ON THE SPECTRUM OF THE METAPLECTIC GROUP
WITH APPLICATIONS TO DEDEKIND SUMS
by
ILAN
VARDI
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY
September, 1982
Q
Ilan Vardi, 1982
Signature of Author .......
Department of Mathematics.
August , 1*82
Certified by......
Ne~ s.auDervisor
Accepted by.............................................
Chairman, Departmental Graduate Committee
MASSACHUSETTS INSTITUTE
Archives
LO68
NOV 231982
IBRARIES
-2-
ABSTRACT
The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and
its applications to arithmetical functions involving
Dedekind sums and Kloosterman sums.
As was discovered by Andre Weil, automorphic forms of
non-integral weight correspond to invariant funtions on
Metaplectic groups. We thus give an explicit description of
Meptaplectic groups corresponding to rational weight
automorphic forms and explain this correspondence.
We also describe the spectral decomposition of
automorphic forms, and use this to find the spectral
decomposition of a class of automorphic forms: the
Poincare series.
The applications center around the fact that for
congruence subgroups of SL(2,Z), the Dedekind
a function
can be used to define multiplier systems of arbitrary
weight, and these involve the Dedekind sum. We can
use the general theory to bound sums of Kloosterman
sums which involve the above multiplier systems, and
therefore Dedekind sums. From this follows results
about the distribution of values of the Dedekind sum.
-3PREFACE
The subject of this thesis is the theory of
non-holomorphic modular forms of non-integral weight,
and its applications to Number Theory.
As was discovered by Andre Weil, modular forms of
non-integral weight correspond to invariant functions
on
Metaplectic groups. This explains the title of the thesis.
Most of the work on modular forms of non-integral
weight has been centered on the case of half-integral
weight. In this work, however, arbitrary rational weight
is considered, yealding some interesting results.
Thus the results of
§4.2 seem to be new, including the
main theorem: Theorem 4.2.4.
Chapter II is a description of the Metaplectic group,
and the exposition has been modelled on the one given by
Gelbart in
[Gel]. However, it would seem that our
decomposition of a more general Metaplectic group has not been
given in the form of
Theorem 2.2.1 .
-)
-
-=----------
~---
-4-
The material of Chapter III has been well known since Selberg's
paper [Sell, however the explicit spectral decomposition of the
Poincare series has never been published. Also, the computation
of the classical integral of Lemma
3.1.1
is new.
Acknowledgments
I would like to thank Peter Sarnak for his many interesting
conversations and suggestions. I should also like to thank Henryk
Iwaniec for his illuminating series of lectures at the Summer
Institute of Analytic Number Theory at the University of Texas.
I give special thanks to my advisor, Dorian Goldfeld, for
introducing me to this subject , and for giving me his invaluable
help and support.
--
-5TABLE
OF CONTENTS
NOTATIONS. .............................................
.
6
INTRODUCTION..........................................
.
7
OUTLINE OF THESIS. ........................................
.15
I- BASIC DEFINITIONS......................................
18
1.1-Basic definitions................................
19
1.2-Automorphic forms............................
. 23
1.3-Fourier expansions..............................
25
1.4-Kloosterman sums ..............................
26
II-METAPLECTIC GROUPS. ..................................
29
2.1-Further properties of multiplier systems ......
30
2.2-Metaplectic groups...............................
36
2.3-Relation to Representation Theory.............
45
III-SPECTRAL THEORY OF AUTOMORPHIC FORMS..................
51
3.1-Eisenstein series................................
53
3.2-Spectral decomposition of M(r,X,G)..............
61
3.3-Poincare series..................................
64
...
IV-APPLICATIONS TO EXPONENTIAL SUMS ....................
72
4.1-The Dedekind rn function and multiplier systems
73
4.2-Application to Dedekind and Kloosterman sums..
76
BIBLIOGRAPHY...........................................
84
-6-
NOTATIONS
R
The set of real numbers
C
The set of complex numbers
z
The set of integers
The set of positive integers
Q
The set of rational numbers
exp (z)
Denotes
ez,
e(z)
Denotes
e2aiz , zCC.
[x]
The integral part of x, xcR.
6y
The Kronecker symbol:
x
f,g
Let
zC.
be
6y
x
=
if
x=y
0
if
xfy
ns IN.
xcR, or
is a positive function, K>0
We write:
a constant with:
f= O(g)
If
f << g
if(x)I<K g(x)
for all sufficiently
large x, or
If(n)I<K g(n)
for all sufficiently
large n.
f= O(g)
If
g
1
complex valued functions defined
all sufficiently large
for
[x]= Max { nx}
ns-Z
lim
+
lim
x+ 0
f(x)
g(x)
=0, or
lim
f(n) =0
n+
g(n)
lim
n+
f(n)
g(n)
g
f g
If
log z
log(jzI)+ i-arg(z): -i< arg(z)4ff
f(x)=1 or
g
=1,
z= IzI-exp(i-arg(z))
,
z#O
The Principal branch of log.
-7-
INTRODUCTION
The subject of this thesis is the theory of modular
forms of non-integral weight, and its applications to arithmetical
functions involving Dedekind sums and Kloosterman sums.
We recall that the Dedekind sum is defined for integers c,d
c
((x))=x-[x]-% for x a real
((j/c))((jd/c)),where
s(d,c)= 1
j=1
number, and
[x]
denotes the integral part of x.
The Dedekind sum plays an important role in Number Theory
and has been extensively studied, the work of Rademacher being
outstanding. Very little, however, is known about the distribution
of values of the Dedekind sum. In
and Grosswald ask if the values of
[Rad-Gro] p.28, Rademacher
s(d,c), as d and c vary over
the integers, are dense on the real line.
In this thesis, the following related result is proved:
Theorem A
Let
( §4.2, p.59)
{x}=x-[x] denote the fractional part of x, for x real.
-8-
Let h and k be integers, then the sequence of fractional parts:
hs(dc)}
kC.
cEO (mod 12k)
O<d<12kc
(d,c)=1,d=l (mod 12k)
is
equidistributed
on
[0,1).
Weyl's criterion for equidistribution (§4.2,p.58)
We now recall
_
which
states that a sequence {E }.
of complex numbers
of
absolute value 1 is equidistributed iff for every positive
integer
m, we have:
.
j<x
=
0(x)
x
as
+- o.
I
Since the study of the fractional part of
is equivalent to
we see by
considering
e(x)=e2xix
x
on [0,1)
on the unit circle,
Weyl's criterion that Theorem A amounts to giving
non-trivial estimates of the sum:
(1)
0<c<x
cEO (mod 12k)
z
0<d<12kc
(d,c)=1
d=l (mod 12k)
e( mhs(d,c)
k
However it turns out that this sum appears naturally in the theory
of modular forms, as a sum of Kloosterman sums.
-9-
This
occurs
as
e( mhs(dc))
for
cEO (mod 12k)
O<d<12kc
(d, c) =1
dEl (mod 12k)
can be expressed as a generalized Kloosterman sum defined below.
The classical Kloostermdn sum is given by:
II
S(m,n,c)=
e( am + dn )
c
O<d<c
(d,c)=1
a-d:l (mod c)
,
for integers m,n,c.
There has been much work on the
culminating in
Weil's
estimate
S(m,n,c) << c +C , for
Magnitude of S(m,n,c)
[Weil]:
m,n fixed and any s>O; and this is
the best possible estimate.
In another direction
7
Kuznetsov
S(m,n,c) «xl/ 6 +E
O<c<x
[Kuzn]
, for m,n fixed
has shown:
and any s>O.
C
Though
Selberg has conjectured that this also holds with
x
replaced by
In
[Sell
x
Selberg generalized the Kloosterman sum, however,
in order to explain this, we will first have to recall the
basic notions of the theory of modular forms.
-10-
It is well known that
SL(2,R)={ a b)a,b,c,dER,ad-bc=l}
acts on the complex upper half plane
Z
az+b
for zEH.
,
It
is known
H={x+iylx,yeR,y>0}
that
d
by:
is a volume
element invariant under this action.
We recall that a subgroup
is a congruence subgroup if
SL(2,Z)j
F(N)={ a
We say that
az 1 +b
with
cz 2 +d
=
a)
If
b)
Every
z ,z2EF,
z2H are
SL(2,Z)={(a b>SL(2,R)ja,b,c,dEZ}
r1(N) C G for some NE IN, where
)(m'd N)}
G-equivalent if there is
Now a Fundamental domain
z2
F C H
open set
z,
b= 0
G of
(
b)EG
F for G, is an
satisfying:
then
z1
is not G-equivalent to z '
2
zEH is G-equivalent to
a
z' EF,
the topological closure
of F.
It is known that every
domain F
congruence subgroup
G has a fundamental
which has finite invariant volume. That is
ffdxdy
F
For example: if G=SL(2,Z),F looks like:
Y
<00
-11-
We let
G be a congruence subgroup of
real number, X:G
-+
SL(2,Z), r
a
T={zeCj zi 2=1},and define zr=exp(r log(z))
where we have chosen the principal branch of log .
We now say that
f:H
and multiplier x for G
-+
if
C
is an
Automorphic form of weight r
f
satisfies:
r
a)
az+b
f(cz+d )=X(g)
b) j f If (z) 1
F
We denote by
(cz+d>
f(z), for zcH, g= a)EG.
-zF+
d1
dxdy < co
yz
M(r,X,G) the space of such functions.
It turns out that there is a correspondence between
automorphic forms of weight
functions on the
of
SL(2,R).
(*)
invariant
Metaplectic group, which is a Covering group
This is the reason for the title of this thesis.
We now say
C"
r and multiplier X and
function of
Aru +u=O,
that
x and y, and there is a XcR
where
invariant Laplacian for
It is known
ucM(r,X,G) is a Modular form if
that
2
2
A= y ( -2
+
32
)2-iy-iry
u
is a
such that:
is
the
M(r,X,G).
(*)
has an orthonormal set of solutions
-12-
K
K
{u }=-v,u EM(r,X,G), with eigenvalues
X.
w
+
as
j
+
C ,
if
K=
0
.
A={X. I0<.<
3
3
The finite set
{X -<...<<=x. < .'''j=-V
}
is called the set of
Exceptional eigenvalues.
The set of eigenvalues has received much attention and is
completely unknown except for a few cases corresponding to
zeta functions of
Quadratic fields.
There has been much work on the problem of whether
X >
,
that is whether the set of exceptional eigenvalues is empty.
This result would have many applications, for example when r=O,
there is
in
[Iwan-Des]. There is a survey article on this problem
[Vigneras].
For our purposes, however, the trivial bound X>O suffices.
We can now define the
S(mjnjcXG)=
O<d<qc
g=
Where
0,
a <1.
