some applications of classical modular forms to number theory

In chapter 2 we use non-holomorphic Eisenstein series for the Hilbertmodular group to obtain a formula for the relative class number of certain abelianextensions of CM number fields.. In

Riad Mohamad Masri

2005

The Dissertation Committee for Riad Mohamad Masri

Certifies that this is the approved version of the following dissertation:SOME APPLICATIONS OF CLASSICAL MODULAR FORMS

SOME APPLICATIONS OF CLASSICAL MODULAR FORMS

TO NUMBER THEORY

by

Riad Mohamad Masri, B.S.; M.S

DISSERTATIONPresented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirementsfor the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

August 2005

UMI Number: 3204214

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by ProQuest Information and Learning Company

To Bonnie

To begin, I want to thank my advisor, Fernando Rodriguez-Villegas, for helpful cussions and encouragement during the last three years I owe much to my friend andcollaborator Jim Kelliher for his patience while listening to me explain my ideas, andMisha Vishik for his constant encouragement I benefited from the advice and sug-gestions of many mathematicians, including Bill Duke, Solomon Friedberg, FarshidHajir, Gergely Harcos, Angel Kumchev, Jeff Lagarias, David Saltman, John Tate,and Jeff Vaaler This list is by no means complete I want to thank Haskell Rosen-thal, who taught me analysis and was instrumental in my coming to the University

dis-of Texas Most importantly, I want to thank my wife, Bonnie Plott, for the love andfulfillment she has brought to my life over the past year

Part of this research was supported by a Joseph Patrick Brannen Fellowship

in Mathematics

v

SOME APPLICATIONS OF CLASSICAL MODULAR FORMS

TO NUMBER THEORY

Publication No

Riad Mohamad Masri, Ph.D

The University of Texas at Austin, 2005

Supervisor: Fernando Rodriguez-Villegas

In this thesis we use classical modular forms to study several problems innumber theory In chapter 2 we use non-holomorphic Eisenstein series for the Hilbertmodular group to obtain a formula for the relative class number of certain abelianextensions of CM number fields In chapter 3 we compute the scattering determi-nant for the Hilbert modular group, and explain how this can be used to provethat the subspace of cuspidal, square integrable eigenfunctions for the Laplacian onproducts of rank one symmetric spaces is infinite dimensional In chapter 4 we usezeta functions of quadratic forms over number fields to sharpen a certain constantappearing in C L Siegel’s lower bound for the residue of the Dedekind zeta function

at s = 1.

vi

Table of Contents

Chapter 2 Relative class numbers of abelian extensions of CM

2.1 Introduction 3

2.2 Fourier expansion of the non-holomorphic Eisenstein series 5

2.3 Taylor expansion of E(s, z; a, b) at s = 0 13

2.4 Analytic and modular properties of log{Ψ(z)} 15

2.5 CM-points on Hilbert modular varieties 18

2.6 The fundamental identity 20

2.7 Proof of Theorem 2.1.3 22

2.8 Proof of Theorem 2.1.4 23

Chapter 3 The scattering determinant for the Hilbert modular group 25 3.1 Introduction 25

3.1.1 Overview 25

3.1.2 The spectral decomposition of ∆ 26

3.1.3 The dimension of the space of cusp forms 29

3.2 Eisenstein series associated to products of Q-rank one symmetric spaces 35 3.3 The scattering determinant for SL2(OK) 41

3.4 The trace of Φ(s) at s = 12 42

3.5 Zeta functions of quadratic forms 43

3.6 Proof of Theorem 4.1.5 44

3.7 Proof of Theorem 3.3.1 52

3.8 The determinant of P 54

vii

3.9 Proof of Theorem 3.3.3 55

3.10 Proof of Theorem 3.4.1 55

Chapter 4 A lower bound for the residue of the Dedekind zeta func-tion at s = 1 59 4.1 Introduction 59

