 n≥1

having deﬁned Ptj (ξ) as in (3.4).

3.1.2 The convexity bound

Apart from questions about the location of the nontrivial zeros of an L-function, we are often interested < 1 its magnitude along the critical line (s) = 2 . It is often the case that other problems in analytic number theory can be reduced to a statement about the growth of an L-function at a point on the critical line as a parameter grows. The trivial estimate, or convexity bound (or convex bound), for L(s, f) at a point < 1 with (s) = 2 follows from the Phragmen-Lindelöf convexity principle and standard bounds for L-functions in vertical strips. For any ε > 0 and −ε ≤ <(s) ≤ 1 + ε, one has

1−σ +ε L(s, f) Q(s, f) 2 ,

and in particular, along the critical line one has

1 +ε 1 1 4 L + it, f Q + it, f . (3.5) 2 2

In fact, the ε appearing in the exponent can be removed along the critical line by the method of Heath-Brown, see [HB08]. The bound (3.5) is quite weak and on its own it is rarely suﬃcient for applications. The Lindelöf hypothesis predicts that the best bound that can be achieved is essentially logarithmic, with the conjectured 1 1 ε L 2 + it, f Q 2 + it, f . Any bound better than (3.5) is a subconvexity bound or subconvex bound.

Example 3.1.3 (The convexity bound and subconvexity bound for L(s, uj)). In the case where f = uj, the convexity bound is 1 1 +ε L + it, u ((|t + t | + 1)(|t − t | + 1)) 4 . 2 j j j 1 1 +ε For fixed values of t, as t → ∞, one therefore has L + it, u t 2 . Meanwhile, for fixed t , as t → ∞ j 2 j t j j 1 1 2 +ε the bound is L 2 + it, uj tj t . Jutila-Motohashi ([JM05]) achieved a uniform subconvexity bound (of “Weyl bound” strength), with 1 1 +ε L + it, u (|t − t| + 1) 3 . 2 j j Notably, this bound is actually weaker than the convexity bound when the spectral parameter and height on the critical line are correlated, where the analytic conductor drops: one of the gamma factors is fixed at

47 3.2. On conductor-dropping behavior of L-functions in the spectral aspect

1 the special point sj = 2 + itj as tj varies, so that the convexity bound is instead 1 1 +ε L + it , u t 4 . 2 j j j

At the special point, the L-function behaves like it is associated to a GL(1) object, rather than a GL(2) object. We discuss this phenomenon more generally in the next section.

3.2 On conductor-dropping behavior of L-functions in the spectral aspect

As mentioned in Section 3.1, it is possible to vary the parameters of an L-function in such a way that some of its gamma factors are fixed or essentially fixed, causing an abnormal drop in the magnitude of its analytic conductor. Consequently, as the parameter of interest varies, the L-function artificially behaves like it has a lower degree. This conductor-dropping behavior can be observed in many aspects, but here we will focus on the spectral aspect. The simplest setting where this conductor-dropping behavior can be observed is with L-functions defined Z 1 2 for Eisenstein series: for example, attached to an SL(2, ) Eisenstein series Et with eigenvalue 4 + t is the degree 2 L-function X τ (n) L(s, E ) = it , for <(s) > 1, t ns n≥1 which factors as the product of two GL(1) L-functions, with

L(s, Et) = ζ(s − it)ζ(s + it). (3.6) 1 1 1 1 Thus, at the point s = 2 + it, we have L 2 + it, Et = ζ 2 ζ 2 + 2it and the L-function literally behaves as a GL(1) L-function as the spectral parameter t varies. This phenomenon is observed in general as the

degree increases; for example, one has L(s, Et ⊗ Q) = L(s − it, Q)L(s + it, Q) for Q an automorphic form for GL(2, R). When considering the features of the L-function in question at the special conductor-dropping point (often referred to as the special point or conductor-dropping point or spectral point), the lengths of sums, convexity bounds, etc are changed accordingly. Below is a table summarizing the sorts of changes that may occur for various families of L-functions.

48 3.2. On conductor-dropping behavior of L-functions in the spectral aspect

Convex Special pt. Sp. pt. L-function Behavior bound behavior convex 1 +ε 1 +ε L (s, uj) GL(2) tj 2 GL(1) tj 4 1 1 ⊗ × 2 +ε 4 +ε L(s, uj χ) GL(2) GL(1) tj GL(1) tj 2 1 +ε 1 +ε L s, sym uj GL(3) tj 2 GL(2) tj 2 2 1 +ε 1 +ε L s, sym uj ⊗ χ GL(3) × GL(1) tj 2 GL(2) tj 2 1+ε 1 +ε L (s, uj ⊗ Q) GL(2) × GL(2) tj GL(2) tj 2 3 3 ⊗ × 2 +ε 4 +ε L (s, uj ψ) GL(2) GL(3) tj GL(3) tj 1 2 ⊗ × 2 +ε × 1+ε Ls, sym uj f GL(3) GL(2) tj GL(2) GL(2) tj 2 3 +ε 3 +ε L s, sym uj ⊗ ψ GL(3) × GL(3) tj 2 GL(2) × GL(3) tj 2

Table 3.1: A table summarizing changes to L-functions when conductor-dropping occurs in the spectral aspect for various GL(2)-associated L-functions. Here uj is a Hecke-Maass cusp form for SL(2, Z), χ is a primitive Dirichlet character of modulus N, Q is a holomorphic Hecke eigencuspform for Γ0(N), and ψ is a Maass cusp form for SL(3, Z).

It is interesting that conductor-dropping has actually been used to great eﬀect, but only ever in situations where there is a convenient factorization involving Eisenstein series like (3.6). For example, Luo-Sarnak ([LS95]) used the factorization to reduce quantum unique ergodicity (QUE) for Eisenstein series on the

modular surface SL(2, Z) \ H to subconvexity bounds of any strength for the L-functions L(s, uj). Similarly, such a factorization was used to prove the analogous equidistribution results for the Picard modular surface Z \ H3 < 1 Z SL(2, [i]) by Koyama ([Koy00]), and for the residues in (s) < 2 of Eisenstein series for SL(2, ) by Petridis-Raulf-Risager ([PRR13]). Similarly, Young [You16] was able to prove QUE for Eisenstein series restricted to the geodesic in SL(2, Z) \ H connecting 0 to i∞ by effectively reducing estimates on shifted convolution sums of divisor functions to subconvexity bounds for GL(2) L-functions of any strength. We describe similar problems of interest in the Part IV. Unfortunately, thus far the subconvexity problem in the presence of conductor-dropping has proven to be very difficult and has resisted dedicated attacks focusing on this aspect. There is only one instance where subconvexity at the special point is known; the monumental work of Michel-Venkatesh ([MV10]) achieves a subconvexity bound for GL(2, R) L-functions that is uniform in all parameters of the analytic conductor, and implicitly yields a special point subconvexity bound. Subsequently the exponent-of-saving was made explicit in [WA18] by Wu. It is unclear what technique—since they make heavy use of measure rigidity results—are strictly needed if at all to achieve special point subconvexity when one ignores the other parameters, or whether the exponent-of-saving can be significantly improved in this situation. There is another type of conductor-dropping that can occur in the spectral aspect: when considering the Rankin-Selberg convolution L-function L(s, f ⊗ g) for two (tempered spherical) automorphic forms f and g for GL(n), the conductor drops if the spectral parameters of f and g are within a fixed range of each other. For this situation, Blomer ([Blo12a]) has established a strong second moment upper bound that is consistent

49 3.2. On conductor-dropping behavior of L-functions in the spectral aspect with the Lindelöf hypothesis. Namely, one has

Z !1+ε X 2 1 ∗ L , f ⊗ g dspecµ . 2 ∥µf −ν∥≤1 ∥µf −µg ∥≤1

Interestingly, he only requires the use of “soft methods”: he makes no use of the trace formula, approximate functional equation, or Voronoi summation.

50 Chapter 4

Preliminaries on the spectral theory of hyperbolic Riemann surfaces and Teichmüller theory

Contents 4.1 Riemann surfaces ...... 51 4.2 Some remarks on Teichmüller theory and hyperbolic 2-surfaces ...... 52 4.2.1 The Teichmüller space of a surface ...... 53 4.2.2 Holomorphic quadratic diﬀerentials ...... 54 4.2.3 Teichmüller’s theorem ...... 54 4.3 The Phillips-Sarnak deformation theory of discrete groups ...... 55

In this chapter we provide the necessary background on the Phillips-Sarnak deformation theory of discrete groups. As a practical matter, this requires us to make a brief digression to Teichmüller theory. However, our focus is on the spectral theory of the Riemann surfaces Γ \ H, so we will not cover any of the rich geometric \ H theory of Tèˇi`c‚hffl(Γ )—there is simply too much to cover. For the sake of providing the appropriate amount of context to the unfamiliar reader for the results described in the introduction, we will at least provide the highlights. The material found in this section primarily follows the book of Gardiner-Lakic ([GL00]). The main result we need is Teichmüller’s description of the quasiconformal mapping between two Riemann surfaces in M(S) in terms of holomorphic quadratic differentials, which explains the connection between the annihilation of cusp forms and the special values L(sj, uj ⊗ Q).

4.1 Riemann surfaces

Let S be a diﬀerentiable manifold of two real dimensions. There are two ways to think deﬁne a C-structure

(marking) on S. One could consider complex atlases {φα} deﬁned on S: these are collections of charts

51 4.2. Some remarks on Teichmüller theory and hyperbolic 2-surfaces

between open subsets of C and S with transition maps that are holomorphic. The atlas endows S with the structure of a Riemann surface. The other perspective is to consider maps to S from surfaces already endowed with C-structure. A C- structure on S is given by a diffeomorphism φ : R → S where R is a Riemann surface; in this case one refers to the structure as (φ, R), and one says that R is marked by S. One may then wish to study the various C-structures that can be assigned to S. Two complex structures ◦ −1 → (φ1,R1) and (φ2,R2) on S are conformally equivalent if φ2 φ1 : R1 R2 is isotopic to a conformal diffeomorphism. On an infinitesimal level, a map f being conformal means that it preserves circles: circles

in the tangent space at z ∈ R1 are mapped to circles in the tangent space at f(z) ∈ R2. The celebrated uniformization theorem of Koebe classiﬁes all Riemann surfaces up to conformal equivalence as belonging to one of ﬁve broad classes of surfaces of constant curvature:

1. Cˆ: the Riemann sphere: positive curvature

2. C: the complex plane: zero curvature

3. S1 × (0, 1): a complex cylinder: zero curvature

4. T2: a complex torus: zero curvature

5. Γ \ H: a ﬁnite volume hyperbolic 2-surface; Γ is a Fuchsian group: negative curvature

Thus the C-structures that can be assigned to a surface S must come from the above classes. As a consequence, the C-structure on the surface is induced by a Riemannian metric with fundamental form ds2 = ρ2|dz|2 = ρ2 dx2 + dy2 ,

with ρ > 0. In order to study the possible C-structures, one considers maps between two structures R1

and R2 that are quasiconformal: the map must send inﬁnitesimal circles in R1 to inﬁnitesimal ellipses in

R2 of bounded eccentricity. The relationship between the various complex structures on S is the subject of Teichmüller theory, and it turns out that the space of C-structures on S is endowed with a metric deﬁned between in terms of quasiconformal mappings. We are interested in the ﬁnal of the above classes, the surfaces of constant negative curvature.

4.2 Some remarks on Teichmüller theory and hyperbolic 2-surfaces

As we have said, a quasiconformal map f sends infinitesimal circles to infinitesimal ellipses of bounded eccentricity. By the Cauchy-Riemann equations, f must essentially (to a first order approximation) satisfy the Beltrami equation

fz = µfz for some constant µ. If µ = 0, then f is holomorphic. When one allows µ = µ(z) to be nonconstant, µ must be essentially bounded; that is, µ ∈ L∞(R). Analytically, quasiconformality is deﬁned on charts as follows.

52 4.2. Some remarks on Teichmüller theory and hyperbolic 2-surfaces

Deﬁnition 4.2.1. Let U, V ⊆ C. A homeomorphism f : U → V is quasiconformal if there is a constant k < 1 such that

1. f has locally integrable (distributional) derivatives fz and fz on U,

2. |fz| ≤ k|fz| almost everywhere. 1+k In this case, for the minimal such k we denote the dilation constant by K(f) = 1−k . The Beltrami fz (z) 1+|µ(z)| coefficient of f is given by µ(z) = , and dilation of f at z ∈ U is Kz(f) = . fz (z) 1−|µ(z)| As is suggested by the above, the quasiconformal homeomorphisms of a Riemann surface R are in bijection with the “Beltrami differentials” µ(z) on R, up to conformal equivalence. That is, the conformal classes of quasiconformal deformations are determined by a Beltrami differential.

dz Definition 4.2.2 ([GL00]). A Beltrami differential µ(z) dz on a Riemann surface R is an assignment to α each chart zα on Uα of an L∞ C-valued function µ defined on zα(Uα) such that dzβ α β dzα µ (zα) = µ (zβ) . dzβ dzα

4.2.1 The Teichmüller space of a surface

One deﬁnes quasiconformal deformations of a surface R up to conformal maps factoring through a homotopy through quasiconformal maps.

Deﬁnition 4.2.3 ([GL00]). Let f1 : R → R1 and f2 : R → R2 be quasiconformal. Then f1 and f2 are

Teichmüller equivalent if there is a conformal map g : R1 → R2 and a homotopy H : R × [0, 1] → R

through quasiconformal self-maps gt : R → R such that

1. g0 = idR,

−1 2. g1 = (f2) ◦ g ◦ f1, and

3. gt(p) = p for 0 ≤ t ≤ 1 and p ∈ ∂R. C The Teichmüller space, T`eˇi`c‚hﬄ(S), of a surface S admitting -structures consists of all (φ, R) marked by S up to Teichmüller equivalence. When we have a Riemann surface R in mind, we may also denote the

Teichmüller surface by Tèˇi`c‚hffl(R). The space Tèˇi`c‚hffl(S) is a metric space where the metric is defined in the following way.

Deﬁnition 4.2.4. Let (φ1,R1) and (φ2,R2) be two Riemann surfaces marked by S. Then the Teichmüller

distance between R1 and R2 is deﬁned by

1 d (R ,R ) = inf {log K(f)} . T`eˇi`c‚hﬄ 1 2 2 f:R1→R2 quasiconformal

That is, d (R ,R ) is the infimum of the dilation of quasiconformal maps from R to R . Tèˇi`c‚hffl 1 2 1 2 The way that d is defined actually only yields a pseudometric on (S); by identifying Riemann Tèˇi`c‚hffl Tèˇi`c‚hffl surfaces marking S up to conformal equivalence it becomes a true metric.

53 4.2. Some remarks on Teichmüller theory and hyperbolic 2-surfaces

4.2.2 Holomorphic quadratic diﬀerentials

The Beltrami coefficients form a Banach space dual to Tèˇi`c‚hffl(R).

Deﬁnition 4.2.5. Let R be a Riemann surface with an atlas of charts {Aα}α. Locally, a holomorphic 2 quadratic diﬀerential φ(dz) is a collection of holomorphic maps {φα} such that for local coordinates z1

and z2, one has

2 dz2 φ1(z1) = φ2(z2) . (4.1) dz1

The integrable holomorphic quadratic diﬀerentials form a Banach space, which we denote by Q(R). This

is a closed subspace Q(R) ⊆ L1(R) of the Banach space of integrable quadratic diﬀerentials on R. ∗ ∼ It is a standard fact that the dual space is L1(R) = L∞(R), where L∞(R) consists of essentially bounded (−1, 1) forms µ on R. This induces the isomorphism

∗ ∼ Q(R) = L∞(R)/N, where N consists of Beltrami differentials µ ∈ L∞(R) such that (φ, µ) = 0 for all φ ∈ Q(R). Furthermore, ∗ ∼ when Q(R) is finite-dimensional, one can show that Q(R) = Tèˇi`c‚hffl(R), so Q(R) can be regarded as the tangent space to Tèˇi`c‚hffl(R) at R.

4.2.3 Teichmüller’s theorem

Teichmüller’s theorem provides the existence of a quasiconformal map f : R → R realizing d (R ,R ); 1 2 Tèˇi`c‚hffl 1 2 furthermore, f is uniquely determined by a (integrable) holomorphic quadratic differential of unit norm in

Q(R1).

Theorem 4.2.1. Let R1 and R2 be Riemann surfaces with ﬁnitely many cusps and such that Q(R1) and

Q(R2) are ﬁnite-dimensional. Suppose that f : R1 → R2 is a quasiconformal map that is not conformal.

Then in the Teichmüller equivalence class of f there is a map f0 : R1 → R2 such that

1 log K(f0) = d (R1,R2), 2 T`eˇi`c‚hﬄ

|φ(z)| 1+k the Beltrami coefficient of f0 is given by µ(z) = k φ(z) , where 0 < k < 1, K(f0) = 1−k , and where 2 φ(z)(dz) is a holomorphic quadratic differential on R1 of unit norm. Put another way, the fundamental form of the Riemannian metric on the deformed surface defining its C-structure can be represented in terms of the original surface via ds2 = ρ(z)|dz + µ(z)dz|2

where ρ > 0.

In the case where R = Γ \ H, where Γ is a Fuchsian group of the ﬁrst kind, this allows us to describe the

54 4.3. The Phillips-Sarnak deformation theory of discrete groups

condition (4.1) in terms of holomorphic modular forms of weight 4. Let φ = {φα}α ∈ Q(Γ \ H). Then the

local holomorphic maps φα lift to a holomorphic function ψ on H, and by (4.1), ψ must satisfy the following: 1. The automorphy condition az + b − ψ(z) = ψ(γ(z)) (γ′(z))2 = ψ (cz + d) 4 cz + d ! a b for any γ = ∈ Γ c d R | | ∞ 2. φ is integrable: Γ\H ψ(z) dµ(z) < . That is, ψ must be a holomorphic modular cusp form of weight 4 for Γ! In the case where Γ is arithmetic,

we have a basis for S4(Γ \ H) consisting of Hecke eigencuspforms. By Teichmüller’s theorem, any path in \ H \ H Tèˇi`c‚hffl(Γ ) starting at Γ can be described in terms of Hecke eigencuspforms for the initial surface. In terms of Beltrami differentials, the automorphy condition can be phrased as ! γ′(z) cz + d 2 µ(z) = µ(γ(z)) = µ(γ(z)) . γ′(z) cz + d

In the next section we describe how this fact can be exploited to get a handle on the relative sizes of NΓ and

MΓ for the deformed surface. The characterization of the quasiconformal deformations in terms of holomorphic quadratic differentials allows us to recover classical results in a simple way. The dimension of the space of holomorphic quadratic differentials is constant as one moves through the Teichmüller space of a given hyperbolic Riemann surface Γ \ H. From the description of the Beltrami differentials, one sees that

\ H \ H − `d˚i‹mﬄ(Q(Γ )) = `d˚i‹mﬄ(S4(Γ )) = 3g 3 + k + ℓ,

where g is the genus, k is the number of cusps, and ℓ is the number of elliptic ﬁxed points. That 3g −3+k +ℓ parameters are necessary to describe all C-structures on a surface of genus g ≥ 2 has been known since Riemann. More can be said: it turns out that the signature of a ﬁnite-volume hyperbolic surface is an invariant up to quasiconformal deformation.

Theorem 4.2.2. Let R1 = Γ1 \ H and R2 = Γ2 \ H be two finite-volume hyperbolic Riemann surfaces. If → ∈ Γ1 and Γ2 have the same signature, then there is a quasiconformal map f : R1 R2, and R2 Tèˇi`c‚hffl(R1). ∈ Conversely, if R2 Tèˇi`c‚hffl(R1), then R1 and R2 have the same signature.

4.3 The Phillips-Sarnak deformation theory of discrete groups

In order to meaningfully study the distribution of the eigenvalues of the Laplacian ∆ on a noncompact surface Γ \ H, one has to consider the contributions from the discrete spectrum and continuous spectrum

55 4.3. The Phillips-Sarnak deformation theory of discrete groups simultaneously; again, Weyl’s law for these surfaces is

\ H 2 NΓ(T ) + MΓ(T ) = ”vˆo˝l(Γ )T + O (T log T ) .

The work of Colin de Verdière ([CdV82], [CdV83]) showed that for perturbations in a compact subset of 1 of congruence surfaces, all but finitely many discrete eigenvalues are destroyed. In order to study the spectrum of ∆ on generic discrete subgroups Γ ⊆ SL(2, R), Phillips-Sarnak ([PS85b], [PS85a]) restricted \ H ≤ ≤ \ H themselves to considering paths Γε , where 0 ε 1, in Tèˇi`c‚hffl(Γ ) beginning at Γ0 = Γ; by the \ H uniformization theorem, a deformed surface along such a path in Tèˇi`c‚hffl(Γ ) corresponds to a subgroup Γε. The behavior of a cusp form under quasiconformal deformations of the underlying surface is real analytic in ε. Phillips-Sarnak considered deformations of the metric given by

1 ds2 = |dz + εµ(z)dz|2, ε y2 where µ(z) is a Beltrami diﬀerential.2 As we have seen in the previous section, the Beltrami diﬀerential can be taken to be µ(z) = y2Q(z), where Q is a weight k holomorphic modular form for Γ. The corresponding Laplacian for the surface essentially3 has the “perturbation series”

∆(ε) = ∆ + ε∆(1) + ε2∆(2) + ··· , where the ∆(i) are certain operators deﬁned on a Hilbert space. In particular, the ﬁrst of these has the explicit form ! ∂ 2 ∆(1) = L = −8< Q y2 . Q ∂z

Through a series of approximations away from cusps, they then reduce the question of whether a cusp form uj remains in the discrete spectrum under deformation to the vanishing of the inner product Z (1) · 1 (1) 1 dxdy ∆ uj,Ea , + itj = ∆ uj (z)Ea z, + itj 2 , 2 Γ\H 2 y 1 \ H where Ea z, 2 + itj is an Eisenstein series attached to one of the ﬁnitely many cusps a of Γ .

Theorem 4.3.1 (Phillips-Sarnak, [PS85b]). Let Γ ⊆ SL(2, R) be a Fuchsian group of the ﬁrst kind that is not cocompact. Suppose that u is a Maass cusp form for Γ with Laplace eigenvalue λ = 1 + t2, and let j j 4 j 1 \H Ea z, 2 + itj be an Eisenstein series attached to a cusp a of Γ . Let Q be a weight-4 cuspidal holomorphic modular form for Γ. Then a suﬃcient condition for uj to be destroyed under quasiconformal deformation of

1These are non-curvature-preserving deformations, and the resulting surface is not a Riemann surface; the spectral problem is of course still interesting on manifolds without complex structure. 2 2 | |2 Actually, the deformations of the metric are given by dsε = ρε(z) dz + εµ(z)dz , where ρε > 0, but the dependence of ρε ∂ ≡ on ε is such that ∂ε [ρε(z)]ε=0 0, so the metrics described above suﬃce. 3One actually needs to ﬁrst make some approximation arguments, since ∆ does not have compact resolvent so perturbation theory does not apply.

56 4.3. The Phillips-Sarnak deformation theory of discrete groups

\ H \ H Γ in the direction generated by Q in T`eˇi`c‚hﬄ(Γ ) is 1 1 L u ,E ·, + it = ∆(1)u ,E ·, + it =6 0. Q j a 2 j j a 2 j

The connection to the special values of Rankin-Selberg L-functions is as follows. Starting with the inner product hLQuj,Ea (·, s)i with <(s) > 1, one unfolds the integral and integrates by parts to obtain an expression in terms of the Rankin-Selberg L-function L(s, uj ⊗ Q). Evaluating this at s = sj then yields ! ! 1 s(s + 1) π−2s s − it + k−1 s + it + k−1 L u ,E ·, + it = Γ j 2 Γ j 2 Q j a 2 j 4 Γ (2 + s) 2 2 ! ! s − it + k+1 s + it + k+1 1 × Γ j 2 Γ j 2 L + it , u ⊗ Q . 2 2 2 j j

The proportion of these special values that are nonzero was studied by Deshouillers-Iwaniec ([DI86]) and Luo ([Luo93], [Luo01]), who showed that in fact, for congruence subgroups of prime level, a positive proportion of 2 the special values are nonzero. That is, one has MΓ(T ) T for those discrete subgroups of SL(2, R) arising

as quasiconformal deformations of Γ0(p) \ H. In Part III we provide a longer overview of these nonvanishing results and the techniques used in their proofs. It is conjectured that, except for the case of the once-punctured torus, for “generic” Γ, there are only finitely many cusp forms, while for arithmetic subgroups cusp forms are known to exist in abundance. ⊗ \ H The connection between the special value of L(s, uj Q) at sj and the actual deformation in Tèˇi`c‚hffl(Γ )

was developed in [PS92a]. The value L(sj, uj ⊗ Q) characterizes the movement of the special point sj oﬀ < 1 < 1 the line (s) = 2 into the resonances—poles of the scattering matrix—of the Laplacian in (s) < 2 for the deformed surface. In reference to classical perturbation theory in quantum physics, this is referred to as Fermi’s golden rule.

Theorem 4.3.2 (Phillips-Sarnak, [PS92a], [PS92b]). Let sj(ε) be the special point associated to a Maass

cusp form uj for Γ under perturbation Γε \ H and let Q ∈ S4(Γ \ H). Then

d2 [<(s (ε))] = −c |L(s , u ⊗ Q)|2 , dε2 j ε=0 j j j where cj > 0 is a constant.

Additional work on the distribution of eigenvalues under deformations of ﬁnite-volume noncompact discrete groups is studied in [BV98], [Wol91], [Wol92], [Wol94], [Jud95], [JP97], [Phi97], and [PR13]. An interesting computational investigation for certain discrete groups is carried out in [FL05].

57 Part II

Upper bounds for moments of L-functions at special points

In this part, we obtain upper bounds for the moments of L-functions at special points. For each of these results, we will need a large sieve type inequality that incorporates additional twisting from the special point. In Chapter 5 we derive a twisted large sieve type inequality over shortened spectral intervals. In Chapter 6 we use this to obtain upper bounds for the short interval second moment of the L-functions

L(sj, uj ⊗ ψ), where ψ is a ﬁxed SL(3, Z) Maass cusp form. In Section 6.4 we modify the proof of the result

for GL(2) × GL(3) L-functions to obtain a short interval sixth moment upper bound on L(sj, uj).

58 Chapter 5

A short interval large sieve inequality with spectral twists

Contents 5.1 Background and survey of existing literature ...... 60 5.1.0.1 The untwisted spectral large sieve inequality ...... 60 5.1.0.2 The twisted spectral large sieve inequality ...... 61 5.1.0.3 The short interval twisted large sieve inequality of Luo ...... 64 5.1.0.4 Some extended remarks on integral transforms appearing in Kuznetsov . 65 5.2 The short interval twisted spectral large sieve inequality ...... 67 5.2.1 Proof of Theorem 5.2.2 ...... 69 5.2.1.1 Application of Kuznetsov trace formula ...... 69 5.2.1.2 Extraction of the problematic main term ...... 71 5.3 Proof of Theorem 5.2.1 ...... 73 5.4 Proof of Lemmas 5.2.4 and 5.2.5 ...... 75 5.4.1 Proof of Lemma 5.2.4: the k = 0 terms ...... 75 d∗”m`a˚i‹nﬄ 5.4.1.1 Extraction of the main term: evaluation of Kℓ M,k=0(A; X) ...... 75 5.4.1.2 Estimation of the error term ...... 76 5.4.2 Proof of Lemma 5.2.5: the k ≠ 0 terms ...... 78 5.4.2.1 Proof of Lemma 5.4.1: properties of the weight function ...... 80

Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z) with the notation from Chapter

2. Let A = (an) ∈ R be a sequence and 1 ≤ M T . In this section we will prove the following short interval mean value estimate for twisted sums of Fourier coeﬃcients of the uj.

Theorem 5.0.1. Let M,N, and T be parameters with 1 M T . Then for any sequence A = (an) ∈ R,

59 5.1. Background and survey of existing literature

we have

2 ! 7 X X 4 3 1 N itj ε 2 anνj(n)n TM + N + M 2 N 2 + T (NT ) kAk . M T −M≤tj ≤T +M n≤N

The proof of Theorem 5.0.1 begins in Section 5.2.1, and is a minor adaptation of the work carried out by Luo in [Luo95] using a slightly different weight function, combined with the refinement of Young in [You13]. Apart from one major difference involving the integral transforms of the weight function, outlined in Section 5.1.0.4, the proof essentially amounts to just checking that the technical features of certain integral transforms of auxiliary weight functions developed in [You13] still hold in this setting. The familiar reader may wish to skip those sections entirely. We include the details, however, for completeness.

5.1 Background and survey of existing literature

The large sieve type inequalities we discuss here all focus on the spectral aspect, although there is an extensive body of work that is dedicated to proving large sieve type inequalities in other various aspects (weight, level, etc.) and for other objects in harmonic analysis. In this section we review the existing literature on spectral large sieve inequalities with additional spectral twists, and we highlight the subtle differences between this problem and the standard large sieve inequality for GL(2, R) automorphic forms. The general large sieve framework is as follows. The goal in the large sieve problem is to efficiently bound 2 the L -mean value of (Fourier) coefficients λf (n) for f in some family F as |F| grows; that is, to obtain bounds of the form

2 X X 2 anλf (n) (F,N) · kAk , (5.1) B f∈F n≤N

A ∈ CN F F where = (an) and B( ,N) is a function depending on the size of and N. By duality, the best bound one can possibly hope to achieve would have

F |F| B( ,N) = + N.

5.1.0.1 The untwisted spectral large sieve inequality

In the spectral aspect, one takes the family F = F(T ) to consist of Maass cusp forms uj restricted to having eigenparameter tj in some range. For the long spectral interval, the range is tj ≤ T for large T , and consequently, by Weyl’s law the size of F(T ) is T 2. For this family, Iwaniec [Iwa80] established the optimal

60 5.1. Background and survey of existing literature

(untwisted) large sieve inequality

2 X X 2 ε 2 anνj(n) T + N (TN) kAk . (5.2)

tj ≤T n≤N

In fact, the family F(T ) can actually be taken to also include the contribution from the continuous spectrum, and one has the following.

