MODULAR. FORMS OF DIMENSION -2 FOR
SUBGROUPS OF THE MODULAR GROUP

Thesis fiat H10 Dome. of Ph. D.
MICWGAH STATE UNEVERSITY
John. Roderick Smart
1961

 

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MODULAR FORMS OF DIMENSION -2 FOR
SUBGROUPS OF THE MODULAR GROUP

presented by
John Roderick Smart

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Date May 10, 1961

 

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ABSTRACT

MODULAR FORMS OF DIMENSION -2
BELONGING TO SUBGROUPS
OF THE MODULAR GROUP

by John Roderick Smart

Joseph Lehner has given a method for defining
Poincare series of dimension -2 on the modular group
['(l) which does not rely on the Hecke method of intro-
ducing a convergence factor. The problem considered in
this thesis is the following: extend the method to con-
gruence subgroups of the modular group; and determine
when the method can be extended to arbitrary subgroups

of finite index.

Let r'be a subgroup of finite index, and let

A3100, 3 = 1, 2, --- , 0(l—) be a complete set of inequiv-

alent parabolic cusps. AJ el’(l). We assume A3100 = 00 if

and only if A3 = I. Let’U’be an abelian character on F-.
- R
Define e(KJ) szAle JA3) where 7&3 is the least positive

_ 7x
integer such that A310 JA36 V and we use the notation

e(z) = exp[2viz]. ‘We define for integers p {'0

 

e((u+A.)V z/‘A )
(*) G(Z,/U,A ,F ,p) = i ‘g _:I_L Cid :2
J c=-m sztm “(A3 vc,d)(cz+d)

CCG(AJ,F) d6 flogAjgr)

John Roderick Smart

The sets of integers QT(AJ,T) and OEKC,AJ,K') are so
defined that vc,d = (a bl c d) runs over a complete set
of matrices in AJT— with different lower row as c runs
over é’(AJ,T ) and d runs over ¢3(c,AJ,i ). The double
series in (*) is not aboslutely convergent, therefore,
we specify that it is to be summed first on d and then
on c. With this convention the functions G defined in
(*) are regular in‘ki, the upper half plane. In order
to show the proper functional equation is satisfied we

must prove a Rademacher lemma.

By a lattice point for AJK' we mean the lower row
(c,d) of a matrix in AJF. Let 0(AJ, l") represent the
set of all lattice points for AJF‘. Furthermore, for
any positive integer K let C}K(AJ,F') be the set of all
lattice points for Aji— contained in the square with
sides u = i K, v = 1 K in the u,v-plane. We define a

class'Vn of matrices such that every Véilll has the form

MUn’X 'X

V=1UmwithM€—Wl, U =(l7\[01)andm

and n integers. Then the Rademacher lemma implies:

e((p+K )vc (ls/Ag:

 

(**) C(zs'U'sAJsr all) 3 11m :‘(UT-J‘

K‘>°° Vc d)(cz+d)2

(c.d)€6‘:(:3.r‘ )M

(EKMJJ' )M = i(c¢+¥d,gc +Sd): (c,d)f-0'K(AJ,i-) i that is

John Roderick Smart

we think of M as acting on the u,v-plane as an affine

transformation.

Using the Lipschitz formula we derive the Fourier
expansion of the functions G. This then shows that they

have the prOper behavior at the cusps.

All of these results required an estimate of

OHcll/2 +8 ) for the Kloosterman sums corresponding to
i-and«yt. We use a result of Petersson's which says

these sums have the prOper estimate if T.is a congruence

subgroup andlv'is identically l on a principal congruence

subgroup.

The problem we considered was solved in the following
generality: whenever the Kloosterman sums have the prOper
estimate the method of Lehner can be extended to subgroups

of finite index.

MODULAR FORMS OF DIMENSION -2 FOR
SUBGROUPS OF THE MODULAR GROUP

BY

John Roderick Smart

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1961

5:", a? z/ //
/..=’/2 2, 5,, /

To Pat

ii

ACKNOWLEDGEMENTS

The author wants to express his sincere thanks
to Professor Joseph Lehner for his patient guidance
on this and other problems. I want, also, to express
my indebtedness to Professor Lehner for his kindness in
letting me read portions of his forthcoming book on
Automorphic Functions. It most certainly clarified my
ideas and the notations which I have used in this paper.

I am indebted to two organizations for financial
support, the National Science Foundation and the
University of Michigan Institute of Science and Tech-
nology.

Finally, I should like to show appreciation to
Professor Hans Rademacher, who showed me his improved
method of partial summation for estimating sums like

(h.28). This was of utmost service to me.

111

51.. when“ h? "'- 1‘
.1 )AM‘H‘NU4-J

TABLE C

l ‘11

Section 1. Introduction ............................. 1
Section 2. Preliminaries ............................ 7
Section 3. Convergence and Regularity ............... 17
Section h. The Rademacher Lemma ..................... 29
Section 5. The Functional Equation .................. 55
Section 6. The Fourier Series ....................... 59
Section 7. The Behavior at the Cusps ................ _6h
Section 8. Kloosterman Sums, Main Theorem and Examples 70
Section 9. [The Inner Product Formula ................ 91

Bibliography OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 102

iv

m
A

to

L1

Figure 3.1
Figure h.1
Figure h.2
Figure h.3
Figure k.h
Figure h.5
Figure h.6
Figure h.7
Figure %.8

OF FIGURES

OOOOOOOOOOOOOOOOOO.
OOOOOOOOOOCOOOOOOOO
COOOOOOOOOCOOOOO...
OCOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOO.
OOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOO
OOOOOCOOCOOOOOOOOOO

20
31

35
#8

1+9
so
51
52
52

LI ST OF TABLES

1. Table 8.1 Generators and Exponent Sums forl'(6) .... 85
2. Table 8.2 Generators and Exponent Sums for FIR) .... 87
3. Table 8.3 Generators and Exponent Sums for r(8) .... 89

vi

1. Introduction

In this section we give a description of the problem
considered and the results of our research. The defini-
tions and results required for this investigation are

given in the next section.