4.1.1 Overview 59

4.1.2 Zeta functions of quadratic forms 61

4.1.3 Functional equations and residues 62

4.1.4 A theorem of Siegel 63

4.1.5 Convexity Bounds 64

4.1.6 Organization 65

4.2 Proof of Theorem 4.1.5 65

4.3 Proof of Theorem 4.1.6 68

4.4 A Hecke type integral representation 69

4.5 Proof of Theorem 4.1.8 71

4.5.1 Upper bound for h K R K 71

4.5.2 Lower bound for h K R K 73

4.6 Proof of Theorem 4.1.1 75

4.7 Proof of Theorem 4.1.9 78

4.8 Proof of Theorem 4.1.10 81

viii

Chapter 1 Introduction

This thesis consists of three self-contained chapters In chapter 2 we prove a formulafor the relative class number of certain abelian extensions of CM number fields Let

H be the Hilbert class field of an imaginary quadratic extension K of a totally real field number field F over Q We obtain a formula which expresses the relative class number h H /h K in terms of the determinant of a matrix whose entries are logarithms

of ratios of a higher analog of the Dedekind eta function evaluated at CM-points

on a Hilbert modular variety This generalizes work of C L Siegel [Si2] in the

case F = Q The ratios obtained by Siegel are elliptic units in the Hilbert class field of K = Q( √ −D) The proof involves evaluating the leading term at s = 0 of the abelian L–function of a non-trivial character χ of Gal(H/K) This vanishes to order |F : Q| at s = 0 The formula we obtain is similar to that appearing in Stark’s conjecture [St2] in the case F = Q.

In chapter 3 we prove that the scattering determinant for the Hilbert modular

group SL2(OK ) over a number field K of degree r1+ 2r2 is (essentially) a ratio of

Dedekind zeta functions of the Hilbert class field of K This generalizes work of Efrat and Sarnak [ES] in the case K imaginary quadratic of discriminant D 6= 1, 3 Given

the appropriate Weyl’s law, we explain how this formula can be used to prove that

the subspace of L2((H2)r1× (H3)r2/SL2(OK)) consisting of cuspidal eigenfunctionsfor the Laplacian ∆ is infinite dimensional

In chapter 4 we study analytic properties of zeta functions of quadratic

forms over number fields We associate a zeta function Z K (Q, s) to each quadratic form Q in the symmetric space of positive n-forms over a number field K We prove a functional equation for Z K (Q, s), and compute the residue at the simple pole at s = n2 We use the functional equation to obtain a Hecke type integral

representation for Z K (Q, s), completed by the appropriate gamma factors Using

the integral representation, we adapt C L Siegel’s orgininal argument [Si1] to obtain

a sharpening of his lower bound for the residue of the Dedekind zeta function of K

1

at s = 1 Finally, we use the functional equation to obtain Phragmen-Lindel¨off type convexity bounds for Z K (Q, s) on vertical lines.

of Gal(H/K) We combine this result with the Frobenius determinant relation to obtain a formula for the relative class number of the extension H/K This extends work of C L Siegel [Si2] in the case n = 1.

The following notation will remain fixed throughout this chapter For a

number field M , let O M denote the ring of integers, U M the units, U M+ the totally

positive units, cl(M ) the (wide) ideal class group, h M the class number, w M the

number of roots of unity, r(M ) the regulator, and d M the absolute value of the

discriminant Given an integral ideal A in M , define the norm by N M/Q (A) =

|O M : A| When A = (α), the norm is given by the product over the embeddings of

M ,

N M/Q ((α)) =Y

σ

|σ(α)|

Let χ be a non-trivial character of Gal(H/K).

Definition 2.1.1 The L–function of χ is defined by

The L-function of χ can be expressed as

L(H/K, χ, s) = X

C∈cl(K)

χ(C)ζ K (s, C), (2.1)

3

where ζ K (s, C) is the Dedekind zeta function of the ideal class C of K,

ζ K (s, C) = X0

A∈C

N K/Q (A) −s , Re(s) > 1.

We now outline our approach to evaluating the leading term of L(H/K, χ, s)

at s = 0 In section 2.2, we compute the Fourier expansion of a non-holomorphic Eisenstein series E(s, z) associated to F This provides a meromorphic continuation

of E(s, z) to C in the s variable In section 2.3, we use the Fourier expansion to compute the Taylor expansion of E(s, z) at s = 0,

E(s, z) = E n−1 s n−1 + E n (z)s n + O(s n+1 ). (2.2)

The number E n−1 is essentially the regulator of F , and the function E n (z) is a multiple of Ψ(z), where

Ψ : Hn → C

is a modular function analogous to the modulus of the Dedekind eta function Here,

H is the complex upper half plane In section 2.4, we show that log{Ψ(z)} is a

potential function for the Laplace-Beltrami operator, and satisfies a transformation

law with respect to a congruence subgroup Γ < GL2(F ).