Generalized Kloosterman sum:
a(m-a)+d(n-a)
qc
X
d)EG
q>0 is such that
G
={(
) EG}= 0
)nZ)e(-)X((
)
-13-
This generalizes the classical Kloosterman sum, for it is
seen that: S(m,n,c)=S(m,n,c,1,SL(2,Z)).
Kuznetsov's result (1) generalizes to:
Theorem B. (Kuznetsov-Goldfeld-Sarnak-Proskurin)
S(mn.G)
A.
C
O<c<x
Where c.=2/r.-T , the
J
J
S=Inf
6>0
c>0
x
),
+r(x
for any c>0.
Xc
A.'s are constants, and
J
JIS(m,n,c,x,G)<cO
1+6
The relation between the generalized Kloosterman sum and
Dedekind sum arises
Dedekind
n-function: n(z)=e(z/24)VV (1-e(nz)).
n=1
az+b
log n( cz+d
for
from the transformation law of the
log
n(Z)+%[lg9(
log n(z)+
2(
This is:
a+d
CZ
czd+d )+2ri( 12c -s(d,c))], for c#O.
2172ib
for c=0.
~frc
a b)sSL(2,Z).
2r
This gives that n (z) satisfies an automorphy condition:
2r
az+d )Xr(
)N
(cz+d)r
2r (z),
(
ESL(2,Z)
where:
-14-
is therefore a multiplier system of weight r for SL(2,Z),
Xr
and therefore for all congruence subgroups.
We can explicitly calculate:
e[r( 12c -s(d,c)
(
Xr
b))=
-
)],
if
c>O
12
cd/
if
te(rb/12),
C=O
From this, it follows that for
h,m,k
positive integers
we have:
r(12k))=
e(-hm )S
e4kS(hm~hmlC9Xhm/kp
e( hmsd,c
O<d<12kc
(d,c)=1
d=l (mod 12k)
And it is now clear
using
how
and
cEO (mod 12k).
follows from Theorem B
partial summation to give non-trivial estimates
S(hmhmcXhm/k,F(1 2 k))
O <c <x
Theorem A
),
of.:
-15-
OUTLINE OF THESIS
Chapter I
is
an outline of the prerequisites to understanding
the following chapters.
SL(2,R) on
In
§1.1
we recall the action of
the complex upper half plane, and also describe
some properties of the discrete action of congruence subgroups
on
H.
In
§1.2
we define our space of automorphic forms,
while §1.3 gives us the Fourier expansions of automorphic
forms at the cusps.
In §1.4
we recall some theorems about
Kloosterman sums.
Chapter
how
II
is independent of the others, and
explains
automorphic forms of non-integral weight correspond
to invariant functions on the Metaplectic group
which is a covering group of SL(2,R).
§2.1
technical facts about multiplier systems.
SL(2,R),
proves some
In
§2.2
we construct the Metaplectic group, and find an explicit
decomposition of it. We also give the correspondence
between automorphic forms and invariant forms on the
Metaplectic group.
-16-
In §2.3
define
we take the approach of Representation Theory
to
modular forms. We then review some facts about the
eigenvalues corresponding to modular forms.
Chapter III
spectral decomposition of
deals with the
automorphic forms. In
:1
we introduce the Eisenstein series
§3.1
which are a fundamental tool in the theory of automorphic
forms.
In '§3.2
we describe
Selberg's spectral decomposition
of the space of automorphic forms.
In §3.3
we find the
explicit spectral decomposition of a class of automorphic
forms: the Poincare series.
Chapter IV
is an application of the theory developped
in the previous chapter
weight automorphic forms
In
§4.1
we show how the
to construct
In §4.2
we
to
a special case involving rational
and special congruence subgroups.
Dedekind n function can be used
multiplier systems of arbitrary weight.
restrict ourselves
to
rational weight
and to the congruence subgroup P(12k),
and show how
mh/k,
-17-
the generalized
Kloosterman sum reduces to a simple
sum involving Dedekind sums. This fact, combined with
a theorem of Kuznetsov, Proskurin, Goldfeld and Sarnak,
allows us to prove the main theorem of this work: Theorem 4.2.4.
-18-
CHAPTER
BASIC
I
DEFINITIONS
-19BASIC DEFINITIONS
1.1
Let R be the set of real numbers, we denote by
H={x+iyly>O} the complex upper half plane. If we write -=
then it is well known that
acts on H and on
_
Rw{o}
az+b
and
Note that
)
a,b,c,dcR,ad-bc=l}
by
where
cz+d
SL(2,R)={ (
a b)SL(2,R), zcH or zRV{co}
1
0) have the same action, and
the group of transformations is PSL(2,R)=SL(2,R)/±l.
However we shall denote a transformation gePSL(2,R) by one of
its associated matrices. We thus write:
CYZ=
az+b
cz+d
for
a= a bSL(2,R)
dx2 +Ax 2 is an SL(2,R)
2
y
invariant metric on H, and the invariant volume element is
The classical theory gives that
given by
dxd
y
Thus considered, the complex upper half plane
becomes a Riemann surface of constant negative curvature,
and is called the Poincare or Lobachevski plane.
Let G be a subgroup of SL(2,R), we say that
are
G-equivalent if there is a
gcG such that
zi,z
2
zl=gz 2 '
H
-20-
We say that G is a discrete subgroup of SL(2,R) if G
is a subgroup and for any
zEH, the set {gz~zcG}
has no
limit point in H.
Let Z be the set of integers
ad-bc=1}. It is well known that
subgroup of
and
SL(2,Z)={
ab
SL(2,Z) is a discrete
SL(2,R).
Now for N a positive integer we define r(N)={ a)
ad
(
d
?)(mod N)}
,
SL(2,Z)i
where the congruence is defined componentwise.
We call F(N) the Principal congruence subgroup of level N.
It can be shown that r(N) has finite index in SL(2,Z),
in fact:
[P(N):SL(2,Z)]= N3 -T (1_
pIN
Defining a subgroup
of
SL(2,Z) if
12
p
GC SL(2,Z) to be a Congruence subgroup
F(N)C G for some N>O, we conclude that:
G is a discrete subgroup of SL(2,R) and G has finite index
in
SL(2,Z).
We will restrict ourselves to
of
SL(2,Z), so in this section
stbgroup of
SL(2,Z)
G a oongruence subgroup
G will aways denote a congruence
-21-
Fundamental domain F for G is an open set
A
a) If
z1 ,z2 cF ; z1
#
FCH satisfying:
zi,z2 are not
z 2 implies that
G-equivalent.
b) Every
z6H is
G-equivalent to a
z0 sF, the topological
closure of F.
It is well known that a fundamental domain for SL(2,Z)
can be given by
F={x+iylx2 +y2 >1,
-<x<}
and an easy calculation shows that
F
F
R
y
23
thus
Vol(SL(2,Z)\ H)<.
G has finite index in SL(2,Z)
Therefore
a) The fundamental domain F of
implies:
G is a finite union of
translates of a fundamental domain of SL(2,Z), and can thus
be chosen to be a simply connected set.
b) Vol(G\H)=[G:SL(2,Z)]Vol(SL(2,Z)\H)< 00
The fundamental domain for G looks like:
F
t%11
nQn
AA
R
-22-
We say that an element
.
g=
la+dl=2, where
if
gESL(2,R) is parabolic
Let G be as before, then we say that KcHvRAoO}
cusp of G, if
K
is left fixed by a parabolic element in G.
It turns out that all cusps are in
Rv{} .
then it is clear that
IxeR}
N={
Let
is a
a) N consists of parabolic elements.
b) N is the subgroup of SL(2,R) of elements fixing c .
c) m is a cusp of G
Further, as
(
is always a cusp of
We note that
and
G
j)r(M), we have
Thus
and we denote this by
G
have that
such
# 0, and
njn}Z
Z
is a cyclic group
has a unique generator
(0
1
q>09
q=q(G).
It can be shown that {co}
SL(2,Z).
G
M
G.
SL(2,Z), ={(1
C SL(2,Z) ,
cusps of
GCO=GON #0
G is a congruence subgroup, there is an
F(M)C G. So as
that
iff
is a complete set of inequivalent
So as G has finite index in
SL(2,Z), we
G has a finite set of inequivalent cusps:
-23-
K,...
and there are
, Kh
such that
a ,... ,ahE SL(2,Z), ai=id
a . (o)=K..
JJ
1.2
AUTOMORPHIC FORMS
Let
G be any subgroup of
j:G x H-+ C
if for
SL(2,R) then we say that
is a Factor of automorphy for
g 1 ,g2sG, zEH:
j(g1 g 2 ,z)=j(glg
2 z)j(g 2 ,z)
j(g,z)=cz+d,
Direct computation immediately gives that
where
g=c b)e SL(2,R), is a factor of automorphy for SL(2,R).
From this it follows that
for
G
SL(2,R), for
j(g,z) r
is a factor of automorphy
r any real number.
Let log z be the principal branch of
zr=exp(r log z)
Jr (gz)=
cz+)
we defime:
r;
We recall that
of the circle.
X:G+ T is a
g1 ,2 cG, zcH:
If
where
log and let
r (g,z)=(cz+d)r
rcR and
T={tcCl I|t
g=(
b)SL(2,R).
2=1} be the multiplicative group
G is a subgroup of SL(2,R), we say that
Factor of automorphy of weight r for G
X(g 1 g 2 )
X(g1 )X(g2 )
ir(
r
1
2 z)jr(g 2
jrC( 1
2
,z)
,z)
if for
-24-
this can also be written:
(1.2.1)
X(g1 g 2 ) =
X(g 1 ) X(g 2 )
as 1jr(g,z)lr
r
l'g2z)Jr(92,z)
r(9 1 92,z)
is a factor of automorphy.
We now define our space of automorphic forms for
a congruence subgroup:
For
9=
r for
f:H+ C
M' (r,x,G) of functions
a)
r be a real number and X a
Le t
multiplier system of weight
b) £ G
G
G. We define the space
satisfying:
we have
f(gz)=X(g)jr (g,z)f(z)=x(g)(cz+d)rf(z)
b)
fJ
r If2(z)d
F
<
y2
F is a fundamental domain for G.
where
We will find it more natural to consider the space
M(r,X ,G) of functions of the form
If we let Im(z)= y, where
gives
Im(gz)=
Y
bi'+d2 2
it becomes clear that
a)
yr/2f(z)
where
fEM'(rX,G).
z=x+iy; x,ycR, then a simple computation
g=(b
b eSL(2,R).
fc M(r,XG)
f(gz)=x (g)Jr (g,z) f (z)=x (g)
Using this fact,
iff:
cz+d
( cZ+d
f(z), g=a
)
-25-
b)
ffIf(z) 12 dxdy
F
<
2
y
For example:
f (Z)=yr/
if
f(z)= Im(z) r/2 f 1 (z)
2
f1 E M' (r, X, G) ; then
f 1 (z) ,
ge G
and if
)/2
d
f(gz)= Im(gz) r/2 f(gz)
X(g)(cz+d)/2
cz+d 12
f1 (z)
yr/2 f (z).
