Sum Formula for SL2 over Imaginary Quadratic Number Fields · PDF fileFormula za sumiraǌe za...

Sum Formula for SL2 overImaginary Quadratic Number Fields

Somformule voor SL2 overimaginair quadratisch getallenlichamen

(met een samenvatting in het Nederlands)

Formula za sumiraǌe za SL2 nadimaginarno kvadratiqno brojno pole

(so rezime na makedonski jazik)

Proefschriftter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezagvan de Rector Magnificus, Prof. dr. W. H. Gispen, ingevolge het besluitvan het College voor Promoties in het openbaar te verdedigen op woensdag3 november 2004 des ochtends te 10:30 uur

doorHristina Lokvenec-Guleska

geboren op 26 juni 1974 te Skopje, Macedonie

Promotor: Prof. dr. J. J. DuistermaatFaculteit der Wiskunde en Informatica,Universiteit Utrecht

Copromotor: Dr. R. W. BruggemanFaculteit der Wiskunde en Informatica,Universiteit Utrecht

2000 Mathematics Subject Classification:11F03, 11F12, 11F30, 11F55, 11F70, 11F72, 11L05, 22E30, 42A16, 44A15

Lokvenec-Guleska, HristinaSum Formula for SL2 over Imaginary Quadratic Number FieldsProefschrift Universiteit Utrecht - met een samenvatting in het Nederlands enin het Macedonisch.

ISBN 90-393-3824-8

Cover designed by Mitko Hadzi-Pulja

To Alek

for his patience

Contents

Introduction v

1 Preliminaries 11.1 Three-Dimensional Hyperbolic Space . . . . . . . . . . . . . . . . . 11.2 Some Discrete Subgroups . . . . . . . . . . . . . . . . . . . . . . . 21.3 Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Representation theory 92.1 Structure of SL2(C) . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Lie group SL2(C) . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Lie algebra sl2(C) . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Irreducible representations of K . . . . . . . . . . . . . . . . . . . . 132.3 Irreducible unitary representations of G . . . . . . . . . . . . . . . 17

3 Automorphic forms and automorphic representations 213.1 Automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Automorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Fourier expansion of automorphic functions . . . . . . . . . 233.2.2 Spectral parameter . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Automorphic representations . . . . . . . . . . . . . . . . . . . . . 31

4 N-equivariant eigenfunctions 334.1 Jacquet integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Goodman-Wallach operator . . . . . . . . . . . . . . . . . . . . . . 42

5 Fourier coefficients 515.1 Fourier expansion of Eisenstein series . . . . . . . . . . . . . . . . . 515.2 Fourier expansion of automorphic representations . . . . . . . . . . 52

ii CONTENTS

6 Kloosterman sums 576.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Poincare series 617.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Fourier expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.3 Scalar product of Poincare series . . . . . . . . . . . . . . . . . . . 65

8 Spectral decomposition of the space L2(Γ\G,χ) 69

9 Auxiliary test functions 739.1 Lebedev transformation . . . . . . . . . . . . . . . . . . . . . . . . 739.2 Choice of Poincare series . . . . . . . . . . . . . . . . . . . . . . . . 83

10 Preliminary sum formula 8910.1 Spectral description . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.2 Geometric description . . . . . . . . . . . . . . . . . . . . . . . . . 9310.3 Preliminary sum formula . . . . . . . . . . . . . . . . . . . . . . . . 95

11 Spectral sum formula 9711.1 Bessel transformation . . . . . . . . . . . . . . . . . . . . . . . . . 9711.2 Extension method . . . . . . . . . . . . . . . . . . . . . . . . . . . 10011.3 Extension of the sum formula . . . . . . . . . . . . . . . . . . . . . 10511.4 Discussion of the spectral sum formula . . . . . . . . . . . . . . . . 110

11.4.1 Comparison with the case of Gaussian number field . . . . 11011.4.2 Remarks concerning general number fields . . . . . . . . . . 112

11.5 Application of the spectral sum formula . . . . . . . . . . . . . . . 112

12 Bessel inversion 12312.1 Inverse Bessel transformation . . . . . . . . . . . . . . . . . . . . . 12312.2 One-sided Bessel inversion . . . . . . . . . . . . . . . . . . . . . . . 13012.3 Kloosterman sum formula . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography 143

Indices 147Selective Index of Terminology . . . . . . . . . . . . . . . . . . . . . . . 147Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Samenvatting 153

Rezime 159

CONTENTS iii

Acknowledgments 165

Curriculum Vitae 167

Introduction

The Bruggeman-Kuznetsov sum formula, see preprint [24], published in [25],and [2], obtained by the authors independently, gives a relation between the Fouriercoefficients of cuspidal real-analytic modular forms on the upper half-plane andKloosterman sums. We shall give a very short overview of it.

Let H2 be the upper half-plane with PSL2(R) acting by fractional linear trans-formations, and let Γ = PSL2(Z). On H2 we put the standard metric of constantcurvature −1. Let −∆ = −y2(∂2

x+∂2y) be the corresponding Laplacian on H2. Let

ψj(z) ⊂ L2(Γ\H2) be a complete orthonormal system of cuspidal Maass formswith spectral parameter νj ∈ i(0,∞), that is, ψj are eigenfunctions of −∆ whichare also eigenfunctions for the Hecke operators, indexed according to increasingeigenvalue λj = 1

4 − ν2j . Let the Fourier expansion of ψj(z) be

ψj(x+ iy) =∑

0 6=n∈Zρj(n)y1/2Kνj

(2π|n|y)e2πinx, (1)

where Kν is the K-Bessel function (1.30); this defines the ρj(n) ∈ C. Let E(ν, z)be the non-analytic Eisenstein series with parameter ν. The Fourier coefficients ofthe Eisenstein series involve divisor sums σν(n) =

∑d|n d

ν and the Riemann-zetafunction ζ(ν). Let h be an even holomorphic function on

ν ∈ C : |Re ν| < 1

2 + ε

for some ε > 0 such that |h(ν)| (1 + |ν|)−2−δe−π| Im ν| for some δ > 0 and allν with |Re ν| 6 1

2 + ε. The sum formula states that for such a test function hthe following equality holds for integers n,m > 1, with absolute convergence of allsums and integrals in the various terms:∞∑j=1

ρj(n)ρj(m)h(νj)

cos(πνj)+

1πi

∫Re s=0

h(s)( nm

)−s σ2s(n)σ−2s(m)|ζ(1 + 2s)|2

ds =

= δn,mi

π2

∫Re s=0

s tan(πs)h(s)ds+∞∑c=1

S(n,m; c)c

ϕ

(4π√nm

c

), (2)

where

ϕ(x) =2πi

∫Re s=0

s J2s(x)cos(πs)

h(s) ds, (3)

vi Introduction

for x > 0, Jν denotes the classical Bessel function (1.25), S(n,m; c) is the classicalKloosterman sum (6.2), and δn,m is the Kronecker delta symbol.

The left hand side of a formula (2) comes from the spectral decomposition of theHilbert space L2(PSL2(Z)\H2) and it is therefore called the spectral side. The twoterms correspond to the discrete and continuous spectrum of the Laplace operator−∆. The right hand side is related to the geometry of the space

(10∗1

)\PSL2(Z)

induced by the Bruhat decomposition of the group PSL2(R) and it is thereforecalled the geometric side. Its first term, called delta term, comes from the repre-sentatives

(acbd

)of the cosets in

(10∗1

)\PSL2(Z) with c = 0. The matrices with

c 6= 0 coming from the big cell in the Bruhat decomposition of PSL2(Z)∩PSL2(R)give rise to the second term called for obvious reasons, the Kloosterman term.

Summation formulas of this type, like (2), may be used in two different ways.On one hand, in the given form (2) with the independent test function on thespectral side, it is a tool to obtain results concerning spectral data. For example,in Proposition 4.1 of [2] one finds the following distribution result

∞∑j=1

e−vλjρj(n)ρj(m)cos(πνj)

=δn,mπ2

∣∣∣ nm

∣∣∣1/2 v−1 +O(v−1/2−ε), (4)

as v ↓ 0, with ε > 0.On the other hand, if we know how to invert the Bessel transformation (3), we

have the independent test function on the geometric side. We can then use thesum formula to obtain estimates for sums of Kloosterman sums. In [25] Kuznetsovgave a one-sided inverse of the Bessel transformation and proved that for m,n > 1and X →∞,

X∑c=1

S(n,m; c)c

n,m X1/6(logX)1/3. (5)

The Weil bound for the Kloosterman sums implies the estimate O(X1/2+ε) forthe sum above (see [44]), and the Linnik hypothesis predicts that it is O(Xε), see[29].

Generalizations of the sum formula (2) are obtained in various ways. BothBruggeman and Kuznetsov consider the full modular group SL2(Z), weight zeroand trivial multiplier system. In [38] Proskurin used Kuznetsov approach to gene-ralize the sum formula to cofinite discrete subgroups of SL2(R), non-trivial multi-plier system and general weights. He works with Fourier terms of positive order.Fourier terms of arbitrary order are treated by Bruggeman in [3], which in contrastto [25] and [38], takes more a representational point of view.

In [35] Miatello and Wallach give a formula of the same type for real connectedsemi-simple Lie groups of R-rank one. There the upper half-plane is replaced byany complete Riemannian non-compact symmetric space of rank one, the discrete

vii

subgroup of isometries has finite co-volume, and only trivial K-types are consid-ered. In [42], the authors extend the formula in [35] to products of rank onegroups. The formula is very explicit when the group is a product of groups of theform SL2(R) or SL2(C). Bruggeman and Miatello [4] use the sum formula in [35]to study sums of generalized Kloosterman sums for this class of groups. By takinga suitable test function, they obtain an estimate of type (5) for those sums (see[4], Theorem 1 in 4.3.). In [5], the same authors give a sum formula for SL2 overan arbitrary number field restricted to trivial K-types. The case of a totally realnumber field is described by Bruggeman, Miatello, and Pacharoni in [6] taking intoaccount all K-types.

In his book [36], Motohashi gives an explicit formula for the fourth powermoment of the Riemann zeta-function using the sum formula for SL2(R). Anal-ogous reasoning leads to an extension of these results to a higher-dimensionalsituation. Bruggeman and Motohashi [8] show how one can perform the samepreparatory work with the group SL2(C) in place of SL2(R). Their ultimate goalis to give a spectral decomposition for the fourth power moment of the Dedekindzeta-function. The sum formula for SL2 over the Gaussian number field includingnon-trivial K-types, as well as an explicit formula for the fourth power momentof the Dedekind zeta-function, are derived by the same authors in [9]. There thecase of even functions is treated.

In this thesis we generalize the sum formula from [9] by considering a generalimaginary quadratic field F and an arbitrary congruence subgroup Γ = Γ0(I), withI ⊂ O a non-zero ideal in the ring O of integers of F . We consider χ-automorphicfunctions with respect to Γ, where χ is a character of Γ trivial on Γ1(I) ⊂ Γ. Wealso consider the case of odd functions.

In Chapters 1 and 2, we describe some fundamental facts about the geometryof the three-dimensional hyperbolic space, transformation groups on this space,and Bessel functions, as well as a small part of the representation theory of the Liegroups SL2(C) and SO(2). In Chapter 3 we introduce automorphic functions andautomorphic representations. Central in their Fourier expansion are Jacquet andGoodman-Wallach operators treated in Chapter 4. A more detailed descriptionof the Fourier coefficients of Eisenstein series and cuspidal automorphic repre-sentations is given in Chapter 5. Chapters 6, 7, and 8 contain known resultsconcerning Kloosterman sums, Poincare series, and spectral decomposition of thespace L2(Γ\SL2(C)). They are stated in a form appropriate for our purposes.

In Chapter 9 we introduce the Lebedev transformation, its inverse on a certainclass of test functions, and give some of its properties. This transform will bethe building block for the Poincare series used to derive the sum formula. MyLebedev transformation is an extension of the classical Lebedev transformationf 7→

∫∞0f(r)Kν(r)drr which plays a significant role in the theory of sum formulas

for rational Kloosterman sums. The square-integrability and boundedness of thechosen Poincare series depend heavily on the results of Miatello and Wallach, [34],

viii Introduction

as well as on Lemma 5.2.1.The derivation of the preliminary sum formula is explained in Chapter 10. It

involves computation of the inner product of two special Poincare series in two dif-ferent ways: the spectral description where the Fourier coefficients CV (ω; νV , pV )of cusp forms appear and the geometric description where the sums of Klooster-man sums appear. This is also the method used by Bruggeman [2], Kuznetsov [25]and Proskurin [38].

Our main result is the spectral sum formula, Theorem 11.3.3 in Chapter 11.It is obtained by extending the class of test functions in its preliminary versionProposition 10.3.1. The extension method is described in Section 11.2. It isanalogue to the method of Miatello and Wallach, [].

In Section 11.5, we apply this formula to obtain weighted density resultsconcerning the cuspidal automorphic representations in L2(SL2(O)\SL2(C)) witheigenvalue λV not exceeding X and prescribed spectral parameter pV . Namely,for arbitrary ω ∈ O′\0, p ∈ 1

2Z and X →∞ we have∑V :pV =±p

λV 6X

|CV (ω; νV , pV )|2 ∼ 2εp3π3√|dF |

X3/2, (6)

where ε0 = 1 and εp = 2 if p 6= 0.In Chapter 12, Theorem 12.2.1, we give a one-sided inversion of the Bessel

transformation (11.1), which allows us to prove the sum formula in a reversedform, see Theorem 12.3.2. Applications of this formula in obtaining estimatesfor the sum of Kloosterman sums (11.15) might be possible, but the limited timeavailable for the present work forces us to postpone their derivation.

Chapter 1

Preliminaries

Convention. We use the notation N = Z>0 for the non-negative integers, andwe call the elements of the set 1

2 + Z half-integers.

1.1 Three-Dimensional Hyperbolic Space

Three-dimensional hyperbolic space is the unique three-dimensional connectedand simply connected Riemannian manifold with constant sectional curvatureequal to −1. This space has certain concrete models which all have certain advan-tages. For our purposes, the upper half-space

H3 =(z, r) | z ∈ C, r > 0

= C× (0,∞) (1.1)

in Euclidean three-space gives a convenient model of the three-dimensional hyper-bolic space which in its properties closely resembles the well-known upper half-plane as a model of plane hyperbolic geometry. It is useful for computations tothink of H3 as a subset of Hamilton’s quaternions H(−1,−1). (See [11], §10.1.)As usual we write 1, i, j, k for the standard R-basis of H(−1,−1). The notationfor points in H3 is

(z, r) = z + rj = x+ yi+ rj (1.2)

where j = (0, 0, 1) and x, y ∈ R. For a discussion of this space, other models, itsRiemannian metric

r−2(dx2 + dy2 + dr2

), (1.3)

its SL2(C)-invariant measure

r−3dx dy dr, (1.4)

2 Preliminaries

and its Laplace-Beltrami operator

L := r2(∂2x + ∂2

y + ∂2r

)− r∂r, (1.5)

see [11], Chapter 1.

The group G = SL2(C) acts on H3, and the action of the matrix g =(acbd

)∈ G

is given by

g(z, r) =(

(az + b)(cz + d) + acr2

|cz + d|2 + |c|2r2,

r

|cz + d|2 + |c|2r2

). (1.6)

This action looks complicated, but actually it is very natural if it is looked at inthe spirit of the theory of topological transformation groups and of Lie theory.Considered as a Lie group over R, G has dimension 6. The special unitary groupK = SU(2) is one of its maximal compact subgroups. The symmetric spaceassociated to G is the quotient space G/K with G acting by left multiplication.The general theory of this situation is contained in [16]. The map

π : G −→ H3, π(g) = g · j, (1.7)

gives rise to an isomorphism, as topological spaces, between the symmetric spaceof SL2(C) and H3 given by the explicit formula

G/K 3( √

r z/√r

0 1/√r

)←→ (z, r) ∈ H3. (1.8)

The equation (1.6) describes in fact the natural action of G on G/K by multipli-cation of cosets from the left. See [11], §1.6, [31], p. 6–7 or [23] for more details onthe computations of this correspondence.

1.2 Some Discrete Subgroups

Let F = Q(√D), with D < 0 a square-free integer, be an imaginary quadratic

number field. We write O = OF for the ring of integers in F . The discriminant ofF is denoted by dF . The negative integer dF satisfies

dF =

4D , D ≡ 2, 3 (mod 4)D , D ≡ 1 (mod 4), ≡

0 (mod 4)1 (mod 4), (1.9)

The ring of integers O has the Z-basis consisting of 1 and β, where

β =dF +

√dF

2=dF2

+ i

√|dF |2

. (1.10)

1.2 Some Discrete Subgroups 3

The group SL2(O) is a discrete cofinite, but not cocompact subgroup of G =SL2(C). (See [11], Theorem 7.1.1. The result stated is actually for PSL2(O), butit holds for SL2(O) as well.)

Given a non-zero ideal I ⊂ O, the principal congruence subgroup of level I isdefined by:

Γ(I) =(

a bc d

)∈ SL2(O)

∣∣∣∣ ( a bc d

)≡(

1 00 1

)mod I

. (1.11)

Any discrete subgroup Γ ⊂ SL2(O) which is G-conjugate to a group containingΓ(I) for some non-zero ideal I ⊂ O is called a congruence subgroup with respectto SL2(O). Examples of congruence subgroups are:

Γ1(I) =(

a bc d

)∈ SL2(O)

∣∣∣∣ c ∈ I, a, d ≡ 1 mod I, (1.12)

Γ0(I) =(

a bc d

)∈ SL2(O)

∣∣∣∣ c ∈ I . (1.13)

The smaller the subgroup Γ ⊂ SL2(O) is, the bigger is the class of automorphicfunctions on Γ\H3. Therefore we want to consider a principal congruence subgroupΓ(I), for I ⊂ O a non-zero ideal. But, conjugation with an element g =

(A0

0A−1

)∈

G gives for(acbd

)∈ Γ1(I):

g

(a bc d

)g−1 =

(a A2b

A−2c d

)∈ Γ(I),

for a suitably chosen A in some principal ideal contained in I such that A2b ∈ I.Therefore, up to conjugation, it is enough for our purpose to consider a congruencesubgroup Γ1(I).

On the other hand, one can easily see that Γ1(I) =⋂χ0

kerχ0, where theintersection runs over all characters χ0 of Γ0(I) which are trivial on Γ1(I). Wenote that since P as in (1.16) is contained in SL2(O/I) and SL2(O) → SL2(O/I)is surjective (see [17], p. 249), any γ ≡

(aobd

)(mod I) is congruent to a product(

10b/d1

)(1/d0

0d

)(mod I), with

(10b/d1

)∈ Γ1(I). Then χ0(γ) = χ0

((1/d0

0d

)),

which means that any character χ0 of Γ0(I) which is trivial on Γ1(I) actuallydepends only on d. Hence, considering Γ0(I) and characters on Γ0(I) of the form(

a bc d

)7→ χ(d), (1.14)

with χ being a character of (O/I)∗, we are actually dealing with Γ1(I).

4 Preliminaries

From now on, we fix a congruence subgroup Γ := Γ0(I), with I ⊂ O a non-zeroideal, and consider characters on Γ of the form (1.14). We shall use the samenotation χ both for a character of (O/I)∗ and the corresponding character of Γindicated in (1.14).

1.3 Cusps

We consider the Riemannian sphere P1(C) = C ∪ ∞ as the boundary ofthe upper half-space H3. Elements in P1(C) are represented by [z1, z2], wherez1, z2 ∈ C and (z1, z2) 6= (0, 0). Here∞ = [1, 0]. The action of G on P1(C) is givenby:

g[z1, z2] = [az1 + bz2, cz1 + dz2], (1.15)

for g =(acbd

)∈ G. See [11], §1.1 for more details.

For each element ζ ∈ H3 ∪ P1(C) we call Γζ = γ ∈ Γ | γζ = ζ, the subgroupof Γ fixing ζ, the stabilizer of ζ in Γ.

Let

P =(

u z0 u−1

) ∣∣∣∣ u ∈ C∗, z ∈ C

(1.16)

be the set of upper-triangular matrices in G, and

N =(

1 z0 1

) ∣∣∣∣ z ∈ C. (1.17)

its unipotent radical. The group N is isomorphic to the additive group C+, andthe group P is isomorphic to the semi-direct product of C+ and C∗. The group Palso appears as the stabilizer of ∞ in the action of G on P1(C).

If ζ ∈ P1(C) and gζ ∈ G with ζ = gζ · ∞ we further define

Γ′ζ = Γ ∩ gζNgζ−1 = Γζ ∩ gζNgζ−1. (1.18)

Note that Γ′ζ consists of the parabolic elements in Γζ together with the identity.By Definition 2.1.10 in [11], the cusps are those elements in P1(C) whose sta-

bilizer in Γ under the action (1.15) contains a free abelian group of rank 2. SinceΓ ⊂ G is a discrete subgroup of finite covolume, it follows from Theorem 2.5.1 in[11], that there are only finitely many Γ-inequivalent cusps. We denote by C(Γ)the set of representatives of the equivalence classes of cusps for Γ.

For each representative κ ∈ C(Γ), we fix gκ ∈ G such that κ = gκ · ∞. For theclass of ∞ we choose ∞ as the representative and g∞ = 1.

For each κ ∈ C(Γ), we have the stabilizer of κ

Γκ = Γ ∩ gκPgκ−1, (1.19)

1.4 Bessel functions 5

and its unipotent subgroup

Γ′κ = Γ ∩ gκNgκ−1 = Γκ ∩ gκNgκ−1. (1.20)

For κ =∞ we have

Γ∞ = Γ ∩ P =(

ε−1 ξ0 ε

) ∣∣∣∣ ξ ∈ O, ε ∈ O∗

=: ΓP ,

Γ′∞ = Γ ∩N =(

1 ξ0 1

) ∣∣∣∣ ξ ∈ O

=: ΓN .

The possibilities for ΓP and ΓN are summarized in [11], Theorem 1.8. The unipo-tent subgroup Γ′κ has finite index in Γκ. (See [11], p.100). For κ = ∞ it is easilyseen that

[ΓP : ΓN ] = |O∗| =

2 if F = Q(√D), D 6= −1,−3

4 if F = Q(i)6 if F = Q(

√−3).

If κ ∈ C(Γ) is a cusp for Γ, then the discrete subgroup g−1κ Γ′κgκ corresponds to

a lattice Λκ in C in the following way:

g−1κ Γ′κgκ =

(10λ

1

) ∣∣∣∣ λ ∈ Λκ

, (1.21)

see Theorem 2.1.8, (3) in [11]. Let Rκ ⊂ C be a fundamental domain for the actionof g−1

κ Γ′κgκ on P1(C)\∞ = C. We denote its Euclidean area by |Λκ|. For Y > 0,we define

Fκ(Y ) = gκ

(10z

1

)(√r

00

1/√r

) ∣∣∣∣ z ∈ Rκ, r > Y

(1.22)

to be the cusp sector corresponding to κ. Then there exists a compact polyhedronF0 ⊂ H3 such that

F = F0 ∪⋃κ∈Cχ

Fκ(Y ) (1.23)

is a fundamental domain for Γ\H3.We denote by Cχ the set of cusps κ ∈ C(Γ) for which the character χ :

(acbd

)7→

χ(d) is trivial on Γ′κ. It is clear that ∞ ∈ Cχ.

1.4 Bessel functions

The Bessel functions of order ν are solutions of Bessel’s differential equation

z2f ′′(z) + zf ′(z) + (z2 − ν2)f(z) = 0, (1.24)

6 Preliminaries

where ν and z are arbitrary complex numbers. Since this equation is singular atz = 0, cutting the complex z-plane along the segment (−∞, 0] gives for ν 6∈ Z twolinearly independent solutions Jν(z) and J−ν(z) of (1.24) which are holomorphicin z ∈ C\(−∞, 0]. Here Jν(z) is given by the power series

Jν(z) =∞∑m=0

(−1)m(z/2)ν+2m

m!Γ(ν + 1 +m), (1.25)

which converges absolutely in the whole complex plane. If ν = n ∈ Z, the solutionsare no longer linearly independent, and we have the relation

J−n(z) = (−1)nJn(z). (1.26)

For fixed z, considered as function of ν, Jν(z) represents an entire function, andJν(z)(z/2)−ν is an even entire function of z for given ν.

Rotating the variable by an angle π/2, that is z 7→ iz, the equation (1.24)transforms into the so called modified Bessel’s differential equation

z2f ′′(z) + zf ′(z)− (z2 + ν2)f(z) = 0. (1.27)

The solutions of this equations are called modified Bessel functions. As before,for ν 6∈ Z, we have two linearly independent solutions holomorphic in C\(−∞, 0].One of these is given by the power series

Iν(z) =∞∑m=0

(z/2)ν+2m

m!Γ(ν + 1 +m), (1.28)

and the other one is I−ν(z). If ν = n ∈ Z, we have the relation

I−n(z) = In(z). (1.29)

Setting

Kν(z) =π

2 sinπνI−ν(z)− Iν(z)

(1.30)

yields a pair Iν(z), Kν(z) of linearly independent solutions to (1.27) for all ν ∈ C.The K-Bessel function Kν(z) has a holomorphic extension to integral values ofthe parameter ν, and it is even with respect to ν:

K−ν(z) = Kν(z). (1.31)

The Bessel functions of different order are related by many recursion formulas.For a short overview of these formulas, other properties, integral representations orestimates of the Bessel functions we refer to [32]. A far more extended treatmentof the Bessel functions can be found in [43].

1.4 Bessel functions 7

Here we give selected results that are going to be used in our further work.The power series expansions (1.28)–(1.30) show that

Iν(z)z0,σ zRe νe

π2 | Im ν|(1 + | Im ν|)−Re ν− 1

2 , (1.32)

and

Kν(z)z0,σ z−|Re ν|−εe−

π2 | Im ν|(1 + | Im ν|)|Re ν|− 1

2 , (1.33)

uniformly for |Re ν| 6 σ, z ∈ (0, z0) with σ > 0, z0 > 0, ε > 0. The ε in theexponent of z in (1.33) is added to take care of the logarithmic contributions atintegral values of ν.

The asymptotic expressions for the functions Iν(z) and Kν(z) on page 139 in[32], for large argument z and fixed order, tell us that the function z 7→ Iν(z) isexponentially increasing as z → ∞, while z 7→ Kν(z) is exponentially decreasingas z →∞. For real positive y > 1 + |ν|2 we have

Iν(y) = (2πy)−1/2ey(

1 +O

(1 + |ν|2

y

)), (1.34)

and

Kν(y) =(π

2y

)1/2

e−y(

1 +O

(1 + |ν|2

y

)). (1.35)

To get an estimate uniform in the parameter ν, we use Basset’s integral forKν , see [43], §6.16:

Kν(z) =Γ(ν + 1

2

)2√π(z/2)ν

∫ ∞

−∞e−izt(1 + t2)−ν−

12 dt. (1.36)

The integral in (1.36) converges absolutely if Re ν > − 12 , and yields an estimate

Kν(z)σ1,σ2 z−Re νe−

π2 | Im ν|(1 + | Im ν|)Re ν

uniformly for Re ν ∈ [σ1, σ2] ⊂ (0,∞). After k-fold partial integration, Basset’sformula becomes

Kν(z) = Γ(ν + 1

2

)(z/2)−ν−k

∫ ∞

−∞pk(t, ν)e−izt(1 + t2)−ν−

12−kdt

where pk(u, ν) is a polynomial in u and ν with degree at most k in both variables.This new integral representation is valid for Re ν > −k2 , and yields an estimate

Kν(z)σ1,σ2,k z−Re ν−ke−

π2 | Im ν|(1 + | Im ν|)Re ν+k (1.37)

uniformly for Re ν ∈ [σ1, σ2] ⊂(−k2 ,∞

)for each integer k > 1.

Chapter 2

Representation theory

2.1 Structure of SL2(C)

In this section we state certain facts concerning the structure of the Lie groupSL2(C), consisting of complex 2 × 2 matrices of determinant 1, as well as its Liealgebra sl2(C).

2.1.1 Lie group SL2(C)

We shall use the following notation for explicit matrix elements:

n[z] =(

1 z0 1

), h[u] =

(u 00 u−1

), k[α, β] =

(α β−β α

),

a[r] = h[√r], w =

(0 −11 0

),

v[t] =

(cosh t

2 sinh t2

sinh t2 cosh t

2

), w[t] =

(cos t2 sin t

2

− sin t2 cos t2

),

for z, α, β, t ∈ C, u ∈ C∗, r > 0. The elements w and n[z], z ∈ C are generatorsfor the group SL2(C) (See [11], Proposition 1.2).

For t ∈ R, we have sinh it = i sin t, cosh it = cos t, and therefore

v[it] =

(cos t2 i sin t

2

i sin t2 cos t2

), w[it] =

(cosh t

2 i sinh t2

−i sinh t2 cosh t

2

).

We shall also use the following subgroups of G:

H =h[u] | u ∈ C∗

, A =

a[r] | r > 0

,

M = H ∩K =h[u] | |u| = 1

.

10 Representation theory

The group P has the decomposition P = NH = NAM .With the Euler angles ϕ, θ, ψ ∈ R, each element of K = SU(2) can be written

as

k[α, β] = h[eiϕ/2]v[iθ]h[eiψ/2]. (2.1)

The Iwasawa decomposition of the Lie group G is G = NAK in the sense thateach element g ∈ G can be written in a unique way as

g = n[z]a[r]k[α, β], (2.2)

for some z, α, β ∈ C, r > 0. By Iwasawa coordinates we mean

(z, r, ϕ, θ, ψ) ∈ C× (0,∞)× R× [0, π]× R, (2.3)

corresponding to g = n[z]a[r]h[eiϕ/2]v[iθ]h[eiψ/2]. On an open dense subset of Gthese coordinates are unique, provided that θ ∈ [0, π) and ϕ± ψ ∈ [0, 4π).

The Haar measures on subgroups of G are given by:

N : dn = d+z = dRe z ∧ d Im z , with n = n[z]

A : da = r−1dr , with a = a[r]

K : dk = (16π2)−1 sin θ dϕ dθ dψ , with k = h[eiϕ/2]v[iθ]h[eiψ/2].

(2.4)

Introducing the notation |a| := |r|2 = r2, for a = a[r] ∈ A, Haar measure on G isgiven by:

G : dg = |a|−1dn da dk, with g = nak. (2.5)

We can easily calculate that∫K

dk =1

16π2

∫ ∫ϕ±ψ∈[0,4π)

dϕ dψ

∫ π

0

sin θ dθ = 1. (2.6)

Since G/K ∼= NA ∼= H3, we have∫Γ\G

dg =∫

FG

dg =∫

F

|a|−1dn da

∫K1/2

dk =vol(Γ\H3)

2, (2.7)

where F is a fundamental domain for Γ\H3, K1/2 is a fundamental domain for1,h[−1]\K, and FG = n[z]a[r]k | (z, r) ∈ F, k ∈ K1/2 is a fundamental domainfor Γ\G.

In particular, for Γ = PSL2(O), Theorem 7.1.1 in [11] yields∫Γ\G

dg =|dF |3/2

8π2ζF (2), (2.8)

where ζF is the zeta function associated to the field F . Fundamental domains ofPSL2(O)\H3 for all imaginary quadratic number fields are explicitly described in[11], §7.3.

2.1 Structure of SL2(C) 11

2.1.2 Lie algebra sl2(C)

The real Lie algebra of G is the real vector space sl2(C) consisting of complex2× 2 matrices with trace equal to zero,

sl2(C) = X ∈M2×2(C) | Tr(X) = 0 (2.9)

with the Lie bracket [X,Y ] := XY − Y X. It has a R-basis consisting of sixelements

H1 = 12

(1 00 −1

), V1 = 1

2

(0 11 0

), W1 = 1

2

(0 1−1 0

),

H2 = 12

(i 00 −i

), V2 = 1

2

(0 i−i 0

), W2 = 1

2

(0 ii 0

). (2.10)

We identify the elements in sl2(C) with real right differentiations on G, by

(Xf)(g) =d

dtf(g exp(tX))

∣∣∣∣t=0

, X ∈ sl2(C), f ∈ C∞(G), g ∈ G.

The variable t is real. A short calculation gives, for example,

(H2f)(g) = ∂ψf(g). (2.11)

Tensoring with C over R gives an identification of the complex Lie algebrag = sl2(C) ⊗R C of G with all left-invariant first-order differential operators withcomplex coefficients. Hence the universal enveloping algebra U(g), as a C-algebra,can be identified with the set of all left-invariant differential operators on G.

The Lie algebra sl2(C) can also be viewed as complexification of the real Liealgebra sl2(R) of SL2(R). As such, it is embedded in g in two ways: X 7→ X andX 7→ X. Here the bar corresponds to complex conjugation in sl2(C) with respectto sl2(R). In this way we see that

g ∼= sl2(C)⊕ sl2(C) (2.12)

as complex Lie algebras. Hence, the center Z(g) of the universal enveloping algebraU(g) is generated by the Casimir elements of both factors. In the action of g byright differentiation in C∞(G), one of the summands corresponds to holomorphicdifferentiation

(Xf)(g) = ∂tf(g exp(tX))|t=0, t ∈ C,

and the other summand then corresponds to anti-holomorphic differentiation

(Xf)(g) = ∂tf(g exp(tX))|t=0, t ∈ C.

We fix a basis of sl2(C)

H = H1 − iH2, V = V1 − iW2, W = W1 − iV2, (2.13)


where the factor i means the complexification of respective elements. The Killingform B(X,Y ) on sl2(C) is the non-degenerate symmetric bilinear form given by

B(X,Y ) = Tr(ad(X)ad(Y )),

where ad(X)Y = [X,Y ]. The chosen basis (2.13) is orthogonal for the Killingform, and

B(H,H) = B(V,V) = −B(W,W) = 8

This leads to the Casimir element Ωsl2(C) = 18

(H2 + V2 −W2

)of sl2(C). (See

[22], Chapter VIII, §3). Hence the element

Ω+ =18(H2 + V2 −W2

)(2.14)

is the Casimir element of the first summand in (2.12), and the Casimir element ofthe second summand is its complex conjugate

Ω− =18(H2 + V2 − W2

)(2.15)

The center Z(g) of U(g) is the polynomial ring C[Ω+,Ω−].As differential operators the Casimir elements Ω± are given by the following

formulas in Iwasawa coordinates

Ω+ =12r2∂z∂z +

12reiϕ cot θ∂z∂ϕ −

12ireiϕ∂z∂θ −

− reiϕ

2 sin θ∂z∂ψ +

18r2∂r

2 − 14ir∂r∂ϕ −

18∂ϕ

2 − 18r∂r +

14i∂ϕ. (2.16)

and

Ω− =12r2∂z∂z +

12re−iϕ cot θ∂z∂ϕ +

12ire−iϕ∂z∂θ −

− re−iϕ

2 sin θ∂z∂ψ +

18r2∂r

2 +14ir∂r∂ϕ −

18∂ϕ

2 − 18r∂r −

14i∂ϕ. (2.17)

Real generators of the ring Z(g) are Ω+ + Ω− and −i(Ω+ − Ω−). It turns outthat the Casimir element of the real Lie algebra sl2(C) of G is Ω+ + Ω−. Appliedto K-invariant functions on G, Ω+ +Ω− gives a constant multiple of the Laplacianon H3:

(Ω+ + Ω−)|Γ\H3 = 14L.

We now consider the maximal compact subgroup K = SU(2). Its real Liealgebra su(2) is generated by H2,W1 and W2. This basis is orthogonal for theKilling form of su(2), and

B(H2,H2) = B(W1,W1) = B(W2,W2) = −2.

2.2 Irreducible representations of K 13

This leads to the Casimir element

Ωk = −12(H2

2 + W12 + W2

2)

=12(H2

2 ± iH2 + E±E∓)

(2.18)

in Z(k), where k = su(2)⊗R C is the complex Lie algebra of K.Here E± = W1 ± iW2 in k. These two elements generate k, and satisfy

[H2,E±] = ∓iE±. Note that E+ = E−.In terms of Iwasawa coordinates we have

E± = e∓iψ(− 1

sin θ∂ϕ ± i∂θ + cot θ∂ψ

)(2.19)

and

Ωk =1

2 sin2 θ

(∂2ϕ + sin2 θ∂2

θ + ∂2ψ − 2 cos θ∂ϕ∂ψ + sin θ cos θ∂θ

). (2.20)

The center Z(k) of U(k) is the polynomial ring C[Ωk].

2.2 Irreducible representations of K

The irreducible representations of the group K = SU(2) are described in detailin [41], §6.2 and §6.3. They are obtained as restrictions of finite-dimensionalrepresentations Tn on G to K, and are uniquely determined, up to equivalence, bythe non-negative integer n ∈ N. We shall use the same notation for the restrictionsto K.

A representation Tn of G is given by the formula

Tn(g)f(z) = (bz + d)nf(az + c

bz + d

), (2.21)

for all g =(acbd

)∈ G and f running through the space Vn of polynomials in one

variable of degree at most n. Each of these representations Tn is irreducible, andunitary for a suitably chosen scalar product.

In particular, g = h[−1] in (2.21) gives Tn(h[−1]) = (−1)n. As we shall seelater in Section 3.1, a χ-automorphic function f of K-type Tn satisfies

f(−g) = χ(−1)f(g)(3.1)= (−1)nf(g),

and therefore it is even (odd) if and only if n is even (odd). We would like todistinguish between the even and the odd case. We write n = 2l and considerfurther l ∈ 1

2N as a parameter to characterize the irreducible representations ofK. We put σl = T2l.


Let l ∈ 12Z. The set

zl−q : q ≡ l (mod 1), |q| 6 l (2.22)

is a basis for V2l, and the representation σl on K is given by

σl (k[α, β]) zl−q = (αz − β)l−q(βz + α)l+q, k[α, β] ∈ K (2.23)

where zl−q is element of the basis (2.22).For the elements of the one-parameter subgroup M = h[eit] : t ∈ R ⊂ K

we have

σl(h[eit]

)zl−q = e−2qitzl−q , q ≡ l (mod 1), |q| 6 l.

So, the space V2l is the direct sum of (2l + 1) one-dimensional weight spaces forthe subgroup M acting with integer or half-integer weights q:

h[eit] σl7−→ e−2qit, |q| 6 l. (2.24)

The number l determines a K-type, that is, an eigenvalue class of irreduciblerepresentations ofK. The representation σl represents this equivalence class. With-in the underlying vector space V2l of σl, there are one-dimensional weight spacesfor M , parameterized by q (q ≡ l (mod 1), |q| 6 l). So the number q describes theM -types of a vector in a particular K-type.

Since K is compact, there exists an invariant scalar product (·, ·) on the spaceV2l. It can be normalized such that

(zl−q, zl−k) =

0 , k 6= q(l − q)!(l + q)! , k = q.

(2.25)

(See [41], §6.2.3).The representation σl is completely determined by the derived action of the

Lie algebra k of K:

σl(X)f(z) = ∂tσl(exp(tX))f(z)|t=0 for all X ∈ k, f ∈ V2l. (2.26)

A simple computation gives

σl(H2)zl−q = −iqzl−q,σl(E±)zl−q = (q ∓ l)zl−(q±1), (2.27)

σl(Ωk)zl−q = −12(l2 + l)zl−q.

Let L2(K) be the Hilbert space of all functions on K which are square inte-grable over K with respect to the Haar measure dk. We denote by (·, ·)K and

2.2 Irreducible representations of K 15

‖ · ‖K the scalar product and the norm on K obtained by integration with respectto the Haar measure dk in (2.4). We want to describe the structure of L2(K), andhence the unitary representations of the compact group K.

Let l, q, p ∈ 12Z, l > 0, l ≡ q ≡ p (mod 1). We define a function Φlp,q on K as

the coefficient of zl−p in the polynomial expansion of σl(k[α, β])zl−q, where σl isan irreducible representation of K given by (2.23), i.e.∑

|p|6l

Φlp,q(k[α, β])zl−p = (αz − β)l−q(βz + α)l+q, for |q| 6 l, (2.28)

and Φlp,q ≡ 0 whenever one of the numbers p or q violate the condition |p|, |q| 6 l.The functions Φlp,q are (by construction) matrix coefficients of the representa-

tion σl, since

Φlp,q(k) =1

(l − p)!(l + p)!(σl(k)zl−q, zl−p

). (2.29)

(See [22], p.16). In particular, we have

Φlp,q(k1k2) =∑|m|6l

Φlp,m(k1)Φlm,q(k2), k1, k2 ∈ K. (2.30)

One easily checks that the following properties are satisfied:

(P1) Φlp,q(v[−iθ]) = (−1)p+qΦl−p,−q(v[iθ]),

(P2) Φlq,p(v[iθ]) =(l − p)!(l + p)!(l − q)!(l + q)!

Φlp,q(v[iθ]),

(P3) Φlp,q(k[α, β]) = e−ipϕ−iqψΦlp,q(v[iθ]) with Euler angles (2.1),

(P4) Φlp,q = (−1)p+qΦl−p,−q.

For any X ∈ k, and k ∈ K, we have

XΦlp,q(k) = ∂tΦlp,q(k exp(tX))|t=0

(2.29)= (l − p)!(l + p)!−1

∂t(σl(k exp(tX))zl−q, zl−p

)|t=0

= (l − p)!(l + p)!−1 (σl(k)∂tσ(exp(tX))zl−q|t=0, z

l−p)(2.26)= (l − p)!(l + p)!−1 (

σl(k)σl(X)zl−q, zl−p).

Using this and (2.27), we immediately get for any k ∈ K:

H2Φlp,q(k) = −iqΦlp,q(k)E±Φlp,q(k) = (q ∓ l)Φlp,q±1(k), (2.31)

ΩkΦlp,q(k) = −12(l2 + l)Φlp,q(k).


This proves the following

Lemma 2.2.1. The functions Φlp,q with l, p, q ∈ 12Z such that l > 0, l ≡ q ≡ p

(mod 1), are simultaneous eigenfunctions of the differential operators H2 and Ωk

with eigenvalues −iq and − 12 (l2 + l), respectively.

Let us now calculate the scalar products(Φlp,q,Φ

l1p1,q1

)K

. We have

(Φlp,q,Φ

l1p1,q1

)K

=∫K

Φlp,q(k)Φl1p1,q1(k)dk

(2.29)= (l − p)!(l + p)!(l1 − p1)!(l1 + p1)!−1 ·

·∫K

(σl(k)zl−q, zl−p

)(σl1(k)zl1−q1 , zl1−p1). (2.32)

Since the Haar measure dk on K is normalized so that∫Kdk = 1, see (2.6), we can

apply Schur’s orthogonality relations (Corollary 1.10 in [22], p.15), and concludethat

(Φlp,q,Φ

l1p1,q1

)K

= 0 if l 6= l1 or q 6= q1 or p 6= p1. Otherwise

(Φlp,q,Φ

lp,q

)K

= (l − p)!(l + p)!−2 (zl−q, zl−q)(zl−p, zl−p)dimV2l

(2.25)=

12l + 1

(l − q)!(l + q)!(l − p)!(l + p)!

=1

2l + 1

(2ll − p

)(2ll − q

)−1

. (2.33)

So, we see that the setΦlp,q

∣∣ l, p, q ∈ 12Z, p ≡ q ≡ l (mod 1), |p|, |q| 6 l

(2.34)

is an orthogonal system in L2(K), with norms

‖Φlp,q‖K =1√

2l + 1

(2ll − p

)1/2( 2ll − q

)−1/2

. (2.35)

It is actually an orthogonal basis of L2(K), see [41], §6.2.3.If we denote by L2

even(K), and L2odd(K) the subspaces of L2(K) consisting of

even, and odd functions respectively, thenΦlp,q

∣∣ l, p, q ∈ Z, |p|, |q| 6 l,

is an orthogonal basis of L2even(K), and

Φlp,q∣∣ l, p, q ∈ 1

2 + Z, |p|, |q| 6 l

is an orthogonal basis of L2odd(K).

2.3 Irreducible unitary representations of G 17

We arrange the basis (2.34) so that we have the following decomposition ofL2(K) into irreducible subspaces

L2(K) =⊕l,q∈Z|q|6l

L2(K; l, q) ⊕⊕

l,q∈ 12 +Z

|q|6l

L2(K; l, q), (2.36)

L2(K; l, q) =⊕

p≡l (mod 1)|p|6l

CΦlp,q. (2.37)

The space

L2(K; l, q) =f ∈ L2(K) | Ωkf = − 1

2 (l2 + l)f, H2f = −iqf,

is called the subspace of type (l, q).In general, in a space in which K acts, we shall say an element is of type (l, q) if

it is a simultaneous eigenvector of H2 and Ωk with eigenvalues −iq and − 12 (l2 + l),

respectively.

2.3 Irreducible unitary representations of G

The principal series of SL2(C) is the family of representations P2p,2ν of Gdiscussed in [22], Chap.II, §4, indexed by the spectral parameter (ν, p) ∈ C× 1

2Z.The unitary principal series representations are then indexed by (ν, p) ∈ iR× 1

2Z,while the complementary series are indexed by (ν, 0) for ν ∈ (−1, 1)\0.

Let H∞(ν, p) be the space of functions f ∈ C∞(G) that satisfy

f(na[r]h[eit]g) = r1+νe−2pitf(g), (2.38)

for all g ∈ G, n ∈ N , a[r] ∈ A, h[eit] ∈M . The Iwasawa decomposition shows thatsuch functions are determined by their behavior on K. The vector space H∞(ν, p)is isomorphic to the space

C∞p (K) :=f ∈ C∞(K) |f(h[eit]k) = e−2pitf(k)

(2.39)

via f 7→ f , where f(na[r]k) := r1+νf(k) for f ∈ C∞p (K).The group G acts in H∞(ν, p) by right translation:

gf(x) = f(xg), for x, g ∈ G.