Remark 2.1.2 A function similar to Ψ(z) was studied in [A].

Let Φ be a CM-type for K/F In section 2.5, we construct for each C ∈ cl(K)

a CM-point Φ(zC) on a Hilbert modular variety X0 := Hn /Γ0(aC) arising from the

decomposition AC = aCω1+ OF ω2 of a fixed integral ideal AC ∈ C −1 Here, aC is

an integral ideal in F , ω1 ∈ a −1C OK , ω2 ∈ O K , and zC := ω2/ω1 In section 2.6,

we express L(H/K, χ, s) as a linear combination of the functions E(s, Φ(zC)) Insection 2.7, we combine this expression with the Taylor expansion (2.2) to obtainthe evaluation formula

For a vector z ∈ H n , let N (y(z)) denote the product of the imaginary parts

of its components The evaluation formula is given in the following result

Theorem 2.1.3 Let H be the Hilbert class field of an imaginary quadratic extension

K of a totally real number field F of degree n over Q Then L(H/K, χ, s) has a zero

²(C −1) =¡N (y(Φ(zC)))N F/Q(aC)−1¢E n−1

Ψ(Φ(zC)).

Stark in [St2] formulated a conjecture which expressed the derivative of

certain abelian L–functions at s = 0 as a linear combination of logarithms of absolute

values of algebraic numbers Stark proved his conjecture for complex quadraticextensions of Q (see Theorem 2, pg 199) In this case, Theorem 2.1.3 should

be compared with Stark’s result Rubin in [R] formulated a similar conjecture for

abelian L–functions with higher order zeros at s = 0 It is unclear as to whether

Theorem 2.1.3 is related to the formula predicted by Rubin’s conjecture, althoughthis is a question we plan to investigate

Our main result is the following formula for the relative class number of

the extension H/K, which will be obtained by combining Theorem 2.1.3 with the

Frobenius determinant relation

Theorem 2.1.4 Let H be the Hilbert class field of an imaginary quadratic extension

K of a totally real number field F of degree n over Q Then

2.2 Fourier expansion of the non-holomorphic Eisenstein series

Let {σ1, , σ n } denote the n real embeddings of F Suppose that a and b are integral ideals in F , and define

where the sum is over a set of representative pairs (a, b) 6= (0, 0) which are associate mod U F Recall that (a, b) and (a 0 , b 0 ) are associate mod U F if there exists

non-a unit ² ∈ U F such that a = ²a 0 and b = ²b 0

Theorem 2.2.2 The Eisenstein series E(s, z; a, b) has a meromorphic continuation

to C with a simple pole at s = 1 with residue

f (z) = X

a∈a ∗

h a (y, s)e 2πiT (ax) ,

where the Fourier coefficients are given by the formula

h a (y, s) = 1

vol(P )

Z

P

f (z)e −2πiT (ax) dx.

In the following proposition, we compute the Fourier coefficients h a (y, s) Proposition 2.2.3 With notation as above,

h0(y, s) = √ N (y) 1−2s

d F N F/Q(a)

" √

πΓ¡s −12¢Γ(s)

#n

,

is the K-Bessel function.

Proof Let d(a) be the absolute value of the discriminant of a Then

d(a) = d F N F/Q(a)2,

so that

vol(P ) =pd(a) =pd F N F/Q (a).

Using the definition of f (z) and that P is a fundamental parallelotope for the lattice

|N (1 − ix)| −2s e −2πiT (ayx) dx.

Define the 1-dimensional integral

Thus, to compute h a (y, s), it suffices to compute h(y, s).

When y = 0, we find from [L2], pg 272, that

It follows that the zeroth Fourier coefficient is

Suppose y 6= 0 Since 1 + t2 is even, we can write

µ2

2 + s

¶Γ

µ1

(|σ j (a)| y j)s−1 = N (y) s−1N F/Q ((a)) s−1.