Theorem 5.1.1 (Iwaniec, [Iwa80]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for

SL(2, Z). Let A = (an) be a sequence of complex numbers, and let N ≥ 2. Then as T → ∞,

2 Z 2 X X 4 T 1 X a ν (n) + a τ (n) dt T 2 + N (TN)εkAk2. (5.3) n j 2 | |2 n it π −T ζ(1 + 2it) tj ≤T n≤N n≤N

Jutila-Motohashi ([Jut00], [JM05]) later extended (5.2) to shortened spectral intervals, with

2 Z 2 X X 4 T +M 1 X a λ (n) + a τ (n) dt (TM + N)(TN)εkAk2. (5.4) n j 2 | |2 n it π T −M ζ(1 + 2it) T −M≤tj ≤T +M n≤N n≤N

Again, this (untwisted) spectral large sieve inequality is optimal because Weyl’s law shows that the size of the family is (Z Z ) −(T +M) T +M φ′ 1 + it { − ≤ ≤ } 2 # uj : T M tj T + M + + 1 dt TM. −(T −M) T −M φ 2 + it

5.1.0.2 The twisted spectral large sieve inequality

itj If one includes additional spectral twists by replacing νj(n) with νj(n)n on the lefthand side of (5.2), complicating new features arise. In [DI86], Deshouillers-Iwaniec implicitly established a nonoptimal upper bound of the following form.

Theorem 5.1.2 (Deshouillers-Iwaniec, [DI86]). Let {uj} be an orthonormal basis of Hecke-Maass cusp

forms for SL(2, Z). Let A = (an) be a sequence of complex numbers, and let N ≥ 2. Then for T 1,

2 Z 2 X X 4 T 1 X a ν (n)nitj + a τ (n) dt T 2 + N 2+ε kAk2 (5.5) n j 2 | |2 n t π −T ζ(1 + 2it) tj ≤T n≤N n≤N for A ∈ R.

61 5.1. Background and survey of existing literature

The issue arises from the additional twisting factors in the continuous spectrum, whence for that contribution they are only able to make the bound

Z 2 T X 4 1 it kAk2 2 2 anτt(n)n dt NT , π − |ζ(1 + 2it)| T n≤N and upon application of Kuznetsov trace formula, the resulting sum of Kloosterman sums is O N 2kAk2 . With the additional assumption of square-root cancellation for additive twists of the sequence A,(5.5) is 3 1 2 ε 2 2 N kAk2 improved to O T + N T + T (NT ) . The original application of Deshouillers-Iwaniec was

to establish a lower bound on the nonvanishing of the special values L(sj, uj ⊗ Q), with a view towards Phillips-Sarnak’s deformation theory of discrete groups. We provide further discussion of this result in Part III. For arbitrary A, in [Luo95], Luo was able to partially circumvent the issue of having ruinously large contributions from the continuous spectrum and Kloosterman sums by recognizing and extracting the large contribution from the continuous spectrum in (5.5) and exactly cancelling it with a corresponding contribution from the sum of Kloosterman sums on the geometric side of the Kuznetsov trace formula. He established the following.

Theorem 5.1.3 (Luo, [Luo95]). For any sequence A = (an)n∈N ∈ R, we have

2 ! ! X X 3 3 1 N 2 itj 2 ε 2 anνj(n)n T + T 2 N 2 + (NT ) kAk . (5.6) T tj ≤T n≤N

In fact, the term extracted by Luo’s method can be shown to be bounded below by TNkAk2 for certain sequences, so that an optimal bound of the same shape as the untwisted large sieve inequality (5.2) is simply not true when the additional spectral twists are included. It is easier to work with smoothed versions of the spectral sums above; denote the smoothed versions of the discrete spectrum average by

2 X X −tj /T itj ST (A) = e anνj(n)n ,

tj n≤N

and of the continuous spectrum average by

Z 2 ∞ −t/T X 2 e it TT (A) = anτit(n)n dt. π |ζ(1 + 2it)|2 0 n≤N

Theorem 5.1.3 is implied by the following asymptotic expansion of the smoothed sums that is implicit in

62 5.1. Background and survey of existing literature

[Luo95].

Theorem 5.1.4 (Luo, [Luo95]). For any sequence A = (an)n∈N ∈ R and 1 ≤ X ≤ T , the smoothed averages satisfy

2T X X 1 S A T A S ˜bˆa`dﬄ A T ( ) + T ( ) = aman 2 S(0, m; r)S(0, n; r) + T ( ; X) (5.7) π ≤ ≥ r m,n N r 1 ! ! 3 NT N 2 + O T 2 + + (NT )εkAk2 , (5.8) X T

≥ ≤ 1 where, for a smooth weight function η satisfying η(x) = 1 for x 1 and η(x) = 0 for x 2 , 1 X X X sn + νm S ˜bˆa`dﬄ(A; X) = 2 cos a a |m − n| e T 2T m n r m≠ n r

Furthermore, the continuous spectrum average has the expansion

2T X X 1 T (A) = a a S(0, m; r)S(0, n; r) + O N + T 2 N εkAk2 , (5.9) T π m n r2 2 m,n≤N r≥1

S ˜bâ`dffl A while T ( ; X) satisfies the bound

3 1 S ˜bˆa`dﬄ A 2 2 εkAk2 T ( ; X) T N (NT ) . (5.10)

The diﬀerence between Theorem 5.1.3 and Theorem 5.1.1 is really quite subtle. While (5.5) is true for

all sequences A, a bound for ST (A) + TT (A) of the same form as (5.2) in fact does not hold at all; if one were to include the contribution from the continuous spectrum on the lefthand side of (5.6), the statement would no longer be true. The matching main terms in (5.7) and (5.9) are a special feature of the spectral theory of automorphic

forms: the origin of the main term of (5.9) is the Ramanujan expansion of σ1−s(n); namely,

X S(n, 0; r) σ − (n) = ζ(s) , for <(s) > 1. 1 s rs r≥1

Meanwhile, the main term of (5.7) comes from analysis of certain additive character sums that arise from the product of Kloosterman sums and an integral transform in Kuznetsov—see Section 5.2.1. The properties of the character sums that we require are collected in Appendix C. The error terms of (5.9) and (5.7) would be admissible for an optimal large sieve inequality like (5.2) for

2+ε A S ˜bˆa`dﬄ A N T . However, the current best bound for general on the problematic term T ( ; X) stops short

63 5.1. Background and survey of existing literature

of an inequality of such strength for general A. There are, however, certain assumptions that can be made on A that allow for substantial improvements—for example, as mentioned earlier the assumption of square

root cancellation for additive twists of the coeﬃcients an results in an improved bound. In [You13], Young revisits Luo’s proof of Theorem 5.1.3 and adapts it to a form that is conducive to application of GL(3)-Voronoi summation; the application is an upper bound for the second moment of

GL(2)×GL(3) L-functions L(sj, uj ⊗ψ) that is consistent with the Lindelöf hypothesis, described in Chapter 6. By Fourier-Poisson method, he instead shows that the problematic term takes the form 1 X X 1 X |m − n| S ˜bˆa`dﬄ(A; X) = 2 cos a a S(k, m; r)S(k, n; r) T 2T m n r2 r m≠ n r

S ˜bâ`dffl A Application of GL(3)-Voronoi summation to on T ( ; X) actually shows that it is negligible when the coefficients are A (ℓ, n) n a = ψ√ ω , n n N

where Aψ(ℓ, n) are the Fourier coeﬃcients of a GL(3) Maass cusp form, and ω is a smooth weight function with dyadic support. The advantage of taking this approach is that Voronoi-type summations are often available in situations

when one does not have any power saving cancellation. for example, when an are deﬁned in terms of the

ternary divisor function d3(n), one can apply the generalized Voronoi-type summation 2.2.3. Young was able to use this to achieve an optimal bound for the sixth moment of GL(2) L-functions at the special point. We discuss this result and other power moment estimates implied by Theorem 5.1.3 in Chapter 6.

5.1.0.3 The short interval twisted large sieve inequality of Luo

Before continuing, we remark that there is already a short interval twisted large sieve inequality due to Luo

([Luo96]) for the ﬁxed length interval T ≤ tj ≤ T + 1.

Theorem 5.1.5 (Luo, [Luo96]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z).

Let A = (an) be a sequence of complex numbers, and let N ≥ 2. Then for T 1,

2 Z 2 X X T +1 1 X a ν (n)nitj + a τ (n)nit dt (T + N)(TN)εkAk2. (5.11) n j | |2 n it T ζ(1 + 2it) T ≤tj ≤T +1 n≤N n≤N

64 5.1. Background and survey of existing literature

Furthermore, if one takes an interval of length 1 M T , then

2 Z 2 X X T +M 1 X a ν (n)nitj + a τ (n)nit dt (TM + NM)(TN)εkAk2. n j | |2 n it T −M ζ(1 + 2it) T −M≤tj ≤T +M n≤N n≤N (5.12)

Theorem 5.1.5 is optimal for N T 1+ε. However, for N T 1+ε one encounters exactly the issues outlined above for the long interval. Indeed, for M T ε the extra twists do not really play a role, and one can see that (5.11) is equivalent to Jutila-Motohashi’s untwisted version by partial summation.

5.1.0.4 Some extended remarks on integral transforms appearing in Kuznetsov

A major difference between working over the long interval as opposed to a short interval is that the integral transforms appearing in Kuznetsov for the long interval weight function are readily computable in terms of elementary functions. Starting with the long interval spectral weight function m it g (t, m, n) = e−t/T T n on the spectral side of Kuznetsov in form 2.1.4, one finds that it is easily expressed as the Titchmarsh coefficients of the test function φ = φβ now described. We have

x sinh (β) φ(x) = − eix cosh(β) π for <(β) > 0 and 0 < =(β) < π. The integral transforms are then X φS(x) = (2k)Nφ(k − 1)Jk−1(x) k>0 2|k Z 1 − 3 sinh(2β) 2 2 2 = − xξJ0(xξ) cosh (β) − ξ dξ, 2π 0 where the Neumann coeﬃcients have the explicit formula

i−k N (k − 1) = e−(k−1)β, φ π and the Titchmarsh integral Z tdt φT(x) = J (x)T (t) 2it φ cosh(πt) R Z 1 − 3 sinh(2β) 2 2 2 = − xξJ0(xξ) cosh (β) − ξ dξ, 2π 0

65 5.1. Background and survey of existing literature

where the Titchmarsh coeﬃcients are explicitly given by Z π ∞ dy Tφ(t) = (J2it(y) − J−2it(y)) φ(y) 2i sinh(πt) 0 y sinh((π + 2iβ)t) = sinh(πt) −A = gT (t, m, n) + O T

for T 1. Thus, by appealing to the full spectral form of the Kuznetsov trace formula, or Sears-Titchmarsh inversion, one can reverse the integral transform weights on the geometric side to get a nicer explicit expan-

sion. √ 4π mn In particular, at the point x = c , one has √ 4π mn 2|m − n| m + n − 2π|m−n| N φ = e e cT + O . (5.13) c c c T m+n This is crucial, because as mentioned earlier, the linear phases e c combine with the Kloosterman sums to yield certain character sums of Iwaniec-Li that are well-understood. Manipulations of these character sums along the lines of [IL07], followed by Poisson summation, or Euler-MacLaurin summation in Luo [Luo95], then allows one to extract the main term in (5.7) that exactly cancels with part of the continuous spectrum contribution. For a short interval with additional spectral twists, the natural spectral weight function is

− −( t−T )2 m it −( t+T )2 m it hT,M (t, m, n) = e M + e M n n m it m −it = à˚u¯sfi¯sfi˚ià‹nffl(t) + à˚u¯sfi¯sfi˚ià‹nffl(−t) . GT,M n GT,M n

But, hT,M does not have such easily computable Sears-Titchmarsh inverses that would give us a corresponding test function like φ. We have no option but to apply Kuznetsov in form of Theorem 2.1.4. To get around this apparent difficulty, in Appendix B.0.2 we apply stationary phase to the Titchmarsh integral, and we find that for m > n, √ √ 4π mn m π(m − n) n + m π(m − n) ∗ 4π mn m h+ ; = e à˚u¯sfi¯sfi˚ià‹nffl + ρ ; , (5.14) T,M c n c c GT,M c T,M c n

√ ∗ 4π mn m where ρT,M c ; n consists of certain lower order terms that we bound trivially. Notably, the phase and weight correspond exactly to those for the long interval test function φ, and so we may view this as a

pseudo-Titchmarsh inverse like (5.13). However, there is no corresponding portion√ of the Eisenstein series ∗ 4π mn m contribution with which to cancel the lower order terms coming from ρT,M c ; n . Unfortunately this restricts the range of M = M(N) to which Theorem 5.0.1 is gainfully applied for general sequences A.

66 5.2. The short interval twisted spectral large sieve inequality

5.2 The short interval twisted spectral large sieve inequality

Let M,N, and T be parameters with 1 ≤ M T . Denote the smoothed discrete spectrum mean value by

( ) 2 X − 2 X − tj T M itj ST,M (A) = e anνj(n)n ,

tj n≤N

and the continuous spectrum mean value by

2 Z ∞ − t−T 2 ( M ) X 2 e it TT,M (A) = τit(n)n dt. π |ζ(1 + 2it)|2 0 n≤N

The continuous spectrum mean value has the following asymptotic expansion, whose proof begins in Section 5.3.

Theorem 5.2.1. Let A = (an)n∈N be an arbitrary sequence with an ∈ R. Let 1 M T . Then for any N T , T A M˜lá˚r`gé A 1+εkAk2 T,M ( ) = T,M ( ) + O N , (5.15) where the main term is M X X 1 M˜lá˚r`gé (A) = √ a a S(0, m; r)S(0, n; r). (5.16) T,M π m n r2 m,n r≥1

The expansion for the combined spectral sum ST,M (A) + TT,M (A) is as follows; the proof begins in Section 5.2.1.

Theorem 5.2.2. Let A = (an)n∈N be an arbitrary sequence with an ∈ R. Let 1 ≤ X ≤ T , N T , and 1 M T . Then we have ! ! 7 4 ˜lá˚r`gé NM N S (A)+T (A) = M (A)+S ˜bâ`dffl (A; X)+O TM + + TX + T T εkAk2 , (5.17) T,M T,M T,M T,M X M

M˜l´a˚r`g´e A where T,M ( ) is the same main term (5.16) that appears in Theorem 5.2.1, and the “bad” term is

X X 1 X S ˜bâ`dffl A T,M ( ; X) = aman 2 S(k, m; r)S(k, n; r) ̸ ≥ r ∈Z m=n r 1Z k |m − n| η(x) π|m − n| −kx × à˚u¯sfi¯sfi˚ià‹nffl 2 GT,M e dx, (5.18) r R x xr r

67 5.2. The short interval twisted spectral large sieve inequality

≤ 1 ≥ where η is a smooth weight function satisfying η(x) = 0 for x 2 and η(x) = 1 for x 1. Furthermore, the “bad” term satisﬁes the bound

Z 2 X X M X 1 1 T 1+ε un S ˜bˆa`dﬄ A T,M ( ; X) M anS(k, n; r)e du. (5.19) r |k| − M rM r

Remark 5.2.1. Note that for N = T 1+ε, the error term of (5.17) is comparable to the error term of (5.7) 7 +ε 3 +ε 21 +ε when M T 11 . When N = T 2 , (5.20) is comparable to (5.6) when M T 22 .

Along the lines of [You13], choosing X appropriately for given length N yields the following corollary, which in turn implies Theorem 5.0.1 by positivity.

Corollary 5.2.3. Let A,M,N, and T be as described in Theorem 5.2.2. Then we have ! 7 4 3 1 N ε 2 S (A) TM + N + M 2 N 2 + T (NT ) kAk . (5.20) T,M M

Remark 5.2.2. Note that for N = T 1+ε, the bound (5.20) is comparable to Luo’s bound (5.6) when M 5 +ε 3 +ε 5 +ε T 9 . And when N = T 2 , (5.20) is comparable to (5.6) when M T 6 .

Proof of Corollary 5.2.3. Combining (5.16) and (5.17), the two main terms exactly cancel, so it remains to

S ˜bˆa`dﬄ A A ε bound T,M ( ; X) for arbitrary . Starting with the bound 5.19, we split the k-sum into T complete sums modulo r, and then open the square appearing in the integral, yielding

Z 2 X X X M X 1 1 T 1+ε un S ˜bˆa`dﬄ A T,M ( ; X) M anS(k, n; r)e du M r ε ℓr + 1 − rM r

2 Z M X X 1+ε X ε 1 T un MT anS(k, n; r)e du r − M rM r

Opening the Kloosterman sums, by orthogonality of additive characters we see that the sum modulo r is X X∗ b(m − n) S(k, m; r)S(k, n; r) = r e . r k (mod r) b(r)

68 5.2. The short interval twisted spectral large sieve inequality

Closing the square, we arrive at

Z 2 X X M X ∗ T 1+ε an un S ˜bˆa`dﬄ A ε T,M ( ; X) MT ane e du, − M r rM r

to which we apply Young’s extension of Gallagher’s large sieve inequality (see [You13]), yielding X2M S ˜bˆa`dﬄ (A; X) MT ε + MX kAk2. T,M T q ≤ 3 N 3 For N M , we take X = M . For N > M , we take X = M. This completes the proof of Corollary 5.2.3.

5.2.1 Proof of Theorem 5.2.2

In this section we shall reduce the proof of Theorem 5.2.2 to two lemmas, whose proofs begin in Sections 5.4.1 and 5.4.2. In order to apply Kuznetsov trace formula (Theorem 2.1.4), we require an even analytic weight function decaying rapidly in a strip; as described in Section 5.1.0.4, we shall use

− − t−T 2 m it − t+T 2 m it h (t, m, n) = e ( M ) + e ( M ) . T,M n n

|= | ≤ 1 ≥ ε Then hT,M is holomorphic in t in the strip (t) 2 , where it clearly decays rapidly. Note that for t T , − 2 it − t T m −A h (t, m, n) = e ( M ) + O T T,M n for any A > 0.

5.2.1.1 Application of Kuznetsov trace formula

Opening the square and exchanging the weight function for hT,M (t, m, n), we pick up a negligible error, and we have X X ST,M (A) + TT,M (A) = aman νj(m)νj(n)hT,M (tj, m, n) ≤ m,n N tj Z X 2 ∞ h (t, m, n) + a a T,M τ (m)τ (n)dt + O T −A . m n π |ζ(1 + 2it)|2 it it m,n≤N 0

69 5.2. The short interval twisted spectral large sieve inequality

By the Kuznetsov trace formula in the form (2.8), we ﬁnd that the spectral sums are Z X 2 ∞ h (t, m, n) ν (m)ν (n)h (t , m, n) + T,M τ (m)τ (n)dt j j T,M j | |2 it it π 0 ζ(1 + 2it) tj X 1 4π √ = δ(n = m)(h ) + S(m, n; c)h+ mn . T,M 0 c T,M c c≥1

The diagonal integral is Z

(hT,M )0 = hT,M (t, m, m)t tanh(πt)dt R Z − t−T 2 − t+T 2 = t tanh(πt) e ( M ) + e ( M ) dt R √ = πT M + O T −A , yielding an overall contribution of X √ 2 −A amanδ(m = n)(hT,M )0 = πT MkAk + O T . m,n≤N

Meanwhile, by Appendix B.0.2 the integral transform appearing in the contribution from Kloosterman sums has the expansion Z 2i tdt h+ (x) = J (x)h (t, m, n) T,M π 2it T,M cosh(πt) R |m − n| m + n π|m − n| T 1+εN = e à˚u¯sfi¯sfi˚ià‹nffl + O . c c GT,M c cM 2

By the Weil bound for Kloosterman sums and the estimate X 1+ε 2 (n, m)aman N kAk , n,m 7 1+ε N 4 the error terms yield of contribution of O T M . Accordingly, we write ! 7 N 4 S (A) + T (A) = c TM + Kℓ∗ (A) + O T 1+ε , T,M T,M 1 M M

where we have denoted X X ∗ 1 m + n |m − n| π|m − n| Kℓ (A) = a a S(m, n; c)e à˚u¯sfi¯sfi˚ià‹nffl . M m n c c c GT,M c m≠ n ≤ ≪ NT ε 1 c T

K ∗ A Next we rearrange the Kloosterman sum and additive character to express ℓM ( ) in a form that is

70 5.2. The short interval twisted spectral large sieve inequality

more amenable to Poisson summation. One has n + m X∗ (1 − d)m + (1 − d)n S(m, n; c)e = e . c c d(c)

Fixing each q = (d−1, c), we split the c-sum so that c = qr, with d = 1−qs and (s(q−s), r) = 1. Accordingly, we have X X X ∗ 1 1 π|m − n| Kℓ (A) = a a |m − n| V− (m, n; r)g , M m n r2 q2 q T,M qr m≠ n r≥1 q≥1 where Vd(m, n; r) are the character sums deﬁned by Iwaniec-Li [IL07], X sn − d + sm V (m, n; r) = e . d r s(r) (s(d+s),r)=1

5.2.1.2 Extraction of the problematic main term

K ∗ A M˜l´a˚r`g´e A Next we extract the main term from ℓM ( ) that we will cancel with the main term T,M ( ) coming from TT,M (A). We formalize this process in a series of lemmas that will be proved later in the chapter. ≤ ≤ K ∗ A For a parameter 1 X T to be determined later, we split ℓM ( ) into

∗ ∗ ∗ K A K ,˛hèá`dffl A K ,˚tá˚i˜l A ℓM ( ) = ℓM ( ; X) + ℓM ( ; X),

where the ﬁrst sum consists of the terms with r < X, and the second sum those with r ≥ X. By the exact same proof as in [Luo95] (see (39) on p.386), we have

ε ∗ NMT Kℓ ,˚t´a˚i˜l(A; X) kAk2. M X

∗ K ,˛hèá`dffl A To deal with ℓM ( ,X), we will apply Poisson summation to the q-sum. Following [You13], let R → R ≥ ≤ 1 η : be a smooth even weight function such that η(t) = 1 for t 1 and η(t) = 0 for t 2 that is ∗ K ,˛hèá`dffl A nondecreasing on t > 0. Inserting η into ℓM ( ; X) does not change the sum. The character sums Vd(m, n; r) are periodic in d mod r. Thus, splitting the q-sum over congruence classes modulo r and then applying Poisson summation, we have X X X ∗, 1 Kℓ ˛hèá`dffl(A; X) = a a |m − n| V− (m, n; r) (5.21) M m n r3 a m≠ n r≥1 Z a(r) X ka η(x) π|m − n| −kx × à˚u¯sfi¯sfi˚ià‹nffl e 2 GT,M e dx. (5.22) r R x xr r k∈Z

71 5.2. The short interval twisted spectral large sieve inequality

By the special factorization of the character sums X ak V− (m, n; r)e = S(k, m; r)S(k, n; r), a r a(r)

we have X X X ∗ 1 K ,˛hèá`dffl A | − | ℓM ( ; X) = aman m n 3 S(k, m; r)S(k, n; r) ̸ ≥ r ∈Z m=nZ r1 k η(x) π|m − n| −kx × à˚u¯sfi¯sfi˚ià‹nffl 2 GT,M e dx R x xr r d∗ d∗ = Kℓ M,k=0(A; X) + Kℓ M,k=0̸ (A; X)

d∗ d∗ where Kℓ M,k=0(A; X) denotes the k = 0 term of 5.21 and Kℓ M,k=0̸ (A; X) the terms with k =6 0. d∗ d∗ We shall prove the following two lemmas regarding the sizes of Kℓ M,k=0(A; X) and Kℓ M,k=0̸ (A; X), Kd∗ A M˜l´a˚r`g´e A whose proofs we defer until Section 5.4. The term ℓ M,k=0( ; X) provides the large main term T,M ( ) that will cancel with the contribution from the continuous spectrum.

d∗ Lemma 5.2.4. The terms of Kℓ M (A; X) with k = 0 have the expansion Kd∗ A M˜l´a˚r`g´e A εkAk2 ℓ M,k=0( ; X) = T,M ( ) + O MXT . (5.23)

d∗ Lemma 5.2.5. The terms of Kℓ M (A; X) with k =6 0 satisfy

2 Z M X X 1+ε X d∗ 1+ε 1 1 T un Kℓ M,k=0̸ (A; X) T X + M anS(k, n; r)e du. (5.24) r |k| − M rM r

Collecting the above and applying Lemma 5.2.4, we obtain the main term of (5.17) and an admissible error term. Applying Lemma 5.2.5 and changing notation to

Kd∗ A ˜bâ`dffl A ℓ M,k=0̸ ( ) = ST,M ( ; X) X X 1 X = aman 2 S(k, m; r)S(k, n; r) ̸ ≥ r ∈Z m=n r 1Z k |m − n| η(x) π|m − n| −kx × à˚u¯sfi¯sfi˚ià‹nffl 2 GT,M e dx, r R x xr r

yields (5.17). This completes the proof of Theorem 5.2.2. □

72 5.3. Proof of Theorem 5.2.1

5.3 Proof of Theorem 5.2.1

M˜l´a˚r`g´e In this section we show how the main term T,M is extracted from using a certain Ramanujan expansion of the divisor function.

Proof of Theorem 5.2.1. It is a remarkable fact that the divisor function can be expressed as the product of < ∈ N R two Dirichlet series. For (¯s) > 1 and n , let n(¯s) denote the Ramanujan sum Dirichlet series

X S(0, n; r) Rn( ) = . ¯s r¯s c≥1

By the identity X r S(0, n; r) = µ d · δ(d|n), d d|r for n ∈ N and s ∈ C with <(s) < 0 one has the Ramanujan expansion

σs(n) = ζ(1 − s)Rn(1 − s),

and this continues to the line <(s) = 0 where both Dirichlet series are conditionally convergent. < R Following Titchmarsh [Tit86], along the line (¯s) = 1, n(¯s) can be approximated by an extremely long Dirichlet polynomial—one having subexponential length. For t T 2, we have

X S(0, n; r) R (s) = + O N −2 (5.25) n r1+it r≤exp(N ε)

s for any ε > 0. Since σ (n) = n 2 τ s (n), and s 2

T −ε ζ(1 + 2it) T ε, it follows that the continuous spectrum contribution has the expansion Z ∞ it 2 − t−T 2 m τit(n)τit(m) e ( M ) dt π n |ζ(1 + 2it)|2 0 Z ∞ X 2it ε 2 − t−T 2 1 r2 MT = e ( M ) S(0, n; r )S(0, m; r )dt + O . (5.26) π r r r 1 2 N 2 0 ε 1 2 1 r1,r2≤exp(N )

Switching the order of the sum and the integral, we observe that the integral can be evaluated up to a negligible error; explicitly,

Z ( ( )) ∞ 2it 2iT 2 − 2 √ r − t T r2 r2 − M log 2 −A e ( M ) dt = πM e r1 + O T . (5.27) 0 r1 r1

73 5.3. Proof of Theorem 5.2.1

Replacing (5.27) back in (5.26), we arrive at the expansion MT ε T (A) = M (A) + O kAk2 , T,M T,M N

where we have denoted

( ( )) X X 2iT r 2 M 1 r2 − M log 2 M (A) = √ a a S(0, n; r )S(0, m; r )e r1 . T,M π m n r r r 1 2 ε 1 2 1 m,n≤N r1,r2≤exp(N )

The terms with r1 = r2 in MT,M (A) contribute the problematic term

M X X 1 √ a a S(0, n; r)S(0, m; r) = M˜l´a˚r`g´e (A) + O T −A , (5.28) π m n r2 T,M m,n≤N r≤exp(N ε) where we have extended the r-sum using

|S(0, n; r)| ≤ (n, r)τ2(n), (5.29)

the mean-value estimate of Ramanujan sums (C.4) X 1 +ε anS(0, n; r) N 2 τ2(r). n≤N

To ﬁnish we must bound the oﬀdiagonal contribution with r1 =6 r2. Without loss of generality, assume 2iT that r1 < r2 (we will take absolute values anyways so the twists (r2/r1) do not play a role). Note that M r2 the terms with r1 ≤ make a negligible contribution, since then M log log T , so that log T r1

( ( )) X X X 2 |S(0, n; r )| |S(0, m; r )| − r2 1 2 M log r −A aman e 1 T , r1 r2 m,n≤N ≤ M r

M r2 log T for any A 1. Similarly, the contribution from the terms with r1 > and ≥ 1+ is also negligible. log T r1 M M r2 log T Consider the terms with r1 > and 1 < < 1 + . By (C.4), these terms contribute log T r1 M

( ( )) X X X 2iT 2 S(0, n; r ) S(0, m; r ) r − r2 1 2 2 M log r M aman e 1 ε r1 ε r2 r1 m,n≤N r1 <1+ log T r1 M ( ( )) X X 2 X X 1 1 − r2 M log r M e 1 amS(0, m; r2) anS(0, n; r1) ε r1 ε r2 r1 <1+ log T r1 M ( ( )) X X r 2 1+ε 2 1 1 − M log 2 MN kAk e r1 . ε r1 ε r2 r1 <1+ log T r1 M

74 5.4. Proof of Lemmas 5.2.4 and 5.2.5 An elementary argument shows that the remaining (r , r )-sum is O 1 , so that the final contribution from 1 2 M these terms is O N 1+ε . Thus we have seen that T A M˜lá˚r`gé A 1+ε T,M ( ) = T,M ( ) + O N .

This completes the proof of Theorem 5.2.1.

5.4 Proof of Lemmas 5.2.4 and 5.2.5

In this section we prove Lemmas 5.2.4 and 5.2.5, which are required to complete the proof of Theorem 5.2.2.

5.4.1 Proof of Lemma 5.2.4: the k = 0 terms

In this section we are going to show that Z X X 1 |n − m| η(x) π|m − n| Kd∗ A à˚u¯sfi¯sfi˚ià‹nffl ℓ M,k=0( ; X) = aman 2 S(0, m; r)S(0, n; r) 2 GT,M dx r r R x xr m≠ n r

Denoting the integral by Z η(x) y Hr (0, y) = 2 gT,M dx, (5.30) R x xr we split into two pieces that will contribute the main term and the error term of (5.23), respectively; we have Z Z 1 y 1 y − Hr(0, y) = 2 gT,M dx + 2 gT,M (η(x) 1)dx R x xr R x xr

”mà˚i‹nffl èˇr˚rò˘rffl = Hr (0, y) + Hr (0, y).

d∗ Accordingly, denote the corresponding pieces of Kℓ M,k=0(A; X) by

Kd∗ A Kd∗”mà˚i‹nffl A Kd∗èˇr˚rò˘rffl A ℓ M,k=0( ; X) = ℓ M,k=0( ; X) + ℓ M,k=0( ; X).