This is a representation of G in the space H∞(ν, p) and it is actually the in-duced representation U(MAN,ψp, ν, ·) given in Knapp [22], Chap.VII, §1, whereψp(h[eit]) = e−2pit and ν(h[t]) = 2νt. More precisely, our space H∞(ν, p) is adense subspace of the space of functions F on G satisfying

F (gman) = e−(νknapp+ρ) log aψp(m)−1F (g)


for all g ∈ G, m ∈ M , a ∈ A, k ∈ K, given on p. 168 in [22], via f 7→ F (g) :=f(g−1) with f ∈ H∞(ν, p). Here νknapp = ν, v = −2iν, and ρ is the half-sum ofthe positive roots of (g, a) for N . Knapp works with left translations of G.

Hence, the “induced picture” in [22] corresponds to our space H∞(ν, p), the“compact picture” corresponds to the restrictions to K, i.e. the space C∞p (K),and in the “non-compact picture” a function f ∈ H∞(ν, p) corresponds to thefunction z 7→ f

(1z

01

)on C.

There is a duality between the spaces H∞(ν, p) and H∞(−ν,−p) given by thebilinear form

〈f1, f2〉 :=∫K

f1(k)f2(k)dk

for f1 ∈ H∞(ν, p), f2 ∈ H∞(−ν,−p). This duality satisfies

〈gf1, gf2〉 = 〈f1, f2〉, for all g ∈ G. (2.40)

(See [22], Chap.VII §2).The restriction to K of each ϕ ∈ H∞(ν, p) is square integrable on K. Con-

versely, each f ∈ L2(K) that satisfies f(h[eit]k) = e−2pitf(k) almost everywherefor k ∈ K, for some p ∈ 1

2Z, can be extended to a function on G satisfying (2.38).The space consisting of such functions we denote by H2(ν, p). The action of G byright translation and the bilinear form 〈·, ·〉 extend to H2(ν, p). All the statementshere about H2(ν, p) are equivalent to the statements concerning P2p,2ν . Theorem16.2 in [22] gives a complete list of all the irreducible unitary representations of Gup to unitary equivalence. In our terms it reads:

• the trivial representation,

• unitary principal series: H2(ν, p) with ν ∈ iR, p ∈ 12Z,

• complementary series: completion of H∞(ν, 0), with ν ∈ (0, 1), with respectto another inner product.

The spaces H2(ν, p) and H2(−ν,−p), with ν ∈ iR, are isomorphic as G-modules.Taking g = exp tX, X ∈ g in (2.40), and differentiating with respect to t, we

get for f1 ∈ H∞(ν, p), f2 ∈ H∞(−ν,−p):

〈Xf1, f2〉+ 〈f1, Xf2〉 = 0, for all X ∈ g. (2.41)

A vector in a representation space for G is called K-finite if its K translatesspan a finite-dimensional space. (See [22] for details.) We denote by H(ν, p) thespace of K-finite vectors in H2(ν, p). Right translations by elements of K leaveH(ν, p) invariant, but the G-action does not. However, it is known that H(ν, p)is preserved by the action of the Lie algebra g. As K is connected, the action of

2.3 Irreducible unitary representations of G 19

K is determined by the action of k ⊂ g. Furthermore, unitarity of the g-action inH∞(ν, p) means that there exists a scalar product (·, ·) on H∞(ν, p) such that

(Xf1, f2) + (f1, Xf2) = 0, for all X ∈ g (2.42)

holds. The space H2(ν, p) is then the completion of H∞(ν, p) with respect to thatscalar product, and the action of G in H2(ν, p) is unitary. The irreducibility of theG-module H2(ν, p) is equivalent to the irreducibility of the g-module H∞(ν, p),and hence the space H(ν, p) as a g-module.

For ν ∈ C, l, p, q ∈ 12Z, l > 0, p ≡ q ≡ l (mod 1), we put

ϕl,q(ν, p)(na[r]k) := r1+νΦlp,q(k), (2.43)

for n ∈ N , r > 0, k ∈ K. Clearly ϕl,q(ν, p) is a left N -invariant function on G oftype (l, q).

Since H2,E±,Ωk ∈ k ⊂ g, it is clear from (2.31) and the definition (2.43), that

H2ϕl,q(ν, p) = −iqϕl,q(ν, p),E±ϕl,q(ν, p) = (q ∓ l)ϕl,q(ν, p), (2.44)

Ωkϕl,q(ν, p) = − l2 + l

2ϕl,q(ν, p).

A direct but lengthy calculation, using (2.16)–(2.17), definition (2.43), and theproperties of the functions Φlp,q, gives

Ω±ϕl,q(ν, p) =18((ν ∓ p)2 − 1

)ϕl,q(ν, p), (2.45)

which means that H(ν, p) are simultaneous eigenspaces of Ω±. See [9], (3.29).It follows from the discussion in Section 2.2 that for given p ∈ 1

2Z, the functionsΦlp,q with l ≡ q ≡ p (mod 1), l > |p|, and |q| 6 l, form an orthogonal basis for thespace

L2p(K) := f ∈ L2(K) | f(h[eit]k) = e−2pitf(k).

This immediately gives an orthogonal basis of the space of K-finite vectors in theunitary principal series representations

H(ν, p) = linear combination of ϕl,q(ν, p) , (2.46)

consisting of the functions ϕl,q(ν, p) for l ≡ q ≡ p (mod 1), l > |p|, and |q| 6 l.Their H(ν, p)-norms are determined as follows: Assuming that we are not in thecase ν ∈ Z, |ν| > |p|, the irreducible spaces H(ν, p) and H(−ν,−p) are isomorphicvia

ι(ν, p)ϕl,q(ν, p) =Γ(l + 1− ν)Γ(l + 1 + ν)

ϕl,q(−ν,−p). (2.47)


One g-morphism from H(ν, p) to H(−ν,−p) is given by the Jacquet integral J0 =J0(ν, p), to be described in (4.23), Section 4.1. It is an isomorphism of g-modulesfor ν 6∈ Z. Indeed, the integral gives a continuous function for Re ν > 0, it hasa meromorphic continuation in ν ∈ C with poles at ν ∈ Z60, commutes withthe action of g, and J0(−ν,−p)J0(ν, p) : ϕl,q(ν, p) 7→ π2

p2−ν2 ϕl,q(ν, p). We chooseι(ν, p) to be a multiple of J0(ν, p) such that ι(−ν,−p)ι(ν, p) is the identity onH(ν, p).

Complex conjugation gives a linear map ¯ : H(ν, p)→ H(ν,−p) which satisfiesXf = Xf for all X ∈ g.

We shall write (·, ·)ps for the scalar product in H(ν, p) with respect to whichthe completion is done, if (ν, p) parameterizes the principal series H(ν, p), and(·, ·)cs if (ν, p) parameterizes the complementary series H(ν, 0).

In the case ν ∈ iR, we have H(ν,−p) = H(−ν,−p), and a scalar product onH(ν, p) is given by

(f1, f2)ps = 〈f1, f2〉, for all X ∈ g. (2.48)

In particular, for ϕl,q(ν, p), ϕl′,q′(ν, p) ∈ H(ν, p) we have

(ϕl,q(ν, p), ϕl′,q′(ν, p))ps =

=⟨ϕl,q(ν, p), ϕl′,q′(ν, p)

⟩=∫K

Φlp,q(k)Φl′p,q′(k)dk = δl,l′δq,q′‖Φlp,q‖2K .

Hence

‖ϕl,q(ν, p)‖ps = ‖Φlp,q‖K . (2.49)

In the case ν ∈ (0, 1), the space H(ν,−p) = H(ν,−p) is isomorphic to H(−ν, p)via ι(ν,−p). For p = 0, a scalar product that gives a unitary structure to H(ν, 0)is given by

(f1, f2)cs = 〈f1, ι(ν, 0)f2〉, for all X ∈ g. (2.50)

In particular, for ϕl,q(ν, 0), ϕl′,q′(ν, 0) ∈ H(ν, 0) we have

(ϕl,q(ν, 0), ϕl′,q′(ν, 0))cs =⟨ϕl,q(ν, 0), ι(ν, 0)ϕl′,q′(ν, 0)

⟩=

Γ(l′ + 1− ν)Γ(l′ + 1 + ν)

∫K

Φl0,q(k)Φl′0,q′(k)dk = δl,l′δq,q′

Γ(l + 1− ν)Γ(l + 1 + ν)

‖Φl0,q‖2K .

Hence

‖ϕl,q(ν, 0)‖cs =

√Γ(l + 1− ν)Γ(l + 1 + ν)

‖Φl0,q‖K . (2.51)

Chapter 3

Automorphic forms andautomorphic representations

3.1 Automorphic forms

Let σ be a finite-dimensional representation of K on the vector space Vσ. Letχ denote a character of (O/I)∗. Also let the corresponding character on Γ be givenby (1.14).

Definition 3.1.1. A χ-automorphic form of K-type σ is a smooth function f :G→ Vσ satisfying

(i) f(γg) = χ(d)f(g), for all γ =(∗∗∗d

)∈ Γ,

(ii) f(gk) = σ(k)−1f(g), for all k ∈ K,

(iii) f is an eigenfunction of all elements in Z(g).

The transformation rule given in condition (i) is called χ-automorphic behaviorof the function f with respect to Γ. We assume that σ is a left representation,and therefore the inverse in condition (ii) is necessary to obtain equivariance. Incondition (iii), we identify elements of Z(g) with the corresponding differentialoperator. As Z(g) is a commutative ring, (iii) implies that there exists a characterΥ of Z(g) such that Xf = Υ(X)f for all X ∈ Z(g). This character is determinedby its values on Ω±, since Ω+ and Ω− generate the ring Z(g).

We denote the space of all χ-automorphic forms for a given representation σand a given character Υ of Z(g) by Aχ(Υ;σ).

Definition 3.1.2. A χ-automorphic form f has polynomial growth if it satisfies

f(gκna[r]k) = O(rbκ) as r →∞

22 Automorphic forms and automorphic representations

for some bκ ∈ R, at each cusp κ ∈ Cχ, uniformly for n ∈ N and k ∈ K.

By Apolχ (Υ;σ) we denote the linear subspace of functions f ∈ Aχ(Υ;σ) that

have polynomial growth.

Definition 3.1.3. A χ-automorphic form f is square integrable if it determinesan element of L2(Γ\G;χ)⊗C Vσ.

The measure on Γ\G in the definition of L2(Γ\G;χ) is induced by the Haarmeasure dg on G, fixed in (2.5). The choice of the norm on Vσ is not importantsince the space Vσ is finite-dimensional and all the norms are equivalent.

The central element h[−1] is contained in G∩Γ, hence the following consistencycondition must be satisfied:

χ(−1) = σ(h[−1]). (3.1)

Each finite-dimensional representation of K is the direct sum of irreducibleones. This reduces the study of χ-automorphic forms on G to those of type σ,where σ is an irreducible representation of K.

3.2 Automorphic functions

Definition 3.2.1. Let l, q ∈ 12Z such that |q| 6 l and q ≡ l (mod 1). Let χ be a

character of (O/I)∗ and Υ a character of Z(g). A χ-automorphic function of type(l, q) with character Υ is a smooth function f : G→ C such that

(i) f(γg) = χ(d)f(g), for all γ =(∗∗∗d

)∈ Γ,

(ii) Ωkf = − l2+l2 f, H2f = −iqf ,

(iii) Ω±f = Υ(Ω±)f .

The automorphic functions are just components of the vector valued automor-phic forms for a suitably chosen basis of the space Vσ.

We denote by Aχ(Υ; l, q) the space of all χ-automorphic functions of type (l, q)with character Υ, and

Apolχ (Υ; l, q) = f ∈ Aχ(Υ; l, q) | f of polynomial growth . (3.2)

We define A2χ(Υ; l, q) as the space of f ∈ Aχ(Υ; l, q) that are square-integrable on

Γ\G, i.e. satisfy∫Γ\G |f(g)|2dg <∞.

3.2 Automorphic functions 23

3.2.1 Fourier expansion of automorphic functions

The ring of integers O ⊂ C is a lattice in C. Let

O′ = z ∈ F | Tr(zz′) ∈ Z, ∀z′ ∈ O

be the dual lattice of O. So, O′ is a fractional ideal containing the ring of integersO. The unitary characters on N are of the form

χω : n[z] 7→ e2πiTr(ωz), ω ∈ C. (3.3)

The characters χω with ω ∈ O′ are precisely the characters of N that are trivial onΓN = N ∩Γ. These will appear in the Fourier expansion of automorphic functions.

For any continuous function f on G having a χ-automorphic transformationbehavior, the function z 7→ f(n[z]g) is periodic on C for the lattice O. Thus, thereexists a Fourier expansion of f at the cusp ∞:

f(g) =∑ω∈O′

Fωf(g), (3.4)

where the Fourier term of order ω is given by

Fωf(g) =1

vol(ΓN\N)

∫ΓN\N

χω(n)−1f(ng)dn. (3.5)

Here vol(ΓN\N) =√|dF |2 is the Euclidean area of the fundamental parallelogram

of O.The function Fωf transforms via the character χω with respect to the left

action of N :

Fωf(ng) = χω(n)Fωf(g), ∀n ∈ N. (3.6)

For any X ∈ U(g), we have

XFωf(g) = ∂tFωf(g exp(tX))|t=0

= ∂t

2√|dF |

∫ΓN\N

χω(n)−1f(ng exp(tX))dn

∣∣∣∣∣t=0

=2√|dF |

∫ΓN\N

χω(n)−1∂tf(ng exp(tX))|t=0 dn

=2√|dF |

∫ΓN\N

χω(n)−1Xf(ng)dn = FωXf(g), (3.7)

that is, the operator Fω commutes with every element of U(g).


For ω ∈ O, let C∞(N\G,ω), be the space of all smooth functions f on G suchthat f(ng) = χω(n)f(g) for all n ∈ N . Identities (3.6) and (3.7) imply that iff ∈ Aχ(Υ; l, q), then Fωf belongs to the space

Wl,q(Υ, ω) = h ∈ C∞(N\G,ω) | h is of type (l, q) with character Υ . (3.8)

In general, the functions occurring in the spectral decomposition of L2(N\G,ω),for some ω ∈ C∗, are called Whittaker functions, generalizing the Whittaker func-tions Wκ,µ that turn up for G = SL2(R). This explains the letter W in (3.8).

We already mentioned in Section 3.1, the character Υ is determined by itsvalues on Ω± since they generate the ring Z(g). Not every point in C2 can occuras(Υ(Ω+),Υ(Ω−)

)for some character Υ of Z(g). We shall now investigate which

complex values can appear as such.

Lemma 3.2.2. If Wl,q(Υ, ω) 6= 0 then there exist ν ∈ C and p ∈ 12Z, |p| 6 l,

uniquely determined modulo (ν, p) 7→ (−ν,−p), such that Υ = Υν,p. Here Υν,p isthe character of Z(g) defined by:

Υν,p(Ω±) =18((ν ∓ p)2 − 1

).

Proof. Let h ∈ Wl,q(Υ, ω) and h 6= 0. We note that for any fixed g ∈ G, thefunction k 7→ h(gk) belongs to L2(K; l, q). In particular, for g = n[z]a[r],

h(n[z]a[r]k) =∑|p|6l

hp(z, r)Φlp,q(k). (3.9)

The formulas (2.16), (2.17) and the property (P3) on page 15 imply that thecondition Ω±h = Υ(Ω±)h is equivalent to the system of equations

Υ(Ω+)hp = 12 (l − p)r∂zhp+1+

+ 18

(4r2∂z∂z + r2∂2

r − (2p+ 1)r∂r + p(p+ 2))hp

Υ(Ω−)hp = − 12 (l + p)r∂zhp−1+

+ 18

(4r2∂z∂z + r2∂2

r + (2p− 1)r∂r + p(p− 2))hp

(3.10)

where hp ≡ 0 if |p| > l.For ω = 0, the function h is N -invariant, which means that the functions

hp = hp(r) depend only on r. We write Υ(Ω±) = 18 (a± − 1), with some a± ∈ C.

Then (3.10) is equivalent tor2h′′p − rh′p +

(p2 + 1− a++a−

2

)hp = 0

p(rh′p − hp

)= a−−a+

4 hp

(3.11)


• If p = 0, then a+ = a− or h0 = 0. For non-zero h0 the first equation of(3.11) is

r2h′′0 − rh′0 + (1− a+)h0 = 0.

– If a+ = 0, this equation has a fundamental system of solutions: r andr log r. Then Υ(Ω+) = Υ(Ω−) = − 1

8 .

– If a+ 6= 0, this equation has a fundamental system of solutions: r1+ν

and r1−ν , where ν2 = a+. Then Υ(Ω+) = Υ(Ω−) = 18

(ν2 − 1

).

• If p 6= 0, the second equation in (3.11) has a solution r1+ν , with ν = a−−a+4p ,

and the first equation therefore implies a+ + a− = 2(ν2 + p2). Hence a± =(ν ∓ p)2, and Υ(Ω±) = 1

8

((ν ∓ p)2 − 1

).

For ω 6= 0, hp(z, r) = e2πi(ωz+ωz)hp(r), and (3.10) becomesr2h′′p − (2p+ 1)rh′p +

((p+ 1)2 + (4πi|ω|r)2 − µ2

+

)hp =

= 8(p− l)πiωrhp+1

r2h′′p + (2p− 1)rh′p +((p− 1)2 + (4πi|ω|r)2 − µ2

−)hp =

= 8(l + p)πiωrhp−1,

(3.12)

where we have written Υ(Ω±) = 18

(µ2± − 1

), for some µ± ∈ C. Without loss of

generality, we may assume that

0 6 Reµ+ 6 Reµ−. (3.13)

Since hp ≡ 0 if |p| > l, we see from the first equation for p = l, and from thesecond equation for p = −l that h±l are solutions of the differential equations

r2d2w

dr2+ (1− 2(l + 1))r

dw

dr+((4πi|ω|r)2 + (l + 1)2 − µ2

±)w = 0. (3.14)

The equations (3.14) have solutions h±l(r) = rl+1Zµ±(4πi|ω|r), with Zµ being anarbitrary solution of the modified Bessel’s differential equation of order µ. Thisfact can be found in [32], p. 77. The function Zµ(4πi|ω|r) can be written as a linearcombination of the fundamental system Kµ(4π|ω|r), Iµ(4π|ω|r). So, the generalsolutions of (3.14) are

h±l(r) = A±rl+1Kµ±(4π|ω|r) +B±r

l+1Iµ±(4π|ω|r), (3.15)

where A±, B± are constants.Let us assume that µ± 6∈ Z. Inductively applying the second equation of (3.12)

to hl(r), we see that in the expansion of h−l(r) all terms are multiples of eitherr−l+1+µ++2m or r−l+1−µ++2n with integers m,n > 0.


• If A− 6= 0 in (3.15), h−l contains a term equal to a multiple of rl+1−µ− .Thus we have either −l + 1 + µ+ + 2m = l + 1− µ− or −l + 1− µ+ + 2n =l + 1− µ−. The first identity gives µ+ + µ− = 2(l −m), which implies thatµ± = ±ν + (l −m) with some ν ∈ C, and 0 6 m 6 l because of (3.13). Thesecond identity gives µ+ = µ−−2(l−n), which implies that µ± = ν∓ (l−n)with some ν ∈ C, and 0 6 n 6 l because of (3.13). In both situations wehave Υ(Ω±) = 1

8

((ν ∓ p)2 − 1

), with some p ∈ 1

2Z, |p| 6 l.

• If A− = 0 in (3.15), we have h−l(r) = B−rl+1Iµ−(4π|ω|r). Inductively

applying the first equation of (3.12) to h−l(r), in the same way as before,we obtain that there exist some ν ∈ C and p ∈ 1

2Z, |p| 6 l, such thatΥ(Ω±) = 1

8

((ν ∓ p)2 − 1

).

We note that µ+ ∈ Z implies µ− ∈ Z, and vice versa. If µ− ∈ Z, then byapplying inductively the first equation of (3.12) to h−l(r), we see that all terms inthe expansion of hl(r) are multiples of either r−l+1+µ−+2m log r or r−l+1+µ−+2n

with integers m,n > 0.

• If A+ 6= 0, then hl contains a term which is multiple of rl+1+µ+ log r, so itmust be µ+ = µ− − 2(l −m), with 0 6 m 6 l.

• If A+ = 0, then hl contains a term which is multiple of rl+1+µ+ , so µ+ =µ− − 2(l − n), with 0 6 n 6 l.

In both cases, we again arrive at the same conclusion.

Remark 1. The argument of the functions (3.15) is twice the corresponding ar-gument in [9], p. 21. This is due to the use of Tr(ωz) = TrF/Q(ωz) as a scalarproduct on C, instead of Re(ωz) = 1

2Tr(ωz) used in [9].The above discussion for ω = 0 actually shows that

Wl,q(Υ0,0, 0) = Cϕl,q(0, 0)⊕ C ∂νϕl,q(ν, p)|ν=0,

Wl,q(Υν,p, 0) = Cϕl,q(ν, p)⊕ Cϕl,q(−ν,−p), if (ν, p) 6= (0, 0). (3.16)

Thus

dimWl,q(Υν,p, 0) = 2. (3.17)

From this discussion we only know that

dimWl,q(Υν,p, ω) 6 2 (3.18)

when ω 6= 0.If a function f is of polynomial growth, then its Fourier terms given by (3.5)

inherit this growth property. We put

W poll,q (Υν,p, ω) =

h ∈Wl,q(Υν,p, ω)

∣∣∣∣ h of polynomial growth,uniformly on K

. (3.19)


The function Kν(x) decreases exponentially and Iν(x) increases exponentiallyas x → ∞. So, for any function h ∈ W pol

l,q (Υν,p, ω), the I-Bessel term in (3.15)does not occur. Recalling the asymptotic expansion of the K-Bessel function

Kν(x) =√π

2x−1/2e−x

(1 +O(|x|−1)

)(see [32], p.139), we obtain the following

Lemma 3.2.3. Let ω 6= 0. If W poll,q (Υν,p, ω) is not empty, then it is spanned by a

unique element h of exponential decay, and

h(na[r]k) = O(|ωr|be−4π|ω|r

), as r →∞

for a certain b ∈ R.

In the next chapter, we shall actually prove that dimW poll,q (Υν,p, ω) = 1 for any

ω 6= 0, and use the Jacquet integral to construct an explicit basis for this space.Moreover, from (3.17) and (3.18) we see that dimWl,q(Υν,p, ω) = 2 always. Weshall use the Goodman-Wallach operator to construct an element of exponentialgrowth in Wl,q(Υν,p, ω), ω 6= 0.

Next we introduce the notion of a cusp form.

Definition 3.2.4. Let κ ∈ Cχ be a cusp for Γ. For a function f ∈ L2(Γ\G), thefunction on N\G given by

aκ(g) =1|Λκ|

∫Rκ

f(gκn[z]g)d+z, (3.20)

is called the Fourier term of order zero of f at κ. Here |Λκ| is the area of thefundamental domain Rκ used in (1.22).

Definition 3.2.5. An automorphic function whose zeroth order Fourier term aκis identically equal to zero for all cusps κ is called a cusp form.

We denote by A0χ(Υν,p; l, q) the space of all χ-automorphic cusp forms of type

(l, q) and character Υν,p. The next lemma is an immediate corollary of Lemma5.2.1 in Section 5.2.

Lemma 3.2.6. All cusp forms f are real-analytic and of exponential decay at eachcusp κ ∈ Cχ:

f(gκna[r]k) = O(e−αr), as r →∞

uniformly over N and K, for some α > 0.


3.2.2 Spectral parameter

Let f ∈ Aχ(Υ; l, q) be a non-zero χ-automorphic function of type (l, q) withcharacter Υ. It has a Fourier expansion f =

∑ω∈O′ Fωf . We assume that f 6= F0f .

Since f 6≡ 0, there exists ω ∈ O′ such that 0 6≡ Fωf ∈ Wl,q(Υ, ω). According toLemma 3.2.2, there are ν ∈ C and p ∈ 1

2Z, |p| 6 l uniquely determined modulo(ν, p) 7→ (−ν,−p) such that Υ = Υν,p. This immediately gives

Lemma 3.2.7. If Aχ(Υ; l, q) 6= 0 then Υ = Υν,p, for certain ν ∈ C and p ∈ 12Z,

|p| 6 l, which are uniquely determined modulo (ν, p) 7→ (−ν,−p).

The pair of numbers (ν, p) is called the spectral parameter of the automorphicfunction f ∈ Aχ(Υν,p; l, q).

The numbers (ν, p) appear in the parameterization of the irreducible unitaryrepresentations of G, so they might also be referred to as the spectral parameterof a representation. The unitary principal series is indexed by pairs (ν, p) withν ∈ iR, and the complementary series is indexed by (ν, 0) with ν ∈ (−1, 1)\0.The pairs (ν, p) and (−ν,−p) correspond to the same representation. Hence, wemay assume that

(ν, p) ∈ i[0,∞)× 12Z or (ν, p) ∈ (0, 1)× 0. (3.21)

For some congruence subgroups of SL2(C), like SL2(Z[i]) or SL2(Z[√−2]) for

example, it is known that there are no complementary series due to the absenceof exceptional eigenvalues of the Laplacian on Γ\H3. (See [11], Proposition 7.6.2.)Generalization of Selberg’s conjecture states that the smallest positive eigenvalueof the Laplacian is > 1 for all Γ0(I) ⊂ SL2(C).

3.3 Eisenstein series

Eisenstein series form a very important example of automorphic functionswhich have polynomial growth, but are not cusp forms. The functions ϕl,q(ν, p)defined in (2.43) are going to be the building blocks for the Eisenstein series. Someof their properties are given in (2.44)–(2.45) and the rest of Section 2.3.

We need to investigate the behavior of the functions ϕl,q(ν, p) under the leftaction of P = NH. For that purpose, it suffices to consider elements h[u] ∈ H,since the functions ϕl,q(ν, p) are N -invariant.

For h[t] ∈M (that is |t| = 1), we have

Φlp,q(h[t]k

)= Φlp,q

(h[ei(2 argt+ϕ)/2]v[iθ]h[eiψ/2]

)= e−ip(2 arg t+ϕ)−iqψΦlp,q(v[iθ])

= e−2ip arg tΦlp,q(h[eiϕ/2]v[iθ]h[eiψ/2]

)= t−2pΦlp,q(k). (3.22)

3.3 Eisenstein series 29

For u, z ∈ C, u 6= 0, r > 0, the identity

h[u]n[z]a[r] = n[zu2]a[r|u|2]h[u/|u|] (3.23)

holds. Therefore, if g = n[z]a[r]k ∈ G, we have for h[u] ∈ H:

ϕl,q(ν, p)(h[u]g) = ϕl,q(ν, p)(n[zu2]a[r|u|2]h[u/|u|]k)

= (r|u|2)1+νΦlp,q(h[u/|u|]k) (3.22)= |u|2(1+ν) (u/|u|)−2p

ϕl,q(ν, p)(g). (3.24)

Hence ϕl,q(ν, p) is a function on N\G that satisfies

ϕl,q(ν, p)(nh[u]g) = |u|2(1+ν) (u/|u|)−2pϕl,q(ν, p)(g) (3.25)

for all nh[u] ∈ P . It makes sense to form the sum∑γ∈ΓP \Γ χ(γ)−1ϕl,q(ν, p)(γg),

provided that the summands are invariant under the left action of ΓP . That isequivalent to the following condition:

χ(ε) = ε2p, ∀ε ∈ O∗. (3.26)

Definition 3.3.1. Let ν ∈ C, l, p, q ∈ 12Z, such that l ≡ p ≡ q (mod 1), and

|p|, |q| 6 l. Let χ be a character of (O/I)∗. We define the Eisenstein series of type(l, q) at the cusp ∞ by

El,q(ν, p;χ)(g) :=∑

γ∈ΓP \Γ

χ(γ)−1ϕl,q(ν, p)(γg). (3.27)

Because of (3.25) the Eisenstein series El,q(ν, p;χ), whenever it converges ab-solutely, will have a χ-automorphic transformation behavior with respect to thediscrete subgroup Γ.

We now generalize the Definition 3.3.1 by introducing an Eisenstein series atany cusp κ ∈ Cχ. Let gκ ∈ G is such that κ = gκ · ∞. We define a function

hκ(g) := ϕl,q(ν, p)(g−1κ g).

For γ ∈ Γκ, we have that γ = gκ(u0

∗u−1

)g−1κ for some

(u0

∗u−1

)∈ P , and

hκ(γg) = ϕl,q(ν, p)(g−1κ γg) = ϕl,q(ν, p)

((u0

∗u−1

)g−1κ g

)(3.24)= |u|2(1+ν)(u/|u|)−2pϕl,q(ν, p)(g−1

κ g) = |u|2(1+ν) (u/|u|)−2phκ(g). (3.28)

Again, we want to consider the sum∑γ∈Γκ\Γ χ(γ)−1hκ(γg). In order to make

the terms in this sum invariant under Γκ, we impose the following compatibilitycondition on Γκ:

χ(gκ

(u0∗u−1

)g−1κ

)= |u|2(1+ν) (u/|u|)−2p

, (3.29)


for all γ = gκ(u0

∗u−1

)g−1κ ∈ Γκ. At the cusp ∞, the condition (3.29) coincides

with (3.26).We note that, by Theorem 2.1.8 in [11], gκ

(u0

∗u−1

)g−1κ ∈ Γ implies that |u| = 1,

which means that the condition (3.29) simplifies to

χ(gκ

(u0∗u−1

)g−1κ

)= u−2p. (3.30)

Definition 3.3.2. Let ν ∈ C, l, q, p ∈ 12Z, such that l ≡ p ≡ q mod 1, and

|p|, |q| 6 l. Let κ ∈ Cχ with χ a character of (O/I)∗ that satisfies the condition(3.29). We define the Eisenstein series of type (l, q) at the cusp κ by

Eκl,q(ν, p;χ)(g) :=∑

γ∈Γκ\Γ

χ(γ)−1ϕl,q(ν, p)(g−1κ γg). (3.31)

Because of (3.28), the Eisenstein series Eκl,q(ν, p;χ), whenever it converges ab-solutely, will have a χ-automorphic transformation behavior with respect to thediscrete subgroup Γ.

We have the isomorphism between G/K and H3 given by (1.8). Therefore wecan identify the right K-invariant functions on G with the functions on the upperhalf space. In this way, using the convention r(g) = r for g = n[z]a[r]k ∈ G, ourEisenstein series

Eκ0,0(ν, 0; 1)(g) =∑

γ∈Γκ\Γ

r(g−1κ γg)1+ν

corresponds to the Eisenstein series 1[Γκ:Γ′κ]Eg−1

κ(z + rj, ν), with EA(z + rj, ν)

defined in [11], §3.3.2.The absolute convergence of the Eisenstein series Eκl,q(ν, p;χ), follows then from

the absolute convergence of Eg−1κ

(z + rj,Re ν). Namely,

|Eκl,q(ν, p;χ)(g)| 6∑

γ∈Γκ\Γ

|ϕl,q(ν, p)(g−1κ γg)| 6 ‖Φlp,q‖∞

∑γ∈Γκ\Γ

|r(g−1κ γg)1+ν |

=‖Φlp,q‖∞[Γκ : Γ′κ]

∑γ∈Γ′κ\Γ

r(g−1κ γg)1+Re ν =

‖Φlp,q‖∞[Γκ : Γ′κ]

Eg−1κ

(z + rj,Re ν), (3.32)

where ‖Φlp,q‖∞ = maxk∈K|Φlp,q(k)|. The series Eg−1κ

(z + rj, s) converges ab-solutely and uniformly on compact subsets of H3 × s | Re s > 1. (See [11],Propositions 3.2.1 and 3.1.3).

Hence, for Re ν > 1, the Eisenstein series Eκl,q(ν, p;χ) converges absolutely anduniformly on compact subsets of G.

3.4 Automorphic representations 31

3.4 Automorphic representations

Let CK(Γ\G;χ) be the space of K-finite elements in the space C∞(Γ\G;χ) ofsmooth χ-automorphic functions on G with respect to Γ. The action of the realLie algebra sl2(C) on C∞(Γ\G;χ) by right differentiation extends C-linearly to g,and hence to the universal enveloping algebra U(g).

An intertwining operator T : H(ν, p) → CK(Γ\G;χ) for the action of the Liealgebra g is called automorphic representation of H(ν, p). By AR(ν, p) we denotethe linear space of all automorphic representations of H(ν, p).

If the spectral parameter (ν, p) is such that H(ν, p) is irreducible, we knowthat H(ν, p) and H(−ν,−p) are isomorphic. This gives a linear bijection T 7→T ι(−ν,−p) between the spaces AR(ν, p) and AR(−ν,−p).

Let T ∈ AR(ν, p). For all integers or half integers l > |p|, p ≡ l (mod 1) andq ≡ l (mod 1), |q| 6 l, the function f = Tϕl,q(ν, p) ∈ CK(Γ\G;χ) satisfies:

f(γgh[eit]) = χ(γ)f(g)e−2iqt, for γ ∈ Γ, g ∈ G, t ∈ R

Ωkf = − l2 + l

2f,

Ω±f =(ν ∓ p)2 − 1

8f.

This follows immediately from (2.44), (2.45), and the intertwining property of T . Itshows that there are linear maps from automorphic representations to automorphicfunctions:

AR(ν, p)→ Aχ(Υν,p; l, q) : T 7→ Tϕl,q(ν, p), (3.33)

with Υν,p determined by its values (ν∓p)2−18 on Ω±.

We define ARpol(ν, p), respectively AR2(ν, p) as the subspaces of automorphicrepresentations T ∈ AR(ν, p) for which Tϕl,q(ν, p) ∈ Apol

χ (Υν,p; l, q), respectivelyTϕl,q(ν, p) ∈ A2

χ(Υν,p; l, q) for all l > |p| and all q ≡ l (mod 1), |q| 6 l. We call theelements of ARpol(ν, p) automorphic representations of polynomial growth, andthe elements of AR2(ν, p) square-integrable automorphic representations.

The maps from AR2(ν, p) to A2χ(Υν,p; l, q) given by (3.33) are surjective for all

l ≡ q ≡ p (mod 1), l > |p|, |q| 6 l.

Chapter 4

N-equivariant eigenfunctions

The Whittaker functions for SL2(C) will be of interest in the derivation of thesum formula. More precisely, we shall be interested in the spaces of eigenfunctionsof the Casimir operators Ω± that are N -equivariant, that is, they transform via acertain character under the left action of N .

4.1 Jacquet integral

One way of constructing Whittaker functions is the use of the Jacquet integral.Also, the Jacquet integral turns up in the computation of the Fourier expansionof automorphic functions and automorphic representations.

Definition 4.1.1. For ω ∈ C, we define an integral operator on the space offunctions f ∈ C∞(G) satisfying an estimate of the form

f(na[r]k) = O(r1+σ) as r ↓ 0 (4.1)

for some σ > 0, uniformly for n ∈ N , k ∈ K, by

Jωf(g) :=∫N

χω(n)−1f(wng)dn. (4.2)

The integral (4.2) is called the Jacquet integral. It was studied by Jacquet [19],for more general groups than SL2(C).

Remark 2. The definition (4.2) of Jω is the same as the definition of Aω in [9],(5.7) except that the character ψω on N defined in [9] equation (4.5) satisfiesψ2ω(n) = χω(n), due to the use of Re(ωz) in the definition, instead of Tr(ωz) =TrF/Q(ωz). This implies that Jω = A2ω.

34 N-equivariant eigenfunctions

We note the following relation in G:

wn[z]a[r] = n[−z

r2 + |z|2

]a[

r

r2 + |z|2

]k

[z√

r2 + |z|2,

−r√r2 + |z|2

]. (4.3)

With the help of this relation, one concludes that the integral in (4.2) convergesabsolutely and uniformly on compact subsets of G.

It is obvious that

Jωf(ng) = χω(n)Jωf(g), for all n ∈ N (4.4)

whenever the integral converges absolutely.Let X ∈ sl2(C), and suppose that Xf also satisfies (4.1). Then

XJωf = JωXf, (4.5)

that is, the integral Jω is an intertwining operator for the action of the universalenveloping algebra U(g).

If lt is the left translation given by

ltf(g) = f(h[t]g), (4.6)

and f satisfies the condition in Definition 4.1.1, then Jω has the following property

ltJωlt = |t|4Jt2ω for any t ∈ C∗. (4.7)

Indeed,

ltJωltf(g) =∫N

χω(n)−1f(h[t]wnh[t]g)dn

=∫

Ce−2πiTr(ωz)f(wn[zt−2]g)d+z

(z 7→t2z)=

∫Ce−2πiTr(t2ωz)f(wn[z]g)t2t 2d+z

= |t|4∫N

χt2ω(n)−1f(wng)dn = |t|4Jt2ωf(g).

This means that actually we may restrict our attention to J0 and J1.Condition (4.1) holds for ϕl,q(ν, p) provided that Re ν > 0. Therefore the

Jacquet integral (4.2) with f = ϕl,q(ν, p) converges absolutely for Re ν > 0. SinceH(ν, p), defined in (2.46), is a U(g)-module, (4.5) implies the smoothness of thefunction Jωϕl,q(ν, p)(g), and (4.4) implies that Jωϕl,q(ν, p) has the right trans-formation behavior under the action of N . Hence, Jωϕl,q(ν, p) ∈ C∞(N\G,ω).The property (4.5) implies in particular that Jωϕl,q(ν, p)(g) is of type (l, q) andcharacter Υν,p. Hence Jωϕl,q(ν, p) ∈Wl,q(Υν,p, ω).

4.1 Jacquet integral 35

We want to compute Jωϕl,q(ν, p) explicitly. We have

Jωϕl,q(ν, p)(n[z]a[r]k) =∫N

χω(n)−1ϕl,q(ν, p)(wnn[z]a[r]k)dn

(4.3)=∫

Ce−2πiTr(ωu)ϕl,q(ν, p)

(n[

−z−ur2+|z+u|2

]·

· a[

rr2+|z+u|2

]k[

z+u√r2+|z+u|2

, −r√r2+|z+u|2

]k

)d+u

(change : u 7→ ru− z)

=∫

Ce−2πiTr(ω(ru−z))ϕl,q(ν, p)

(n[

−ur(1+|u|2)

]·

· a[

1r(1+|u|2)

]k[

u√1+|u|2

, −1√1+|u|2

]k

)r2d+u

= χω(n[z]) r1−ν∫

C

e−2πiTr(ωru)

(1 + |u|2)1+νΦlp,q

(k[

u√1+|u|2

, −1√1+|u|2

]k

)d+u.

Because of (2.30) we have that

Φlp,q

(k[

u√1+|u|2

, −1√1+|u|2

]k

)=∑|m|6l

Φlp,m

(k[

u√1+|u|2

, −1√1+|u|2

])Φlm,q(k),

which implies

Jωϕl,q(ν, p)(na[r]k) = χω(n)∑|m|6l

vlm(r, ω)Φlm,q(k) (4.8)

with

vlm(r, ω) := r1−ν∫

C

e−2πiTr(ωrz)

(1 + |z|2)1+νΦlp,m

(k[

z√1+|z|2

, −1√1+|z|2

])d+z. (4.9)

After a change of variables z = ueiφ in the integral in (4.9), we have

vlm(r, ω) = r1−ν∫ ∞

0

∫ π

−π

e−2πiTr(ωrueiφ)

(1 + u2)1+νΦlp,m

(k[ue−iφ√

1+u2 ,−1√1+u2

])u dφ du.

Now, we use the relation

k[e−iφα, β] = h[e−iφ/2]k[α, β]h[e−iφ/2]

and the property (P3) on page 15, to obtain

vlm(r, ω) = r1−ν∫ ∞

0

u

(1 + u2)1+νΦlp,m

(k[

u√1+u2 ,

−1√1+u2

])I(ω) du (4.10)


where

I(ω) :=∫ π

−πe−2πiTr(ωrueiφ)+i(p+m)φdφ (4.11)

with p+m ∈ Z. We consider the cases ω = 0 and ω 6= 0 separately.

For ω = 0, the fact that

I(0) =∫ π

−πei(p+m)φdφ =

2π if m = −p,0 if m 6= −p,

implies the equality

vlm(r, 0) = δm,−p 2πr1−ν∫ ∞

0

u

(1 + u2)1+νΦlp,m

(k[

u√1+u2 ,

−1√1+u2

])du. (4.12)

By definition of the function Φlp,−p we have

Φlp,−p(k[

u√1+u2 ,

−1√1+u2

])= (−1)p−l

l−|p|∑i=0

(−1)i(l − |p|i

)(l + |p|i

)u2i

(1 + u2)l,

and the integral in (4.12), for m = −p, is equal to

= (−1)p−ll−|p|∑i=0

(−1)i(l − |p|i

)(l + |p|i

)∫ ∞

0

u2i+1du

(1 + u2)1+l+ν

= (−1)p−ll−|p|∑i=0

(−1)i(l − |p|i

)(l + |p|i

)Γ(i+ 1)Γ(l + ν − i)

2Γ(1 + l + ν)

= (−1)p−ll−|p|∑i=0

(−1)i(l − |p|i

)(l + |p|)!Γ(l + ν − i)

2(l + |p| − i)!Γ(1 + l + ν)

= (−1)p−lΓ(l + |p|+ 1)2Γ(1 + l + ν)

l−|p|∑j=0

(−1)l−|p|−j(l − |p|j

)Γ(|p|+ ν + j)Γ(2|p|+ 1 + j)

= (−1)p−|p|Γ(l + |p|+ 1)Γ(|p|+ ν)2Γ(1 + l + ν)Γ(2|p|+ 1)

l−|p|∑j=0

(−1)j(l − |p|j

)(|p|+ ν)j(2|p|+ 1)j

. (4.13)

Here (α)j = α(α+ 1) . . . (α+ j − 1) is Pochhammer’s symbol. The identity

k∑j=0

(−1)j(k

j

)(α)j(β)j

=(β − α)k

(β)k(4.14)


can be proved by induction. We continue in (4.13):

= (−1)p−|p|Γ(l + |p|+ 1)Γ(|p|+ ν)2Γ(1 + l + ν)Γ(2|p|+ 1)

(|p|+ 1− ν)l−|p|(2|p|+ 1)l−|p|

= (−1)p−|p|Γ(l + 1− ν)Γ(|p|+ ν)

2Γ(l + 1 + ν)Γ(1 + |p| − ν).

So, we get

vlm(r, 0) = δm,−p(−1)p−|p|πΓ(1 + l − ν)Γ(|p|+ ν)

Γ(1 + l + ν)Γ(|p|+ 1− ν)r1−ν . (4.15)

For ω 6= 0, we use the integral representation for the Bessel function of integerorder

Jn(z) =12π

∫ π

−πeiz cos tein(t−

π2 )dt, n > 0

(see [32], p. 79). The above formula and the fact (1.26) yield for p+m > 0

I(ω) =∫ π

−πe−4πi|ω|ru cos(φ+argω)+i(p+m)φdφ

= e−i(p+m)(argω−π2 ) · 2πJp+m(−4π|ω|ru)

= 2π (iω/|ω|)−p−m Jp+m(4π|ω|ru),

and similarly, for p+m < 0

I(ω) = e−i(p+m)(argω+ π2 ) · 2πJ−p−m(−4π|ω|ru)

= 2π(−1)p+m (iω/|ω|)−p−m J−p−m(4π|ω|ru).

We combine the last two results in:

I(ω) = 2πi−|p+m| (ω/|ω|)−p−m J|p+m|(4π|ω|ru). (4.16)

According to sgn (p+m) = ±1, the definition of Φlp,m gives for α, β ∈ R

Φlp,m(k[α, β]) =

= (−1)xminl∓m,l∓p∑

i=0

(−1)i(l ∓mi

)(l ±ml ∓ p− i

)α2i+|p+m|β2l−2i−|p+m|,


where x = l − 12 |m+ p| − 1

2 (m− p). Thus,

Φlp,m(k[

u√1+u2 ,

−1√1+u2

])=

= (−1)x−2l+|m+p| u|p+m|

(1 + u2)l

minl∓m,l∓p∑i=0

(−1)iu2i

(l ∓mi

)(l ±ml ∓ p− i

)= (−1)−l+

12 |m+p|− 1

2 (m−p) u|p+m|

(1 + u2)l·

·minl∓m,l∓p∑

i=0

i∑j=0

(i

j

)(−1)j(1 + u2)j

(l ∓mi

)(l ±ml ∓ p− i

).

Let A = l ∓m, B = l ∓ p, c = |m+ p|. Changing the order of summation we get

Φlp,m(k[

u√1+u2 ,

−1√1+u2

])=

= (−1)−l+c2−

12 (m−p) uc

(1 + u2)l

minA,B∑j=0

(−1)j(1 + u2)jξlp(m, j), (4.17)

where ξlp(m, j) =∑minA,Bi=j

(ij

)(Ai

)(A±2mB−i

). We rewrite ξlp(m, j) in the fol-

lowing way

ξlp(m, j) =A!(A± 2m)!

j!

minA,B∑i=j

1(i− j)!(A− i)!(B − i)!(c+ i)!

.

The sum is symmetric in A and B, so without loss of generality we may assumethat A 6 B. Then we have

A∑i=j

1(i− j)!(A− i)!(B − i)!(c+ i)!

=

=(1

(A− j)!(B − j)!(c+ j)!

A−j∑k=0

(−1)k(A− jk

)(j −B)k

(c+ j + 1)k

(4.14)=

(A+B + c− j)!(A− j)!(B − j)!(c+A)!(c+B)!

,

and thus

ξlp(m, j) =A!(A± 2m)!

j!(A+B + c− j)!

(A− j)!(B − j)!(c+A)!(c+B)!

=j!(2l − j)!

(l − p)!(l + p)!

(A

j

)(B

j

).


Noting that l− 12 (|m+ p|+ |m− p|) = minA,B and l− 1

2 (|m+ p| − |m− p|) =maxA,B we get

ξlp(m, j) =j!(2l − j)!