1
= X(g) (
It is seen from a) that
X(g) Jr (g,z)1=1. Thus
is G-invariant, as
b) makes sense.
We also note that
M(0 ,x,G).
Vol(F)< o implies that the constant functions are in
Condition
b)
gives that
M(r,X,G) is a
Hilbert space
fg dxdy
inner product
F
y
with
f,ge M(r,X,G).
2
We call this the Petersson Inner Product.
FOURIER EXPANSIONS
1.3
Let
G,
={g c Gjgco =g}
Now let X
and
G be a congruence subgroup, we recall that
, q=q(G)
>0
is such that
Goo={ (o
be a multiplier system of weight. r for G,
q = e(-a),
0 < a<1-
We
write a=a(X,G).
1) In
re R
Z}.
-26-
If
fe M(r,X,G) we see that
So
f(x+iy) has a
f(z+q)=X(
q (0-z+l)rf(z)=e(-a)f(z).
w:
Fourier expansion at
n=o
f(x+iy)= Z an(y) e( (n-a) x)
n=-co
q
Similarily if
K.
is any cusp of
G K ={gc GlgK=K}, so
We let
h(z)=Jr(a ,z)~
f(z)
a
G, with
G K a = G0
satisfies
as SL(2,Z),
q(c)=
K.
We then have that
h(z+q)=e(-a)f(z);
Fourier expansion at the cusp K:
We thus have the
n=oo
r(a ,z)
(1.3.1)
f(a z)=E
an,
n=-o
We define the space of
subspace of functions
of
G, then
e(
(y)
q
Cusp forms
fE M(r,X,G)
a0,K(y)=O, where
(n-a) x).
S(r,X,G) to be the
such that for any cusp
a ,K(y)
is the zero'th Fourier
coefficient
at the cusp
Remark:
In this work, we will be working exclusively
K,
with Fourier expansions at
a n,K(y)
K
as in
(1.3.1).
0.
KLOOSTERMAN SUMS
1.4
The classical
Kloosterman sum
is defined for integers m,n,c
-27-
and is given by: S(m,n,c)= d(mod c)
e( ma+nd )
c
(d,c)=1
a-d=l(mod c)
The Kloosterman has been extensively studied, with much work
on finding upper bounds on IS(m,n,c)I
for
m,n fixed.
The trivial estimate is IS(m,n,c)|<c.
obtained
Estermann and Salie
S(m,n,c)<<c 3/4+e,for any c >0, m,n fixed. Davenport
improved this to S(m;in,c)<< c 2/3+c
Andre Weil found that
,any E>0.
S(m,n,c)<< c
,
Finally.
for E>O, and m,n fixed;
and this is also the best possible.
In another direction Selberg studied the sum
and conjectured that
c~c<x
c
Cc
< cX
S(m,n,c)
,,c
for any c >0
The best result known,however, is due to Kuznetsov and gives:
(1.4.2)
Oc< x
S(m,n,c) <
c
/6+F
for any E >0.
In his paper on the Fourier coefficients of modular forms [Selberg 1],
Selberg showed that the estimation of the sum
(1.4.1) is
inextricably related to the theory of modular forms.
-28-
Selberg also showed how the Kloosterman sum could be generalized
to a congruence subgroup G of SL(2,Z), and a multiplier
system X of weight r for G, rc R, by:
X(g)
S(m,n,c,x,G)=
e(
gE G\o
X(g)
d (mod c)
(m-ct)a + (n-c)d )
qc
'
g=(a
b)
c d/
e( (m-a)a + (n-a)d )
qc
O<d<qc
where
, q=q(G), a
Go
The sum
/
c<x
(1.4.1)
c)
)(G,adEl(mod
g=(*
are
as in section 1.3.
can be generalized to:
S(m,n,c,x,G)
c
,
and the following result has been proved:
Theorem 1.4 (Kuznetsov, Goldfeld, Sarnak, Proskurin)
c<x
Where
S(m,n,c,X,G) =
j=1
c
A. are constants,
3
to the set A of
and
J
T
+
2V-.,
where
4 j
S(m,n,G)
c1+6
G=SL(2,Z), r=0,X=l
), for any
L(xP/3+
exceptional eigenvalues
3= lim inf {
o<c
6
In the case
T.=
A. x
J
O<X.<4
J
>O.
belong
defined in section 2.3,
<
it turns out that
S(m,n,c,XG)
=S(m,n,c). Also there are no exceptional eigenvalues,and
Weil's estimate. Thus Kuznetsov result (1.4.2) follows.
=
by
-29-
CHAPTER
II
METAPLECTIC GROUPS
-30-
2.1
FURTHER PROPERTIES OF MULTIPLIER SYSTEMS
We first characterize factors of automorphy and
multiplier systems.
Proposition 2.1.1
Let G be a subgroup of
SL(2,R)
and
then the existence of a non-vanishing
f(gz)=j (g,z)f(z)
implies that
j
,
f:H+ C
satisfying
gsG
(g,z) is a factor of automorphy for
Proof:
Let
f(gig 2 z)=j(g
j:G x H+ C,
1 g2 ,z)f(z)=f(g 1
=j(g1)92z)j(g2,z)f(z)
g1 ,g 2 c G
(g2 z))=
and
j
G.
then
(g1 ,g 2 z)f(g 2 z)
f(z)#O
give the result
//
We next have.
Lemma 2.1.1
Let
then
X
G be a subgroup of
is a multiplier system
X(g)jr(g,z)
Proof :
SL(2,R)
and
of weight
rs R,
X:G-+ T,
r
for
G
iff
and X(g)J r(gz) are factors of automorphy for G.
The proof follows by cross multiplying in
-31-
(2.1.1)
=
X(g1 g 2 )
X(gp)X(g2 )
r(g 1 2'2 z)jr(g2'z)
Jr(g
-
ir(g 1g 2 'z)
g2 z)Jr(g2 z)
V g 1 'g2 G
Jr ( 1g 2 z)
where the last equality is a consequence of the fact that
lijr(g,z)l was shown in section 1.2
to be a factor of automorphy
for SL(2,R), and in fact is true for any
g1 ,g2 ESL(2,R)
//
Corollary 2.1.1
Let
G be a subgroup of
SL(2,R),
existence of a novanishing function
f(gz)=X(g)jr (g,z)f(z) ,
and X:G+ T. Then the
f:H+ C satisfying
ge G
implies thatX is a multiplier system of weight r for G.
Proof:
Follows from above.
//
Definition 2.1.1
We will say that a multiplier system of weight r for G
is non-trivial
if there exists a function f satisfying
the conditions of Corollary 2.1.1.
We will restrict ourselves to such multiplier systems, and
x
is always assumed to be non-trivial.
-32-
Proposition 2.1.2
XlX2 be multiplier systems for G of weight
Let
xlX
respectively, then
2
ri,r 2
is a multiplier system of weight r +r 2 for G.
d)
Follows if we note that for g=(c dgzj(~)(zd
r+ 2
~~~~~~~
Proof:
j~
jr Ig
jr 2(g,z)=(cz+d) r(cz+d)
a)
Abelian character of
X:G+ T to be an
We define
X(g1 g 2 )=X(g1 )X(g2 )
G
//
if:
( 192 cG
implies X -0 0 =
-')c G
b)
=j(r +r )(g~z)
=(cz+d)
Proposition 2.1.3
neZ, then a multiplier system X of weight
Let
G is an abelian character of
for
Proof:
G.
in section 1.2
As
j 1(g, z) =cz+d,
is a factor of automorphy for SL(2,R). Thus, so is
x(gg 2 )
=
nI"(g'g
X(g1 )X(g2 )
Also if
gives
(
2 z)jn( 2
,z)
a b
d)
g= (c
jl(g,z), hence
=1
in(g1 g 2 ,z)
_c G, then letting
f(z)=f( ~+u)=
-(
n
)
f:H+ C
f(z),
be as in Definition 2.1.1
so X( 0
)-
/
-33-
Proposition 2.1.4
Let
r be a real number, and
system of weight
x'
G is of the form
XO is and Abelian character of
Proof:
be a multiplier
r for G, then any other multiplier system
r for
of weight
X
X'=X O X , where
G.
It follows from Proposition 2.1.2
that
X/X'
is a multiplier system of weight zero, which is an Abelian
character by
Proposition 2.1.3.
//
We now recall a Theorem of Maass.
Theorem 2.1. (Maass)
Let
G be a congruence subgroup of
group of abelian characters of
SL(2,Z)
G is isomorphic to
then the
G/K*,
G
K* is the group generated by the commutator of
where
and by
~ -i).
Proof:
Moreover this group is finite of order n(G).
[Maass 1]
page 117.
//
Corollary 2.1.2
Let G be a congruence subgroup, r a real number, then there
-34-
are exactly
G.
n(G) multiplier systems of weight r for
proof:
By theorem 2.1
and proposition 2.1.4, the
existence of one multiplier system of weight r for G
implies that there are exactly n(G) such.
(Corollary 4.1.2)
of weight
But in Chapter
IV
we will construct a multiplier system
r for any
rER, and any congruence subgroup.
//
Corollary 2.1.3
Let
r=m/k
be a rational number with
is a multiplier system of weight
subgroup G, then
X(g) is aways a
r for G for the congruence
k-n(G)
Xk
By proposition 2.1.4
Proof:
root of unity.
is a multiplier system
of weight m, and is thus an abelian character of
so is an
X
m,kEZ. If
G, and
n(G) root of unity by theorem 2.1
//
Let
nth
T n={tCItn=l}
be the multiplicative group of
roots of unity, then we can express the result of
Corollary 2.1.3
as
X:G+ Tn
,
We will require the following result:
n=k-n(G).
-35-
Proposition 2.1.5
Let
r=m/kc
Q;
m,ks Z, then for
jr(gl'g2z)jr(92, z)
is a
g1 ,g2 sSL(2,R)
kth root of
unity.
jr(9 1 9 2 ,z)
Proof:
as in
( jm/k(g 1'E2z)jm/k.(92
jm/k(9192,z)
as in the proof of
Proposition 2.1.2
,z)
j (gl g2z)jm(92,Z)
jm(9
1
2
= 1
,z)
proposition 2.1.3
//
Corollary 2.1.4
With notation as in proposition 2.1.5, we have that
m/kflg2z)Jm/k 2,z)
im/k(9192,z)
Proof:
is aways a kth root of unity.