Kd∗”mà˚i‹nffl A 5.4.1.1 Extraction of the main term: evaluation of ℓ M,k=0( ; X) 1 yu−T 6 Changing variables u = x and subsequently w = M , we find that for y = 0, the first integral is Z √ Mr − 2 πMr ”mà˚i‹nffl w Hr (0, y) = e dw = , y R y

75 5.4. Proof of Lemmas 5.2.4 and 5.2.5

and this contributes the main term in (5.26). We have

X X | − | d∗”mà˚i‹nffl 1 n m Kℓ (A; X) = a a S(0, m; r)S(0, n; r) H ”mà˚i‹nffl(0, π|m − n|) M,k=0 m n r2 r r m≠ n r

5.4.1.2 Estimation of the error term Kd∗èˇr˚rò˘rffl A It remains to bound ℓ M,k=0( ; X). First we evaluate the integral. For y > 0, we see that

( ) y 2 Z ∞ −T − xr 1 M èˇr˚rò˘rffl − Hr (0, y) = 2 2 e (η(x) 1) dx 0 x ( ) y 2 Z −T 1 1 − xr M − = 2 2 e (η(x) 1) dx 0 x ( ) yw 2 Z ∞ −T − r = 2 e M ψ(w)dw, (5.31) 1 1 − 1 writing ψ(w) = η w 1 after making the change of variables x = w , and in the second line we are using that η(x) = 1 for x ≥ 1. Integrating by parts once, the first term of (5.31) is

( ) yw 2 Z ∞ −T Z ∞ − r e M ψ(w)dw = lim (erfc (w)ψ(w)) + erfc (w)ψ′(w)dw, (5.32) →∞ T,M T,M 1 w 1

76 5.4. Proof of Lemmas 5.2.4 and 5.2.5

where erfcT,M (w) denotes the adjusted complementary error function

( ) yt 2 Z ∞ −T − r M erfcT,M (w) = e dt w Z ∞ rM 2 = e−u du y yw − T rM M rM yw T = erfc − . y rM M

We easily that erfcT,M (w) → 0 as w → ∞, so that the ﬁrst term of (5.32) is

lim erfcT,M (w)ψ(w) = 0. w→∞

Since ψ(w) is constant for w ≥ 2, it follows that Z 2 2rM yw T ′ èˇr˚rò˘rffl − Hr (0, y) = erfc ψ (w)dw. (5.33) y 1 rM M

Kd∗èˇr˚rò˘rffl A Returning to the full contribution of ℓ M,k=0( ; X), by a dyadic partition of unity and (5.33), we have

X X | − | d∗èˇr˚rò˘rffl 1 π n m Kℓ (A; X) = a a S(0, m; r)S(0, n; r) H èˇr˚rò˘rffl(0, π|m − n|) M,k=0 m n r2 r r m≠ n r

| | ≤ Again, extending the (m, n)-sum to the diagonal m = n using the bound S(0, n; r) r yields an acceptable εkAk2 7→ rw error term of O MXT . Thus, by changing variables w R , we have

Kd∗èˇr˚rò˘rffl A ℓ M,k=0( ; X) X X Z 2R X M r ′ rw π|m − n|w T = ψ amanS(0, m; r)S(0, n; r)erfc − dw rR R R RM M R=2j ≤X R

Z X X 1 X | − | M π m n w − T 2 amanS(0, m; r)S(0, n; r)erfc dw R 1 RM M R=2j ≤X R

To bound the innermost sum in (5.34), we use the Fourier integral r ! − ξ2 Z ξ 2 e 4 2 2 erfcd (ξ) = et dt . (5.35) π ξ 0

77 5.4. Proof of Lemmas 5.2.4 and 5.2.5

Kd∗èˇr˚rò˘rffl A Since erfc(x) is Schwarz, by Fourier inversion the estimate of ℓ M,k=0( ; X) reduces to estimating ! Z 2 Z 2 X − ξ ξ X M e 4 2 2 πmξ t 2 e dt amS(0, m; r)e dξ. (5.36) R R ξ RM R=2j ≤X 0 m

By the rapid decay of the erfc,d we can truncat the integral to ξ T ε. Opening the Ramanujan sum, and then applying Young’s extension of Gallagher’s large sieve inequality, we have

2 Z ε X T X X∗ X Kd∗èˇr˚rò˘rffl A M ℓm πmξ ℓ M,k=0( ; X) ame e dξ R − ε r RM R=2j ≤X T R

To conclude, we have shown that Kd∗ A M˜l´a˚r`g´e A εkAk2 ℓ M,k=0( ; X) = T,M ( ) + O MXT .

This completes the proof of Lemma 5.2.4. □

5.4.2 Proof of Lemma 5.2.5: the k =6 0 terms

S ˜bâ`dffl A Kd∗ A In this section we will prove that the “bad” sum T,M ( ; X) = ℓ M,k=0̸ ( ; X) satisfies the bound

2 Z M X X 1+ε X d∗ 1+ε 1 1 T un Kℓ M,k=0(A; X) T X + M anS(k, n; r)e du. r |k| − M rM r

d∗ Consider the integral weight function appearing in the deﬁnition of Kℓ M,k=0̸ (A; X). We have Z |m − n| ∞ η(t) π|m − n| −kt 2 gT,M e dt r 0 t tr r Z ∞ M 1 π|m − n|t − 1 − T 2 −πkt|m − n| ( t M ) = 2 η e e 2 dt. π 0 t rM r M

Following Young, write x = m − n, and

rM r2M A = and B = . π πk

We may then express the weight function as Z 1 |x|t − 1 − T 2 −|x|t ( t M ) WA,B(x) = 2 η e e dt, (5.37) R t A B and WA,B has the following technically useful features, the proof of which we defer to the end of the section.

78 5.4. Proof of Lemmas 5.2.4 and 5.2.5

Lemma 5.4.1. The weight function defined in (5.37) is Fourier-invertible, with Z 1 c u ux WA,B(x) = WA,B e du, (5.38) R A A A and satisfies − − B K A 1 K W (x) 1 + T ε (5.39) A,B K |x| |x| for any K ≥ 0. Furthermore, the Fourier transform has the exact formula Z ∞ 2 c T tuT tuT − (tu) t t WA,B(ξ) = 2 cos + tu sin e 2 η e − dt (5.40) 0 M M M A B and also satisfies ( ) − u2 1 u T |B| e 8 Wc min , . (5.41) A A,B A MA |u|

To complete the proof of Lemma 5.2.5, we apply Fourier inversion to WA,B(x) and reduce the length of integration by using the bounds provided by Lemma 5.4.1. In the notation outlined above, by (5.38) we have Z X X X d∗ M 1 1 c u ux −A K ̸ A ℓ M,k=0( ; X) = aman 2 S(k, m; r)S(k, n; r) WA,B e du + O T . 2 r R A A A n≠ m r

Extending the sum to include the diagonal m = n costs X X X Z M | |2 1 | |2 1 c u ux an 2 S(k, n; r) WA,B e du 2 r R A A A n≤N r

by the Weil bound for Kloosterman sums. From this we see that the k =6 0 contribution has the bound

d∗ Kℓ ̸ (A; X) M,k=0 Z X 1 X 1 u X u(m − n)π 1+ε c XT + M 2 W anamS(k, m; r)S(k, n; r)e du r R A A rM r

Z 2 X X X 1+ε 1 1 c u unπ = XT + M W anS(k, n; r)e du. r R A A rM r

| | ≤ M Next we show that we only need to integrate over the range u T 1+ε ; the other terms yield an admissible error term, but this range will require additional input about the sequence A to get the optimal bound for

79 5.4. Proof of Lemmas 5.2.4 and 5.2.5

the large sieve inequality. From the estimate (5.41), we see that the contribution from |u| T ε is negligible because of the rapid M ≤ | | ≤ ε ≤ | | ≤ decay of the Gaussian. So consider a dyadic subdivision of the range T 1+ε u T , say with U u 2U. Then by (5.41), the contribution for this range is bounded by

Z 2 X X X 1 1 −u2/8 unπ M 2 e anS(k, n; r)e du r ≤| |≤ |u| rM r

Z 2 X X X 1 1 unπ M 2 anS(k, n; r)e du r U ≤| |≤ rM r

Z 2 X X X ε 1 1 unπ MT 2 anS(k, n; r)e du r U ≤| |≤ rM r

Z 2 X X∗ X ε 1 bn unπ = MT ane e du, (5.42) U ≤| |≤ r rM r

M Thus we see that we can take U as small as T 1+ε and still get an admissible error term. This completes the proof of Lemma 5.2.5. □

5.4.2.1 Proof of Lemma 5.4.1: properties of the weight function

The proof of Lemma 5.4.1 is essentially the same as the proof of the estimates for the analogous weight

à˚u¯sfi¯sfi˚ià‹nffl function in [You13], but is somewhat simpler owing to the better decay properties of GT,M . That WA,B is Fourier-invertible follows from it being smooth with rapid decay. Repeated application of integration by parts yields Z B K ∞ ∂K 1 t x |x|t W (x) = η e − dt (5.43) A,B | | K 2πi x 0 ∂t t A t B from which (5.39) follows. c Next we will derive the explicit formula for WA,B and use this to prove (B.8). Since WA,B is even, we may write Z ∞ c c c WA,B(ξ) = WA,B(x)(e(xξ) + e(−xξ)) dx = W+(ξ) + W−(ξ). 0

80 5.4. Proof of Lemmas 5.2.4 and 5.2.5

7→ t c Changing variables t ξ , then W+ is the double integral

Z ∞ Z ∞ c 1 t ξ − ξ − T 2 t W (ξ) = e(uξ) η e ( t M ) e − dtdu + t A t B 0 0 Z Z ∞ ∞ t = e(xξ) f(ξ, t)e − dtdu, (5.44) 0 0 B − ξ − T 2 1 t ξ ( t M ) where we have set f(ξ, t) = t η A t e . As in [You13], (5.44) does not converge absolutely. However, since 2 3 ∂ ξ ξ 2ξ t ξ ′ t − ξ − T 2 f(ξ, t) = − + + η + η e ( t M ) , ∂t t3 t4 t5 A At2 A

after one application of integration by parts to (5.44) the inner integral converges absolutely, so we may switch the order of the integrals and also the order of the integration and diﬀerentiation, yielding Z Z ∞ B t ∞ ∂ Wc (u) = e − e(uξ) f(ξ, t)dξdt + 2πi B ∂t Z0 0 Z ∞ B t ∂ ∞ = e − e(uξ)f(ξ, t)dξ dt 0 2πi B ∂t 0 Z ∞ Z ∞ B t ∂ 1 t ξe(uξ) − ξ − T 2 = e − η e ( t M ) dξ dt. (5.45) 0 2πi B ∂t t A 0 t

Using Fourier inversion for Gaussian functions, the inner integral of (5.45) evaluates to

Z ∞ 2 2 ξe(uξ) − ξ − T tT 2 tuT − (tu) e ( t M ) dξ = + it u e e 2 . 0 t M M c A similar calculation for W− then leads to the explicit formula Z ∞ 2 c T tuT tuT − (tu) t t WA,B(ξ) = 2 cos + tu sin e 2 η e − dt. (5.46) 0 M M M A B

The ﬁrst bound in (5.41) follows by integration by parts, and the second bound by application of the van der Corput bound B.0.1. By the explicit formula (5.40), to bound the terms involving cosine, it suﬃces to consider Z ∞ 2 T ±i xT − x x xA e M e 2 η e − dx. (5.47) Mu u u Bu 2

By the van der Corput bound, we see that (5.47) is

T Mru 1 Mu T ru − k |u|

81 5.4. Proof of Lemmas 5.2.4 and 5.2.5

since k rN ε. For the terms involving sine, it suﬃces to consider

Z ∞ 1 xT 2 x xA x sin e−x /2η e − dx. (5.48) A u M u Bu 2 1 We easily see that these terms satisfy a much better bound of O rM . □

82 Chapter 6

The second moment of GL(2) × GL(3) L-functions at special points over a shortened spectral interval

Contents 6.1 Background and survey of existing literature ...... 84 6.2 Preliminaries on GL(2) × GL(3) Rankin-Selberg L-functions ...... 86 6.3 The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points ...... 88 6.3.0.1 Initial cleaning and preparation ...... 88 6.3.0.2 Application of GL(3) Voronoi summation ...... 91 6.4 The shortened spectral interval sixth moment of GL(2) L-functions at the special point ...... 93

In this section we use Theorem 5.2.2 to obtain the following upper bound for the second moment of GL(2)×GL(3) L-functions at the conductor-dropping point over shortened spectral intervals. The main

S ˜bˆa`dﬄ A technique is to use GL(3) Voronoi summation to obtain extra savings in the problematic term T,M ( ; X) in Theorem 5.2.2. We prove the following.

Theorem 6.0.1 (Restatement of Theorem 1.0.3). Let {uj} be an orthogonal basis of Hecke-Maass cusp forms 21 +ε for SL(2, Z), and let ψ be an Maass cusp form for SL(3, Z) of type (ν1, ν2). Then for T 22 M T and T 1, we have X 2 1+ε |L(sj, uj ⊗ ψ)| T M. (6.1)

T −M≤tj ≤T +M

3 As mentioned in Chapter 5, the cubic L-function L(s, uj) can be treated essentially as though it were

83 6.1. Background and survey of existing literature

L(s, uj ⊗ E3), where E3 is the degenerate minimal parabolic Eisenstein series

∂3 E3(z) = [E(z, ν1, ν2)] 1 . 2 ν1=ν2= 3 ∂ν1∂ν2

By the same techniques used to prove Theorem 6.0.1, namely, the GL(3)-type Voronoi summation (Theorem

2.2.3) for the ternary divisor function τ3, we will also obtain the following shortened spectral interval upper bound for the sixth moment of GL(2) L-functions at the special point.

Theorem 6.0.2 (Restatement of Theorem 1.0.4). Let {uj} be an orthogonal basis of Hecke-Maass cusp 21 +ε forms for SL(2, Z). Then for T 22 M T , we have X 6 1+ε |L(sj, uj)| T M. (6.2)

T −M≤tj ≤T +M

6.1 Background and survey of existing literature

Let us first survey the existing literature on power moment bounds for L-functions at special points. We will not describe the existing literature on asymptotic expansions of moments of L-functions at the special point: this will be discussed in Part III. In general, when studying a family of L-functions F, there is often little one can do to say something meaningful about an individual member of the family; it seems that there is a certain uncertainty principle obeyed by L-functions, in that by restricting attention to a single L-function, very little can be said, but one can say something about the behavior of the L-functions in question when they are considered all at once or on average. Usually, one might seek to obtain a bound of the form X |L(s, f)|n |F|1+ε, f∈F and by the Lindelöf hypothesis this is the best one can expect to be able to do. Typically one uses the Kuznetsov trace formula to study moments of L-functions in the family F(T ) consisting of L-functions attached to GL(2) automorphic forms with spectral parameter in a range defined 1 in terms of T . When the L-functions are taken at the special point sj = 2 + itj, which also involves the spectral parameter, then the problem becomes more subtle—see the background section in Chapter 5. To obtain upper bounds needs a special large sieve type inequality that incorporates the additional twists coming from the special point. Luo established the following. In [Luo95] and [You13], it is noted that as a consequence of Theorem 5.1.3, one can derive the following special point power moment estimates. Here Q is a fixed holomorphic newform of weight 2k and level p and ψ is a fixed SL(3, Z) Maass cusp form.

84 6.1. Background and survey of existing literature

tj ≤T tj ≤T

Table 6.1: Bounds implied by Luo’s twisted large sieve inequality (Theorem 5.1.3).

Only the bounds in the ﬁrst row are as strong as predicted by the Lindelöf hypothesis. Young was able to improve the bounds on the L-functions in the second row by using GL(3)-type Voronoi summation to extract additional savings from a certain problematic term that Luo bounds trivially..

Theorem 6.1.1 (Young, [You13]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z). Let ψ be a ﬁxed Hecke-Maass cusp form for SL(3, Z). Then for T 1, X 2 2+ε |L(sj, uj ⊗ ψ)| T .

tj ≤T

Furthermore, for the sixth moment, one has X 6 2+ε |L(sj, uj)| T .

tj ≤T

In an upcoming paper, Xiannan Li and Vorrapan Chandee have remarkably obtained an optimal upper bound on the long interval second moment of GL(2) × GL(4) L-functions at special points. Their proof can also be modiﬁed to obtain the same bound upper bound on the eighth moment of GL(2) L-functions at special points. Namely, they show the following.

Theorem 6.1.2 (Chandee-Li, [CL20]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z). Let ψ be a ﬁxed Hecke-Maass cusp form for SL(4, Z).1Then for T 1, X 2 2+ε |L(sj, uj ⊗ ψ)| T .

tj ≤T

Furthermore, for the eighth moment, one has X 8 2+ε |L(sj, uj)| T .

tj ≤T

85 6.2. Preliminaries on GL(2) × GL(3) Rankin-Selberg L-functions

Chandee-Li’s proof of Theorem 6.1.2 uses the same basic setup as does Young’s proof of Theorem 6.1.1. However, whereas application of GL(3) Voronoi summation results in a very short dual sum that is therefore negligible in Young’s case, application of GL(4) Voronoi summation in Chandee-Li’s case results in a dual sum of essentially the same length as the original, and thus requires additional cancellation in the new character sums that arise. Furthermore, it seems likely that the methods of Chandee-Li can be modified to obtain the analogous bound for the long interval fourth moment of GL(2)×GL(2) L-functions at special points, although we have made no effort to do so. One expects that for Q a fixed holomorphic Hecke eigencuspform of weight 2k for

Γ0(p), for T 1 we expect that X 4 2+ε |L(sj, uj ⊗ Q)| T .

tj ≤T

In Chapter 8 we describe some consequences of this in terms of the simultaneous nonvanishing of GL(2) × GL(2) L-fucntions at special points.

6.2 Preliminaries on GL(2) × GL(3) Rankin-Selberg L-functions

The Rankin-Selberg L-function of Godement-Jacquet is deﬁned by

X λ (n)A (m, n) L(s, u ⊗ ψ) = j ψ for <(s) > 1. j (m2n)s m,n≥1

Writing εj = 0, 1 if uj is even or odd, respectively, the six gamma factors for L(s, uj ⊗ ψ) are s + ε + it − α s + ε + it − β s + ε + it − γ γ(s, u ⊗ ψ) = π−3sΓ j j Γ j j Γ j j j 2 2 2 s + ε − it − α s + ε − it − β s + ε − it − γ × Γ j j Γ j j Γ j j 2 2 2 Y s + ε + ηit + α∗ = π−3s Γ j j , 2 η=±1, α∗=α,β,γ where the spectral parameters of ψ are α = 1−ν1 −2ν2, γ = −1+2ν1 +ν2, and β = −ν1 +ν2. The completed L-function

Λ(s, uj ⊗ ψ) = γ(s, uj ⊗ ψ)L(s, uj ⊗ ψ)

continues analytically to all s ∈ C and satisﬁes the functional equation

e Λ(s, uj ⊗ ψ) = ε(uj ⊗ ψ)Λ(1 − s, uj ⊗ ψ),

e where ψ is the dual form of ψ of type (ν2, ν1) and ε(uj ⊗ ψ) is the root number with |ε(uj ⊗ ψ)| = 1.

1We have not deﬁned automorphic forms for GL(4, R). The reader should see Goldfeld’s book [Gol06] for a deﬁnition; it is similar to the theory laid out earlier.

86 6.2. Preliminaries on GL(2) × GL(3) Rankin-Selberg L-functions

3 We will also need the following expansion of L(s, uj) as a (multiple) Dirichlet series. It is a simple exercise with the GL(2) Hecke relations, but we have been unable to ﬁnd a satisfactory derivation in the literature.

Proposition 6.2.1. Let uj be a Hecke-Maass cusp form for SL(2, Z). Then for <(s) > 1, we have

X µ(f)τ (g) X τ (n)λ (fn) L(s, u )3 = 3 3 j . (6.3) j (fg)2s (fn)s f,g≥1 n≥1

Proof. Two applications of the Hecke relations yield X 3 1 L(s, uj) = s λj(a)λj(b)λj(c) ≥ (abc) a,b,c 1 ′ ′ X X a b c 1 λj e2 = , d2s (a′b′c)s d,e≥1 a′,b′,c≥1 e|(a′b′,c)

′ ′ ′ having written a = a d and b = b d and changing the order of summations. Writing e = e1e2, with a = e1α ′ and b = e2β and (e2, β) = 1, and c = γe, we then have

X 1 X X λ (αβγ) L(s, u )3 = j . (6.4) j (de)2s (αβγ)s d,e≥1 e=e1e2 α,β,γ≥1 (e2,β)=1

We can then detect the condition that (e2, β) = 1 using Mobius inversion; we have X δ((e2, β) = 1) = µ(f).

f|(e2,β)

′ ′ Writing e = e1e2 f and β = β f, and switching the order of the summations in 6.4, we arrive at

X µ(f) X λ (fαβ′γ) L(s, u )3 = j j ′ 2s ′ s ′ (de1e2 f) (fαβ γ) d,e1,e2 ,f≥1 α,β,γ≥1 X µ(f)τ (g) X τ (n)λ (fn) = 3 3 j . (fg)2s (fn)s f,g≥1 n≥1

87 6.3. The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points

6.3 The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points

Now we prove Theorem 6.0.1. Many of Young’s estimates and methods will be the same for our purposes, so we will summarize them rather than reproduce them when necessary.

6.3.0.1 Initial cleaning and preparation

1 By Theorem 3.1.1, at the conductor-dropping point sj = 2 + itj, for X > 0, the balanced form of the approximate functional equation for L(sj, uj ⊗ ψ) takes the form X 2 X λj(n)Aψ(m, n) m n λj(n)Aψ(m, n) 2 L(sj, uj ⊗ ψ) = s W1,s + ε(uj ⊗ ψ) W2,s m nX , (m2n) j j X 2 sj j m,n≥1 m,n≥1 (m n) (6.5) for any X > 0. The weight functions are given by Z 1 γ s + 1 + it , u ⊗ ψ ds W (ξ) = G(s) 2 j j ξ−s 1,sj 1 ⊗ 2πi (σ) γ 2 + itj, uj ψ s

and Z 1 e γ s + − itj, uj ⊗ ψ 1 2 −s ds W2,sj (ξ) = G(s) ξ , 2πi (σ) 1 − ⊗ e s γ 2 itj, uj ψ where G(s) is an even holomorphic function that has rapid decay in any vertical strip as =(s) → ∞ and 1+ε that satisﬁes G(0) = 1. The weight functions eﬀectively restrict the two sums to have essential length X 3 1+ε tj and X , respectively. Along the lines of [You13], by Stirling’s approximation the weight functions have expansions

 s   Z 3 3σ −1+ε 1 t 2 ds t 2 W (ξ) =  j  h (s) + O  j  (6.6) 1,sj 1 | | 2πi (σ) ξ s ξ     3 3σ −1+ε t 2 t 2 := V  j  + O  j  (6.7) 1 ξ |ξ| and

 s   Z 3 3σ −1+ε 1 t 2 ds t 2 W (ξ) =  j  h (s) + O  j  2,sj 2 | | 2πi (σ) ξ s ξ     3 3σ −1+ε t 2 t 2 := V  j  + O  j  , 2 ξ |ξ|

88 6.3. The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points

where h1 and h2 are entire functions of rapid decay in vertical strips. Next we reduce our analysis to the one problematic term appearing in Theorem 5.2.2. By positivity, we

introduce the short interval spectral weight function to the moment and split over uj even and odd; we have

T −M≤tj ≤T +M tj tj uj even uj odd

The treatment of the sum over uj odd is similar to uj even, so we will only describe the latter sum. By Cauchy-Schwarz and the approximate functional equation (6.5), we have

( ) ( ) 2 X t −T 2 X t −T 2 X 2 − j − j λj(n)Aψ(m, n) m n M | ⊗ |2 M e L(sj, uj ψ) e s W1,sj (6.8) (m2n) j X tj tj m,n≥1 u even u even j j ( ) 2 X − 2 X − tj T λ (n)A (m, n) M j ψ 2 + e W2,sj m nX . (6.9) (m2n)sj tj m,n≥1 uj even

The treatment of either sum in (6.8) is similar, so let us only consider the ﬁrst. Introducing a dyadic partition of unity, split the (m, n)-sum and apply Cauchy-Schwarz, yielding

( ) 2 X − 2 X − tj T λ (n)A (m, n) m2n M j ψ e s W1,sj (m2n) j X tj m,n≥1 u even j ( ) 2 X X t −T 2 X 2 2 − j λj(n)Aψ(m, n) m n m n M e s ω W1,sj , (m2n) j N X 3 +ε tj m,n≥1 N≪T 2 u even dyadic j

where ω is a smooth weight function with compact support in [1, 2].

Next, we need to remove the dependence of the weight function W1,sj on tj. By the asymptotic expansion (6.6), it suﬃces to expand to ﬁrst order (we could include lower order terms if needed); integrating the

89 6.3. The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points

resulting terms in X, we have   ( ) 2 X t −T 2 X 2 2 − j λj(n)Aψ(m, n) m n m n e M ω V   2 sj 1 3 (m n) N 2 tj m,n≥1 Xtj u even j   ( ) Z 2 X t −T 2 2 X 2 2 − j λj(n)Aψ(m, n) m n m n dX e M ω V   2 sj 1 3 1 (m n) N 2 X tj m,n≥1 Xtj u even j   ( ) Z 3 2 X t −T 2 2(T/tj ) 2 X 2 2 − j λj(n)Aψ(m, n) m n m n dX e M ω V   3 2 sj 1 3 (T/t ) 2 (m n) N 2 X tj j m,n≥1 Xtj u even j Z ( ) 2 2 X t −T 2 X 2 2 − j λj(n)Aψ(m, n) m n m n dX e M ω V 2 sj 1 3 1 (m n) N XT 2 X tj m,n≥1 uj even

3 2 where in the ﬁnal line we made the changes of variables X 7→ (T/tj) X. Similarly, for the dual sum we have   ( ) 2 X t −T 2 X 2 2 − j λj(n)Aψ(m, n) m n m nX e M ω V   2 sj 2 3 (m n) N 2 tj m,n≥1 tj u even j Z ( ) 2 2 X t −T 2 X 2 2 − j λj(n)Aψ(m, n) m n m nX dX e M ω V . 2 sj 2 3 1 (m n) N T 2 X tj m,n≥1 uj even

For both sums, by positivity we extend the sum to all uj, and then apply Cauchy-Schwarz: we consider

( ) 2 X − 2 X − tj T λ (n)A (m, n) m2n M j ψ e s ω (m2n) j P tj m,n≥1 u even j ( ) 2 X X t −T 2 X 2 1 − j λj(n)Aψ(m, n) m n log(T ) e M ω (6.10) m nsj P 1 +ε tj n≥1 ≪ 2 m P uj even

3 2 +ε P where ω is a smooth weight function with compact support in [1, 2] and P T . Write N = m2 for each 2 m and introduce the weights |νj(1)| , which by the bound of Hoﬀstein-Lockhart [HL94] satisfy

−ε ε tj νj(1) tj .

90 6.3. The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points

In order to apply Theorem 5.2.2, we split the sum over real and imaginary parts; by Cauchy-Schwarz,

( ) 2 X t −T 2 X − j νj(n)Aψ(m, n) n e M ω nsj N tj n≥1 u even j ( ) 2 X − 2 X − tj T < (A (m, n)) n M ψ e νj(n)ω nsj N tj n≥1 u even j ( ) 2 X − 2 X − tj T = (A (m, n)) n M ψ + e νj(n)ω . nsj N tj n≥1 uj even

Restricting the sum to n ≤ 2N is redundant to the support of ω. Finally, since Aψ(m, n) = Aψ(n, m), in the notation of Theorem 5.2.2, we are thus reduced to studying the sums

( ) 2 X − 2 X − tj T M ± itj ST,M (A±) = e a νj(n)n , n tj n≤2N uj even

with the sequence A± = A±(m) being given by A (m, n) A (n, m) n a± = ψ √ ψ ω . n n N

6.3.0.2 Application of GL(3) Voronoi summation

1 N 3 By Theorem 5.2.2, taking X = 1 (X in the notation of Theorem 5.2.2, not the variable of integration m 3 T ε 21 +ε X from earlier), for the range T 22 M T , we have ! 1 7 4 2 1 TN 3 N S A ε 3 2 3 kA k2 |S ˜bˆa`dﬄ A | T,M ( ±) T TM + N + N M m + 1 + T ± + T,M ( ±; M) , (6.11) m 3 M with

Z 2 X X M X 1 1 T 1+ε Aψ(m, n) Aψ(m, n) n un S ˜bˆa`dﬄ A √ T,M ( ; M) M ω S(k, n; r)e du. r |k| − M n N rM r

3 +ε 2 21 P T 22 +ε Note that the other terms in (6.11) are all admissible, since N = m2 m2 and M T . We use the S ˜bˆa`dﬄ same analysis as in [You13] to show that the contribution from T,M is actually negligible.

91 6.3. The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points

Lemma 6.3.1. We have ! 1 3 N −ε −A S ˜bˆa`dﬄ A±; T T T,M m

3 +ε for any A > 0 and N T 2 .

Assuming Lemma 6.3.1, let us wrap up the proof of Theorem 6.0.1. By the Rankin-Selberg bound, we have X | |2 X | |2 2 Aψ(m, n) Aψ(n, m) Aψ(m, n) 1+ε kA±k = mN . n n n≤N n≤N Returning to the sum (6.10), we see that X 2 1+ε |L(sj, uj ⊗ ψ)| T M.