(l − p)!(l + p)!·

·(l − 1

2 (|m+ p|+ |m− p|)j

)(l − 1

2 (|m+ p| − |m− p|)j

). (4.18)

Substitution of (4.16) and (4.17) into (4.10) gives

vlm(r, ω) = 2π(−1)l−p (iω/|ω|)−p−m ·

· r1−νl− 1

2 (|m+p|+|m−p|)∑j=0

(−1)jξlp(m, j)Y (j), (4.19)

with

Y (j) :=∫ ∞

0

u1+|p+m|

(1 + u2)l+1+ν−j J|p+m|(4π|ω|ru)du. (4.20)

The formula∫ ∞

0

uτ+1

(1 + u2)µ+1Jτ (au)du =

(a/2)µ

Γ(1 + µ)Kτ−µ(a) (a > 0),

holds for −1 < Re τ < 2 Reµ + 32 . (See [32], p.105, line 12). Applying it to the

integral (4.20) gives

Y (j) =(2π|ω|r)ν+l−j

Γ(l + 1 + ν − j)Kν+l−|m+p|−j(4π|ω|r). (4.21)

Now, substituting (4.21) into (4.19) yields

vlm(r, ω) = (−1)l−p(2π)ν |ω|ν−1 (iω/|ω|)−p−ml− 1

2 (|m+p|+|m−p|)∑j=0

(−1)j ·

· ξlp(m, j)(2π|ω|r)l+1−j

Γ(l + 1 + ν − j)Kν+l−|m+p|−j(4π|ω|r). (4.22)

From the discussion above we have proved

Lemma 4.1.2. We have for ω = 0

J0ϕl,q(ν, p) = (−1)p−|p|πΓ(1 + l − ν)Γ(|p|+ ν)

Γ(1 + l + ν)Γ(|p|+ 1− ν)ϕl,q(−ν,−p), (4.23)


and for ω 6= 0

Jωϕl,q(ν, p)(na[r]k) = (−1)l−p(2π)ν |ω|ν−1 ·

· χω(n)∑|m|6l

(iω

|ω|

)−p−mwlm(ν, p; |ω|r)Φlm,q(k), (4.24)

where

wlm(ν, p; r) :=l− 1

2 (|m+p|+|m−p|)∑j=0

(−1)j ·

· ξlp(m, j)(2πr)l+1−j

Γ(l + 1 + ν − j)Kν+l−|m+p|−j(4πr), (4.25)

and ξlp(m, j) is given by (4.18).

For Re ν > 0, the function Jωϕl,q(ν, p) is well defined and equal to the expres-sion in the right hand-sides of (4.23) and (4.24). These expressions are meromor-phic in ν if ω = 0, and holomorphic in ν if ω 6= 0. Thus, (4.23) and (4.24) give ameromorphic, respectively holomorphic, extension of Jωϕl,q(ν, p) to ν ∈ C.

In the case ω 6= 0, the exponential decay of Kµ(r) as r →∞ (see (1.35)) impliesthat wlm(ν, p; r), and hence Jωϕl,q(ν, p)(na[r]k), is of polynomial growth. Hence,Jωϕl,q(ν, p) ∈ W pol

l,q (Υν,p, ω). Moreover, Lemma 3.2.3 tells us that Jωϕl,q(ν, p)spans the space W pol

l,q (Υν,p, ω), for all values of ν ∈ C, p ∈ 12Z.

In this way, (4.24) defines a linear operator, called the Jacquet operator, fromH(ν, p) → W pol(Υν,p, ω), where W pol(Υν,p, ω) is the space spanned by all sub-spaces W pol

l,q (Υν,p, ω), |p|, |q| 6 l. It is an intertwining operator for the action ofU(g), see (4.5). We note that the term Jacquet operator is limited to its applica-tion to the space H(ν, p), whereas the term Jacquet integral will be used whereverit is defined.

In particular, since the space W poll,q (Υ−ν,−p, ω) is identical to W pol

l,q (Υν,p, ω),the function Jωϕl,q(−ν,−p) is a multiple of Jωϕl,q(ν, p). Checking the coefficientsof Φll,q in these functions, we find the functional equation

(2π|ω|)−ν (−iω/|ω|)p+ξ Γ(l + 1 + ν)Jωϕl,q(ν, p) =

= (2π|ω|)ν (−iω/|ω|)−p+ξ Γ(l + 1− ν)Jω ϕl,q(−ν,−p), (4.26)

where

ξ =

0 , p ∈ Z12 , p ∈ 1

2 + Z (4.27)

Note that the number ξ above actually parameterizes the central characterh[±1] 7→ (±1)2ξ, and because of the relation (3.1) we have χ(−1) = (−1)2ξ.


Remark 3. For l ∈ Z (which implies that p, q are also integers), Lemma 4.1.2reduces to the Lemma 5.1 in [9], where wlm(ν, p; r) = αlm(ν, p; 2r). Also the func-tional equation (4.26) coincides with (5.29) in [9].

Lemma 4.1.3. Let σ > 0. For ω 6= 0, ν ∈ C, and l ∈ 12N, p, q ≡ l (mod 1),

|p|, |q| 6 l, the following estimates of the Jacquet integral are satisfied:

Jωϕl,q(ν, p)(na[r]k)ω,ε r1−|Re ν|−ε(1 + | Im ν|)2|Re ν|−1, (4.28)

uniformly for |Re ν| 6 σ, r 6 r0, r0 > 0, n ∈ N , k ∈ K, for each ε > 0, and

Jωϕl,q(ν, p)(g)ω,g (1 + | Im ν|)2|Re ν|−1, (4.29)

for fixed g = na[r]k ∈ G.

Proof. The expressions (4.24) and (4.25) in Lemma 4.1.2 and the estimate(1.33) imply that

Jωϕl,q(ν, p)(na[r]k)

ω maxm,j

∣∣∣Γ(l + 1 + ν − j)−1rl+1−jKν+l−|m+p|−j(4π|ω|r)∣∣∣

ω maxm,j

rl+1−j−

∣∣Re ν+l−|m+p|−j∣∣−ε·

· (1 + | Im ν|)−Re ν−l−1+j+ 12+∣∣Re ν+l−|m+p|−j

∣∣− 12

, (4.30)

where |m| 6 l, 0 6 j 6 l − 12 |m+ p| − 1

2 |m− p|. On one side,

Re ν + l − |m+ p| − j 6 Re ν + l − j 6 |Re ν|+ l − j,

and on the other side, since 2j 6 2l − |m+ p| − |m− p| 6 2l − |m+ p|,

Re ν + l − |m+ p| − j > Re ν − l + j > −|Re ν| − l + j.

Hence∣∣Re ν + l − |m+ p| − j

∣∣ 6 |Re ν|+ l − j, which implies

l + 1− j −∣∣Re ν + l − |m+ p| − j

∣∣− ε > 1− |Re ν| − ε,

and−Re ν − l − 1 + j +

∣∣Re ν + l − |m+ p| − j∣∣ 6 −1 + 2|Re ν|.

Estimate (4.28) follows from (4.30) and the last two inequalities. For fixed g =na[r]k ∈ G, the power of r in (4.30) is not present, and we get (4.29).


4.2 Goodman-Wallach operator

As we saw in the previous section, with the help of the Jacquet integral weconstructed an explicit element in Wl,q(Υν,p, ω) of at most polynomial growth.In this section we shall use the Goodman-Wallach operator to construct an ex-plicit element in Wl,q(Υν,p, ω) of exponential growth for all ν that are not integraland strictly smaller than −|p|. In that way we shall show that also for ω 6= 0,dimWl,q(Υν,p, ω) = 2. (See (3.17), Section 3.2.1).

In [13], Goodman and Wallach construct operators that in the context ofSL2(C) are g-morphisms Mω : H(ν, p)→ C∞(G) of the form

Mωϕ(g) =∑m,n>0

a(m,n;ω) ∂mz ∂nz ϕ(wn[z]w−1g)

∣∣z=0

. (4.31)

The coefficients a(m,n;ω) are to be determined, depending only on ν, p, ω, so that

Mωϕ(ng) = χω(n)Mωϕ(g), ∀n ∈ N (4.32)

is satisfied. This is equivalent to∂tMωϕ(n[t]g)|t=0 = 2πiωMωϕ(g)

∂tMωϕ(n[t]g)|t=0 = 2πiωMωϕ(g)with t ∈ C. (4.33)

We still do not know the growth of the coefficients a(m,n;ω), so we work formally.From the definition (4.31), we have

∂tMωϕ(n[t]g)|t=0 =∑m,n>0

a(m,n;ω) ∂t∂mz ∂nz ϕ(wn[z]w−1n[t]g)

∣∣z=0t=0

, (4.34)

∂tMωϕ(n[t]g)|t=0 =∑m,n>0

a(m,n;ω) ∂t∂mz ∂nz ϕ(wn[z]w−1n[t]g)

∣∣z=0t=0

. (4.35)

Simple calculation shows that the following equalities hold:

wn[z]w−1 =(

1−z

01

)= exp

(−z(

01

00

)),

wn[z]w−1

(00t

0

)(wn[z]w−1)−1 = t

(00

10

)+ tz

(10

0−1

)− tz2

(01

00

).

The exponential of the right side of the latter equality, as t→ 0, is

exp(t

(00

10

)+ tz

(10

0−1

)− tz2

(01

00

))= exp

(t

(00

10

))exp

(tz

(10

0−1

))exp

(−tz2

(01

00

))+O(t2)

= n[t]h[etz]wn[tz2]w−1 +O(t2). (4.36)

4.2 Goodman-Wallach operator 43

On the other hand, the matrix identity exp(BAB−1) = B(expA)B−1, for A =(00t0

)and B =

(1−z

01

)yields for the exponential of the left side:

exp(

wn[z]w−1

(00t

0

)(wn[z]w−1)−1

)= wn[z]w−1n[t](wn[z]w−1)−1. (4.37)

Comparing (4.36) and (4.37) we get for t in the neighborhood of 0:

wn[z]w−1n[t] = n[t]h[etz]wn[tz2 + z]w−1 +O(t2). (4.38)

Thus

∂tϕ(wn[z]w−1n[t]g)∣∣t=0

= ∂tϕ(h[etz]wn[tz2 + z]w−1g)∣∣t=0

= ∂t

(e(1+ν−p)tz+(1+ν+p)tzϕ(wn[tz2 + z]w−1g)

)∣∣∣t=0

= (1 + ν − p)zϕ(wn[z]w−1g) + z2∂zϕ(wn[z]w−1g), (4.39)

and similarly

∂tϕ(wn[z]w−1n[t]g)∣∣t=0

=

= (1 + ν + p)zϕ(wn[z]w−1g) + z2∂zϕ(wn[z]w−1g). (4.40)

By induction, from (4.39) and (4.40), we obtain

∂t∂mz ∂

nz ϕ(wn[z]w−1n[t]g)

∣∣z=0t=0

=

= m(ν − p+m) ∂m−1z ∂nz ϕ(wn[z]w−1g)

∣∣z=0

, (4.41)

∂t∂mz ∂

nz ϕ(wn[z]w−1n[t]g)

∣∣z=0t=0

=

= n(ν + p+ n) ∂mz ∂n−1z ϕ(wn[z]w−1g)

∣∣z=0

. (4.42)

Substituting (4.41) and (4.42) respectively into (4.34) and (4.35), we get that(4.33) is equivalent to

2πiω a(m,n;ω) = (m+ 1)(ν − p+ 1 +m) a(m+ 1, n;ω),2πiω a(m,n;ω) = (n+ 1)(ν + p+ 1 + n) a(m,n+ 1;ω).

(4.43)

Hence

a(m,n;ω) =(2πiω)m(2πiω)n

m!n! (ν − p+ 1)m(ν + p+ 1)na(0, 0;ω).

Choosing a(0, 0;ω) = Γ(ν − p+ 1)Γ(ν + p+ 1)−1, we obtain

a(m,n;ω) =(2πiω)m(2πiω)n

m!n! Γ(ν − p+ 1 +m)Γ(ν + p+ 1 + n). (4.44)


With this choice of the vector a(m,n;ω) | m,n > 0, the sum in (4.31)converges absolutely for any element ϕ ∈ H(ν, p). Indeed, the fact that z 7→ϕ(wn[z]w−1g) is an analytic function provides us with the necessary bound of thederivatives.

For any X ∈ U(g), we have

XMωϕ(g) = ∂t

∑m,n>0

a(m,n;ω) ∂mz ∂nz ϕ(wn[z]w−1getX)

∣∣z=0

∣∣∣∣∣∣t=0

=∑m,n>0

a(m,n;ω) ∂mz ∂nz

(∂tϕ(wn[z]w−1getX)|t=0

)∣∣z=0

=∑m,n>0

a(m,n;ω) ∂mz ∂nzXϕ(wn[z]w−1g)

∣∣z=0

= MωXϕ(g). (4.45)

Because of (4.32) and (4.45), the intertwining operator Mω maps ϕl,q(ν, p) intoWl,q(Υν,p, ω). Hence, there should be an expansion Mωϕl,q(ν, p) in terms of Φlm,q,|m| 6 l. It is described explicitly in the following

Lemma 4.2.1. For any ω 6= 0, we have

Mωϕl,q(ν, p)(na[r]k) =

= (2π|ω|)−ν−1χω(n)∑|m|6l

(−iω|ω|

)p−mµlm(ν, p; |ω|r)Φlm,q(k), (4.46)

where

µlm(ν, p; r) =l− 1

2 (|m+p|+|m−p|)∑j=0

ξlp(m, j)(2πr)l+1−j

Γ(l + 1 + ν − j)Iν+l−|m+p|−j(4πr). (4.47)

We also have the functional equation

π−2(2π|ω|)−ν (−iω/|ω|)p−ξ Γ(1 + l + ν)Jωϕl,q(ν, p) =

= − (−1)p−ξ

sinπ(ν − p)(2π|ω|)ν (iω/|ω|)−p−ξ Γ(1 + l + ν)Mωϕl,q(ν, p) +

+(−1)−p−ξ

sinπ(ν − p)(2π|ω|)−ν (iω/|ω|)p−ξ Γ(1 + l − ν)Mωϕl,q(−ν,−p), (4.48)

with ξ given by (4.27).

Remark 4. For l, p, q ∈ Z, we have µlm(ν, p; r) = βlm(ν, p; 2r), where βlm(ν, p; r)is defined with (6.14) in [9]. Comparing our (4.46) with (6.13) in [9] we see thatMω = B2ω. Using this, we also see that the functional equation (4.48) in this casesimplifies to (6.15) in [9].


Proof. Let us suppose that ν 6∈ Z. In the definition (4.25) of wlm(ν, p; r) wereplace the K-Bessel function by its defining expression (1.30). Then wlm(ν, p; r) isa difference of two parts; one is r−ν times a power series in r, and the other one isrν times another power series in r. Since wlm(ν, p; r) satisfies the system (3.10) andthe terms of both parts are not mixed under differentiation, we conclude that eachpart satisfies (3.10). The part which contains rν is equal to (−1)1+|m+p|π

2 sinπ(ν+l) µlm(ν, p; r),whence the right side of (4.46) belongs to Wl,q(Υν,p, ω). The other part yieldsanother member of Wl,q(Υν,p, ω), and they are linearly independent. In Section3.2.1, equation (3.18), we have seen that dimWl,q(Υν,p, ω) 6 2. Therefore we findthat

dimWl,q(Υν,p, ω) = 2. (4.49)

On the other hand, there is a power series P (r) such that

M1ϕl,p(ν, p)(a[r]) = a(0, 0; 1)r1+νP (r). (4.50)

Indeed, from the fact that

wn[z]w−1a[r] = n[−r2z

1 + r2|z|2

]a[

r

1 + r2|z|2

]k

[1√

1 + r2|z|2,

rz√1 + r2|z|2

],

it follows that

M1ϕl,q(ν, p)(a[r]) =∑m,n>0

a(m,n; 1) ·

· ∂mz ∂nz ϕl,q(ν, p)(

a[

r1+r2|z|2

]k[

1√1+r2|z|2

, rz√1+r2|z|2

])∣∣∣∣z=0

= a(0, 0; 1)r1+νP (r),

where

P (r) :=∑m,n>0

a(m,n; 1)a(0, 0; 1)

·

· ∂mz ∂nz

(1 + r2|z|2)−ν−1 Φlp,q

(k[

1√1+r2|z|2

, rz√1+r2|z|2

])∣∣∣∣z=0

.

The term with m = n = 0 in P (r) is equal to Φlp,q(1), which is 1 if p = q, and0 otherwise. In the other terms, after differentiation we get an expression whichhas powers of r2, r2z or r2z in the numerator, and powers of 1 + r2|z|2 in thedenominator. When z = 0, we are only left with the powers of r. Hence P (r) isindeed a power series in r with P (0) = 1 if p = q.


Now, M1ϕl,p(ν, p) ∈ Wl,p(Υν,p, 1) and (4.50) imply that M1ϕl,p(ν, p) must bea constant multiple of the right hand side of (4.46). Entering the defining sum ofIν+l−|m+p|−j(z) into the right hand side of (4.46) for ω = 1, we see that lowestpower of r, for p = q, is equal to

(2π)−ν−1ξlp(p, l − |p|)(2πr)ν+1

Γ(ν + 1 + |p|)Γ(ν + 1− |p|)= a(0, 0; 1)rν+1,

which immediately implies that the constant is 1. Hence (4.46) is true for ω = 1.Next we observe that

ltMωl−1t = Mt2ω for any t ∈ C∗. (4.51)

Indeed, for any ϕ ∈ H(ν, p), we have

ltMωl−1t ϕ(g) =

∑m,n>0

a(m,n;ω) ∂mz ∂nz ϕ(h[1/t]wn[z]w−1h[t]g

)∣∣z=0

=∑m,n>0

a(m,n;ω) ∂mz ∂nz ϕ(wn[t2z]w−1g

)∣∣z=0

=∑m,n>0

a(m,n;ω)t2mt 2n∂mz ∂

nz ϕ(wn[z]w−1g

)∣∣z=0

=∑m,n>0

a(m,n; t2ω) ∂mz ∂nz ϕ(wn[z]w−1g

)∣∣z=0

= Mt2ωϕ(g)

Since (4.46) is true for ω = 1, relation (4.51) implies that (4.46) holds for generalnon-zero ω.

As for equation (4.48), we note that Mωϕl,q(ν, p) and Mωϕl,q(−ν,−p) arelinearly independent elements of Wl,qΥν,p, ω) because of (4.46)–(4.47) and theappearance of Iν(4πr), respectively I−ν(4πr) in the expressions for Mωϕl,q(ν, p),respectively Mωϕl,q(−ν,−p). Thus Jωϕl,q(ν, p) ∈ Wl,q(Υν,p, ω) must be a linearcombination of them. Comparing the coefficients of Φll,q in these three elementswe obtain (4.48).

The case ν ∈ Z is settled by analytic continuation, since both sides of (4.46)are entire in ν. Similarly for (4.48).

Another property of the Goodman-Wallach operator which we shall use lateris the following:

lτMωϕl,q(ν, p) = |τ |2(1+ν)(τ/|τ |)−2pMτ2ωϕl,q(ν, p), (4.52)

for any τ ∈ C∗.


Indeed, direct computation gives for any g ∈ G

lτMω2ϕl,q(ν, p)(g) =

=∑m,n>0

a(m,n;ω2)∂mz ∂nz ϕl,q(ν, p)(wn[z]w−1h[τ ]g)|z=0

=∑m,n>0

a(m,n;ω2)∂mz ∂nz ϕl,q(ν, p)(h[τ ]wn[τ2z]w−1g)|z=0

(3.24)= |τ |2(1+ν) (τ/|τ |)−2p

∑m,n>0

a(m,n;ω2)τ2mτ2n ·

· ∂mz ∂nz ϕl,q(ν, p)(wn[z]w−1g)|z=0

= |τ |2(1+ν) (τ/|τ |)−2pMτ2ω2ϕl,q(ν, p)(g).

We now investigate the behavior of r 7→ Mω(na[r]k) as the argument ap-proaches 0 or ∞. Note that

Mωϕl,q(ν, p)(a[r]g) =

=∑m,n>0

a(m,n;ω)r1+ν+m+n ∂mz ∂nz ϕl,q(ν, p)(wn[z]w−1g)

∣∣z=0

= ϕl,q(ν, p)(a[r]g) (a(0, 0;ω) +O(r)) , as r ↓ 0. (4.53)

Since for Re ν > 0, ϕl,q(ν, p) satisfies condition (4.1) with σ = Re ν, the estimate(4.53) implies

Mωϕl,q(ν, p)(na[r]k) = O(r1+Re ν), as r ↓ 0 (4.54)

uniformly for k ∈ K, n ∈ N .Later we shall need a refined version of the estimate above. Namely, Lemma

4.2.1 and the estimate (1.32) give

Mωϕl,q(ν, p)(na[r]k) max

p,m,j

∣∣Γ(l + 1 + ν − j)−1rl+1−jIν+l−|m+p|−j(4π|ω|r)∣∣

maxp,m,j

r1+Re ν+2l−2j−|m+p|(1 + | Im ν|)−2 Re ν−1−2l+2j+|m+p|

eπ| Im ν|

r1+Re ν(1 + | Im ν|)−2 Re ν−1eπ| Im ν|, (4.55)

uniformly for |Re ν| 6 σ, 0 < r 6 r0 with σ > 0, r0 > 0, and n ∈ N , k ∈ K.As to the behavior of Mω(na[r]k) as r → ∞, we see from Lemma 4.2.1 and

the estimate (1.34) that r 7→Mω(na[r]k) increases exponentially.Because of (4.54) it makes sense to apply the Jacquet integral Jω1 to the

function Mω2ϕl,q(ν, p) and the integral will be absolutely convergent.


Lemma 4.2.2. Let ω2 6= 0 and Re ν > 0. Then

J0Mω2ϕl,q(ν, p) =sinπ(ν − p)ν2 − p2

Γ(1 + l − ν)Γ(1 + l + ν)

ϕl,q(−ν,−p), (4.56)

and for ω1 6= 0

Jω1Mω2ϕl,q(ν, p) = J∗ν,p(4π√ω1ω2)Jω1ϕl,q(ν, p), (4.57)

with

J∗ν,p(z) = J∗ν−p(z)J∗ν+p(z), (4.58)

where J∗ν (z) is the even entire function of z which is equal to Jν(z)(z/2)−ν forz > 0.

Remark 5. In the case when p ∈ Z, J∗ν,p(z) = |z/2|−2ν(z/|z|)2pJν,p(z) with Jν,pthe Bessel function given by [9], (6.21). Obviously, the function Jν,p(

√z) is of

importance to us, and for p ∈ 12 + Z this function is no longer continuous in z.

This discontinuity is actually neutralized by the discontinuity of the factor (z/|z|)pin the expression |z/

√2|−ν(z/|z|)pJν,p(

√z), but to avoid the complications arising

from the choices of square roots, we have decided to introduce the new notationJ∗ν,p(√z) for this expression. In this way, the function J∗ν,p is continuous for all p,

integer or half-integer. The use of the letter J in this notation is to indicate therelation with the Bessel functions, although in general J∗ν,p is not a Bessel function.

Proof. For Re ν > 0, Jω1Mω2ϕl,q(ν, p) is given by the integral (4.2) applied tothe sum (4.31). By absolute convergence, we may change the order of integrationand summation:

Jω1Mω2ϕl,q(ν, p)(g) =∫

Ce−2πiTr(ω1u)Mω2ϕl,q(ν, p)(wn[u]g)d+u

=∑m,n>0

a(m,n;ω2)∫

Ce−2πiTr(ω1u) ∂mz ∂

nz ϕl,q(ν, p) (wn[z + u]g)|z=0 d+u

(change : z 7→ z − u)

=∑m,n>0

a(m,n;ω2)∫

Ce−2πiTr(ω1u) ∂mz ∂

nz ϕl,q(ν, p) (wn[z]g)|z=u d+u

=∑m,n>0

a(m,n;ω2)∫

Ce−2πiTr(ω1u)∂mu ∂

nuϕl,q(ν, p) (wn[u]g) d+u. (4.59)

We transform the last integral by deforming the integration area and using inte-gration by parts. If SR = z ∈ C : |z| 6 R is a closed disk in R2 with radius R,then limR→∞

∫SR

=∫

C holds. Integration by parts gives∫SR

e−2πiTr(ω1u)∂uϕl,q(ν, p) (wn[u]g) d+u =


=∫∂SR

e−2πiTr(ω1u)ϕl,q(ν, p) (wn[u]g) du−

−∫SR

(−2πiω1)e−2πiTr(ω1u)ϕl,q(ν, p) (wn[u]g) d+u,

and by letting R→∞, we obtain∫Ce−2πiTr(ω1u)∂uϕl,q(ν, p) (wn[u]g) d+u =

= 2πiω1

∫Ce−2πiTr(ω1u)ϕl,q(ν, p) (wn[u]g) d+u. (4.60)

Similarly∫Ce−2πiTr(ω1u)∂uϕl,q(ν, p) (wn[u]g) d+u =

= 2πiω1

∫Ce−2πiTr(ω1u)ϕl,q(ν, p) (wn[u]g) d+u. (4.61)

By induction, from (4.60) and (4.61), it follows∫Ce−2πiTr(ω1u)∂mu ∂

nuϕl,q(ν, p) (wn[u]g) d+u =

= (2πiω1)m(2πiω1)n∫

Ce−2πiTr(ω1u)ϕl,q(ν, p) (wn[u]g) d+u. (4.62)

We insert (4.62) in (4.59), and get

Jω1Mω2ϕl,q(ν, p)(g) =

=∑m,n>0

a(m,n;ω2)(2πiω1)m(2πiω1)n∫N

χω1(n)ϕl,q(ν, p) (wng) dn

=∑m,n>0

a(m,n;ω2)(2πiω1)m(2πiω1)n Jω1ϕl,q(ν, p)(g). (4.63)

For ω1 = 0, all the summands in (4.63) are zero, except for the term withm = n = 0, which is equal to a(0, 0;ω2). Using (4.23) we get

J0Mω2ϕl,q(ν, p)(g) = a(0, 0;ω2)J0ϕl,q(ν, p)(g)(4.23)= a(0, 0;ω2)(−1)p−|p|π

Γ(1 + l − ν)Γ(|p|+ ν)Γ(1 + l + ν)Γ(|p|+ 1− ν)

ϕl,q(−ν,−p)(g)

= (−1)p−|p|sinπ(ν − |p|)ν2 − p2

Γ(1 + l − ν)Γ(1 + l + ν)

ϕl,q(−ν,−p)(g),

which proves (4.56), since sinπ(ν − |p|) = (−1)p−|p| sinπ(ν − p).


For ω1 6= 0, we transform the sum (4.63) as follows:∑m,n>0

a(m,n;ω2)(2πiω1)m(2πiω1)n =

=∑m,n>0

(−4π2ω1ω2)m(−4π2ω1ω2)n

m!n!Γ(ν − p+m+ 1)Γ(ν + p+ n+ 1)

=∑m>0

(−1)m(2π√ω1ω2)2m

m!Γ(ν − p+m+ 1)

∑n>0

(−1)n(2π√ω1ω2)n

n!Γ(ν + p+ n+ 1)

= J∗ν−p(4π√ω1ω2)J∗ν+p(4π

√ω1ω2) = J∗ν,p(4π

√ω1ω2).

This proves (4.57).

Chapter 5

Fourier coefficients

5.1 Fourier expansion of Eisenstein series

Let κ, η ∈ Cχ be two cusps of Γ (not necessarily distinct) with correspondinggκ, gη ∈ SL2(C) as in §1.3. Direct generalization of Section 3.4 in [11] to non-trivialK-types gives the explicit Fourier expansions of the Eisenstein series Eκl,q(ν, p;χ)at the cusp η.

Let Λη ⊂ C be the lattice that corresponds to the discrete subgroup gη−1Γ′ηgη,see (1.21). Let |Λη| be the Euclidean area of a fundamental domain for Λη and letΛ′η = z ∈ C | Tr(zλ) ∈ Z, ∀λ ∈ Λη be its dual lattice.

For Re ν > 1 the Λη-invariant function g 7→ Eκl,q(ν, p;χ)(gηg), that is

Eκl,q(ν, p;χ)(gηn[λ]g) = Eκl,q(ν, p;χ)(gηg), for all λ ∈ Λη, (5.1)

has Fourier expansion

Eκl,q(ν, p;χ)(gηg) = δκ,η ϕl,q(ν, p)(g)

+(−1)p−|p| π|Λη|[Γκ : Γ′κ]

Γ(l + 1− ν)Γ(|p|+ ν)Γ(l + 1 + ν)Γ(|p|+ 1− ν)

Dκ,ηχ (0; ν, p)ϕl,q(−ν,−p)(g)

+1

|Λη|[Γκ : Γ′κ]

∑0 6=ω∈Λ′η

Dκ,ηχ (ω; ν, p)Jωϕl,q(ν, p)(g). (5.2)

where the Fourier coefficient of order ω 6= 0 is given by

Dκ,ηχ (ω; ν, p) :=

=∑

( ∗c ∗d )∈R

χ(gκ

(∗c

∗d

)gη−1)−1

|c|−2(1+ν) (c/|c|)2p e2πiTr(ωd/c), (5.3)

52 Fourier coefficients

and R is a system of representatives of the double cosets in

gκ−1Γ′κgκ\gκ−1Γgη/gη−1Γ′ηgη

such that c 6= 0.The Eisenstein series Eκl,q(ν, p;χ), p ∈ 1

2Z, have a meromorphic continuationin ν to the whole complex plane, and moreover, this continuation is holomorphicon the line Re ν = 0. The same holds for the Fourier coefficients Dκ,∞

χ (ω; ν, p).

5.2 Fourier expansion ofautomorphic representations

In this section, we restrict ourselves to the Fourier expansion of an automorphicrepresentation at the cusp ∞, since that is sufficient for our purpose of deriving aspectral sum formula.

One can also develop a sum formula using the Fourier expansion of automorphicrepresentations at other cusps. The Fourier coefficients will then depend on twodata: their order and the cusp at which the expansion is done. The ideas stay thesame as in the case restricted to the cusp ∞, only the necessary book keeping ismore complicated. In the case of SL2(R), such a formula is derived by Proskurin,[38], and by Bruggeman, [3], with more emphasis on representation theory.

Let ω ∈ O and Fω be the operator giving the Fourier term of order ω definedin (3.5). If T ∈ AR(ν, p) is an automorphic representation of H(ν, p), then FωTgives an intertwining operator from H(ν, p) to C∞(N\G,ω). Let W(ν, p;ω) bethe linear space of intertwining operators from H(ν, p) to C∞(N\G,ω). So ifS ∈W(ν, p;ω), then Sϕl,q(ν, p) ∈Wl,q(Υν,p, ω) for each type (l, q).

Let ω = 0. We recall that we have defined the space H(ν, p) as a subspaceof C∞(N\G, 0) (see (2.46)), and therefore the identity map Id(ν, p) : H(ν, p) →H(ν, p) is an element in W(ν, p; 0). If the pair (ν, p) is such that H(ν, p) is irre-ducible, then ι(ν, p) given by (2.47) gives an other element in W(ν, p; 0). Thesetwo elements are linearly independent for (ν, p) 6= (0, 0), which together with thefact that dim W(ν, p; 0) 6 dimWl,q(Υν,p, 0) = 2 (see (3.17)), implies that Id(ν, p)and ι(ν, p) form a basis for W(ν, p; 0) in this case. If (ν, p) = (0, 0), then a basisis given by Id(0, 0) and an other linear operator which acting on the elements ofH(0, 0) gives a function with logarithmic term in r.

Let T ∈ ARpol(ν, p), and (ν, p) such that H(ν, p) is irreducible. For each ω ∈ O,the intertwining operator FωT is an element in W(ν, p;ω). If ω = 0, the Fourierterm F0T is a linear combination of the bases discussed above.

Next we consider ω 6= 0. The operators Jω and Mω give an explicit basis for thespace W(ν, p;ω), since Jωϕl,q(ν, p) and Mωϕl,q(ν, p) form a basis of Wl,q(Υν,p, ω)for each type (l, q) satisfying l > |p|, |q| 6 l. One sees this from (4.49) and thebehavior of the functions Jωϕl,q(ν, p)(na[r]k) and Mωϕl,q(ν, p)(na[r]k) as r →∞.

5.2 Fourier expansion of automorphic representations 53

(See §4.1 and §4.2.) So, FωT must be a linear combination of Jω and Mω. SinceT ∈ ARpol(ν, p), the function Tϕl,q(ν, p) ∈ Apol

χ (Υν,p; l, q) for all types (l, q),and its Fourier term of order ω belongs to the space Wl,q(Υν,p, ω). Moreover,FωTϕl,q(ν, p) has polynomial growth inherited from Tϕl,q(ν, p). Therefore, wecannot have a contribution from Mω in FωT . Hence, FωT must be a multiple ofthe Jacquet operator:

FωT = cT (ω)Jω. (5.4)

The coefficients cT (ω) depend only on the order ω and the automorphic represen-tation T , since both T and Jω commute with the action of g.

The automorphic representations T such that F0T = 0 are called cuspidal.The Fourier expansion of a cuspidal automorphic representation of an irreducibleH(ν, p) is given by

T =∑

0 6=ω∈O′

cT (ω)Jω, (5.5)

In particular, if T is a cuspidal representation, then Tϕl,q(ν, p) ∈ A0χ(Υν,p; l, q),

and we have

Tϕl,q(ν, p) =∑

0 6=ω∈O′

cT (ω)Jωϕl,q(ν, p). (5.6)

From this we see that the Fourier coefficients cT (ω) of a cuspidal automorphic rep-resentation T with spectral parameter (νT , pT ) also appear as Fourier coefficientsof the cups forms with the same spectral parameter and type (l, q), for all l > |pT |,|q| 6 l.

Since lεJωϕl,q(ν, p) = ε2pJε2ωϕl,q(ν, p) for a unit ε ∈ O∗, we conclude that theFourier coefficients must satisfy

cT (ε2ω) = ε2pcT (ω), ∀ε ∈ O∗. (5.7)

Let now κ ∈ Cχ be a cusp for Γ. The following lemma describes more closelythe behavior near the cusps of any Γ′κ-invariant function with an expansion of thetype (5.6).

Lemma 5.2.1. Let κ ∈ Cχ be a cusp, p ∈ 12Z, and l ∈ 1

2N such that l ≡ p (mod 1),l > |p|. Let Λκ ⊂ C be the lattice corresponding to the discrete subgroup gκ−1Γ′κgκwith dual Λ′κ and set ω0 := min

|ω| : ω ∈ Λ′κ\0

.

(i) Any Γ′κ-invariant function f on G of type (l, q) (q ≡ l (mod 1), |q| 6 l) andspectral parameter (ν, p) which has an expansion

f(gκna[r]k) =∑

0 6=ω∈Λ′κ

c(ω)Jωϕl,q(ν, p)(na[r]k), (5.8)


satisfies

f(gκna[r]k) = O(r l+12 e−2πω0r) as r →∞, (5.9)

with the implicit constant depending on l, ν, and p.

(ii) Any Γ′κ-invariant family of functions f(ν) ∈ L2(Γ\G) of type (l, q) and spec-tral parameter (ν, p), where ν runs through a compact set N ⊂ C, with anexpansion of type (5.8) satisfies

f(ν; gκna[r]k) = O(r l+12 e−2πω0r) as r →∞, (5.10)

uniformly for ν ∈ N.

Proof. The analysis on the compact group K, see Section 2.2, implies that forfixed n ∈ N and a ∈ A the function k 7→ f(gκnak) is of type (l, q). The space ofsuch functions is finite-dimensional with basis Φlm,q, |m| 6 l. Thus, using Lemma4.1.2, we have

f(gκna[r]k) =∑

0 6=ω∈Λ′κ

∑|m|6l

c(ω)jωm(ν; r)χω(n)Φlm,q(k), (5.11)

withjωm(ν; r) = (−1)l−p(2π)ν |ω|ν−1 (iω/|ω|)−p−m wlm(ν, p; |ω|r),

and wlm(ν, p; r) as given in (4.25), where the series∑0 6=ω∈Λ′κ

c(ω)jωm(ν; r)χω(n) (5.12)

converges absolutely.In order to estimate the double sum (5.11), we use the asymptotic estimate of

the K-Bessel function for large argument, see (1.35),

Kν+l−|m+p|−j(4π|ω|r) ∼ (8|ω|r)− 12 e−4π|ω|r as r →∞, (5.13)

for all |m| 6 l and 0 6 j 6 l− 12 (|m+ p|+ |m− p|) appearing in wlm(ν, p; r). This

gives

limr→∞

(2π|ω|r)−l−1(8π|ω|r) 12 e4π|ω|rwlm(ν, p; |ω|r) =

= limr→∞

ξlp(m, 0)

Γ(l + 1 + ν)(8π|ω|r) 1

2 e4π|ω|rKν+l−|m+p|(4π|ω|r)

+l− 1

2 (|m+p|+|m−p|)∑j=1

ξlp(m, j)(−2π|ω|r)−j

Γ(l + 1 + ν − j)(8π|ω|r) 1

2 ·

· e4π|ω|rKν+l−|m+p|−j(4π|ω|r)

=(

2ll − |p|

)Γ(l + 1 + ν)−1 + lim

r→∞O(r−1) =

(2l

l − |p|

)Γ(l + 1 + ν)−1.

5.2 Fourier expansion of automorphic representations 55

That is,

wlm(ν, p; |ω|r) ∼ 12

(2l

l − |p|

)(2π|ω|r)l+ 1

2

Γ(l + 1 + ν)e−4π|ω|r as r →∞. (5.14)

If ν is fixed, then the last estimate and the absolute convergence of the series(5.12) at any point (0, t) with t > 1+(l+|ν|)2

4πω0, imply that the coefficients c(ω) must

satisfy the following estimate

c(ω) = O(|ω|−Re ν−l+ 1

2 e4π|ω|t). (5.15)

So if r > 2t,

f(gκna[r]k) <∑ω∈Λ′κ

∑|m|6l

∣∣c(ω)∣∣∣∣jωm(ν; r)

∣∣∣∣Φlm,q(k)∣∣

∑0 6=ω∈Λ′κ

∣∣c(ν;ω)∣∣∣∣jωm(ν; r)

∣∣l,ν,p rl+ 1

2

∑0 6=ω∈Λ′κ

e−4π|ω|(r−t)

l,ν,p rl+ 1

2 e−2πω0r as r →∞,

which proves (5.9).Let now ν run over a compact set N ⊂ C. Suppose that each f = f(ν) is

square integrable on Γ\G, and that its L2- norm ‖f(ν)‖Γ\G is bounded uniformlyfor ν ∈ N. Then this bound holds also for the L2-norm of each term in theexpansion (5.11) when integrated over a cusp sector at κ:∫

Rκ×(r0,∞)

∫K1/2

∣∣∣c(ν;ω)jωm(ν; r)e2πiTr(ωz)Φlm,q(k)∣∣∣2 dk d+z

dr

r3< C1, (5.16)

where C1 = maxν∈N ‖f(ν)‖2Γ\G and c(ν;ω) = c(ω). This inequality remains validwhen we shrink the cusp sector by increasing r0. The left side of (5.16) is equal to∫ ∞

r0

∫Rκ

∣∣c(ν;ω)∣∣2∣∣jωm(ν; r)

∣∣2 ∫K1/2

∣∣Φlm,q(k)∣∣2dk d+zdr

r3=

=12‖Φlm,q‖2K |Rκ|

∣∣c(ν;ω)∣∣2 ∫ ∞

r0

∣∣jωm(ν; r)∣∣2 drr3

=12‖Φlp,q‖2K |Rκ|

∣∣c(ν;ω)∣∣2(2π)2 Re ν |ω|2(Re ν−1)

∫ ∞

r0

∣∣wlm(ν, p; |ω|r)∣∣2 drr3.

If µ = min|m|6l‖Φlp,q‖K

, then (5.16) implies that

∣∣c(ν;ω)∣∣2 < C2(2π)−2 Re ν |ω|−2(Re ν−1)

(∫ ∞

r1

∣∣wlm(ν, p; |ω|r)∣∣2 drr3

)−1

, (5.17)


for all r1 > r0, where C2 = 2C1|Rκ|−1µ−2. It is essential that the asymptoticequality (5.13), and hence (5.14), is uniform for ν ∈ N. Since (5.14) implies∣∣wlm(ν, p; |ω|r)

∣∣ > 14

(2l

l − |p|

)(2π|ω|r)l+ 1

2

Γ(l + 1 + ν)e−4π|ω|r

for large enough r, we have for suitably large r1 > r0∫ ∞

r1


>41−2l

16

(2l

l − |p|

)2 (2π|ω|)2

Γ(l + 1 + ν)2Γ(2l − 1, 8π|ω|r1).

For any a ∈ Z>1, the incomplete gamma function has an expansion of the formΓ(a, z) = e−zza−1

(1 +O(z−1)

)as z →∞, which implies that Γ(a, z) > 1

2e−zza−1

for suitably large z. So, for all r2 > r1 suitably large, we get∫ ∞

r2


>1

128

(2l

l − |p|

)2 (2π|ω|)2l

Γ(l + 1 + ν)2r2

2l−2e−8π|ω|r2 . (5.18)

We substitute (5.18) into (5.17) with r1 replaced by r2 and obtain

∣∣c(ν;ω)∣∣ < 8

√2C2

(2l

l − |p|

)−1 |Γ(l + 1 + ν)|(2π)l+Re ν

r21−l|ω|−l−Re ν+1e4π|ω|r2 . (5.19)

From (5.14) we obtain the asymptotic estimate for jωm(ν; r), which for r largeenough yields∣∣jωm(ν; r)

∣∣ < ( 2ll − |p|

)(2π)l+1+Re ν

|Γ(l + 1 + ν)|r l+

12 |ω|Re ν+l− 1

2 e−4π|ω|r. (5.20)

For r > 2r2, inequalities (5.20) and (5.19) give∑0 6=ω∈Λ′κ

∣∣c(ν;ω)∣∣∣∣jωm(ν; r)

∣∣ << 16π

√2C2 r2

1−l r l+12

∑0 6=ω∈Λ′κ

|ω| 12 e−4π|ω|(r−r2)

r l+12

∑0 6=ω∈Λ′κ

|ω| 12 e−2π|ω|r r l+12 e−2πω0r, for all |m| 6 l. (5.21)

Expansion (5.11) and estimate (5.21) above imply, for all ν ∈ N,∣∣f(ν; gκna[r]k)∣∣ < ∑

ω∈Λ′κ

∑|m|6l

∣∣c(ν;ω)∣∣∣∣jωm(ν; r)

∣∣∣∣Φlm,q(k)∣∣

∑0 6=ω∈Λ′κ

∣∣c(ν;ω)∣∣∣∣jωm(ν; r)

∣∣N r l+12 e−2πω0r as r →∞,

which proves (5.10).

Chapter 6

Kloosterman sums

The spectral sum formula connects the Fourier coefficients of cuspidal auto-morphic representations with Kloosterman sums.

6.1 Definition and properties

Definition 6.1.1. Let F be an imaginary number field, and O its ring of integers.For c ∈ O \ 0, ω, ω′ ∈ O′ \ 0, and a character χ of (O/(c))∗, we define aKloosterman sum associated to F by

Sχ(ω, ω′; c) =∑∗

dmod (c)

χ(d)−1 e2πiTr((dω+dω′)/c), (6.1)

Here∑∗ means that d runs over representatives of (O/(c))∗ and dd ≡ 1 mod (c).

Definition 6.1.1 generalizes the classical Kloosterman sum over Q:

S(m,n; c) =∑

d mod cdd≡1 mod c

e2πi(dm+dn)/c, for m,n, c ∈ Z\0. (6.2)

Kloosterman sums over Q were introduced by Kloosterman, [21], in the study ofquadratic forms.

Using the symmetries d 7→ −d and d 7→ d respectively, we obtain some simpleproperties for a Kloosterman sum (6.1):

Sχ(ω, ω′;−c) = χ(−1)Sχ(ω, ω′; c), (6.3)

Sχ(ω′, ω; c) = Sχ−1(ω, ω′; c), (6.4)

Sχ(ω, ω′; c) = χ(−1)Sχ−1(ω, ω′; c) = χ(−1)Sχ(ω′, ω; c). (6.5)

58 Kloosterman sums

6.2 Estimates

We have the following trivial estimate for the Kloosterman sums

|Sχ(ω, ω′; c)| 6 |N(c)|, (6.6)

where N(c) = |O/(c)| = |c|2.To derive the sum formula, we do not need an estimate for Kloosterman sums.

However, it will turn out in §11.2 that a nontrivial estimate for a Kloostermansum allows us to enlarge the class of test functions.

The classical Kloosterman sum (6.2) satisfies the Salie-Weil type estimate

S(m,n; c)m,n,δ c12+δ

for each δ > 0. See [39], [44], [12]. In [4], §5, the authors worked out the general-ization to Kloosterman sums S1(ω, ω′; c) over an arbitrary algebraic number field.Following closely the approach of Estermann and using the results in [4], we shallprove similar estimate for Sχ(ω, ω′; c), with χ not necessarily trivial.

For each c ∈ I\0, we have (c) =∏j p

mj

j , with pj running through differentprime ideals of O. Let Ic =

∏j:pj |I p

mj

j . Then (c) = Ic · ((c)/Ic) is a decompositionof (c) in relatively prime ideals.

