some applications of classical modular forms to number theory

Copyright
by
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
TO NUMBER THEORY
Committee:
Fernando Rodriguez-Villegas, Supervisor
David Boyd
Sean Keel
David J. Saltman
John Tate
SOME APPLICATIONS OF CLASSICAL MODULAR FORMS
TO NUMBER THEORY
by
Riad Mohamad Masri, B.S.; M.S.
DISSERTATION
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT AUSTIN
August 2005
UMI Number: 3204214
3204214
2006
UMI Microform

Copyright
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, MI 48106-1346
by ProQuest Information and Learning Company.
To Bonnie
Acknowledgments
To begin, I want to thank my advisor, Fernando Rodriguez-Villegas, for helpful dis-
cussions and encouragement during the last three years. I owe much to my friend and
collaborator Jim Kelliher for his patience while listening to me explain my ideas, and
Misha Vishik for his constant encouragement. I benefited from the advice and sug-
gestions of many mathematicians, including Bill Duke, Solomon Friedberg, Farshid
Hajir, 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
of Texas. Most importantly, I want to thank my wife, Bonnie Plott, for the love and
fulfillment 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 in

number theory. In chapter 2 we use non-holomorphic Eisenstein series for the Hilbert
modular group to obtain a formula for the relative class number of certain abelian
extensions 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 prove
that the subspace of cuspidal, square integrable eigenfunctions for the Laplacian on
products of rank one symmetric spaces is infinite dimensional. In chapter 4 we use
zeta functions of quadratic forms over number fields to sharpen a certain constant
appearing in C. L. Siegel’s lower bound for the residue of the Dedekind zeta function
at s = 1.
vi
Table of Contents
Acknowledgments iv
Abstract v
Chapter 1. Introduction 1
Chapter 2. Relative class numbers of abelian extensions of CM num-
ber fields 3
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 SL
2
(O
K
) . . . . . . . . . . . . . . . . . . 41
3.4 The trace of Φ(s) at s =
1
2
. . . . . . . . . . . . . . . . . . . . . . . . . . 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
Bibliography 82
Vita 84
viii
Chapter 1
Introduction
This thesis consists of three self-contained chapters. In chapter 2 we prove a formula
for 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 SL
2
(O
K
) over a number field K of degree r
1
+ 2r
2
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 = 1, 3. Given
the appropriate Weyl’s law, we explain how this formula can be used to prove that
the subspace of L
2
((H
2
)
r
1
×(H
3
)
r
2

/SL
2
(O
K
)) consisting of cuspidal eigenfunctions
for the Laplacian ∆ is infinite dimensional.
In chapter 4 we study analytic properties of zeta functions of quadratic
forms over numb er 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 =
n
2
. 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.
2
Chapter 2

Relative class numbers of abelian extensions of CM
number fields
2.1 Introduction
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. In this chapter, we prove a Stark type
formula for the leading term at s = 0 of the L–function of a non-trivial character
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)) =

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
L(H/K, χ, s) =

p

1 −
χ(p)
N
K/Q
(p)
s

−1
, Re(s) > 1,
where the product is taken over all primes p of K.
The L-function of χ can be expressed as
L(H/K, χ, s) =

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) =


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 : H
n
→ 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 < GL
2
(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(z
C
) on a Hilbert modular variety X
0
:= H
n
/Γ
0
(a
C

) arising from the
decomposition A
C
= a
C
ω
1
+ O
F
ω
2
of a fixed integral ideal A
C
∈ C
−1
. Here, a
C
is
an integral ideal in F , ω
1
∈ a
−1
C
O
K
, ω
2
∈ O
K
, and z

C
:= ω
2
/ω
1
. In section 2.6,
we express L(H/K, χ, s) as a linear combination of the functions E(s, Φ(z
C
)). In
section 2.7, we combine this expression with the Taylor expansion (2.2) to obtain
the 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
of order n at s = 0, and
L
(n)
(H/K, χ, 0)
n!
=
1
|U
K
: U
F
|


C∈cl(K)
χ(C) log

(C
−1
)

,
4
where
(C
−1
) =

N(y(Φ(z
C
)))N
F/Q
(a
C
)
−1

E
n−1
Ψ(Φ(z
C
)).
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 quadratic
extensions 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, although
this 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
h
H
h
K
=
w
H
w
K
r(K)
r(H)
1
|U
K
: U
F
|
h
K