T −M≤tj ≤T +M

This completes the proof of Theorem 6.0.1. □

Proof of Lemma 6.3.1. Split the sum over by Cauchy-Schwarz and denote the resulting innermost sum of (6.12) inside the square by X A (m, n) n un B (N, u; ψ) = ψ√ ω S(k, n; r)e . M n N rM n≤N

Opening the Kloosterman sum and applying GL(3) Voronoi summation (Theorem 2.2.1), we have X A (m, n) n un ψ√ ω S(k, n; r)e n N rM n≤N X∗ bk X A (m, n) bn un n = e ψ√ e e ω r n r rM N b(r) n≤N X∗ X X X 2 bk Aψ(n2, n1) mr ± n2n1 = r e S bm, ηn2; Φψ∗ 3 , r ± n1n2 n1 r m b(r) n1|rm n2≥1 where the integral transform is

Z ∞ ± ∓ 1 3 Φψ∗ (ξ) = φ(y)Jψ∗ 2π (ξy) dy. 0

n n2 ∓ 2 1 ε For ξ = r3m , we have ξN T , so we can apply the asymptotic expansion (2.14) of the Bessel kernel Jψ∗ . ± We will only consider the ﬁrst term in the expansion of Φψ∗ (ξ); the analysis of the lower order terms is similar. We are reduced to analyzing Z 1 ∞ 3 e 3(ξy) uy y 1 1 e ω √ dy. (6.13) 0 2π(ξy) 3 rM N y

92 6.4. The shortened spectral interval sixth moment of GL(2) L-functions at the special point

The phase function 1 2πuy f(y) = 3π(ξy) 3 + rM has a stationary point at 1 3 ξ 2 (rM) 2 y0 = 3 . |u| 2 Since the weight function in (6.13) has support in [N, 2N], by Theorem B.0.2 the integral transform will be 3 M 2 +ε 2 negligibly small unless y0 N. We have u T 1+ε and N T m , so (6.13) is only not negligible when

N 2|u|3 1 ξ , r3M 3 r3m4T ε

2 1 and hence n2n1 m3T ε . Thus ! 1 3 N −ε −A S ˜bˆa`dﬄ A±; T T T,M m

for any A > 0. This completes the proof of Lemma 6.3.1.

6.4 The shortened spectral interval sixth moment of GL(2) L- functions at the special point

Proof of Theorem 6.0.2. Let X > 0. By Proposition 6.2.1 and Theorem 3.1.1, for uj even, the approximate 3 functional equation for L(sj, uj) takes the form X µ(f)τ (g) X τ (n)λ (fn) f 3g2n 3 −3itj 3 3 j L(sj, uj) = π V (fg)2sj (fn)sj X f,g≥1 n≥1 X µ(f)τ (g) X τ (n)λ (fn) 3itj 3 3 j ∗ 3 2 + π Vj f g nX , (fg)2sj (fn)sj f,g≥1 n≥1 where the weight functions are 1 1 Z s+ 2 s+ 2 +2itj 1 Γ 2 Γ 2 ds V (ξ) = G(s) ξ−s j 1 +2it 2πi (σ) 1 2 j s Γ 4 Γ 2

and 1 1 Z s+ 2 s+ 2 +2itj 1 Γ 2 Γ 2 ds V ∗(ξ) = G(s) ξ−s , j 1 −2it 2πi (σ) 1 2 j s Γ 4 Γ 2 where G(s) is an even holomorphic function satisfying G(0) = 1 and rapidly decaying in vertical strips as

=(s) → ∞. The approximate functional equation for uj odd is similar.

93 6.4. The shortened spectral interval sixth moment of GL(2) L-functions at the special point

Following the same steps as in the proof of Theorem 6.0.1, we see that

X X 1 |L(s , u )|6 T ε H (f), j j fg N T −M≤t ≤T +M 1 +ε j f,g≪L 2

where the HN (f) are sums of the form

( ) 2 X − 2 X − tj T M 2 itj HN (f) = e |νj(1)| an,f λj(n)n ,

tj N

3 L 2 +ε where N = f 2g2 , L T , and   √1 n n | n τ3 f ω N if f n, an,f =  0 otherwise,

and ω is a smooth weight function with compact support in [1, 2]. By Theorem 5.2.2, we ﬁnd that

2 Z − 2 X X T ε X 1+ε M 1 1 un HN (f) T M + an,f S(k, n; r)e du. (6.14) T r |k| − −ε rT r

To estimate the bad term, we will use the GL(3)-type Voronoi summation provided in Theorem 2.2.3. Opening the sum inside the square in (6.14), we have X un 1 X τ (n) ufn nf a S(k, n; r)e = √ 3√ S (k, nf; r) e ω n,f rT f n rT N n≤N n≤ N f 1 X∗ ka X τ (n) anf ufn nf = √ e 3√ e e ω f r n r rT N a(r) n≤ N f 1 X∗ ka X τ (n) anf ′ uf ′n nf = √ e 3√ e e ω f r n r′ r′T N a(r) n≤ N f 1 X∗ ka X anf ′ uf ′n nf = √ e τ3 (n) e e ω1 N r r′ r′T N a(r) ≤ N n f q ′ r ′ f nf N nf where we set r = (f,r) and f = (f,r) and ω1 N = nf ω N Applying Theorem 2.2.3 to the innermost

94 6.4. The shortened spectral interval sixth moment of GL(2) L-functions at the special point

sum then yields X anf ′ uf ′n nf τ (n) e e ω 3 r′ r′T 1 N n≤ N f r′ X X X 1 X X n r′ mn2 = σ , m S m, a; Φ± 3 0,0 E3 ′3 2 2π ± ′ nm n n1n2 n r n|r m≥1 n1|n n2| n1 ′ ′ ′ + ∆0(a, r ; φ) + ∆1(a, r ; φ) + ∆2(a, r ; φ).

The analysis of the integral transform Φ± is exactly the same as in Section 6.3, so we will not repeat it here. E3 We need to analyze the main terms that appear after applying Theorem 2.2.3. The analysis of each of these is largely the same, so let us focus on the ﬁrst term, which is given by 1 X nτ (n) r ∆ (a, r; φ) = 2 S 0, a; T (n, r)φe(1), 0 2 r2 n 3,0 n|r

where

2 9 2 2 T3,0(n, r) = (log n) − 5 (log r) (log r) + (log r) + 3γ + 7γ log n − 9γ log r − 3γ1 2 1 X 3 + log(d) (log(nr) − 5γ) − (log d)2 . τ2(n) 2 d|n

Note that Z Z ∞ ξf ufξ N ω (ξ) uNξ N uN φe(1) = ω1 e dξ = √ e dξ = ωb . 0 N rT f R ξ rT f rT Making a trivial bound, we then see that this term contributes

2 Z −ε 2 X T X X∗ X M N 1 1 ak nτ2(n) r b uN 2 e 2 S 0, a; T3,0(n, r)ω du T f r − −ε |k| r r n rT r

2 Z −ε 2 ε X T X X 2 X∗ 2 M T 1 1 1 n ak r b uN 2 e S 0, a; ω du. TN r − −ε |k| r r r n rT r

We have Z − Z T ε 2 2 uN uN rT ωb du ωb du , −T −ε rT R rT N and splitting into T ε complete sums modulo r, we have

2 X X∗ 1 ak r 1+ε ε e S 0, a; r T . |k| r n 0≠ |k|≪rT ε a(r) 2 ε Combining these estimates, it follows that the full contribution from the term ∆0(a, r; φ) is O M T . This

95 6.4. The shortened spectral interval sixth moment of GL(2) L-functions at the special point completes the proof of Theorem 6.0.2.

96 Part III

Asymptotic expansions of moments of L-functions at special points

In this part we will prove the results (Theorem 1.0.1, Theorem 1.0.2) regarding the relative number of

cusp forms in Weyl’s law for the congruence surfaces Γ0(p) \ H that are destroyed under quasiconformal

deformation, where p ≥ 5 is prime. In this case, Γ0(p) \ H is a noncompact Riemann surface of genus    p − 1 if p ≡ 1 (mod 12),  12 (Γ (p) \ H) = p ≡ `gé›n˚u¯s 0  12 if p 5, 7 (mod 12),   p ≡ 12 + 1 if p 11 (mod 12), with two inequivalent cusps at 0 and ∞, and the weight 4 cuspidal holomorphic modular forms form a vector space with j k p (S (Γ (p) \ H)) = . `d˚i‹mffl 4 0 4 \H \H As outlined in Chapter 4, the space S4(Γ0(p) ) generates a basis for Tèˇi`c‚hffl(Γ0(p) ), and the condition

that L(sj, uj ⊗ Q) =6 0 determines whether the cusp form uj is destroyed under quasiconformal deformations \ H \ H of Γ0(p) in the initial directions in T`eˇi`c‚hﬄ(Γ0(p) ) generated from Q. Phillips-Sarnak ([PS85b]) computationally showed that the Rankin-Selberg L-function√ is nonzero for a particular class of uj and Q, namely, those uj arising as lifts of Hecke characters of Q( 2). Deshouillers- Iwaniec took an alternative approach to addressing the nonvanishing of these L-functions. Instead of focusing on whether an individual L-function is nonzero, they showed that a large number of the L-functions with spectral parameter up to height T are nonzero.

Theorem III.1 (Deshouillers-Iwaniec, [DI86]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms Z ∈ ”n`e›w \ H for SL(2, ), and let Q Sk (Γ0(p) ). Then X −tj /T 2 2 2 2 e |νj(1)| |L(sj, uj ⊗ Q)| = c1T log T + O T log log T . (6.15)

97 Assuming the Ramanujan-Petersson conjecture, it follows that

9 −ε #{tj ≤ T : L(sj, uj ⊗ Q) =6 0} T 8 .

In [Luo93], Luo substantially improved the lower bound on the order of nonvanishing of L(sj, uj ⊗ Q) and removed the assumption of the Ramanujan-Petersson conjecture. He obtained an asymptotic expansion of the (smoothed) ﬁrst moment of GL(2) × GL(2) L-functions at the special point with a power-saving error term.

Theorem III.2 (Luo, [Luo93]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z), ∈ ”n`e›w \ H and let Q Sk (Γ0(p) ). Then X 7 −tj /T 2 2 +ε e |νj(1)| L(sj, uj ⊗ Q) = c1T + O T 4 . (6.16)

Applying Cauchy-Schwarz and the second moment result of Deshouillers-Iwaniec 6.152, he showed that for any ε > 0, one has 2−ε #{tj ≤ T : L(sj, uj ⊗ Q) =6 0} T .

This is the nearly optimal order predicted by Phillips-Sarnak, but because of the second moment estimate one cannot remove the ε in the exponent. The purpose of studying both the first and second moments is that if one obtains essentially the same order of magnitude for the asymptotic expansion of the first moment as the upper bound for the second moment, then by Cauchy-Schwarz one can translate this to a statement regarding the number of values in question that are nonzero. Later, in [Luo01] Luo studied the mollified moments of the same L-functions to delicately improve the order of nonvanishing to 2 #{tj ≤ T : L(sj, uj ⊗ Q) =6 0} T .

Thus, under the assumption of the multiplicity conjecture one sees that the conjecture of Selberg is false

for generic subgroups of SL(2, R); a positive proportion of the cuspidal spectrum of Γ0(p) \ H is destroyed under quasiconformal deformation in the Teichmüller directions induced by Q. In [Luo12], improved the second moment asymptotic expansion of Deshouillers-Iwaniec to a power-saving error term.

2One actually only needs an upper bound for the second moment; the result of Deshouillers-Iwaniec is stronger.

98 Theorem III.3 (Luo, [Luo12]). Let {uj} be an orthonormal basis of Hecke-Maass cusp forms for SL(2, Z), ∈ ”n`e›w \ H and let Q Sk (Γ0(p) ). Then X 11 −tj /T 2 2 2 2 +ε e |νj(1)| |L(sj, uj ⊗ Q)| = c1T log T + c2T + O T 6 ,

tj where c1 and c2 are constants that depend only on Q.

Additional work by Xu on the nonvanishing of L-functions at the special point sj appears in [Xu14]. Xu

established short interval analogues of Luo’s molliﬁed moments results for the GL(2) L-functions L(sj, uj); namely, he showed that 2 #{tj ≤ T : L(sj, uj) =6 0 and uj is even} T .

One important diﬀerence, as we have previously observed in Chapter 5, between working over the long interval versus the short interval, is that the integral transforms appearing in the Kuznetsov trace formula are easily made explicit in the long interval case. In contrast, for the short interval one must analyze the integral transforms appearing in the Kuznetsov trace formula delicately. In Xu’s case, this includes dealing with the integral transform involving the K-Bessel function, for which the asymptotics are made more diﬃcult since the behaviour of the K-Bessel function is unclear in the so-called transition range. This issue is present since

the weight functions in the approximate functional equation for L(sj, uj) are deﬁned diﬀerently depending

on whether uj is even or odd, so one must use the form of the Kuznetsov trace formula that restricts the spectral sums to the appropriate forms.

99 Chapter 7

The nonvanishing of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Contents 7.1 Properties of the Rankin-Selberg L-function ...... 102 7.2 An asymptotic expansion of the ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals ...... 103 7.2.1 Proof of Theorem 7.0.2 ...... 103

7.2.2 Proof of Lemma 7.2.1: contribution from the sum Σ1(T,M,N; Q) ...... 104

`d˚ià`gó“nà˜l 7.2.2.1 Contribution to Σ1(T,M,N; Q) of M ...... 105

7.2.2.2 Contribution to Σ1(T,M,N; Q) of ME˚i¯sfiè›n¯sfi˚téˇi‹nffl ...... 105 `géó“mèˇtˇr˚i`c 7.2.2.3 Contribution to Σ1(T,M,N; Q) of M ...... 106 7.2.2.3.1 Contribution from small L: application of the Hecke bound ... 106 7.2.2.3.2 Contribution from large L: application of GL(2) Voronoi summation ...... 107

7.2.3 Proof of Lemma 7.2.2: contribution from the dual sum Σ2(T,M,N; Q) ...... 110

`d˚ià`gó“nà˜l 7.2.3.1 Contribution to Σ2(T,M,N; Q) of M ...... 110 `d˚uà˜l 7.2.3.2 Contribution to Σ2(T,M,N; Q) of ME˚i¯sfiè›n¯sfi˚téˇi‹nffl ...... 111 `d˚uà˜l `géó“mèˇtˇr˚i`c 7.2.3.3 Contribution to Σ2(T,M,N; Q) of M ...... 112 `d˚uà˜l 7.2.3.3.1 Contribution of small c to M`géó“mèˇtˇr˚i`c ...... 112 `d˚uà˜l 7.2.3.3.2 Contribution of large c to M`géó“mèˇtˇr˚i`c ...... 113 `d˚uà˜l

In this chapter we will establish the lower bound on the order of nonvanishing of the special values

L(sj, uj ⊗ Q) over shortened spectral intervals.

100 ≥ ∈ N ∈ ”n`e›w \ Theorem 7.0.1 (Restatement of Theorem 1.0.1). Let p 5 be prime, k even, and let Q Sk (Γ0(p) 1 +ε H). Then for T 2 M T , we have

1−ε # {T − M ≤ tj ≤ T + M : L(sj, uj ⊗ Q) =6 0} T M. (7.1)

Theorem 7.0.1 is implied by the following ﬁrst moment asymptotic expansion combined with an upper bound for the second moment.

≥ ∈ N ∈ ”n`e›w \ Theorem 7.0.2 (Restatement of Theorem 1.0.6). Let p 5 be prime, k even, and let Q Sk (Γ0(p) 1 +ε H). Then for T 2 M T , we have

( ) X t −T 2 − j 2TM 3 M | |2 ⊗ 2 +ε e νj(1) L(sj, uj Q) = 3 + O T . (7.2) π 2 tj

Remark 7.0.1. In Luo’s result, the power-saving error term suggests that one can reduce the length of the 3 +ε 1 +ε spectral interval to M T 4 via a Tauberian argument. That we are able to improve this to M T 2 is a consequence of our use of GL(2)-Voronoi at a critical step, instead of applying the Hecke bound.

The proof of Theorem 7.0.2 occupies the majority of this chapter. Let us ﬁrst provide a short proof of Theorem 7.0.1.

Proof of Theorem 7.0.1. By Theorem 7.0.2, the triangle inequality, and Hölder’s inequality, we have the lower bound X 2 TM |νj(1)| |L(sj, uj ⊗ Q)| (7.3)

T −M log T ≤tj ≤T +M log T   1 2   1  X  X 2  4  2  |νj(1)|  |L(sj, uj ⊗ Q)| . (7.4)

T −M log T ≤tj ≤T +M log T T −M log T ≤tj ≤T +M log T L(sj ,uj ⊗Q)=0̸

As mentioned in the introduction of Part II, Luo’s twisted large sieve inequality implies the optimal upper bound X 2 1+ε |L(sj, uj ⊗ Q)| T M.

T −M≤tj ≤T +M −ε ε Combining these with (7.3), and by Hoﬀstein-Lockhart’s bound tj νj(1) tj , it follows that

1−ε # {T − M log T ≤ tj ≤ T + M log T : L(sj, uj ⊗ Q) =6 0} T M.

101 7.1. Properties of the Rankin-Selberg L-function

7.1 Properties of the Rankin-Selberg L-function

Let Q ∈ Sk(Γ0(p) \ H) be a fixed Hecke newform of level p with trivial nebentypus, and let Q be normalized so that its first Fourier coefficient is νQ(1) = 1. Then νQ(n) = λQ(n) for (n, p) = 1, and Q has the Fourier expansion X k−1 Q(z) = λQ(n)n 2 e(nz). n≥1 As in [AL70], the Atkin-Lehner operator ! 1 Wp = : S”nè›w(Γ0(p)) −→ S”nè›w(Γ0(p)) p k k is an endomorphism, since the nebentypus is trivial. Additionally, Q is an eigenfunction of W , and it has √ p Atkin-Lehner eigenvalue ε(Q) = −λQ(p) p.

The Rankin-Selberg L-function associated to an SL(2, Z) Maass cusp form uj and Q is deﬁned by the Dirichlet series ε(Q) X λ (n)λ (n) L(s, u ⊗ Q) = 1 − ζ(2s) j Q j p2s ns n≥1 X α (ℓ) = j,Q , (7.5) ℓs ℓ≥1 for <(s) > 1, where we have denoted the coeﬃcients by X v αj,Q(ℓ) = (−ε(Q)) λj(n)λQ(n). pv m2n=ℓ v=0,1

The completed L-function √ p 2s Λ(s, u ⊗ Q) = Γ(k + s − s )Γ(k + s − s )L(s, u ⊗ Q) j 2π j j j satisﬁes the functional equation

Λ(s, uj ⊗ Q) = Λ(1 − s, uj ⊗ Q), (7.6) whence L(s, uj ⊗ Q) extends to an entire function of C. Crucially, because the root number is 1 for GL(2) ×

GL(2) L-functions, we do not need to treat the even and odd uj separately.

102 7.2. An asymptotic expansion of the ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals 7.2 An asymptotic expansion of the ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

In this section we will derive an asymptotic expansion of the short interval ﬁrst moment of GL(2) × GL(2)

∈ ”n`e›w \ H L-functions at the special point. For Q Sk (Γ0(p) ), we consider the moments of the form ( ) X − 2 − tj T M 2 e |νj(1)| L(sj, uj ⊗ Q). (7.7)

7.2.1 Proof of Theorem 7.0.2

1 The unbalanced form of the approximate functional equation for L(sj, uj ⊗ Q) takes the form X X v 2 ⊗ − v λj(n)λQ(n) p m n L(sj, uj Q) = ( ε(Q)) 1 H v 2 2 +itj N v=0,1 m,n≥1 (p m n) X X − v λj(n)λQ(n) 2 v 2 + ( ε(Q)) 1 − Ptj π p m nN , (7.8) v 2 2 itj v=0,1 m,n≥1 (p m n) where N = T 1+θ for some 0 < θ < 1, and the weight functions are Z 1 Γ(k + s) ds H(x) = x−s (7.9) 2πi (σ) Γ(k) s and Z 1 Γ(k + s) Γ(k + s − 2it ) ds P (x) = j x−s . (7.10) tj − 2πi (σ) Γ(k) Γ(k s + 2itj) s

Applying (7.8) to (7.7), we have

M¯sfi˛hò˘r˚t,1 (GL(2)×GL(2),Q) T,M X − v X v 2 X ( ε(Q)) λQ(n) p m n −itj = √ H à˚u¯sfi¯sfi˚ià‹nffl(t )ν (n)ν (1) pvm2n pv/2 m n N GT,M j j j v=0,1 m,n≥1 tj X − v X X ( ε(Q)) λQ(n) itj + √ à˚u¯sfi¯sfi˚ià‹nffl(t )ν (n)ν (1) pvm2n P π2pvm2nN pv/2 m n GT,M j j j tj v=0,1 m,n≥1 tj

= Σ1(T,M,N; Q) + Σ2(T,M,N; Q).

We will handle the two pieces Σ1(T,M,N; Q) and Σ2(T,M,N; Q) separately; the bounds on these are collected below. Their proofs begin shortly.

1Remark: it is important that the root number is 1 for GL(2) × GL(2) by result of Ramakrishnan, so that we can treat the cases where uj is even or odd identically, rather than needing to use the two diﬀerent versions of Kuznetsov in that case.

103 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Lemma 7.2.1. In the notation above, we have the asymptotic expansion ! 5 +ε 2 1 T 2 3 +ε Σ1(T,M,N; Q) = 3 TM + O T M + 2 . π 2 M

Lemma 7.2.2. In the notation above, the dual sum satisﬁes

7 +ε T 2 Σ (T,M,N; Q) . 2 N

2−ε 1 +ε Theorem 7.0.2 follows upon taking N = T and restricting M T 2 .

7.2.2 Proof of Lemma 7.2.1: contribution from the sum Σ1(T,M,N; Q)

For the first sum Σ1(T,M,N; Q), in order to apply Kuznetsov trace formula, we exchange the weight function −itj à˚u¯sfi¯sfi˚ià‹nffl v 2 GT,M (tj) p m n for the even weight function

( ) ( ) t −T 2 t +T 2 − j −it − j it v 2 M v 2 j M v 2 j hT,M (tj, 1, p m n) = e p m n + e p m n ,

at the cost of a negligible error. We have X Z v 2 δ(n = 1) v 2 hT,M (tj, 1, p m n)νj(n)νj(1) = 2 t tanh(πt)hT,M (t, 1, p m )dt (7.11) π R t j Z 2 ∞ h (t, 1, pvm2n) − T,M τ (n)dt (7.12) | |2 it π 0 ζ(1 + 2it) X Z √ 1 2i 4π n v 2 tdt + S(n, 1; c) J2it hT,M (t, 1, p m n) , c π R c cosh(πt) c≥1 (7.13) where again Accordingly, we expand Σ1(T,M,N; Q) as `d˚ià`gó“nà˜l `géó“mèˇtˇr˚i`c −A Σ1(T,M,N; Q) = M − ME˚i¯sfiè›n¯sfi˚téˇi‹nffl + M + O T , for each of the terms (7.11), (7.12), and (7.13), respectively.

104 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

`d˚ià`gó“nà˜l 7.2.2.1 Contribution to Σ1(T,M,N; Q) of M

For the diagonal term M`d˚ià`gó“nà˜l, a straightforward computation yields Z √ 2 v 2 v 2 −iT − 1 (M log(pv m2)) 3 v 2 −A t tanh(πt)hT,M (t, 1, p m )dt = π p m e 4 2TM − iM log p m + O T . R

Accordingly, the overall contribution is Z X 2 λQ(1) 1 m M`d˚ià`gó“nà˜l v 2 = 2 H t tanh(πt)hT,M (t, 1, p m )dt π m N R m≥1 2 −A = 3 TM + O T , π 2 with the main term coming from v = 0 and m = 1.

7.2.2.2 Contribution to Σ1(T,M,N; Q) of ME˚i¯sfiè›n¯sfi˚téˇi‹nffl

To deal with the contribution from the continuous spectrum, one may either apply a mean-value estimate of Deshouillers-Iwaniec under the assumption of square-root cancellation of the coeﬃcients A described in

Chapter 5, or the factorization special factorization of L(s, Et ⊗ Q). The main difference between this situation and the large sieve problem is that the coefficients A have extra features that yield additional cancellation. Consequently there is no need to carefully extract a large contribution from the continuous contribution. Because we later wish bound the continuous spectrum contribution for sequences that do not have the full strength of square-root cancellation, we provide an alternative argument that makes use of the factorization of L(s, Et ⊗ Q). Switching the order of the sum and the integral, and then using the definition of the weight function H, we see that

Z t−T 2 − X v X v 2 ∞ −( ) v 2 it 2 (−ε(Q)) λQ(n) p m n e M p m n ME˚i¯sfiè›n¯sfi˚téˇi‹nffl = H τ (n)dt v/2 1 2 it π p v 2 2 N |ζ(1 + 2it)| v=0,1 m,n≥1 (p m n) 0 + O T −A

Z ∞ − t−T 2 Z 2 e ( M ) 1 1 ds = γ (s)L + s + it, E ⊗ Q N s dt + O T −A | |2 k t π 0 ζ(1 + 2it) 2πi (σ) 2 s Z ∞ − t−T 2 Z 2 e ( M ) 1 1 1 ds = γ (s)L + s, Q L + s + 2it, Q N s dt. | |2 k π 0 ζ(1 + 2it) 2πi (σ) 2 2 s

Since Q is cuspidal, L(s, Q) is entire. Thus, shifting the contour integral to <(s) = −σ, we pick up a residue at s = 0 that contributes a lower order term, and by the t-aspect subconvexity bound of Meurmann, and 1 | | the estimate log |τ| ζ(1 + iτ) log τ on the edge of the critical strip, we may bound it by Z ∞ − t−T 2 1 ( M ) 2L 2 ,Q e 1 1 +ε L + 2it, Q dt T 3 M. | |2 k,p π 0 ζ(1 + 2it) 2

105 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

The remaining contour integral can be estimated trivially.

`géó“mèˇtˇr˚i`c 7.2.2.3 Contribution to Σ1(T,M,N; Q) of M

It remains to bound the contribution from Kloosterman sums, M`géó“mèˇtˇr˚i`c. First we use a dyadic partition of unity to break the (m, n)-sum into pieces of length L N 1+ε, since we can ignore the pieces of larger length because of the decay of the weight function in the approximate functional equation. It suffices to consider sums of the form X v X 2 v 2 X (−ε(Q)) λQ(n) m n p m n 1 M`géó“mèˇtˇr˚i`c(L) := √ ω H S(n, 1; c) v/2 p ≥ m n L N ≥ c v=0,1 Z m,n√1 c 1 2i 4π n v 2 tdt × J2it hT,M t, 1, p m n . π R c cosh(πt)

LT ε LT ε By Proposition B.0.5, the Titchmarsh coeﬃcients are negligibly small for c mT ; that is, for c mT we Z √ have 4π n v 2 tdt −A J2it hT,M t, 1, p m n T R c cosh(πt) for any arbitrary A > 0. We now separate into two cases: when L T 1+ε and when N 1+ε L T 1+ε. In the latter case, we will apply Voronoi summation to show the sum is actually negligible. For the former, we will apply the 1 +ε Hecke bound to get an acceptable error term for M T 2 .

7.2.2.3.1 Contribution from small L: application of the Hecke bound

Suppose that L T 1+ε. In this case the c-sum is very short and will not play a role. Applying Proposition B.0.4 with √ 4π n 1 x = L 2 T, c for this range of c, the Titchmarsh coeﬃcients have the expansion Z √ 4π n v 2 tdt J2it hT,M t, 1, p m n R c cosh(πt)     2 2 π(m n−1) π m2n − 1 m2n + 1  − T  T 2+ε = e − exp −  cm   + O (7.14) cm cm M M 2 for m2n > 1. By the Weil bound, the contribution from the error term is bounded by

2+ε X X 2+ε 1 5 +ε T |λ (n)| |S(n, 1; c)| T L 2 T 2 Q√ , M 2 m n c M 2 M 2 m2n≪LT ε ≪ LT ε c mT

106 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

1 +ε which will be a power-saving error term provided that M T 2 . The contribution from the main term, on the other hand, is2 X (−ε(Q))v X λ (n) m2n M`géó“mèˇtˇr˚i`c(L) = π Q√ H ”mà˚i‹nffl pv/2 m n N v=0,1 m,n≥1 ! X S(n, 1; c) m2n + 1 m2n − 1 m2n π m2n − 1 × e − ω à˚u¯sfi¯sfi˚ià‹nffl . c cm cm L GT,M cm ≪ LT ε c T

Making use of the Fourier inversion Z − 2 √ − x T 3 −(uMπ)2 xe ( M ) = π TM − iπuM e(−uT )e e(xu)du, R and opening the Kloosterman sum, we see that Z X v X X −1 √ (−ε(Q)) 1 2 e −πξ M`géó“mèˇtˇr˚i`c(L) = π TM − iπuM 3 e(−uT )e−(uMπ) cm e ”mà˚i‹nffl v/2 p ε c R m cm v=0,1 c≪ LT m≥1 T X λ (n) −mn πmnξ m2n m2n × Q√ S(n, 1; c)e e ω H dξ n c c L N n≥1 Z X v X X −1 √ (−ε(Q)) 1 2 e −πξ = π TM − iπuM 3 e(−uT )e−(uMπ) cm e v/2 p ε c R m cm v=0,1 c≪ LT m≥1  T    X∗ a X λ (n) an −mn πmnξ m2n m2n × e Q√ e e e ω H dξ. c  n c c c L N  a(c) n≥1

Applying the Hecke bound to the innermost sum and making a trivial bound on everything else, these terms satisfy M`géó“mèˇtˇr˚i`c 1+ε ”mà˚i‹nffl (L) T .

7.2.2.3.2 Contribution from large L: application of GL(2) Voronoi summation

Suppose that N 1+ε L T 1+ε. In this case, rather than explicitly evaluating the integral transform, we ﬁrst apply GL(2) Voronoi summation to the n-sum. Switching the order of the sum and the integral and

2We include that m2n = 1 contribution at no cost, since the main term there is zero, and the actual contribution is trivially 1 bounded by N 2 MT ε.