Proposition 6.2.1. Let ω, ω′ ∈ O′\0, and let χ be a character of (O/I)∗. Forc ∈ I\0 and δ > 0 we have

Sχ(ω, ω′; c) = O(N(Ic)|N((c)/Ic)|

12+δ). (6.7)

Proof. For an ideal J ⊂ I and characters ϕ and ψ of the additive group O/J ,there is the generalized Kloosterman sum

Sχ(ϕ,ψ; J) =∑

d∈(O/J)∗

χ(d)−1ϕ(d)ψ(d), (6.8)

with dd ≡ 1 mod J .If J =

∏rj=1 p

mj

j is expanded as a product of prime ideals pj in O, then O/J ≡∏rj=1 O/p

mj

j , and correspondingly ϕ = ⊗rj=1ϕj , ψ = ⊗rj=1ψj , and χ = ⊗rj=1χj ,where ϕj and ψj are characters of the additive group O/p

mj

j , and χj are charactersof (O/pmj

j )∗. The assumption that χ is a character modulo I implies that χj = 1if pj does not divide I. Thus we have the multiplicative property

Sχ(ϕ,ψ; J) =r∏j=1

Sχj(ϕj , ψj ; p

mj

j ). (6.9)

6.2 Estimates 59

We estimate the factors indexed by j such that pj |I with the trivial bound(6.6). For the other factors, Proposition 9 in [4] gives:

|S1(ϕj , ψj ; pmj

j )| 6 c(pj)N(pj)mj−Nj/2 (6.10)

with Nj minimal such that pmj

j is contained in both ker(ϕj) and ker(ψj). Theconstants c(pj) = 2 if 2 - N(pj) and c(pj) = 2N(pj)vj(2)+

12 if vj(2) > 1, where vj

is the valuation at the prime pj .We now take J = (c), ϕ(d) = e2πiTr(ωd), ψ(d) = e2πiTr(ω′d), and obtain

Sχ(ϕ,ψ; J) = Sχ(ω, ω′; c). In [4], §5.2, it is shown that

Nj = max0,−vj(ω)− dj + vj(c),−vj(ω′)− dj + vj(c),

where dj is the order at pj of the different of O. This gives

|Sχ(ω, ω′; c)| 6∏j:pj |I

N(pj)mj ·∏j:pj -I

c(pj)N(pj)vj(c)−Nj/2 6

6 |dF |∏j:pj |I

N(pj)mj ·∏j:pj -I

c(pj)N(pj)12vj(c)+

12 maxvj(ω),vj(ω

′)

F,I,ω,ω′ N(Ic)|N((c)/Ic)|1/22r, (6.11)

where r is the number of prime ideals dividing (c). By the last remark (iv) in §5.2,[4], we have 2r = O(|N(c)|δ) for each δ > 0, which proves the proposition.

A consequence of Proposition 6.2.1 is the following non-trivial estimate for aKloosterman sum:

Sχ(ω, ω′; c)ω,ω′,ε |N(c)| 12+ε, (6.12)

for each ε > 0.

Chapter 7

Poincare series

7.1 Definition and properties

Let χ denote a character of (O/I)∗ and also the corresponding unitary characteron Γ of the form (1.14). Let L2(Γ\G,χ), be the space of all square-integrable χ-automorphic functions on G.

Definition 7.1.1. Let f ∈ C∞(N\G,ω), ω ∈ O′. The Poincare series Pχfgenerated by f , is defined by

Pχf(g) =1

[ΓP : ΓN ]

∑γ∈ΓN\Γ

χ(γ)−1f(γg). (7.1)

Since f is a ΓN -invariant function, the series Pχf has χ-automorphic behaviorwith respect to Γ, provided that the sum converges absolutely.

The convergence is determined by the behavior of the function f(na[r]k) asr ↓ 0. Let us impose the following condition:

f(na[r]k) r1+σ0 , as r ↓ 0 for some σ0 > 0, (7.2)

uniformly for n ∈ N , k ∈ K.On the basis of the Bruhat decomposition Γ ∩ G = Γ ∩ (P t PwN), we can

decompose the series Pχf into two sub-sums

Pχf(g) = Σ1(g) + Σ2(g). (7.3)

The first sum Σ1(g) = 1[ΓP :ΓN ]

∑γ∈ΓN\ΓP

χ(γ)−1f(γg) is finite, hence convergent,and the second sum Σ2(g) = 1

[ΓP :ΓN ]

∑γ∈ΓN\(Γ∩PwN) χ(γ)−1f(γg) is estimated

by the corresponding sum in the Eisenstein series E0,0(σ0, 0; 1). Specifically, the

62 Poincare series

elements γg that appear in the sum belong to a region G(r0(g)) := Na[r] : r 6r0(g)K. In this region |f(γg)| r1+σ0 , and thus

|Σ2(g)| 61

[ΓP : ΓN ]

∑γ∈ΓN\(Γ∩PwN)

|f(γg)| = O(r1−σ0) (7.4)

is convergent. Therefore the Poincare series Pχf converges absolutely if the func-tion f satisfies (7.2) with σ0 > 1.

We now examine the square integrability of the function Pχf on Γ\G. Weassume that f has polynomial decay near infinity, that is,

f(na[r]k) r1−σ∞ , as r →∞ for some σ∞ > 0. (7.5)

We saw in §1.3 that the fundamental domain of Γ\H3 is of the form (1.23), seealso Proposition 2.3.9 in [11]. Near the cusp∞, we estimate the two sums in (7.3)as follows:

|Σ1(na[r]k)| 6 1[ΓP : ΓN ]

∑ε∈O∗

|f(h[ε]na[r]k)|

=1

[ΓP : ΓN ]

∑ε∈O∗

|f(n′a[r]k′)| (7.5)= O(r1−σ∞), as r →∞, (7.6)

which is square-integrable on the interval [1,∞) with respect to the measurer−3dr. If r > c, all γna[r]k that appear in the sum Σ2(na[r]k) belong to a regionG(rc). Because of (7.5), we have in this region |f(γna[r]k)| rc

1+σ∞ , and weestimate the second sub-sum by the corresponding sum in the Eisenstein seriesE0,0(σ∞, 0; 1)(na[r]k):

|Σ2(na[r]k)| 6 1[ΓP : ΓN ]

∑γ∈ΓN\(Γ∩PwN)

|f(γna[r]k)| = O(r1−σ∞), (7.7)

which is square-integrable on [1,∞). The estimates (7.6) and (7.7) show thatPχf(g) is square-integrable on the cusp sector at ∞.

Near a cusp κ 6= ∞, Σ1(gκg) disappears since Γgκ ∩ P = ∅. Therefore, forg = na[r]k with r > r′, all γgκg belong to a region G(r(r′, κ)), and we estimatethe Poincare series Pχf(gκg) by the whole Eisenstein series E0,0(σ∞, 0; 1)(gκg):

|Pχf(gκg)| 61

[ΓP : ΓN ]

∑γ∈ΓN\Γ

|f(γgκg)| = O(r1−σ∞), as r →∞.

Hence Pχf is also square-integrable on the cusp sectors at κ 6=∞. We summarizethe discussion above into

Proposition 7.1.2. If the function f ∈ C∞(N\G,ω), ω ∈ O′, satisfies the fol-lowing growth conditions

7.2 Fourier expansion 63

(i) f(na[r]k) r1+σ0 , as r ↓ 0 for some σ0 > 1,

(ii) f(na[r]k) r1−σ∞ , as r →∞ for some σ∞ > 0,

uniformly for n ∈ N , k ∈ K, then Pχf ∈ L2(Γ\G,χ).

7.2 Fourier expansion

Let f ∈ C∞(N\G,ω) with some ω ∈ O′, satisfies conditions (i) and (ii) inProposition 7.1.2. Since the function z 7→ Pχf(n[z]g) is periodic on C for thelattice O, the Poincare series Pχf has a Fourier expansion at the cusp ∞

Pχf(g) =∑ω′∈O′

Fω′Pχf(g), (7.8)

with a Fourier term given by

Fω′Pχf(g) =2√|dF |

∫ΓN\N

χω′(n)−1Pχf(ng)dn. (7.9)

We want to obtain an explicit expression for the Fourier term of order ω′.Toward this goal, we substitute (7.1) into (7.9), and use the absolute convergenceof the series to interchange the order of summation and integration:

Fω′Pχf(g) =2

[ΓP : ΓN ]√|dF |

∑γ∈ΓN\Γ

χ(γ)−1

∫ΓN\N

χω′(n)−1f(γng)dn. (7.10)

From the Bruhat decomposition G = P t PwN , we have

Γ = ΓP t

⊔c∈Oc 6=0

⊔d mod (c)〈c,d〉=O

ΓN(∗c

∗d

)ΓN

(7.11)

where we use the notation 〈c, d〉 = c · O + d · O. The sum in (7.10) splits into twosub-sums:

Fω′Pχf(g) =2

[ΓP : ΓN ]√|dF |

∑γ∈ΓN\ΓP

+∑

γ∈ΓN\(Γ∩PwN)

. (7.12)

Using the fact that ΓP has a structure of a semi-direct product, we take el-ements h[1/ε] with ε ∈ O∗ as representatives for ΓN\ΓP , and the first sub-sum

64 Poincare series

then equals∑ε∈O∗

χ(ε)−1

∫ΓN\N

χω′(n)−1f(h[1/ε]ng)dn =

=∑ε∈O∗

χ(ε)−1

∫C mod O

e−2πiTr(ω′z)f(n[zε−2]h[1/ε]g)d+z

=∑ε∈O∗

χ(ε)−1f(h[1/ε]g)∫

C mod O

e2πiTr(zωε−2−zω′)d+z

=∑ε∈O∗

χ(ε)−1δε2ω′,ωl1/εf(g) · vol(C modO)

=

√|dF |2

∑ε∈O∗

δε2ω′,ω χ(ε)−1l1/εf(g), (7.13)

where lt is the left translation given by (4.6).Let us denote the big cell in the Bruhat decomposition by C = PwN . The

second sub-sum is equal to∑γ∈ΓN\(Γ∩C)

χ(γ)−1

∫ΓN\N

χω′(n)−1f(γng)dn =

=∑

γ∈ΓN\(Γ∩C)/ΓN

∑δ∈ΓN

χ(γδ)−1

∫ΓN\N

χω′(n)−1f(γδng)dn

=∑


χ(γ)−1∑δ∈ΓN

∫ΓN\N

χω′(δn)−1f(γδng)dn

=∑


χ(γ)−1

∫N

χω′(n)−1f(γng)dn. (7.14)

Using (7.11), for γ ∈ ΓN\ (Γ ∩ C) /ΓN , we get γ = n[a/c]h[1/c]wn[d/c], where cruns over the non-zero elements in I, d runs over representatives of (O/(c))∗, anda is such that ad ≡ 1 mod (c). As χ is a character of (O/(c))∗, we continue with(7.14)

=∑′

c∈I

∑∗

dmod (c)

χ(d)−1

∫N

χω′(n)−1f(n[a/c]h[1/c]wn[d/c]ng)dn

=∑′

c∈I

∑∗

dmod (c)

χ(d)−1

∫Ce−2πiTr(ω′z)+2πiTr(ωa/c)f(h[1/c]wn[d/c+ z]g)d+z

( change: z 7→ z − d/c )

=∑′

c∈I

∑∗

dmod (c)

χ(d)−1e2πiTr(aω/c)

∫Ce−2πiTr(ω′(z−d/c))f(h[1/c]wn[z]g)d+z

7.3 Scalar product of Poincare series 65

=∑′

c∈I

∑∗

dmod (c)

χ(d)−1 e2πiTr((dω′+aω)/c)∫N

χω′(n)−1f(h[1/c]wng)dn

=∑′

c∈ISχ(ω′, ω; c)Jω′ l1/cf(g). (7.15)

Here Sχ(ω′, ω; c) is the Kloosterman sum defined by (6.1), and Jω′f is the Jacquetintegral described in §4.1.

Finally, substituting (7.13) and (7.15) into (7.12), we obtain the following ex-plicit form of the Fourier term of order ω′ of the Poincare series Pχf :

Fω′Pχf =1

[ΓP : ΓN ]

∑ε∈O∗

δε2ω′,ω χ(ε)−1l1/εf +

+2

[ΓP : ΓN ]√|dF |

∑′

c∈ISχ(ω′, ω; c)Jω′ l1/cf. (7.16)

7.3 Scalar product of Poincare series

Under the conditions of Proposition 7.1.2, the Poincare series are square-integrable functions on Γ\G. So, it makes sense to consider the inner productof such Poincare series with a square-integrable function on Γ\G. We may alsoconsider the inner product of such Poincare series with a smooth χ-automorphicfunction on Γ\G that is not square-integrable, provided that its product with thefunction that generates the Poincare series is integrable on Γ\G.

Let ω 6= 0. Denote by Pl,q(N\G,ω) the space of functions f ∈ C∞(N\G,ω)that have type (l, q) and satisfy the following growth conditions:

f(na[r]k) =O(r1+σ0) as r ↓ 0 for some σ0 > 0,O(r1−σ∞) as r →∞ for some σ∞ > 0. (7.17)

The numbers σ0 and σ∞ may depend on the function f .The following lemma is a well known result for absolutely convergent Poincare

series:

Lemma 7.3.1. Let ω ∈ O′ \ 0 and l, q ∈ 12Z, l ≡ q (mod 1), |q| 6 l. Let

f ∈ Pl,q(N\G,ω) satisfy conditions (7.17) with σ0 > 1, and suppose that φ ∈C∞(Γ\G,χ; l, q) is such that Pχf · φ ∈ L1(Γ\G). Then

〈Pχf, φ〉Γ\G =

√|dF |

2[ΓP : ΓN ]〈f, Fωφ〉N\G.

Proof. A simple computation yields

[ΓP : ΓN ]〈Pχf, φ〉Γ\G = [ΓP : ΓN ]∫

Γ\GPχf(g)φ(g)dg

66 Poincare series

=∫

Γ\G

∑γ∈ΓN\Γ

χ(γ)−1f(γg)φ(g)dg =∑

γ∈ΓN\Γ

∫Γ\G

f(γg)φ(γg)dg

=∫

ΓN\Gf(g)φ(g)dg =

∫N\G

∫ΓN\N

f(ng)φ(ng)dn dg

=∫N\G

f(g)∫

ΓN\Nχω(n)φ(ng)dn dg

= vol(ΓN\N)∫N\G

f(g)Fωφ(g)dg =

√|dF |2〈f, Fωφ〉N\G .

Using Lemma 7.3.1 and the expression (7.16) for the Fourier coefficient of aPoincare series, we can obtain an explicit expression for the scalar product of twosquare-integrable Poincare series, which explains how and why Kloosterman sumsappear.

Lemma 7.3.2. Let ω1, ω2 ∈ O′ \ 0 and fi ∈ Pl,q(N\G,ωi), for i = 1, 2, be twofunctions that satisfy the conditions (7.17) with σ0,i > 1.

We have the following expression for the scalar product of the square-integrablePoincare series Pχf1 and Pχf2:

〈Pχf1, Pχf2〉Γ\G =

√|dF |

2[ΓP : ΓN ]2∑ε∈O∗

δε2ω1,ω2 χ(ε)〈f1, l1/εf2〉N\G

+χ(−1)

[ΓP : ΓN ]2∑′

c∈ISχ(ω2, ω1; c)〈f1,Jω1 l1/cf2〉N\G. (7.18)

Proof. First, Lemma 7.3.1 gives the relation between the Poincare series andFourier terms, and then (7.16) gives the explicit result:

〈Pχf1, Pχf2〉Γ\G =

√|dF |

2[ΓP : ΓN ]〈f1, Fω1Pχf2〉N\G

=

√|dF |

2[ΓP : ΓN ]2∑ε∈O∗

δε2ω1,ω2 〈f1, χ(ε)−1l1/εf2〉N\G

+1

[ΓP : ΓN ]2∑′

c∈I〈f1, Sχ(ω1, ω2; c)Jω1 l1/cf2〉N\G

=

√|dF |

2[ΓP : ΓN ]2∑ε∈O∗

δε2ω1,ω2 χ(ε)〈f1, l1/εf2〉N\G

+χ(−1)

[ΓP : ΓN ]2∑′

c∈ISχ(ω2, ω1; c)〈f1,Jω1 l1/cf2〉N\G.

In the last line, we used the property (6.5) of a Kloosterman sum.

7.3 Scalar product of Poincare series 67

To see the convergence of the inner products in (7.18) it suffices to note thefollowing: If we replace the functions f1, f2 by |f1|, |f2|, and work with χ = 1 andω1 = ω2 = 0, then the conditions in Proposition 7.1.2 are satisfied by |f1|, |f2|, thePoincare series P1|f1| and P1|f2| are square-integrable over Γ\G, so their scalarproduct 〈P1|f1|, P1|f2|〉Γ\G is finite. By Fubini’s theorem, the scalar products〈|f1|, l1/ε|f2|〉N\G and 〈|f1|,J0l1/c|f2|〉N\G are finite as well, and they provide amajorant for the right side of (7.18). This implies the absolute convergence withoutthe necessity of providing any estimates for the Kloosterman sums.

Chapter 8

Spectral decomposition ofthe space L2(Γ\G, χ)

Let L2(Γ\G,χ) be the Hilbert space of χ-automorphic functions with respectto the subgroup Γ which are square-integrable on Γ\G with respect to the measureinduced by dg. We have the following orthogonal decomposition:

L2(Γ\G,χ) = C⊕ L2,cusp(Γ\G,χ)⊕ L2,cont(Γ\G,χ). (8.1)

Here C stands for the one-dimensional subspace of constant functions on Γ\G,L2,cusp(Γ\G,χ) for the cuspidal subspace, and L2,cont(Γ\G,χ) for the orthogonalcomplement of C ⊕ L2,cusp(Γ\G,χ). The spectral decomposition (8.1) is a con-sequence of the general theory of Eisenstein series due to Langlands [28]. In ourarithmetical situation, it can be shown that the residues of the Eisenstein seriescontribute only the constant functions in the spectral decomposition.

The closed subspace L2,cont(Γ\G,χ) is generated by integrals of Eisensteinseries.

The cuspidal subspace L2,cusp(Γ\G,χ) is spanned by the cusp forms, functionson L2(Γ\G,χ) with vanishing constant terms in their Fourier expansions at allcusps. It can be decomposed into at most countably many cuspidal subspaces Virreducible with respect to the action of G:

L2,cusp(Γ\G,χ) =⊕

V . (8.2)

The two Casimir elements Ω± given by (2.16) and (2.17) act as multiplication bya constant in each V . There are numbers νV , pV , specified in (3.21), such that

Ω±|V = ΥνV ,pV(Ω±) · 1,

70 Spectral decomposition of the space L2(Γ\G, χ)

with the character ΥνV ,pVon g as in Lemma 3.2.2. Therefore, each subspace V is

characterized by the spectral parameter (νV , pV ).According to the right action of the group K, the space V decomposes into

K-irreducible subspaces

V =⊕

l>|pV |,|q|6l

Vl,q, Vl,q = CTV ϕl,q(νV , pV ), (8.3)

where Vl,q has dimension one. It consists of cuspidal χ-automorphic forms of type(l, q) and spectral parameter (νV , pV ). One can choose a unitary isomorphismof g-modules TV : H(νV , pV ) → V . The image consists of the K-finite vectorsin V , and it is dense in V . For each (l, q) in (8.3) the automorphic functionTV ϕl,q(νV , pV ) spans Vl,q. (The space H(νV , pV ) has been defined in (2.46), seealso Section 2.3).

If pV = 0, then V contains a one-dimensional space of K-trivial vectors inL2,cusp(Γ\G,χ). The set TV ϕ0,0(νV , pV ) | pV = 0 corresponds to an orthogonalsystem in L2(Γ\H3, χ), where χ(−1) = 1. If pV 6= 0, then in V occur only theK-types with l > |pV |. The unitarity of TV implies that

‖TV ϕl,q(νV , pV )‖ =

‖ϕl,q(νV , pV )‖ps if νV ∈ i[0,∞)‖ϕl,q(νV , 0)‖cs if νV ∈ (0, 1),

(8.4)

where (νV , pV ) are as in (3.21) and the norms ‖ · ‖ps in the unitary principal seriesand ‖ ·‖cs in the complementary series are given by (2.49) and (2.51), respectively.

We restrict the decomposition (8.1) to the subspace L2(Γ\G,χ; l, q) spannedby all square-integrable χ-automorphic functions of type (l, q).

Theorem 8.1. Let f1, f2 ∈ L2(Γ\G,χ; l, q) be represented by bounded functionsin C∞(Γ\G). We denote their inner product by

〈f1, f2〉Γ\G =∫

Γ\Gf1(g)f2(g)dg.

Then, the inner products 〈fj , Eκl,q(ν, p;χ)〉Γ\G for j = 1, 2, are defined andsquare-integrable as functions of ν ∈ iR, and the following equality holds

〈f1, f2〉Γ\G = δl,01

vol(Γ\G)〈f1, 1〉Γ\G〈1, f2〉Γ\G

+∑V

1‖ΦlpV ,q‖

2K

〈f1, TV ϕl,q(νV , pV )〉Γ\G〈TV ϕl,q(νV , pV ), f2〉Γ\G

71

+∑V ′

Γ(l + 1 + νV ′)Γ(l + 1− νV ′)

1‖Φl0,q‖2K

〈f1, TV ′ϕl,q(νV ′ , 0)〉Γ\G ·

· 〈TV ′ϕl,q(νV ′ , 0), f2〉Γ\G

+1

4πi

∑κ∈Cχ

[Γκ : Γ′κ]|Λκ|

∑χ

|p|6l

1‖Φlp,q‖2K

∫(0)

〈f1, Eκl,q(ν, p;χ)〉Γ\G ·

· 〈Eκl,q(ν, p;χ), f2〉Γ\G dν. (8.5)

Here V , respectively V ′, runs through the subset of unitary principal series,respectively complementary series, in the orthogonal system chosen above,

∑χ|p|6l

means that the sum runs through all p ∈ 12Z such that |p| 6 l with the condition

χ(ε) = ε2p satisfied for all ε ∈ O∗, and |Λκ| is the Euclidean area of a period par-allelogram for the lattice Λκ ∈ C corresponding to the discrete subgroup gκ−1Γ′κgκ.

Remark 6. The normalization factors 14πi [Γκ : Γ′κ]|Λκ|−1 in the last term of (8.5)

are taken from [11], Theorem 6.3.4, (3.12). Although the authors there workwith trivial character χ and trivial K-type, their results are applicable in our casetoo, since the normalization factors depend only on the geometry of the part ofthe fundamental domain near the cusp κ, and not on the character χ or on theparameter p ∈ 1

2Z.

Chapter 9

Auxiliary test functions

In order to build the Poincare series that will be used in the proof of the sumformula, we shall employ auxiliary test functions. More precisely, we shall considerthe Lebedev transforms of a certain class of functions as building blocks for thePoincare series.

The first section of this chapter is devoted to the introduction of the Lebedevtransformation Lωl,q, its one-sided inverse Lωl,q, and some of their properties. In thesecond section we shall choose the auxiliary test functions.

9.1 Lebedev transformation

Definition 9.1.1. Let f ∈ Pl,q(N\G,ω), and ξ ∈ 0, 12 as in (4.27). Let σ0 > 0

such that (7.17) is satisfied for f . We define the Lebedev transform Lωl,qf(ν, p) off , with |Re ν| 6 σ0 and p ∈ ξ + Z, by

Lωl,qf(ν, p) :=(−iω/|ω|)−p+ξ

π2‖Φlp,q‖K(2π|ω|)ν ·

· Γ(l + 1− ν)∫N\G

f(g)Jωϕl,q(−ν, p)(g)dg, (9.1)

where dg is the quotient measure on N\G, which corresponds to the measurer−3drdk under the isomorphism N\G ∼= AK.

From (4.24) and (4.25) we see that the absolute convergence of the integral in(9.1) is no problem for r > 1, as Kν is exponentially decreasing. For r ∈ (0, 1],using the estimate (4.28), we see that the contributions to the integral are: r1+σ0

from the function f , r1−σ0−ε from the Jacquet integral, and r−3 from the measure.This implies the convergence of the integral on the interval (0, 1]. The functionν 7→ Lωl,qf(ν, p) is holomorphic on the strip |Re ν| 6 σ0.

74 Auxiliary test functions

Remark 7. In the case l ∈ Z, the transform (9.1) is a multiple of the Lebedevtransform in [9] given by (7.4). Namely, Lωl,q =

√2L2ω

l,q . The factor√

2 comes fromthe different normalizations of the Haar measure on K here and in [9]; see (2.33)and [9], (3.23).

We transform the integral in (9.1) in terms of J0f as follows:∫N\G

f(g)Jωϕl,q(−ν, p)(g)dg =

=∫N\G

∫N

f(g)χω(n)ϕl,q(−ν, p)(wng)dn dg

=∫G

f(g)ϕl,q(−ν, p)(wg)dg

(g 7→w−1g)=

∫G

f(w−1g)ϕl,q(−ν, p)(g)dg. (9.2)

Note that f(w−1g) = f(wgh[−1]) = χ(−1)f(wg), since f has K-type l and theconsistency relation (3.1) holds. Thus, we continue in (9.2):

= χ(−1)∫N\G

∫N

f(wng)ϕl,q(−ν, p)(g)dn dg

= χ(−1)∫N\G

J0f(g)ϕl,q(−ν, p)(g)dg. (9.3)

The behavior f(na[r]k) = O(r1+σ0) as r ↓ 0 implies that J0f(g) convergesabsolutely, and as r →∞ it satisfies

J0f(a[r]k) =∫

Cf(wn[z]a[r]k)d+z

(4.3)=∫

Cf(n[

−zr2+|z|2

]a[

rr2+|z|2

]k′)d+z

(z=ρeiφ)= 2π

∫ ∞

0

f(n[−ρe−iφ

r2+ρ2

]a[

rr2+ρ2

]k′)ρ dρ

∫ ∞

0

r1+σ0(r2 + ρ2)−1−σ0ρ dρ = r1+σ0r−2σ0

2σ0=

12σ0

r1−σ0 . (9.4)

As r ↓ 0, the function J0f with f satisfying (7.17) is estimated as follows:

J0f(a[r]k) =∫

Cf(wn[z]a[r]k)d+z

(z 7→rz)= r2

∫Cf(wn[rz]a[r]k)d+z

(4.3)= r2

∫Cf(n[

−zr(1+|z|2)

]a[

1r(1+|z|2)

]k′)d+z

9.1 Lebedev transformation 75

(z=ρeiφ)= 2πr2

∫ ∞

0

f(n[−ρe−iφ

r(1+ρ2)

]a[

1r(1+ρ2)

]k′)ρ dρ

r2∫ ∞

0

min(r(1 + ρ2))−1+σ∞ , (r(1 + ρ2))−1−σ0

ρ dρ

= r2∫ √1/r−1

0

r−1+σ∞(1 + ρ2)−1+σ∞ρ dρ

+ r2∫ ∞

√1/r−1

r−1−σ0(1 + ρ2)−1−σ0ρ dρ 61

2σ∞r +

12σ0

r r. (9.5)

The properties of the intertwining operator J0 together with estimates (9.4) and(9.5) show that J0f is a continuous function satisfying the growth conditions

J0f(na[r]k) =O(r) as r ↓ 0,O(r1−σ0) as r →∞. (9.6)

Any such function has an expansion in terms of Φlm,q, |m| 6 l. Thus, we maywrite

J0f(na[r]k) =∑|m|6l

um(r)Φlm,q(k), (9.7)

where the functions um(r) are continuous and satisfy

um(r)r , r ↓ 0r1−σ0 , r →∞ for all m. (9.8)

The integral (9.3) then equals

= χ(−1)∫ ∞

0

∑|m|6l

um(r)r−2−ν∫K

Φlm,q(k)Φlp,q(k)dk dr

= χ(−1)‖Φlp,q‖2K∫ ∞

0

up(r)r−ν−2dr = χ(−1)‖Φlp,q‖2KMup(−ν − 1). (9.9)

Here Mφ is the Mellin transform of a function φ given by

Mφ(s) =∫ ∞

0

φ(r)rs−1dr. (9.10)

(See e.g. [41], p.125). We recall that the inversion formula for this transformationis given by

φ(r) =1

2πi

∫Re s=σ

Mφ(s)r−sds. (9.11)

Applying the Mellin transformation to the function up imposes a condition onRe s. Namely, from the estimates (9.8) we see that Mup(s) exists on the strip−1 < Re s < σ0 − 1. Thus, we have proved


Lemma 9.1.2. Let f ∈ Pl,q(N\G,ω) and let J0f be expanded as in (9.7). Then,for ν such that −σ0 < Re ν < 0, the Lebedev transform (9.1) of f is linked to theMellin transform in the following way:

Lωl,qf(ν, p) = χ(−1)π−2‖Φlp,q‖K (−iω/|ω|)−p+ξ ·· Γ(l + 1− ν)(2π|ω|)ν Mup(−ν − 1). (9.12)

Next, we shall see that the Lebedev transformation is invertible on a suitablespace of functions.

Definition 9.1.3. Let σ > 0 and l ∈ 12N. We denote by Tlσ the linear space of

functions η defined on the set(ν, p) ∈ C× 1

2Z : |Re ν| 6 σ, p ≡ l (mod 1), |p| 6 l

(9.13)

such that

(i) η(ν, p) is holomorphic on a neighborhood of the strip |Re ν| 6 σ,

(ii) η(ν, p) e−π2 | Im ν|(1 + | Im ν|)−A for any A > 0,

(iii) η(ν, p) = η(−ν,−p).

Theorem 9.1.4. Let σ ∈(1, 3

2

)and ξ ∈ 0, 1

2 as in (4.27). For η ∈ Tlσ we definethe following transform:

Lωl,qη(g) :=1

2π3i

∑|p|6l

(iω/|ω|)p−ξ

‖Φlp,q‖K

∫(0)

η(ν, p)(2π|ω|)−ν ·

· Γ(l + 1 + ν)Jωϕl,q(ν, p)(g)νε(p) sinπ(ν − p)dν, (9.14)

with ε(0) = 1, ε(p) = −1 for p ∈ Z \ 0, and ε(p) = 0 for p ∈ 12 + Z.

Then Lωl,qη(g) ∈ Pl,q(N\G,ω), and we have

Lωl,qLωl,qη(ν, p) =

= − 2π2

(−1)p−ξΓ(l + 1− ν)Γ(l + 1 + ν)sinπ(ν − p)ν2 − p2

νε(p)η(ν, p) (9.15)

on any strip |Re ν| < α with 0 < α < 1.

Remark 8. For l ∈ Z, Lωl,qη =√

2M2ωl,qη, where Mω

l,q is the transform given by(7.9) in [9]. Also the equation (9.15) reduces to [9], (7.10) with the differentnormalizations of dk in mind.


Proof. First we prove that Lωl,qη(g) ∈ Pl,q(N\G,ω). The relations (4.24)–(4.25) express Jωϕl,q(ν, p)(na[r]k) in terms of K-Bessel functions. For the con-vergence of the integral, we use estimate (1.37). This estimate also shows thatLωl,qη(na[r]k) is of rapid decay with respect to r as r → ∞. To investigate thebehavior as r ↓ 0 we observe that by (1.37) the contour in (9.14) can be shifted toRe ν = α with 0 < α < 1. Then the functional equation (4.48) and condition (iii)give

Lωl,qη(g) =1

2πi

∑|p|6l

1‖Φlp,q‖K

∫(α)

η(ν, p) ·

·

−(2π|ω|)ν

(iω

|ω|

)−p−ξΓ(l + 1 + ν)Mωϕl,q(ν, p)(g)

+(−1)2ξ(2π|ω|)−ν(iω

|ω|

)p−ξΓ(l + 1− ν)Mωϕl,q(−ν,−p)(g)

νε(p)dν

=−12πi

∑|p|6l

(iω/|ω|)−p−ξ

‖Φlp,q‖K

∫(α)

+∫

(−α)

η(ν, p) ·

· (2π|ω|)νΓ(l + 1 + ν)Mωϕl,q(ν, p)(g)νε(p)dν

=i

π

∑|p|6l

(iω/|ω|)−p−ξ

‖Φlp,q‖K

∫(α)

η(ν, p)(2π|ω|)ν ·

· Γ(l + 1 + ν)Mωϕl,q(ν, p)(g)νε(p)dν

+ l!∑p∈Z

16|p|6l

(iω/|ω|)−p

‖Φlp,q‖Kη(0, p)Mωϕl,q(0, p)(g). (9.16)

We note that the second term after the last equality sign in (9.16) is only presentif l is integral and l > 1.

The estimate (4.54) implies that, as r ↓ 0, the first sum after the last equalitysign in (9.16) is O(r1+α). Using the estimate (7.15) in [9] we get

l!∑p∈Z

16|p|6l

(iω/|ω|)−p

‖Φlp,q‖Kη(0, p)Mωϕl,q(0, p)(g) =

= B(η)χω(n)r2Φl0,q(k) +O(r3) = B(η)Mωϕl,q(1, 0)(g) +O(r3), (9.17)

where l ∈ Z, l > 1, and B(η) = 2πl · l! η(0, 1)|ω|‖Φl1,q‖−1K . Collecting these, we

have the following estimates

Lωl,qη(na[r]k) =O(r1+α) as r ↓ 0 for 0 < α < 1,O(r−k) as r →∞ for all k > 1, (9.18)


and we conclude that Lωl,qη ∈ Pl,q(N\G,ω).In order to use Lemma 9.1.2 in computing Lωl,qL

ωl,qη, we need to express J0Lωl,qη

in the form (9.7). We apply J0 to the last expression in (9.16) and by absoluteconvergence (obtained using the estimate (4.54)) we have:

J0Lωl,qη(na[r]k) =i

π

∑|p|6l

(iω/|ω|)−p−ξ

‖Φlp,q‖K

∫(α)

η(ν, p)(2π|ω|)ν ·

· Γ(l + 1 + ν)J0Mωϕl,q(ν, p)(g)νε(p)dν

+ l!∑p∈Z

16|p|6l

(iω/|ω|)−p

‖Φlp,q‖Kη(0, p)J0Mωϕl,q(0, p)(g).

Note that (4.56) implies J0Mωϕl,q(0, p) = 0 for p ∈ Z\0. So, the second termin the above expression vanishes, and furthermore by (4.56) we have:

=i

π

∑|p|6l

(iω/|ω|)−p−ξ

‖Φlp,q‖K

∫(α)

η(ν, p)(2π|ω|)νΓ(l + 1− ν) sinπ(ν − p)ν2 − p2

·

· νε(p)r1−νΦl−p,q(k)dν

( p7→−p )=

i

π

∑|p|6l

(iω/|ω|)p−ξ

‖Φlp,q‖K

∫(α)

η(−ν, p)(2π|ω|)νΓ(l + 1− ν) sinπ(ν + p)ν2 − p2

·

· νε(p)r1−νdν · Φlp,q(k)

=∑|p|6l

up(r)Φlp,q(k),

where

up(r) :=i

π‖Φlp,q‖K(iω/|ω|)p−ξ

∫(α)

η(−ν, p) ·

· (2π|ω|)νΓ(l + 1− ν) sinπ(ν + p)ν2 − p2

νε(p)r1−νdν. (9.19)

The equality (9.19) can be written in the following form:

r−1up(r) =1

2πi

∫(α)

g(ν)r−νdν,

with

g(ν) := −2(iω/|ω|)p−ξ

‖Φlp,q‖Kη(−ν, p)(2π|ω|)νΓ(l + 1− ν) sinπ(ν + p)

ν2 − p2νε(p). (9.20)


From (9.18) and (9.6) we get

J0Lωl,qη(na[r]k) =O(r) , as r ↓ 0O(r1−α) , as r →∞ for 0 < α < 1, (9.21)

The functions up(r) given by (9.19) inherit these growth conditions, and thereforethe Mellin inversion formula (9.11) gives for 0 < Re ν < α

g(ν) =∫ ∞

0

r−1up(r)rν−1dr = Mup(ν − 1).

Hence, for −α < Re ν < 0, we have

Lωl,qLωl,qη(ν, p) =

(9.12)=

χ(−1)π2‖Φlp,q‖K (−iω/|ω|)−p+ξ Γ(l + 1− ν)(2π|ω|)ν g(−ν)

(9.20)= − 2

π2(−1)p−ξΓ(l + 1− ν)Γ(l + 1 + ν)

sinπ(ν − p)ν2 − p2

νε(p)η(ν, p). (9.22)

Estimate (9.18) and Definition 9.1.1, imply that the integral defining Lωl,qLωl,qη(ν, p)

is absolutely convergent and holomorphic as a function in ν on |Re ν| < α, for allα ∈ (0, 1). The right side of (9.22) is holomorphic as a function in ν on the widerstrip |Re ν| < α for all α ∈ [0, 1]. The equality (9.22) holds on −α < Re ν < 0. Byholomorphic continuation, the equality stays valid on the strip |Re ν| < α, withα ∈ (0, 1).

The fact that Lωl,qη ∈ Pl,q(N\G,ω) implies in particular that Lωl,qη is square-integrable on N\G. Related to this we have a Parseval property of the transfor-mation Lωl,q:

Lemma 9.1.5. Let σ ∈(1, 3

2

), and η, θ ∈ Tlσ. Then we have

〈Lωl,qη, Lωl,qθ〉N\G =1π3i

∑|p|6l

∫(0)

η(ν, p)θ(ν, p) ·

· Γ(l + 1− ν)Γ(l + 1 + ν)sin2 π(ν − p)p2 − ν2

ν2ε(p)dν. (9.23)

Remark 9. In case l ∈ Z, this lemma reduces to Lemma 7.1 in [9].Proof. We replace Lωl,qθ(g) by its defining expression (9.14). The resulting

double integral over N\G × iR is absolutely convergent (as r ↓ 0 use estimates(4.28) and (9.18) for Re ν = 0, and as r → ∞ use (1.35), (4.24)–(4.25), and thesecond part of (9.18)), and we have∫

N\GLωl,qη(g)Lωl,qθ(g)dg =

i

2π3

∑|p|6l

(iω/|ω|)−p+ξ

‖Φlp,q‖K

∫(0)

θ(ν, p)(2π|ω|)ν ·

· Γ(l + 1− ν)∫N\G

Lωl,qη(g)Jωϕl,q(ν, p)(g)dg sinπ(ν − p)νε(p)dν. (9.24)


Here we have used the fact that, for Re ν = 0,

sinπ(ν − p)νε(p) = sinπ(−ν − p)(−ν)ε(p)

= − sinπ(ν − p+ 2p)(−1)ε(p)νε(p)

= −(−1)ε(p)+2p sinπ(ν − p)νε(p) = sinπ(ν − p)νε(p). (9.25)

Using the definition (9.1) of the Lebedev transform, and then the property (9.15),we continue in (9.24):

=i

2π

∑|p|6l

(−1)p−ξ∫

(0)

θ(ν, p)Lωl,qLωl,qη(ν, p) sinπ(ν − p)νε(p)dν

=1π3i

∑|p|6l

∫(0)

η(ν, p)θ(ν, p)Γ(l + 1− ν)Γ(l + 1 + ν)sin2 π(ν − p)p2 − ν2

ν2ε(p)dν,

which ends the proof.

We now define a function that shall later appear as the kernel of the Besseltransformation (11.1).

Definition 9.1.6. For ν ∈ C, p ∈ 12Z, and z ∈ C∗, we define

K∗ν,p(z) :=

1sinπ(ν − p)

|z/2|−2ν (iz/|z|)2p−2ξ

J∗−ν,−p(z)−

− |z/2|2ν (iz/|z|)−2p−2ξJ∗ν,p(z)

, (9.26)

with J∗ν,p as given in (4.58) and ξ ∈ 0, 12 as in (4.27).

Directly from the definition we obtain:

(K1) K∗−ν,−p(z) = K∗

ν,p(z),

(K2) K∗ν,p(−z) = K∗

ν,p(z),

(K3) K∗ν,p(z) = (z/|z|)4ξK∗

ν,−p(z) = (z/|z|)4ξK∗−ν,p(z).

Some further properties are given in the following

Lemma 9.1.7. Let σ ∈(1, 3

2

), l ∈ 1

2Z, l > |p|, z ∈ C∗. The function K∗ν,p(z) is

holomorphic in ν ∈ C and satisfies

K∗ν,p(z)W (1 + | Im ν|)2σ−1 (9.27)

on the strip |Re ν| < σ. The estimate (9.27) is uniform in z ∈W , for an arbitrarycompact subset W ⊂ C∗.


Proof. We have |z/2|2ν (z/|z|)−2pJ∗ν,p(z) = Jν−p(z)Jν+p(z) if we choose thebranches suitably. Thus we see that the zeros of the difference

|z/2|−2ν (iz/|z|)2p−2ξJ∗−ν,−p(z)− |z/2|

2ν (iz/|z|)−2p−2ξJ∗ν,p(z)

appear exactly when ν − p ∈ Z. So, the poles of 1sinπ(ν−p) are cancelled by these

zeros, which for fixed z makes the function K∗ν,p(z) holomorphic in ν ∈ C.

Assuming that µ ∈ C \ Z6−1, from the power series expansion of J∗µ(z), wehave for fixed z

|J∗µ(z)| 6 e|z|2/4 max

m>0

|Γ(µ+ 1 +m)|−1

,

and thus J∗µ(z)z |Γ(µ+ 1)|−1 uniformly for Reµ > −ε, ε > 0.If Reµ < 0, then

|Γ(µ+ 1 +m)| = |Γ(µ+ 1)|m−1∏j=0

|µ+ 1 + j| > |Γ(µ+ 1)|m−1∏j=0

|Reµ+ 1 + j|,

where only two factors |Reµ + 1 + j| are in the neighborhood of the origin, andall the others satisfy |Reµ+ 1 + j| > 1. This means that for all m > 0,

|Γ(µ+ 1 +m)| > |Γ(µ+ 1)|ε(1− ε),

for some ε > 0. Hence J∗µ(z) z,ε |Γ(µ + 1)|−1 uniformly for Reµ < 0 withdistance greater than ε from elements in Z6−1.

We now look at the functions J∗ν±p(z), p ∈ 12Z, for ν in a vertical strip with

distance greater than ε from elements in 12Z6−1 that belong to that strip. Then,

J∗ν±p(z)z,ε |Γ(ν ± |p|+ 1)|−1. (9.28)

We note that

|Γ(ν + |p|+ 1)|−1 (1 + | Im ν|)−Re ν−|p|− 12 e

π2 | Im ν|, (9.29)

Γ(ν − |p|+ 1)|−1 =| sinπ(ν − |p|)||Γ(|p| − ν)|

π

(1 + | Im ν|)−Re ν+|p|− 12 e

π2 | Im ν|. (9.30)

Thus, for fixed z, we have

J∗ν±p(z)ε (1 + | Im ν|)−Re ν∓|p|− 12 e

π2 | Im ν|,

which implies the estimate

J∗ν,p(z)W (1 + | Im ν|)−2 Re ν−1eπ| Im ν| (9.31)


uniformly for p ∈ 12Z, z ∈ W for any compact subset W ⊂ C∗, and |Re ν| < σ,

σ ∈ (1, 32 ), such that |ν − k| > ε for k ∈ 0,− 1

2 ,−1. Definition 9.1.6 now gives

K∗ν,p(z)W (1 + | Im ν|)−1

((1 + | Im ν|)2 Re ν + (1 + | Im ν|)−2 Re ν

)W (1 + | Im ν|)2|Re ν|−1, (9.32)

which implies (9.27).

Lemma 9.1.8. For any non-zero ω1, ω2, τ ∈ C, we define the map

κ∗(ω1, ω2, τ) : η 7→ K∗ν,p (4πτ

√ω1ω2) η, (9.33)

for η ∈ Tlσ and K∗ν,p(z) given by (9.26).

Then, κ∗(ω1, ω2, τ) is a linear operator in the space of functions Tlσ, and wehave

Jω1 lτ Lω2l,qη = |πτ |2

(iτω1

|τω1|

)2ξ

Lω1l,q

(κ∗(ω1, ω2, τ)η

). (9.34)

We note that the choice of the square root√ω1ω2 in the definition of the

operator κ∗(ω1, ω2, τ) does not matter since K∗ν,p is even function on C∗.

Remark 10. For integer values of p, K∗ν,p(z) = Kν,p(z) and κ∗(ω1, ω2, τ) =

κ(2ω1, 2ω2, τ) = κ(ω1, ω2, 2τ), where Kν,p and κ(ω1, ω2, τ) are the functions de-fined in [9], (7.21) and (7.20), respectively. In that sense, this lemma generalizesLemma 7.2 in [9].

Proof. Lemma 9.1.7 shows that κ∗(ω1, ω2, τ) is indeed an operator on theset Tlσ, given in Definition 9.1.3, which acts by multiplication with the functionK∗ν,p

(4πτ√ω1ω2

). Hence the right side of (9.34) is well-defined. Because of the

property (4.52), we transform the left side of (9.34) using (9.16), where the ex-change of the order of integrals is allowed because of the estimate (4.55). So, forα ∈ (1, 3

2 ), we obtain:

Jω1 lτ Lω2l,qη(g) =

i

π|τ |2

(τ

|τ |

)2ξ ∑|p|6l

(iτ2ω2/|τ2ω2|

)−p−ξ‖Φlp,q‖K

∫(α)

η(ν, p) ·

· (2π|τ2ω2|)νΓ(l + 1 + ν)Jω1Mτ2ω2ϕl,q(ν, p)(g)νε(p)dν

+ l! |τ |2∑p∈Z

16|p|6l

(iτ2ω2/|τ2ω2|

)−p‖Φlp,q‖K

η(0, p)Jω1Mτ2ω2ϕl,q(0, p)(g).