Follows as above
//
-36-
2.2
METAPLECTIC GROUPS
In this section we fix a rational number
with
m,kEZ and fix nEZ
we will write
Definition
J(g,z)
kin.
for
Also for
=
gbSL(2,R)
Jm/k(gz)- cz+d)m/k,
cz+d )
zsH.
2.2.1
We define the Metaplectic Group
the set of pairs
(2.2.1)
(b
r=m/k;
SL( 2 ,R)m,k,n
{(g,t)jgESL(2,R), tT n}
to be
with multiplication law
(glptl)(glt2)=(192,a(glpg2)tlt2)
(2.2.2)
a(glg
2 )=
J(g 1 9'2 z)J(g 2 ,z)
J(g1 g 2 ,z)
We note that, by corollary 2.1.4,
so
a(g 1
2 )T
k
Tn
as
kin,
(2.2.1) is well defined.
We will supress
Note:
SL(2 ,R)
for
m,k,n and henceforth write
SL( 2 ,R)m,k,n
Remark 2.2.1
SL(2,R)
is an
is an
exact sequence
n-fold cover of
SL(2,R). That is, there
1i T n-* SL(2,R)+ SL(2,R)+ 1
-37-
is a factor set , that is
Also a(g 1 ,g )
2
a(g 1 g2 ,g3 ) c(g 1 ,g2 )=a(g1 ,g2 g 3 )ca(g
Since
map
the
SL(2,R)+ SL(2,R)
is a homomorphism, we see that
plane H by
for any
g,g2'9 3 eSL(2,R)
2 ,g3 )
given by
(gt)t+ g
SL(2,R) acts on the upper half
gz=(g, t)z=gz, where we have written g=(g,t).
We also extend
J(g,z) to
J:SL(2,R) x'.H+ C
by
where again g=(g,t). We have then :
J(g,z)=t J(g,
Proposition 2.2.1
J(g,z) is a factor of automorphy for
Proof:
Let
g.=(g ,t )c
SL(2,R)
SL(2,R).
for j=1,2
J(g 1 g 2 'z)=J( (g1 ,t1 )(g2 ,t2 ),z)= J((g 1 g 2 ,J
then
l'g 2 z)J(g 2 z) t 1 t
J(g 1 g 2 ,z)
J( 1',g2 z)J(g 2 ,z)
t 1 t2 J(9 1 9 2 ,z)= t1
gJ(,g2 z)t 2 J(g 2 ,z)
J(g 1 g 2 ,z)
=t 1 J(g, g2 z)t2 J(g 2 ,z)=J(g 1 1g2 z)J(g 2,z)
//
We will now obtain a decomposition of
SL(2,R).
Lemma 2.2.1
Let
N={
(
1) IxcR}
,={(
y
0
y2
)4)
ly >0, ycR}
2
) ,z)
-38-
then
N, A
are subgroups of
this follows from the fact that
Proof:
J(
SL(2,R).
(
l),z)= 1 -
X
(0
Z+
0
y ~k r
=1 ,
for
y >0
2
y
Z -iy4
We now examine how
/1
SL(2,R)
//
acts on the point
i.
Proposition 2.2.2
a)
(
(
1)(
[y2
i= x+iy
k,0
0
y >0
0<2Tr,
K= {gcSL(2,R)jgi=i}=
b)
where
r()=
Proof:
os
tcT n
-sinO
(sinO cose
Follows from direct computations.
//
We require the following simple result:
Lemma 2.2.2
J(r(6),i)= exp(ire)
Proof:
Sexp(ir)
1
we have
J(r(O),i)
= exp(irO),
as
sin +Cos
isin
*
+os)
ecR
gives
r
exp~
r
lexp(ire)1=1.
//
We now have:
-39-
Proposition 2.2.3
}:KT x Tn/k
There is an isomorphism
given by
k
T(r(E3),t)=(texp(ire),t )
Proof:
We
first show that
K- T,K+Tn/k
for T
and
are homomorphisms
re spectively. We will then find inverses
T2'
Now
7) t
(r
T
=T 1[(r( 1 )r(O 2 ),t1t
Letting
given by
T2 (r(8),t)=tk
T 1 (r(O),t)=texp(ir6),
of
S1,V
, 2
) (r
) .,t2
J(r( 1 ),r(6 2 )z)J(r(0 2 ,z) )
J(r( 1 )r(62 ),z)
2
z=i we appeal to Lemma 2.2.2, and noting that
r(0 1 )r(62 )=r(
the above becomes:
1 +e 2 )
exp(ir( 1 +0 2 ))t
1 t 2 exp(ire )exp(ire 2
exp(ir(
=T(r(e ),t
1 +62 ))
1 T (r( e2, t 2 ))
Also we have
T 2 (rre
1
) ,t1 ) (r(0 2 ) ,t2 )
=T 2(r(O 1 +e2),tl t 2a[(r(O ),t
=tk tk
(a[(r(O ),t
1),(r(e 2 ),t2D
)(r(O2),t2M k= tktk=
((r(a 1),t
)T 2((r(O 2),t2)
-40-
as we have noted that a(gl,g 2 )k=1
To show that
maps to
0
define: r(O)=exp(iO/k) for
We first
Then let:
: <) e <27Tk}
x=[xl+{x}
x respectively.
+
,
t
r([e/2TI));
He(r(e))=(r(2{/20}
where
g 1 ,g 2.
T is one to one and onto, we find inverse
V, '2.
e<2nk.
for any
E2
n/k
(e(jk/n))=(r(-i
k),e(j/n)),j<
denote the integral and fractional parts of
We then have that
Y
=id.T,
T2 -E=id.Tn/k'
These results follow from a direct computation, and the
proposition follows.
//
Corollary 2.2.1
ueK can be uniquely written as
Every element
where
Proof:
0, e<27k and
We
u=rT(7t,
tk =.
use E
Tn/k with their images in
to
identify
{ rTe)0,<e<27k} and
K. The result then follows from
Proposition 2.2.3.
//
We next prove:
-41-
Lemma 2.2.3
Let
ueE,
and
u=r(e)*t as above, then:
J(u,i)=exp(irO)
Proof:
As
J(g,z) is a factor of automorphy for
SL(2,R), we have:
J(u,i)=J(r(),ti)J(t,i)=J(r(6),i)J(t,i)
Now by Propositon 2.2.3 there is a
t=(r(-
mn ),e(m/n)),
jcZ such that
so by Lemma 2.2.2
J(t,i)=e(m/n)exp(-2 ijk m)=e(m/n)e(-
Also Proposition 2.2.3
gives
m/n)=
r(O)=(r(2{6/27},e(r[G/27])),
J(r(e),i)=e(r[0/21)exp(2ire/2})=exp(2ir([1/2]+{e/2}))
=exp (27ir O/27)=exp(ire)
//
If we combine Proposition 2.2.2 and Corollary 2.2.1
we obtain:
Theorem 2.2.1
SL(2,R) has the decomposition
and every
SL(2,R)=N
geSL(2,R) can be uniquely written
as:
K
so:
-42-
g=
(2.2.3)
x)y
) r(6)-t,
where
xeR,y>0,0<<2Trk,t n/k=1
and we have made the obvious identifications for
N,
A.
Remark 2.2.2
SL(2,R), and
(2.2.2) gives us local coordinates for
SL(2,R).
enables us to carry out analysis on
G be a congruence subgroup of
We now let
SL(2,Z),
r=m/kcQ; m,keZ,X a multiplier system of weight r for
as in Theorem 2.1
and
n=k-n(G) (so
considering the Metaplectic group
we again denote by
SL(2,R).
G, n(G)
We will be
kin).
SL( 2 ,R)mk,n , which
We first prove:
Proposition 2.2.4
The map
G + SL(2,R) given by
g F- (g,X(g)) is an isomorphism.
Proof:
The map is well defined as we showed in Corollary 2.1.3
that X(g)cTn.
X(gg
2)
X(g 1 )X(g2 )
SJ(
By definition we have that
1 ,g2 z)J(g 2
J(g 1 92 ,z)
,z)
for
g 1 ,g2 G
so:
-43-
(g1 ,X(g1 ))(g2 ,X(g2 ))=(g 1 g 2 'X(g1 )X(g2 )
X(91 g 2 ) )=(g1 g 2 ,X(g1 g 2 ))
X(g1 )X(g2 )
//
We will use this map to identify
SL(2,R).
G with its image in
Theorem 2.2.2
There is an isomorphism
a)
where
f t+ $
L2 (G\SL(2,R)) given by
+
gESL(2,R)
$f (g)=f (gi) J(g, i)
The image of this map is the
b)
of
M(r,XG)
L(r,G) C L2 G\SL(2,R))
space
gcSL(2,R)
functions 4 satisfying $(g-r(e)-t)=$(g)exp(-ire),
where we have used the representation of
We first show that Pf
Proof:
letting
E=(h,X(h))eG
,
=f(gi)X(h)J(h,gi)J(h-g,i)
=f(gi)J(Egi)J(E,gi)
of automorphy
=f(gi)J(gi) -
is left
Corollary 2.2.1.
G-invariant:
-g=(g,t)cSL(2,R) we have
and
$f(h-g)=f(h-gi)J(h-g,i)
K given in
=f(h-gi) J(h-g,i)-1
1
i) -1
f(gi)J(igi)J(-
J(, i)
as
J(-,-)
is a factor
for SL(2,R). The above is thus equal to
f(g).
We next show that in terms of x,y,e,t
we have:
-44-
4gf(xyet)=f(x+iy)exp(-ire).
This will imply the result of
the above proof shows that
b),
for if
f (x+iy)=$(x,y,0,1) satisfies
f (gz)=X(g)J(gtz)f
the automorphy condition
so
We thus write
=
J(g,i)=J[ (
Y
$cL(r,G), then
,1)
r
geG.
gi=x+iy
and
y)
t
= [
,i
26)-t ,Y)
(z),
( 0
,Y) , r (
)
6
- t)-
i
by the proof of Lemma 2.2.1 and by Lemma 2.2.3.
=1-exp(ire)
Our result then follows.
Finally we note that the decomposition of
gives a product measure on
given in Theorem 2.2.1
dg=
dxdy 2 de
T7Tk g
y
2
SL(2,R)
IZ
k
tTf
G SL(2,R)
1
ink
SL(2,R):
so $g is square integrable
iff
2Tk
G |f(x+iy)exp(-ire)12
G\H
d
y dO
2
2Fk
k
/k 2T~k
0
G
f(x+iy)1 2 d.xyde= ( f (x+iy)1 2dxdy < c
2
F
y
as fcM(r,X,G).