107 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

opening the Kloosterman sums, we have 2i X (−ε(Q))v X 1X∗ a X 1 M`géó“mèˇtˇr˚i`c(L) = e ”mà˚i‹nffl v/2 π p ε c c m v=0,1 c≪ LT a(c) m≥1 Z T √ X 2 v 2 λQ(n) an m n p m n 4π n v 2 tdt × √ e ω H J2it hT,M t, 1, p m n R n c L N c cosh(πt) n≥1 Z ∞ X X − v X X∗ X − 2 2i ( ε(Q)) 1 1 − t T v 2 ∓it = e ( M ) p m v/2 π p ε c m 0 ± v=0,1 c≪ LT a(c) m≥1  T  √ X 2 v 2  × λQ(n) an 4π n m n p m n tdt −A 1 ± e J±2it ω H + O T .  2 it c c L N  cosh(πt) n≥1 n

In the second line we used the evenness of hT,M and the rapid decay of the Gaussian to get the negligible error term. Consider the innermost sum. Applying GL(2) Voronoi summation (Theorem 2.1.6), we have √ X 2 v 2 k X λQ(n) an 4π n m n p m n 2πi ηQ(p2) ap2r r e J± ω H = √ λ ∗ (r)e − Φ , 1 ±it 2it Q Q 2 2 c c L N c p2 c c p2 n≥1 n r≥1

p where p2 = (c,p) and ηQ(p2) = 1 if p2 = 1 and ηQ(p2) = ε(Q) if p2 = p, and where the weight function on the righthand side is given by Z r √ ∞ 2 v 2 r + r 4π rx − 1 ∓it 4π x m x p m x − 2 ± ΦQ 2 = ΦQ 2 = Jk 1 x J 2it ω H dx. c p2 c p2 0 c p2 c L N

For r ≥ 1, we have r 1−ε 1−ε 4π rx T T ε 1 1 T , c p2 L 2 N 2 provided that N T 2−ε, say. Since k is ﬁxed, we expand the k-order J-Bessel function according to (A.8) as iy −iy Jk−1(y) = e Wk−1(y) + e Wk−1(y),

where Wk−1(y) is the Watson-Whittaker function, which has the integral representation

iπk iπ 1 Z − 1 − 2 ∞ k 2 e 2 4 2 −u iu Wk(y) = 1 e u 1 + du, Γ k + 2 πy 0 2y

(j) 1 and which for y 1 satisﬁes Wk−1(y) k,j 1 . yj (1+y) 2 For t 1 we also use the uniform asymptotic expansion (A.11) (reproduced below) to expand the

imaginary order J-Bessel function J2it(y) according to   πr − iπ +iω (y) X e 2 4 r t − L+1 √  ℓ 2 2 2  Jir(y) = 1 ℓ + O r + x , 2 2 4 2π (r + y ) 0≤ℓ≤L (r2 + y2) 2

108 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

where the phase is given by p ! p r2 + y2 − r ω (y) = r2 + y2 + r log . r y

For J−2it(y), we can use the relation Jν (z) = Jν (z), so in the discussion that follows we will only consider the

contribution from J2it(y) as the analysis of the other is similar. For our purposes, it will suﬃce to consider the ℓ = 0 in (A.11), since the other terms are similar but smaller in magnitude. Thus we focus our study on the integral transform

Z ∞ r − πi πt 4 iψ±(x) ΦQ 2 = e ρ(x)e dx. (7.15) c p2 0 where the weight function is

r !− 1 2 4 2 v 2 4π rx 2 4π m x p m x ρ(x) = ρ(x; t) = Wk−1 4t + x ω H c p2 c L N

and the phase function is √ r 4π x 4π rx ψ±(x) = ψ±(x; t) = ω2t − t log x . c c p2

We record the properties of Φ0 in the following.

ε − 1 LT 1+ε 2 ε 2 Lemma 7.2.3. Let c mT with N L T , N T , and m L . Then we have r −A ΦQ 2 T c p2

for any A > 0.

Proof of Lemma 7.2.3. Note that   s 1 4π 2 4π √ ψ′ (x) =  4t2 + x − 2t rxp  . ± 2x c c 2

p ′ ′ 1 r √1 For ψ , note that ψ (x) ≥ for all x > 0. For ψ−, note that for r ≥ p2, we have |ψ−(x)| ≥ . + + c x √ c x 2 2−ε 1+ p | − | ≥ √ For r < p2, as long as m x L T , we have ψ (x) c x . The higher order derivatives satisfy 1 (j) r 2 2 ψ± (x) j j− 1 for m x L. cL 2 (j) 1 j+1 On the other hand, the weight function satisﬁes ρ (x) j c 2 L . Thus by Theorem B.0.2, it follows r that ΦQ 2 is negligible. c p2

109 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

7.2.3 Proof of Lemma 7.2.2: contribution from the dual sum Σ2(T,M,N; Q)

Next we will bound the contribution from the dual sum

Σ2(T,M,N; Q) ( ) v − 2 X − X X tj T ( ε(Q)) λQ(n) − itj = √ e M ν (n)ν (1) m2n P π2pvm2nN pv/2 m n j j tj v=0,1 ≥ t m,n 1 j Z X v X (−ε(Q)) λ (n) 1 −s = Q√ γ (s) (2π)4pvm2nN v/2 k p ≥ m n 2πi (σ) v=0,1 m,n 1   ( ) X t −T 2 − j it Γ(k + s − 2it ) ds  M v 2 j j  × e νj(n)νj(1) p m n . (7.16) Γ(k − s + 2itj) s tj

In order to apply the Kuznetsov trace formula, at the cost of a negligible error we exchange the weight function in the spectral sum for the even weight function

( ) ( ) − 2 2 tj T − tj +T − itj Γ(k + s 2it ) − −itj Γ(k + s + 2it ) `d˚u`a˜l v 2 M v 2 j M v 2 j hT,M (t; p m n, 1; s) = e p m n + e p m n . Γ(k − s + 2itj) Γ(k − s − 2itj)

Applying Kuznetsov to the spectral sum yields Z X δ(n = 1) `d˚uà˜l v 2 `d˚uà˜l v 2 νj(n)νj(1)hT,M (tj; p m n, 1; s) = 2 t tanh(πt)hT,M (t; p m , 1; s)dt (7.17) π R tj Z ∞ 2 h`d˚uà˜l (t; pvm2n, 1; s) − T,M τ (n)dt (7.18) π |ζ(1 + 2it)|2 it 0 Z √ X S(n, 1; c) 2i 4π n tdt `d˚uà˜l v 2 + J2it hT,M t; p m n, 1; s . c π R c cosh(πt) c≥1 (7.19)

Accordingly, we expand Σ2(T,M,N; Q) as `d˚ià`gó“nà˜l `géó“mèˇtˇr˚i`c −A Σ2(T,M,N; Q) = M − ME˚i¯sfiè›n¯sfi˚téˇi‹nffl + M + O T , `d˚uà˜l `d˚uà˜l `d˚uà˜l for each of the terms (7.17), (7.18), and (7.19), respectively.

`d˚ià`gó“nà˜l 7.2.3.1 Contribution to Σ2(T,M,N; Q) of M `d˚uà˜l We will show that the diagonal integral is admissible by applying van der Corput’s second derivative bound for oscillatory integrals (Theorem B.0.1). We first truncate the contour integral to |=(s)| ≤ T ε using the

`d˚u`a˜l v 2 rapid decay of γk(s). Since hT,M (t; p m , 1; s) is even and since tanh(πt) = sgn(t) + O T −A ,

110 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

for t T ε, we have Z Z ∞ − `d˚u`a˜l v 2 `d˚u`a˜l v 2 A t tanh(πt)hT,M (t; p m , 1; s)dt = 2 thT,M (t; p m , 1; s)dt + O T R Z0 ∞ − t−T 2 it Γ(k + s − 2it) − ( M ) v 2 A = 2 te p m − dt + O T . 0 Γ(k s + 2it)

Stirling’s approximation yields an expansion of the gamma factors, namely

ℑ − Γ(k + s − 2it) 2t − =(s) 2i( (s) 2t) 1 + |s|4 = i(2t)2s 1 + O . Γ(k − s + 2it) e |t|

We then apply the van der Corput bound (B.0.1) (second derivative test) to see that Z ∞ k − 2 − − t T v 2 it Γ 2 + s 2it 1+2σ+ε te ( M ) p m dt T , k − 0 Γ 2 s + 2it

1 2 ε 1 1 since m N T . Accordingly, taking σ = 2 + log T , the diagonal contribution satisﬁes

5 +ε T 2 M`d˚ià`gó“nà˜l 1 . `d˚uà˜l N 2

7.2.3.2 Contribution to Σ2(T,M,N; Q) of ME˚i¯sfiè›n¯sfi˚téˇi‹nffl `d˚uà˜l The contribution from the continuous spectrum is handled similar to how the corresponding contribution 1 k was dealt with for Σ1(T,M,N; Q). For σ > 2 (and σ < 2 ), we have Z X 1 4 −s v λQ(n) ME˚i¯sfiè›n¯sfi˚téˇi‹nffl = γk(s) (2π) N (−ε(Q)) `d˚uà˜l 1 2πi v 2 2 +s (σ) m,n≥1 (p m n) v=0,1 Z − 2 ∞ −( t T ) k 2 e M it Γ + s − 2it ds × τ (n) m2n 2 dt + O T −A π |ζ(1 + 2it)|2 it k − s 0 Γ 2 s + 2it Z Z − 2 ∞ − t T k 1 γ (s) 2 e ( M ) 1 Γ + s − 2it ds = k L − it + s, E ⊗ Q 2 dt + O T −A 2πi ((2π)4N)s π |ζ(1 + 2it)|2 2 t k − s (σ) 0 Γ 2 s + 2it Z Z − 2 ∞ − t T k 1 γ (s) 2 e ( M ) 1 1 Γ + s − 2it ds = k L + s, Q L − 2it + s, Q 2 dt 4 s | |2 k 2πi (σ) ((2π) N) π 0 ζ(1 + 2it) 2 2 Γ − s + 2it s 2 + O T −A

Z − 2 Z ∞ − t T k 2 e ( M ) 1 γ (s) 1 1 Γ + s − 2it ds = k L + s, Q L − 2it + s, Q 2 dt | |2 4 s k π 0 ζ(1 + 2it) 2πi (σ) ((2π) N) 2 2 Γ − s + 2it s 2 + O T −A .

111 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

≥ < − 1 − Since L(s, Q) is entire, and since k 2, shifting the contour to (s) = 2 ε, we cross a pole at s = 0, yielding

Z − 2 ∞ − t T k 2 e ( M ) 1 1 Γ − 2it ME˚i¯sfiè›n¯sfi˚téˇi‹nffl = L ,Q L − 2it, Q 2 dt `d˚uà˜l π |ζ(1 + 2it)|2 2 2 k 0 Γ 2 + 2it Z − 2 Z ∞ − t T k 2 e ( M ) 1 γ (s) 1 1 Γ + s − 2it ds + k L + s, Q L − 2it + s, Q 2 dt | |2 4 s k π 0 ζ(1 + 2it) 2πi (− 1 −ε) ((2π) N) 2 2 Γ − s + 2it s 2 2 + O T −A .

Estimating the spectral integral trivially by using the subconvexity bound of Meurmann, we see that term contributes Z ∞ − t−T 2 ( M ) k − 2 e 1 1 Γ 2 2it 1 +ε L ,Q L − 2it, Q dt T 3 M. π |ζ(1 + 2it)|2 2 2 k 0 Γ 2 + 2it MT ε The remaining contour integral, on the other hand, contributes O 1 by Stirling’s approximation and a T 2 1 +ε trivial bound. Thus ME˚i¯sfiè›n¯sfi˚téˇi‹nffl T 3 M. `d˚uà˜l

`géó“mèˇtˇr˚i`c 7.2.3.3 Contribution to Σ2(T,M,N; Q) of M `d˚uà˜l Bounds for the geometric contribution depend on the size of the argument of the J-Bessel function. We split √ √ into two cases: when c ≤ 2π n and when c ≥ 2π n.

7.2.3.3.1 Contribution of small c to M`géó“mèˇtˇr˚i`c `d˚uà˜l √ For c ≤ 2π n, we make use of the Poisson integral formula (A.4), which yields

√ 2it √ 2π n Z π √ 2 4π n c 4π n 4it J2it = 1 1 cos cos θ sin (θ)dθ. (7.20) c Γ 2 + 2it Γ 2 0 c

Since the weight function is even, and by the rapid decay of the Gaussian, we have Z √ 4π n tdt J h`d˚u`a˜l t; pvm2n, 1; s 2it c T,M cosh(πt) R Z √ X ∞ 4π n tdt `d˚u`a˜l v 2 = ( ) J±2it hT,M t; p m n, 1; s ± 0 c cosh(πt) Z √ X ∞ 4π n − t−T 2 it Γ(k + s − 2it) tdt − ( M ) 4 v 2 A = ( ) J±2it e (2π) p m n − + O T . ± 0 c Γ(k s + 2it) cosh(πt)

By the rapid decay of the gamma factors in the s-integral, we can truncate to =(s) T ε after shifting to < 1 (s) = σ = 2 + ε. Applying (7.20) and Stirling’s approximation

ℑ − Γ(k + s − 2it) 2t − =(s) 2i( (s) 2t) 1 + |s|4 = i(2t)2s 1 + O , Γ(k − s + 2it) e |t|

112 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

we are reduced to estimating

Z π Z ∞ √ 2it −2it −2i(2t−ℑ(s)) 2 2π n 2t 2t − =(s) tdt 4it 4 v 2 it (sin θ) (2π) p m n dθ; (7.21) 0 0 c e e cosh(πt) the error term in Stirling’s approximation yields a term that is O T 2σ+εM . By the second derivative √ 3 +2σ+ε estimate of van der Corput, we see that (7.21) is O T 2 . Thus the full contribution from c ≤ 2π n is bounded by

3 +2σ+ε X v X X 3+ε T 2 |ε(Q)| |λ (n)| |S(n, 1; c)| T Q . σ v/2 1 +σ 3 N p 2 2 √ c N 4 v=0,1 n≪ T 2+ε (m n) c≤2π n m N

7.2.3.3.2 Contribution of large c to M`géó“mèˇtˇr˚i`c `d˚uà˜l √ For large values of c with c > 2π n, we use the first few terms of the power series definition of the J-Bessel function. We apply a contour-shifting argument for the first term to show that the c-sum is absolutely convergent, which will come at the cost of the strength of the length M. Using the power series expansion (A.1), we split the integral into Z tdt `d˚uà˜l v 2 I ˛hèá`dffl I ˚tá˚i˜l J2it (x) hT,M t; p m n, 1; s = T,M (x; s) + T,M (x; s), R cosh(πt)

where we have defined Z it x 2it (2π)4pvm2n Γ(k + s − 2it) tg(t) I ˛hèá`dffl T,M (x; s) = dt R 2 Γ(1 + 2it) Γ(k − s + 2it) cosh(πt)

and   Z X ℓ x 2it+2ℓ (−1) it Γ(k + s − 2it) tg(t) I ˚t´a˚i˜l  2  4 v 2 T,M (x; s) = (2π) p m n dt. R ℓ!Γ(1 + ℓ + 2it) Γ(k − s + 2it) cosh(πt) ℓ≥1

I ˛hèá`dffl < 1 We begin with T,M (x; s), for which we use the contour-shifting argument. For 1 > (s) > 2 , we cross

113 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

= 1 no poles by moving the t-integral contour to (t) = 4 + ε, yielding √ Z √ 2it `d˚uà˜l v 2 4π n 2π n hT,M (t, p m n, 1; s) tdt I ˛hèá`dffl T,M ; s = c R c Γ(1 + 2it) cosh(πt) Z √ w w h`d˚uà˜l , pvm2n, 1; s −1 2π n T,M 2i wdw = πw 4 (0) c Γ(1 + w) cosh 2i Z √ w w h`d˚uà˜l , pvm2n, 1; s −1 2π n T,M 2i wdw = πw 4 1 c Γ(1 + w) cosh ( 2 +ε) 2i Z √ 1 +ε+2it 1 ε 1 2π n 2 h`d˚uà˜l t − i + , pvm2n, 1; s 1 + ε + 2it dt = T,M 4 2 2 3 − 1 ε 2i R c Γ 2 + ε + 2it cosh πt π 4 + 2 √ 1 +ε √ √ 1 2π n 2 4π n − 4π n = T ˛hèá`dffl,+ ; s + T ˛hèá`dffl, ; s , 2i c hT,M c hT,M c

√ ± I ˛hèá`dffl, 4π n where we define T,M c ; s corresponding to the integral with t < 0 and t > 0, respectively. For each of these, note that in the essential range of the integral, the weight function has the expansion

( ) 2 t−i 1 + ε −T − ( 4 2 ) 1 1 1 ε M Γ k + s − − ε − 2it √ 2it+ 2 +ε h`d˚u`a˜l t − i + , pvm2n, 1; s = e 2 pv/2m n T,M 4 2 Γ k − s + 1 + ε + 2it 2 1 1 √ 2it+ +ε − 2 − − − v/2 2 − t T Γ k + s 2 ε 2it log T = p m n e ( M ) 1 + O − 1 Γ k s + 2 + ε + 2it M (7.22) for t > 0, and for t < 0, 1 √ −2it− −ε 2 1 1 ε v 2 v/2 2 − t+T Γ k + s + 2 + ε + 2it log T h`d˚u`a˜l t − i + , p m n, 1; s = p m n e ( M ) 1 + O . T,M − − 1 − − 4 2 Γ k s 2 ε 2it M (7.23)

Accordingly, we make a trivial estimate on the contribution with t > 0, applying Stirling’s approximation and (7.22) to get

√ Z √ − 2 ∞ 2it − t T 1 1 4π n 2π n e ( M ) Γ k + s − − ε − 2it + ε + 2it dt I ˛hèá`dffl,+ ; s = 2 2 T,M c c Γ 3 + ε + 2it Γ k − s + 1 + ε + 2it cosh πt − π 1 + ε 0 2 2 4 2 √ 1 +ε + O m n 2 T ε

√ 1 +ε m n 2 MT ε.

MT 2+ε By the Weil bound, we see that the ﬁnal contribution of these terms is 3 . N 2 The gamma factors for contribution with t < 0 shifted in the opposite direction, so they make a much

114 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

larger contribution; Stirling’s approximation yields

− | |−ℑ Γ k + s + 1 + ε + 2it 2|t| − =(s) 2i(2 t (s)) 1 2 = i(2|t| − =(s))2σ+1+2ε 1 + O − − 1 − − | | Γ k s 2 ε 2it e t and 3 √ 2|t| 2it 1 Γ + ε + 2it = 2πe−π|t|(2|t|)1+ε 1 + O . 2 e |t| Combining the above estimates, we see that we must bound the integral

√ Z √ t+T 2 − 0 2it −( ) 2 it 1 1 − 4π n 2π n e M m n Γ k + s + + ε + 2it + ε + 2it dt I ˛hèá`dffl, 2 2 T,M ; s = 3 1 1 ε c −∞ c 2iΓ + ε + 2it Γ k − s − − ε − 2it cosh πt − iπ + 2 ! 2 4 2 T 2+ε + O 1 +ε (m2n) 4 − 1 ε Z πi( + ) 0 − −e 4 2 ,− ˛hèá`dffl, √ ˛hèá`dffl iψT,M (t) = ρT,M (t)e dt + O (?) , 21+ε 2π −∞ where the weight function is 2σ+1+2ε ,− − t+T 2 (2|t| − =(s)) ρ˛hèá`dffl (t) = e ( M ) , T,M |t|ε and the phase is − 2π 2|t| 2|t| − =(s) ψ(t) = ψ˛hèá`dffl, (t) = 2t log − 2t log − 2(2|t| − =(s)) log . T,M cm e e

7 +ε T 4 Applying the second derivative method (van der Corput lemma),√ the overall contribution is N . 4π n I ˚t´a˚i˜l Lastly, we need to bound the contribution from T,M c ; s . For this, the functional equation for the gamma function yields

Γ(1 + ℓ + 2it) = (2 + 2it)(3 + 2it) ··· (ℓ + 2it)Γ(2 + 2it). √ √ 4π n Hence, for c > 2π n, the tail of the power series expansion of J2it c satisﬁes

√ 2 2π n X ℓ x 2it+2ℓ (−1) c 2 . ℓ!Γ(1 + ℓ + 2it) |Γ(2 + 2it)| ℓ≥1 √ 5 +ε 4π n 2 I ˚tá˚i˜l T M Accordingly, making a trivial bound on T,M c ; s yields a contribution that is O 3 . N 2 Combining the above bounds, we see that the contribution of M`géó“mèˇtˇr˚i`c is bounded by `d˚uà˜l

2+ε 2+ε 5 +ε 7 +ε 7 +ε T M MT T 2 M T 4 T 4 M`géó“mèˇtˇr˚i`c + 3 + 3 + . `d˚uà˜l N N 2 N 2 N N

115 7.2. The ﬁrst moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Thus the dual sum Σ2(T,M,N; Q) satisﬁes

7 +ε T 4 Σ (T,M,N; Q) . 2 N

This completes the proof of Lemma 7.2.2. □

116 Chapter 8

The simultaneous nonvanishing of GL(2) × GL(2) L-functions at special points

Contents 8.1 Proof of Theorem 8.0.1: an asymptotic expansion of the second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals . 119 `d˚ià`g 8.1.1 Proof of Lemma 8.1.1: expansion of the diagonal Σ1 (T,M,N,Q1,Q2) ...... 121 ò˝f¨f´d˚ià`g 8.1.2 Proof of Lemma 8.1.2: bounds for the offdiagonal sum Σ1 (T,M,N,Q1,Q2) .. 123 8.1.2.1 Proof of Lemma 8.1.4: the unbalanced case ...... 124 8.1.2.2 Proof of Lemma 8.1.5: the balanced case ...... 127 8.1.2.3 The contribution from the diagonal terms M`d˚ià`gó“nà˜l ...... 128 8.1.2.4 Contribution from ME˚i¯sfiè›n¯sfi˚téˇi‹nffl: the continuous spectrum ...... 128 8.1.2.5 Contribution from M`géó“mèˇtˇr˚i`c: the Kloosterman sums terms ...... 130

8.1.3 Proof of Lemma 8.1.3: the contribution from Σ2(T,M,N,Q1,Q2) ...... 132

`d˚ià`gó“nà˜l 8.1.3.1 The contribution from M (T,M,N,Q1,Q2): the diagonal terms ... 133 `d˚uà˜l 8.1.3.2 The contribution from ME˚i¯sfiè›n¯sfi˚téˇi‹nffl(T,M,N,Q1,Q2): the diagonal terms .. 134 `d˚uà˜l `géó“mèˇtˇr˚i`c 8.1.3.3 The contribution from M (T,M,N,Q1,Q2): the Kloosterman sums 134 `d˚uà˜l 8.1.3.3.1 The contribution from large c ...... 135 8.1.3.3.2 The contribution from small c ...... 136

In this chapter we will prove Theorem 1.0.21. As we outlined earlier, the nonvanishing is in general implied by a first moment type asymptotic expansion combined with a second moment type upper bound. In the case of simultaneous nonvanishing, the first moment type asymptotic expansion is actually an asymptotic expansion of the second moment, while the second moment upper bound is replaced by a fourth moment upper bound. The first of these is the following.

1We should note that the tools to prove this have been available since Luo’s twisted large sieve inequality became available in [Luo95].

117 Theorem 8.0.1 (Restatement of Theorem 1.0.7). Let {uj} be an orthonormal basis of Hecke-Maass cusp 5 Z ∈ ”n`e›w \ H ∈ N 6 +ε forms for SL(2, ). Let Q1,Q2 Sk (Γ0(p) ) with k even. Then for T M T one has ( ) X t −T 2 − j 11 M | |2 ⊗ ⊗ 6 +ε e νj(1) L(sj, uj Q1)L(sj, uj Q2) = c1,Q1,Q2 TM log T + c2,Q1,Q2 TM + O T , (8.1)

where c1,Q1,Q2 and c2,Q1,Q2 depend only on Q1 and Q2.

One obtains a nonoptimal upper bound for the fourth moment problem by Luo’s long interval twisted large sieve inequality over the long interval (Theorem 5.1.3), which yields X 4 5 +ε |L(sj, uj ⊗ Q)| T 2 .

tj ≤T

Or, if one prefers, by Cauchy-Schwarz the second moment is

tj tj tj

These are combined similar to the proof of Theorem 1.0.1 to yield a lower bound on the order of nonvanishing by Hölder’s inequality. Over the long interval, we have X 2 tj /T T log T e |L(sj, uj ⊗ Q1)| · |L(sj, uj ⊗ Q2)|

tj 1 1+ε 2 # tj ≤ T : L(sj, uj ⊗ Q1) =6 0 and L(sj, uj ⊗ Q2) =6 0   1   1 X 4 X 4  4  4 × |L(sj, uj ⊗ Q1)| |L(sj, uj ⊗ Q2)|

tj tj

1 5 1+ε 2 +ε # tj ≤ T : L(sj, uj ⊗ Q1) =6 0 and L(sj, uj ⊗ Q2) =6 0 T 4 ,

2 where again we use the bound of Hoﬀstein-Lockhart to remove the weights |νj(1)| . Thus 1+ε 3 −ε # tj ≤ T : L(sj, uj ⊗ Q1) =6 0 and L(sj, uj ⊗ Q2) =6 0 T 2 .

Remark 8.0.1. If one prefers, the fourth moment of a GL(2) × GL(2) L-function can be regarded as a second moment of a GL(2) × GL(4) L-function. One has

X 1 X L(s, u ⊗ Q)2 = λ (n )λ (n )λ (n )λ (n ), for <(s) > 1, j n2 j 1 j 2 Q 1 Q 2 n≥1 n1n2=n

which is essentially a GL(2) × GL(4) L-function. If one follows a similar approach as with GL(2) × GL(3) L-functions studied in Chapter 6 by applying GL(2) Voronoi summations, the extra divisor twists cause an

118 8.1. Proof of Theorem 8.0.1: an asymptotic expansion of the second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

issue that results in dual sums that are the same length as the starting sums, but with a diﬀerent type of additive twist. One needs to extract additional cancellation from the character sums that arise. At this point we refer the reader to the upcoming paper of Chandee-Li ([CL20]) in which they establish an optimal upper bound for the second moment of GL(2) × GL(4) L-functions at the special point. We have made no eﬀort to adapt their proof using GL(4) Voronoi summation to this setting but it seems reasonable to expect the details to carry through without any substantial changes. Such a result would yield an essentially optimal order of simultaneous nonvanishing.

Remark 8.0.2. We also note that any short interval analogue for the fourth moment of GL(2) × GL(2) L-functions is doomed from the outset, since already the large sieve inequality has a term of size T 2+ε arising from the length of the sums in the balanced form of the approximate functional equation.

8.1 Proof of Theorem 8.0.1: an asymptotic expansion of the second moment of GL(2)×GL(2) L-functions at special points over shortened spectral intervals

From the functional equation for the individual L-functions L(s, uj ⊗ Q1) and L(s, uj ⊗ Q2) presented in Chapter 7, we are able to expand their product in the approximate functional equation X α (n )α (n ) n itj n n ⊗ ⊗ j,Q1 1 j,Q2 2 2 1 2 L(sj, uj Q1)L(sj, uj Q2) = 1 H 2 n1 N n1,n2≥1 (n1n2) − X α (n )α (n ) n itj (2π)8n n N j,Q1 1 j,Q2 2 2 √ 1 2 + 1 Ptj , (8.2) 2 n1 p n1,n2≥1 (n1n2) where the weight functions are given by Z 2 1 Γ (k + s) −s ds H (ξ) = 2 ξ (8.3) 2πi (σ) Γ (k) s and Z 1 Γ2(k + s) Γ(k + s − 2it)Γ(k + s + 2it) ds P (ξ) = ξ−s , (8.4) t 2 − − − 2πi (σ) Γ (k) Γ(k s 2it)Γ(k s + 2it) s

< 1 2+θ where (s) = σ > 2 , and where N = T is the essential length of the first sum; the dual sum has essential T 4 2−θ length N = T . We recall that the Dirichlet series coefficients are defined by X − v αj,Qi (n) = ( ε(Q)) λj(ℓ)λQi (ℓ). pv d2ℓ=n

To save space, from time to time we may denote the ratio of gamma factors that are independent of t by

Γ2(k + s) γ (s) = . k Γ2(k)

119 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Expanding L(sj, uj ⊗ Q1)L(sj, uj ⊗ Q2) by (8.2), we break the spectral sum into two pieces,

( ) X − 2 − tj T M 2 e |νj(1)| L(sj, uj ⊗ Q1)L(sj, uj ⊗ Q2) = Σ1(T,M,N,Q1,Q2) + Σ2(T,M,N,Q1,Q2), (8.5)

tj where we have denoted the ﬁrst sum by

( ) X t −T 2 X itj − j αj,Q (n1)αj,Q (n2) n2 n1n2 M | |2 1 2 Σ1(T,M,N,Q1,Q2) = e νj(1) 1 H 2 n1 N tj n1,n2≥1 (n1n2)

and the dual sum by

( ) − X t −T 2 X itj 8 − j αj,Q (n1)αj,Q (n2) n2 (2π) n1n2N M | |2 1 2 √ Σ2(T,M,N,Q1,Q2) = e νj(1) 1 Ptj . 2 n1 p tj n1,n2≥1 (n1n2)

To provide a clearer outline of what pieces of the above ultimately contribute which pieces of (8.1), we further separate the ﬁrst sum into

`d˚ià`g ò˝f¨f´d˚ià`g Σ1(T,M,N,Q1,Q2) = Σ1 (T,M,N,Q1,Q2) + Σ1 (T,M,N,Q1,Q2),

`d˚ià`g ò˝f¨f´d˚ià`g 6 where Σ1 denotes the terms of Σ1(T,M,N,Q1,Q2) with n1 = n2, and Σ1 those terms with n1 = n2. In the remainder of this section we will extract main terms from Σ1(T,M,N,Q1,Q2) and Σ2(T,M,N,Q1,Q2) and bound the remaining parts to prove Theorem 8.0.1. Explicitly, we will show the following.

Lemma 8.1.1. The diagonal terms of the sum Σ1(T,M,N,Q1,Q2) have the asymptotic expansion `d˚i`a`g 1 − θ 2 4 +ε 1+ε Σ1 (T,M,N,Q1,Q2) = c0(θ)TM log T + c1TM + O T M + T , (8.6) where the constants are   4(2 + θ) Res (L(s, Q ⊗ Q)) if Q1 = Q2 = Q, c (θ) = s=1 0 3  π 2 0 otherwise,

 and ′  4Γ (k) Res (L(s, Q ⊗ Q)) + γQ if Q1 = Q2 = Q, 3 s=1 c1 = π 2 Γ(k)  L(1,Q1 ⊗ Q2) otherwise.