Substituting these relations in (2.3) yields

We will need the following lemma, which can be proved in a manner similar

h a (by, s)e 2πiT (abx) (2.5)

From the formula for h0(y, s) in Proposition 2.2.3,

Then substituting (2.5) in (2.4) yields

E(s, z; a, b) = N (y) s N F/Q(a)−2s ζ F (2s, [a −1])

h a (by, s)e 2πiT (abx) (2.6)

Using the definition of the trace, we find that

σ˜a (y, s)e 2πiT (˜ ax) , (2.7)

where the Fourier coefficients are given by the following sum of divisors:

σ˜a (y, s) = X

˜

a=ab a∈a ∗

From the formula for h a (y, s), a 6= 0, in Proposition 2.2.3, we can express

the Fourier coefficients as

K s−1(2π |σ j (ab)| y j ).

Finally, by substituting (2.7) into (2.6), and using the formula (2.8), weobtain the Fourier expansion

E(s, z; a, b) = N (y) s N F/Q(a)−2s ζ F (2s, [a −1])

+√ N (y) 1−s

d F N F/Q(a)

" √

πΓ¡s − 12¢Γ(s)

σ˜a (y, s)e 2πiT (˜ ax)

converges uniformly on compact subsets of C, and hence defines an entire function

on C Therefore, C(s) is entire on C.

The function ζ F (s, C) has a meromorphic continuation to C with a simple pole at s = 1 Therefore, A(s) and B(s) have meromorphic continuations to C We conclude that E(s, z; a, b) = A + B + C has a meromorphic continuation to C.

We want to determine the poles of E(s, z; a, b) The function B(s) has a pole at s = 1 To compute the residue, recall the Laurent expansion (see [L1], pg.

and Γ(1/2) = √ π, we find that the residue of the pole of B(s) at s = 1 is

+ O(1) = N (y) √ 1/22n−1 r(F )

d F w F N F/Q(a)

1

s − 1 2

2.3 Taylor expansion of E(s, z; a, b) at s = 0

We now use the Fourier expansion (2.9) to compute the first two terms in the Taylor

expansion of E(s, z; a, b) at s = 0,

E(s, z; a, b) = E n−1 s n−1 + E n (z)s n + O(s n+1 ).

We will compute the Taylor expansions of A, B, and C, separately.

First, observe that

N (y) s N F/Q(a)−2s= 1 + log©N (y)N F/Q(a)−2ªs + O(s2),

and, arguing as in section 2.2, we determine that

ζ F (2s, [a −1]) = 2n−1 r(F )

w F s

n−1 + O(s n ).

N (y)N F/Q(a)−2ªs n + O(s n+1 ).

Second, using the expansion

·1

b∈b/U F

e −2πS(aby)

N F/Q ((a)) e

2πiT (abx) s n + O(s n+1 ).

Finally, from the sum A + B + C, we conclude that

E n−1 = −2 n−1 r(F )

w F ,

b∈b/U F

e −2πS(aby)

N F/Q ((a)) e

2πiT (abx)

2.4 Analytic and modular properties of log{Ψ(z)}

Define the Laplace-Beltrami operators

Proof Let σ be an embedding of F , (a, b) ∈ a × b, and z ∈ H It can be shown by

direct computation that

y2∆(y s |σ(a) + σ(b)z| −2s ) = s(s − 1)y s |σ(a) + σ(b)z| −2s

Using the relation

Thus, E(s, z; a, b) is an eigenfunction for the operator D j , with eigenvalue s(s − 1).

From section 2.3, E(s, z; a, b) has the expansion

E(s, z; a, b) = E n−1 s n−1 + E n (z)s n + O(s n+1 ).

Substitute this expansion into the RHS of (2.14), expand, and equate coefficients toobtain the recurrence relation

D j E k = E k−2 − E k−1 , for k = 0, 1, ,

where

E k = 0 for k = −2, −1, , n − 2.

From the definition of E n (z), we see that

log{Ψ(z)} = E n (z) − E n−1log©N (y)N F/Q(a)−2ª. (2.15)

Also, a straightforward computation yields

D jlog©N (y)N F/Q(a)−2ª= −1.