The theorem now follows
//
-45-
2.3
RELATION TO REPRESENTATION THEORY
In this section we fix a congruence subroup G, r=m/kEQ
with
m,kc., and let
n=n(G)-k, SL( 2 ,R)=SL(2 ,R)mkn
be as in the last section.
We presently recall that
by right multiplication:
and this is called the
SL(2,R)
Rg($(h))
acts on
(h g)
L 2 (\SL(2,R))
EL2
SL(2,R)),
(G
Right regular representation.
It has been the approach of Representation Theory to regard
the theory of Modular Forms as the analysis of the decomposition
of
R-.
g
We shall use this approach to motivate the definition
of Modular Forms.
We note that
and that
K
u= r(E)-t
We see that
t+
is a
Maximal compact subgroup of SL(2,R),
exp(irO)
L(r,G), and so,by
corresponds to decomposing
Rg
is a character of
Theorem 2.2.2, also
for
SL(2,R), for it is well known that
under the invariant subspaces of
M(r,X,G),
according to characters of
For further decomposition, we must appeal to the
operator A
K.
A.
K.
Casimir
Rg
decomposes
-46-
SL(2,R) is a covering group of
We recall that
Lie Algebra
and thus has the same
that
SL(2,R). It follows
as
has the same Casimir operator as
SL(2,R)
SL(2,R)
SL(2,R).
Using the local coordinates given in Theorem 2.2.1,
can be written as:
A$(x,y,e,t)=y2
for $ a
22
-
Co function of
2
+
2
2- ) + y
A
x,ye.
a
is
to
We now analyse the restriction of
Proposition
Let
write
SL(2,R) invariant.
L(r,G).
2.3.1
$ be a
$=f where
C"
x,y,O in
function of
fcM(r,X,G), then-
L(r,G)
and
where
A f=$A
f
r
f
)-iry Dy
f + a
A f-y 2
r
D
A~
:MrDG)+Mr,,)
Ar:M(rXG)
C"
M(rX,G).
C
As
function of
h:H- C
if
is
Arh(z)= Ar(h(gz)J(g,z)
$f=f(x+iy)exp(-ir8)
x and y.
We
to
and this can be extended
Further
function of x,y then:
Proof:
a
+
,
,
compute:
we see that
any
x(g),
f
is
gEG.
-47-
A
+ y
As
f
aexp(-irO)
as
Arf- exp(-irO) =
CW function are dense in
A:L2 (G\SL(2,R))
Finally
as
A is an
f
Ar :M(rXG)
+
M(r,X,G)
L 2 (G\SL(2,R)).
SL(2,R)
invariant, we see that the
given
$o(g)= f(gi)J(g,i)
correspondence
Theorem 2.3.1
+
+
M(r,X,G), we see that Af=$A f
r
feM(r,X,G) and so
is true for any
(-ir))
exp(-ir))
)=y
AI=
A[f(x+iy)exp(-ir
gives that
Ar
in.
h as the required invariance property.
//
With the decomposition of A
r
in mind, we define:
Definition 2.3.1
fsM(rXG) is a
We say that
Ar f + Xf=O
Modular
f is C" and
for some XeR.
There has been extensive study of the
First of all
form if
\>Q
unless
eigenvalues X.
there are negative eigenval-ues :
corresponding to holomorphic functions
y-r/2 u(z), uEM(r,X,G).
-48-
These eigenvalues correspond to the
of
Discrete series representation
SL(2,R).
We say that X is an Exeptional eigenvalue, if
and denote by A
Now if
X
the
does not correspond to the discrete series representation
X
if
series representation of
For
,
finite set of such eigenvalues.
then we have that X<
to the
O<x<
to the
correspond
SL(2,R), and
if
X>
Continuous series representation of
r=O , the
not occur, and
Complementary
X
corresponds
SL(2,R).
discrete series representation does
Selberg has conjectured:
Conjecture 2.3.1 (Selberg)
For r=O, G a congruence subgroup of SL(2,Z), there are
no exceptional eigenvalues.
This can be generalized to:
Problem-2.3.1
Let G be a congruence subgroup, and
SL(2,R)
as before. For'which
representation
not
occur
r
in
rcQ,
0< r <2,
and
does the complementary series
R-
?
R-,
g
-49-
is true
Selberg showed that Conjecture 2.3.1
for
G = SL(2,Z). For G a general congruence subgroup, the best
that is known is the result of Jacquet-Gelbart, which gives
that
X>3/16,
if
X O.
has many application, for
The truth of conjecture 2.3.1
example in the work of Iwaniec and Deshouiller [Iwan-Des]. There
is a survey article
Now let
[Vigneras] on the work on this conjecture.
6(z)=
e(n z),
famous Jacobi theta
zcH, be the
n=-co
function.
It is well known that
e(gz)=X (g)(cz+d) 21 O(z)
for
1d£
c d
Where
)ESL(2,z)f 4jc}.
(4={(a
,ge'r
satisfies:
6(z)
c O
c=O
(4)
the Jacobi symbol, and
is
Since it is known that
that
X
so called
is
e(z)#O
d=
for
d=l (mod 4)
1
if
1
if d=3 (mod 4)
zEH, Corollary 2.1.1
a multiplier system of weight
%
implies
for r 0 (4)-the
theta multiplier system.
Now it turns out that
3/16
is an exceptional eigenvalue
-50-
corresponding
to
Goldfeld and Sarnak have shown that
However
not corresponding to
For general
y-4 (z),
real weight
it must be that
for X
X>15/64.
r, the best known result
0<r<2, X an arbitrary multiplier, G any congruence
is that for
subgroup:
e(z) is holomorphic.
yk0(z), and
X> %r(l-4r).
This result is due to [Roelcke], and was actually proved
for G any
of
Fuchsian group. That is, a discrete subgroup G
SL(2,R) such that
J
F
domain of
G.
dxd
yZ
<
, where
F is a fundamental
-51-
CHAPTER
III
SPECTRAL THEORY OF AUTOMORPHIC FORMS
-52-
In this chapter we will be considering a fixed congruence
subgroup
G,
of weight
a real number and
r
r
for
G.
We recall from
X a multiplier system
we have a complete set of inequivalent cusps of
00
=K
, K 2' '. .
Further if
with
So
X
0
every
'p
GC={ 0
=e(-).
G:
id.=a ,. .. ,9hcSL( 2 ,Z) satisfying a j
with
Kh
that
Section 1.1
ncZ}, q >0, then we choose
We
let
G
={gGIgK =K }=a G
a=a(XG), 0<O1,
GO.
f M(r,X,G) has a Fourier expansion at w given by:
n=co
f(x+iy)=
nwe also write
an(y)e( (n-a)x)
q
j(gz)=X(g)( cz+d
9=(
g
b)EG
=K .
-53-
3.1
EISENSTEIN SERIES
Definition 3.1
For
(3.1.1)
j=1,...,h
Eisenstein series
we define the
E (zgs)=gF
K
G
[Im(or.- gz)1 s j(g~z) -1
zcH,
sEC.
Proposition 3.1.1
Re(s)>l,
converges absolutely for
E. (z,s)
and satisfies
E. (gz,s)=j(g,z)E. (z,s), for geG, in this domain.
S
Proof:
We note
5
that [Im(gz)] =
s
Icz+di
9= a
for
lj(g,z)1=1. So for example:
while
2 Re(s)
Icz+d-
Re(s)
c :)
GQ
which is well known to converge for
Similarily
Re(s)>l.
nverges absolutely for
E. (z,s)
J
It is then obvious that
for
2s
E (z,s)
satisfies
Re(s)>l.
E (gz,s)=j(g,z)E (z,s)
gcG,Re(s) >1,
//
Remark 3.1.1
If
we define
[Im(gz)]s,
E(z,s)=
CG\ G
then it
can be shown
-54-
that for
Re(s)>1, s fixed,
From this it follows that
in other words
|E(z,s)-ys|
E(z,s)
is bounded
is not
for all zeH.
square integrable,
E(zs) M(rXG).
Selberg has proved the following:
Theorem 3.1.1
has an analytic continuation in s to the whole
E . (z,s)
complex plane, except for a possible finite set of simple poles
on the interval
(%,1].
at these poles p,...
,py
Furthermore the
residues
1,.. .e
are square integrable automorphic
forms which are not cusp forms.
Also, the poles of
E.(z,s)
coincide with the poles of the constant term in the Fourier
expansion
Proof:
of
E.(zs).
[Kubota]
//
We will compute the
but first we need
Fourier expansion of
E .(z,s),
some special functions.
Definition 3.1.2
Let
a,beC, we define
the Whittaker function for
a-b-%
-
,,
6.
Mi
.1--l-
-
-
--
-
-
- -
-
-
- -
-
-
----
-
-55-
not a
negative integer by:
Wa,b (y)=
and
1 F(a-b+%)exp(-%y)
2ai
t(+-
(1
2 exp (-t) dt
)a+b-ex-
C
is the contour given by:
C
-t
(3.1.2)
(-t) b-a-%
C:
:
-e(
t+E)
t< -E
:
-E:
for any
<t<E;
s>Q
2E;
t
:t>C
and looks like
C
It is known that
Wa,b (y)
set of solutions of
(3.1.3)
As
u
u"(Y)+
W - a,b (-y)
and
Whittaker's equation:
k-b 2
+
y +
2 )u(Y)=0
y
k
(- 4
are a fundamental
W a,b(z)% exp(- z)
as
y >0.
, we see that only
z +
W a,b(y), y >0, satisfies the regularity condition at
o.
We can now prove:
Theorem 3.1.2
n=co
Let
Bn,j (y,s) e((n-aI)x)
E (z,s)=
n=-o
B
.(y,s)=6n ys0
T(n-
q
a)
s-1
D
.(s)
W
3.
then
F (s+r/2)
r
-sgn (n) ,s-%
(4nn-aj
q
y)
-56-
Here
D n.
D nj(s)
is the
Dirichlet series:
c-2s
(s)=
n,3
(n-a)d
qc
e(
X(g)
O<d<qc
C>0
* c-'G
9=*
g=cd
jG
Proof:
n-=
q
We write
q
Bn
, and note that for
-1
ys+
(y, s)=0
s
.z)I
Re(s)>l
(gz)-1.e(- x)dx
G .
J.
q
0i
-1
s+
z
0
[Im(gz)]
as
1 e(-Sx) dx,
-1
G K. =a Ji G 0ja.