By (4.57) we further have

=i

π|τ |2

(τ

|τ |

)2ξ ∑|p|6l

(iτ2ω2/|τ2ω2|


∫(α)

η(ν, p)(2π|τ2ω2|)ν ·

· Γ(l + 1 + ν)J∗ν,p(4πτ√ω1ω2)Jω1ϕl,q(ν, p)(g)ν

ε(p)dν

9.2 Choice of Poincare series 83

+l! |τ |2(τ

|τ |

)2ξ ∑l,p∈Z

16|p|6l

(iτ2ω2/|τ2ω2|

)−p‖Φlp,q‖K

η(0, p) ·

·J∗0,p(4πτ√ω1ω2)Jω1ϕl,q(0, p)(g). (9.35)

The estimates (4.29) and (9.31) allow us to shift the contour (α) of one half ofthe integral to (−α); then the last sum vanishes. In the integral over (−α), wemake the change (ν, p) 7→ (−ν,−p) and apply the functional equation (4.26). Wecontinue in (9.35) with some rearrangement:

=i

2π|τ |2

(τ

|τ |

)2ξ ∑|p|6l

(iτ2ω2/|τ2ω2|


∫(α)

+∫

(−α)

η(ν, p) ·

·(2π|τ2ω2|)νΓ(l + 1 + ν)J∗ν,p(4πτ√ω1ω2)Jω1ϕl,q(ν, p)(g)ν

ε(p)dν

=i

2π|τ |2

(τ

|τ |

)2ξ ∑|p|6l

(iω1/|ω1|)p+ξ

‖Φlp,q‖K

∫(α)

η(ν, p)Γ(l + 1 + ν)(2π|ω1|)−ν ·

·

(4π2|τ2ω1ω2|)ν

(−τ2ω1ω2

|τ2ω1ω2|

)−p−ξJ∗ν,p(4πτ

√ω1ω2)− (4π2|τ2ω1ω2|)−ν ·

·(−τ2ω1ω2

|τ2ω1ω2|

)p−ξJ∗−ν,−p(4πτ

√ω1ω2)

Jω1ϕl,q(ν, p)(g)ν

ε(p)dν

=|τ |2

2πi

(τ

|τ |

)2ξ (iω1

|ω1|

)2ξ ∑|p|6l

(iω1/|ω1|)p−ξ

‖Φlp,q‖K

∫(0)

K∗ν,p(4πτ

√ω1ω2)η(ν, p) ·

·(2π|ω1|)−νΓ(l + 1 + ν)Jω1ϕl,q(ν, p)(g)νε(p) sinπ(ν − p)dν

= |πτ |2(iτω1

|τω1|

)2ξ

Lω1l,q

(κ∗(ω1, ω2, τ)η

)(g),

and the lemma has been proved.

9.2 Choice of Poincare series

We shall use the Lebedev transform Lωl,qη of functions η ∈ Tlσ, to generate aPoincare series, which will later be used for deriving the preliminary sum formula.

Let us consider the Poincare series

PχLωl,qη(g) :=1

[ΓP : ΓN ]

∑γ∈ΓN\Γ

χ(γ)−1Lωl,qη(γg) (9.36)

with non-zero ω ∈ O′.


The behavior of Lωl,qη near 0 is important for the absolute convergence of thesum in (9.36). We use (9.16), where the line of integration is allowed to be moved toany α ∈ (0, σ] with σ > 0, because of the estimate (4.55). The contribution of thefirst term in (9.16) to Lωl,qη(na[r]k) is then O(r1+σ). This term is determining thebehavior of Lωl,qη(na[r]k) as r ↓ 0 if σ < 1. However, for the absolute convergenceof the Poincare series (9.36) we need σ > 1. If σ > 1, the first term in (9.16) has theright behavior, but the second term causes problems with convergence. Namely,we see from (9.17) that it has a contribution O(r2), and it therefore determines thebehavior of Lωl,qη(na[r]k) as r ↓ 0, which is not enough for the absolute convergenceof PχLωl,qη.

To solve this problem, we shall use results of Miatello and Wallach in [34] con-cerning meromorphic continuation of Poincare series. Their results are derived forthe trivial character χ = 1, but for any cofinite discrete subgroup. In order toapply those results we need to work with Poincare series for Γ1(I). So, we shallwrite the Poincare series (9.36) as a linear combination of (shifted) Poincare seriesover Γ1(I) with trivial character χ = 1, use results in [34] about the meromor-phic continuation of those series, and obtain a meromorphic continuation of theoriginally defined series. Let

Pf(g) :=∑

γ∈ΓN\Γ1(I)

f(γg),

denote a Poincare series generated by the function f over the group Γ1(I) andtrivial character.

Each γ ∈ ΓN\Γ can be written as a product γ = ζδ, with ζ running over a setof representatives for ΓN\Γ1(I) and δ over a set of representatives for Γ1(I)\Γ.Then, we may rewrite the Poincare series Pχf in the following way

Pχf(g) :=1

[ΓP : ΓN ]

∑γ∈ΓN\Γ

χ(γ)−1f(γg)

=1

[ΓP : ΓN ]

∑ζ∈ΓN\Γ1(I)

∑δ∈Γ1(I)\Γ

χ(ζδ)−1f(ζδg)

=1

[ΓP : ΓN ]

∑δ∈Γ1(I)\Γ

χ(δ)−1Pf(δg). (9.37)

Note that the character χ is trivial on Γ1(I).If T > 0 is fixed, we introduce a so called cut off function ρ ∈ C∞(G) defined

as a left N -invariant, right K-invariant function such that ρ(a[r]) = 1 if r ∈ (0, T ],ρ(a[r]) = 0 if r ∈ (T + 1,∞), and 0 6 ρ 6 1. For ω ∈ O′\0, we consider thefollowing Poincare series

PχMωϕl,q(ν, p)(g) =1

[ΓP : ΓN ]

∑γ∈ΓN\Γ

χ(γ)−1Mωϕl,q(ν, p)(γg), (9.38)


and

PχρMωϕl,q(ν, p)(g) =1

[ΓP : ΓN ]

∑γ∈ΓN\Γ

χ(γ)−1ρ(γg)Mωϕl,q(ν, p)(γg), (9.39)

for Re ν > 1. From (9.39) we see that PχρMωϕl,q(ν, p) can be estimated at infinityby a finite sum over the units in O plus a part of the Eisenstein series E0,0(Re ν, 0; 1)corresponding to the big cell in the Bruhat decomposition. This gives

PχρMωϕl,q(ν, p)(na[r]k) = O(r1−Re ν), as r →∞, (9.40)

for Re ν > 1. At a cusp κ ∈ Cχ which is not Γ-equivalent to ∞, we estimate theseries (9.39) by the whole Eisenstein series E0,0(Re ν, 0; 1)(gκg), and get

PχρMωϕl,q(ν, p)(gκna[r]k) = O(r1−Re ν), as r →∞, (9.41)

for Re ν > 1.Because of (9.37) we have

PχMωϕl,q(ν, p)(g) =1

[ΓP : ΓN ]

∑δ∈Γ1(I)\Γ

χ(δ)−1PMωϕl,q(ν, p)(δg), (9.42)

and

PχρMωϕl,q(ν, p)(g) =1

[ΓP : ΓN ]

∑δ∈Γ1(I)\Γ

χ(δ)−1PρMωϕl,q(ν, p)(δg). (9.43)

Here the Poincare series PMωϕl,q(ν, p) and PρMωϕl,q(ν, p) correspond respec-tively to the Poincare series M(ξp, ν, g,Φlp,q) and M(ξp, ν, g,Φlp,q) for Γ1(I) in[34], §2, with ξp : h[eit] 7→ e−2pit a character of M = H ∩K.

Returning to (9.16) and (9.17), we define Lω,∗l,q η by

Lωl,qη = Lω,∗l,q η +B(η)ρMωϕl,q(1, 0). (9.44)

By construction the function Lω,∗l,q η satisfies Lω,∗l,q η(na[r]k) = O(r1+σ) as r ↓ 0with σ ∈

(1, 3

2

). This implies absolute convergence of the Poincare series PχL

ω,∗l,q η.

We shall also want this series to be square-integrable, which is determined bythe behavior of the function Lω,∗l,q η near infinity. The estimate (9.18) implies thatLω,∗l,q η(na[r]k) = O(r−k) as r →∞ for all k > 0, and by Proposition 7.1.2 we havethat PχL

ω,∗l,q η ∈ L2(Γ\G; l, q).

Theorem 2.5 in [34] implies that for any Γ the function ν 7→ PρMωϕl,q(ν, 0)can be analytically continued as meromorphic function in ν ∈ C, where the singu-larities in the region Re ν > 0 occur only at values of ν for which (ν, 0) is a spectral


parameter. The pole at ν = 1, which might occur if l = 0, does not concern us sincein that case the problematic term in (9.16) is not present. The spectral parametersν, where (ν, 0) characterize the complementary series, are called exceptional andthey form a discrete subset of the interval (0, 1). So, there exists ε > 0 such thatthe neighborhood |ν − 1| 6 ε of 1 does not contain exceptional spectral param-eters. At present, it is known that exceptional spectral parameters, if present,are concentrated in a small subinterval of (0, 1). Hence the family of functionsν 7→ PρMωϕl,q(ν, 0) has an analytic continuation as a holomorphic function in νfor Re ν > 1− ε. Because of (9.43) the same holds for ν 7→ PχρMωϕl,q(ν, 0).

Remark 11. This ε can be almost 1. The best results up until now known to meare those of Kim and Shahidi, [20]. However, for our purposes it will be sufficientto consider a small ε > 0.

For the derivation of the sum formula (see next chapter) we shall also needthat PχρMωϕl,q(ν, 0) is an element of L2(Γ\G,χ; l, q) for ν in a neighborhood of1. We derive this conclusion using the results of Miatello and Wallach in [34]. Forreasons of convenience let us write ψω(ν) := ρMωϕl,q(ν, p) and assume for thetime being that l > 1 is integral. (In the case l = 0 or l ∈ 1

2 + Z, the offendingterm in the behavior of Lωl,qη does not occur.) We consider general p although wewill need the results only for p = 0. The function ψω(ν) has the following growthbehavior, for Re ν > 0:

ψω(ν)(na[r]k) =O(r1+Re ν) as r ↓ 0

0 if r is sufficiently large, (9.45)

uniformly in n ∈ N , k ∈ K. For Re ν > 1, these estimates and Proposition 7.1.2imply that the Poincare series Pχψω(ν) is square-integrable on Γ\G.

To see that ν 7→ Pχψω(ν) extends to an L2-holomorphic map with values inL2(Γ\G,χ; l, q) also for Re ν > 1 − ε, we may restrict our reasoning to Pψω(ν)because of (9.37). We then obtain the desired conclusion from the results in [34].We consider the function

(4(Ω+ + Ω−)− ν2 − p2 + 1

)Pψω(ν). For Re ν > 1, we

may differentiate within the Poincare series, and find(4(Ω+ + Ω−)− ν2 − p2 + 1

)Pψω(ν) = Pψω(ν),

where the function ψω(ν) :=(4(Ω+ + Ω−)− ν2 − p2 + 1

)ψω(ν) is holomorphic

pointwise in g for at least Re ν > 0. The Poincare series Pψω(ν) correspondsto the series M(ξp, ν, g,Φlp,q) in [34] for Γ1(I). The fact that the linear operator(4(Ω+ + Ω−)− ν2 − p2 + 1

)applied to the function Mωϕl,q(ν, p) yields zero, im-

plies that ψω(ν)(na[r]k) is non-zero only for T 6 r 6 T + 1. Thus, the Poincareseries Pψω(ν) is locally given by a finite sum. This sum converges absolutely forRe ν > 0, uniformly for g in a fundamental domain for Γ\G. Hence, Pψω(ν)converges absolutely for all ν with Re ν > 0. Since

∣∣ψω(ν)(na[r]k)∣∣ = O(1) for


T 6 r 6 T + 1 uniformly for |ν − 1| 6 ε, the Poincare series Pψω(ν) can be ana-lytically continued to a C∞–function in ν and g with compact support in Γ1(I)\Gindependent of ν and it is square-integrable on Γ1(I)\G uniformly for |ν − 1| 6 ε.For Re ν > 1, the series Pψω(ν) has a spectral expansion, see p. 424 in [34], and thespectral expansions of Pψω(ν) and Pψω(ν) are linked, see relations on p. 428–429in [34]. Miatello and Wallach use these relations between the spectral coefficientsto show that the Poincare series Pψω(ν) is square-integrable over Γ1(I)\G also for|ν−1| 6 ε, with a uniformly bounded norm. Hence, we conclude that the Poincareseries Pχψω(ν) = PχρMωϕl,q(ν, p) is square-integrable on Γ\G for Re ν > 1− ε.

The estimates (9.40) and (9.41) can be extended to |ν − 1| 6 ε in order todescribe the behavior of PχρMωϕl,q(ν, p) at the cusps also for ν in a neighborhoodof 1 when the series (9.39) does not converge absolutely. For that purpose weobserve that if we choose the truncation parameter T large enough, then for g =na[r]k ∈ G with r > T , we have

PχρMωϕl,q(ν, p)(gκg) = PχMωϕl,q(ν, p)(gκg)

+δκ,∞

[ΓP : ΓN ]

∑ε∈O∗

χ(ε)−1ε2p(ρ(g)− 1

)Mω/ε2ϕl,q(ν, p)(g). (9.46)

Obviously, the series PχρMωϕl,q(ν, p) and PχMωϕl,q(ν, p) are equal near all thecusps κ that are not Γ-equivalent to∞. From the Fourier expansion of the Poincareseries PχMωϕl,q(ν, p) at a cusp κ ∈ Cχ

PχMωϕl,q(ν, p)(gκg) =

= aκ0 (ν, p)ϕl,q(−ν,−p)(g) +∑

0 6=ω′∈Λ′κ

aκω′(ν, p)Jω′ϕl,q(ν, p)(g)

+δκ,∞

[ΓP : ΓN ]

∑ε∈O∗

ω′ε2=ω

χ(ε)−1ε2pMω/ε2ϕl,q(ν, p)(g), (9.47)

and (9.46), we see that for g = na[r]k with r →∞

PχρMωϕl,q(ν, p)(gκg) =

= aκ0 (ν, p)ϕl,q(−ν,−p)(g) +∑

0 6=ω′∈Λ′κ

aκω′(ν, p)Jω′ϕl,q(ν, p)(g)

+δκ,∞

[ΓP : ΓN ]

∑ε∈O∗

ω′ε2=ω

χ(ε)−1ε2pρ(g)Mω/ε2ϕl,q(ν, p)(g). (9.48)

If κ is not Γ-equivalent to ∞ then δκ,∞ = 0, while if κ =∞ then ρ(na[r]k) = 0 asr → ∞, so the last term in (9.48) vanishes at all cusps. For |ν − 1| 6 ε, the sumover non-zero ω′ ∈ Λ′κ is absolutely convergent and square-integrable over Γ\G, soaccording to Lemma 5.2.1, (ii) it has exponential decay in r as r →∞ uniformly


in ν. This means that ϕl,q(−ν,−p)(na[r]k) = r1−νΦl−p,q(k) determines the growthof PχρMωϕl,q(ν, p) at each cusp. Hence

PχρMωϕl,q(ν, p)(gκna[r]k) = O(r1−Re ν), as r →∞, (9.49)

uniformly for Re ν > 1− ε, at each cusp κ ∈ Cχ.Since the Poincare series PχρMωϕl,q(ν, 0) is square-integrable on Γ\G for

Re ν > 1 − ε, we denote by PχρMωϕl,q(1, 0) its value at ν = 1, and define aPoincare series PχLωl,qη by

PχLωl,qη = PχLω,∗l,q η +B(η)PχρMωϕl,q(1, 0). (9.50)

The discussion above implies square-integrability of the Poincare series PχLωl,qη onΓ\G. Moreover, it is a bounded function on Γ\G.

Indeed, for ν = 1 and p = 0, estimate (9.49) gives

PχρMωϕl,q(1, 0)(gκna[r]k) 1 as r →∞, (9.51)

for all κ ∈ Cχ. Hence

PχρMωϕl,q(1, 0)(g) 1, for all g ∈ Γ\G. (9.52)

On the other hand, since Lω,∗l,q η(na[r]k) is O(r−b) as r → ∞ for all b > 0, wehave PχL

ω,∗l,q η(gκna[r]k) r−b as r → ∞ at each cusp κ ∈ Cχ. This implies

boundedness of the Poincare series Lω,∗l,q η on Γ\G. Hence, we conclude

PχLωl,qη(g) = O(1), for all g ∈ Γ\G. (9.53)

Chapter 10

Preliminary sum formula

In this chapter we shall derive the preliminary sum formula via spectral andgeometric computations of the inner product of two Poincare series as discussedin Section 9.2. We shall first carry out some preparations.

Lemma 10.1. Let f be a continuous χ-automorphic function with respect to Γwhich is integrable over Γ\G, f(na[r]k) is at least O(1) as r ↓ 0, and f(gκna[r]k)is at most O(r) as r →∞ at each cusp κ. Then, the following equality holds:

〈PχLωl,qη, f〉Γ\G =

√|dF |

2[ΓP : ΓN ]〈Lωl,qη, Fωf〉N\G, (10.1)

Proof. From the definition (9.50), we see that the function PχLωl,qη is a sum oftwo functions. The function Lω,∗l,q η satisfies the growth conditions from Proposition7.1.2 and therefore, by Lemma 7.3.1, we have

〈PχLω,∗l,q η, f〉Γ\G =

√|dF |

2[ΓP : ΓN ]〈Lω,∗l,q η, Fωf〉N\G. (10.2)

For Re ν > 1, the equality

〈PχρMωϕl,q(ν, 0), f〉Γ\G =

√|dF |

2[ΓP : ΓN ]〈ρMωϕl,q(ν, 0), Fωf〉N\G (10.3)

holds by Lemma 7.3.1. Because of the growth conditions imposed on f , the rightside of (10.3) converges absolutely to a holomorphic function for Re ν > 0. Alsothe left side of (10.3) is holomorphic for Re ν > 1 − ε with ε > 0; it followsfrom estimate (9.49) and the growth of f on cusp sectors. Note that f is notnecessarily square-integrable; so the holomorphy does not follow from the L2-holomorphy of PχρMωϕl,q(ν, 0). On compact subsets of Γ\G the integrability of

90 Preliminary sum formula

PχρMωϕl,q(ν, 0) · f is not a problem. Estimate (9.49) implies that we also haveintegrability of PχρMωϕl,q(ν, 0) · f on cusp sectors. Hence, the equality (10.3)extends holomorphically to Re ν > 1− ε. In particular, it holds for ν = 1.

Combining (9.50), (10.2), and (10.3) for ν = 1, we obtain(10.1).

10.1 Scalar product of Poincare series,spectral description

The cuspidal automorphic functions TV ϕl,q(νV , pV ) are continuous and inte-grable over Γ\G with exponential decay at the cusps and appropriate growth oncompact subsets of Γ\G. So, we may apply Lemma 10.1 with f = TV ϕl,q(νV , pV ).

By (5.6) we have the Fourier expansion

TV ϕl,q(νV , pV ) =∑

0 6=ω∈O′

cTV(ω)Jωϕl,q(νV , pV ). (10.4)

It will be convenient to use the normalization of the Fourier coefficients cTV(ω)

given by

CV (ω; νV , pV ) := (2π|ω|)νV (ω/|ω|)−pV +ξcTV

(ω), (10.5)

Equality (10.1) yields

〈PχLωl,qη, TV ϕl,q(νV , pV )〉Γ\G =cTV (ω)

√|dF |

2[ΓP : ΓN ]

∫N\G

Lωl,qη(g)Jωϕl,q(νV , pV )(g)dg,

which by definition of the Lebedev transform (9.1) is further equal to

=π2√|dF |

2[ΓP : ΓN ]‖ΦlpV ,q‖K i

−pV +ξ

Γ(l + 1 + νV )CV (ω; νV , pV )Lωl,qL

ωl,qη(−νV , pV )

(9.15)= −

√|dF |

[ΓP : ΓN ]i pV −ξ‖ΦlpV ,q‖K CV (ω; νV , pV ) ·

· Γ(l + 1− νV )sinπ(νV − pV )νV

2 − p2V

νVε(pV )η(−νV , pV ). (10.6)

Taking the complex conjugate of the last expression yields

〈TV ϕl,q(νV , pV ), PχLωl,qη〉Γ\G = −√|dF |

[ΓP : ΓN ]i−pV +ξ‖ΦlpV ,q‖K ·

· CV (ω; νV , pV )Γ(l + 1− νV )sinπ(νV − pV )

ν2V − p2

V

νVε(pV )η(−νV , pV ). (10.7)

10.1 Spectral description 91

Let θ ∈ Tlσ be another function with the same properties as η.If V is in the unitary principal series, that is νV ∈ i[0,∞), then we recall (9.25),

and equalities (10.6)–(10.7) give

〈PχLω1l,qη, TV ϕl,q(νV , pV )〉Γ\G〈TV ϕl,q(νV , pV ), PχLω2

l,qθ〉Γ\G =

=|dF |

[ΓP : ΓN ]2‖ΦlpV ,q‖

2KCV (ω1; νV , pV ) ·

· CV (ω2; νV , pV )λl(νV , pV )η(νV , pV )θ(νV , pV ), (10.8)

where

λl(ν, p) := Γ(l + 1 + ν)Γ(l + 1− ν) sin2 π(ν − p)(ν2 − p2)2

ν 2ε(p). (10.9)

If V is in the complementary series, i.e. νV ∈ (0, 1) ⊂ R, then pV = 0, andfrom (10.6)–(10.7) we get

〈PχLω1l,qη, TV ϕl,q(νV , 0)〉Γ\G〈TV ϕl,q(νV , 0), PχLω2

l,qθ〉Γ\G =

=|dF |

[ΓP : ΓN ]2Γ(l + 1− νV )Γ(l + 1 + νV )

‖Φl0,q‖2KCV (ω1; νV , 0) ·

· CV (ω2; νV , 0)λl(νV , 0)η(νV , 0)θ(νV , 0). (10.10)

We now turn to the Eisenstein series. The fact that the functions Eκl,q(ν, p;χ)with Re ν = 0 are continuous and integrable functions over Γ\G, and O(r) on cuspsectors as well as on compact subsets of Γ\G, allow us to apply Lemma 10.1 withf = Eκl,q(ν, p;χ).

The expression (5.2) gives, in particular, the Fourier expansion of Eκl,q(ν, p;χ) atinfinity. Again, for convenience, we normalize the Fourier coefficients Dκ,∞

χ (ω; ν, p)of the Eisenstein series as follows

Bκ,χ(ω;λ, p) := (2π|ω|)λ (ω/|ω|)−p+ξDκ,∞χ (ω;λ, p). (10.11)

We mentioned at the end of Section 5.1 that the Fourier coefficients Dκ,∞χ (ω; ν, p),

and hence Bκ,χ(ω; ν, p), of the Eisenstein series are meromorphic over C withrespect to ν, and holomorphic on the line Re ν = 0. So, on the line Re ν = 0, wehave by (10.1)

〈PχLωl,qη,Eκl,q(ν, p;χ)〉Γ\G =

=

√|dF |

2 [ΓP : ΓN ]Dκ,∞χ (ω; ν, p)[Γκ : Γ′κ]

∫N\G

Lωl,qη(g)Jωϕl,q(ν, p)(g)dg

(9.1)=

π2√|dF |

2[ΓP : ΓN ]‖Φlp,q‖K[Γκ : Γ′κ]

Dκ,∞χ (ω; ν, p)

Γ(l + 1− ν)(−iω/|ω|)p−ξ

(2π|ω|)νLωl,qL

ωl,qη(ν, p).


Equality (9.15) and normalization (10.11) yield

〈PχLωl,qη,Eκl,q(ν, p;χ)〉Γ\G = −√|dF |

[ΓP : ΓN ]i p−ξ‖Φlp,q‖K

[Γκ : Γ′κ]·

·Bκ,χ(ω; ν, p) Γ(l + 1 + ν)sinπ(ν − p)ν2 − p2

νε(p)η(ν, p). (10.12)

Taking the complex conjugate of the last expression, using ν = −ν and (9.25),gives

〈Eκl,q(ν, p;χ), PχLωl,qη〉Γ\G = −√|dF |

[ΓP : ΓN ]i−p+ξ‖Φlp,q‖K

[Γκ : Γ′κ]·

·Bκ,χ(ω; ν, p)Γ(l + 1− ν) sinπ(ν − p)ν2 − p2

νε(p)η(ν, p). (10.13)

Then, from (10.12) and (10.13) we get

〈PχLω1l,qη,E

κl,q(ν, p;χ)〉Γ\G〈Eκl,q(ν, p;χ), PχLω2

l,qθ〉Γ\G =

=|dF |

[ΓP : ΓN ]2‖Φlp,q‖2K[Γκ : Γ′κ]2

Bκ,χ(ω1; ν, p) ·

·Bκ,χ(ω2; ν, p)λl(ν, p)η(ν, p)θ(ν, p), (10.14)

with λl(ν, p) as in (10.9).For the constant function f ≡ 1, the fact that ω 6= 0 implies Fω1 ≡ 0, and

therefore we have 〈PχLωl,qη, 1〉Γ\G = 0.

Since the square-integrable Poincare series PχLωl,qη is bounded, see (9.53), wemay take f1 = PχLω1

l,qη and f2 = PχLω2l,qθ in Theorem 8.1. Using (10.8), (10.10),

and (10.14) we obtain the spectral side of the sum formula:

〈PχLω1l,qη, PχL

ω2l,qθ〉Γ\G =

|dF |[ΓP : ΓN ]2

∑V

CV (ω1; νV , pV ) ·

· CV (ω2; νV , pV )λl(νV , pV )η(νV , pV )θ(νV , pV )

+|dF |

[ΓP : ΓN ]2∑κ∈Cχ

14πi [Γκ : Γ′κ]|Λκ|

∑χ

|p|6l

∫(0)

Bκ,χ(ω1; ν, p) ·

·Bκ,χ(ω2; ν, p)λl(ν, p)η(ν, p)θ(ν, p) dν. (10.15)

Here V runs over an orthogonal system of irreducible cuspidal subspaces of thespace L2(Γ\G,χ) that intersect L2(Γ\G,χ; l, q) non-trivially.

10.2 Geometric description 93

10.2 Scalar product of Poincare series,geometric description

We now proceed with the geometric computation of the same inner product asin the previous section.

For ω1 ∈ O′, ω1 6= 0, we observe that (9.50) implies

Fω1PχLω2l,qη = Fω1PχL

ω2,∗l,q η +B(η)Fω1PχρMω2ϕl,q(1, 0). (10.16)

The functions Lω2,∗l,q η and ρMω2ϕl,q(ν, 0), Re ν > 1, satisfy the growth conditions

in Proposition 7.1.2. (See the discussion just after (9.44), and the estimate (9.45),respectively.) Thus, we have by (7.16):

Fω1PχLω2,∗l,q η =

1[ΓP : ΓN ]

∑ε∈O∗

δε2ω1,ω2 χ(ε)−1l1/εLω2,∗l,q η

+2

[ΓP : ΓN ]√|dF |

∑′

c∈ISχ(ω1, ω2; c)Jω1 l1/cL

ω2,∗l,q η (10.17)

as well as

Fω1PχρMω2ϕl,q(ν, 0) =1

[ΓP : ΓN ]

∑ε∈O∗

δε2ω1,ω2 χ(ε)−1l1/ερMω2ϕl,q(ν, 0)

+2

[ΓP : ΓN ]√|dF |

∑′

c∈ISχ(ω1, ω2; c)Jω1 l1/cρMω2ϕl,q(ν, 0), (10.18)

for Re ν > 1. Using again the notation ψω(ν) = ρMωϕl,q(ν, 0) (here with p = 0),by the definition of the Jacquet operator, we have

Jω1 l1/cψω2(ν)(g) =∫N

χω1(n)−1ψω2(ν)(g)(h[1/c]wng)dn

<

∫N∗

∣∣∣ψω2(ν)(n[

zc2(r2+|z|2)

]a[

r|c|2(r2+|z|2)

]k′)∣∣∣ d+z, (10.19)

where N∗ := z ∈ C : r 6 (T + 1)|c|2(r2 + |z|2) with T > 0 and g = a[r]k. Forthe last inequality we used the relation (4.3) and the fact that the cut off functionρ is non-zero if its argument is in (0, T + 1]. Estimate (9.45), for z ∈ N∗, yields

ψω2(ν)(n[

zc2(r2+|z|2)

]a[

r|c|2(r2+|z|2)

]k′)(

r

|c|2(r2 + |z|2)

)1+Re ν

, (10.20)

uniformly for Re ν > 0 and r > 0. Here the implicit constant does not depend oneither c or r. Thus, for Re ν > 1− ε we get the estimate

Jω1 l1/cρMω2ϕl,q(ν, 0)(a[r]k) r−1−Re ν |c|−2(1+Re ν) rε|c|−4+2ε, (10.21)


uniformly for r > 0 and c ∈ I \ 0. This means that the second term in theright side of (10.18) converges absolutely to a holomorphic function for Re ν >1 − ε. Hence, the expression (10.18) extends holomorphically to Re ν > 1 − ε. Inparticular, it may be evaluated at ν = 1.

Substitution of (10.17) and (10.18) with ν = 1 into (10.16) gives

Fω1PχLω2l,qη =

1[ΓP : ΓN ]

∑ε∈O∗

δε2ω1,ω2 χ(ε)−1l1/εLω2l,qη

+2

[ΓP : ΓN ]√|dF |

∑′

c∈ISχ(ω1, ω2; c)Jω1 l1/cL

ω2l,qη. (10.22)

From (9.16) and (4.52) we get that

lτ Lω2l,qη = |τ |2(τ/|τ |)2ξ Lτ

2ω2l,q η for any τ ∈ C∗,

which together with Lemma 9.1.8 implies

Fω1PχLω2l,qη =

1[ΓP : ΓN ]

∑ε∈O∗

δε2ω1,ω2 χ(ε)−1ε−2ξLω1l,qη

+2π2 (iω1/|ω1|)2ξ

[ΓP : ΓN ]√|dF |

∑′

c∈I

(c

|c|

)−2ξSχ(ω1, ω2; c)|c|2

Lω1l,q

(κ∗(ω1, ω2, 1/c)η

). (10.23)

On the other hand, since the square-integrable function PχLω2l,qθ is also bounded

on Γ\G, see (9.53), we may apply Lemma 10.1 with f = PχLω2l,qθ and obtain


ω2l,qθ〉Γ\G =

√|dF |

2[ΓP : ΓN ]〈Lω1,

l,q η, Fω1PχLω2l,qθ〉N\G. (10.24)

We now insert (10.23) into (10.24), change the order of summation and integration,and get:


ω2l,qθ〉Γ\G =

=

√|dF |

2[ΓP : ΓN ]2∑ε∈O∗

δε2ω1,ω2 χ(ε)ε2ξ〈Lω1l,qη, L

ω1l,qθ〉N\G

+ χ(−1)π2(iω1/|ω1|)−2ξ

[ΓP : ΓN ]2∑′

c∈I

(c

|c|

)2ξSχ(ω2, ω1; c)|c|2

·

· 〈Lω1l,qη, L

ω1l,q

(κ∗(ω1, ω2, 1/c)η

)〉N\G, (10.25)

In the last line we used property (6.5) of a Kloosterman sum.

10.3 Preliminary sum formula 95

Finally, an application of Lemma 9.1.5 gives the geometric side of the sumformula


ω2l,qθ〉Γ\G =

√|dF |

2π3i [ΓP : ΓN ]2∑ε∈O∗

δε2ω1,ω2 χ(ε)ε2ξ ·

·∑|p|6l

∫(0)

η(ν, p)θ(ν, p)λl(ν, p)(p2 − ν2)dν

+(iω2/|ω2|)2ξ

πi [ΓP : ΓN ]2∑′

c∈I

(c

|c|

)−2ξSχ(ω2, ω1; c)|c|2

·

·∑|p|6l

∫(0)

K∗ν,p

(4πc

√ω1ω2

)η(ν, p)θ(ν, p)λl(ν, p)(p2 − ν2)dν, (10.26)

where we used properties (K1) and (K3) of the function K∗ν,p with Re ν = 0, see

page 80.

10.3 Preliminary sum formula

The equality of the expressions in (10.15) and (10.26) is the basis for the sumformula. Equating their right sides proves:

Proposition 10.3.1. Let η, θ ∈ Tlσ with fixed σ ∈ (1, 32 ). Then, for any ω1, ω2 ∈

O′ \ 0 we have:∑V

CV (ω1; νV , pV )CV (ω2; νV , pV )η(νV , pV )θ(νV , pV )λl(νV , pV )

+1

4πi

∑κ∈Cχ

1[Γκ : Γ′κ]|Λκ|

∑χ

|p|6l

∫(0)

Bκ,χ(ω1; ν, p) ·

·Bκ,χ(ω2; ν, p)η(ν, p)θ(ν, p)λl(ν, p) dν =

=1

2π3i√|dF |

∑ε∈O∗

δε2ω1,ω2 χ(ε)ε2ξ ·

·∑|p|6l

∫(0)

η(ν, p)θ(ν, p)λl(ν, p)(p2 − ν2)dν

+(iω2/|ω2|)2ξ

|dF |πi∑′

c∈I

(c

|c|

)−2ξSχ(ω2, ω1; c)|c|2

∑|p|6l

∫(0)

K∗ν,p

(4πc

√ω1ω2

)·

· η(ν, p)θ(ν, p)λl(ν, p)(p2 − ν2)dν, (10.27)

where |Λκ| is the Euclidean area of a period parallelogram for the lattice Λκ in Ccorresponding to the discrete subgroup gκ−1Γ′κgκ, V runs over a maximal orthogo-


nal system of irreducible cuspidal subspaces of L2(Γ\G;χ) such that the type (l, q)occurs in V for one (hence for all) q ∈ 1

2Z satisfying q ≡ l (mod 1), |q| 6 l, and∑χ|p|6l means that the sum runs through all p ∈ 1

2Z such that |p| 6 l with thecondition χ(ε) = ε2p satisfied for all ε ∈ O∗.

Chapter 11

Spectral sum formula

Based on the preliminary version (10.27), we shall extend the validity of thesum formula to a wider class of test functions. (See Theorem 11.3.3.) Beforewe enlarge the class of test functions, we have to know more about the Besseltransformation appearing in the geometric side of the formula.

11.1 Bessel transformation

For σ ∈ (0, 32 ), σ 6= 1, and a, b ∈ R, we define Hσ(a, b) to be the space of func-

tions h : ν ∈ C : |Re ν| 6 σ × 12Z −→ C that satisfy the following conditions

(i) h(ν, p) = h(−ν,−p),

(ii) ν 7→ h(ν, p) is holomorphic on a neighborhood of the strip |Re ν| 6 σ,

(iii) h(ν, p) (1 + | Im ν|)−a(1 + |p|)−b,

(iv) If σ > 12 and p ∈ 1

2 +Z, then the function ν 7→ h(ν, p) has at ν = ± 12 zeros of

at least order 2, and if σ > 1 and p ∈ Z \ 0, then the function ν 7→ h(ν, p)has at ν = ±1 zeros of at least order 2.

The functions h built from η, θ ∈ Tlσ in (11.8) are elements of Hσ(a, b) for alla, b ∈ R.

Lemma 11.1.1. Let σ ∈ (0, 32 ), σ 6= 1 and h ∈ Hσ(a, b) with a, b ∈ R. We define

the Bessel transform Bh by

Bh(u) :=1

2πi

∑p∈ 1

2 Z

∫(0)

K∗ν,p(u)h(ν, p)(p

2 − ν2)dν, (11.1)

with K∗ν,p defined in (9.26).

If a > 2 and b > 3, then the sum and the integral in (11.1) converge absolutely.

98 Spectral sum formula

Proof. For Re ν = 0, we have

|p2 − ν2| 6 (|p|+ | Im ν|)2 6 (1 + |p|+ | Im ν|)2 6 (1 + | Im ν|)2(1 + |p|)2.

This and the estimate (9.32), for Re ν = 0 and fixed u ∈ C∗, gives


2 − ν2) (1 + | Im ν|)1−a(1 + |p|)2−b. (11.2)

This estimate further implies

Bh(u)∑p∈ 1

2 Z

(1 + |p|)2−b∫

(0)

(1 + | Im ν|)1−adν, (11.3)

from which we clearly see that the integral converges absolutely if a > 2, and thesum is absolutely convergent if b > 3.

Remark 12. For integer values of p, and in case of the Gaussian number fieldQ(i), the Bessel transform defined in (11.1) is four times the Bessel transform Bhdefined in [9], (10.2).

Let a > 2, b > 3. The estimate (9.32) allows us to shift the line of integrationto the line Re ν = α1 for any |α1| 6 σ < a

2 − 1:

Bh(u) =1

2πi

∑p∈ 1

2 Z

∫(α1)


2 − ν2)dν.

We then use (9.26) to split this integral into sum of two integrals with J∗ν,p(u) andJ∗−ν,−p(u). This is possible because of the estimate (9.31). The same estimatemakes the resulting integrals absolutely convergent, and also the sum over p ∈12Z. Condition (iv) in the definition of Hσ(a, b) ensures that the integrands areholomorphic inside the strip |Re ν| < σ except for singularities at ν = 0 in theterms with p ∈ Z\0. We now rearrange the sum, shift one of the lines ofintegration, and pick up the residues, to obtain:

Bh(u) =1πi

∑p∈ 1

2 Z

∫(α1)

|u/2|2ν(iu/|u|)−2p−2ξ

sinπ(ν − p)J∗ν,p(u)h(ν, p)(ν

2 − p2)dν

+1π

∑p∈Z\0

p2 (u/|u|)−2pJ∗0,p(u)h(0, p). (11.4)

For σ ∈ (0, 32 ), σ 6= 1, we take α1 = σ in (11.4), and estimate both terms

separately in order to obtain an estimate for the Bessel transform Bh(u) as |u| ↓ 0.In case p ∈ Z, we know that (u/|u|)−2pJ∗0,p(u) = J0,p(u). (See Remark 5 on

page 48). Using the estimate J0,p(u) (|u|/2)2|p|/(|p|!)2, see (10.8) in [9], weconclude that

(u/|u|)−2pJ∗0,p(u)u(|u|/2)2|p|

(|p|!)2, as |u| ↓ 0. (11.5)

11.1 Bessel transformation 99

Therefore the second term in the right side of (11.4) is O(|u|2), as |u| ↓ 0.For Re ν = σ, we have uniformly for p ∈ 1

2Z∣∣∣∣∣ |u/2|2ν(iu/|u|)−2p−2ξJ∗ν,p(u)sinπ(ν − p)

∣∣∣∣∣ 6 |u|2σ|J∗ν−p(u)||J∗ν,p(u)|| sinπ(ν − p)|(9.28) |u|2σ

| sinπ(ν − p)||Γ(ν + |p|+ 1)|−1|Γ(ν − |p|+ 1)|−1

=|u|2σ

π

|Γ(|p| − ν)||Γ(ν + |p|+ 1)|

σ |u|2σ|ν|−2σ−1, as |u| ↓ 0. (11.6)

In the case p ∈ Z, the last estimate reduces to (10.9) in [9], and in the case p ∈ 12 +Z

it is obtained as follows: We write |p| = 12 + n with n ∈ N and get

|Γ(|p| − ν)||Γ(ν + |p|+ 1)|

=|Γ(−ν − 1/2)||Γ(ν + 1/2)|

n∏j=0

∣∣∣∣j − ν − 1/2j + ν + 1/2

∣∣∣∣ |Γ(−ν − 1/2)||Γ(ν + 1/2)|−1 |ν + 1/2|−2σ−1e2σ+1 σ |ν|−2σ−1.

The estimate (11.6) implies that the first term in the right side of (11.4) is O(|u|2σ),as |u| ↓ 0.

Collecting these results we have proved

Lemma 11.1.2. For σ ∈ (0, 32 ), σ 6= 1, a > 2, b > 3. For each h ∈ Hσ(a, b) we

have

Bh(u)σ |u|2 minσ,1, as |u| ↓ 0.

We need the following lemma in the proof of the sum formula in Section 11.3.

Lemma 11.1.3. For σ ∈ (0, 32 ), σ 6= 1, a > 2, b > 3. Take α ∈ (0, σ]. Let (fn)

be a sequence of elements in Hσ(a, b) converging pointwise to f ∈ Hσ(a, b) for all(ν, p) with Re ν ∈ (0, α], p ∈ 1

2Z, and moreover

sup06Re ν6α,p∈ 1

2 Z,n∈N(1 + | Im ν|)a(1 + |p|)b

∣∣fn(ν, p)− f(ν, p)∣∣ <∞.

Then, the integral defining Bf(u) converges absolutely, and

limn→∞

|u|−2 minσ,1Bfn(u) = |u|−2 minσ,1Bf(u) (11.7)

uniformly on each set u ∈ C∗ : |u| 6 r0 with r0 > 0.


Proof. Using the integral representation (11.5) and (11.6) we get∣∣∣|u|−2 minσ,1Bfn(u)− |u|−2 minσ,1Bf(u)∣∣∣ 6

|u|−2 minσ,1∑p∈ 1

2 Z

∫(α)

|u|2σ|ν|−2σ−1|h(ν, p)||ν2 − p2|dν

+ |u|−2 minσ,1∑

p∈Z\0

p2|u|2|p|(|p|!)−2|h(0, p)|.

If we denote S := supν,p,n(1 + | Im ν|)a(1 + |p|)b∣∣fn(ν, p)− f(ν, p)

∣∣, the differencecan be further estimated as follows

S|u|−2 minσ,1

|u|2σ ∑p∈ 1

2 Z

(1 + |p|)2−b∫

(α)

(1 + | Im ν|)−2σ+1−adν

+|u|2∑

p∈Z\0

(1 + |p|)2−b |u|−2 minσ,1 · |u|min2σ,2 1.

Applying dominated convergence gives the result.

11.2 Extension method

In the preliminary sum formula, Proposition 10.3.1, the test functions η and θdo not occur separately, but always as a product η · θ. We also have the presenceof the function λl containing Γ-factors. We want our test functions to be as simpleas possible, and certainly not built as a product of different functions. We alsowant to eliminate the dependence of l in the test functions. The latter is easilyhandled by defining functions on the set ν ∈ C : |Re ν| 6 σ × 1

2Z vanishing for|p| > l.

We define a function h built from η, θ ∈ Tlσ, with σ ∈ (1, 32 ), and λl as in (10.9),

in the following way:

h(ν, p) =η(ν, p)θ(ν, p)λl(ν, p) , |p| 6 l0 , |p| > l,

(11.8)

where θ(ν, p) := θ(−ν, p). For each such function h, we obtain from (10.27):∑V

CV (ω1; νV , pV )CV (ω2; νV , pV )h(νV , pV )

+1

4πi

∑κ∈Cχ


∑χ

p∈ 12 Z

∫(0)

Bκ,χ(ω1; ν, p)Bκ,χ(ω2; ν, p)h(ν, p)dν =

11.2 Extension method 101

=1

2π3i√|dF |

∑ε∈O∗

δε2ω1,ω2 χ(ε)ε2ξ∑p∈ 1

2 Z

∫(0)

h(ν, p)(p2 − ν2)dν

+2|dF |

(iω2

|ω2|

)2ξ∑′

c∈I

(c

|c|

)−2ξSχ(ω2, ω1; c)|c|2

·

· 12πi

∑p∈ 1

2 Z

∫(0)

K∗ν,p

(4πc

√ω1ω2

)h(ν, p)(p2 − ν2)dν, (11.9)

where all sums and integrals in the relation above converge absolutely.It is convenient to introduce notation for the various terms in the sum formula.

On the spectral side, the choice of the spectral parameter (νV , pV ) is as indicatedin (3.21), and we introduce a measure dσω,ω′ on the set

Y :=((i[0,∞) ∪ (0, 1)

)× 0

)⋃(i[0,∞)×

(12Z\0

) )⊂ C× 1

2Z (11.10)

by ∫Y

f(ν, p)dσω,ω′(ν, p) :=∑V

CV (ω; νV , pV )CV (ω′; νV , pV )f(νV , pV )

+1

2πi

∑κ∈Cχ


∑χ

p∈ 12 Z

∫i[0,∞)

Bκ,χ(ω; ν, p) ·

·Bκ,χ(ω′; ν, p)f(ν, p)dν. (11.11)

It is clear that for ω = ω′ the measures dσω,ω are non-negative.If the function f is integrable for dσω,ω and for dσω′,ω′ , then it is also integrable

for dσω,ω′ , and∣∣∣∣∫Y

f dσω,ω′

∣∣∣∣ 6 (∫Y

|f | dσω,ω)1/2(∫

Y

|f | dσω′,ω′)1/2

. (11.12)

On the geometric side, the delta term can be described by introducing anothermeasure dδω,ω′ on the same set Y :∫

Y

f(ν, p)dδω,ω′(ν, p) :=α(χ, ξ;ω, ω′)π3i√|dF |

∑p∈ 1

2 Z

∫i[0,∞)

f(ν, p)(p2 − ν2)dν, (11.13)

where

α(χ, ξ;ω, ω′) :=

2χ(ε)ε2ξ , if ω′ = ε2ω with ε ∈ O∗/±1,0 , otherwise. (11.14)

Actually, the support of dδω,ω′ is the subset i[0,∞) × 12Z ⊂ Y . This measure is

positive if ω′ = ε2ω with some ε ∈ O∗/±1.


If f is an even function on iR, then the integrals over i[0,∞) can be replacedby one half of the integrals over the imaginary axis.

To describe the Kloosterman term, we note that for a test function h(ν, p)discussed above, the expression defining the Bessel transform Bh converges abso-lutely.

Now, for any even function f : C∗ −→ C we define the following sum ofKloosterman sums:

Kl(ω, ω′; f) :=2|dF |

(iω′

|ω′|

)2ξ∑′

c∈I

(c

|c|

)−2ξSχ(ω′, ω; c)|c|2

f(

4πc

√ωω′

). (11.15)

Absolute convergence of Kl(ω, ω′; f) follows from the estimate (6.12) as soon asthe function f satisfies f(u) |u|2α with α > 1

2 . Indeed,

|Kl(ω, ω′; f)| 6 2|dF |

∑′

c∈I

|Sχ(ω′, ω; c)||c|2

∣∣∣f ( 4πc

√ωω′

)∣∣∣ω,ω′,ε

∑′

c∈I|c|−1+ε−2α.

Since I ⊂ O is a lattice in C, the last sum may be estimated by an integral, andthe convergence will follow for −1 + ε− 2α < −2.

For h(ν, p) as above, the absolute convergence of Kl(ω, ω′;Bh) is clear becauseof Lemma 11.1.2.