−1
det
C,C

=1
log



C(C

)
−1

 (C)

.
For n = 1, Theorem 2.1.4 was proved by Siegel in [Si2]. In his result the
ratios 

C(C

)
−1

/ (C) are elliptic units in H (see [L2], pg. 166). It would be
interesting to determine whether for n > 1 these ratios are units, or even algebraic
numbers, in H.
2.2 Fourier expansion of the non-holomorphic Eisenstein series
Let {σ

1
, . . . , σ
n
} denote the n real emb eddings of F . Suppose that a and b are
integral ideals in F , and define
N(a + bz) =
n

j=1
(σ
j
(a) + σ
j
(b)z
j
)
for (a, b) ∈ a ×b.
Definition 2.2.1. The non-holomorphic Eisenstein series associated to (a, b) is
defined by
E(s, z; a, b) =


(a,b)∈a×b/U
F
N(y)
s
|N(a + bz)|
−2s
, Re(s) > 1, z ∈ H
n

,
5
where the sum is over a set of representative pairs (a, b) = (0, 0) which are non-
associate mod U
F
. Recall that (a, b) and (a

, b

) are associate mod U
F
if there exists
a unit  ∈ U
F
such that a = a

and b = b

.
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
Res
s=1
E(s, z; a, b) =
2
n−1
π
n
r(F)
d

F
w
F
N
F/Q
(ab)
.
For a = b = O
F
, the Eisenstein series E(s, z; O
F
, O
F
) satisfies the functional equa-
tion
G(1 −s)E(s, z; O
F
, O
F
) = G(2(1 −s))E(1 − s, z; O
F
, O
F
),
where G(s) is the gamma factor
G(s) = d
s/2
F

π

−s/2
Γ

s
2

n
.
To prove Theorem 2.2.2, we will need formulas for the Fourier coefficients of
the function
f(z) =

a∈a
|N(z + a)|
−2s
, Re(s) > 1.
Let a
∗
be the dual lattice of a, T be the trace, and vol(P ) be the volume of a
fundamental parallelotope P for a. The function f(z) is holomorphic on H
n
and
periodic with respect to a, and thus has a Fourier expansion
f(z) =

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 )

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,
h
0
(y, s) =
N(y)
1−2s
√
d
F
N
F/Q
(a)

√

πΓ

s −
1
2

Γ(s)

n
,
6
and for a = 0,
h
a
(y, s) =
2
n
N(y)
1
2
−s
√
d
F
N
F/Q
(a)

π
s

Γ(s)

n
n

j=1
K
s−
1
2
(2π |σ
j
(a)|y
j
)N
F/Q
((a))
s−
1
2
,
where
K
v
(z) =

∞
0
e
−z cosh(t)

cosh(vt)dt, t > 0,
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 ) =

d(a) =

d
F
N
F/Q
(a).
Using the definition of f(z) and that P is a fundamental parallelotope for the lattice
a, we find that
h
a
(y, s) =
N(y)
1−2s
√
d
F

N
F/Q
(a)

R
n
|N(1 −ix)|
−2s
e
−2πiT(ayx)
dx.
Define the 1-dimensional integral
h(y, s) =

R
|1 −it|
−2s
e
−ity
dt =

R

1 + t
2

−s
e
−ity
dt, Re(s) >

1
2
.
Using the definition of the trace, we find that
h
a
(y, s) =
N(y)
1−2s
√
d
F
N
F/Q
(a)
n

j=1
h(2πσ
j
(a)y
j
; s).
Thus, to compute h
a
(y, s), it suffices to compute h(y , s).
When y = 0, we find from [L2], pg. 272, that
h(0, s) =

R

(1 + t
2
)
−s
dt =
√
πΓ

s −
1
2

Γ(s)
.
It follows that the zeroth Fourier coefficient is
h
0
(y, s) =
N(y)
1−2s
√
d
F
N
F/Q
(a)

√
πΓ


s −
1
2

Γ(s)

n
.
7
Suppose y = 0. Since 1 + t
2
is even, we can write
h(y, s) =

R

1 + t
2

−s
e
−ity
dt = 2

∞
0
(1 + t
2
)
−s

cos(ty)dt.
Further, from [GR], pg. 426, we have the formula

∞
0
(1 + t
2
)
−s
cos(ty)dt =
1
√
π

2
|y|

1
2
−s
cos

π

1
2
− s

Γ(1 −s)K
s−

1
2
(|y|),
where
K
v
(z) =

∞
0
e
−z cosh(t)
cosh(vt)dt, t > 0,
is the K-Bessel function. It follows that
h(y, s) =
2
√
π