Lemma 8.1.2. Let δ > 0. Then oﬀdiagonal terms of the sum Σ1(T,M,N,Q1,Q2) satisfy the bound

`o˝f¨f´d˚i`a`g − 3δ θ 2 δ+ε 1+ 2 + 4 +ε Σ1 (T,M,N,Q1,Q2) T + T . (8.7)

120 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Lemma 8.1.3. The dual sum Σ2(T,M,N,Q1,Q2) has the asymptotic expansion 2− θ +ε Σ2(T,M,N,Q1,Q2) = c0(θ)TM log T + c1δ(Q1 = Q2)TM + O T 2 , (8.8) where c0(θ) and c1 are constants that depend on Q1, Q2, and θ.

7 5 1 1 3 +ε 6 +ε Taking θ = 3 and δ = 2 , so that N = T then yields a power-saving error term for T M T .

`d˚i`a`g 8.1.1 Proof of Lemma 8.1.1: expansion of the diagonal Σ1 (T,M,N,Q1,Q2)

Expanding the Dirichlet series coeﬃcients αj,Q1 (n1) and αj,Q2 (n2), we arrive at ( ) − 2 X tj T X `d˚i`a`g − αj,Q (n1)αj,Q (n2) n1n2 M | |2 1 2 Σ1 (T,M,N,Q1,Q2) = e νj(1) 1 H 2 N tj n1,n2≥1 (n1n2) n1=n2 X X − i1 − i2 i1+i2 2 2 ( ε(Q1)) ( ε(Q2)) λQ1 (ℓ1)λQ2 (ℓ2) p d1 d2 ℓ1ℓ2 = 1 H (i1+i2)/2 2 2 p (d1 d2 ℓ1ℓ2) 2 N i1=0,1 d1,d2≥1 i2=0,1 ℓ1,ℓ2≥1 i1 2 i2 2 p d1ℓ1=p d2ℓ2 ( ) X − 2 − tj T M × e νj(ℓ1)νj(ℓ2).

In the innermost spectral sum appearing on the second line, we replace the weight function by an even weight function, with

( ) ( ) ( ) X − 2 X − 2 2 − tj T − tj T − tj +T 2 2 2 −A e M νj(ℓ1)νj(ℓ2) = e M + e M νj(ℓ1)νj(ℓ2) + O T (8.9)

tj tj for arbitrary A > 0. Applying the Kuznetsov trace formula to the main term on the right-hand side of (8.16) yields

( ) ( ) X − 2 2 − tj T − tj +T 2 2 e M + e M νj(ℓ1)νj(ℓ2)

tj Z δ(ℓ1 = ℓ2) − t−T 2 − t+T 2 ( M2 ) ( M2 ) = 2 t tanh(πt) e + e dt (8.10) π R Z − 2 2 ∞ −( t T ) −( t+T ) 2 e M2 + e M2 − τ (ℓ )τ (ℓ )dt (8.11) π |ζ(1 + 2it)|2 it 1 it 2 0 √ X Z 2i 1 4π ℓ1ℓ2 −( t−T )2 −( t+T )2 tdt + S(ℓ1, ℓ2; c) J2it e M2 + e M2 . (8.12) π c R c cosh(πt) c≥1

121 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

By a ﬁrst derivative estimate on the Kloosterman integral, using the asymptotic expansion (A.11), for small values of c we have Z √ − 2 2 4π ℓ1ℓ2 −( t T ) −( t+T ) tdt 1+ε J2it e M2 + e M2 T , R c cosh(πt)

whereas the integral is negligible by Proposition B.0.5 for large values of c. It follows that the Kloosterman sums contribution (8.12) satisﬁes XX 2 2 (−ε(Q ))i1 (−ε(Q ))i2 λ (ℓ )λ (ℓ ) pi1+i2 d d ℓ ℓ 1 2 Q1 1 Q2 2 H 1 2 1 2 (i +i )/2 2 2 1 p 1 2 (d d ℓ ℓ ) 2 N pi1 d2ℓ =pi2 d2ℓ 1 2 1 2 1 1 2 2 √ X Z 1 4π ℓ1ℓ2 −( t−T )2 −( t+T )2 tdt × S(ℓ1, ℓ2; c) J2it e M2 + e M2 c R c cosh(πt) c≥1 T 1+ε.

Similarly, one can make a trivial bound to show that the continuous spectrum (8.11) contributes XX 2 2 (−ε(Q ))i1 (−ε(Q ))i2 λ (ℓ )λ (ℓ ) pi1+i2 d d ℓ ℓ 1 2 Q1 1 Q2 2 H 1 2 1 2 (i +i )/2 2 2 1 p 1 2 (d d ℓ ℓ ) 2 N i1 2 i2 2 1 2 1 2 p d1ℓ1=p d2ℓ2 Z − 2 2 ∞ −( t T ) −( t+T ) e M2 + e M2 × τ (ℓ )τ (ℓ )dt | |2 it 1 it 2 0 ζ(1 + 2it) MT ε.

Meanwhile, by Gradshteyn-Rhyzhik [GR94], the diagonal integral evaluates directly to Z − 2 2 √ −( t T ) −( t+T ) −A t tanh(πt) e M2 + e M2 dt = 4 πT M + O T , (8.13) R

i1 2 i2 2 where A > 0 is arbitrary. These terms contribute the main term: since p d1ℓ1 = p d2ℓ2 and ℓ1 = ℓ2, it

follows that i1 = i2 and d1 = d2. Thus, switching the sums and the integral deﬁning H(ξ), the main term of (8.13) contributes X X i1 2i1 4 2 4TM (ε(Q1)ε(Q2)) λQ1 (ℓ)λQ2 (ℓ) p d ℓ −A 3 i 2 H 2+θ + O T π 2 p 1 d ℓ T i1=0,1 d,ℓ≥1 Z ! 2 4TM 1 Γ (k + s) s ⊗ ds −A = 3 2 N L (1 + 2s, Q1 Q2) + O T . π 2 2πi (σ) Γ (k) s

6 ⊗ < − 1 If Q1 = Q2, then L(s, Q1 Q2) is entire. Thus, shifting the contour to (s) = 4 , it follows by the residue

122 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

theorem that Z 1 Γ2(k + s) ds T (2+θ)sL (1 + 2s, Q ⊗ Q ) 2πi Γ2(k) 1 2 s (σ) Z 2 ⊗ 1 Γ (k + s) s ⊗ ds = L(1,Q1 Q2) + 2 N L (1 + 2s, Q1 Q2) . 2πi − 1 Γ (k) s ( 4 )

On the other hand, if Q1 = Q2 = Q, then L(s, Q ⊗ Q) has a simple pole at s = 1. In this case, shifting the < − 1 contour to (s) = 4 , we have Z 1 Γ2(k + s) ds T (2+θ)sL (1 + 2s, Q ⊗ Q) 2πi Γ2(k) s (σ) 2 + θ Γ′(k) = log T + Res (L(s, Q ⊗ Q)) 2 Γ(k) s=1 Z 2 1 Γ (k + s) s ⊗ ds + γQ + 2 N L (1 + 2s, Q Q) , 2πi − 1 Γ (k) s ( 4 )

where γQ is the zeroth coeﬃcient in the Laurent series expansion of L(s, Q ⊗ Q) at s = 0. To bound the contribution of the shifted contour integral in either of the above two cases, we truncate to = ε (s) = τ T , since Γ(s) is rapidly decaying in the imaginary direction. Regardless of whether Q1 = Q2 1 ⊗ or not, by the convexity bound for L 2 + iτ, Q1 Q2 , the remaining integral satisﬁes Z 2 − 1 Γ (k 4 + iτ) − 1 +iτ 1 dτ − 1 − θ +ε N 4 L + 2iτ, Q ⊗ Q T 2 4 . 2 1 2 − 1 |τ|≪T ε Γ (k) 2 4 + iτ

Thus we see that `d˚i`a`g 1 − θ 2 4 +ε 1+ε Σ1 (T,M,N,Q1,Q2) = c0TM log T + c1TM + O T M + T , where explicitly the constants are   4(2 + θ) Res (L(s, Q ⊗ Q)) if Q1 = Q2 = Q, c = s=1 0 3  π 2 0 otherwise,

and  ′  4Γ (k) Res (L(s, Q ⊗ Q)) + γQ if Q1 = Q2 = Q, 3 s=1 c1 = π 2 Γ(k)  L(1,Q1 ⊗ Q2) otherwise. This completes the proof of 8.1.1. □

ò˝f¨f´d˚ià`g 8.1.2 Proof of Lemma 8.1.2: bounds for the offdiagonal sum Σ1 (T,M,N,Q1,Q2) There is a slight issue that arises from using the combined version of the approximate functional equation for the product L-function: in many of the sums we will later consider, we will only care about the size of

123 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals itj n2 one of the twisting factors derived from , and we will require the size of n1 or n2 to be smaller than n1 2 T . Fortunately, this issue arises only when one of n1 and n2 is very large and the other consequently very small. In this case one can apply the functional equation to the longer sum to force it to be short, at the cost of exchanging the spectral weight function for one that is twisted by gamma factors. 2−δ 2+θ+ε θ+δ+ε The unbalanced case occurs, say, when T ≤ n1 ≤ T and n2 T , where δ > 0 is a constant to be determined in the course of the proof.

Lemma 8.1.4. Let δ > 0. With all the notation as previously laid out in this section, we have

( ) X t −T 2 X itj j 3δ θ − αj,Q1 (n1)αj,Q2 (n2) n2 n1n2 M | |2 1+ 2 + 4 +ε e νj(1) 1 H T . 2 n1 N t 2−δ 2+θ+ε (n1n2) j T ≤n1≤T δ+θ+ε n2≤T

2 2−δ The balanced case occurs when both n1 and n2 are smaller than T : that is, when n1, n2 ≤ T .

Lemma 8.1.5. Let δ > 0. With all the notation as previously laid out in this section, we have

( ) X t −T 2 X itj − j αj,Q (n1)αj,Q (n2) n2 n1n2 − M | |2 1 2 2 δ+ε e νj(1) 1 H T . 2 n N 2−δ (n n ) 1 tj n1,n2≤T 1 2

8.1.2.1 Proof of Lemma 8.1.4: the unbalanced case

2−δ 2+θ+ε θ+δ+ε Without loss of generality, suppose that T ≤ n1 ≤ T , and thus n2 ≤ T . By a dyadic partition of unity, it suﬃces to consider sums of the form X α (n )α (n ) n itj n n n n U j,Q1 1 j,Q2 2 2 1 2 1 2 tj (A, B) = 1 H f g , 2 n1 N B A n1,n2≥1 (n1n2)

where f and g are smooth weight functions compactly supported in [1, 2], and T 2−δ ≤ B ≤ T 2+θ+ε and A ≤ T θ+δ+ε, with δ > 0 to be determined later. Denote the full spectral sum by

( ) X − 2 − tj T U M | |2U T,M (A, B) = e νj(1) tj (A, B).

Denote the auxiliary weight function by ξBn w(ξ) = w (ξ) = g(ξ)H 2 . n2 N

124 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

By Mellin inversion and the functional equation Λ(s, uj ⊗ Q1) = Λ(1 − s, uj ⊗ Q1), we will only need to < 1 consider the terms where n1 and n2 are both small; for (s) = σ > 2 , we have X Z αj,Q1 (n1) n1 1 s w = we(s)L(s + sj, uj ⊗ Q1)B ds sj ≥ n1 B 2πi (σ) n1 1 Z 1 s = we(s)L(s + sj, uj ⊗ Q1)B ds 2πi (−σ) Z √ − p 4s 4itj − 1 e − ⊗ Γ(k + s)Γ(k + s 2itj) −s = w( s)L(s + sj, uj Q1) − − B ds 2πi (σ) 2π Γ(k s)Γ(k s + 2itj) X αj,Q1 (n1) = s W (n1B; tj), n1 j n1≥1 where we have deﬁned

Z √ − p 4s 4itj − 1 e − Γ(k + s)Γ(k + s 2itj) −s W (u; t) = w( s) − − u ds. 2πi (σ) 2π Γ(k s)Γ(k s + 2itj)

By Stirling’s approximation, shifting the contour very far to the right shows that the partial sum with δ+ε δ+ε n1 ≥ T is negligible. For n1 ≤ T , we use the rapid decay of the gamma factors in the imaginary direction to truncate the weight function to

Z √ − p 4s 4itj − 1 e − Γ(k + s)Γ(k + s 2itj) −s −A W (u; t) = w( s) − − u ds + O T . 2πi ℜ(s)=σ,|ℑ(s)|≪T ε 2π Γ(k s)Γ(k s + 2itj)

Hence,

UT,M (A, B) Z 1 X 1 n Γ(k + s) p2 s = g 2 we(−s) 1 − 4 2πi 2 A ℜ(s)=σ,|ℑ(s)|≪T ε Γ(k s) (2π) n1B n2≥1 (n1n2) ≤ δ+ε n1 T  ( ) √ − X t −T 2 4itj  − j p Γ(k + s − 2itj) × M | |2 itj e νj(1) αj,Q1 (n1)αj,Q2 (n2)(n1n2) ds + O(1).  2π Γ(k − s + 2itj) tj

125 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Opening the Dirichlet series coeﬃcients αj,Q1 (n1) and αj,Q2 (n2), and applying the Hecke relations, we have

UT,M (A, B) X i2 X 1 (−ε(Q ))i1 (−ε(Q )) 1 pi2 d2rℓ λ (rℓ )λ (rℓ ) = 1 2 g 2 4 Q1 3 Q2 4 (i +i )/2 2 1 2πi p 1 2 d1d2r A 2 i1,i2=0,1 d2,ℓ4≥1 (ℓ3ℓ4) pi1 d2rℓ ≤T δ+ε Z 1 3 2 s × e − Γ(k + s) p w( s) 4 i 2 ε Γ(k − s) (2π) p 1 d rℓ B ℜ(s)=σ,|ℑ(s)|≪T 1 3  ( )  − 2  X tj T − − − itj Γ(k + s 2itj) × M | |2 4 i1+i2 2 2 2 2 e νj(1) λj(ℓ3ℓ4) (2π) p d1d2r ℓ3ℓ4 ds + O(1).  Γ(k − s + 2itj) tj (8.14)

Now denote the exponential twisting factor in the innermost spectral sum by

4 i1+i2−2 2 2 2 C = (2π) p d1d2r ℓ3ℓ4,

and deﬁne the spectral weight function by

− 2 − 2 − t T Γ(k + s 2it) it − t+T Γ(k + s + 2it) −it W (t; C) = e ( M ) C + e ( M ) C . Γ(k − s + 2it) Γ(k − s − 2it)

Exchanging the weight function in (8.14) for W (t; C) introduces a negligible error. Applying the Kuznetsov trace formula, we have Z Z X δ(ℓ ℓ = 1) 2 ∞ W (t; C) ν (ℓ ℓ )ν (1)W (t ; C) = 3 4 t tanh(πt)W (t; C)dt − τ (ℓ ℓ )dt j 3 4 j j 2 | |2 it 3 4 π R π 0 ζ(1 + 2it) tj √ X Z 2i 1 4π ℓ3ℓ4 tdt + S(ℓ3ℓ4, 1; c) J2it W (t; C) , π c R c cosh(πt) c≥1

and accordingly, we write `d˚ià`gó“nà˜l E˚i¯sfiè›n¯sfi˚téˇi‹nffl `géó“mèˇtˇr˚i`c −A UT,M (A, B) = UM − UM + UM + O T .

Simple estimates suffice for the contributions of UM`d˚ià`gó“nà˜l and UME˚i¯sfiè›n¯sfi˚téˇi‹nffl. By the first derivative bound of van der Corput, the diagonal integral satisfies Z t tanh(πt)W (t; C)dt T 1+ε. R

1+δ+ θ +ε Thus, UM`d˚ià`gó“nà˜l T 2 . And a trivial bound on the continuous spectral integral yields the bound δ+ θ +ε UM`d˚ià`gó“nà˜l MT 2 . The only term we need to deal with is the Kloosterman sums contribution; this has essentially already been done in Chapter 7 for the first moment problem, and one finds that UM`géó“mèˇtˇr˚i`c 1+ 3δ + θ +ε T 2 4 .

126 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

8.1.2.2 Proof of Lemma 8.1.5: the balanced case

Now we deal with the oﬀdiagonal terms with n1 =6 n2. Without loss of generality, assume that n1 < n2.

Write ( ) X t −T 2 X itj − j αj,Q (n1)αj,Q (n2) n2 n1n2 B M | |2 1 2 (T,M) = e νj(1) 1 H . 2 n1 N tj n1,n2≥1 (n1n2) n1

Expanding the Dirichlet series coeﬃcients αj,Q1 (n1) and αj,Q2 (n2), we arrive at X X 2 2 (−ε(Q ))i1 (−ε(Q ))i2 λ (ℓ )λ (ℓ ) pi1+i2 d d ℓ ℓ B 1 2 Q1 1 Q2 2 1 2 1 2 (T,M) = 1 H (i1+i2)/2 2 2 p (d1 d2 ℓ1ℓ2) 2 N i1=0,1 d1,d2≥1 i2=0,1 ℓ1,ℓ2≥1 i1 2 i2 2 p d1ℓ1

In the innermost spectral sum, we replace the weight function, at the cost of introducing a negligible error, by

( ) ( ) − 2 itj 2 −itj − tj T pi2 d2ℓ − tj +T pi2 d2ℓ i2 2 i1 2 2 2 2 2 2 2 h tj, p d ℓ2, p d ℓ1 = e M + e M (8.16) 2 1 i1 2 i1 2 p d1ℓ1 p d1ℓ1 ( ) − 2 itj tj T i2 2 − p d2ℓ2 −A = e M2 + O T , (8.17) i1 2 p d1ℓ1 for tj > 0 and arbitrary A > 0. By Kuznetsov trace formula, up to a negligible error the innermost spectral sum in (8.15) is X i2 2 i1 2 νj(ℓ1)νj(ℓ2)h tj, p d2ℓ2, p d1ℓ1 tj Z 4δ(ℓ = ℓ ) 1 2 i2 2 i1 2 = 2 t tanh(πt)h t, p d2, p d1 dt π R Z ∞ 2 h t, pi2 d2ℓ , pi1 d2ℓ − 2 2 1 1 τ (ℓ )τ (ℓ )dt π |ζ(1 + 2it)|2 it 1 it 2 0 Z √ 2i X 1 4π ℓ ℓ tdt 1 2 i2 2 i1 2 + S(ℓ1, ℓ2; c) J2it h t, p d2ℓ2, p d1ℓ1 , π c R c cosh(πt) c≥1

and accordingly, we write − B(T,M) = M`d˚ià`gó“nà˜l − ME˚i¯sfiè›n¯sfi˚téˇi‹nffl + M`géó“mèˇtˇr˚i`c + O T A .

127 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

8.1.2.3 The contribution from the diagonal terms M`d˚ià`gó“nà˜l

The contribution from these terms is negligible under mild conditions on M. Write ℓ1 = ℓ2 = ℓ, and without

i2 2 i1 2 loss of generality, suppose that p d2 > p d1. The diagonal integral can be evaluated as Z i2 2 i1 2 t tanh(πt)h t, p d2, p d1 dt R ( ( )) 2 pi2 d2 2iT √ − M log 2 i2 2 i2 2 i1 2 p d2 p d2 3 −A = 2 πe p d1 2TM + i log M + O T . i1 2 i1 2 p d1 p d1 pi2 d2 pi2 d2 − 2 2 2 2 The weight factor exp M log i1 2 will be negligibly small when M log i1 2 log T , say. Since p d1 p d1 i1+i2 2 2 2 ε p d1d2ℓ NT , we have pi2 d2 pi2/2d 1 log 2 = 2 log 2 ≥ 2 log 1 + pi1 d2 i1/2 i1/2 1 p d1 p d1 ≥ 1 2 log 1 + 1 N 4 T ε ≥ 1 1 . N 4 T ε

1 ε Thus the contribution from M`d˚ià`gó“nà˜l is negligible when M N 4 T .

8.1.2.4 Contribution from ME˚i¯sfiè›n¯sfi˚téˇi‹nffl: the continuous spectrum

Along the lines of [Luo93] we can show that the continuous spectrum contributes n o 1 + θ +ε 1 1+ θ − δ ME˚i¯sfiè›n¯sfi˚téˇi‹nffl max MT 2 4 ,M 2 T 4 4 ,

1 + θ provided that M T 2 4 . Denote the Dirichlet series coeﬃcients by X i0 αt,Q(n) = (−ε(Q)) λQ(ℓ)τit(ℓ). pi0 d2ℓ=n

128 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Using a dyadic partition of unity, we will remove the condition that n1 < n2. Let f, g be smooth weight functions with compact support in [1, 2]. By Cauchy-Schwarz, the continuous spectrum contribution satisfies − ME˚i¯sfiè›n¯sfi˚téˇi‹nffl − O T A  

Z ∞ − t−T 2 it ( M )  X  2 e  αt,Q1 (n1)αt,Q2 n2 n1n2  = 2  1 H  dt π |ζ(1 + 2it)| 2 n N 0 2−δ (n1n2) 1 n1,n2≤T n1

− 2 Z s Z ∞ − t T it ( M )  X  1 γk(s) 2 e  αt,Q1 (n1)αt,Q2 n2 n1n2  ds = 2  1 H  dt 2πi N π |ζ(1 + 2it)| 2 n N s (σ) 0 2−δ (n1n2) 1 n1,n2≤T n1

Next we bound the terms under the square-roots. Since both sums are similar, it will suﬃce to consider only one. By Cauchy-Schwarz, we have

2 Z ∞ − t−T 2 X e ( M ) α (n ) n t,Q1 1 f 1 dt | |2 1 +it 0 ζ(1 + 2it) 2 A n1≥1 n1

2 − 2 X X Z ∞ − t T X |ε(Q )|i1 1 e ( M ) λ (ℓ )τ (ℓ ) pi1 d2ℓ ε 1 Q1 1 it 1 1 1 T 1 f dt i1/2 2 +it+s p d1 |ζ(1 + 2it)| 2 A i =0,1 1 0 ℓ ≥1 ℓ 1 d ≪A 2 1 1 1 2 Z ∞ − t−T 2 X | |i1 X ( M ) X ε ε(Q1) 1 e −2it = T r c(r) dt, i1/2 | |2 p d1 0 ζ(1 + 2it) i1=0,1 1 r≥1 d1≪A 2 where in the last line we opened the divisor function τit(ℓ1), and where we set X i1 2 λQ1 (qr) p d1qr c(r) = 1 f . 2 +s A q≥1 (qr)

By the mean value theorem for Dirichlet polynomials, it follows that

2 Z ∞ − t−T 2 X X X X e ( M ) α (n ) n |ε(Q )|i1 1 t,Q1 1 f 1 dt T ε 1 (M + r) |c(r)|2. 2 1 i /2 |ζ(1 + 2it)| 2 +it A 1 d 0 p 1 1 n1≥1 n1 i1=0,1 r≪ A d1≪A 2 i1 2 p d1

129 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

By the Hecke bound, we have  √  ε i1/2 |s|T if r ≤ A/p d1,  √ ε √ ε c(r) T A if A/pi1/2d < r AT ,  rpi1/2d 1 pi1 d2  1 1  −K AT ε T for any K > 0 if r i1 2 . p d1

Thus it follows that

2 Z − 2 ∞ − t T X √ e ( M ) α (n ) n t,Q1 1 f 1 dt M A + A T ε. | |2 1 +it 0 ζ(1 + 2it) 2 A n1≥1 n1

Combining all of the above, we see that the contribution from ME˚i¯sfiè›n¯sfi˚téˇi‹nffl satisfies n o 1 + θ +ε 1 1+ θ − δ ME˚i¯sfiè›n¯sfi˚téˇi‹nffl max MT 2 4 ,M 2 T 4 4 .

8.1.2.5 Contribution from M`géó“mèˇtˇr˚i`c: the Kloosterman sums terms

We bound the contribution from the Kloosterman terms using the Hecke bound. Explicitly, we have X X i1 i2 i1+i2 2 2 2i (−ε(Q1)) (−ε(Q2)) λQ (ℓ1)λQ (ℓ2) p d1 d2 ℓ1ℓ2 M`géó“mèˇtˇr˚i`c 1 2 = 1 H (i1+i2)/2 2 2 π p (d1 d2 ℓ1ℓ2) 2 N i1=0,1 d1,d2≥1 i2=0,1 ℓ1,ℓ2≥1 pi1 d2ℓ

1−δ+ε 1−δ+ε We can restrict our attention to c T , since the contribution√ from c T is negligible by 4π ℓ1ℓ2 Proposition B.0.5. And by Proposition B.0.4, setting x = c , we see that the integral transform has the asymptotic expansion Z √ 4π ℓ1ℓ2 tdt J h t, pi2 d2ℓ , pi1 d2ℓ 2it c 2 2 1 1 cosh(πt) R ! i1 i2 i1 i2 π p 2 d ℓ p 2 d ℓ p 2 d ℓ p 2 d ℓ = 1 1 − 2 2 e 1 1 + 2 2 c i2 i1 i2 i1 p 2 d2 p 2 d1 p 2 d2 p 2 d1   !2 ( ) ! i1 i2 1 ε 1 π p 2 d ℓ p 2 d ℓ N 2 T × exp − 1 1 − 2 2 − T  + O max ,T . M 2 c i2 i1 c M 2 p 2 d2 p 2 d1 (8.18)

130 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals 3+ θ −δ+ε 2 2 θ − δ T 3 + 6 3 +ε The error term contributes O M 2 overall, which will be an error term provided that M T . Denote the main term contribution to M`géó“mèˇtˇr˚i`c by X X 2 2 2i (−ε(Q ))i1 (−ε(Q ))i2 λ (ℓ )λ (ℓ ) pi1+i2 d d ℓ ℓ M`géó“mèˇtˇr˚i`c 1 2 Q1 1 Q2 2 1 2 1 2 = 1 H ”mà˚i‹nffl (i1+i2)/2 2 2 π p (d1 d2 ℓ1ℓ2) 2 N i1=0,1 d1,d2≥1 i2=0,1 ℓ1,ℓ2≥1 i1 2 i2 2 ≤ 2−δ p d1ℓ1,p d2ℓ2 T pi1 d2ℓ

We handle the main term contribution as follows. By Fourier inversion, we have Z − 2 √ − t T 3 −(πξM)2 te ( M ) = π TM − iπM ξ e(−ξT )e e (tξ) dξ. (8.19) R

Thus, applying (8.18) and (8.19), we have Z X √ 2 1 M`géó“mèˇtˇr˚i`c − 3 − −(πξM) ”mà˚i‹nffl = π TM iπM ξ e( ξT )e R c c≪T 1−δ+ε X X X (−ε(Q ))i1 (−ε(Q ))i2 λ (ℓ ) λ (ℓ ) × 1 2 Q√1 1 Q√2 2 S(ℓ , ℓ ; c) (i +i )/2 1 2 p 1 2 d1d2 ℓ1 ℓ2 i ,i =0,1 ≥ i2 2 1 2 d1,d2,ℓ2 1 p d2ℓ2 ℓ1≤ i2 2 ≤ 2−δ pi1 d2 p d2ℓ2 T 1 i1 2 2−δ p d ℓ1≪T ! 1 ! ! ! i2 i1 i1 i2 i +i 2 2 p 2 d ℓ p 2 d ℓ p 2 d ℓ ξ p 2 d ℓ ξ p 1 2 d d ℓ ℓ × e 2 2 e 1 1 e 1 1 e − 2 2 H 1 2 1 2 dξ. (8.20) i1 i2 i2 i1 N cp 2 d1 cp 2 d2 p 2 d2cM p 2 d1cM

Opening the Kloosterman sums and applying the double Hecke bound (Propostion 2.1.9), we see that the contribution from these terms is T 2−δ+ε.