Each function h defined as in (11.8), based on η, θ ∈ Tlσ with σ ∈ (1, 32 ), is well-

defined on Y and the support of the measure dσω,ω′ is contained in the domain inh, so we may reformulate the result (11.9) with the new notations:

Lemma 11.2.1. Let σ ∈ (1, 32 ) and ω, ω′ ∈ O′\0. Let l ∈ 1

2N and suppose thath is defined as in (11.8), based on η, θ ∈ Tlσ.

Then h is integrable for the measures dσω,ω′ and dδω,ω′ , the Bessel transfor-mation as defined in (11.1) converges absolutely, and∫

Y

h dσω,ω′ =∫Y

h dδω,ω′ + Kl(ω, ω′;Bh). (11.16)

By the phrase the sum formula with (ω, ω′) holds for f we mean that all sumsand integrals involved in (11.16) converge absolutely and the equality (11.16) holdsfor f .

It is clear that any function f ∈ Hσ(a, b) with σ ∈ (1, 32 ) is well-defined on

Y , since Y is contained in the domain of f , and the integral∫Yf dσω,ω′ makes

sense. We would like to extend the sum formula to functions f ∈ Hσ(a, b) withσ ∈ ( 1

2 , 1), but in this case the domain of f does not contain the support Y of themeasure dσω,ω′ , and the integral

∫Yfdσω,ω′ does not make sense. To “fix” this,

for σ ∈ ( 12 , 1), we define

Y σ :=((i[0,∞) ∪ (0, σ]

)× 0

)⋃(i[0,∞)×

(12Z\0

) ), (11.17)

11.2 Extension method 103

and recall that the results of Kim and Shahidi, [20], imply that there are noexceptional spectral parameters in the interval [σ, 1) ⊂ ( 1

2 , 1); see Section 9.2.This means that supp dσω,ω′ ⊂ Y σ, and hence the integral

∫Y σ fdσω,ω′ makes

sense. We extend the definition of the set Y σ for σ ∈ (1, 32 ) by setting Y σ := Y .

We shall now give two lemmas which will be used in the extension of the sumformula.

Lemma 11.2.2. Let σ > 12 and ω ∈ O′\0. Suppose the function f and the

sequence of functions (fn) on Y σ satisfy the following conditions:

(i) The sum formula with (ω, ω) holds for each fn.

(ii) The integral defining Bf(u) converges absolutely, and

limn→∞

|u|−2 minσ,1Bfn(u) = |u|−2 minσ,1Bf(u)

uniformly on each set u ∈ C∗ : |u| 6 4π|ω|.

(iii) f is integrable for dδω,ω.

(iv) fn > 0 on Y σ, and limn→∞ fn = f pointwise on Y σ.

Then the sum formula with (ω, ω) holds for f , and

limn→∞

∫Y σ

fn dσω,ω =∫Y σ

f dσω,ω, (11.18)

limn→∞

∫Y σ

fn dδω,ω =∫Y σ

f dδω,ω, (11.19)

limn→∞

Kl(ω, ω;Bfn) = Kl(ω, ω;Bf). (11.20)

Proof. We consider the Kloosterman term. The arguments uc := 4πc |ω| oc-

curring in (11.15) all satisfy |uc| 6 4π|ω|, since |c| > 1 for all c ∈ I\0. If ε > 0then, from the condition (ii), we have for all sufficiently large n:∣∣∣Bfn ( 4π

c |ω|)−Bf

(4πc |ω|

) ∣∣∣ < ε(

4π|ω||c|

)2 minσ,1.

Using this, Lemma 11.1.2, and the non-trivial estimate of a Kloosterman sum(6.12), we get

|Kl(ω, ω; f)| 6 |Kl(ω, ω;Bfn)|+ |Kl(ω, ω;Bfn)−Kl(ω, ω;Bf)|

62|dF |

∑′

c∈I

|Sχ(ω, ω; c)||c|2

∣∣Bfn ( 4πc |ω|

)∣∣+ ∣∣Bfn ( 4πc |ω|

)−Bf

(4πc |ω|

)∣∣ ω,ε

∑′

c∈I|c|−1+ε−2 minσ,1.


The expression Kl(ω, ω;Bf) converges absolutely since minσ, 1 > 12 , and (11.20)

holds.Conditions (iii) and (iv) allow us to apply dominated convergence to see that

limn→∞∫Y σ fn dδω,ω =

∫Y σ f dδω,ω. As the sum formula with ω = ω′ holds for

each fn, we have a sequence (fn) of dσω,ω-integrable functions on Y σ for which theintegrals tend to a limit equal to

∫Y σ f dδω,ω+Kl(ω, ω;Bf). So, by Fatou’s lemma

(see e.g. [1], p. 105, [27], p. 141), the limit function f is also integrable for dσω,ω.Again by dominated convergence we see that limn→∞

∫Y σ fn dσω,ω =

∫Y σ f dσω,ω,

and the sum formula with ω = ω′ must hold for f .

Lemma 11.2.3. Let σ > 12 and ω, ω′ ∈ O′\0. Suppose the functions f , h and

the sequence of functions (fn) on Y σ satisfy the following conditions:

(i) The sum formula with (ω, ω′) holds for each fn.

(ii) The sum formula holds for h with (ω, ω) as well as with (ω′, ω′).

(iii) The integral defining Bf(u) converges absolutely, and

limn→∞

|u|−2 minσ,1Bfn(u) = |u|−2 minσ,1Bf(u)

uniformly on each set u ∈ C∗ : |u| 6 r0 with r0 > 0.

(iv) limn→∞ fn = f pointwise on Y σ.

(v) |fn| 6 h on Y σ.

Then the sum formula with (ω, ω′) holds for f , and

limn→∞

∫Y σ

fn dσω,ω′ =∫Y σ

f dσω,ω′ , (11.21)

limn→∞

∫Y σ

fn dδω,ω′ =∫Y σ

f dδω,ω′ , (11.22)

limn→∞

Kl(ω, ω′;Bfn) = Kl(ω, ω′;Bf). (11.23)

Proof. The absolute convergence of the sum of Kloosterman sums and thecorresponding limit formula follow as in the proof of the previous lemma.

The definition (11.14) gives that α(χ, ξ;ω, ω) 6= 0, and for ω′ = ε2ω with ε ∈O∗, we have α(χ, ξ;ω, ω′) = χ(ε)ε2ξα(χ, ξ;ω, ω). This implies that h is integrablefor dδω,ω′ . By dominated convergence, using h as a majorant, we see that (11.19)is satisfied.

From (11.12) we see that h is integrable for the measure dσω,ω′ as well. There-fore it can be used as majorant in the application of dominated convergence forthe spectral side, which implies that (11.18) holds.

11.3 Extension of the sum formula 105

11.3 Extension of the sum formula

From Lemma 11.2.1 we know that the sum formula holds for functions h of theform (11.8). We shall approximate the elements f ∈ Hσ(a, b) with σ ∈ (1, 3

2 ) bysuch functions and then conclude that the sum formula holds also for f .

Lemma 11.3.1. Let σ ∈ (1, 32 ), ω, ω′ ∈ O′\0, a, b ∈ R, and f ∈ Hσ(a, b). The

sequence of functions fn, n > 1, defined by

fn(ν, p) :=f(ν, p)e2ν

2/n , if |p| 6 n,0 , if |p| > n

(11.24)

converges to the function f pointwise on Y (see (11.10)), and the sum formulawith (ω, ω′) holds for each fn.

Proof. The pointwise convergence on Y is clear from the definition (11.24) offn. To prove the latter statement, we put

Λn(ν, p) :=λl(ν, p) , if n 6 l,0 , if n > l,

ηn(ν, p) :=

Λn(ν, p)−1f(ν, p)eν2/n , if |p| 6 n,

0 , if |p| > n,

θn(ν, p) :=eν

2/n, if |p| 6 n,0 , if |p| > n.

For n 6 l, it is obvious that θn ∈ Tlσ. To conclude the same for ηn, we note that thefunction Λ−1

n is meromorphic on the strip |Re ν| 6 σ with double poles at ν = ± 12

if p ∈ 12 + Z. Those poles are cancelled by the double zeros of the function f(ν, p)

at ν = ± 12 if p ∈ 1

2 + Z, prescribed in the definition of f . This means that theproduct Λ−1

n f , and hence ηn, are holomorphic on the strip |Re ν| 6 σ. Directlyfrom the definition of λl(ν, p) we get that Λ−1

n (−ν,−p) = Λ−1n (ν, p), which implies

that also ηn remains unchanged for (−ν,−p) 7→ (ν, p). Some work shows that wehave the estimate Λn(ν, p)−1 (1 + | Im ν|)−2l+1e−π| Im ν|, which implies that ηnhas indeed the necessary growth behavior. So, ηn ∈ Tlσ.

In addition, the constructions above ensure that θn = θn, and ηnθnΛn = fn.So, the functions fn are of the form (11.8), and by Lemma 11.2.1 the sum formulawith (ω, ω′) holds for each fn, n 6 l.

We shall now conclude that the sum formula holds for elements of Hσ(a, b),σ ∈ (1, 3

2 ) in the following “step by step” way:Define a function fa,b, for some a, b ∈ R, with

fa,b(ν, p) :=((1− ξ)2 − ν2

)2 (4− ν2

)−2−a/2 (1 + p2

)−b/2. (11.25)


Here ξ depends on p and it is defined in (4.27). The conditions that the numbersa, b should satisfy will be determined later. By construction, the function fa,b isan element of Hσ(a, b).

1. We take f = fa,b and construct a sequence of functions (fn) as in Lemma11.3.1. The lemma then tells us that each fn satisfies the condition (i) bothin Lemma 11.2.2 and Lemma 11.2.3.

2. Using Lemma 11.1.3 we see that the conditions (ii) in Lemma 11.2.2 and (iii)in Lemma 11.2.3 are satisfied provided that a > 2 and b > 3.

3. We have fa,b(ν, p)(p2 − ν2) (1 + | Im ν|)2−a(1 + |p|)2−b, which impliesintegrability of the function fa,b for the measure dδω,ω′ under the conditionsa > 3 and b > 3.

4. A check in (11.25) convinces us that fa,b(ν, p) is chosen in such a way thatis non-negative on the set Y defined in (11.10). The factor e2ν

2/n is alsopositive on Y . Hence, fn > 0, and fn(ν, p)→ fa,b(ν, p) on Y for n→∞, asrequired in condition (iv) in Lemma 11.2.2.

5. Let ω ∈ O\0. Steps 1–4 tell us that all conditions in Lemma 11.2.2 aresatisfied. Therefore we conclude that the sum formula with (ω, ω) holds forfa,b provided that a > 3 and b > 3.

6. We take a suitable multiple of fa,b as a majorant in Lemma 11.2.3. Leth = Cfa,b and ω, ω′ ∈ O\0. Step 5 implies that if a > 3, b > 3, the sumformula holds for h with (ω, ω) as well as for (ω′ω′). This gives condition(ii) in Lemma 11.2.3.

7. If we choose the constant C in h such that C > e2σ2, where |Re ν| 6 σ, then

|fn| 6 Cfa,b = h on Y . This is condition (v) in Lemma 11.2.3.

8. Steps 1, 2, 4, 6, and 7 imply that all the conditions in Lemma 11.2.3 are sat-isfied, which means that the sum formula with (ω, ω′) holds for fa,b providedthat a > 3 and b > 3.

The discussion above proves

Proposition 11.3.2. Let ω, ω′ ∈ O\0, σ ∈ (1, 32 ), and a, b ∈ R such that a > 3,

b > 3. Each function h ∈ Hσ(a, b) is integrable for the measures dσω,ω′ anddδω,ω′ , the Bessel transformation Bh in (11.1) converges absolutely, and the sumKl(ω, ω′;Bh) in (11.15) converges absolutely. Moreover∫

Y

h dσω,ω′ =∫Y

h dδω,ω′ + Kl(ω, ω′;Bh). (11.26)


Our final goal is to extend the class of test functions in Proposition 11.3.2 toh ∈ Hσ(a, b) with σ ∈ ( 1

2 , 1).Let Σ ∈ (1, 3

2 ) and σ ∈ ( 12 , 1). We consider a function f ∈ Hσ(a, b) with

a > 3, b > 3. We shall prove that the sum formula holds for f approximating itby functions fn ∈ HΣ(a, b).

The function f0,0 as defined in (11.25) has the minimal number of zeros pre-scribed in the definition of Hσ(a, b), see p. 97, and it is holomorphic on any strip|Re ν| 6 σ with σ < 2. We can write the function f as a product f = f0,0 g, witha function g satisfying (i)–(iii) in the definition of the set Hσ(a, b).

We define gn by convolution of g with a Gauss kernel:

gn(ν, p) := −i√n

π

∫(0)

g(µ, p)en(µ−ν)2dµ. (11.27)

The integral converges absolutely for each ν ∈ C and defines a holomorphic func-tion ν 7→ gn(ν, p) on C invariant under (ν, p) 7→ (−ν,−p). The line of integrationin (11.27) may be moved to any line Re ν = α1, α1 ∈ [−σ, σ].

Let ν = α + ir with α ∈ [−Σ,Σ], r ∈ R. To estimate the functions gn, wefirst rewrite the integral (11.27), by putting µ = it, t ∈ R, and use the fact that gsatisfies estimate (iii) in the definition of the set Hσ(a, b):

gn(ν, p) =√n

π

∫ ∞

−∞g(it, p)enα

2−n(r−t)2+2nα(r−t)idt

n1/2enα2(1 + |p|)−b

∫ ∞

−∞(1 + |t|)−ae−n(r−t)2dt. (11.28)

For any a > 0 and n > 1, the integral is estimated as follows:∫ ∞

−∞(1 + |t|)−ae−n(r−t)2dt

∫|t|6|r|/2

1 · e−n(t−r)2dt+ (1 + |r|)−a∫|t|>|r|/2

e−n(t−r)2dt

∫ 3|r|/2

|r|/2e−nt

2dt+ (1 + |r|)−a

∫ ∞

−∞e−nt

2dt

n−1/2

∫ ∞

|r|/2e−t

2dt+ n1/2(1 + |r|)−a

n−1/2

1 , |r| 6 1e−|r|

√n/2 , |r| > 1

+ n1/2(1 + |r|)−a

n−1/2(1 + |r|)−a. (11.29)

From (11.28) and (11.29) we get

gn(ν, p)n (1 + |p|)−b(1 + | Im ν|)−a. (11.30)


This implies that all functions gn, n > 1, satisfy the conditions (i)–(iii) in thedefinition of HΣ(a, b). Hence, the product fn := f0,0 gn is an element of HΣ(a, b).By Proposition 11.3.2 the sum formula holds with (ω, ω′) for all fn, which meansthat the condition (i) in Lemma 11.2.3 is satisfied. We want to show that thesequence of functions (fn) approximates f in such a way that Lemma 11.2.3 canbe applied.

For all |α| 6 σ, we can move the line of integration:

gn(ν, p) = −i√n

π

∫(α)

g(µ, p)en(ν−µ)2dµ =√n

π

∫ ∞

−∞g(α+ it, p)e−n(t−r)2dt

√n

∫ ∞

−∞(1 + |p|)−b(1 + |t|)−ae−n(t−r)2dt

(11.29) (1 + |p|)−b(1 + |r|)−a.

This estimate is uniform in n > 1. It shows that it is possible to choose a constantC > 0 such that |gn(ν, p)| 6 C

|fa,b(ν,p)||f0,0(ν,p)| for all (ν, p) with |Re ν| 6 σ. Then the

function h := Cfa,b is the necessary majorant for condition (v) in Lemma 11.2.3to be satisfied.

On the other hand, h ∈ HΣ(a, b) and Proposition 11.3.2 implies that condition(ii) in Lemma 11.2.3 is also satisfied.

To obtain the pointwise convergence of gn(ν, p) for |Re ν| 6 σ, (Re ν = α), wecarry out the following change of variables:

gn(ν, p) =√n

π

∫ ∞

−∞g(α+ it, p)e−n(t−r)2dt(

change : t 7→ r + t/√n)

=1√π

∫ ∞

−∞g

(ν +

it√n, p

)e−t

2dt. (11.31)

The function ν 7→ g(ν, p) is bounded on vertical lines, and thus we have an inte-grable majorant. We are allowed to take the limit as n → ∞ inside the integral,which gives

limn→∞

gn(ν, p) =1√π

∫ ∞

−∞limn→∞

g

(ν +

it√n, p

)e−t

2dt = g(ν, p).

This further implies pointwise convergence of fn:

limn→∞

fn(ν, p) = f0,0(ν, p) limn→∞

gn(ν, p) = f0,0(ν, p)g(ν, p) = f(ν, p), (11.32)

for all (ν, p) with |Re ν| 6 σ, and hence also for (ν, p) ∈ Y σ as required in thecondition (iv) in Lemma 11.2.3.

Finally, we use Lemma 11.1.3 to establish validity of condition (iii) in Lemma11.2.3. For that purpose we need an estimate for the difference g

(ν + it√

n, p)−

g(ν, p), which we shall obtain by estimating the partial derivative ∂νg.


Let α ∈ (−σ, σ) and γ = µ ∈ C : |µ−ν| = σ−|α| be a small circle around thepoint ν. Using the Cauchy integral formula and differentiating under the integralsign yields

∂νg(ν, p) =1

2πi

∫γ

g(µ, p)(µ− ν)2

dµ (1 + |p|)−b∫γ

(1 + | Imµ|)−a(µ− ν)−2dµ

(1 + |p|)−b(1 + | Im ν| − σ + |α|)−a(σ + |α|)−1

α (1 + |p|)−b(1 + | Im ν|)−a. (11.33)

We use this estimate with α ∈ [0, σ] to obtain from (11.31):

gn(ν, p)− g(ν, p) =1√π

∫ ∞

−∞

[g

(ν +

it√n, p

)− g(ν, p)

]e−t

2dt

(11.33)

∫ ∞

−∞

∣∣∣∣ it√n∣∣∣∣ (1 + |p|)−b(1 + | Im ν|)−ae−t

2dt

6 (1 + |p|)−b(1 + | Im ν|)−a∫ ∞

−∞|t|e−t

2dt = (1 + |p|)−b(1 + | Im ν|)−a.

Multiplication by f0,0 does not change this estimate. Therefore, we may applyLemma 11.1.3 for f and fn to see that the condition (ii) in Lemma 11.2.3 is alsosatisfied.

Hence, by Lemma 11.2.3, the sum formula holds for f . This proves

Theorem 11.3.3. (Spectral sum formula) Let σ ∈ ( 12 , 1), and h be a function

defined over the set ν ∈ C : |Re ν| 6 σ × 12Z such that

(i) h(ν, p) = h(−ν,−p),

(ii) ν 7→ h(ν, p) is holomorphic on a neighborhood of the strip |Re ν| 6 σ,

(iii) h(ν, p) (1 + | Im ν|)−a(1 + |p|)−b with a > 3, b > 3,

(iv) the function ν 7→ h(ν, p) has at least double zeros at ν = ± 12 if p ∈ 1

2 + Z.

Then, for any non-zero ω, ω′ ∈ O′, we have:∑V

CV (ω; νV , pV )CV (ω′; νV , pV )h(νV , pV )

+1

4πi

∑κ∈Cχ


∑χ

p∈ 12 Z

∫(0)

Bκ,χ(ω; ν, p)Bκ,χ(ω′; ν, p)h(ν, p) dν =

=1

2π3i√|dF |

∑ε∈O∗

δε2ω,ω′ χ(ε)ε2ξ∑p∈ 1

2 Z

∫(0)

h(ν, p)(p2 − ν2)dν

+2|dF |

(iω′

|ω′|

)2ξ∑′

c∈I

(c

|c|

)−2ξSχ(ω′, ω; c)|c|2

Bh(

4πc

√ωω′

), (11.34)


where∑χ

p∈ 12 Z means that the sum runs through all p ∈ 1

2Z such that the condition

χ(ε) = ε2p is satisfied for all ε ∈ O∗, V runs over a maximal orthogonal systemof irreducible cuspidal subspaces of L2(Γ\G;χ), |Λκ| is the Euclidean area of aperiod parallelogram for the lattice Λκ ∈ C corresponding to gκ

−1Γ′κgκ, and Bhis the Bessel transform in (11.1). Convergence of these expressions is absolutethroughout.

11.4 Discussion of the spectral sum formula

With condition (iv) in Theorem 11.3.3, we have one extra restriction in theclass of test functions in the case of half-integer p, making this class smaller thanthe corresponding class in the case of integer p. Even if we somehow manageto prove that there are no exceptional eigenvalues for Γ0(I) in the interval (0, 1

2 )and choose σ < 1

2 , with σ determining the width of the strip where the spectralparameter ν belongs, the estimation of the sum of Kloosterman sums will cause aproblem. More precisely, for such σ, we will not have the absolute convergence ofKl(ω, ω′;Bh).

As a consequence, (iv) is a strong condition which seems to be unavoidable,and it causes complications in some applications of the sum formula.

11.4.1 Comparison with the case of Gaussian number field

In [9], Bruggeman and Motohashi prove the spectral sum formula in the case ofthe Gaussian number field F = Q(i), trivial character χ = 1 and Γ = PSL2(Z[i]).In this case we have

dF = −4, ξ = 0, l, p, q ∈ Z, O∗ = ±1,±i.

Viewing ΓP and ΓN as subgroups of PSL2(Z[i]) we have [ΓP : ΓN ] = 2. There isonly one cusp for Γ in this case, i.e. Cχ = ∞.

It is known (see [11], Proposition 7.6.2) that in this case there are no com-plementary series due to the absence of exceptional eigenvalues of the Laplacian.Hence the sum in the first term in the left side of (11.34) runs only over V ’s thatare isomorphic to a principal series representation of G.

Recalling that the dual lattice to Z[i] is 12Z[i], we return to (5.6) and see that

TV ϕl,q(νV , pV ) =∑

ω∈ 12 Z[i]

ω 6=0

cTV(ω)Jωϕl,q(νV , pV ) =

∑ω∈Z[i]ω 6=0

cTV

(ω2

)Aωϕl,q(νV , pV ),

because of the relation Jω = A2ω. This mean that our Fourier coefficients cTVand

the Fourier coefficients cV in [9], (8.8) satisfy

cTV(ω) = cV (2ω).

11.4 Discussion of the spectral sum formula 111

If tV (ω) ∈ R is the eigenvalue of the Hecke operator Tω given by [9], (8.12), therelation (8.17) in [9] gives

CV (ω, νV , pV ) = πνV cV (1)tV (2ω).

On noting that νV ∈ iR, we have that the first term in the left side of (11.34)equals: ∑

V

|cV (1)|2tV (2ω)tV (2ω′)h(νV , pV ), (11.35)

where V runs over all Hecke invariant right-irreducible cuspidal subspaces ofL2(Γ\G) that intersect the space L2(Γ\G; l, q) non-trivially.

For the Fourier coefficients of the Eisenstein series El,q(ν, p; 1) we have:

B∞,1(ω; ν, p) = (2π|ω|)ν(ω/|ω|)−pD∞,∞1 (ω; ν, p),

withD∞,∞

1 (ω; ν, p) =12

∑c6=0

|c|−2(1+ν) (c/|c|)2p SF (2ω, 0; c),

where SF (ω, ω′; c) is the Kloosterman sum defined in [9], (1.2). Here p must satisfythe condition i2p = 1, i.e. p ∈ 2Z. We note that in this case, I = Z[i]/±1, andhence the sum

∑′

c∈I = 12

∑c6=0. Using the Ramanujan identity for Q(i), see (2.18)

in [9], we get

B∞,1(ω; ν, p) = 2(2π|ω|)ν(ω/|ω|)−p σ−ν(2ω, p/2)ζF (1 + ν, p/2)

,

for ω ∈ 12Z[i]\0. Here ζF (s, p/2) is the Hecke L-function of Q(i) associated with

the Grossencharakter (ω/|ω|)2p, and σν(ω, p) is the divisor sum given by (2.4) in[9].

Since |Λ∞| = 1, the second term in the left side of (11.34) is equal to

12πi

∑p∈2Z

(ωω′

|ωω′|

)p ∫(0)

σν(2ω,−p/2)σν(2ω′,−p/2)|4ωω′|ν |ζF (1 + ν, p/2)|2

h(ν, p) dν. (11.36)

The units in PSL2(Z[i]) form the set O∗ = 1, i, so∑ε∈O∗

δω′,ωε2 χ(ε)ε2ξ = δω,ω′ + δω,−ω′ ,

and therefore the delta term in (11.34) equals

δω,ω′ + δω,−ω′

4π3i

∑p∈Z

∫(0)

h(ν, p)(p2 − ν2)dν. (11.37)


We have K∗ν,p(z) = Kν,p(z), where Kν,p(z) is the Bessel kernel in [9]. This

implies for the Bessel transform

Bh(u) :=1

2πi

∑p∈Z

∫(0)

Kν,p(u)h(ν, p)(p2 − ν2)dν = 4Bh(u),

where Bh(u) is the Bessel transform given by [9], (10.2). The Kloosterman sumsSF (ω, ω′; c) in [9] and S1(ω, ω′; c) are related by

S1(ω, ω′; c) = SF (2ω, 2ω′; c) = SF (2ω′, 2ω; c).

Recall that∑′

c∈I = 12

∑c6=0, so the Kloosterman term in (11.34) is equal to

∑c6=0

SF (2ω, 2ω′; c)|c|2

Bh(

2πc

√4ωω′

). (11.38)

Collecting (11.35)–(11.38), replacing (2ω, 2ω′) by (ω, ω′), and noting that (iv)in Theorem 11.3.3 is an empty condition, we conclude that our Theorem 11.3.3reduces in the case of the Gaussian number field and trivial character to Theorem10.1 in [9].

11.4.2 Remarks concerning general number fields

Comparing our sum formula given in Theorem 11.3.3 with the sum formula forSL2 over a totally real number field given in [6], Theorem 2.7.1, as well as all thenecessary ingredients (Fourier expansion of automorphic forms on the upper half-space, Kloosterman sums, test functions, Bessel transformations etc.), we see thatthey are completely analogous. We will not enter into detailed comparisons, butwe would like to mention that our result might enable one to extend the validityof the sum formula to SL2 over an arbitrary number field.

11.5 Application of the spectral sum formula

We shall use the spectral sum formula, as stated in Theorem 11.3.3 with fixedω = ω′, to obtain a density result for the cuspidal automorphic representations bychoosing a suitable test function depending on some a > 0 and then exploring theasymptotic behavior of all the terms as a ↓ 0. We will show that the delta termproduces the leading contribution.

Let a > 0. We fix a number p ∈ 12Z, integer or half-integer, and choose the

following test function

h(ν, q) :=mp(ν)e−aλ , q = ±p

0 , q 6= ±p (11.39)

11.5 Application of the spectral sum formula 113

where λ = 1−ν2−p2 is the eigenvalue of the real Casimir operator −4(Ω+ +Ω−),and

mp(ν) := (

14 − ν

2)2 if p ∈ 1

2 + Z1 if p ∈ Z

is the factor providing the necessary zeros of the test function in the odd case.With this function and ω = ω′, formula (11.34) becomes∑

V :pV =±p|CV (ω; νV , pV )|2mpV

(νV )e−a(1−ν2V −p

2V ) =

=i

4π

∑κ∈Cχ


∑±p

∫(0)

|Bκ,χ(ω; ν, p)|2mp(ν)e−a(1−ν2−p2)dν

+1

π3i√|dF |

∑±p

∫(0)

mp(ν)e−a(1−ν2−p2)(p2 − ν2)dν

+2|dF |

(iω

|ω|

)2ξ∑′

c∈I

(c

|c|

)−2ξSχ(ω, ω; c)|c|2

Bh(

4π|ω|c

), (11.40)

where

Bh(

4π|ω|c

)=

12πi

∑±p

∫(0)

K∗ν,p

(4π|ω|c

)mp(ν)e−a(1−ν

2−p2)(p2 − ν2)dν. (11.41)

Note that the Eisenstein term in (11.40) appears only if the chosen p satisfies thecondition χ(ε) = ε2p for all units ε ∈ O∗.

We now consider the terms on the right side of (11.40) separately.Delta term. Let us define εp = 2 if p 6= 0, and ε0 = 1. We have

1π3i√|dF |

∑±p

∫(0)

mp(ν)e−a(1−ν2−p2)(p2 − ν2)dν =

=2εp

π3√|dF |

e−a(1−p2)

∫ ∞

0

mp(it)e−at2(p2 + t2)dt. (11.42)

An easy computation gives for the integral

∫ ∞

0

mp(it)e−at2(p2 + t2)dt =

15√π

16 a−7/2 +Op(a−5/2) if p ∈ 12 + Z,

√π

4 a−3/2 +Op(a−1/2) if p ∈ Z.

The index p in the notations Op(a−5/2) and Op(a−1/2), means that the implicit


constants depend on the fixed number p. Hence, as a ↓ 0, the delta term is

1π3i√|dF |

∑±p

∫(0)

mp(ν)e−a(1−ν2−p2)(p2 − ν2)dν =

=

εp

2π5/2√|dF |

a−3/2 +Op(a−1/2) if p ∈ Z,15

4π5/2√|dF |

a−7/2 +Op(a−5/2) if p ∈ 12 + Z.

(11.43)

Kloosterman term. We start with estimation of the Bessel transformation.From (11.4), with α1 = σ ∈ ( 1

2 , 1) and h as in (11.39), we have

Bh(u) =2πp2 (u/|u|)−2p

J∗0,p(u)e−a(1−p2)

+1πi

∑±p

∫(σ)

|u/2|2ν(iu/|u|)−2p−2ξ

sinπ(ν − p)J∗ν,p(u)mp(ν)e−a(1−ν

2−p2)(ν2 − p2)dν,

where the first term comes from the sum of residues at ν = 0, and it is presentonly if the chosen p is a non-zero integer. We have used that for p ∈ Z, the relation(u/|u|)−2p

J∗0,p(u) = (u/|u|)2p J∗0,−p(u) holds.For p ∈ Z\0, estimate (11.5) implies uniformly in p

2πp2 (u/|u|)−2p

J∗0,p(u)e−a(1−p2)

p2 |u|2|p|

(|p|!)2e−a(1−p

2) 6 |u|2|p|e−a(1−p2) p |u|2|p|, as |u| ↓ 0.

Estimate (11.6) implies, as |u| ↓ 0,

1πi

∑±p

∫(σ)

|u/2|2ν(iu/|u|)−2p−2ξ

sinπ(ν − p)J∗ν,p(u)mp(ν)e−a(1−ν

2−p2)(ν2 − p2)dν

∫

(σ)

|u|2σ∣∣ν + |p|∣∣ |ν|−2σ|mp(ν)|

∣∣∣e−a(1−ν2−p2)∣∣∣ |ν2 − p2||dν|. (11.44)

Since σ > 12 , both 2σ and 2σ−1 are positive, and using |ν| > |Re ν| = σ we obtain∣∣ν2 − p2∣∣∣∣ν + |p|

∣∣|ν|2σ 6 |p||ν|−2σ + |ν|−(2σ−1) < |p|σ−2σ + σ−(2σ−1) = σ−2σ(|p|+ σ).

Thus, by writing ν = σ + it, we continue in (11.44)

< 2 |u|2σe−a(1−p2−σ2)σ−2σ(|p|+ σ)

∫ ∞

0

|mp(σ + it)|e−at2dt. (11.45)


If p ∈ Z, then mp(σ + it) = 1, and the last integral in (11.45) is√π

2 a−1/2. Ifp ∈ 1

2 + Z, then∣∣mp(σ + it)

∣∣ = ( 14 − σ

2 + t2)2 + 4σ2t2, and the integral is∫ ∞

0

|mp(σ + it)|e−at2dt =

3√π

8a−5/2

(1 +

1 + 4σ2

3a+

1− 4σ2

12a2

).

Hence, as a ↓ 0,

Bh(u)p,σ

ςp|u|2|p| + |u|2σa−1/2 if p ∈ Z,|u|2σa−5/2 if p ∈ 1

2 + Z.(11.46)

with ςp = 1 if p ∈ Z\0, and ς0 = 0.The estimate (6.12), for ω = ω′ 6= 0 and any c ∈ I\0, yields

Sχ(ω, ω; c)ω,δ |c|1+δ, (11.47)

for each δ > 0. From (11.46) and (11.47), we obtain the following estimate for theKloosterman term:∣∣∣∣∣ 2|dF |

(iω

|ω|

)2ξ∑′

c∈I

(c

|c|

)−2ξSχ(ω, ω; c)|c|2

Bh(

4π|ω|c

)∣∣∣∣∣F,ω,δ

∑′

c∈I|c|δ−1

∣∣∣Bh( 4π|ω|c

)∣∣∣F,ω,δ,σ,p

ςp∑′

c∈I |c|δ−1−2|p| +(∑′

c∈I |c|δ−1−2σ)a−1/2 if p ∈ Z,(∑′

c∈I |c|δ−1−2σ)a−5/2 if p ∈ 1

2 + Z.

If p = 0 or p ∈ 12 + Z, the sum

∑′

c∈I |c|δ−1−2|p| is absent and the choice of δ such

that δ < 2α−1 implies convergence of the sum∑′

c∈I |c|δ−1−2σ. If p ∈ Z\0, thenby choosing δ < min2|p| − 1, 2α − 1, both sums converge. Hence, for any fixedp ∈ 1

2Z, the Kloosterman term, as a ↓ 0, is estimated as follows,

F,ω,δ,σ,p

a−1/2 if p ∈ Z,a−5/2 if p ∈ 1

2 + Z.(11.48)

Eisenstein term. The Eisenstein series Eκl,q(ν, p;χ) for Γ = Γ0(I) with acharacter χ are linear combinations of Eisenstein series for the principal congru-ence subgroup Γ(I) with trivial character. Thus, also the Fourier coefficients ofthe Eisenstein series for Γ are linear combinations of the Fourier coefficients ofEisenstein series for Γ(I). The Fourier coefficients of the Eisenstein series for theprincipal congruence subgroup, on the other hand, may be written as a sum of


quotients of a bounded expressions and certain L-series. A lower bound for thoseL-series on the critical line will hence give an estimate for the Fourier coefficientDκ,∞χ (ω; ν, p). We use the following claim:

For each cusp κ ∈ Cχ, Re ν = 0 and ε > 0, we have

Dκ,∞χ (ω; ν, p) |ω|ε

log7(2 + | Im ν|) + log7 |p| if p 6= 0,

log7(2 + | Im ν|) if p = 0.(11.49)

The implicit constants in the estimates depend on the field F , the ideal I, thechoice of the elements gκ describing cusps. We do not work out all details of theproof here. The reasoning to reach estimate (76) in [7] can be also carried out inthe present situation.

Remark 13. Lower powers of the logarithm can be obtained in a much moredirect way. The technique in Sections 3.10–3.11 in [40] is applicable for all L-series with Euler products (this is the important point). Motohashi has done thecalculations in the case of the Gaussian number field and the ideal I = Z[i]. Hisidea is that an upper bound analogue to the first condition in [40], Theorem 3.10,is a consequence of the functional equation satisfied by the L-series in questionand the Phragmen-Lindelof convexity principle. Then, via Landau’s lemmas onecan obtain a lower bound for the L-series.

Estimate (11.49) implies that

Bκ,χ(ω; ν, p)F,I,ε,κ |ω|ε+Re ν

log7(2 + | Im ν|) + ςp log7 |p|,

where ςp = 1 if p 6= 0, and ς0 = 0. Let C := |Cχ||dF |πi

minκ∈Cχ [Γκ : Γ′κ]|Λκ|

−1. TheEisenstein term in (11.40) is then estimated as follows

14πi

∑κ∈Cχ


∑±p

∫(0)

|Bκ,χ(ω; ν, p)|2mp(ν)e−a(1−ν2−p2)dν

2C∫

(0)

|ω|2(ε+Re ν)

log14(2 + | Im ν|) + ςp (log |p|)14·

·∣∣mp(ν)

∣∣ ∣∣∣e−a(1−ν2−p2)∣∣∣ |dν|

|ω|2εe−a(1−p2)

∫ ∞

0

log14(2 + t) + ςp (log |p|)14

∣∣mp(it)∣∣e−at2dt

ω,p a−1/2

∫ ∞

0

log14(2√a+ t)

∣∣mp(ia−1/2t)∣∣e−t2dt

+ a−1/2| log a|14 + ςp (log |p|)14

∫ ∞

0

∣∣mp(ia−1/2t)∣∣e−t2dt. (11.50)


Density results. Finally, collecting (11.43), (11.48), (11.53), and (11.54) intothe formula (11.40), we obtain∑


(νV )e−a(1−ν2V −p

2V ) = Kp a

−sp + rest(a),

where

Kp =

εp

2π5/2√|dF |

if p ∈ Z15

4π5/2√|dF |

if p ∈ 12 + Z

, sp =

32 , p ∈ Z72 , p ∈ 1

2 + Z,, (11.55)

and

|rest(a)| =

O(| log a|14a−sp+1) if p s.t. χ(ε) = ε2p, ∀ε ∈ O∗

O(a−sp+1) otherwise.(11.56)

Since obviously, both O(| log a|14a−sp+1) and O(a−sp+1) are o(a−sp) as a ↓ 0, thisproves

Proposition 11.5.1. Let p ∈ 12Z and ω ∈ O′\0 be fixed, χ a character of Γ0(I),

and εp = 2 if p ∈ Z\0 and ε0 = 1. Let V run through all the χ-automorphicrepresentations with spectral parameter (νV ,±p), let CV (ω; νV , pV ) be the Fouriercoefficient of order ω in the expansion (10.4) normalized as in (10.5), and let λVbe the eigenvalue of the real Casimir operator −4(Ω+ + Ω−).

The following estimates are true, for each ε > 0, as a ↓ 0:(i) if p ∈ Z, then∑

V :pV =±p|CV (ω; νV , pV )|2e−aλV =

εp

2π5/2√|dF |

a−3/2 +O(a−1/2−ε),

(ii) if p ∈ 12 + Z, then∑

V :pV =±p|CV (ω; νV , pV )|2

(14 − ν

2V

)2e−aλV =

154π5/2

√|dF |

a−7/2 +O(a−5/2−ε).

Applying a Tauberian argument to the result in the proposition above, weobtain the following density result:

Theorem 11.5.2. Let p ∈ 12Z and ω ∈ O′\0 be fixed, χ a character of Γ0(I),

and εp = 2 if p ∈ Z\0 and ε0 = 1. Let V run through all the χ-automorphicrepresentations with spectral parameter (νV ,±p), let CV (ω; νV , pV ) be the Fouriercoefficient of order ω in the expansion (10.4) normalized as in (10.5), and let λVbe the eigenvalue of the real Casimir operator −4(Ω+ + Ω−).

The following estimates, concerning the cuspidal representations V with eigen-value λV not exceeding X, hold:


(i) if p ∈ Z, then∑V :pV =±p

λV 6X

|CV (ω; νV , pV )|2 ∼ 2εp3π3√|dF |

X3/2, as X →∞,

(ii) if p ∈ 12 + Z, then∑

V :pV =±p

λV 6X

|CV (ω; νV , pV )|2(

14 − ν

2V

)2 ∼ 47π3√|dF |

X7/2, as X →∞.

Proof. We set the non-negative function α(t) to be equal to the step-function

α(t) :=∑

V :pV =±p

λV 6t

|CV (ω; νV , pV )|2mpV(νV ), t > 0,

where λV = 1−ν2V −p2. Then the Stieltjes integral f(x) :=

∫∞0e−xtdα(t) becomes

f(x) =∑


(νV )e−xλV ,

which by Proposition 11.5.1 is approximated as follows: f(x) ∼ Kp x−sp as x ↓ 0,

with Kp and sp as in (11.55). The Tauberian result, Theorem 4.3 on page 192 in[45], then gives

α(X) ∼ Kp

Γ(sp + 1)Xsp , as X →∞

which is (i) for p ∈ Z, and (ii) for p ∈ 12 + Z.

It would be nice to get rid of the factor(

14 − ν

2V

)2 in (ii), in order to getasymptotic for the same sum both for integer and half-integer p. For that purposewe shall use Theorem 11.5.2, (ii) and a partial summation formula. We could notfind precisely the needed result in the literature so we give a proof here withoutclaiming any originality.

Let us denote µV := 14 −ν

2V . Since we consider the odd case, i.e. p ∈ 1

2 +Z, weknow that there are no complementary series, and hence νV ∈ iR. This impliesthat µV = 1

4 + |νV |2 > 14 , and λV 6 X is equivalent to µV 6 X + p2 − 3

4 .Let A(t) :=

∑V :pV =±p

µV 6t

|CV (ω; νV , pV )|2µ2V . The result (ii) of Theorem 11.5.2

can be then rewritten as follows

A(Y ) ∼ β Y γ , as Y →∞, (11.57)

where

β := 4

7π3√|dF |

, γ := 72 > 2, and Y := X + p2 − 3

4 . (11.58)


We take g(x) := x−2 in the partial summation formula (A.21) on page 489 in [18]and get

∑V :pV =±p

µV 6Y

|CV (ω; νV , pV )|2 =A(Y )Y 2

+ 2∫ Y

1/4

A(t)t3

dt. (11.59)

From (11.57) we have that for any ε > 0, there is nε > 14 such that for all t > nε

holds

β(1− ε) tγ < A(t) < β(1 + ε) tγ . (11.60)

Since the set νV , and hence also µV , is discrete in R with finite multiplicities,the function A(t) is a step-function and the interval [ 14 , nε) is contained in a finiteunion

⋃k−1i=0 [ti, ti+1) for some k ∈ N, t0 := 1

4 and tk > nε, where A is constant oneach subinterval. If we denote

M := maxi=0,...,k

∣∣∣∣A(ti)βti

γ

∣∣∣∣ and m := mini=0,...,k

∣∣∣∣ A(ti)βti+1

γ

∣∣∣∣ ,then, for all t ∈ [ 14 , nε), we have

βm tγ 6 A(t) 6 βM tγ . (11.61)

From (11.60) and (11.61), for Y > max 2, nε, we get

β

γ − 2

1− ε+ (m− 1 + ε)

(nεY

)γ−2

− m

(4Y )γ−2

6

61

Y γ−2

∫ Y

1/4

A(t)t3

dt 6β

γ − 2

1 + ε+ (M − 1− ε)

(nεY

)γ−2

− M

(4Y )γ−2

.

This means that both lim and lim as Y → ∞ of the quantity Y −γ+2∫ Y1/4

A(t)t3 dt

are equal to the constant βγ−2 . Hence

limY→∞

1Y γ−2

∫ Y

1/4

A(t)t3

dt =β

γ − 2. (11.62)

Thus, (11.59) yields

limY→∞

1Y γ−2

∑V :pV =±p

µV 6Y

|CV (ω; νV , pV )|2 =

= limY→∞

A(Y )Y γ

+ 2 limY→∞

1Y γ−2

∫ Y

1/4

A(t)t3

dt(11.57),(11.62)

= β +2βγ − 2

,


that is, ∑V :pV =±p

µV 6Y

|CV (ω; νV , pV )|2 ∼ βγ

γ − 2Y γ−2, as Y →∞.

Returning to the notations (11.58), this proves

Theorem 11.5.3. Let p ∈ 12 + Z and ω ∈ O′\0 be fixed, and χ a character

of Γ0(I). Let V run through all the χ-automorphic representations with spectralparameter (νV ,±p), let CV (ω; νV , pV ) be the Fourier coefficient of order ω in theexpansion (10.4) normalized as in (10.5), and let λV be the eigenvalue of the realCasimir operator −4(Ω+ + Ω−).

The following estimate, concerning the odd cuspidal representations V witheigenvalue λV not exceeding X, holds:∑

V :pV =±p

λV 6X

|CV (ω; νV , pV )|2 ∼ 43π3√|dF |

X3/2, as X →∞.

Comparing Theorem 11.5.2, (i) with Theorem 11.5.3, we see that the samedensity result holds for all p 6= 0, integer or half-integer.

One of the symmetries of interest for automorphic forms on Γ\G is derivedfrom the Galois action on the group G, g → g. This involution maps Γ ontoitself and induces the involution on L2(Γ\G) given by f∗(g) = f(g). It sends evenfunctions to even functions and odd functions to odd functions, it preserves theproperties of polynomial growth, square-integrability, orthogonality, and it satisfies(Fωf)∗ = Fωf

∗. For all g = na[r]k ∈ G, we have

ϕl,q(ν, p)∗(g) = ϕl,q(ν, p)(na[r]k) = r1+νΦlp,q(k)

= (−1)p+qr1+νΦl−p,−q(k) = (−1)p+qϕl,−q(ν,−p)(g). (11.63)

Each ϕl,q(ν, p) generates the space of automorphic forms on Γ\G of type (l, q) withspectral parameter (ν, p), and therefore the equality (11.63) means that ϕl,q(ν, p)∗

belongs to the space of automorphic forms on Γ\G of type (l,−q) with spectralparameter (ν,−p).

The symmetry in p means that whenever a representation with pV = p 6= 0occurs in the sum

∑V :pV =±p

λV 6X

, a representation with pV = −p occurs also. Having

in mind that we only consider a Fourier coefficients of fixed order ω and the choice(3.21), this implies that the sum over representations with pV = p 6= 0 is one halfof the sum over representations with pV = ±p, provided that ν 6= 0.

An immediate consequence of Theorem 11.5.2, (i) and Theorem 11.5.3, is

Corollary 11.5.4. For each p ∈ 12Z, there is an infinite orthogonal system of

cuspidal automorphic representations in L2(SL2(O)\SL2(C)) with pV = p.

Moreover, all cuspidal automorphic representations in L2(SL2(O)\SL2(C)) withpV = p are nicely spread with the same density for different p’s.

Chapter 12

Bessel inversion

The sum formula (11.34) has the independent test function h on the spec-tral side and therefore it is useful to obtain information concerning spectral data:the spectral parameter (νV , pV ) and the coefficients CV (ω; νV , pV ) of the cuspi-dal representations. The Bessel transform Bh in the sum of Kloosterman sumsKl(ω, ω′;Bh) on the geometric side of the formula depends on the test functionh. If we can treat the function Bh as an independent test function, then the sumformula becomes more useful: it can be used to investigate sums of Kloostermansums Sχ(ω, ω′; c). To do that, we have to invert the Bessel transformation B. Thesum formula with the independent test function on the geometric side is called theKloosterman sum formula.