G~ jG
r
s
=6n ys+ c,d
g= cc
j(gz)
Icz+d I
r
Icz+dI 2s
e(- x)dx,
since, by
X (g) (cz+d)
G,c>O
a
by absolute convergence, we can interchange summation and integration,
and we have
d=qr c+d
1
=6
ys+ ys
expanded
(0,<d
1
<qc)
and obtain:
e(-
g)e (
o
O<d<qc
(s) ys
We now write
j(gz).
I X
g= (
=n ys+ D
0n.
Im(gz),
a
co
G
dx
e(- x)
2s-r
r
z
r dx
-
zr
-57-
The theorem will therefore follow
if we prove:
Lemma 3.1.1
s
1
-e(x)
dx=
Y 2s-r r
(4 IT
Wr
y)
7sgn(n) ,s-
i r(s+r/2)
Proof:
e (-
e (- Bx)
= y
z 2s-r r dx
Ys
-00Iz I
now
we have
x2 +2=(y+ix)(y-ix),
i (y-ix)=(x+iy)
(y-ix)s-r (Y+ix)s-r
We now
let
1
ir2(4y)s-1
x
exp(-
2 TrBy)
r
s-1
5s+ix)
+r (.+ix)
dx,
s-kr
1, exp(47rr y( +ix))
( +ix)s+ r
1
2T1i
f
CI
we let
dx.
x
+
-x
We put z=-
and expand:
to get:
-ix
_2iX) s-%r
exp(4 3y(-z))
dz
(-Z)s+ r(1+z) s-%r
+
Where
C' is the contour
C':
- -it,
t=-0
C'
to t=0.
-2
Now let
I = C
J
r dx
and obtain:
2yx
(-
-00
so above is:
e(-rx)
( ix) s+-2-r (Y+ ix) s
ir
COe(2pxy)
exp(-27iry)
i 2(4y)s-1
_
+
5'
dx=
(y_-ix) r
dx
(X+y)r
(X2 +y2 )s-r
-00
e (-Bx)
1 CO
x)
exp(47T y(-z))
dz
(-Z)s+r(1+z)s-r
Where
4-
C
are given by
-- , T>4,
/ --- -iT
..
Cl
- +iT
C4
C. : j=
0
;
C
C2
O<
-<,12.
,4
-58-
C2 , C3
are segments of the circle
is now seen that
where
IC',IC
as
T
-+
>:
IC
(4
1
2Tri
C
s--
- (-z)-2r--+
3
C'
14 * IC'
I., j=l, ...,4.
IC +C +C2+C3=0,
exp(-4Tr yz)
(-~+
y)s+ r-1 7exp(-27 y)
r (4y) s-i
~
(4y
i 4s-lF (s+%r)
C3,C4
we conclude
The above quantity is thus:
IC' .
Trexp(-27y)
r
s1
I1 ,I2
defined similarily to
are
As the residue theorem gives that
that
while
,
C, C' defined above, respectively.
are segments of
It
{Iz+Ik=T}
2'rri
r ( - ( s -%)
fexp(-z)
C
+%I+%1r)exp
)-+
)(1+
r dz , we let
r(1zs-%r
s+-)
y) (47,
z
4a y
-s+ r dz
z
(-z) -s- r
(--2-7
z+
y),-k
exp(-z)dz
}
But, by Definition 3.1.2 , the expression in brackets is
W 2rsgn(n),-s+%
sgn(3)=sgn(n), and also
%rsgn(n),s--(4Tlj
y),
as Osa<l
Wa,b=Wa,-b'
The result follows directly.
//
implies
-59-
Corollary 3.1.1
Let 0(z)
then
be the residue of
O(z) has the
E (zs) at p, as in Theorem 3.1.1 ,
Fourier expansion at w given by:
n=w
0(x+iy)=
ny
an y) e(
A=-cq
7
0
x),
(n--a) p-i Res
) 4+
where
Dn
(s) W rsgn(n),p-%(4- 1n-oIy)
q
1 rIr(p+kr)
Proof:
Follows by taking the
residue of
B
.(s)
at
s=p
and then applying Theorem 3.1.1
//
We conclude this section by proving that 0.: j=1,...,y
are
modular forms.
Proposition 3.1.1
Let
6(z)
correspond
A re+p(l-p)=0O,
Proof:
We show that
to
Res E.(z,s)
s=p
i
thus
for
ArE (z,s)+s(j-s)E (z,s)=0.
follow by analytic continuation.
then
corresponds to the eigenvalue p(1-p).
Re(s)>l
The result
will then
-60-
2s
s = Y2
2
Ar7s
-
Ar [(Iq)S
+
2 s
S)-iry
ax2
=s(s-l)y s-2y 2=s(s-l)y .
We now use the invariance property of Ar given in Proposition 2.3.1
to get
Arl(Im(gz) sj(g,z) -1 ]=s(s-l)(Imz) s
Summing over
gE G
Kj
G
j
(g,z)
-1
gives the result.
//
Remark 3.1.2
For
as
O<p
p.fl, 0
3
<1
We will
corresponds to an exceptional eigenvalue,
C)
implies that
0< pj(l-pj)<
adopt the notation
0.(z)= Res E. (z,s).
s=v
3
v.
4.
to denote E
(z,s) where
-61-
3.2
SPECTRAL DECOMPOSITION OF M(r,x,G)
Definition 3.2.1
Let
C(r,X,G)
be the space of functions:
( -,v
fE
c=1
u~z)-
+it)
>E i (z, +it)dt,
for fcM(r,X,G)
C(rXG) is the Continuous spectrum.
Definition 3.2.2
0 1,..,y
Let
Theorem 3.1.1.
as in
of
b e the residues of
We denote by
M(r,X,G) generated by
We also recall that
of f at
.(y) e(
R (rXG)the subspace
S(rXG)
(n-a) x)
q
is the subspace of cusp forms,
a0,j(y)=0
is the
for
j=l,...,h
where
Fourier expansion
K..
Selberg's spectral decomposition is then:
Theorem 3.2.1
,
6 1,...,e
that is, function satisfying
a
all E (z,s) at pl,...,p
(Selberg)
M(rXG)=S(rXG)9C(r,XG)@R(rXG)
-62-
We further have:
Theorem 3.2.2 (Selberg)
S(r,X,G)
has. a complete orthonormal set given by
{u }
eigenfunctions of Ar.
Xj-*co as
X0=O<x 1 <X 2 *.'
_
,
Sjj-v
to holomorphic functions
Aru +X u =0,
r j
j-O
.
y-
u (Z).
jj
x.,
And
j<O
where, if
K=m,
then
correspond
Putting these results together yealds:
Theorem 3.2.3
Let
fcM(r,X,G)
then
-1
<f,u.>u.(z)+'7<f,u.>u.(z)+
<f,e.>6.(z)
J J
J J
j=1
J
(3.2.1)
+
(-,
1<f,E
J=0
+it)>E
We now compute the
U
(z,2+it)dt
-
j)Q, given in
Fourier expansion at
of the eigenfunctions
Theorem 3.2.2
Proposition 3.2.1
For
a
j >0
let
u.(x+iy)= Ia
n#O
.(y)=an~j.W1-~rsgn(n)v/T- 4rn-y)
n~j
-A
q
.(y) e( (n-a) x)
n,q
then:
-63-
Proof:
Equating Fourier coefficients in
(3.2.1)
al
.(y)+(-4T 2 2 + irr
ny
+
j
(3.2.1)
is just
n,j (y =
)a
Whittaker's equation
So after a change of variables we get the
aA[ Wr,/4-:
n~j
I/T41n A.(47rfy) + a"
.(y)
a" .=0
n,
+
if
0
as
>O,
y
+ <o.
(3.1.3).
solution
(-4Tr
y)
W- -2r ,
But we note that the square integrability
a
gives:
a _ n-a
q
where again we have written
We see that
A u.+k.u.=0
rj
jj3
of
u.
So as in Section 3.1.2
that is n >0, as a<l. And
implies
we have
a' .=O if
n,
n<0.
The result then follows,
//
-64-
POINCARE
3.3
SERIES
Definition 3.3.1
Let
(3.3.1)
zcH,
m >0 be an integer, then for
P m(Z' s)
=
seC
E
Im(gz)s j(g,z)~
gEG\0G
Non-holomorphic Poincare series.
is the
Proposition 3.3.1
For
Proof:
then
Re(s)>1,
(3.3.1) converges absolutely and Pm z,s)cM(r,X,G).
It is clear that if
P (z,s)
(3.3.1) converges absolutely
satisfies the automorphy condition.
7
We note that
which converges abolutely for
(Imz)s e( (m-a)
q
v=
z)
, (Imgz))Re(s)
IP (Z's)g<
Re (s) >1
m
g G
as in the
proof of
Further, we have that:
Proposition 3.1.1.
maximum
so
qe( gz)l<l
(m-a)
yRe(s)
((Re(s)-1)g
exp(-2n(M-a)
y)
exp(-2~q
exp(1-Re(s))
2 (m-c)
at
which has a
y= (Re(s)-1)q
2 (m-a)
So in fact:
IP m(z,s)I<
j(Im(z)s e( (m-a)
q
z)I
+ IE(z,s)-ys j< yi+JE(z,s)-ysI<Cst.
-65-
E(z,s)=
Where, as before,
is bounded for
G
and |E(z,s)-y s
(Im(gz))s
zeH, s fixed.
Pm(z,s)
We therefore have that
Re(s)>1.
g E,\
ifF
Hence
<w3
dy
2
y
is
bounded for zEH, s fixed,
implies that
Pm(z,s)
2
F
dxdy <o
2
y
and the proposition follows
//
Pm(z,s)
The important property of
the
is that it
"picks"
f automorphic forms:
n th Fourier coefficient
Proposition 3.3.2
Let
fEM(r,x,G)
f(x+iy)=
exp (-2
Tr
(M-a)
0
Proof:
q
y) am(y) y
<f,P m(-,s)>=
F
= -
f(z)
F
Im(gz) s
00
then
(n-ca) x)
q
an (Y) eC
<f,Pm(-,s)>=
Fourier expansion at
have the
j(g,z)
s-2
dy
f(z)Pm(z, s) dxdy
2
y
e(_ m-a) gz)
q
dxdy , by
yZ
absolute convergence we have interchanged integration and summation.