The claim now follows from these two facts

Define the group

and the subgroup of matrices stabilizing (a, b),

Γ(a, b) =©M ∈ GL(2, F ) : det(M ) ∈ U F+, (a, b) · M = (a, b)ª.

We can embed the subgroup

Γ(a, b) ,→ GL2(R)n

Theorem 2.4.2 The function log{Ψ(z)} satisfies the transformation law

log{Ψ(M (z))} = log{Ψ(z)} + log

for each M ∈ Γ(a, b).

Proof From the definition of Γ(a, b), we have the invariance property

E(s, M (z); a, b) = E(s, z; a, b) for all M ∈ Γ(a, b) Then using the Taylor expansion

E(s, z; a, b) = E n−1 s n−1 + E n (z)s n + O(s n+1 ),

we see that E n (M (z)) = E n (z) for all M ∈ Γ(a, b).

Write

E n (z) = log{Ψ(z)} + E n−1log©N (Im(z))N F/Q(a)−2ª, (2.16)

where we have set N (y) = N (Im(z)) Then using (2.16) and the invariance of E n (z) under Γ(a, b), we compute

log{Ψ(M (z))} = E n (M (z)) − E n−1log©N (Im(M (z)))N F/Q(a)−2ª

= E n (z) − E n−1log©N (Im(M (z)))N F/Q(a)−2ª

= log{Ψ(z)} + E n−1log©N (Im(z))N F/Q(a)−2ª

− E n−1log©N (Im(M (z)))N F/Q(a)−2ª

Finally, we obtain the transformation law

log{Ψ(M (z))} = log{Ψ(z)} + log

2.5 CM-points on Hilbert modular varieties

Here, we follow in part [M], section 3.2 Let D K/F be the relative different Since K

is a quadratic extension of F , K = F ( √ α) for some α ∈ F By considering prime

ideal factors, it can be shown that (√ α) D −1 K/F = ˜aOK for some ideal ˜a in F The ideal class [˜a] is independent of the choice of α.

Lemma 2.5.1 (Chevalley [C]) Let A be an integral ideal in K Then the relative norm N K/F (A) lies in the ideal class of the form a[˜a], a being an integral ideal in

F , if and only if there exist ω1 ∈ a −1OK and ω2 ∈ O K such that A = aω1+ OF ω2.

Choose a complete set of representatives {a j } j∈J of ideal classes of F Among the ideal classes {a j [˜a]} j∈J of F , choose the sub-collection {a i [˜a]} i∈I of ideal classes

which contain the relative norm of an ideal in K We may assume that the ideals {a i } i∈I are integral

Let C be an ideal class of K, and let A be an integral ideal in C Then

NK/F (A) ∈ a i [˜a] for some i ∈ I It follows from Lemma 2.5.1 that there is a position A = a i ω1+ OF ω2, where ω1 ∈ a −1 i OK and ω2 ∈ O K Up to multiplication

decom-by a unit in F , we may assume that the imaginary parts of the components of

zC:= ω2/ω1 under a given choice of n real embeddings of F are positive.

Since K is a totally imaginary quadratic extension of F , there are 2n beddings of K, occurring in complex conjugate pairs Let Φ = {σ1, , σ n } be a CM-type for K/F , which by definition is a choice of one embedding for each complex

em-conjugate pair Define the CM-point

Φ(zC) := (σ1(zC), , σ n (zC)) ∈ H n

Let Γ0(ai) := Γ(ai , O F), and define the map

©

C ∈ cl(K) : N K/F(C) = ai[˜a]ª,→ X0(ai) := Hn /Γ0(ai)by

C) (see [M], Lemma 1) Thus, the map is well-defined

Finally, let A = l.c.m.{a i } i∈I Using the covering maps

X0(A) → X0(ai ),

we obtain a map

cl(K) ,→ X0(A)defined by

C 7→ Φ(zC) mod Γ0(A).