12.1 Inverse Bessel transformation

For any function f on C∗ with suitable growth behavior near 0 and∞, we havethe transforms:

Jf(ν, p) :=∫

C∗f(u)|u/2|2ν(iu/|u|)−2p+2ξJ∗ν,p(u)d∗u, (12.1)

Kf(ν, p) :=π

2

∫C∗f(u)(u/|u|)4ξK∗

ν,p(u)d∗u, (12.2)

with the measure d∗u = |u|−2d+u on C∗, ξ ∈ 0, 12 as in (4.27), and

J∗ν,p(z) = J∗ν−p(z)J∗ν+p(z),

K∗ν,p(z) =

(iz/|z|)−2ξ

sinπ(ν − p)

∣∣∣z2

∣∣∣−2ν(iz

|z|

)2p

J∗−ν,−p(z)−∣∣∣z2

∣∣∣2ν ( iz|z|)−2p

J∗ν,p(z)

as in (4.58) and (9.26), respectively.

124 Bessel inversion

We call the transformation K the inverse Bessel transformation simply becauseof the order of appearance. We started with the spectral sum formula, where theBessel transformation B appeared, and then by inverting B we shall obtain theKloosterman sum formula, where the inverse Bessel transformation K appears.

Some simple properties of these transforms are

Kf(ν, p) =(−1)2ξπ

2 sinπ(ν − p)

Jf(−ν,−p)− Jf(ν, p)

, (12.3)

Kf(−ν,−p) = Kf(ν, p), (12.4)

Jf(ν, p) = (−1)2ξJf(ξ)(ν,−p), (12.5)

Kf(ν, p) = (−1)2ξKf(ξ)(ν,−p) = (−1)2ξKf(ξ)(−ν, p). (12.6)

Here f(ξ)(u) := (u/|u|)4ξf(u).

Remark 14. In the case p ∈ Z, we have Jf(ν, p) = (−1)pJf(ν, p) and Kf(ν, p) =π2 Kf(ν, p), where Jf(ν, p) and Kf(ν, p) are the functions in [9] given by (11.6) and(11.1), respectively.

For most of this section, we shall consider these transforms for f ∈ C∞c,ev(C∗),the space of smooth, compactly supported, even functions on C∗.

Each function f ∈ C∞c,ev(C∗) has a polar Fourier expansion

f(reiϕ) =∑n∈Z

e2inϕfn(r), fn(r) =12π

∫ 2π

0

f(reiϕ)e−2inϕdϕ.

The Fourier coefficients fn(r) are smooth, compactly supported functions on theinterval (0,∞); their support is contained in a compact set not depending onn. Moreover, a multiple application of integration by parts gives, for any integerA > 0,

fn(r) CAf (1 + |n|)−A, (12.7)

where the constant Cf depends on the compact support of f , and the implicitconstant only on A and f .

For a smooth, compactly supported function f on (0,∞), the Mellin transformMf , see (9.10) and (9.11), is holomorphic on C and satisfies for each a ∈ N:

Mf(s)f,a C|Re s|f

1|(s)a|

.

The constant Cf depends on the compact support of f , and the implicit constantdepends on the support of f and the supremum norm of f and its derivatives upto order a.

12.1 Inverse Bessel transformation 125

The complex Mellin transform of f ∈ C∞c,ev(C∗) is given by

Mcf(ν, q) :=∫

C∗f(u)|u|2ν(u/|u|)−2qd∗u (12.8)

for ν ∈ C and q ∈ Z. We carry out the change of variables u = reiϕ with r ∈ (0,∞),ϕ ∈ [0, 2π), and get

Mcf(ν, q) =∫ ∞

0

∫ 2π

0

f(reiϕ)r2νe−2qiϕr−2r dϕ dr = 2πMfq(2ν).

Again, by repeated integration by parts, we get for integers A,B > 0

Mcf(ν, q) C2|Re ν|+A+Bf (1 + |q|)−A 1

|(2ν)B |. (12.9)

The constant Cf depends on the compact support of f , and the implicit constantdepends on the support of f and the supremum norms of derivatives of f up toorder A+B.

The inversion formula for the complex Mellin transform is

f(u) =1

2π2i

∑q∈Z

∫(σ)

Mcf(ν, q)|u|−2ν(u/|u|)2qdν, (12.10)

and the Parseval-Plancherel formula yields∫C∗f(u)g(u)d∗u =

12π2i

∑q∈Z

∫(σ)

Mcf(ν, q)Mcg(−ν,−q)dν. (12.11)

From the power series expansion of the function J∗ν,p, we express the transformJf(ν, p) as a linear combination of complex Mellin transforms:

Jf(ν, p) =∑m,n>0

bm,n(ν, p)Mcf(ν +m+ n, p− ξ −m+ n), (12.12)

with

bm,n(ν, p) :=(−1)m+n+p−ξ 4−ν−m−n

m!n!Γ(ν − p+m+ 1)Γ(ν + p+ n+ 1). (12.13)

In order to get an estimate for Jf(ν, p), we estimate the series of absolutevalues corresponding to (12.12). We write Re ν = σ, | Im ν| = t, P :=

[|p−ξ|

2

], and

assume that t > 1 and p > ξ. Estimate (12.9) implies∑m,n>0

|bm,n(ν, p)||Mcf(ν +m+ n, p−m+ n)|

C2|σ|f

∑m,n>0

(C2f/4)m+n(1 + |p− ξ −m+ n|)−A|(2(ν +m+ n))B |−1

m!n! |Γ(ν − p+m+ 1)||Γ(ν + p+ n+ 1)|

= S1 + S2, (12.14)


where S1 is the sum over m,n > 0 such that |p − ξ −m + n| > P , and S2 is thesum over m,n > 0 such that |p− ξ −m+ n| 6 P .

For all m,n > 0 we have

|(2(ν +m+ n))B | > |2ν|B > (2| Im ν|)B > (1 + t)B ,

as well as

|Γ(ν − p+m+ 1)| > |Γ(ν − p+ 1)|, |Γ(ν + p+ n+ 1)| > |Γ(ν + p+ 1)|.

If |p− ξ −m+ n| > P , then (1 + |p− ξ −m+ n|)−A < (1 + P )−A (1 + |p|)−A,and hence the first sub-sum in (12.14) is estimated by

S1 C2|σ|f

∑m,n>0

|p−ξ−m+n|>P

(C2f/4)m+n(1 + |p|)−A(1 + t)−B

m!n! |Γ(ν − p+ 1)||Γ(ν + p+ 1)|

f(1 + |p|)−A(1 + t)−B

|Γ(ν − p+ 1)||Γ(ν + p+ 1)|. (12.15)

If |p− ξ −m+ n| 6 P , then trivially (1 + |p− ξ −m+ n|)−A < 1, and the secondsub-sum in (12.14) is

S2 C2|σ|f

∞∑n=0

(C2f/4)n(1 + t)−B

n! |Γ(ν + p+ n+ 1)|

n+p−ξ+P∑m=n+p−ξ−P

(C2f/4)m

m!|Γ(ν − p+m+ 1)|

f

∞∑n=0

(C2f/4)n(1 + t)−B

n!|Γ(ν + p+ n+ 1)|(C2

f/4)n+p−ξ−P

(n+ p− ξ − P )! |Γ(ν + n− ξ − P )|·

·2P∑k=0

(C2f/4)k(n+ p− ξ − P )! |Γ(ν + n− ξ − P )|

(n+ p− ξ − P + k)! |Γ(ν + n− ξ − P + k)|.

To estimate the sum over k, we use that |Γ(ν+n−ξ−P )| 6 |Γ(ν+n−ξ−P +k)|,for all k > 0, and by Stirling’s formula

(n+ p− ξ − P )!(n+ p− ξ − P + k)!

ek(n+ p− ξ − P + 1 + k)−k(

n+ p− ξ − P + 1n+ p− ξ − P + 1 + k

)n+p−ξ−P+k+ 12

.

Since n+p−ξ−P +1 6 n+p−ξ−P +1+k and n+p−ξ−P + 12 > n+P + 1

2 > 0,we have (

n+ p− ξ − P + 1n+ p− ξ − P + 1 + k

)n+p−ξ−P+k+ 12

< 1.


Furthermore, n+ p− ξ − P + 1 + k > n+ P + 1 + k > 1 + k, so

(n+ p− ξ − P + 1 + k)−k 6 (1 + k)−k.

Collecting these gives (n+p−ξ−P )!(n+p−ξ−P+k)!

(e

1+k

)k, which implies

2P∑k=0

(C2f/4)k(n+ p− ξ − P )! |Γ(ν + n− ξ − P )|

(n+ p− ξ − P + k)! |Γ(ν + n− ξ − P + k)|

2P∑k=0

(C2f

4

)k (e

1 + k

)k· 1 1,

as P →∞. Hence,

S2 f

∞∑n=0

(C2f/4)n(1 + t)−B

n!|Γ(ν + p+ n+ 1)|(C2

f/4)n+p−ξ−P

(n+ p− ξ − P )! |Γ(ν + n− ξ − P )|

f(1 + t)−B

|Γ(ν + p+ 1)|(C2

f/4)p−ξ−P

(p− ξ − P )! |Γ(ν − ξ − P )|. (12.16)

From the definition of P we have 0 6 P 6 p−ξ−P , which implies (p−ξ−P )! > P !and |Γ(ν − ξ − P )| > |Γ(ν − p)|. Thus

(C2f/4)p−ξ−P

(p− ξ − P )!|Γ(ν − ξ − P )|

(C2f/4)P

P ! |Γ(ν − p)|

(C2f

4eP

)P|ν − p|

|Γ(ν − p+ 1)|,

where(C2f

4eP

)P (1 + P )−P

1 if P 6 2(

1 + |p|2

)−|p|/2if P > 3

(1 + |p|)−A

for all A ∈ N. This and (12.16) imply that

S2 f(1 + t)1−B(1 + |p|)−A

|Γ(ν + p+ 1)||Γ(ν − p+ 1)|, for each A,B ∈ N. (12.17)

Because of (12.15) and (12.17), the total sum in (12.14) is estimated by

S1 + S2 f(1 + |p|)−A(1 + t)1−B

|Γ(ν + p+ 1)||Γ(ν − p+ 1)|. (12.18)

The restriction p > ξ is not essential, since for p < ξ the estimate (12.15) still


holds, and instead of (12.17), we have

S2 C2|σ|f

∞∑m=0

(C2f/4)m(1 + t)−B

m!|Γ(ν − p+m+ 1)|

m−p+ξ+P∑n=m−p+ξ−P

(C2f/4)n

n!|Γ(ν + p+ n+ 1)|

f

∞∑m=0

(C2f/4)m(1 + t)−B

m!|Γ(ν − p+m+ 1)|(C2

f/4)m−p+ξ−P

(m− p+ ξ − P )! |Γ(ν +m+ ξ − P )|

f(1 + t)−B

|Γ(ν − p+ 1)|(C2

f/4)−p+ξ−P

(−p+ ξ − P )! |Γ(ν + ξ − P )|.

Since in this case 0 6 P 6 −p+ξ−P , we have (−p+ξ−P )! > P ! and |Γ(ν+ξ−P )| >|Γ(ν + p)|, which implies

(1 + t)−B

|Γ(ν − p+ 1)|(C2

f/4)P

P ! |Γ(ν + p)| (1 + |p|)−A(1 + t)1−B

|Γ(ν − p+ 1)||Γ(ν + p+ 1)|,

for each A ∈ N. Hence, (12.18) is still valid.Let us now consider the product |Γ(ν + p + 1)|−1|Γ(ν − p + 1)|−1. First, we

consider the case of small σ. On the strip |Re ν| 6 Σ with some Σ > 0, for all|p| 6 2Σ + 2, we have the estimate

|Γ(ν + p+ 1)|−1|Γ(ν − p+ 1)|−1 (1 + t)−2σ−1eπt. (12.19)

For |p| > 2Σ + 2, we use Stirling’s formula in the following way:Γ(ν + p+ 1)Γ(ν − p+ 1)

−1 =− sinπ(ν − |p|)Γ(|p| − ν)

π Γ(ν + |p|+ 1)

∣∣|p| − ν∣∣|p|−σ− 1

2 e−t arctant

|p|−σ∣∣|p|+ ν + 1∣∣|p|+σ+ 1

2 e−t arctant

|p|+σ+1

eπt.

Since∣∣∣ |p|−ν|p|+ν+1

∣∣∣2 =∣∣∣1− (2σ+1)(2|p|+1)

(|p|+σ+1)2+t2

∣∣∣ 6 1 + CΣ|p| for some CΣ > 0, we get∣∣∣∣ |p| − ν|p|+ ν + 1

∣∣∣∣|p| Σ 1, for each |p| > 2Σ + 2.

By the mean value theorem, we know that

arctan t|p|+σ+1 − arctan t

|p|−σ =−t(2σ + 1)

(|p| − σ)(|p|+ σ + 1)· 11 + (t/(|p|+ ζ))2

6 (2Σ + 1)t

(|p|+ ζ)2 + t2· (|p|+ ζ)2

(|p| − σ)(|p|+ σ + 1)

6 (2Σ + 1)t

(|p| − Σ)2 + t2(|p|+ Σ + 1)2

(|p| − Σ)(|p| − Σ + 1)< DΣ

t

(|p| − Σ)2 + t2,


for some ζ ∈ [−Σ,Σ + 1] and some DΣ > 0. Thus, in the expression∣∣∣∣ |p| − ν|p|+ ν + 1

∣∣∣∣|p| |(|p| − ν)(|p|+ ν + 1)|−σ− 12 eDΣ

t2

(|p|−Σ)2+t2 ,

the exponential factor is O(1), while

|(|p| − ν)(|p|+ ν + 1)|−σ− 12 (1 + |p|)−σ− 1

2 (1 + t)−σ−12 .

Hence the product of the Γ-factors, for |p| > 2Σ + 2 and |σ| 6 Σ, is estimated asfollows:

|Γ(ν − |p|+ 1)|−1|Γ(ν + |p|+ 1)|−1 (1 + |p|)−σ− 12 (1 + t)−σ−

12 . (12.20)

Next we consider large σ. For σ > 2|p|+ 1, Stirling’s formula gives

|Γ(ν − p+ 1)|−1|Γ(ν + p+ 1)|−1

e2σ+2etarctant

σ+|p|+1+arctan tσ−|p|+1

|ν + |p|+ 1|σ+|p|+ 12 |ν − |p|+ 1|σ−|p|+ 1

2

e2σ(σ − |p|+ 1)−2σ−1eπt, (12.21)

where we have used that | arctanx| 6 π2 for any x ∈ R.

Collecting the results (12.18)–(12.21), and changing the A and B above, weobtain part of the following statement:

Lemma 12.1.1. Let σ > 0, and let f ∈ C∞c,ev(C∗). For A,B ∈ N, p ∈ 12Z, ξ as

in (4.27), and ν ∈ C, we have the estimates:

Jf(ν, p) f,A,B,σ

C2|Re ν|f (1 + |p|)−A(1 + | Im ν|)−B

(Re ν − |p|+ 1)2 Re ν+1eπ| Im ν| (12.22)

for |Re ν| > 2|p|+ 1,f,A,B,σ (1 + |p|)−A(1 + | Im ν)−Beπ| Im ν| (12.23)

for |Re ν| 6 σ,

Kf(ν, p) f,A,B,σ (1 + |p|)−A(1 + | Im ν|)−B (12.24)for |Re ν| 6 σ.

Proof. The estimates (12.22) and (12.23) are proved by the discussion abovein the case | Im ν| > 1. The statement (12.24) in this case follows immediatelyfrom the property (12.3) of Kf(ν, p).

If we take B = 0 in the computations on page 125, the factor (2(ν +m+ n))Bin the denominator is absent, and the reasoning goes through for | Im ν| 6 1. Theestimate of Kf(ν, p) follows, for ν staying away from the integers in the strip|Re ν| 6 σ. By representing Kf(ν, p) by an integral over a small circle on whichthe estimate holds already, we can extend the estimate of Kf(ν, p) to integers.


12.2 One-sided Bessel inversion

The main result in this section is the following

Theorem 12.2.1. For any function f that is even, smooth and compactly sup-ported on C∗, the transformation

Kf(ν, p) :=π

2

∫C∗f(u)(u/|u|)4ξK∗

ν,p(u)d∗u

is a one-sided inverse of the Bessel transform B defined in (11.1), i.e.

BKf = f. (12.25)

Proof. The theorem is an immediate corollary of Proposition 12.2.2 below.Namely, for any f, g ∈ C∞c,ev(C∗), we have∫

C∗BKf(u) g(u)(u/|u|)4ξd∗u =

1π2i

∑p∈ 1

2 Z

∫(0)

Kf(ν, p)Kg(ν, p)(p2 − ν2)dν.

The change in the order of integration is allowed because of the estimates (9.27)and (12.24). Proposition 12.2.2 implies that∫

C∗

(BKf(u)− f(u)

)g(u)(u/|u|)4ξd∗u = 0, ∀g ∈ C∞c,ev(C∗).

So, it must be BKf(u) = f(u) for all u ∈ C∗, which proves the theorem.

Proposition 12.2.2. For all f, g ∈ C∞c,ev(C∗), the functions Kf and Kg aresquare integrable for the Plancherel measure (p2 − ν2)dν, and the equality∑

p∈ 12 Z

∫(0)

Kf(ν, p)Kg(ν, p)(p2 − ν2)dν = π2i

∫C∗f(u)g(u) (u/|u|)4ξ d∗u

holds.

Proof. We fix f, g ∈ C∞c,evC∗). The estimate (12.24) implies that the left sideof the equality in Proposition 12.2.2 converges absolutely, and therefore we maymove the line of integration to (σ), for a suitably chosen σ 6∈ 1

2Z. Then the leftside of the equality in Proposition 12.2.2 is equal to∑

p∈ 12 Z

∫(σ)

Kf(ν, p)Kg(ν, p)(p2 − ν2)dν =

=π2

4

∑±

∑p∈ 1

2 Z

∫(±σ)

Jf(ν, p)Jg(ν, p)p2 − ν2

sin2 π(ν − p)dν

− π2

4

∑±

∑p∈ 1

2 Z

∫(±σ)

Jf(ν, p)Jg(−ν,−p) p2 − ν2

sin2 π(ν − p)dν. (12.26)

12.2 One-sided Bessel inversion 131

In the second term on the right side of (12.26), we replace Jf(ν, p) and Jg(−ν,−p)by the expansion (12.12) and get:

π2

4

∑±

∑p∈ 1

2 Z

∫(±σ)

Jf(ν, p)Jg(−ν,−p) p2 − ν2

sin2 π(ν − p)dν =

=π2

4

∑±

∑p∈ 1

2 Z

∫(±σ)

b0,0(ν, p)b0,0(−ν,−p)p2 − ν2

sin2 π(ν − p)·

·Mcf(ν, p− ξ)Mcg(−ν,−p− ξ)dν

+π2

4

∑±

∑p∈ 1

2 Z

∑′

m,n,k,l>0

∫(±σ)

bm,n(ν, p)bk,l(−ν,−p)p2 − ν2

sin2 π(ν − p)·

·Mcf(ν +m+ n, p− ξ −m+ n)Mcg(−ν + k + l,−p− ξ − k + l)dν,

where the prime in the∑′

m,n,k,l>0 means that the term with m = n = k = l = 0is to be omitted. The absolute convergence, implied from (12.9), allows us toexchange the order of integration and summation. Let us denote

Bp+(σ; f, g) :=

∫(σ)

Jf(ν, p)Jg(ν, p)p2 − ν2

sin2 π(ν − p)dν (12.27)

and

Bp−(σ; f, g) :=

∑′

m,n,k,l>0

∫(σ)


sin2 π(ν − p)·

·Mcf(ν +m+ n, p− ξ −m+ n)Mcg(−ν + k + l,−p− ξ − k + l)dν. (12.28)

A simple calculation shows that b0,0(ν, p)b0,0(−ν,−p) = − sin2 π(ν−p)π2(p2−ν2) . Therefore

the right hand side of (12.26) further equals

π2

4

∑±

∑p∈ 1

2 Z

Bp+(±σ; f, g)− π2

4

∑±

∑p∈ 1

2 Z

Bp−(±σ; f, g) +

+14

∑±

∑p∈ 1

2 Z

∫(±σ)

Mcf(ξ)(ν, p+ ξ)Mcg(−ν,−p− ξ)dν,

where we have used Mcf(ν, p − ξ) = Mcf(ξ)(ν, p + ξ). The Parseval-Plancherelformula (12.11) finally gives∑

p∈ 12 Z

∫(0)

Kf(ν, p)Kg(ν, p)(p2 − ν2)dν = π2i

∫C∗f(u)g(u)(u/|u|)4ξd∗u

+π2

4

∑±

∑p∈ 1

2 Z

Bp+(±σ; f, g)− π2

4

∑±

∑p∈ 1

2 Z

Bp−(±σ; f, g).


From this we see that in order to prove Proposition 12.2.2, we need to show∑±

∑p∈ 1

2 Z

Bp+(±σ; f, g) =

∑±

∑p∈ 1

2 Z

Bp−(±σ; f, g), (12.29)

for all f, g ∈ C∞c,ev(C∗).Let us first consider B

p+. Estimate (12.22) shows that B

p+(σ; f, g) tends to zero

as σ →∞. Moving off the line of integration to the right leaves us with a sum ofresidues

Bp+(σ; f, g) = −2πi

∑ρ>σ

Resν=ρJf(ν, p)Jg(ν, p)p2 − ν2

sin2 π(ν − p),

where ρ ∈ 12Z such that ρ ≡ p (mod 1). Choosing σ ∈ (0, 1

2 ), the only possibleresidue between −σ and σ is at ν = 0. So, defining a0 = 1 and aρ = 2 for ρ 6= 0,we get

∑±

Bp+(±σ; f, g) = −2πi

∑06ρ≡p (1)

aρResν=ρJf(ν, p)Jg(ν, p)p2 − ν2

sin2 π(ν − p)

=2πi

∑06ρ≡p (1)

aρ ∂ν

((p2 − ν2)Jf(ν, p)Jg(ν, p)

)∣∣∣ν=ρ

.

From the definition of the transformation J we get that

Jf(ν, p)Jg(ν, p) =∫

C∗×C∗f(u)g(w)(uw/|uw|)4ξJν,p(u)Jν,p(w)d∗u d∗w,

where Jν,p(u) := |u/2|2ν(iu/|u|)−2p−2ξJ∗ν,p(u). Note that, for suitably chosenbranches,

Jν,p(u) = (−1)p+ξ(u/|u|)−2ξJν−p(u)Jν+p(u). (12.30)

Therefore we can easily derive the following properties for p, ρ ∈ 12Z such that

ρ ≡ p (mod 1):

J−ρ,−p = Jρ,p, and Jρ,p = Jp,ρ. (12.31)

From the power series expansion (1.25) of the J-Bessel function, we easily getthe following estimate

Jn(u)|u/2||n|

|n|!e|u|

2/4, for any n ∈ Z. (12.32)


From the asymptotic expansion of the logarithmic derivative ψ(z) of Γ(z), forz →∞ and |arg z| < π, given in [32] on page 18, we have for z = n+ 1:

ψ(n+ 1) = log(1 + n)− 12(n+ 1)

+O(n−2), as n→∞.

By termwise differentiation of (1.25), and using the estimate of ψ above, we obtain

∂νJν(u)|ν=n

|u/2|nn! e|u|

2/41 + log(1 + n)

, n > 0,

e|u|2/4|u/2||n||n|! + (|n| − 1)!

∣∣u2

∣∣−|n| , n < 0.(12.33)

The estimates (12.32) and (12.33) together with (12.30) imply for any p, ρ ∈ 12Z

such that ρ ≡ p (mod 1):

Jρ,p(u)∣∣∣u2

∣∣∣|ρ−p|+|ρ+p| e|u|2/2

|ρ− p|!|ρ+ p|!, (12.34)

and

∂ν Jν,p(u)|ν=ρ

∣∣∣u2

∣∣∣|ρ−p|+|ρ+p| e|u|2/2 1 + log(1 + |ρ+ p|) + log(1 + |ρ− p|)|ρ− p|!|ρ+ p|!

(12.35)

+∣∣∣u2

∣∣∣|ρ+p|−|ρ−p| e|u|2/2 (|ρ− p| − 1)!|ρ+ p|!

only if ρ− p 6 −1

+∣∣∣u2

∣∣∣|ρ−p|−|ρ+p| e|u|2/2 (|ρ+ p| − 1)!|ρ− p|!

only if ρ+ p 6 −1.

We note that the last two terms in the estimate (12.35) are quite bigger than thefirst one, and they appear only in the indicated cases.

The estimates (12.34) and (12.35) further imply the following estimate, uniformin (u,w) ∈W , for each compact subset W ⊂ C∗ × C∗:

∂ν

((p2 − ν2)Jν,p(u)Jν,p(w)

)∣∣∣ν=ρ

W

CW(p2 + ρ2 + log(1 + |ρ+ p|) + log(1 + |ρ− p|)

)(|ρ− p|!|ρ+ p|!)2

. (12.36)

So, the sum

Bp+(u,w) :=2πi

∑06ρ≡p (1)

aρ∂ν

((p2 − ν2)Jν,p(u)Jν,p(w)

)∣∣∣ν=ρ

(12.37)


converges absolutely on each W . The expression (12.37) defines a continuousfunction Bp+(u,w) on C∗ × C∗ such that∑

±Bp+(±σ; f, g) =

∫C∗×C∗

f(u)g(w)(uw/|uw|)4ξBp+(u,w)d∗u d∗w. (12.38)

We define a function Bp−(u,w) by

Bp−(u,w) :=2πi

∑06ρ≡p (1)

aρ∂ν

((p2 − ν2)Jν,p(u)J−ν,−p(w)

)∣∣∣ν=ρ

. (12.39)

The estimates (12.34) and (12.35) imply that an estimate as (12.36) holds alsofor ∂ν

((p2 − ν2)Jν,p(u)J−ν,−p(w)

)∣∣∣ν=ρ

. This means that the sum in (12.39) con-

verges absolutely on each compact subset W ⊂ C∗ ×C∗, and defines a continuousfunction Bp−(u,w) on C∗ × C∗. We shall prove that this function satisfies∑

±Bp−(±σ; f, g) =

∫C∗×C∗

f(u)g(w)(uw/|uw|)4ξBp−(u,w)d∗u d∗w. (12.40)

Going back to (12.28), we denote by τ±(m,n, k, l) the term of order (m,n, k, l)in B

p−(±σ; f, g). It is an integral over the line Re ν = ±σ. Estimate (12.9) implies

the absolute convergence of the integral. We may deform the path of integrationto C±, going up from −i∞ to −iσ, moving via ±σ to iσ along a half circle centeredat 0, and finally going up to i∞. Let C±(T ) be the part of C± cut off between−iT and iT , with T 6∈ 1

2Z. We then have

τ±(m,n, k, l) =∫

C∗×C∗f(u)g(w)(uw/|uw|)4ξK±(u,w) d∗u d∗w, (12.41)

where

K±(u,w) := |u|2(m+n)(u/|u|)−2(p+ξ−m+n)|w|2(k+l)(w/|w|)−2(−p+ξ−k+l) ·

· limT→∞

∫C±(T )

bm,n(ν, p)bk,l(−ν,−p)∣∣∣ uw

∣∣∣2ν p2 − ν2

sin2 π(ν − p)dν. (12.42)

Assume that |u| 6 |w|. We shift both integration lines C±(T ) to the right intoCT , the right half of a circle centered at 0 with radius T . After picking up theresidues, we have∑

±

∫C±(T )


∣∣∣2ν p2 − ν2


= 2∫CT


∣∣∣2ν p2 − ν2


− 2πi∑

ρ≡p (1)06ρ<T

aρResν=ρ

bm,n(ν, p)bk,l(−ν,−p)

∣∣∣ uw

∣∣∣2ν p2 − ν2

sin2 π(ν − p)

,


with a0 = 1 and aρ = 2 for ρ ∈ 12Z \ 0. We simplify the expression inside the

integral over CT :


sin2 π(ν − p)= − 1

π2

(−1)m+n+k+l4−(m+n+k+l)

m!n! k! l!λ(ν, p;m,n, k, l)

where λ(ν, p;m,n, k, l) := (ν− p+ 1)m(ν+ p+ 1)n(−ν+ p+ 1)k(−ν− p+ 1)l, andget

2∫CT


∣∣∣2ν p2 − ν2


= − 2π2

(−1)m+n+k+l4−(m+n+k+l)

m!n! k! l!

∫CP,T

|u/w|2ν

λ(ν, p;m,n, k, l)dν. (12.43)

Since (m,n, k, l) 6= (0, 0, 0, 0), we have λ(ν, p;m,n, k, l) T and the contributionof the integral in (12.43) is

∫ π/2

0

|u/w|2T cosϕ

TTdϕ min

1,

1T | log |u/w||

.

Integrating this bound over any compact region in C∗ × C∗ where the condition|u/w| 6 1 is satisfied leads to a bound O(T−1 log T ), which is o(1) as T → ∞.Hence∫

|u|6|w|f(u)g(w)(uw/|uw|)4ξ

∑±K±(u,w) d∗u d∗w =

=∫|u|6|w|

f(u)g(w)(uw/|uw|)4ξ(− 2πi |u|2(m+n)(u/|u|)−2(p+ξ−m+n) ·

· |w|2(k+l)(w/|w|)−2(−p+ξ−k+l)∑

06ρ≡p (1)

aρResν=ρ

bm,n(ν, p) ·

· bk,l(−ν,−p)∣∣∣ uw

∣∣∣2ν p2 − ν2

sin2 π(ν − p)

)d∗u d∗w. (12.44)

Summing in (12.44) over all m,n, k, l > 0 such that (m,n, k, l) 6= (0, 0, 0, 0) gives∑′

m,n,k,l>0

∫|u|6|w|

f(u)g(w)(uw/|uw|)4ξ∑±K±(u,w) d∗u d∗w =

=∫|u|6|w|

f(u)g(w) (uw/|uw|)4ξ(− 2πi

∑06ρ≡p (1)

aρResν=ρ

∑′

m,n,k,l>0

|u|2(m+n)(u/|u|)−2(p+ξ−m+n)|w|2(k+l)(w/|w|)−2(−p+ξ−k+l) ·

· bm,n(ν, p)bk,l(−ν,−p)∣∣∣ uw

∣∣∣2ν p2 − ν2

sin2 π(ν − p)

)d∗u d∗w.


The formula for residues of a function at a double pole gives further

=∫|u|6|w|

f(u)g(w)(uw/|uw|)4ξ(

2πi

∑06ρ≡p (1)

aρ∂ν

((p2 − ν2) ·

·∑′

m,n,k,l>0

|u|2(m+n)(u/|u|)−2(p+ξ−m+n)|w|2(k+l)(w/|w|)−2(−p+ξ−k+l) ·

· bm,n(ν, p)bk,l(−ν,−p) |u/w|2ν)∣∣∣ν=ρ

)d∗u d∗w.

Inserting the defining expressions for bm,n(ν, p) and bk,l(−ν,−p), see (12.13), andsome rearrangement gives

=∫|u|6|w|

f(u)g(w)(uw/|uw|)4ξ 2πi

∑06ρ≡p (1)

aρ ·

· ∂ν(

(p2 − ν2)Jν,p(u)J−ν−p(w)− (p2 − ν2)T0

)∣∣∣∣ν=ρ

d∗u d∗w,

where

T0 =(−1)2ξ|u/w|2ν(u/|u|)−2p−2ξ(w/|w|)2p−2ξ

Γ(ν − p+ 1)Γ(ν + p+ 1)Γ(−ν + p+ 1)Γ(−ν − p+ 1).

Simplification in the expression for T0 gives

−(p2 − ν2)T0 = π−2 sin2 π(ν − p)|u/w|2ν(u/|u|)−2p−2ξ(w/|w|)2p−2ξ,

and thus ∂ν(− (p2 − ν2)T0

)∣∣ν=ρ

= 0. Therefore

∑′

m,n,k,l>0

∫|u|6|w|


=∫|u|6|w|

f(u)g(w)(uw/|uw|)4ξ ·

· 2πi

∑06ρ≡p (1)

aρ∂ν

((p2 − ν2)Jν,p(u)J−ν,−p(w)

)∣∣∣ν=ρ

d∗u d∗w. (12.45)

If |u| > |w| we go back to (12.42) and now shift the both integration linesC±(T ) to the left into C ′T , the left half of a circle centered at 0 with radius T .


After picking up the residues, we have∑±

∫C±(T )


∣∣∣2ν p2 − ν2


= 2∫C′T


∣∣∣2ν p2 − ν2


+ 2πi∑ρ∈ 1

2 Z06ρ<T

aρResν=ρ

bm,n(−ν, p)bk,l(ν,−p)

∣∣∣ uw

∣∣∣−2ν p2 − ν2

sin2 π(ν − p)

.

In the same way as before we show that the integral over C ′T is negligible asT →∞, and that∑′

m,n,k,l>0

∫|u|>|w|


=∫|u|>|w|


·(− 2πi

) ∑06ρ≡p (1)

aρ∂ν

((p2 − ν2)J−ν,p(u)Jν,−p(w)

)∣∣∣ν=ρ

d∗u d∗w

=∫|u|>|w|


· 2πi

∑06ρ≡p (1)

aρ∂ν

((p2 − ν2)Jν,p(u)J−ν,−p(w)

)∣∣∣ν=ρ

d∗u d∗w. (12.46)

For the last equality, we have carried out the change of variables ν 7→ −ν and wethen changed the summation over −ρ instead of ρ. Summing the two identities(12.45) and (12.46), we obtain∑′

m,n,k,l>0

∑±τ±(m,n, k, l) =

∫C∗×C∗


· 2πi

∑06ρ≡p (1)

aρ∂ν

((p2 − ν2)Jν,p(u)J−ν,−p(w)

)∣∣∣ν=ρ

d∗u d∗w, (12.47)

which is exactly (12.40).Because of (12.38) and (12.40), proving equality (12.29) is equivalent to proving∫

C∗×C∗f(u)g(w)(uw/|uw|)4ξ ·

· limP→∞

P∑p=−P

(Bp+(u,w)−Bp−(u,w)

)d∗u d∗w = 0, (12.48)


for all f, g ∈ C∞c,ev(C∗).Let us consider the difference Bp+(u,w)−Bp−(u,w). We have

Bp+(u,w)−Bp−(u,w) =

=2πi

∑06ρ≡p (1)

aρ ∂ν

((p2 − ν2)Jν,p(u)

[Jν,p(w)− J−ν,−p(w)

])∣∣∣ν=ρ

= − 2πi

∑06ρ≡p (1)

aρ(p2 − ρ2)Jρ,p(u) ∂ν(J−ν,−p(w)− Jν,p(w)

)∣∣∣ν=ρ

= 2i∑

06ρ≡p (1)

aρ(−1)ρ−p(p2 − ρ2)Jρ,p(u)K∗ρ,p(w) =: Cp(u,w). (12.49)

Here we have used that for ρ, p ∈ 12Z with ρ ≡ p (mod 1),

K∗ρ,p(w) = lim

ν→ρ

J−ν,−p(w)− Jν,p(w)sinπ(ν − p)

=(−1)ρ−p

π∂ν

(J−ν,−p(w)− Jν,p(w)

)∣∣∣ν=ρ

,

as well as the property J−ρ,−p = Jρ,p in (12.31).The expression of K∗

ρ,p(w) as a derivative of the difference J−ν,−p(w)− Jν,p(w)at ν = ρ and the estimate (12.35) imply

K∗ρ,p(w)

∣∣∣w2

∣∣∣|ρ−p|+|ρ+p| e|w|2/2 1 + log(1 + |ρ+ p|) + log(1 + |ρ− p|)|ρ− p|!|ρ+ p|!

+∣∣∣u2

∣∣∣|ρ+p|−|ρ−p| e|u|2/2 (|ρ− p| − 1)!|ρ+ p|!

only if ρ− p 6= 0

+∣∣∣u2

∣∣∣|ρ−p|−|ρ+p| e|u|2/2 (|ρ+ p| − 1)!|ρ− p|!

only if ρ+ p 6= 0. (12.50)

The estimates (12.35) and (12.50) suffice to obtain an estimate that is uniform oncompact subsets in C∗ × C∗, and gives the convergence in ρ of the series definingCp(u,w) in (12.49).

Before we consider the sum of Cp(u,w) over p, we shall prove a property ofthe Bessel kernel K∗

ν,p which we shall use later.

Lemma 12.2.3. For any p, ρ ∈ 12Z with ρ ≡ p (mod 1), we have

K∗ρ,p(w) = K∗

|p|,ρ sign(p)(w) = K∗p,ρ(w).

Proof. The identity follows from the following expression:

K∗ρ,p(w) = − (−1)p+ξ

π(u/|u|)−2ξ

Jp−ρ(u)Yp+ρ(u) + Yp−ρ(u)Jp+ρ(u)

,


where

Yν(u) := 2πeiπνJν(u) cosπν − J−ν(u)

sin 2πνis the Hankel function, which for n ∈ Z is

Yn(u) = limν→n

Jν(u)− (−1)nJ−ν(u)ν − n

= ∂νJν(u)|ν=n − (−1)n∂νJ−ν(u)|ν=n,

and satisfies Y−n = (−1)nYn; see [43], p. 57–63. We now have

P∑p=−P

Cp(u,w) = 2i∑|p|6P

∑ρ≡p (1)

(−1)ρ−p(p2 − ρ2)Jρ,p(u)K∗ρ,p(w)

(change : ρ− p = a, ρ+ p = b)

= −2i∑a,b∈Z

|b−a|62P

(−1)aab J a+b2 , b−a

2(u)K∗

a+b2 , b−a

2(w).

Of course the term with a = 0 or b = 0 can be omitted. The contribution of thepairs (a, b) with |a + b| 6 2P and |b − a| 6 2P is zero because of (12.31) andLemma 12.2.3. Moreover, the symmetry (a, b) 7→ (−a,−b) following from (12.31)and (K1) on p. 80, brings us to the following result:

P∑p=−P

Cp(u,w) = −4i∑

a,b>1|b−a|62P <a+b

(−1)aab J a+b2 , b−a

2(u)K∗

a+b2 , b−a

2(w). (12.51)

We use (12.34) and (12.50) to get an estimate for the latter sum in (12.51). For(u,w) ∈W , with W ⊂ C∗ ×C∗ arbitrary compact subset, the sum is bounded by

P∑p=−P

Cp(u,w)∑

a,b>1|b−a|62P <a+b

ab1 + log(1 + a) + log(1 + b)

(a! b!)2

=∞∑

n=2P+1

∑a>1

|n/2−a|6P

a(n− a) 1 + log(1 + a) + log(1 + n− a)(a! (n− a)!)2

.

Since x! > x for all non-negative x, and log(1 + a) < log(1 + n) as well as log(1 +n− a) < log(1 + n), this is further

∞∑

n=2P+1

1 + log(1 + n)n!

n∑a=1

(na

)=

∞∑n=2P+1

1 + log(1 + n)n!

(2n − 1)

log(2P )Γ(2P + 1)

22P (2P )−1/2 log(2P )( eP

)2P

.


Taking the limit as P →∞ yields

limP→∞

P∑p=−P

Cp(u,w) limP→∞

((2P )−1/2 log(2P )

) ( eP

)2P

= 0, (12.52)

uniformly for (u,w) in any compact subset of C∗×C∗. This implies (12.48), whichfinishes the proof of the proposition.

12.3 Kloosterman sum formula

We are now ready to give another form of the sum formula given in Theorem11.3.3, which has an independent test function in the Kloosterman term. Thefollowing proposition shows that the delta term in the sum formula (11.34) vanishesfor a test function of the form Kf , with f ∈ C∞c,ev(C∗).

Proposition 12.3.1. If f ∈ C∞c,ev(C∗), then

∑p∈ 1

2 Z

∫(0)

Kf(ν, p)(p2 − ν2)dν = 0.

Before we prove the proposition, it is worthwhile to mention that although∫Y

Kf dδω,ω′ = 0, for an even, smooth and compactly supported f , there are testfunctions ϕ ∈ Hσ(a, b) for which the integral

∫Yϕdδω,ω′ is positive. This means

that B : C∞c,ev(C∗) −→ T, with T the class of test functions from Theorem 11.3.3,is not surjective. Therefore the transformation K is only a one-sided inversion ofthe Bessel transformation B.

Proof. The estimates (12.22) and (12.24) allow us to shift the integration lineRe ν = 0 to Re ν = P + 1

4 for any P ∈ 12Z, P > 1, use the property (12.3) of the

transform Kf , and picking up the residues to write:∑p∈ 1

2 Z

∫(0)

Kf(ν, p)(p2 − ν2)dν =

= −2π(−1)2ξ limP→∞

∑|p|6P

∫(P+ 1

4 )

Jf(ν, p)p2 − ν2

sinπ(ν − p)dν

+ 2πi(−1)2ξ∑

p,q∈ 12 Z

ρ≡p (1)

(−1)q−pJf(q, p)(q2 − p2). (12.53)

The sum of residues vanishes because of the relation Jf(q, p) = Jf(p, q), for allp, q ∈ 1

2Z such that p ≡ q (mod 1), which follows from (12.31) and the definition

12.3 Kloosterman sum formula 141

(12.1) of the transform Jf . The estimate (12.22) implies that the latter integralin (12.53) is

C2Pf (1 + |p|)2−A

∫ ∞

0

(1 + t)2−Cdt

for any fixed large A,C > 0. For |p| 6 P , we have (1 + |p|)−1 > (2P )−1, whichimplies that for any large enough A > 0 we have∑

|p|6P

(1 + |p|)2−A ∑|p|6P

(2P )−2P =1 + 2P(2P )2P

(2P )−2P .

Therefore ∑p∈ 1

2 Z

∫(0)

Kf(ν, p)(p2 − ν2)dν limP→∞

(Cf2P

)2P

= 0.

Theorem 12.3.2. (Kloosterman sum formula) Let ω, ω′ ∈ O′\0 and ξ asin (4.27). Let f be an even, smooth, and compactly supported function on C∗ suchthat Kf(ν, p) has at least double zeros at ν = ±1/2 if p ∈ 1

2 + Z. Then,(iω′

|ω′|

)2ξ∑′

c∈I

(c

|c|

)−2ξSχ(ω′, ω; c)|c|2

f(

4πc

√ωω′

)=

=|dF |2

∑V

CV (ω; νV , pV )CV (ω′; νV , pV )Kf(νV , pV )

+1

2πi

∑κ∈Cχ


∑χ

p∈ 12 Z

∫(0)

Bκ,χ(ω; ν, p) ·

·Bκ,χ(ω′; ν, p)Kf(ν, p) dν, (12.54)

where the transformation K is defined in (12.2), V runs over a maximal orthogonalsystem of irreducible cuspidal subspaces of L2(Γ\G;χ), and |Λκ| is the Euclideanarea of a fundamental domain for the lattice Λκ ∈ C corresponding to gκ−1Γ′κgκ.Convergence of the expressions is absolute throughout.

Proof. For f ∈ C∞c,ev(C∗), the inverse Bessel transform Kf satisfies the condi-tions in Theorem 11.3.3. Indeed, property (12.4) is condition (i), estimate (12.24)gives condition (iii), the holomorphy of the Bessel kernel K∗

ν,p in ν (see Lemma9.1.7) implies condition (ii), and condition (iv) is the requirement for f to have atleast double zeros at ν = ± 1

2 in the half-integer case. Therefore, for any non-zeroω, ω′ ∈ O′, we have∫

Y

Kf dσω,ω′ =∫Y

Kf dδω,ω′ + Kl(ω, ω′;BKf).

Proposition 12.3.1 implies that∫Y

Kf dδω,ω′ = 0, and Theorem 12.2.1 then gives∫Y

Kf dσω,ω′ = Kl(ω, ω′; f). This is exactly (12.54).


Remark 15. In the case of Gaussian number field Q(i), trivial character χ, andthe discrete group PSL2(Z[i]), Theorem 12.2.1 reduces to Theorem 11.1 in [9],and the result in Theorem 12.3.2 simplifies exactly to (13.1) in [9]. The class oftest functions in [9], Theorem 13.1, is wider than our class of even, smooth andcompactly supported functions on C∗.

At the very end we note that a possible application of the formula given inTheorem 12.3.2 is deriving an estimate for the sum of Kloosterman sums

∑c∈I\0|N(c)|6X

(c

|c|

)−2ξSχ(ω′, ω; c)|N(c)|

as X →∞.

The extra condition on the test function f concerning the zeros of Kf at firstsight seems insurmountable. However, we may choose the even, smooth, compactlysupported function f arbitrarily on |u| > 4π|ωω′|1/2. The limited time for thisthesis did not allow us to investigate whether this enables us to satisfy the conditionKf( 1

2 , p) = 0 for all p ≡ 12 (mod 1) without changing a given f ∈ C∞c,ev(C∗) on

|u| 6 4π|ωω′|1/2. We therefore stop here, and leave it as an open problem to returnto at a future date.