2
|y|

1
2
−s
cos

π

1

2
− s

Γ(1 −s)K
s−
1
2
(|y|),
from which we obtain
n

j=1
h(2πσ
j
(a)y
j
; s) = 2
n

π
s−1
cos

π

1
2
− s

Γ(1 −s)


n
× (2.3)
n

j=1
(|σ
j
(a)|y
j
)
s−
1
2
K
s−
1
2
(2π |σ
j
(a)|y
j
).
From the identity
Γ

1
2
+ s


Γ

1
2
− s

=
π
cos(πs)
,
we determine the relation
π
s−1
cos

π

1
2
− s

Γ(1 −s) =
π
s
Γ(s)
.
Further, we determine the relation
n

j=1

(|σ
j
(a)|y
j
)
s−
1
2
= N(y)
s−
1
2
N
F/Q
((a))
s−
1
2
.
Substituting these relations in (2.3) yields
h
a
(y, s) =
2
n
N(y)
1
2
−s
√

d
F
N
F/Q
(a)

π
s
Γ(s)

n
n

j=1
K
s−
1
2
(2π |σ
j
(a)|y
j
)N
F/Q
((a))
s−
1
2
.
8

We will need the following lemma, which can be proved in a manner similar
to equation (2.17).
Lemma 2.2.4. Let [a] denote the ideal class of F containing a. Then
ζ
F
(2s, [a
−1
]) = N
F/Q
(a)
2s


α∈a/U
F
|N(α)|
−2s
,
where a/U
F
denotes the sum over a collection of α ∈ a which are non-associate mod
U
F
.
Proof of Theorem 2.2.2. Using Lemma 2.2.4 and the change of summation

(a,b)∈a×b/U
F
b=0
=


a∈a/U
F


b∈b
,
we compute
E(s, z; a, b) =


a∈a/U
F
N(y)
s
|N(a)|
−2s
+

(a,b)∈a×b/U
F
b=0
N(y)
s
|N(a + bz)|
−2s
= N(y)
s
N
F/Q

(a)
−2s
ζ
F
(2s; [a
−1
]) +

a∈a/U
F


b∈b
N(y)
s
|N(a + bz)|
−2s
= N(y)
s
N
F/Q
(a)
−2s
ζ
F
(2s; [a
−1
]) +



b∈b/U
F

a∈a
N(y)
s
|N(a + bz)|
−2s
. (2.4)
Write z = x + iy, so that bz = bx + i(by). Letting z → bz in f(z) yields

a∈a
|N(a + bz)|
−2s
=

a∈a
∗
h
a
(by, s)e
2πiT(abx)
= h
0
(by, s) +


a∈a
∗
h

a
(by, s)e
2πiT(abx)
. (2.5)
From the formula for h
0
(y, s) in Proposition 2.2.3,
h
0
(by, s) =

N(y)N
F/Q
((b))

1−2s
√
d
F
N
F/Q
(a)

√
πΓ

s −
1
2


Γ(s)

n
.
9
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
])
+
N(y)
s
|N(y)|
1−2s
√
d
F
N
F/Q
(a)

√

πΓ

s −
1
2

Γ(s)

n


b∈b/U
F
N
F/Q
((b))
1−2s
+ N(y)
s


b∈b/U
F


a∈a
∗
h
a
(by, s)e

2πiT(abx)
= 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 −
1
2

Γ(s)


n
N
F/Q
(b)
−(2s−1)
ζ
F
(2s −1, [b
−1
])
+ N(y)
s


b∈b/U
F


a∈a
∗
h
a
(by, s)e
2πiT(abx)
. (2.6)
Using the definition of the trace, we find that
h
a
(by, s) =


N(y)N
F/Q
((b))

1−2s
√
d
F
N
F/Q
(a)
n

j=1
h(2πσ
j
(ab)y
j
; s).
Then combining terms with fixed ab yields


b∈b/U
F


a∈a
∗
h

a
(by, s)e
2πiT(abx)
=
N(y)
1−2s
√
d
F
N
F/Q
(a)