2.6 The fundamental identity

Let C ∈ cl(K), and fix an integral ideal AC ∈ C −1 Then, as A runs over all integral ideals in C, the ideal A · AC = (α) runs over all principal ideals (α) with (α) ≡ 0 mod AC It follows that

From section 2.5, there is a decomposition AC= aCω1+OF ω2, ω1 ∈ a −1C OK , ω2 ∈

OK , for some integral ideal aC∈ {a i } Therefore, we can express (2.18) as

We will need the following lemma

Lemma 2.6.1 Let Φ be a CM-type for K/F Then

N K/Q ((a + bzC)) = |N (a + bΦ (zC))|2.

Proof For an embedding σ of K, let ˜ σ denote its restriction to F Then

that the map C 7→ Φ(zC) mod Γ0(aC) is well-defined with the invariance of the

Eisenstein series E(s, z; aC, O F) with respect to the group Γ0(aC)

Let G be a finite abelian group and χ ∈ b G := Hom(G, C × ) Let f be

a complex-valued function on G The following formula is a consequence of the

Frobenius determinant relation (see [L2], pg 283):

Y

χ∈ b G χ6=1

z is in the complex upper half-plane H2, and y(z) = Im(z) Because E(z, s) is

SL2(Z)-automorphic, it is invariant under the lattice at infinity Z corresponding tothe stabilizer Γ∞ The Fourier expansion is of the form

E(z, s) = y s + φ(s)y 1−s+X

n6=0

a(n, s)y1K s−1(2π |n| y)e 2πinx ,

where if ζ ∗ (s) := π − s2Γ(s/2)ζ(s) is the completed Riemann zeta function,

φ(s) = ζ ∗ (2s − 1)

ζ ∗ (2s) . This setup can be generalized as follows Let K be a number field and O K

be its ring of integers The Hilbert modular group SL2(OK) acts on the projectiveline P1(K) by linear fractional transformations, dividing P1(K) into h cusp classes [x1], , [x h ], where h is the order of the (wide) ideal class group cl(O K) To each

cusp x i , one can associate an Eisenstein series E [x i](see section 4.1.3) Because E [x i]

is SL2(OK )-automorphic, it is invariant under the lattice at infinity L jcorresponding

25

to the stabilizer Γx j Thus, E [x i] has a Fourier expansion in each cusp x j, and onecan use the zeroth Fourier coefficients of the Fourier expansions in the various cusps

to form the h × h Eisenstein matrix Φ(s) In this chapter we compute the scattering determinant φ(s) := det(Φ(s)) for SL2(OK) (see Theorem 3.3.3)

We should remark that the matrix Φ(s) was computed for congruence groups of the group P SL2(Z) by Hejhal [He] and Huxley [Hu] The matrix Φ(s) and determinant φ(s) were computed for the Bianchi groups P SL2(OD ), D 6= 1, 3,

sub-by Efrat and Sarnak in [ES] Finally, the matrix Φ(s) was computed for congruence subgroups of the group P SL2(OK ) for K totally real by Efrat in [E].

3.1.2 The spectral decomposition of ∆

Let Hn+1 be hyperbolic n + 1-space and Γ be a discrete subgroup of the orientation

preserving isometries Isom+(Hn+1) such that Hn+1 /Γ is non-compact and finite

volume The scattering determinant is closely related to the problem of existence

of cuspidal eigenfunctions for the Laplacian ∆ on L2(Hn+1 /Γ) In order to explain

this connection, we briefly review some background from the spectral theory ofautomorphic forms, following closely the lecture notes [CS]

We assume here for simplicity that n = 2, noting that all of what follows holds for arbitrary n Accordingly, let H3 be the hyperbolic three-space and ΓD =

P SL2(OD) be the Bianchi group corresponding to the imaginary quadratic field

K = Q( √ −D) Fix an embedding of Q( √ −D) into C Then Γ D is a discretesubgroup of the orientation preserving isometries Isom+(H3) = P SL2(C) such that

H3/Γ D is a non-compact, finite volume arithmetic orbifold

There is an identification of H3 with the quaternionic upper half-plane

Hc = {w = x1+ ix2+ jy : z = x1+ ix2 ∈ C, y > 0}.