=- ff
gF
G
gF
(Im (z))
f(z) e(_
M-a)
q
Z)
dxdy
yz
-66(
s f (z) e(- (
=
H
G\"
=5 yi
z) dxdy
yT
q
exp(-2r (mt~)
0
a (y)
mr
0
y-2
s
M-a)
f(z) e(-
z) dxdy
0 0
f(z) e(- (M-a) x)dx dy
q
0
c
s-2
exp(-27 (M-a)
q
dy
//
For further investigation of
Pm(z, s) we will
require
o.
its Fourier expansion at
Proposition 3.3.3
n=o
(n-a) x),
Q n(y,s) e(
q
Let Pm (z,s)=
n=(y~s)
s) =Sn y n,s ,
,
Qn
Y
c- 2s S(mjncXG)
c>0
0e[(- (n-a)x--a)
-
X(g)
g £ \G//G
g=c
dx
z2s-r zr
--
S(m,n~c, XG
Where
then
e[(m-ct)a+(n-a)d]
qc
b)
is the generalized Kloosterman sum of Section 1.4 .
We have
Proof:
=ns +
=
-G
+
+gi-
q
f e
0
x)dx
)= Prz,s)e(-n-a)
q
(IMn(y,s=fPm(,)e
(-a gz)(Im(gz)) s
q
by absolute convergence for
x(g)
Re (s) >1.
r
(cz+d rr
(cz+d)r
e(_(n-a) x) dx
q
-67-
Now
writing
for
g= a
Im(gz)= jcz+dj 2
c
2
e[
s
(m
__
x)dx
q
ce
-cc-z-(z+d/c)
0
z+d cz|
As in the proof of Theorem 3.1.2
0<d 1 <qc.
and
]z+dlr
(
22
Y)
\G
glid.
CEZ
c c(z+d/c)
And the above is:
Sys
Sge
gz
and
X(g)
we write:
(z+
-)
d= qc+d
,
where
The result then follows directly.
//
Remark 3.3.1
We see from the above proposition that the generalized
Kloosterman sum occurs naturally in the Fourier expansion
of
In fact it occurs as a coefficient in the
Pm(zs),
Dirichlet series:
which is the
Using
Z(s,m,nXG)=
c >0
S (m,n,c,x ,G)
2s
Kloosterman-Selberg Zeta function
2+LOO s s
Xds=
1
Perron's formula:
2Ti
2-i
0
for
x >1
for
x <1
2+io
we have:
0<c<x
S(m,n,x,G)
c
=2
2
f
2-ic
Z(l+s,m,n,X,G) x s ds
s
2
and this is the key to the Goldfeld-Sarnak proof of Theorem 1.4.
-68-
We will give the spectral decomposition of
Pm(zs)
for which we will need:
Proposition 3.3.4
(3.3.2)
<P (-,s),E.(-,t)>=ir
j
m
q-a)-
4
-s
(m-a
-1D4q .(t)
.r(s-t)r(s+t-1)
1 (s- r)F (t+ r)
Proposition 3.3.5
For
j
u. as in Theorem 3.2.2, anj as in Proposition 3.2.1:
J
1-s
>0,
<P (-,s),u.)>=a
m
j
mj
(4
) )(m
q
S(s-/ -X.-%)F(s+/-
.
)
F (s- r)
Before proving these propositions, we note that Proposition 3.3.4
gives:
Corollary 3.3.1
Let 8(z)= Res E.(z,s)
s=p i
<P (-,s),O>=
M
Proof:
(47
-
q ) 1
be as in
Theorem 3.1.1
(m-~
4 a))
q
Res D
s=P
m,n
then:
(s) F(s-p)F(s+p-1)
(s-%r)F (p+%r)
Follows by taking the residue at s=p in
(3.3.2), and by noting that pc(k,l]
and so is a real number.
//
-69-
Proof of Proposition 3.3.4:
Writing
E
(Zs)=
q(y,s)e(
x)
B
n= -co
as before, we have, as in Proposition 3.3.2, with
q_a
00
<P (-,s),E(-,t)>=<E(-,t),Pm M-,s)>=0
0
S
=50
(
B
.(y,s)exp(-2nry)ys-2dy
t-
(M- a)
D
4q
exp (-27rf-y)
(t)
y
s-2
dy
r (s+%r)
r-
Tr(m-c)a)-
-
4q
D
(t)
W
-%(4T
0
F (t+ r)
2r
.
m-a y)
q
exp(-2T (mq
Real analytic.
as all function are
We now require:
Theorem 3.3.1 (Barnes)
oi
W
r (w-a)r(-w-b+%)
)F(-w+b+)
1 (-a-b- )r(-a+b+ )
b(y)=exp(-y)
27ri
_
[Whittaker and Watson
Proof :
We
j
.exp ( -2f y)2Tr i
-
300
have:
yw dw
//
§16. 4]
Wr,-%( 47Ty)
(w-%r) F ( -w-t+1) F ( -w+t) OTTTy) w dw
(
Substituting in the above gives:
Tr
)E-1
r D
.(t)
F(E+kr) F (T-%r) F (-t-%r+1)
Ci
exp (- -4f y)
0
ys-2
e(w-r) F(-w-t+l) F (-w+t) ( 4 7TSy)w dw dy
2 dy
s-)y)y
-70-
We interchange the order of integration in the above and note that:
00
f0 exp(-4fry)(4ffy)w y s-2
. So we obtain the integral:
dy- r(w+s-1)
(47T)1
ooi
- i
2Tr
_--0
r (w-
r)Ir(-w-E+1)Tr(-w+t)r(w+s-1)
_ P(-%r-E+1)r(t-r)r(s-t)r(s+t-l)
r (s - %r)
by
dw
Barne's Lemma
[Whittaker
14.52].
and Watson,
The result now follows directly
//
Proof of Proposition 3.3.5:
This is similar to the previous
proof, as we have:
<P (-,s),u
>=
m
j
am,3
amqj
m-)
q
1-s
0
r ,/y-
F(s-/-X
(m-a)y)exp(-2m
qr,
-
S)(s+'1-_
)
-%)
F(s-4r)
By the same computation as above.
//
Putting the above three results together yealds:
Theorem 3.3.2
P m( z Vs )
has the spectral
decomposition:
s-2 dy
-71-
P (z,s)=( 4qM-a
m
+
q
j=1
If
Ar
( M-a
;
D
a
m,
.
~-eA-)~+eA-)
3
u. (z)
3
J
.
02 -it)
r(s-
2+it)
r(S-';--it)
E(z,%+it)dt
_00
t
/
- -it
J(
i
4
+
1
r(s- r)
r (k-it+kr)
)
Pj~
r(s-p.)1(s+p.-1)
r (p +%r)
has no negative eigenvalues in M (r, X,G) .
Res D
m, v
S=
J
j
(s) 6.(z)
J
}
-72-
CHAPTER
IV
APPLICATIONS TO EXPONENTIAL SUMS
A
i
-73-
4.1
THE DEDEKIND
n FUNCTION
The classical Dedekind
T
r(z)=e(z/24)
(4.1.1)
n
AND MULTIPLIER SYSTEMS
function is given by:
(1-e(nz)),
for
z6H.
n=1
If
we recall that the Dedekind sum
s(d, c)=
_((j/c)) -((jd/c)),
where
is given by:
c,dEZ
and ((x))=x-[x]-%, xsR.
j=l
Then
we have the following transformation law:
Theorem 4.1.1
Let
b)
(Dedekind)
log z denote the principal branch of log, then for
cSL(2,Z)
we have:
log rl(z)+ k[log(
(4.1.2)
cz+d
)+27rri
a+d
- s(d,c))] ,
for
log(n(gz))=
+(122Tib ) ,
log n(z)
c=O
There are many proofs of this, an elegant one given
Proof:
in
for
[Siegel].
//
an
n2r(z)=exp(2rlog q(z))
will show that
We
automorphy condition:
Where
g=
E)SL(2,Z),
and
n2r (gz)=xr
jr(gz)
j r(g,z)=(cz +d)r
satisfies
2r (z),
c#O
-74-
For
this we require:
4.1.2:
Theorem
b
wa
Let
and
(4.1.3)
j
1
g1 ,g2 '93 ESL(2,R),
(gj,z)= c z+d .
log(j 1 (g1 ,z))+
where
Then
for
gj= (c.
g 3 =9 2 9 1
log(j 1 (g2 ' 1 z))= log(j 1 (
3 ,z))
d.,
j=1,2,3.
we have:
+ 27TiW(g 2 'g ), where
1
[sgn(c 1 )+sgn(c 2 )-sgn(c 3 )-sgn(c1 c2 c3 );
- (1-sgn(c))(1-sgn(c2);
Proof:
C1 c 2 0,
c 3 =0
(1+sgn(c1 ))(1-sgn(d2)),
*c c3
(1-sgn(a1 ))(1+sgn(c2));
C2 c 3 #O,c1 =0
(1-sgn(a1 ))(1-sgn(d 2 ));
c 1 = c 2=c3=0
[ Maass]
Theorem
c 1 c 2 c3 00
,c2= 0
16, p. 1 1 5
//
Corollary 4.1.1
Let the notation be as in Theorem 4.1.2, and define
ag
(4.1.4)
2 ,g)=
e(r-W(g 2 'g1 ),
j r(g1 ,z)jr(
Proof:
2
'
1
It is seen
then we have:
)= jr(3,z)
that
(g2 '91 )
(4.1.4)
is obtained by
e(r-)
equation (4.1.3).
//
of
-75-
Proposition
4.1.1.
Let f(z)= exp(2rlog n(z)), then for
f(gz)= Xrg
jr g~z)
(4.1.5)
where
f(z),
e [ r(
+ a+d
12c
-1/4
Xr(g)=
if
Pe (rb/12),
s(d,c))]
,
c
>0
c=0
This fo llows by applying
Proof:
ESL(2,Z)
=gccd/~
e(2r-) to equation (4.1.2)
r
and appealing to Corollary 4.1.1 to get that: (cz+d
i
i- 1 =e(-1/4) and
where
we can always choose
(
c >0.
)
with
=(cz+d)re(-r/4),
As we have identified
g and -g,
c ,0.
Corollary 4.1 .2
Let G be a congruence subgroup of SL(2,Z), r a real number
then
X
given by
multiplier system of weight r for G.
By restriction, we see that (4.1.5) holds for any
Proof:
ge G,
(4.1.5) is a
r
a congruence subgroup of
that n(z) 0 for zEH , therefore
Therefore
Corollary 2.1.1
system of weight r for
SL(2,Z).
2r(Z)=f(z)
implies that *
Now we see from (4.1.1)
O,
for zeH.
is a multiplier
G.
//
-76-
4.2
APPLICATIONS TO DEDEKIND AND KLOOSTERMAN SUMS
We
GC=
{( 0
recall
that for G a congruence subgroup
1nq
1 )IneZ}, where
Furthermore
F(N)={ (
it is
denote
q=q(G)>O
clear that
)ESL(2,Z)l(a
Let us now fix
given
b),(l 0)
r=h/kE
Q,
the multiplier system
in
Xr( c d)~=
+
q(P(N))=N, where
Ns IN.