131 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

8.1.3 Proof of Lemma 8.1.3: the contribution from Σ2(T,M,N,Q1,Q2)

Now we need to bound the contribution from the dual sum Σ2(T,M,N,Q1,Q2). Recall that

Σ2(T,M,N,Q1,Q2) ( ) X − 2 X − tj T (−ε(Q ))i1 (−ε(Q ))i2 M 2 1 2 = e |νj(1)| p(i1+i2)/2 tj i1=0,1 i =0,1 2 X itj 2 2 λ (ℓ )λ (ℓ ) pi1 d2ℓ (2π)8pi1+i2 d d ℓ ℓ N × Q1 1 Q2 2 1 1 1 2 1 2 1 Pt √ 2 2 i2 2 j (d1 d2 ℓ1ℓ2) 2 p d2ℓ2 p d1,d2≥1 ≥ ℓ1,ℓ2 1 Z X i i X (−ε(Q )) 1 (−ε(Q )) 2 λ (ℓ )λ (ℓ ) 1 −s 1 2 Q1 1 Q2 2 4 i1+i2 2 2 = 1 γk(s) (2π) p d d ℓ1ℓ2N (i1+i2)/2 2 2 1 2 p (d1 d2 ℓ1ℓ2) 2 2πi (σ) i1=0,1 d1,d2≥1 ≥ i2=0,1  ℓ1,ℓ2 1  ( ) X − 2 itj  − tj T pi1 d2ℓ Γ(k + s − 2it )Γ(k + s + 2it ) ds M 1 1 j j × e νj(ℓ1)νj(ℓ2) .  i2 2 − − −  p d2ℓ2 Γ(k s 2itj)Γ(k s + 2itj) s tj

We replace the spectral weight function up to a negligible error by the even weight function

h`d˚u`a˜l(t; s) = h`d˚u`a˜l(t; pi1 d2ℓ , pi2 d2ℓ ; s) 1 1 2 2 ! it −it − 2 i1 2 2 i1 2 − − t T p d1ℓ1 − t+T p d1ℓ1 Γ(k + s 2it)Γ(k + s + 2it) = e ( M ) + e ( M ) . i2 2 i2 2 − − − p d2ℓ2 p d2ℓ2 Γ(k s 2it)Γ(k s + 2it)

Applying the Kuznetsov trace formula, we arrive at X `d˚uà˜l i1 2 i2 2 νj(ℓ1)νj(ℓ2)h (tj; p d1ℓ1, p d2ℓ2; s) tj Z 2δ(ℓ = ℓ ) 1 2 `d˚uà˜l i1 2 i2 2 = 2 t tanh(πt)h (t; p d1ℓ1, p d2ℓ2; s)dt π R Z ∞ 2 h`d˚uà˜l(t; pi1 d2ℓ , pi2 d2ℓ ; s) − 1 1 2 2 τ (ℓ )τ (ℓ )dt π |ζ(1 + 2it)|2 it 1 it 2 0 Z √ X S(ℓ , ℓ ; c) 2i 4π ℓ ℓ tdt 1 2 1 2 `d˚uà˜l i1 2 i2 2 + J2it h (t; p d1ℓ1, p d2ℓ2; s) , c π R c cosh(πt) c≥1

and accordingly we write the dual sum as

Σ2(T,M,N,Q1,Q2) `d˚ià`gó“nà˜l `géó“mèˇtˇr˚i`c −A = M (T,M,N,Q1,Q2) − ME˚i¯sfiè›n¯sfi˚téˇi‹nffl(T,M,N,Q1,Q2) + M (T,M,N,Q1,Q2) + O T . `d˚uà˜l `d˚uà˜l `d˚uà˜l

We will show that `d˚ià`gó“nà˜l 2− δ +ε M (T,M,N,Q1,Q2) = c0TM log T + c1TM + O T 2 , `d˚uà˜l

132 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals while the nondiagonal terms satisfy

1− θ +ε ME˚i¯sfiè›n¯sfi˚téˇi‹nffl(T,M,N,Q1,Q2) T 2 M `d˚uà˜l and `géó“mèˇtˇr˚i`c 2− 3θ +ε M (T,M,N,Q1,Q2) T 4 . `d˚uà˜l

`d˚ià`gó“nà˜l 8.1.3.1 The contribution from M (T,M,N,Q1,Q2): the diagonal terms `d˚uà˜l

`d˚ià`gó“nà˜l We extract the main term from M (T,M,N,Q1,Q2). We have `d˚uà˜l

`d˚ià`gó“nà˜l M (T,M,N,Q1,Q2) `d˚uà˜l X X 2 (−ε(Q ))i1 (−ε(Q ))i2 λ (ℓ)λ (ℓ) = 1 2 Q1 Q2 2 (i +i )/2 π p 1 2 d1d2ℓ i1=0,1 d1,d2≥1 ≥ i2=0,1Z ℓ 1 Z 1 −s ds × 4 i1+i2 2 2 2 `d˚uà˜l i1 2 i2 2 γk(s) (2π) p d1d2ℓ N t tanh(πt)h t; p d1, p d2; s dt . 2πi (σ) R s

Consider the spectral integral. Since the weight function is even, by applying Stirling’s approximation the gamma factors, we have

Z Z ∞ it i1 2 − t−T 2 p d Γ(k + s − 2it)Γ(k + s + 2it) `d˚u`a˜l i1 2 i2 2 ( M ) 1 t tanh(πt)h t; p d1, p d2; s dt = 2 t tanh(πt)e dt i2 2 − − − R 0 p d2 Γ(k s 2it)Γ(k s + 2it) Z ∞ it − 2 i1 2 − t T p d1 4s ε = 2 t tanh(πt)e ( M ) (2t) dt + O (MT ) . i2 2 0 p d2

i1 2 i2 2 i1 2 6 i2 2 We split into two cases: where p d1 = p d2 and where p d1 = p d2. i1 2 i2 2 When p d1 = p d2, we must have i1 = i2, so that d1 = d2. Thus the full contribution of these terms is Z Z X j X ∞ − t−T 2 (ε(Q1)ε(Q2)) λQ1 (ℓ)λQ2 (ℓ) 1 4 2j 4 2 s − 4s ds γ (s) (2π) p d ℓ N t tanh(πt)e ( M ) (2t) dt (j 2 k p ≥ d ℓ 2πi (σ) 0 s j=0,1 d,ℓ 1 ( ) Z ∞ Z − 2 −( t T ) 1 4 −s −2s ds = t tanh(πt)e M γk(s) (2π) N L (1 + 2s, Q1 ⊗ Q2) (2t) dt. 0 2πi (σ) s

< − 1 6 Shifting the contour integral to (s) = 4 , if Q1 = Q2, then we have Z 1 −s ds γ (s) (2π)4N L (1 + 2s, Q ⊗ Q ) (2t)4s 2πi k 1 2 s (σ) Z 1 4 −s ds = L(1,Q1 ⊗ Q2) + γk(s) (2π) N L (1 + 2s, Q1 ⊗ Q2) . 2πi − 1 s ( 4 )

On the other hand, if Q1 = Q2 = Q, then L(s, Q ⊗ Q) has a simple pole at s = 1. In this case, shifting the

133 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

< − 1 contour to (s) = 4 yields Z 1 −s ds γ (s) (2π)4N L (1 + 2s, Q ⊗ Q) (2t)4s 2πi k s (σ) 4 4 ′ 1 2p t Γ (k) ⊗ = log 8 + Res (L(s, Q Q)) 2 (2π) N Z Γ(k) s=1 1 8 −s 4s ds + γQ + γk(s) (2π) N L (1 + 2s, Q ⊗ Q) (2t) , 2πi − 1 s ( 4 )

where γQ is the zeroth coeﬃcient in the Laurent series expansion of L(s, Q ⊗ Q) at s = 0. The spectral integral can be approximated as follows:

Z ∞ Z ∞ Z ∞ 4 − 2 − 2 4 − 2 t − t T 2−θ − t T p − t T t log e ( M ) dt = t log T e ( M ) dt + t log e ( M ) dt N (2π2)4 0 0 Z 0 ∞ 4 t − t−T 2 + t log e ( M ) dt T 4 0 √ √ p = (2 − θ) πT M log T + 4 π log TM + O M 2T ε . 2π2

i1 2 6 i2 2 In the case where p d1 = p d2, we estimate the integral using the van der Corput bound. This results in 2− θ +ε a contribution that is O T 2 . Actually it seems likely that we can improve this estimate by evaluating the t-integral directly in terms of parabolic cylinder functions. At any rate, shifting the contour integral to <(s) = ε, we have

Z ∞ it 2 i1 2 1+ε − t−T p d1 4s T t tanh(πt)e ( M ) (2t) dt . i2 2 pi1 d2 0 p d2 1 log i2 2 p d2

2 2 2−θ+ε Since the dual sum is eﬀectively truncated to d1d2 T , it follows that the contribution from the terms − θ i1 2 6 i2 2 2 2 +ε with p d1 = p d2 is O T .

8.1.3.2 The contribution from ME˚i¯sfiè›n¯sfi˚téˇi‹nffl(T,M,N,Q1,Q2): the diagonal terms `d˚uà˜l We make a trivial estimate on the continuous spectrum contribution. After shifting the s-integral to <(s) = ε, this yields Z ∞ h`d˚uà˜l(t; pi1 d2ℓ , pi2 d2ℓ ; s) 1 1 2 2 τ (ℓ )τ (ℓ )dt M. | |2 it 1 it 2 0 ζ(1 + 2it)

1− θ +ε Thus the overall contribution is ME˚i¯sfiè›n¯sfi˚téˇi‹nffl(T,M,N,Q1,Q2) T 2 M. `d˚uà˜l

`géó“mèˇtˇr˚i`c 8.1.3.3 The contribution from M (T,M,N,Q1,Q2): the Kloosterman sums `d˚uà˜l

`géó“mèˇtˇr˚i`c The analysis of the contribution from M (T,M,N,Q1,Q2) is largely the same as appears in [Luo12]. `d˚uà˜l The only difference is that the stationary phase arguments actually become simpler because we do not need to worry about small values of t, since the weight function effectively restricts t to t T . As usual, we treat 1− θ +ε 1− θ +ε the cases with c T 2 and c T 2 by two different approaches.

134 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

8.1.3.3.1 The contribution from large c

1− θ +ε Suppose that c T 2 . First we shift the s-contour to <(s) = ε, and by the rapid decay of the gamma = ε = − 1 − ε factors we can truncate the integral to height (s) T . Shifting the t-integral to (t) = 4 2 , by the power series expansion of the J-Bessel function, we see that Z √ 4π ℓ ℓ tdt 1 2 `d˚uà˜l i1 2 i2 2 J2it h (t; p d1ℓ1, p d2ℓ2; s) R c cosh(πt) Z √ 1 1 4π ℓ1ℓ2 1 ε + ε + 2it dt `d˚uà˜l − i1 2 i2 2 2 = J 1 +ε+2it h t i + ; p d1ℓ1, p d2ℓ2; s 2 − 1 ε 2i R c 4 2 cosh πt iπ 4 + 2 Z √ 1 +ε+2it 1 ε 1 2 h`d˚uà˜l t − i + ; pi1 d2ℓ , pi2 d2ℓ ; s + ε + 2it dt 1 2π ℓ1ℓ2 4 2 1 1 2 2 2 = 3 1 ε 2i R c Γ + ε + 2it cosh πt − iπ + !!2 4 2 1 1 M d ℓ 2 d ℓ 2 + O 1 1 + 2 2 . (8.21) 1 +ε c 2 T d2 d1 The contribution from the error term is O T 1−θ+εM . We will apply stationary phase to (8.21), and then the Hecke bound to the resulting sum. By Stirling’s formula, it suffices to consider the integral √ Z ∞ ±2it ∓2it it Z ∞ − 2 i1 2 − 2 2π ℓ1ℓ2 − t T 2t p d1ℓ1 − t T iψ(t) e ( M ) dt = e ( M ) e dt, i2 2 0 c e p d2ℓ2 0

with the phase function being √ i1 2 2t 2π ℓ1ℓ2 p d1ℓ1 ψ±(t) = ∓2t log 2t log + t log . i2 2 e c p d2ℓ2

We will consider only ψ+, since the analysis for ψ− is similar. We have √ 2π ℓ ℓ pi1 d2ℓ ψ′ (t) = −2 log(2t) + 2 log 1 2 + log 1 1 , + i2 2 c p d2ℓ2

and there is a stationary point at i1−i2 d1ℓ1 t0 = πp 2 . d2c At the stationary point we have i1−i2 d1ℓ1 ψ+(t0) = 2πp 2 , d2c and ′′ − 2 ψ+(t0) = . t0 ∈ − 6∈ − There are two cases: if t0 [T M log T,T + M log T ] or not. If t0 [T M log T,T + M log T ], then we apply the ﬁrst derivative test to get that the contribution in this case is O T 2−θ+ε .

135 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

Suppose that t0 lies in the range of integration. Then by Theorem B.0.3, we have

√ 1 +ε 1 + ε Z 2 i1 2 4 2 T +M log T − 2 2π ℓ1ℓ2 p d1ℓ1 − t T iψ (t) e ( M ) e + dt i2 2 c p d2ℓ2 T −M log T √ r √ 1 +ε 1 + ε r 2 i1 2 4 2 πi i1−i2 i1−i2 i1−i2 2π ℓ1ℓ2 p d1ℓ1 2π π d1ℓ1 d1ℓ1 à˚u¯sfi¯sfi˚ià‹nffl d1ℓ1 = e 4 p 4 e p 2 G πp 2 i2 2 T,M c p d2ℓ2 i 2 d2c d2c d2c + O (T ε) r √ 1 +ε 1 + ε 2 i1 2 4 2 πi i1−i2 i1−i2 i1−i2 2π ℓ1ℓ2 p d1ℓ1 − d1ℓ1 d1ℓ1 à˚u¯sfi¯sfi˚ià‹nffl d1ℓ1 = πe 4 p 4 e p 2 G πp 2 i2 2 T,M c p d2ℓ2 d2c d2c d2c + O (T ε) . 2−θ+ε The contribution from the error term is then O T . For the main term, we first break up the ℓ1-sum using a dyadic partition of unity, and it suffices to consider sums of the form X p i1−i2 i1−i2 λQ1 (ℓ1) d1ℓ1 ℓ1 à˚u¯sfi¯sfi˚ià‹nffl d1ℓ1 ℓ1S(ℓ1, ℓ2; c)e p 2 ω πp 2 , iτ GT,M ℓ1 d2c L d2c ℓ1≥1

T 2−θ+ε where ω has compact support in [1, 2], L 2 2 , and =(s) = τ. After applying Fourier inversion, we use d1d2ℓ2 the Hecke bound (Proposition 2.1.8) to get that X p i1−i2 i1−i2 λQ1 (ℓ1) d1ℓ1 ℓ1 à˚u¯sfi¯sfi˚ià‹nffl d1ℓ1 ε ℓ1S(ℓ1, ℓ2; c)e p 2 ω πp 2 cLT . iτ GT,M ℓ1 d2c L d2c ℓ1≥1 Making a trivial bound on the remaining sums, we find that the contribution from these terms is O T 2−θ+ε .

8.1.3.3.2 The contribution from small c

1− θ +ε Now suppose that c T 2 . In this case we will use the asymptotic expansion (A.11) of the J-Bessel function. In this case, after applying Stirling’s formula to the gamma factors, it suﬃces to consider the integral

Z ∞ − t−T 2 ( M ) te iψ±(t) 1 e dt, 0 2 4π 2 4 4t + c ℓ1ℓ2

where the phase function is √ 4π ℓ1ℓ2 i1−i2 d1ℓ1 ψ±(t) = ω2t + t log p 2 . c d2c

136 8.1. The second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals

In fact, this integral is essentially the same as appears in the corresponding contribution of Σ1. As we found there, we have ! − 2 Z ∞ − t T i1 i2 i1 i2 te ( M ) π p 2 d ℓ p 2 d ℓ p 2 d ℓ p 2 d ℓ eiψ±(t)dt = 1 1 − 2 2 e 1 1 + 2 2 1 i2 i1 i2 i1 4 c 0 2 4π 2 p 2 d2 p 2 d1 p 2 d2 p 2 d1 4t + c ℓ1ℓ2   !2 i1 i2 2+ε 1 π p 2 d ℓ p 2 d ℓ T × exp − 1 1 − 2 2 − T  + O . M 2 c i2 i1 M 2 p 2 d2 p 2 d1 7 − 3θ +ε 2 4 5 − θ T 6 4 +ε In this case the error term contributes O M 2 , which will be acceptable for M T . The main term, on the other hand, by Fourier inversion and the double Hecke bound (Proposition 2.1.9), contributes O T 2−θ+ε . This completes the proof of Lemma 8.1.3. □

137 Part IV

Future directions

In this part we make some extended remarks on possible future research directions. These include analogues of the results presented so far for diﬀerent families of L-functions, as well as adjacent topics.

138 Chapter 9

Future directions

Contents 9.1 Direct analogues ...... 140 9.2 An asymptotic expansion of the fourth moment of GL(2) L-functions at special points ...... 140 9.3 The ﬁrst moment of GL(2) × GL(3) L-functions at special points ...... 141 9.3.1 The mixed ﬁrst moment of GL(2) × GL(2) and GL(2) L-functions at special points 142 9.4 The prime geodesic theorem ...... 142 9.4.1 The prime geodesic theorem in short intervals ...... 143 9.4.2 The mean value of the error term in short intervals ...... 144 9.4.3 The prime geodesic theorem for hyperbolic 3-surfaces ...... 144 9.5 Special point subconvexity problems ...... 144 9.5.1 Special point subconvexity problems for GL(2)×GL(2) L-functions and an analogue to quantum unique ergodicity ...... 145 9.5.2 Special point subconvexity bounds for GL(2) L-functions ...... 146 9.6 Higher-order Fermi golden rules ...... 147 9.6.1 Spectral deformations and modular symbols ...... 147 9.7 The newform Weyl law ...... 148

In this chapter we collect some thoughts about future directions for the line of inquiry explored this document. There are many natural and seemingly straightforward problems one could consider, some of these more interesting than others, as well as other longer term projects on which one might embark. For example, it would probably not take too much effort to obtain similar nonvanishing results as in Chapters 7 where the holomorphic modular form Q is replaced by another Maass cusp form. Similarly, one could extend the results to square-free level, or to uj taken to have level as well—recall that we were only interested in a lower bound on the order of nonvanishing of the L-functions in question, so it sufficed to consider the cusp forms for Γ0(p) coming from the full modular group. Some of these problems we have already worked on, but encountered a difficulty that we had neither the patience nor the time to resolve. Each of these problems seems quite doable.

139 9.1. Direct analogues

9.1 Direct analogues

It seems reasonable to try to extend of all of the results in this dissertation to Kleinian groups, or for Hilbert modular forms. However, there is no longer a nice application in terms of deformation theory of surfaces, since by the Mostow rigidity theorem the C-structure on a complex manifold (satisfying mild assumptions) of dimension higher than one is unique. The analogue of the Kuznetsov trace formula is slightly more complicated in the Kleinian group setting, since the contribution from the discrete series is no longer as simple, nor is the theory of Bessel functions as well-understood. Another possibility is to try to improve the nonvanishing and simultaneous nonvanishing results by using the molliﬁer method. It would be interesting to see what can be said of the “inversion” that occurs for the Bessel function transforms appearing in the Kuznetsov trace formula with the extra twist in terms of the Kirillov model. Hopefully this would lead to a better theoretical explanation of the appearance of additive characters that replace the spectral twists.

9.2 An asymptotic expansion of the fourth moment of GL(2) L- functions at special points

One might wonder if the methods used in Chapter 8 to obtain an asymptotic expansion of the second moment of GL(2) × GL(2) L-functions at special points could be used to establish an asymptotic expansion of the fourth moment of GL(2) L-functions at special points. That is, can one establish an expansion of the form X −tj /T 2 4 2 2−δ e |νj(1)| |L(sj, uj)| = T P4(log T ) + O T ,

where P4 is some polynomial of degree 4 whose coeﬃcients are all independent of T , and δ > 0 is absolute? The fourth moment of a GL(2) L-function should be compared to be the second moment GL(2) × GL(2) L-function. One has X λ (n)τ (n) L(s, u )2 = ζ(2s) j 2 , for <(s) > 1, j ns n≥1

where τ2 denotes the binary divisor function. In this direction we have essentially carried through the exact same analysis as Luo uses in [Luo12] for the second moment of GL(2)×GL(2) L-functions, with the exception that one can no longer use the Hecke bound to achieve square-root cancellation at a crucial step, and that one needs to treat the cases where uj is even and odd separately. The latter issue does not complicate things much over the long interval, since the relevant integral transforms are all explicitly computable. For the issue of ﬁnding a suitable alternative to the application of the Hecke bound, one might hope to replace this with the GL(2)-type Voronoi summation in Theorem 2.1.7. However, there is a slight diﬃculty with the main term ∆(c, φ)—it is essentially of the size of the main term, but not very clean in appearance. Probably with a modest amount of work, one can either extract a cleaner-appearing main term from this term. The analogous question can be asked for a short interval as well.

140 9.3. The ﬁrst moment of GL(2) × GL(3) L-functions at special points

Remark 9.2.1. In many ways, a bound of this type is comparable to the asymptotic expansions of Moto- hashi ([Mot93], [Mot97], [Mot92]), Bruggeman-Motohashi ([BM01], [BM03], [BM05]), and Ivić-Motohashi ([IM95], [IM94]) of the fourth moment of the Riemann zeta function and Dedekind zeta functions: the simplest of these is essentially1 Z T 4 1 2 +ε ζ + it dt = TP4 (log T ) + O T 3 . 0 2

Similar problems were also considered by Young ([You11]) for Dirichlet L-functions.

9.3 The ﬁrst moment of GL(2)×GL(3) L-functions at special points

Since Young ([You13]) has already obtained a long interval upper bound for the second moment of GL(2) × GL(3) L-functions at special points, it seems natural to try to obtain an asymptotic expansion for the ﬁrst moment as well; doing so would yield an optimal order of nonvanishing of these L-functions. Working over the long interval seems like a more tractable starting point, since the integral transforms in Kuznetsov are

all explicitly computable in some way. This is important since we need to treat the cases where uj is even and odd separately since the root number is not constantly 1 as in the GL(2) × GL(2) setting. One would hope to obtain an expansion of the form X −tj /T 2 2 2−δ+ε e |νj(1)| L(sj, uj ⊗ ψ) = cψT + O T

for some δ > 0 absolute and cψ possibly depending on the GL(3) automorphic form ψ, depending on how one chooses to normalize. Actually, power-saving is not necessary—any saving in the error term will suffice. However, it seems there are some slight difficulties that arise. The natural length of the sums in the 3 2 +ε approximate functional equation for these L-functions are tj , so there is some “room” between the corresponding first moment of GL(2)×GL(2) L-functions and the second moment of GL(2) ×GL(2) L-functions, so one has some hope. However, one sees that the six gamma factors on the dual side will force one to take the original sum to be longer than T 2 to achieve a power-saving error term, while, at the same time, application of GL(3) Voronoi summation to the original sum requires it to be shorter than T 2 for the dual Voronoi sum to be negligible2. One doesn’t need the dual Voronoi sum to be negligible necessarily, but the cost is that one must extract extra cancellation from the character sums that appear. Perhaps there is some input from the work of Conrey-Iwaniec ([CI00]) or Blomer ([Blo12b]) that can be adapted here.

1In [IM95], Ivić-Motohashi provide a more precise estimate of the error term. 2In the second moment result, we need to take the approximate functional equation sums longer than T 2 from the outset, but because one is working with two L-functions, the phases from the gamma factors on the dual side cancel each other out, so that one doesn’t need to take that sum to be so short. Also, one deals with the case where one sum is longer than T 2 by the trick of Luo of applying the functional equation to one of the L-functions.

141 9.4. The prime geodesic theorem

9.3.1 The mixed ﬁrst moment of GL(2) × GL(2) and GL(2) L-functions at special points

If one wishes to study a problem similar to the ﬁrst moment of GL(2) × GL(3) L-functions at special points, one may consider the mixed ﬁrst moment of GL(2) × GL(2) and GL(2) L-functions at special points. One considers sums of the form X −tj /T 2 e |νj(1)| L(sj, uj ⊗ Q)L(sj, uj).

tj uj even

3 +ε ⊗ 2 The natural length of the sums in the approximate functional equation for L(sj, uj Q)L(sj, uj) is tj , and the issue of treating the even and odd forms separately is present. In fact, one sees that in order to avoid diﬃculties with the transition range of the K-Bessel function, one takes the length of one sum to be longer than T 2. The issue of balancing the dual sum, however, is much simpler in the mixed ﬁrst moment case: since we have two L-functions, we can use Luo’s trick to shorten the relevant sum to which we apply Voronoi to be shorter than T 2, and the integral transforms for the dual approximate functional equation sum have better bounds because the phases of the gamma factors are partially balanced.

9.4 The prime geodesic theorem

This is maybe one way to see the utility of the extra spectral twists for higher rank settings. The relationship between the geometry and spectral theory (Selberg trace formula) of the hyperbolic n-surface in question still holds, but there is no longer any deformation theory. The spectral twists control the error term in the

prime geodesic theorem; let πΓ denote the prime geodesic counting function for Γ, namely X πΓ(x) = 1

P0 primitive hyperbolic N(P0)≤x

Here the sum is over primitive hyperbolic conjugacy classes P0 ∈ Γ with norm up to x. In some sense the hyperbolic conjugacy classes determine the geometry of Γ \ H; they describe the length spectrum of the surface.

As with the proof of the prime number theorem, one can reduce statements about the asymptotics of πΓ by partial summation to statements about the asymptotics of the summatory function of the analogue of the von Mangoldt function, deﬁned to be X ψΓ(s) = ΛΓ(P ), P hyperbolic N(P )≤x where ΛΓ denotes the von Mangoldt function   log N(P ) if P is a power of a primitive P0, ΛΓ(P ) =  0 otherwise.

142 9.4. The prime geodesic theorem

By Perron’s formula, one can express ψΓ in terms of the inverse Mellin transform of the logarithmic derivative of the Selberg zeta function for Γ, and then one applies standard methods from complex analysis to derive

an explicit formula for πΓ. The Selberg zeta function has the “Euler product” deﬁnition

!−1 Y Y 1 ZΓ(s) = 1 − . (N(P ))s+k P0 k≥0 0

The explicit formula is X 1 +it 1+ε x 2 j x ψΓ(x) = x + 1 + O , + itj T |tj |≤T 2

1 where 1 ≤ T x 2 . As one can see, the quality of the error term in this case is controlled by the cancellations in the sum over the discrete spectrum; compared to the prime number theorem, one needs to achieve

cancellation in the sum over the zeros of ZΓ(s) since there are many more of them (by Weyl’s law) than there are nontrivial zeros of ζ(s). (The main term comes from the residual spectrum.) Iwaniec achieved the bound X it 11 +ε x j T x 48

tj ≤T by appealing to the Kuznetsov trace formula. This allowed him to establish 35 +ε ψΓ(x) = x + O x 48 . 7 +ε Luo-Sarnak ([LS95]) later improved the error term to O x 10 , and this was extended to congruence 25 +ε subgroups by Luo-Rudnick Sarnak ([LRS99]). The current best bound on the error term is O x 36 , due to Soundararajan-Young ([SY13]).

9.4.1 The prime geodesic theorem in short intervals

If one wishes to study the distribution of primitive hyperbolic conjugacy classes with norm in a shortened interval, one is led to consider the restricted von Mangoldt summatory function X ψΓ(x, y) = ΛΓ(P ) x≤N(P )≤x+y X xsj x + y x = y + + O + , sj T M T −M≤tj ≤T

1 1 where 1 ≤ M < T x 2 and M y 2 . The quality of the error term becomes an exercise in extracting the cancellations in the short interval twisted spectral sum X xitj ,

T −M≤tj ≤T +M

143 9.5. Special point subconvexity problems

and one is led to appeal to the Kuznetsov trace formula, as before.

Alternatively, one can reexpress ψΓ(x) in terms of the class numbers of real quadratic extensions of Q using the class number formula. The primitive hyperbolic conjugacy classes whose norm is the minimal solution√ to Pell’s equation of discriminant D are in bijection with the ideal classes in the ideal class groupp of Q( D), so that πΓ can be interpreted as the summatory function counting the class numbers of Q( (D)) ordered by their fundamental units. Using this approach, Bykovskii ([Byk94]) establishes an asymptotic 1 +ε formula of ψSL(2,Z)(x, y) for x 2 ≤ y ≤ x. In addition to improving the error term over long intervals, 1 2+ε Soundararajan-Young ([SY13]) improve the lower bound on the length to x 2 (log x) ≤ y ≤ x, assuming the generalized Riemann hypothesis.

9.4.2 The mean value of the error term in short intervals

Along the lines of [CG18], one considers the second moment of the error term in the prime geodesic theorem:

Z 2 2A X s s 1 (x + y) j − (x − y) j ψΓ(x, y) − dx, A sj A 1 ≤ 2

δ 1 where A 1 and y = x with δ > 2 . For the modular group, it is relatively straightforward to show that this reduces to estimates of the twisted spectral sums

Z ( ) 2 − 2 2A X tj T 1 − it e M x j dx. A A tj

Using the asymptotic expansion appearing in Proposition B.0.4, and ignoring the analysis of the error terms, the main term of these appears to be admissible, although we have not carried out any of the details.

9.4.3 The prime geodesic theorem for hyperbolic 3-surfaces

The error term in the prime geodesic theorem for hyperbolic 3-surfaces has been studied by Balkanova, Chatzakos, Cherubini, Frolenkov, and Laaksonen ([BCC+19]). It would be interesting to try to extend their results for short intervals.

9.5 Special point subconvexity problems

As mentioned in Chapter 3, the subconvexity problem at the special point becomes more complicated because of the conductor-dropping phenomenon that occurs there: if one chooses to use the moment methods, one must either consider much higher moments or amplify a high moment3. The only known family of L-functions for which a conductor-dropping subconvexity bound is known is for GL(2) L-functions, following the work of Michel-Venkatesh ([MV10]) and Wu ([WA18]).

3If one wants to achieve a special point weak subconvexity estimate, then one may try to push through the methods of Soundararajan-Thorner ([ST19]). For the Hecke-Maass L-functions, one would need a zero-density estimate for L(s, uj ) near sj .

144 9.5. Special point subconvexity problems

9.5.1 Special point subconvexity problems for GL(2)×GL(2) L-functions and an analogue to quantum unique ergodicity

There is an analogue of quantum unique ergodicity (QUE) that can be reduced to subconvexity bounds for GL(2) × GL(2) L-functions at special points. By the spectral theorem, QUE can be reduced to studying the decay of the inner products Z dµ(z) hψ, dµ (z)i = ψ(z)|u (z)|2 , j j k k2 Γ\H uj where ψ is a smooth compactly supported function on SL(2, Z) \ H. By considering incomplete Poincaré series instead, one can reduce the estimates of these inner product to the decay of shifted convolution sums of the Fourier coeﬃcients of GL(2) automorphic forms. In [You16], Young achieves a power-saving error term for the binary shifted convolution of divisor func-

tions τit(n).

Theorem 9.5.1 (Young, [You16]). Let ω be a smooth function on (0, ∞) with support in [T, 2T ], and (j) −j satisfying ω (x) j T . Then for m > 0 we have X τiT (n)τiT (n + m)ω(n) = M(T ) + E(T ), n∈Z

where M(T ) is the main term Z X | |2 ∞ ζ(1 + 2iT ) ∓iT ±iT M(T ) = σ−1(m) (x + m) x ω(x)dx ± ζ(2) 0 Z X 2 ∞ ζ(1 2iT ) ∓iT ∓iT + ∓ σ−1±4iT (m) (x + m) x ω(x)dx ± ζ( 4iT ) 0

θ 5 +ε 11 +ε and the error term satisﬁes E(T ) m T 6 + T 12 , where θ is the best-known exponent towards the 7 Ramanujan-Petersson conjecture for GL(2) (currently θ = 64 ).