The group P SL2(C) acts on Hc by linear fractional transformations Let x1 =

∞, x2, , x h be a complete set of cusps for the fundamental domain F of H3/Γ D

and let Γx i be the stabilizer of the cusp x i Assume that D 6= 1, 3, so that Γ D is

torsion-free One can then choose (not necessarily uniquely) σ i ∈ P SL2(C) such

For a quaternion w ∈ H c, let

The Eisenstein series E i (w, s) satisfies the following properties:

1 E i (w, s) is an eigenfunction for ∆ with eigenvalue s(s − 2).

2 E i (w, s) has a meromorphic continuation to C with a simple pole at s = 2.

3 E i (w, s) is holomorphic on the line s = 1 + it, t ∈ R.

Although the functions E i (w, 1 + it), t ∈ R, are not in L2(H3/Γ D), they

can be used to construct the continuous spectrum for ∆ on L2(H3/Γ D) Assume

that there is one cusp x1 = ∞ (the analysis for the case of h cusps is similar) Let

It can be shown that T extends to an isometry of L2(0, ∞) into L2(H3/Γ D) Let E

denote the image of T in L2(H3/Γ D) Then E is an invariant subspace for ∆, and

the spectrum for ∆ on E is absolutely continuous and equal to (1, ∞).

Let k be a point pair invariant kernel function; thus, k ∈ C ∞

where h(t) := ˆ k(1 + it), ˆ k being the Harish-Chandra-Selberg transform of k It

fol-lows that E (and E⊥ , because L is self-adjoint) is invariant under L It can be shown that the restriction of L to E ⊥ is compact By the spectral theorem for compactself-adjoint operators, E⊥has an (countable) orthonormal basis of eigenfunctions for

∆; that is, there exists a sequence 0 = λ0< λ1 ≤ λ2 ≤ · · · and mutually orthogonal {u j } ⊂ L2(H3/Γ D) such that

∆u j + λ j u j = 0, ||u j ||2= 1, E⊥ = span{u j }.

A major problem of interest is to determine whether or not there exist

in-finitely many distinct λ j

There is a decomposition of E⊥ into the residual and cuspidal spectrum,which we now explain In order to deal with all of the cusps, one forms the vectorEisenstein series

−

→

E (w, s) = (E1(w, s), , E h (w, s)) t Because each Eisenstein series E i (w, s) is Γ D-automorphic, it is invariant under the

lattice at infinity L j corresponding to the stabilizer Γx j The Fourier expansion is

of the form

E i (w (j) , s) = a0(i, y (j) , s) +X

l∈L ∗ j

a(l, s, i, j)y (j) K s−1 (4π |l| y (j) ),

with

a0(i, y (j) , s) = δ ij (y (j))s + φ ij (s)(y (j))2−s

(here L ∗

j is the dual lattice) Form the h×h Eisenstein matrix Φ(s) = (φ ij (s)) Then

it can be shown that the vector Eisenstein series satisfies the functional equation

−

→

E (w, s) = Φ(2 − s) · − → E (w, 2 − s).

The vector Eisenstein series− → E (w, s) has a simple pole at s = 2 whose residue

is an L2-eigenfunction of ∆ with eigenvalue 0 Thus it is harmonic and henceconstant It can be shown that − → E (w, s) has at most finitely many poles in (1, 2],

all of which are simple, and that the residues at these possible simple poles are

L2-eigenfunctions of ∆ Let R ⊂ E ⊥ be the finite dimensional subspace spanned bythe residues of− → E (w, s) in (1, 2].

Finally, given u ∈ C ∞

0 (H3/Γ D), let ˆu i (y, 0) be the zeroth Fourier coefficient

of u in the i-th cusp Let 1 < σ1 < σ2 < · · · < σ N = 2 be the simple poles of

By spectral resolution, hu(·), E i (·, 1 + it)i ∈ L2(H3/Γ D), and by what we have

al-ready observed, Res(E i , s = σ k ) ∈ L2(H3/Γ D) It follows that (3.1) extends to

L2(H3/Γ D ), and thus if u ∈ (E ⊕ R) ⊥, then ˆu i (y, 0) ≡ 0 Conversely, if ˆ u i (y, 0) ≡ 0, growth considerations imply that hu(·), E i (·, 1 + it)i = 0 and hu(·), Res(E i , s = σ K )i =