(mod N))
with
of
h,kE N.
weight
s(d,c)))
c
r
for
c
0
let Xr
r (12k)
,
for
the Generalized Kloosterman sum:
S(m,n,cX r,(12k))=
Xr (g)
O<d<12kc
9= (a
we note
that
q=q(r (12k))=12k,
e(
am + dn )
l2kc
r (12k)
Xr( 0 1)=e(qr/12)=e(12kh/12k)=e(h)=l=e(o)
and
therefore we have ac(Xr j(12k))=O.
We immediately have:
Theorem 4.2.1
Let
We
!e(rb/12), if c=O
We will be studying
as
is uniquely defined.
Proposition 4.1.1:
e(r( -
where
we have
m,n,c
be positive integers, then:
-77-
(4.2.1)
S(m,n,c,Xr,
1 2k))=e(.dr)
O<d<12kc
+ hs(d,c)
e( a(m-h)+d(n-h)
l2kc
k
(a
dI r (12k)
C
(4.2.2)
+d(n-h) + hs (d, c) )
e( e(a(m-h)
l2kc
=e( r)
O<d<12kc
(d,c)=1
d=l(mod 12k)
c=O (mod 12k)
a-d=-l (mod c)
Proof:
Substituting the explicit formula for Xr
Kloosterman sum, and noting that
[ (-4)+
-
O<d<12kc
c
and
We
S(m,npc~,.r(12k))
s(d,c)] + lknc)
(12k)
the first
obtain
equation
(4.2.2) from
(4.2.1) follows.
(4.2.2) by noting that
c=O (mod 12k), d=l (mod 12k)
that
gives
and we have
[(l-sgn(c))(sgn(d)-1)=0,
e(-
c>O
in the
ad=l
and (c,d)=1; also
cd Er (12k)
ad-bc=1 implies
(mod c).
//
Corollary 4.2.1
S~h~~cX
/k~
(12k))=
e(-4 h )
iff
e( h s(d,c))
O<d<12kc
c=O (mod 12k)
dEl (mod 12k)
(c,d)=1
-78We now apply Theorem 1.4,p.10, to S(m,n,c,XrVF(12k)) and get:
Theorem 4.2.2
S(m,n,c,Xr, r(12k))
(4.2.3)
Q<c<x
for
any
0<x <4
c
c>O.
Where
A
XEA
xT
(x /3
+0
+
)
A is the set of exceptional eigenvalues
associated
to
Also
= Inf
6>0
constants.
=
M(r,Xr ,r(12k)), T =2vT-
{7
C>-
IS(m,n,c,x r, r(12k))
1+6
c
I
and A. 's
i
<0), Iso
are
81.
Corollary 4.2.2
For
0<r< 2,
and for every
Sma
Y
S(m,n,c,Xr ,or(12k))= (
c>O,
we have:
2/
-(%r(l-%r)
xc{
1/3} +E
)
m
r<c<x
in particular,
0<c <x
for
S(m,n, c, Xr' r (12k)) «
c
Proof:
which gives
This
that
this gives:
2/3<r<4/3
X>
1/3+F
follows
4r(1-
r)
from
for
for any
Roelcke's
any
>0 .
result [Roelcke],
XcA, and from the
trivial estimate f 41.
//
-79-
We
We
now
let
r= h
k
be an arbitrary rational number, h,kE N.
examine in detail the
case
m=n=h
in
Theorem 4.2.2.
Proposition 4.2.1
There is a 6>0
(4.2.4)
S(h,h,c,x ,r1(12k))
c
0<c<x
Proof:
so
v=
We
as
A
such that:
note
that
1-6
implies that T= 2/kEE <1
0<X<
is a finite set, by Theorem 3.2.2, we have
Max{T. }< 1.
X.cA J
Theorem 4.2.2
Let P'= Max{P,1/3}<l,
that 6=
-11
will give
us
then we see from
the required estimate, if
e<6.
we choose
To prove our next result we require the following well
known technique.
Lemma 4.2.1
Let
for
(Partial Summation)
f(t) be a continuously differentiable function
1<t<x, and
A(x)=
>_1 a(n)
then:
0<n<x
A(t)f'(t)dt
a(n)f(n)= A(x)f(x)-A(l)f(1)1<n<x
Proof:
,
1
A proof can be found in [Apostol]
p.77.
//
-80Proposition 4.2.2
There is a
6'>0
such that:
x2-61'
S(h,h, c,X r , r(12k) ) <<
(4.2.5)
0<c<x
Proof:
We apply partial summation
f(t)= t.
So
there is a constant
Proposition 2.4.1.
K
with
with
a(n)= S(h,h,c,Xr,1r(12k)),
A(x)<K-x 1 -6 , by
We therefore have:
x
S(h,h,c,X r, r(12k))=
0<c<x
<<
0<c<x
<-6
X K1- 1-.
x+f Kt 1 6 dt
=
K(1+
a(c)f(c)= A(x)-x - 0-0+ fA(t)-ldt
1
2 -6 )x
2-6
<<x
2-6
. And we can
take 6'=6 <2.
//
To prove our concluding Theorem, we will require:
Theorem 4.2.3 (Weyl)
Let
{%}
Lf
j1l
be a sequence of complex numbers of
CO
absolute value one, then
iff for every
Z
j<x
the sequence {% } j1
is equidistributed
me N, we have:
= O(x)
E.
i
Proof:
A proof of this is given in [Polya-Szego]
Part II,N*164.
//
-81-
Theorem 4.2.4
Let
{x} = x-[x]
denote the fractional part of x , xs R.
The sequence of fractional parts
h s(d,c)
c>O
k.
c=O (mod 12k)
0<d<12kc
(d,c)=1, d:l (mod 12k)
is equidistributed on [0,1).
Proof:
that
We first note that this is equivalent to the statement
e( h s (dc))
k
}c>0
cO (mod 12k)
0<d<12kc
(d,c)=1, d:l (mod 12k)
the sequence
is equidistributed on the unit circle T.
We next note that:
where
p(n)=
7
0<c<x
c
(mod 12k)
0cscO
0<d<12kc, (d,c)=1
d=l (mod 12kc)
1
U__
1
is
$(c)
c:O (mod
12k)
$ function.
Euler's
(j,n)=1
Now it is well known
for example
in
that
x<<
ZI
$(c)
0<c<x
c:O (mod 12k)
[Apostol] Theorem 3.7.
If we now let m be a positive integer, and we combine Corollary 4.2.1,
Proposition 4.2.2
-
I -
-
--
. -
-
- -
,,
-82-
e( $)YI
mhks(dc))
e(k
O<c<x
cEO(mod 12k)
O<d<12kc, (d,c)=1
dEO (mod 12k)
for
S(hm,hm,X r,r(12k)) <<
x2-6
O<c<x
a 6>0.
Combining this with:
x
<<
1
O<c<x
cEO (mod 12k)
O<d<12k, (d,c)=1
dEO (mod 12k)
the sequence
e(
by
h s(d,c)
above, we see
that
c>O
cEO (mod 12k)
O<d<12kc, (d,c)=1
d=l (mod 12k)
satisfies the conditions of Weyl's criterion Theorem 4.2.3,
and is therefore equidistributed.
//
I --- "-
--- -maj
-83-
BIBLIOGRAPHY
1.
[Apostol]
Apostol, T.,
Introduction to Analytic Number
Theory, Springer-Verlag, New-York-Hedelberg-Berlin,
1976.
2.
[Gel]
Gelbart, S.,
Weil's Representation and the
Spectrum of the Metaplectic Group, Lecture
Notes in Mathematics #530, 1976.
Goldfeld, D., and Sarnak, P., Sums of Kloosterman
3.
Sums, Preprint, 1982.
4.
[Iwan-Des]
Iwaniec, H.,
and Deshouiller, J.-M., Kloosterman
Sums and Fourier Coefficients of Cusp Forms,
Preprint, 1982
Jacquet, H.,
5.
and Gelbart, S.,
A Relation
between Automorphic Representations of GL(2)
and GL(3), Ann. Scientifiques de l'Ecole Normale
6.
[Kubota]
1978 p. 4 7 1 - 5 4 2 .
Superieure
11 (4),
Kubota, T.,
Elementary Theory of Eisenstein Series,
Halsted Press, New York, 1973
7.
[Kuzn]
Kuznetsov, N.V., Peterson's Hypothesis for
Parabolic Forms of Weight Zero. Sums of
Kloosterman Sums, Math. Sbornik III(153),n*3(1980)
p.334-383.
-84-
8.
[Maass]
Maass, H., Lectures on Modular Forms of
One Complex Variable, Tata Institute Lecture
Notes # 29, Bombay, 1964.
Polya, G., Szego, G.,
9.
Problems and Theorems
in Analysis I, Springer-Verlag, New-YorkHeidelberg-Berlin, 1972.
10.
[Rad-Gro]
Rademacher, H.,
and Grosswald, E., Dedekind Sums,
Carus Mathematical monographs
#16,
Mathematical Association of America, 1972.
11.
[Roelcke]
Roelcke, W., Das Eigenwert Problem in der
Hyperbolischen Ebene III, Math. Annalen 167
(1966) p.292-337, ibid. 168(1967) p. 2 6 1 -3 2 4 .
12.
[Sell
Selberg, A., On the Estimation of Fourier
Coefficients of Modular Forms, Proc. of
Symposia in Pure Mathematics VIII 1965 (AMS) p.1-15.
13,
[Vigneras]
Vigneras, M.-F., On the Smallest Eigenvalue X
1
of the Non-Holomorphic
Laplacian, Preprint,1982.
Whittaker, E.T., and Watson, G.N., A Course
14.
of Modern Analysis, Cambridge University Press,
Cambridge, 1973.
15.
[Weil]
Weil, A., On Some Exponential Sums
Nat. Acad. Sci. USA
,
Proc.
34, 1978, p. 2 0 4 -2 0 7 .
-85-
Gelfand, Graev, Piatetski-Shapiro, Representation
16.
Theory and Automorphic Functions, W.B. Saunders Co.,
Philadelphia, 1969
Proskurin, N.V., Summation Formulas for
17.
Generalized Sums (in Russian), Zap. Naucn. Sem.
Leningrad Otdel. Math. Inst. Steklov, 82(1979),103-135.
18.
[Siegel]
Siegel, C.L., A simple Proof of
n(-1/T)=n(T)/T7i,
Mathematika 1 (1954) 4.
19.
Wohlfart, K., Ueber Dedekindsche Summen und
Untergruppen der Modulgruppe, Abh. Math. Sem.
Univ. Hamburg, 23 (195 9 )p.1-10.