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Selective Index ofTerminology

Aanti-holomorphic differentiation, 11automorphic form, 21, 22automorphic function, 22automorphic representation, 31automorphic transformation rule, 21auxiliary test function, 73

BBessel function, 5, 37, 48, 132Bessel inversion, 123, 130Bessel kernel, 80, 138Bessel transformation, 97big cell, 64, 85Bruhat decomposition, 61, 63

CCasimir element, 12, 13Casimir operator, 12, 13center, 9central character, 40character, 3, 21, 23classification of representations, 18complementary series, 17complex Mellin inversion, 125complex Mellin transformation, 125congruence subgroup, 3cusp, 4cusp form, 27cusp sector, 5

cuspidal subspace, 69cut off function, 84

Ddelta term in sum formula, 101, 111,

113, 140differentiation, 11discriminant, 2dual lattice, 23

Eeigenfunction, 33Eisenstein series, 29, 30Euler angles, 10exceptional spectral parameter, 86

Ffinite-dimensional representation, 21Fourier coefficient, 51, 53Fourier expansion, 23, 51, 52, 63Fourier term, 23, 27, 52, 65fundamental domain, 10

Ggeneralized Kloosterman sum, 58geometric side of sum formula, 95Goodman-Wallach operator, 42

HHaar measure, 10half-integer, 1

148 SELECTIVE INDEX OF TERMINOLOGY

Hamilton’s quaternions, 1Hankel function, 139holomorphic differentiation, 11

Iinvolution on L2(Γ\G), 121irreducible representation, 13irreducible unitary representation, 18,

28Iwasawa coordinates, 10Iwasawa decomposition, 10

JJacquet integral, 33Jacquet operator, 40

KKilling form, 12Kloosterman sum, 57Kloosterman sum formula, 123, 141Kloosterman term in sum formula,

102

LLaplace-Beltrami operator, 2Lebedev inversion, 76Lebedev transformation, 73Lie algebra

complex, 11real, 11, 12

Lie bracket, 11Lie group, 9

Mmatrix coefficients, 15maximal compact subgroup, 2Mellin inversion, 75Mellin transformation, 75

Oone-parameter subgroup, 14

PParseval-Plancherel formula, 125

Plancherel measure, 130Poincare series, 61preliminary sum formula, 89, 95principal congruence subgroup, 3principal series, 17

RRiemannian metric, 1Riemannian sphere, 4

SSalie-Weil type estimate, 58space of test-functions, 76spectral decomposition, 69spectral parameter, 28, 70spectral side of sum formula, 92spectral sum formula, 109stabilizer, 4subspace of type (l, q), 17sum of Kloosterman sums, 102

Ttest function, 97, 100

Uunitary principal series, 17, 19, 28universal enveloping algebra, 11

center of, 11–13upper half-space model, 1

Wweight, 14weight space, 14Whittaker function, 24, 33

Notation Index

Symbols(·, ·)K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14(·, ·)cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20(·, ·)ps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28‖Φlp,q‖K . . . . . . . . . . . . . . . . . . . . . . . . . .16‖ϕl,q(ν, 0)‖cs . . . . . . . . . . . . . . . . . . . . . 20‖ϕl,q(ν, p)‖ps . . . . . . . . . . . . . . . . . . . . . 20

AA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Apolχ (Υ;σ) . . . . . . . . . . . . . . . . . . . . . . . 22

Apolχ (Υ; l, q) . . . . . . . . . . . . . . . . . . . . . . 22

Aχ(Υ;σ) . . . . . . . . . . . . . . . . . . . . . . . . . 21Aχ(Υ; l, q) . . . . . . . . . . . . . . . . . . . . . . . 22A0χ(Υν,p; l, q) . . . . . . . . . . . . . . . . . . . . . 27

A2χ(Υ; l, q) . . . . . . . . . . . . . . . . . . . . . . . 22

a(m,n;ω) . . . . . . . . . . . . . . . . . . . . 42, 43aκ0 (ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . . 87aκ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27aκω′(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . 87aρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135a[r] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9(α)j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36α(χ, ξ;ω, ω′) . . . . . . . . . . . . . . . . . . . . 101AR(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . 31AR2(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . 31ARpol(ν, p) . . . . . . . . . . . . . . . . . . . . . . .31

BB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12B(η) . . . . . . . . . . . . . . . . . . . . . . . . . 77, 88Bp+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bp− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Bκ,χ(ω;λ, p) . . . . . . . . . . . . . . . . . . . . . 91Bp+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Bp− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97bm,n(ν, p) . . . . . . . . . . . . . . . . . . . . . . . 125β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

CC ′T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136CK(Γ\G;χ). . . . . . . . . . . . . . . . . . . . . .31C∞(N\G,ω) . . . . . . . . . . . . . . . . . . . . . 24C∞(Γ\G;χ) . . . . . . . . . . . . . . . . . . . . . 31C∞c,ev(C∗) . . . . . . . . . . . . . . . . . . . . . . . 124C∞p (K) . . . . . . . . . . . . . . . . . . . . . . . . . . 17Cp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134CV (ω; νV , pV ) . . . . . . . . . . . . . . . . . . . . 90C± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C±(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . 134C(Γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Cχ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5cTV

(ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . 90cT (ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3χω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

DD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Dκ,ηχ (ω; ν, p) . . . . . . . . . . . . . . . . . . . . . 51

Dκ,∞χ (ω; ν, p) . . . . . . . . . . . . . . . . . . . . .91

dF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2da . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

150 NOTATION INDEX

dδω,ω′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 101δκ,η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51dg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73dg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10dk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10dσω,ω′ . . . . . . . . . . . . . . . . . . . . . . . . . . 101d∗u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123d+z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

EEκl,q(ν, p;χ) . . . . . . . . . . . . . . . 30, 51, 91El,q(ν, p;χ) . . . . . . . . . . . . . . . . . . . . . . 29E± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13ε(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76εp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

FF = Q(

√D) . . . . . . . . . . . . . . . . . . . . . . .2

Fω′Pχf . . . . . . . . . . . . . . . . . . . . . . . . . . 65Fω . . . . . . . . . . . . . . . . . . . . . . . . . . . 23, 52F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5, 10F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5FG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Fκ(Y ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .5f∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121f(ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124fa,b(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . 105fn(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . 105

Gg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11gζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4gn(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . 107Γ′∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Γ′ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Γ(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Γ0(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Γ1(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Γ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Γζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

HH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9H(ν, p) . . . . . . . . . . . . . . . . . . . 18, 19, 31H2(ν, p). . . . . . . . . . . . . . . . . . . . . . . . . .18H∞(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . .17H3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1H(−1,−1) . . . . . . . . . . . . . . . . . . . . . . . . 1Hσ(a, b) . . . . . . . . . . . . . . . . . . . . . . . . . 97H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24hκ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25h[u] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Iν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6ι(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

JJ∗ν±p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81J∗ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Jν±p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Jν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123J0ϕl,q(ν, p) . . . . . . . . . . . . . . . . . . . . . . .39Jω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Jωϕl,q(ν, p) . . . . . . . . . . . . . . . . . . . . . . 40J∗ν,p . . . . . . . . . . . . . . . . . . . . . .48, 80, 123Jν,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

KK = SU(2) . . . . . . . . . . . . . . . . . . . . . . . . 2K1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Kν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6K± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Kp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118K . . . . . . . . . . . . . . . . . . . . . . . . . . 123, 141K∗ν,p . . . . . . . . . . . . . . . . . . . . . 80, 97, 123

K∗ν,p

(4πc

√ω1ω2

). . . . . . . . . . . . . . . . . 95

Kl(ω, ω′; ·) . . . . . . . . . . . . . . . . . . . . . . 102

NOTATION INDEX 151

k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13k[α, β] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9κ∗(ω1, ω2, τ) . . . . . . . . . . . . . . . . . . . . . 82

LL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2L2,cont(Γ\G,χ) . . . . . . . . . . . . . . . . . . .69L2,cusp(Γ\G,χ) . . . . . . . . . . . . . . . . . . 69L2(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14L2(K; l, q) . . . . . . . . . . . . . . . . . . . . . . . 17L2(Γ\G,χ). . . . . . . . . . . . . . . . . . . . . . .61L2(Γ\G,χ; l, q) . . . . . . . . . . . . . . . . . . . 70L2p(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

L2even(K) . . . . . . . . . . . . . . . . . . . . . . . . .16

L2odd(K) . . . . . . . . . . . . . . . . . . . . . . . . . 16

Lωl,qf(ν, p). . . . . . . . . . . . . . . . . . . . . . . .73Lω,∗l,q η . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Lωl,qη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13lt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Λ′η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Λκ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5|Λκ| . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 27λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113λ(ν, p;m,n, k, l) . . . . . . . . . . . . . . . . . 135λV . . . . . . . . . . . . . . . . . . . . . . . . . 118, 121λl(ν, p). . . . . . . . . . . . . . . . . . . . . . . . . . .91

MM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Mω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Mωϕl,q(ν, p) . . . . . . . . . . . . . . . . . . . . . 44Mφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Mcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125mp(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . 113µlm(ν, p; r) . . . . . . . . . . . . . . . . . . . . . . . 44

NN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4N∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1n[z] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9(νV , pV ) . . . . . . . . . . . . . . . . . . . . . . . . . 70

OO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2O′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23O∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Ωk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Ω± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

PP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Pψω(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . 86Pψω(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . 86PχMωϕl,q(ν, p) . . . . . . . . . . . . . . . . . . 84PχLωl,qη . . . . . . . . . . . . . . . . . . . . . . . . . . 83Pχψω(ν) . . . . . . . . . . . . . . . . . . . . . . . . . 86PχρMωϕl,q(ν, p) . . . . . . . . . . . . . . . . . 85Pχf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61Pl,q(N\G,ω) . . . . . . . . . . . . . . . . . . . . . 65Pf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84P1(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Φlp,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15ϕl,q(ν, p) . . . . . . . . . . . . . . . . . . . . . . . . . 19ϕl,q(ν, p)∗ . . . . . . . . . . . . . . . . . . . . . . . 121ψω(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86ψω(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

RRκ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 27ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

SS(m,n; c) . . . . . . . . . . . . . . . . . . . . . . . . 57Sχ(ω, ω′; c) . . . . . . . . . . . . . . . . . . . . . . .57Sχ(ϕ,ψ; J) . . . . . . . . . . . . . . . . . . . . . . . 58sp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118σl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14σl = T2l . . . . . . . . . . . . . . . . . . . . . . . . . . 13ςp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115SL2(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1sl2(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . .11su(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

TT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31, 52

152 NOTATION INDEX

TV ϕl,q(νV , pV ) . . . . . . . . . . . . . . . . . . . 90Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Tlσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76τ±(m,n, k, l) . . . . . . . . . . . . . . . . . . . . 134

UU(g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11U(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Υ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Υν,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24ΥνV ,pV

. . . . . . . . . . . . . . . . . . . . . . . . . . .70

VV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Vl,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Vn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11V1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11V2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11vlm(r, ω) . . . . . . . . . . . . . . . . . . . . . . . . . .35v[t] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

WWl,q(Υ, ω) . . . . . . . . . . . . . . . . . . . . . . . 24Wl,q(Υν,p, ω) . . . . . . . . . . . . . . . . . . . . . 27W poll,q (Υν,p, ω) . . . . . . . . . . . . . . . . 26, 27

W(ν, p;ω) . . . . . . . . . . . . . . . . . . . . . . . . 52W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11W1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11W2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 33wlm(ν, p; r) . . . . . . . . . . . . . . . . . . . . . . . 40w[t] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Xξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40, 123ξlp(m, j) . . . . . . . . . . . . . . . . . . . . . . . . . . 39

YY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Y σ . . . . . . . . . . . . . . . . . . . . . . . . . 102, 103Yν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

ZZ(g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Z(k). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13ζF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Samenvatting

Het onderwerp van dit proefschrift is een generalisatie van de klassieke somfor-mule van Bruggeman en Kuznetsov naar de bovenhalfruimte H3. Ten eerste zalik de klassieke somformule en sommige toepassingen ervan kort uitleggen. Daarnavolgt een overzicht van generalisaties van deze formule in verschillende richtingen.Ten slotte zal ik de somformule die het onderwerp van dit proefschrift is beschrijvenen zowel op de overeenkomsten wijzen als de belangrijkste complicaties veroorzaaktdoor de generalisatie aangeven.

De somformule, zie Kuznetsov [25], en Bruggeman [2], die beide auteurs on-afhankelijk van elkaar verkregen hebben, geeft een verband tussen de Fourier-coefficienten van de reeel-analytische spitsvormen op het bovenhalfvlak en de klas-sieke Kloostermansommen.

De verzameling H2 = (x, y) | x ∈ R, y > 0 wordt het bovenhalfvlak genoemd.De groep PSL2(R) werkt op deze ruimte via gebroken lineaire transformaties. Geefmet Γ = PSL2(Z) de modulaire groep aan. Op H2 is de standaard metriek met eenconstante kromming −1 gegeven. De differentiaal operator −∆ = −y2(∂2

x + ∂2y) is

de Laplace operator op H2 die bij deze metriek hoort. Laat ψj(z) ⊂ L2(Γ\H2)een compleet orthonormaal stelsel van Maass-spitsvormen met spectrale parameterνj ∈ i(0,∞) zijn. Dit zijn eigenfuncties van de Laplace operator die tegelijkertijdeigenfuncties van de Heckeoperatoren zijn, geordend volgens de toenemende ei-genwaarden λj = 1

4 − ν2j . De Fourierontwikkeling van de functies ψj(z) is van de

vorm

ψj(x+ iy) =∑

0 6=n∈Zρj(n)y1/2Kνj

(2π|n|y)e2πinx, (1)

waarbij Kν de K-Besselfunctie is. Met E(ν, z) geef ik de Eisensteinreeks metparameter ν aan. De Fouriercoefficienten van de Eisensteinreeksen hebben eenexpliciete beschrijving in termen van de delersommen σν(n) =

∑d|n d

ν en dezetafunctie van Riemann ζ(ν).

Laat h een even holomorfe functie zijn die gedefinieerd is op de verzamelingν ∈ C : |Re ν| < 1

2 + ε, met ε > 0 zo dat |h(ν)| (1 + |ν|)−2−δe−π| Im ν| voor

een δ > 0 en alle ν met |Re ν| 6 12 + ε. Voor dergelijke testfunctie h geldt de

154 Samenvatting

volgende gelijkheid voor alle gehele getallen n,m > 1 met absolute convergentievan alle sommen en integralen in de verschillende termen:

∞∑j=1

ρj(n)ρj(m)h(νj)

cos(πνj)+

1πi

∫Re ν=0

h(ν)( nm

)−ν σ2ν(n)σ−2ν(m)|ζ(1 + 2ν)|2

dν =

=δn,m i

π2

∫Re ν=0

ν tan(πν)h(ν)dν +∞∑c=1

S(n,m; c)c

ϕ

(4π√nm

c

), (2)

waarbij

ϕ(x) =2πi

∫Re s=0

s J2s(x)cos(πs)

h(s) ds, voor x > 0, (3)

Jν de klassieke J-Besselfunctie is, S(n,m; c) de klassieke Kloostermansommen zijnen δn,m het Kronecker delta symbool is.

De linkerkant van de formule (2) komt uit de spectrale ontbinding van deHilbertruimte L2(SL2(Z)\H2) en wordt daarom de spectrale kant genoemd. Detwee termen corresponderen met het discrete en het continu spectrum van deLaplace operator −∆. De rechterkant van (2) komt van de meetkunde van deruimte

(10∗1

)\SL2(Z) die geınduceerd wordt door de Bruhatdecompositie van de

groep SL2(R), en wordt daarom de meetkundige kant genoemd. De eerste, zo-genaamde delta term, komt uit de representanten

(acbd

)van de nevenklassen in

(

10∗1

)\SL2(Z) met c = 0. Matrices met c 6= 0 die afkomstig zijn uit de grote

cel in de Bruhatdecompositie van SL2(Z) ∩ SL2(R) geven de tweede, zogenaamdeKloosterman term.

Somformules van dit type, zoals (2), kunnen op twee verschillende manierengebruikt worden. Aan de ene kant, in de gegeven vorm (2) met de onafhankelijketestfunctie aan de spectrale kant, is het een middel om resultaten in verband metde spectrale data te krijgen. Bijvoorbeeld, Stelling 4.1 in [2] geeft het volgendedichtheidsresultaat:

∞∑j=1


=δn,mπ2

∣∣∣ nm

∣∣∣1/2 v−1 +O(v−1/2−ε), (4)

als v ↓ 0 en ε > 0.Aan de andere kant, als we weten hoe de Besseltransformatie (3) te inverte-

ren, staat de onafhankelijke testfunctie aan de meetkundige kant. Dan kan desomfomule gebruikt worden om Kloostermansommen te schatten. In [25] geeftKuznetsov een eenzijdig inverse van de Besseltransformatie en bewijst dat voorm,n > 1 en X →∞,

X∑c=1

S(n,m; c)c

n,m X1/6(logX)1/3. (5)

155

Weil’s schatting van Kloostermansommen impliceert een schatting O(X1/2+ε)voor de bovengenoemde som, zie [44], en de hypothese van Linnik voorspelt datde schatting eigelijk O(Xε) is, zie [29].

Generalisaties van de somformule (2) zijn op verschillende manieren verkregen.Zowel Bruggeman als Kuznetsov werken met de volle modulaire groep SL2(Z),gewicht nul en het triviale multiplicator systeem. In [38] heeft Proskurin de aan-pak van Kuznetsov gebruikt om de somformule te generaliseren naar alle coeindigediscrete ondergroepen van SL2(R) met niet-triviaal multiplicator systeem en alge-meen gewicht. Hij werkt met Fouriercoefficienten die positieve orde hebben. DeFouriercoefficienten met algemene orde heeft Bruggeman in [3] behandeld. In te-genstelling tot Kuznetsov [25] en Proskurin [38], heeft de somformule in [3] eenmeer representationeel karakter.

In [35] geven Miatello en Wallach een formule van hetzelfde type voor reelesamenhangende semisimpele Liegroepen met R-rang 1. Daar is het bovenhalfvlakdoor een complete Riemannse niet-compacte symmetrische ruimte van rang eenvervangen en de discrete ondergroep is een groep van isometries met eindig volumevan het quotient. Er worden alleen maar de triviale K-types behandeld. In [42]generaliseren de auteurs de formule van [35] voor producten van groepen met rangeen. De formule wordt nauwkeuriger bepaald als de groep die bekeken wordt eenproduct is van groepen van de vorm SL2(R) of SL2(C). Bruggeman en Miatello,[4], gebruiken de formule in [35] om sommen van gegeneraliseerde Kloosterman-sommen voor deze klasse van groepen te bestuderen. Met een geschikte keuzevan de testfunctie, krijgen ze een schatting van type (5) voor die sommen (zie[4], Hoofdstelling 1 in §4.3). In [5] geven dezelfde auteurs een somformule voorSL2 over een willekeurig getallenlichaam met een beperking tot triviale K-types.Het geval van een totaal reeel getallenlichaam waarin alle K-types in aanmerkingworden genomen, behandelen Bruggeman, Miatello en Pacharoni in [6].

In zijn boek [36] geeft Motohashi een expliciete formule voor het vierde mo-ment van de zetafunctie van Riemann, waarvoor hij de somformule voor SL2(R)gebruikt. Een overeenkomstige redenering leidt tot uitbreiding van deze resultatentot sommige kwadratische getallenlichamen. Bruggeman en Motohashi laten in [8]zien hoe men hetzelfde voorbereidingswerk kan uitvoeren met de groep SL2(C)in plaats van SL2(R). Het uiteindelijke doel is een spectrale ontbinding van hetvierde moment van de zetafunctie van Dedekind te geven. De somformule voorSL2(Z[i])\SL2(C) voor alle K-types en tevens een expliciete formule voor het vier-de moment van de zetafunctie van Dedekind leiden deze auteurs in [9] af, beperkttot even functies.

In dit proefschrift generaliseer ik de somformule gegeven in [9] voor een alge-meen imaginair kwadratisch getallenlichaam F en een willekeurige congruentieon-dergroep Γ = Γ0(I) met I ⊂ O een ideaal in de ring van gehelen van F ongelijk aannul. Ik bekijk χ-automorfe functies met betrekking tot Γ, waarbij χ een karakterop Γ is dat triviaal op Γ1(I) ⊂ Γ is. Ik behandel ook het geval van oneven functies.

156 Samenvatting

In het eerste hoofdstuk beschrijf ik enkele elementaire feiten over de meetkundevan de driedimensionale hyperbolische ruimte H3, transformatiegroepen op dezeruimte en klassieke Besselfuncties. In het tweede hoofdstuk is een klein stuk vande voorstellingstheorie van de Liegroepen SL2(C) en SU(2) beschreven.

In het derde hoofdstuk voer ik de automorfe functies en automorfe voorstellin-gen in. Kies l, q, p ∈ 1

2Z zodanig dat |p|, |q| 6 l en p ≡ q ≡ l (mod 1). Kies verdereen karakter χ op (O/I)∗ dat correspondeert met een karakter op Γ0(I) in de vol-gende manier:

(acbd

)7→ χ(d) en ν ∈ C zodanig dat (ν, p) spectrale parameter is.

De gladde functie f : SL2(C) → C is een χ-automorfe functie met betrekking totΓ0(I) van type (l, q) met spectrale parameter (ν, p) genoemd, als f aan de volgendevoorwaarden voldoet:

(i) f(γg) = χ(d)f(g), voor alle γ =(∗∗∗d

)∈ Γ0(I), g ∈ SL2(C),

(ii) Ωkf = − l2+l2 f, H2f = −iqf ,

(iii) Ω±f = (ν∓p)2−18 f .

Hier Ω± zijn de Casimir elementen van de complexe Lie-algebra g van SL2(C) enΩk is het Casimir element van de complexe Lie-algebra k van SU(2).

Met H(ν, p) duid ik de ruimte van K-eindige vectoren in de hoofdreeks voor-stellingen aan. Elke twijnoperator voor de werking van g vanuit H(ν, p) naar deruimte van K-eindige vectoren in C∞(Γ\SL2(C);χ), gladde χ-automorfe functiesop SL2(C) met betrekking tot Γ, is een automorfe voorstelling voor H(ν, p).

In de beschrijving van de Fourierontwikkeling van de automorfe functies (voor-stellingen) spelen de operatoren van Jacquet en Goodman-Wallach een belangrijkerol. Ze worden apart behandeld in het vierde hoofdstuk.

Een gedetailleerde beschrijving van Fouriercoefficienten van de Eisensteinreek-sen en de spitsvormen geef ik in het vijfde hoofdstuk. De gelijkheid (5.6) laat ziendat de Fouriercoefficienten van een cuspidale automorfe voorstelling V met spec-trale parameter (νV , pV ) tegelijkertijd de Fouriercoefficienten van de spitsvormenmet dezelfde spectrale parameter en type (l, q), voor alle l > |pV |, |q| 6 l zijn.

Het zesde, het zevende en het achtste hoofdstuk bevatten bekende resultatenover Kloostermansommen, Poincarereeksen en spectrale ontbinding van de Hilbert-ruimte L2(Γ\SL2(C)). Ze worden gegeven in het meest geschikte vorm voor mijndoeleinden.

In het negende hoofdstuk definieer ik de Lebedevtransformatie, geef ik de een-zijdige inverse op een bepaalde klasse van testfuncties en beschrijf ik enige ei-genschapen van beide. De inverse Lebedevtransformatie zal een bouwsteen zijnvoor de constructie van de speciale Poincarereeksen die gebruikt zullen worden omde somformule af te leiden. Mijn Lebedevtransformatie is een soort uitbreidingvan de klassieke Lebedevtransformatie f 7→

∫∞0f(r)Kν(r)drr die een belangrijke

rol speelt in de theorie van somformules voor rationale Kloostermansommen. Een

157

van de grootste problemen in dit proefschrift was de kwadratisch integreerbaarheiden begrensdheid van de gekozen Poincarereeksen te bewijzen. Het bewijs steuntaanzienlijk zowel op de resultaten van Miatello en Wallach in [34] als op Lemma5.2.1.

De afleiding van de voorlopige somformule behandel ik in het tiende hoofd-stuk. Het bestaat uit de berekening van hetzelfde inproduct van twee specialePoincarereeksen op twee verschillende manieren: spectrale beschrijving waarin deFouriercoefficienten CV (ω; νV , pV ) van de cuspidale automorfe voorstellingen ver-schijnen en meetkundige beschrijving waarin de sommen van Kloostermansommenverschijnen. Dit is ook de methode die Bruggeman [2], Kuznetsov [25] en Proskurin[38] gebruiken.

Het belangrijkste resultaat in dit proefschrift is de spectrale somformule, zieHoofdstelling 11.3.3 in het elfde hoofdstuk. Deze wordt verkregen door de klassevan testfuncties in de voorlopige versie, Stelling 10.3.1, uit te breiden. De uitbrei-dingsmethode beschrijf ik in §11.2. Deze is analoog aan de methode van Miatelloen Wallach in [35].

In §11.5 pas ik de somformule (11.34) toe om gewogen dichtheidsresultaten tekrijgen voor cuspidale automorfe voorstellingen in L2(SL2(O)\SL2(C)) met eigen-waarde λV 6 X voor een voorgeschreven spectrale parameter pV . Namelijk, voorwillekeurige ω ∈ O′\0, p ∈ 1

2Z en X →∞ geldt∑V :pV =±p

λV 6X

|CV (ω; νV , pV )|2 ∼ 2εp3π3√|dF |

X3/2, (6)

waarbij ε0 = 1 en εp = 2 als p 6= 0.In het laatste, twaalfde hoofdstuk, in Hoofdstelling 12.2.1, geef ik een eenzijdige

inverse van de Besseltransformatie (11.1). Dit maakt het mogelijk om de somfor-mule in de omgekeerde richting te schrijven, zie Hoofdstelling 12.3.2. De explicietetestfunctie is hier aan de meetkundige kant van de formule waar de Kloosterman-sommen verschijnen en daarom is deze vorm van de somformule ook bekend alsKloosterman somformule. Toepassingen van de formule (12.54) om schattingenvoor sommen van de vorm∑′

c∈I(c/|c|)−2ξ Sχ(ω, ω′; c)

|c|2, (7)

waarbij ξ = 0 als p geheel is en ξ = 12 als p ∈ 1

2 + Z te krijgen zijn allicht mogelijk.

Rezime

Celta na ovaa disertacija e da ustanovam analog na klasiqnata for-mula za sumiraǌe na Brugeman i Kuznecov za gorniot poluprostor H3.Najprvo ke dadam kuso objasnuvaǌe na klasiqnata formula za sumiraǌei nekoi nejzini primeni. Potoa sledi pregled na obopxtuvaǌa na ovaaformula vo razliqni nasoki. Na krajot ke ja opixam vo kratki crtiformulata za sumiraǌe koja e predmet na ovoj trud, naglasuvajki gipritoa analogiite so klasiqnata formula kako i osnovnite komlikaciikoi proizleguvaat od obopxtuvaǌeto.

Formulata za sumiraǌe, vidi Kuznecov [25] i Brugeman [2], koja enezavisno izvedena od dvajcata avtori, dava vrska pomegu Furievitekoeficienti na kaspidalnite realno-analitiqki modularni formi nagornata poluramnina i klasiqnite sumi na Klosterman.

Mnoestvoto H2 = (x, y) | x ∈ R, y > 0 se narekuva gorna poluram-nina. Grupata PSL2(R) = SL2(R)/±1 dejstvuva na ovoj prostor prekudrobno-racionalnite linearni transformacii. Neka Γ = PSL2(Z) e mo-dularnata grupa. Na prostorot H2 e dadena standardnata metrika sokonstantna krivina −1, taka nareqena hiperboliqna metrika. Laplaso-viot operator na H2 koj i soodvetstvuva na ovaa metrika e oznaqen so−∆ = −y2(∂2

x + ∂2y). Neka ψj(z) ⊂ L2(Γ\H2) e kompleten ortonormalen

sistem od kaspidalni Mas-formi so spektralen parametar νj ∈ i(0,∞).Funkciite ψj se sopstveni funkcii na Laplasoviot operator koi se is-tovremeno i sopstveni funkcii na Hekeovite operatori, indeksiranispored indeksite na svoite rasteqki sopstveni vrednosti λj = 1

4 − ν2j .

Furieviot razvoj na ψj(z) e daden so

ψj(x+ iy) =∑

0 6=n∈Zρj(n)y1/2Kνj (2π|n|y)e2πinx, (1)

kade xto Kν e K-Beselovata funkcija. Neka so E(ν, z) e oznaqen Ajzen-xtajnoviot red so parametar ν. Furievite koeficienti na Ajzenxtajno-vite redovi se eksplicitno opixani so izrazi vo koi figuriraat sumiteod deliteli σν(n) =

∑d|n d

ν kako i Rimanovata zeta-funkcija ζ(ν).

160 Rezime

Neka h e parna holomorfna funkcija definirana na mnoestvotoν ∈ C : |Re ν| < 1

2 + ε, so ε > 0 takvo xto |h(ν)| (1 + |ν|)−2−δe−π| Im ν|

za nekoe δ > 0 i site ν so |Re ν| 6 12 + ε. Za vakva test-funkcija h

vai slednava formula za sumiraǌe, za site celi broevi n,m > 1, prixto konvergencijata na site sumi i integrali vo pooddelnite qlenovie absolutna:

∞∑j=1

ρj(n)ρj(m)h(νj)

cos(πνj)+

1πi

∫Re ν=0

h(ν)( nm

)−ν σ2ν(n)σ−2ν(m)|ζ(1 + 2ν)|2

dν =

=δn,m i

π2

∫Re ν=0

ν tan(πν)h(ν)dν +∞∑c=1

S(n,m; c)c

ϕ

(4π√nm

c

). (2)

Tuka

ϕ(x) =2πi

∫Re s=0

s J2s(x)cos(πs)

h(s) ds, za x > 0, (3)

Jν e klasiqnata Beselova funkcija, S(n,m; c) e klasiqna suma na Kloster-man i δn,m e Kronekeroviot delta simbol.

Levata strana na formulata (2) proizleguva od spektralnata dekom-pozicija na Hilbertoviot prostor L2(Γ\H2) i zatoa se narekuva spek-tralna strana. Nejziniot prv qlen soodvetstvuva na diskretniot, avtoriot na neprekinatiot del od spektarot na Laplasoviot operator−∆. Desnata strana na ravenstvoto (2) e povrzana so geometrijata naprostorot

(10∗1

)\PSL2(Z) induciran od Bruhatovoto razlouvaǌe na

grupata PSL2(R) i zatoa se narekuva geometriska strana. Prviot, taka

nareqen delta qlen, proizleguva od pretstavnicite(acbd

)na komplek-

site vo (

10∗1

)\PSL2(Z) so c = 0. Matricite so c 6= 0 koi doagaat od

golemata kelija vo Bruhatovoto razlouvaǌe na PSL2(Z) ∩ PSL2(R) godavaat vtoriot, taka nareqen Klostermanov qlen.

Formulite za sumiraǌe od ovoj tip, kako na primer formulata (2),moe da bidat upotrebeni na dva razliqni naqini. Od edna strana vooblikot (2), so nezavisnata test funkcija na spektralnata strana, taa esredstvo za dobivaǌe rezultati vo vrska so spektralnite podatoci. Naprimer, Propozicijata 4.1 od [2] go dava sledniov rezultat za raspre-delba:

∞∑j=1


=δn,mπ2

∣∣∣ nm

∣∣∣1/2 v−1 +O(v−1/2−ε), (4)

koga v ↓ 0 i ε > 0.

161

Od druga strana, ako e poznato kako da se invertira Beselovatatransformacija (3), nezavisnata test funkcija se dobiva na geometris-kata strana i togax formulata za sumiraǌe moe da se koristi za do-bivaǌe ocenki za sumi od Klostermanovi sumi. Vo [25], Kuznecov davaednostrana inverzija na Beselovata transformacija i pokauva deka zam,n > 1 i X →∞,

X∑c=1

S(n,m; c)c

n,m X1/6(logX)1/3. (5)

Granicata na Veil za sumite na Klosterman implicira ocenka O(X1/2+ε)za gore navedenata suma (vidi [44]), dodeka hipotezata na Linik pred-viduva deka ocenkata e vsuxnost O(Xε), vidi [29].

Obopxtuvaǌa na formulata za sumiraǌe (2) se dobieni na razliqninaqini. I Brugeman i Kuznecov ja razgleduvaat potpolnata modularnagrupa SL2(Z), teina nula i trivijalen sistem od multiplikatori. Vo[38], Proskurin go koristi pristapot na Kuznecov obopxtuvajki ja for-mulata za sumiraǌe na site diskretni kofinitni podgrupi na SL2(R),netrivijalen sistem od multiplikatori i opxta teina. Toj raboti sofurievite koeficienti qij reden broj e pozitiven. Brugeman, vo svo-jata kniga [3], gi tretira site furievi koeficienti, no za razlika odKuznecov [25] i Proskurin [38], negovata formula e so poveke reprezen-taciski karakter.

Miatelo i Valah davaat vo [35] formula od istiot tip za realnisvrzani poluednostvani Li-grupi so R-rank 1. Tie ja zamenuvaat gor-nata poluramnina so bilo koj kompleten Rimanov nekompakten simetri-qen prostor od rank 1, diskretnata podgrupa Γ e grupa od izometriiso koneqen volumen na faktor prostorot Γ\H2 i gi razgleduvaat samotrivijalnite K-tipovi. Vo [42], avtorite ja obopxtuvaat formulata od[35] na proizvodi od grupi so realen rank 1. Formulata stanuva mnogupoeksplicitna koga grupata koja se razgleduva e proizvod od grupi odoblikot SL2(R) ili SL2(C). Brugeman i Miatelo vo [4] ja upotrebuvaatformulata za sumiraǌe od [35] za prouquvaǌe na sumi od obopxteniKlostermanovi sumi za ovaa klasa na grupi. Izbirajki pogodna test-funkcija tie izveduvaaat ocenka od tipot (5) za ovie sumi (vidi [4],Teorema 1 vo §4.3). Vo [5], istite avtori davaat formula za sumiraǌeza SL2 nad proizvolno brojno pole, no so ograniquvaǌe na trivijal-nite K-tipovi. Sluqajot nad totalno realno brojno pole zemajki gi vopredvid site K-tipovi e tretiran od Brugeman, Miatelo i Paqaronivo [6].

Motohaxi, vo svojata kniga [36], dava eksplicitna formula za momen-tot od qetvrti stepen na Rimanovata zeta-funkcija koristejki ja formu-lata za sumiraǌe za SL2(R). Analogno razmisluvaǌe vodi kon obopxtu-

162 Rezime

vaǌe na ovie rezultati za nekoi kvadratiqni brojni poliǌa. Brugemani Motohaxi pokauvaat vo [8] kako ednakva podgotvitelna rabota moeda se izvede so grupata SL2(C) namesto SL2(R), so krajna cel da se dadespektralno razlouvaǌe na momentot od qetvrti stepen na Dedekin-dovata zeta-funkcija. Formulata za sumiraǌe za SL2(Z[i])\SL2(C) kojagi vkǉuquva site K-tipovi, kako i eksplicitna formula za momentotod qetvrti stepen na Dedekindovata zeta-funkcija se izvedeni od ovieavtori vo [9]. Tamu e tretiran samo sluqajot na parni funkcii.

Vo ovoj trud ja obopxtuvam formulata za sumiraǌe dadena vo [9]razgleduvajki proizvolno imaginarno kvadratiqno brojno pole F , pro-izvolna kongruenciska podgrupa Γ = Γ0(I) so I ⊂ O nenulti ideal voprstenot od celi broevi na poleto F i χ-avtomorfni funkcii vo odnosna Γ, kade χ e karakter na Γ koj e trivijalen na Γ1(I) ⊂ Γ. Isto takago vkǉuquvam i sluqajot na neparni funkcii voveduvajki taka nareqencentralen karakter.

Vo prvite dve glavi se dadeni nekoi osnovni fakti za geometrijatana trodimenzionalniot hiperboliqen prostor H3, grupata od transfor-macii na ovoj prostor i Beselovite funkcii, kako i mal del od repre-zentaciskata teorija na Li-grupite SL2(C) i SU(2).

Vo tretata glava se vovedeni avtomorfnite funkcii i avtomorfnitereprezentacii koi ke bidat razgleduvani. Neka l, q, p ∈ 1

2Z se takvi xto|p|, |q| 6 l i l ≡ p ≡ q (mod 1). So χ e oznaqen karakter na (O/I)∗ koj

soodvetstvuva na karakter na Γ0(I) na sledniov naqin:(acbd

)7→ χ(d).

Neka ν ∈ C e takov xto (ν, p) e spektralen parametar. Za glatkata funk-cija f : SL2(C)→ C se veli deka e χ-avtomorfna funkcija vo odnos na Γ0(I)od tip (l, q) so spektralen parametar (ν, p) ako gi zadovoluva sledniveuslovi:

(i) f(γg) = χ(d)f(g), za site γ =(∗∗∗d

)∈ Γ0(I), g ∈ SL2(C),

(ii) Ωkf = − l2+l2 f, H2f = −iqf ,

(iii) Ω±f = (ν∓p)2−18 f .

Tuka so Ω± se oznaqeni Kazimirovite elementi na kompleksnata Li-algebra g na SL2(C), dodeka Ωk e Kazimiroviot element na kompleksnataLi-algebra k na grupata SU(2).

So H(ν, p) e oznaqen prostorot od K-koneqni vektori vo glavnotonizovno pretstavuvaǌe na SL2(C). Sekoj linearen operator od prostorotH(ν, p) vo prostorot od K-koneqni elementi vo C∞(Γ\SL2(C);χ), glatkiχ-avtomorfni funkcii na SL2(C) vo odnos na Γ, koj komutira so dejstvotona Li-algebrata g, se narekuva avtomorfna reprezentacija za H(ν, p).

163

Vo eksplicitniot opis na furieviot razvoj na avtomorfnite funk-cii (reprezentacii) centralno mesto zavzemaat operatorite na ake iGudman–Valah koi se oddelno tretirani vo qetvrtata glava.

Podetalen opis na furievite koeficienti na Ajzenxtajnovite redovii kaspidalnite avtomorfni funkcii e daden vo pettata glava. Raven-stvoto (5.6) ukauva na faktot deka furievite koeficienti na kaspi-dalna avtomorfna reprezentacija V so spektralen parametar (νV , pV ) seistovremeno i furievi koeficienti na avtomorfnite funkcii so istiotspektralen parametar i tip (l, q), za site l > |pV |, |q| 6 l.

Xestata, sedmata i osmata glava sodrat veke poznati rezultati vovrska so sumite na Klosterman, Poenkareovite redovi i spektralnatadekompozicija na prostorot L2(Γ\SL2(C)). Tie se dadeni vo oblik koj esoodveten na potrebite na ovoj trud.

Vo devettata glava e definirana Lebedevata transformacija, dadenae nejzina ednostrana inverzija na odredena klasa od test funkcii iopixani se nekoi nivni svojstva. Inverznata Lebedeva transformacijake igra glavna uloga vo konstrukcijata na specijalnite Poenkareoviredovi koi pak podocna ke bidat upotrebeni za izveduvaǌe na formulataza sumiraǌe. Mojata Lebedeva transformacija e eden vid obopxtuvaǌena klasiqnata Lebedeva transformacija f 7→

∫∞0f(r)Kν(r)drr koja e od

ogromno znaqeǌe vo teorijata na formuli za sumiraǌe za racionalnisumi na Klosterman. Eden od najgolemite problemi pri pixuvaǌeto natezava bexe dokazot na kvadratno-integrabilnosta i ograniqenosta naizbranite Poenkareovi redovi. Toj cvrsto se temeli vrz rezultatitena Miatelo i Valah vo [34] kako i vrz Lema 5.2.1.

Izveduvaǌeto na preliminarna formula za sumiraǌe e opixano vodesettata glava. Toa se sostoi od presmetuvaǌe na istiot skalarenproizvod od dva specijalni Poenkareovi redovi na dva razliqni naqini:spektralen opis vo koj se pojavuvaat furievite koeficienti CV (ω; νV , pV )na kaspidalnite avtomorfni reprezentacii i geometriski opis vo koj sepojavuvaat sumi od Klostermanovi sumi. Ova e metodot koj go koristati Brugeman [2], Kuznecov [25] i Proskurin [38].

Glaven rezultat vo ovoj trud e spektralnata formula za sumiraǌekoja e opixana vo Teorema 11.3.3, vo edinaesettata glava. Taa e dobienaso proxiruvaǌe na klasata od test-funkcii vo nejzinata preliminarnaverzija, Propozicija 10.3.1. Metodot na ekstenzija e opixan vo §11.2.Toj e potpolno analogen so metodot na Miatelo i Valah vo [35].

Vo §11.5 ja primenuvam formulata za sumiraǌe (11.34) i dobivamrezultati vo vrska so raspredelbata na kaspidalnite avtomorfni re-prezentacii vo prostorot L2(SL2(O)\SL2(C)) qija sopstvena vrednost λV ,pri daden spektralen parametar pV , ne nadminuva X. Imeno, za proiz-volni ω ∈ O′\0, p ∈ 1

2Z i X →∞ vai

164 Rezime

∑V :pV =±p

λV 6X

|CV (ω; νV , pV )|2 ∼ 2εp3π3√|dF |

X3/2, (6)

kade xto ε0 = 1 i εp = 2 ako p 6= 0.Vo poslednata, dvanaesetta glava, so Teorema 12.2.1 e dadena ednos-

trana inverzija na Beselovata transformacija (11.1). Toa ovozmouvada se dokae formulata za sumiraǌe vo obratna forma kako vo Teorema12.3.2. Eksplicitnata test-funkcija ovoj pat se naoga na geometriskatastrana vo koja figuriraat sumite na Klosterman, poradi xto ovoj ob-lik na formulata za sumiraǌe e uxte poznat pod imeto Klostermanovaformula za sumiraǌe. Primeni na ovaa formula za dobivaǌe ocenki nasumite od Klostermanovi sumi od oblikot∑′

c∈I(c/|c|)−2ξ Sχ(ω, ω′; c)

|c|2, (7)

kade ξ = 0 ako p e cel broj i ξ = 12 ako p ∈ 1

2 + Z, se moni, no ograni-qenosta na raspoloivoto vreme za priprema na ovoj trud ni nalouvada go odloime nivnoto izveduvaǌe za nekoja druga prilika.

Acknowledgments

This thesis is a result of the work that was carried out in the pleasant andstimulating atmosphere at the Mathematical Institute of the Utrecht Universityin The Netherlands.

I express my deepest gratitude to my supervisor Dr. Roelof Bruggeman for hisstimulating advises, patient guidance and constant support and encouragement.Our regular discussions helped me in maintaining my confidence to achieve suchhigh level results in scientific work. Without his contribution this thesis wouldnot have existed. I thank my supervisor Professor Hans Duistermaat for his care-ful reading of my thesis and the pleasant time spent lecturing together at theUniversity College.

I would also like to thank the members of the reading committee, Profes-sor Roberto Miatello, Professor Aleksandar Ivic, Professor Frits Beukers, andDr. Joop Kolk for their valuable comments and suggestions.

Many thanks to Dr. Richard Cushman for his time spent reading this thesisand his comments on my English language as well as Frank van de Wiel whom Ihave bothered with every single computer problem that I had.

Thanks also to my present and former colleagues Jordan, Behruz, Abadi, Clair,Beno, and all the others for making the institute such an enjoyable place to work.

All of my hard work on this thesis would have never been done properly withoutthe love and continuous support and encouragement from my wonderful husbandAleksandar. He knows the best the other side of such a respectful scientific work.

I am thankful to my parents, Martin and Dana, for everything they thoughtme, for their love and support and for always being by my side although far away. Iam so proud of them. And, finally, without my sister Olja, her humor and cheeringme up in some dark moments this period would have been too serious.

∗ ∗ ∗ ∗ ∗

Ovoj trud ne ke bexe ona xto e sega bez ǉubovta, postojanata pod-drxka i bezrezervniot kura na mojot prekrasen soprug Aleksandar.

166 Acknowledgments

Toj najdobro ja znae drugata strana na vakvata visoko poqituvana nauqnarabota. Go sakam i blagodarna sum mu za toa.

Blagodarna sum im i na moite roditeli, Martin i Dana, za se xto menauqija vo ivotot, za nivnata ǉubov i poddrxka i za toa xto sekogaxse so mene duri i koga se daleku. Se gordeam so niv. I koneqno, bezsestra mi Oǉa, nejziniot humor i bodreǌe vo nekoi temni momenti ovojperiod ke bexe premnogu seriozen.

Curriculum Vitae

Hristina Lokvenec-Guleska was born on June 26th, 1974 in Skopje, Macedo-nia. She attended the primary school “Jochan Heinrich Pestalozzi”, as well as theprimary ballet school “Ilija Nikolovski–Luj”, both in Skopje from September 1981till June 1989, and the “Rade Jovcevski Korcagin” high school in Skopje fromSeptember 1989 till June 1993. In September 1993 Hristina started her study intheoretical mathematics at the University “St. Cyril and Methodius” in Skopje. OnMay 22, 1998 she graduated on the Faculty of Natural Sciences and Mathemat-ics, Department of Mathematics, Theoretical Mathematics–Major and obtainedthe title Graduated mathematician. Her final exam was supervised by ProfessorDonco Dimovski and is titled Hyperbolic geometry. During her studies Hristinawas several times as an exchange student and member of the IAESTE studentorganization on practical training at universities around Europe. In September1998 she started Master studies in Mathematics at the University “St. Cyril andMethodius”in Skopje. In the period April, 1999 to August, 1999 she worked asmathematics teacher in the “Rade Jovcevski Korcagin” high school. In September1999 she got accepted for the Master Class in Arithmetic Algebraic Geometry atthe Mathematical Institute of Utrecht University, The Netherlands. She completedit successfully in June 2000, and her finishing test-problem has title Some examplesof modular parameterizations. She married Aleksandar Lokvenec on August 24th,2000 in Skopje. A month later, in September 2000 Hristina started her Ph.D.studies at the Mathematical Institute of Utrecht University as an AiO (assistantin education) under the supervision of Dr. Roelof Bruggeman and Professor HansDuistermaat. During her work, she was a member of the electing committee forthree open post-doc positions at the Utrecht University, participated in the Sum-mer School in Automorphic forms and Shimura varieties at the Fields Institutein Toronto, Canada (June 2003) and attended the International Conference onTopology and Applications in Skopje, Macedonia (September 2004).

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