0 If this holds for each i, then u ∈ (E ⊕ R) ⊥

Let C be the subspace of L2(H3/Γ D) spanned by functions which have zerothFourier coefficient which is identically zero in each cusp By the argument in the

preceding paragraph, C = (E⊕R) ⊥ We have observed that the restriction of L to E ⊥

(and hence to C) is compact, and thus C has an orthonormal basis of eigenfunctions

for ∆ In particular, we can now conclude that L2(H3/Γ D ) = E ⊕ R ⊕ C, each subspace being invariant under ∆ Define a cusp form to be an L2-eigenfunction of

∆ which is in C

3.1.3 The dimension of the space of cusp forms

We now explain how the logarithmic derivative of φ(s) = det(Φ(s)) can be used to study the dimension of the subspace C Normalize the eigenvalue λ j = 1 + r j2, anddefine the counting function

The strongest form of the Roelcke-Selberg conjecture for the quotients H3/Γ D

states that

Z T

−T

φ 0 (1 + it) φ(1 + it) dt = O(T

1+² ) for all ² > 0. (3.3)

Clearly, (3.2) and (3.3) imply that N (T ) is unbounded as a function of T as T → ∞,

or equivalently, that there exist infinitely many L2-eigenfunctions of ∆ on H3/Γ D.Since R is finite dimensional, this implies that C is infinite dimensional, or equiva-lently, that there exist infinitely many cusp forms

We now demonstrate how the analytic properties of φ(s) can be used to

verify (3.3) for the quotients H3/Γ D See also the paper [M2], where we studythe distribution of poles of Eisenstein series associated to products of Q-rank onesymmetric spaces

We begin by summarizing some of the analytic properties of Φ(s) Property

1 can be deduced from Proposition 3.2.9 and Corollary 3.3.2 Properties 2-4 are aconsequence of the Mass-Selberg relations

1 Φ(s) satisfies the functional equation Φ(2 − s)Φ(s) = I h (here Ih denotes the

h × h identity matrix).

2 Φ(s) is symmetric.

3 Φ(s) is holomorphic and unitary on the line Re(s) = 1.

4 Φ(s) is holomorphic and bounded in Re(s) ≥ 1, except for finitely many sible simple poles 1 < σ1 < σ2 < · · · < σ N = 2 (the possible simple poles of

pos-−

→

E (w, s) in (1, 2] are possible simple poles of Φ(s)).

By property 4, φ(s) is holomorphic and bounded in Re(s) ≥ 1, except for the possible simple poles {σ k } N k=1 in (1, 2] Define the function

Then φ ∗ (s) is holomorphic in Re(s) ≥ 1.

Using property 3, it is easily verified that Φ∗ (s) is holomorphic and unitary

on the line Re(s) = 1 Thus |φ ∗ (1 + it)| = 1 By the maximum principle, |φ ∗ (s)| ≤ 1

in Re(s) ≥ 1.

Using property 1, it is easily verified that Φ∗ (2 − s)Φ ∗ (s) = I h, and thus

φ ∗ (2 − s)φ ∗ (s) = 1.

The functional equation φ ∗ (2 − s)φ ∗ (s) = 1 implies that the zeros and poles

of φ ∗ (s) occur symmetrically about the line Re(s) = 1 Factoring out the zeros and poles of φ ∗ (s) yields the product expansion

where g is an entire function and ρ = β + iγ are the poles of φ ∗ (s).

The bound |φ ∗ (s)| ≤ 1 in Re(s) ≥ 1 implies that e g(s−1) is bounded in

Re(s) ≥ 1; in particular, g(s − 1) is bounded in Re(s) ≥ 1.

Substitute (3.5) into the LHS of (3.4) to obtain the Blaskche product

Using (3.6), a detailed but straightforward calculation yields the following

formula for the logarithmic derivative of φ(s) on the line Re(s) = 1.

Theorem 3.1.2 With notation as above,

ρ

µ

1 − β (t − γ)2+ (1 − β)2

¶

.

We have observed that g(s − 1) is entire and bounded in Re(s) ≥ 1, or equivalently, that g(s) is entire and bounded in Re(s) ≥ 0 Using these facts, it can

be shown that g 0 (s) is bounded on the line Re(s) = 0 Let M > 0 be such that

|g 0 (it)| ≤ M for all t ∈ R A simple estimate yields