˜a∈a
∗
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) =

˜a=ab
a∈a
∗
b∈b/U

F
n

j=1
h(2πσ
j
(ab)y
j
; s)N
F/Q
((b))
1−2s
.
From the formula for h
a
(y, s), a = 0, in Proposition 2.2.3, we can express
the Fourier coefficients as
σ
˜a
(y, s) = 2
n
N(y)
s−
1
2

π
s
Γ(s)


n
× (2.8)

˜a=ab
a∈a
∗
b∈b/U
F

N
F/Q
((a))
N
F/Q
((b))

s−
1
2
n

j=1
K
s−
1
2
(2π |σ
j
(ab)|y
j

).
10
Finally, by substituting (2.7) into (2.6), and using the formula (2.8), we
obtain 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 −
1

2

Γ(s)

n
N
F/Q
(b)
−(2s−1)
ζ
F
(2s −1, [b
−1
])
+
2
n
N(y)
1
2
√
d
F
N
F/Q
(a)

π
s
Γ(s)


n
×


˜a∈a
∗

˜a=ab
a∈a
∗
b∈b/U
F

N
F/Q
((a))
N
F/Q
((b))

s−
1
2
e
2πiT(abx)
n

j=1
K

s−
1
2
(2π |σ
j
(ab)|y
j
)
= A(s) + B(s) + C(s). (2.9)
The expression (2.8) provides an analytic continuation of σ
˜a
(y, s) to an entire
function on C. Further, by estimating σ
˜a
(y, s) on compact subsets of C, one can
show that the series

˜a∈a
∗
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.
254)
ζ
F
(s, C) =
κ
s −1
+ O(1),
where the residue is given by
κ =
2
n
r(F)
w
F
√
d
F
.
Using the expansion
ζ
F
(2s −1, [a
−1
]) =
κ/2
s −1
+ O(1),
11

and Γ(1/2) =
√
π, we find that the residue of the pole of B(s) at s = 1 is
Res
s=1
B(s) =
π
n
√
d
F
N
F/Q
(ab)
κ
2
=
2
n−1
π
n
r(F)
d
F
w
F
N
F/Q
(ab)
.

We claim that A(s) and B(s) have simple poles at s = 1/2 with residues
which cancel. Thus, since C(s) is holomorphic at s = 1/2, we can conclude that
E(s, z; a, b) has only the pole at s = 1 coming from B(s), with residue Res
s=1
E(s, z; a, b) =
Res
s=1
B(s).
Using the expansion
ζ
F
(2s, [a
−1
]) =
κ/2
s −
1
2
+ O(1),
we obtain
A(s) =
N(y)
1/2
N
F/Q
(a)
κ
2
1
s −

1
2
+ O(1) =
N(y)
1/2
2
n−1
r(F)
√
d
F
w
F
N
F/Q
(a)
1
s −
1
2
+ O(1).
We know that Γ(s) has a simple pole at s = 0 with residue 1, so
Γ

s −
1
2

n
=

1

s −
1
2

n
+ O

s −
1
2

1−n
. (2.10)
The functional equation for ζ
F
(s, C) is given by
G(s)ζ
F
(s, C) = G(1 −s)ζ
F
(1 −s, C),
where G(s) is the gamma factor
G(s) = d
s
2
F

π

−
s
2
Γ

s
2

n
(see [L1], pg. 254). Using the functional equation, one can show that ζ
F
(s, C) has a
zero of order n −1 at s = 0, with leading term
ζ
(n−1)
F
(0, C)
(n −1)!
= lim
s→0
s
−(n−1)
ζ
F
(s, C) = −
r(F)
w
F
,
and therefore

ζ
F
(s, C) = −
r(F)
w
F
s
n−1
+ O(s
n
).
12
In particular,
ζ
F
(2s −1, [b
−1
]) = −
2
n−1
r(F)
w
F

s −
1
2

n−1
+ O


s −
1
2

n
, (2.11)
so that multiplying (2.10) and (2.11) yields
B(s) = −
N(y)
1/2
2
n−1
r(F)
√
d
F
w
F
N
F/Q
(a)
1
s −
1
2
+ O(1).
We conclude that A(s) and B(s) have simple poles at s = 1/2 with residues
which cancel.
Finally, assume that a = b = O