The proof uses the method of Motohashi to reduce the problem to subconvexity bounds for L 1 + it, E ⊗ u , 2 t ℓ 1 ⊗ 1 ⊗ → ∞ L 2 + it, Et Q , and L 2 + it, Et Er , as t with the spectral parameters of uℓ, Q, and Er in some manageable range. Since these L-functions have a special factorization in terms of the product of two GL(2) L-functions, the problem is reduced to known subconvexity bounds for GL(2) L-functions: for example, one has 1 1 1 L + it, E ⊗ Q = L ,Q L + 2it, Q . 2 t 2 2 The key starting point used to reduce to subconvexity bounds for these L-functions is the special approximate functional equation for the divisor function X X −iT S(m, 0; ℓ) ℓ iT S(m, 0; ℓ) ℓ τ (m) = m f √ + m f− √ , iT ℓ1−2iT 2iT m ℓ1+2iT 2iT m ℓ≥1 ℓ≥1

145 9.5. Special point subconvexity problems

where Z 1 −w G(w) fξ(x) = x ζ(1 − ξ + w) dw, 2πi (ε) w and G(w) = exp(w2). After careful cleaning of the resulting sums, Young applies the functional equation for the Estermann zeta function (amounts to Voronoi summation) and the Kuznetsov trace formula, which results in the appearance of the aforementioned L-functions. Analogously, one may wish to consider the false “inner products” Z h i 1 dµ(z) ψ, dµj(z) = ψ(z)uj(z)E + itj, z h i, Γ\H 2 uj,Etj which, as in the normal situation, can be related to mixed shifted convolution sums of the Fourier coeﬃcients

λj(n) and τitj (n): one is led to consider sums of the form X

λj(n)τitj (n + m)ω(n), n∈Z

where ω is a smooth function as in Theorem 9.5.1. Following the method of Motohashi again, the shifted convolution sums can be further reduced to subconvexity bounds at the special point for the L-functions

L(sj, uj ⊗ uℓ), L(sj, uj ⊗ Q), and L(sj, uj ⊗ Er), with uℓ, Q, and Er as in the binary shifted convolution sums case.

In the past we have studied the subconvexity problem for L(sj, uj ⊗ Q) using the amplification method. By Luo’s short interval large sieve inequality one already has an optimal bound for the unamplified sums, yielding the convexity bound, so one aims to achieve a bound on amplified sums of the form

2 X X 2 1+ε amλj(m) |L(sj, uj ⊗ Q)| T M,

N δ where (am) ∈ C and M = T for some small δ > 0. It seems that some additional input or averaging is needed, since the standard techniques all seem to return the convexity bound.

9.5.2 Special point subconvexity bounds for GL(2) L-functions

One consequence of Chandee-Li’s upcoming result ([CL20]) is the convexity bound for GL(2) L-functions at the special point. It would be interesting to see if there is any way to apply the ampliﬁcation method to their proof to obtain a subconvexity bound. Incidentally, when we began working on the short interval analogue of Young’s upper bound on the sixth moment upper bound for GL(2) L-functions at special points, we had hoped that if one could shorten 1 the interval down to M = T 2 , then one would have the hope of using ampliﬁcation to obtain a special

point subconvexity bound for L(sj, uj). However, we neglected the error terms coming from the stationary phase analysis of the integral transform appearing in the Kuznetsov trace formula. In fact, it seems like if one were to rework the proof of the short interval large sieve inequality and keep the lower order error

146 9.6. Higher-order Fermi golden rules

terms explicit, one could probably improve the result substantially, since the bounds made were done for arbitrary coeﬃcients, and conceivably one could still apply Voronoi summation. Furthermore, the work of Deshouillers-Iwaniec suggests that when additional cancellation is available, the delicate cancellation of the continuous spectrum main term and the geometric main term is no longer necessary.

9.6 Higher-order Fermi golden rules

The condition that L(sj, uj ⊗ Q) =6 0 is suﬃcient for the cusp form uj to be destroyed on quasiconformal

deformation of Γ\H, but it is not necessary. In fact, there are hypotheses that guarantee that L(sj, uj ⊗Q) =

0: for example, if the surface Γ \ H has genus higher than the multiplicity of the eigenvalue λj, then one of the cusp forms in the λj-eigenspace of Γ \ H is guaranteed to satisfy L(sj, uj ⊗ Q) = 0. A natural question is whether there exist additional conditions that are equivalent to the destruction of the cusp form under quasiconformal deformation.

In Chapter 4, we saw that the special values L(sj, uj ⊗Q) appear in the Fermi golden rule for quasiconformal deformations of Γ \ H. Petridis-Risager ([PR13]) study higher order Fermi golden rules that provide the additional conditions that are equivalent to the destruction of cusp forms under other types of deformations.

9.6.1 Spectral deformations and modular symbols

For the moment, instead of quasiconformal deformations of Γ \ H, let us consider the spectral decomposition with respect to ∆ as the multiplier system changes. One considers the characters attached to modular

symbols h·, ·i :Γ × S2(Γ \ H) → C given by Z γz0 hγ, Qi = −2πi Q(z)dz, z0

where Q is a normalized weight 2 holomorphic cusp form for Γ and γ ∈ Γ, and the deﬁnition is independent ε⟨γ,Q⟩ of the choice of z0. These provide a family of characters χε :Γ → C given by χε(γ) = e . One considers the parametrized spaces

2 L (Γ \ H, χε) = {f : kfk < ∞ and f(γz) = χε(γ)f(z) for all γ ∈ Γ}.

The (formally) perturbed Eisenstein series X s Eε(z, s) = χε(γ)(=(γz)) , for <(s) > 1,

γ∈Γ∞\Γ

span the continuous spectrum. By construction, these have the special feature that their variation at zero are the Eisenstein series attached to modular symbols that were studied by Goldfeld. Namely,

d X [E (z, s)] hγ, Qi (=(γz))s = E∗(z, s). dε ε ε=0 γ∈Γ∞\H

147 9.7. The newform Weyl law

Goldfeld conjectured that the distribution of the modular symbol is controlled by the residues of E∗(z, s) at s = 1. Importantly, by the Eichler-Shimura correspondence his conjecture is equivalent to Szpiro’s conjecture for elliptic curves over Q. 2 Petridis showed that in the spectral deformations L (Γ \ H, χε), the perturbed Eisenstein series have scattering poles at the embedded eigenvalues on the condition of nonvanishing of the ﬁrst variation inner product being nonzero. Similar to the Phillips-Sarnak theory, the inner product decomposes in terms of GL(2) × GL(2) L-functions at special points.

Again, the nonvanishing of the L-function at sj implies the cusp form is destroyed, but it is not a necessary condition. In [PR13], Petridis-Risager show that the derivatives of perturbed Eisenstein series yield higher order Fermi golden rules that provide conditions equivalent to the destruction of a cusp form

under deformations by χε. It would be interesting to try to extend the results of Luo and others in this setting.

9.7 The newform Weyl law

Recently Petrow-Young ([PY19]) have developed a version of the Kuznetsov trace formula restricted to newforms. It would be interesting to consider the analogous nonvanishing problems for GL(2) × GL(2) L-functions as both the level and the spectral parameter vary.

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156 Part V

Appendices

157 Appendix A

Compendium of special functions and their properties

Contents A.1 Bessel functions ...... 158 A.1.1 The standard Bessel functions ...... 158 A.1.2 The modiﬁed Bessel functions ...... 159 A.1.3 Spectral theory of D+ on R+ ...... 159 A.1.4 Spectral theory of D− on R+ ...... 161 A.1.5 Properties and integral representations ...... 161 A.1.6 Asymptotic behavior ...... 161

A.1 Bessel functions

The Bessel functions arise naturally in the automorphic setting when ﬁnding expansions of GL(2, R) automorphic forms in rectangular coordinates. Their properties are important for understanding the relationship between an automorphic form and its L-function. Owing to this relationship, they also appear naturally in the various integral transforms in the Kuznetsov trace formula. (the results and framework presented in this section come from the excellent exposition in Iwaniec’s book on spectral theory of automorphic forms)

A.1.1 The standard Bessel functions

The Bessel functions of the ﬁrst kind, the J-Bessel functions {Jν }ν∈C, are the eigenfunctions of the diﬀerential operator d2 d D+ = z2 + z + z2I, dz2 dz

158 A.1. Bessel functions

with eigenvalue ν2, and they have the power series expansion

∞ X ( 1 z2)k J (z) = ( 1 z)ν (−1)k 4 (A.1) ν 2 k!Γ (ν + k + 1) k=0

for z ∈ C \ (−∞, 0]. Meanwhile, for ﬁxed z =6 0, Jν (z) is entire in ν. 2 The two linearly independent subspaces of the ν -eigenspace are spanned by Jν (z) and J−ν (z), respec- n tively, when ν 6∈ Z. When ν ∈ Z, Jν is entire in z and J−n(z) = (−1) Jn(z), so the two solutions are no longer independent. In this case, the other solutions are the modiﬁed Bessel functions of the second kind,

the Y -Bessel functions {Yν }ν∈C, which are deﬁned by

cos(νπ)J (z) − J− (z) Y (z) = ν ν , ν sin(νπ)

and for integer orders n ∈ Z, one deﬁnes Yn(z) = limν→n Yα(z).

A.1.2 The modiﬁed Bessel functions

Analogously, the modified Bessel functions of the first kind, the I-Bessel functions {Iν }ν∈C, are the eigenfunctions of the differential operator

d2 d D− = z2 + z − z2I, dz2 dz with eigenvalue ν2. They have the power series expansion

∞ X ( 1 z2)k I (z) = ( 1 z)ν 4 = i−ν J (iz) ν 2 k!Γ (ν + k + 1) ν k=0 for z ∈ C \ (−∞, 0]. The situation is similar to that for the standard Bessel functions, and one deﬁnes the modiﬁed Bessel functions of the second kind, the K-Bessel functions, by

π I− (z) − I (z) K (z) = ν ν , (A.2) ν 2 sin(νπ)

with the limiting value being taken for ν ∈ Z.

A.1.3 Spectral theory of D+ on R+

Via the Kuznetsov trace formula, one sees that the spectral theory of Kloosterman sums is intimately connected to the spectral theory of D+ on the multiplicative group R+. The Bessel functions of the ﬁrst kind form a complete eigenpacket for D+ on L2(R+). The odd integer order J-Bessel functions form the discrete spectrum of D+; for f ∈ L2(R+) one deﬁnes Neumann series X f S(x) = (2k)Nf (2k − 1)J2k−1(x), k>0

159 A.1. Bessel functions where the Neumann coeﬃcients are deﬁned by Z ∞ dx Nf (ℓ) = f(x)Jℓ(x) . 0 x

Meanwhile, by the orthogonality property A.6, the subspace spanned by {B2it}t>0, where

− J−ν (x) Jν(x) Bν (x) = πν , 2 sin 2 forms the orthogonal complement of the subspace spanned by the odd integer order J-Bessel functions. One defines the Titchmarsh integral by Z tdt f T(x) = J2it(x)Tf (t) , R cosh(πt) where the Titchmarsh coefficients are defined by

Z ∞ J−2it(ξ) − J2it(ξ) dξ Tf (t) = f(ξ) . 0 sin(πt) ξ

One has the Sears-Titchmarsh inversion formula (also regarded as a consequence of Hankel inversion)

f(x) = f S(x) + f T(x) for suﬃciently nice f. In fact, the above is really a consequence of Hankel inversion; denoting the Hankel transform (of order 0) Hf (u) by Z ∞ Hf (u) = f(y)J0(uy)dy, 0 one has Z ∞ f(x) = Hf (y)J0(xy)xydy, 0 from which one sees that Z 1 f S(x) = uxJ0(ux)Hf (u)du, 0 and Z ∞ f T(x) = uxJ0(ux)Hf (u)du. 1 This framework is important when considering extensions to Kuznetsov trace formula to Kleinian groups, for example.

160 A.1. Bessel functions

A.1.4 Spectral theory of D− on R+

The spectral theory of D− on R+ is comparatively simpler than that of D+. One has the Kontorovich- Lebedev inversion Z sinh(πt) f(x) = Lf (t)Kit(y) 2 tdt, (A.3) R π

where the coeﬃcients are Z ∞ dy Lf (t) = Kit(y)f(y) . 0 y

A.1.5 Properties and integral representations

The J-Bessel function has the Poisson integral representation

ν Z z π 1 2 2ν < − Jν (z) = 1 1 cos (z cos θ) sin (θ)dθ, for (ν) > . (A.4) Γ 2 Γ ν + 2 0 2

Similarly, the I-Bessel function has the integral representation

ν Z z π 1 2 z cos θ 2ν < − Iν (z) = 1 1 e sin (θ)dθ, for (ν) > . (A.5) Γ 2 Γ ν + 2 0 2

For the K-Bessel function, one has Z 1 − tu Kit(x) = πt cos (x sinh(u)) e du. 2 cosh 2 R 2π

Amongst integrals involving two Bessel functions, of particular importance is − Z ∞ π(µ ν) dx 2 sin 2 J (x)J (x) = , (A.6) ν µ 2 − 2 0 x π µ ν

for =(ν + µ) > 0. From this one sees that the J-Bessel functions with orders satisfying µ − ν ∈ 2Z and =(ν + µ) > 0 are orthogonal to one another with respect to the inner product on R+.

A.1.6 Asymptotic behavior

The asymptotic expansions in this section can be found in [Wat95]. For small values of z = x ∈ R+, the first couple terms of the power series defining each of the Bessel functions provide efficient approximations. For large x > 0 and fixed order ν, the J-Bessel function has the approximation

1 2 2 πν π 1 J (x) = cos x − − + O . (A.7) ν πx 2 4 x

161 A.1. Bessel functions

In fact, for integer orders k, we have the explicit form of (A.7),

ix −ix Jk(x) = e Wk(x) + e Wk(x), (A.8) where Wk(x) is the Watson-Whittaker function, which has the integral representation

iπk iπ 1 Z − 1 − 2 ∞ k 2 e 2 4 2 −y iy Wk(x) = 1 e y 1 + dy. (A.9) Γ k + 2 πx 0 2x

For x 1 and j ≥ 0, the Watson-Whittaker function satisﬁes the bound

(j) 1 Wk (x) k,j 1 . (A.10) xj(1 + x) 2

For purely imaginary ν = ir with r > 0, the J-Bessel function has the uniform asymptotic expansion   πr − iπ +iω (x) X e 2 4 r t − L+1 √  ℓ 2 2 2  Jir(x) = 1 ℓ + O r + x (A.11) 2 2 4 2π (r + x ) 0≤ℓ≤L (r2 + x2) 2 where the tℓ are certain absolute constants, with t0 = 1, L > 1, and the phase function ωr(x) is given by √ ! p r2 + x2 − r ω (x) = r2 + x2 + r log . (A.12) r x

For r < 0, the asymptotic expansion follows from Jir(x) = J−ir(x), and we have   πr iπ − + −iω− (x) X e 2 4 r t − L+1 √  ℓ 2 2 2  Jir(x) = 1 ℓ + O r + x . (A.13) 2 2 4 2π (r + x ) 0≤ℓ≤L (r2 + x2) 2

For K-Bessel functions Kir of purely imaginary order, a similar expansion holds, but there is an issue at the transition range where r ∼ x. For x > r and L > 1 one has

√ ! − x2−r2−t arcsin r LX−1 e ( x ) b K (x) = ℓ + O x−L , (A.14) ir 1 1 ℓ 2 2 − 2 4 2 2 2 2 (x r ) ℓ=0 (x − r ) while for x > r, ! − πr LX−1 ℓ LX−1 −ℓ e 2 iπ −iκ (x) i cℓ − iπ +iκ (x) i cℓ −L K (x) = e 4 r + e 4 r + O x , (A.15) ir 1 1 ℓ ℓ 2 2 − 2 4 2 2 2 2 2 2 i2 (r x ) ℓ=0 (r − x ) ℓ=0 (r − x ) where the phase function κr(x) is given by √ ! p r2 − x2 − r κ (x) = r2 − x2 + r log . (A.16) r x

162 A.1. Bessel functions

When x and t are both allowed to vary, the issue arises in the transition range when x ∼ t, for which the asymptotic behavior is unclear.

163 Appendix B

Asymptotic behavoir of oscillatory integral

Contents B.0.1 Tools for analyzing oscillatory integrals ...... 164 B.0.2 A pseudo-Sears-Titchmarsh inverse for short interval weight function ...... 166 B.0.2.1 The behavior of a special Titchmarsh integral ...... 166 B.0.2.2 Bounds for Kloosterman integral for large values of c ...... 169

B.0.1 Tools for analyzing oscillatory integrals

The simplest nontrivial available requires a lower bound on the size of a derivative of the phase function. The van der Corput bound is useful when one does not have the required hypotheses for the nonstationary phase analysis described below, and one does not want or need the full strength of stationary phase expansion, or maybe it isn’t available either.

Theorem B.0.1. Let f be a smooth function on [α, β] such that f (k)(x) λ for all x ∈ (α, β) and for some k ≥ 2 (or k = 1 and f monotone) and λ > 0. Then for any smooth weight function ρ on [α, β], Z Z ! β β if(x) 1 | | | ′ | ρ(x)e dx 1 ρ(β) + ρ (x) dx . (B.1) α λ k α

Stationary phase analysis essentially uses the Fourier transform of the Gaussian, combined with Taylor series expansion of the phase function, which is assumed to satisfy certain technical conditions that guarantee the above, to extract a main contribution of an oscillatory integral; the integral mass accumulates at places

164 where the parametrization of motion around S1 given by the oscillatory factor eiψ(t) slows and reverses direction. The principle of nonstationary phase, on the other hand, is useful in situations where one can guarantee the derivative of the phase function is sufficiently large and the weight function sufficiently flat so that many repeated applications of integration by parts show that the integral is negligible. The following two theorems come directly from [BKY13]; the principle of nonstationary phase is phrased in the following.

Theorem B.0.2 (Blomer-Khan-Young, [BKY13]). Let Y > 1. Suppose that ρ is a smooth function on R with support on [α, β] satisfying X ρ(j)(t) j U j for j ≥ 0. Suppose that h is a smooth function on [α, β] such that

|h′(t)| ≥ R

for some R > 0, and such that Y h(j)(t) j Qj for j ≥ 2. Then the integral deﬁned by Z I = ρ(x)eiψ(x)dx R

satisﬁes ! Y A I (β − α)X + (RU)−A . QR

The version of stationary phase that appears in [BKY13] is convenient because it includes the explicit contributions of lower order terms in the asymptotic expansion, beyond the well-known main term; when cares primarily about the phase of the main term it can be convenient to use this form, since by taking suﬃciently many lower order terms, one obtains a weight function that shares the same essential features as the original weight function, and the error term can be taken to be negligible.

1 Theorem B.0.3 (Blomer-Khan-Young, [BKY13]). Let 0 < ε1 < 10 , X, Y, V, Q > 0,Z := Q+X +Y +V +1, and assume that ε1 QZ 2 ≥ 3ε1 ≥ Y Z and V 1 . Y 2 R (j) X Let ρ be a smooth function on with support on an interval of length V satisfying ρ (x) j V j for all ′ j ≥ 0. Let ψ be a smooth function with a unique point x0 in supp(ρ) where ψ (x0) = 0, and such that

165 (j) Y ≥ ψ (x) j Q2 for all j 1 on supp(ρ). Suppose furthermore that we have the lower bound

1 +η Y 2 ψ′′(x) Q2

1 ≤ 1 for some 3 < η 2 . Then the integral I deﬁned by Z I = ρ(x)eiψ(x)dx R has the asymptotic expansion √ πi iψ(x0)+ X j 2pπe 4 1 i (2j) −B I = ′′ ′′ g (x0) + OB,ε1 Z , j! 2ψ (x0) ψ (x0) j≤B where g(x) = ρ(x)eiτ(x), with

ψ′′(x ) τ(x) = ψ(x) − ψ(x ) − 0 (x − x )2. 0 2 0

B.0.2 A pseudo-Sears-Titchmarsh inverse for short interval weight function

Because several of the results throughout this document use similar spectral weight functions, we perform the relevant analysis here and collect the results in a single theorem. For the shortened interval problems, we denote the weight function by

− 2 2 −( t T ) it −( t+T ) −it Th(t; y) = hT,M (t, y) = e M y + e M y − à˚u¯sfi¯sfi˚ià‹nffl it à˚u¯sfi¯sfi˚ià‹nffl − it = GT,M (t)y + GT,M ( t)y .

Note that hT,M is essentially supported for t ∈ [−T − M log T, −T + M log T ] ∪ [T − M log T,T + M log T ], − log T since outside of this region we have the bound hT,M (t, y) T . Let hT(x, y) denote the corresponding Titchmarsh transform, Z 2i tdt hT(x, y) = J2it(x)hT,M (t, y) . π R cosh(πt) √ 4π mn T Recall that x = c usually. The behavior of h (x, y) depends greatly on the size of x.

B.0.2.1 The behavior of a special Titchmarsh integral

For large values of x, the behavior is similar to that of the Sears-Titchmarsh inverse.

166 − 1 Proposition B.0.4. Let T, M, x > 0 with T ε (max{T, x}) 2 M T . Suppose that y > 0 and x 1 −ε T y + y T . Denote η(y) = sgn(log(y)). If y = 1, for suﬃciently large T we have

−A hT(x, y) A T . (B.2)

Meanwhile, if y =6 1, then for any K 1, we have the asymptotic expansion −K hT(x, y)) = h”m`a˚i‹nﬄ(x, y)pK (x, y) + O T , (B.3)

where the main term is ( ) iη(y)x 1 − 1 x 1 − 1 y 2 +y 2 x 1 − 1 h”mà˚i‹nffl(x, y) = y 2 − y 2 e 2 à˚u¯sfi¯sfi˚ià‹nffl y 2 − y 2 , 4 GT,M 4 and the weight function is KX−1 pK (x, y) = pK,ℓ(x, y) ℓ=0 with pK,0(x, y) = 1 and the lower order terms are defined in terms of the lower order terms of the asymptotic expansion (A.11) of J2it(x) and the lower order functions described in Theorem B.0.3.

By extracting certain problematic terms appearing in the lower order weights pK,ℓ(x; y) with ℓ ≥ 1, we can alternatively express (B.3) as max{T, x} hT(x, y) = h”mà˚i‹nffl(x; y) 1 + h˜bâ`dffl(x; y) + O , (B.4) M 2

with the distinguished problematic term    2 1 − 1 x y 2 − y 2  ix   1 − 1  h˜bˆa`dﬄ(x; y) = exp  − T y 2 + y 2  − 1. (B.5) 8M 4 4

à˚u¯sfi¯sfi˚ià‹nffl Remark B.0.1. The weight function GT,M (t) can be replaced with any suitable weight function that satisfies the requirements of B.0.3. The error term will change accordingly, but the essential features in the main term, namely, the phase, will be exactly analogous with the above.

à˚u¯sfi¯sfi˚ià‹nffl Proof. We begin by performing some preparatory cleaning of the integral. By the rapid decay of GT,M (t) and a change of variables, we have Z X ∞ it tdt −A T ± à˚u¯sfi¯sfi˚ià‹nffl h (x, y) = J 2it(x)GT,M (t)y + O T (B.6) ± 0 cosh(πt)

,+ for any A > 0. We will deal with the integral with kernel J2it, which we denote by hT (x, y); the analysis − of hT, (x, y) is slightly diﬀerent, but analogous. The main contributions when η(y) = 1 will come from

167 ± hT, (x, y), while for η(y) = 0 we will prove (B.2).

By the uniform asymptotic expansion (A.11) of Jir(x) and considering the integral arising from the ℓ = 0 terms there, up to a negligible error, we make a further reduction to studying Z

T,+ iψ+(t;x,y) h0 (x, y) = ρ(t; x)e dt. (B.7) R

Here we have denoted the weight function by

t à˚u¯sfi¯sfi˚ià‹nffl ρ(t; x) = 1 GT,M (t) (4t2 + x2) 4

and the phase function by

iω2t(x)+it log y ψ+(t; x, y) = e ,

where ωr(x) is deﬁned in (A.12). The terms with ℓ ≥ 1 in the expansion (A.11) of Jir(x) are handled analogously and we omit their analysis here; these comprise the lower order terms appearing in (B.3). By Leibniz’s rule, we see that the derivatives of the weight function satisfy

j (j) T (log T ) ρ (t; x) j 1 (B.8) (max{T, x}) 2 M 2j

T,+ in the essential range of h0 (x, y). In fact, we can separate out a distinguished problematic term that yields (B.5), writing ! (−1)jt 2(t − T ) j (log T )j (j) à˚u¯sfi¯sfi˚ià‹nffl ρ (t; x) = 1 2 GT,M (t) + O 1 . (B.9) (4t2 + x2) 4 M (max{T, x}) 2 M 2j

For the phase function ψ+(t) = ψ+(t; x, y), we have √ ! 4t2 + x2 − 2t ψ′ (t) = 2 log + log y, (B.10) + x so the phase function has a stationary point at x 1 − 1 t = y 2 − y 2 . (B.11) 0 4

Additionally, at t0 there is the relation p x 4t 2 + x2 = 2t + √ . (B.12) 0 0 y

The second derivative is −4 ψ′′ (t) = √ , (B.13) + 4t2 + x2

168 and this, combined with (B.10), shows that ψ(t) satisﬁes the relation

−4 ′ ψ+(t) = ′′ + tψ+(t), ψ+(t) as well as the diﬀerential equation ′′ 3 − ′′′ t ψ+(t) = 8ψ+ (t). (B.14)

From (B.14) we see that the derivatives of the phase function satisfy

n−2 (n) T −n+1 ψ+ (t) n 2n−3 T (B.15) 2 nm 2 T + c2 in the support of ρ(t; x). By (B.12), we see that x 1 1 ψ (t ; x, y) = y 2 + y 2 . (B.16) + 0 2 Combining the bounds (B.8) and (B.15) with (B.16), when y > 1 we apply Theorem B.0.3 to obtain

KX−1 − T,+ ”mà˚i‹nffl K h0 (x, y) = h (x; y) pℓ(x; y) + O T (B.17) ℓ=0 with the weight functions being defined as in Theorem B.0.3. Meanwhile, for 0 < y < 1 we see that t0 lies outside the range of integration, and we apply Theorem B.0.2 to see that

− T,+ A h0 (x, y) A T

− T, for any A > 0; when 0 < y < 1 the main contribution will come from h0 (x, y) instead. Similar analysis when η(y) = −1 will yield (B.3) in that case. Furthermore, by (B.9), in both cases we

can extract the problematic terms that appear in each ρℓ(x; y) with ℓ ≥ 1 and we ﬁnd that together these comprise the Taylor series for (B.5). This completes the proof.

B.0.2.2 Bounds for Kloosterman integral for large values of c √ 4π mn ε−1 For small values of x = c , particularly, when c NT , we use a contour-shifting argument to show that I is negligible and satisﬁes I T −A for any A > 0. This is a standard argument for short intervals—see [Liu17], [Luo96], or [Xu14].

Proposition B.0.5. Let M T , and let y > 0. Suppose that

T 1−ε x n o. 1 − 1 max y 2 , y 2

169 Then for arbitrary A > 0, we have I T −A, where the implicit constant depends only on A.

Proof. Let K ∈ N. Shifting the contour deﬁning I to =(t) = −K, we cross poles of sech (πt) at imaginary − i(2k−1) ≤ ≤ K ∈ R half-integers t = 2 , with 1 k 2 . Since Jν (z) is entire in ν for ﬁxed z , by Cauchy’s theorem we have Z −( t−T )2 it −( t+T )2 −it tdt I = J2it (x) e M (y) + e M (y) ℑ(t)=0 cosh(πt) K X2 J (x) tg (t) = Res 2it T,M − t=− i(2k 1) cosh(πt) k=1 Z 2 −( t−T )2 it −( t+T )2 −it tdt + J2it (x) e M (y) + e M (y) ℑ(t)=−K cosh(πt) =: R + I.

Individually, each of the residues satisfy ! à˚u¯sfi¯sfi˚ià‹nffl J2it (x) tG (t) T,M 4 k 1 2 1 Res = (−1) k − k − 1 J2k−1(x)hT,M i − k , y − i(2k−1) cosh(πt) π 2 2 t= 2 2 k− 1 − 1 T 2 ( 2 ) 1 2 −1 k 2 − − k − k − 1 |J − (x)| max{y, y } e M2 e M2 2 2k 1

so their overall contribution satisﬁes R T −A. Meanwhile, the contour integral I can be estimated using the Poisson integral formula (A.4): Z z ν π 2 1 2 2ν < − Jν (z) = 1 1 cos (z cos θ) sin (θ)dθ, for (ν) > . Γ 2 Γ ν + 2 0 2

For t ∈ R, by Stirling’s approximation we have Z x 2it+2K π 2 2 2 4it+4K J2it+2K (x) = 1 1 cos (x cos θ) (sin θ) dθ Γ Γ 2K + + 2it 0 2 2 x 2K eπ|t|. T

170 Taking trivial bounds, we see that the shifted integral satisﬁes Z − − 2 − 2 − −( t iK T ) it+K −( t iK+T ) −it−K (t iK)dt I = J2it+2K (x) e M (y) + e M (y) R cosh(πt − πiK) Z Z −T +M log T T +M log T x 2K x 2K − (y)K |t − iK|dt + (y) K |t − iK|dt −T −M log T T T −M log T T  n o2K 1 − 1 x max y 2 , y 2   T 1+εM. T

Choosing K suﬃciently large, we see that I T −A for any A > 1.

171 Appendix C

Miscellaneous bounds for character sums

Contents C.1 Compendium of properties of Kloosterman sums and Ramanujan sums and special character sums ...... 172

C.1 Compendium of properties of Kloosterman sums and Ra- manujan sums and special character sums

In this section we record some properties of Kloosterman sums for GL(2).

Proposition C.1.1. For any m, n ∈ N, the Kloosterman sums satisfy the Hecke relations X mn c S(m, n; c) = dS , 1; . (C.1) d2 d d|(m,n,c)

Proposition C.1.2. For any m, n ∈ N, the Kloosterman sums satisfy the Weil bound

1 1 |S(m, n; c)| ≤ τ(c)(m, n, c) 2 c 2 . (C.2)

Meanwhile, for n = 0, the Ramanujan sums satisfy

|S(0, m; c)| ≤ max{c, (m, c)τ(m)}. (C.3)

172 C.1. Compendium of properties of Kloosterman sums and Ramanujan sums and special character sums

N By Cauchy-Schwarz and (C.3), for any arbitrary A = (am) ∈ C and N 1, we have

  1   1 X X 2 X 2 kAk  2 2  εkAk  2 amS(0, m; r) (n, r) τ2 (n) N (n, r) . m≤N n≤N n≤N

An elementary argument shows that X 2 1+ε 2 (n, r) N τ2(r) . n≤N so we have X 1 +ε amS(0, m; r) N 2 τ2(r)kAk. (C.4) m≤N

The character sums deﬁned by Iwaniec-Li ([IL07]), X sn − d + sm V (m, n; r) = e d r s(r) (s(d+x),r)=1 satisfy the following: X ak V− (m, n; r)e = S(k, m; r)S(k, n; r). a r a(r)

173