F
. Then by applying the functional equations
G(s)ζ
F
(s, C) = G(1 −s)ζ
F
(1 −s, C)
and
K
−v
(z) = K
v
(z)
in (2.9), one can show that E(s, z; O
F
, O
F
) satisfies the functional equation
G(1 −s)E(s, z; O
F
, O
F
) = G(2(1 −s))E(1 − s, z; O
F
, O
F
).
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)
−2

s + O(s
2
),
and, arguing as in section 2.2, we determine that

ζ
F
(2s, [a
−1
]) =
2
n−1
r(F)
w
F
s
n−1
+ O(s
n
).
13
Then,
A(s) = −2
n−1
r(F)
w
F
s
n−1
− 2
n−1
r(F)
w
F
log


N(y)N
F/Q
(a)
−2

s
n
+ O(s
n+1
).
Second, using the expansion

1
Γ(s)

n
= s
n
+ O(s
n+1
), (2.12)
and Γ(−1/2) = −2
√
π, we find that
B(s) =
(−1)
n
2
n

π
n
N(y)N
F/Q
(b)
√
d
F
N
F/Q
(a)
ζ
F
(−1, [b
−1
])s
n
+ O(s
n+1
).
Third, using
K
−v
(z) = K
v
(z),
and
K
1/2
(z) =


π/2z · e
−z
,
we compute
n

j=1
K
−1/2
(2π |σ
j
(ab)|y
j
) =
N(y)
−1/2
2
n
N
F/Q
((ab))
−1/2
e
−2πS(aby)
, (2.13)
where
S(aby) :=
n


j=1
|σ
j
(ab)|y
j
.
Using (2.12) and (2.13), we find that
C(s) =
1
√
d
F
N
F/Q
(a)


˜a∈a
∗

˜a=ab
a∈a
∗
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
,
14
and
E
n
(z) = log


N(y)N
F/Q
(a)
−2

E
n−1

Ψ(z)

,
where
log{Ψ(z)} =
(−1)
n
2
n
π
n
N(y)N
F/Q
(b)
√
d
F
N
F/Q
(a)
ζ
F
(−1, [b
−1
])
+
1
√
d
F

N
F/Q
(a)


˜a∈a
∗

˜a=ab
a∈a
∗
b∈b/U
F
e
−2πS(aby)
N
F/Q
((a))
e
2πiT(abx)
.
2.4 Analytic and modular prop erties of log{Ψ(z)}
Define the Laplace-Beltrami operators
D
j
= y
2
j

∂

2
∂x
2
j
+
∂
2
∂y
2
j

, j = 1, 2, . . . , n.
Theorem 2.4.1. The function log{Ψ(z)} is a potential function for the operators
D
j
, j = 1, . . . , n.
Proof. Let σ be an embedding of F , (a, b) ∈ a ×b, and z ∈ H. It can be shown by
direct computation that
y
2
∆(y
s
|σ(a) + σ(b)z|
−2s
) = s(s −1)y
s
|σ(a) + σ(b)z|
−2s
.
Using the relation

N(y)
s
|N(a + bz)|
−2s
=
n

j=1
y
s
j
|σ
j
(a) + σ
j
(b)z
j
|
−2s
,
it follows immediately that
D
j
E(s, z; a, b) = s(s −1)E(s, z; a, b). (2.14)
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
).
15
Substitute this expansion into the RHS of (2.14), expand, and equate coefficients to
obtain 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−1
log

N(y)N
F/Q
(a)
−2

. (2.15)
We claim that
D
j

E
n
(z) −E
n−1
log

N(y)N
F/Q
(a)
−2

= 0,
and hence that log{Ψ(z)} is a potential function for the Laplace-Beltrami operator
D
j
.

As a consequence of the recurrence relation,
D
j
E
n
(z) = −E
n−1
.
Also, a straightforward computation yields
D
j
log

N(y)N
F/Q
(a)
−2

= −1.
The claim now follows from these two facts.
Define the group
GL
2
(F ) =

M =

α β
γ δ


: α, β, γ, δ ∈ F and αδ −βγ ∈ F
×

,
and the subgroup of matrices stabilizing (a, b),
Γ(a, b) =

M ∈ GL(2, F ) : det(M) ∈ U
+
F
, (a, b) ·M = (a, b)

.
We can embed the subgroup
Γ(a, b) → GL
2
(R)
n
16