Lehner [6] introduced the series

°° °° e<- -(p-°L)Vk r)
(1.1) r (1:) =2 X ‘m

M m__mg flVk,-m"'1‘kt'm”2

(msk)=1

 

where E is a multiplier system for l—(l) and the dimen-
sion -2, o _<, ot< 1, U = (1 1| 0 1), Vk,-m = (* *l.k,-m)
€l"(1) and chat ) = 6(U). We use the notation

e(z) = exp [2viz].

Furthermore, we write matrices in one line with a bar
separating rows. He proved that for u = l, 2, ... Fp(‘C3
is a mbdular form of dimension -2 for the multiplier system
Eregular in71L== {z = x+iy: y > 0}. Basic to the proof is
an estimation of the Kloosterman sums Ak,“(m) which arise

as 0(k1/2+E' ).

. t; '

Furthermore, the results depend heavily on

a generalization of a lemma due to Rademacher, which

-2-

allows the rearrangement of certain conditionally conver-
gent double series. Lehner derives the Fourier expansion
of these functions at the infinite cusp. These coeffi-
cients are expressed as an infinite series of the
Petersson-Rademacher type, which involve Bessel functions.
The functions F“(1:) are not identically zero since they

have a pole at It: 100.

Our problem can now be stated:
(1) extend the methods and results to congruence
subgroups of the modular group:
(ii) extend the results to arbitrary subgroups of
finite index in l—(l).

We obtain partial solution of these problems.

Let r_be a subgroup of l‘(l) of finite index.
Assume -16 I". Let 13100 = p3, J = 1, 2, a be a set
of inequivalent parabolic cusps of r.. AJELT'(1) and A1 =
I. Suppose 4I=1I(T',-2) is a multiplier system for T’and

the dimension -2. Consider the following series:

 

° ° °“|*"'3’Vc d25/71 )
(1.2) G(z,qr,AJ,f ,p) e Z Z l.

c=-oo dz-cn “RASJ'VC’CQ(cud)2
c 6 (is

guyr) (13¢ij >

where p is a non-zero integer and vc d = (* *1 c d)6 AJF'.
9
The sets Guyr) and fi<c,AJ,r ) are so defined that as c

runs over é:(AJ,r') and d runs over 05(c,AJ,V),VE d runs

-3-
over a complete set of matrices in Aj'Trwith different
lower row. 3‘3 is a positive integer which depends upon
r’ . 0 $»Kj < 1: K3 depends upon’U‘and f'. The series is
not absolutely convergent and we shall have to specify the

order in which the terms are summed.

If the series in (1.2) were absolutely convergent
(which it is if 2 is replaced by r > 2), we could rear-
range the series. We would get easily the following

results:

1) G(z,W3AJ,V‘.u) is a modular form. That is, it
satisfies the functional equation
(1.3) G(VZ{V:A33 roll) = ’U(V)(CZ + (1)2 6(29’15A31r :l‘)
for each V = (* *\ c d)6 F'. Furthermore, it satisfies
the required regularity conditions.

2) We would obtain the Fourier series coefficients

given in (6.10).

3) we would obtain the inner product formula given

in (9.2).

In order for us to obtain these results we must rearrange
conditionally convergent series. In doing so, we must

rely on:
k) a Rademacher lemma: and

5) a non-trivial estimate on the Kloosterman sums

-u-

(1,l+) wc(n+/(,p+xj)=ZIE(A31vc,d)e[(n+k)d/c +(p+KJ)a/C;l.3],
defié(AJDI—)

namely,
8
Wc(n,u) = O(|cll/2 + ) 5’0
where the constant in the C9-symbol does not depend upon n.

Our main result is this: ‘1; r'ig a subgroup of finite
index ig_l"(l), and i£_the Kloosterman sums have the re:

quired estimate then 1), 2) and 3) are obtained.

 

 

 

In 598 we give a series of examples of groups l. and
multipliers systems’U'for which we are able to prove the
validity of the Kloosterman estimate. They include:

F = Y’(l) and all six multiplier systems/U'of dimension
-2; r'= V'z, the unique subgroup‘of index 2 in i—(l), and
all nine multiplier systems: and f‘o(q), q a prime of the
form km + l and the multiplier system which depends
upon the Legendre symbol. In §>9 we return to these ex-
amples and calculate the dimension of the space of cusp
forms belonging to these systems. Here we use results of

Petersson.

The proof that the G of (1.2) satisfies (1.3) depends
heavily on our Rademacher-type lemma. Roughly, this lemma
states that the series (1.2) can be summed over expanding

parallelograms centered at the origin. This is somewhat

-5-
analogous to the standard way in which one proves the con-
vergence of the weierstrass 65-function. To some extent
we follow Lehner [6] in proving our Rademacher lemma:
however, since in general r.will have more generators than
T'(l) we need a more comprehensive lemma. M. I. Knopp
has developed other extensions of the original Rademacher
lemma ([5] , [5.11): still others are in the process of

publication.

For the estimate of the Kloosterman sums we rely on
the researches of Petersson [ll]. Petersson proves the
theorem: if r-is a congruence subgroup and’U’an abelian
character on V which is identically l on some principal
congruence subgroup T'(N) Cl’, then the Kloosterman sums
(1.h) have the required estimate. In § 8 we give an
elaboration of his proof so that the interested reader
may see the neat way in which the complicated sums (l.h)
are reduced to the classical Kloosterman sums. We require
the estimate of the sums (l.h) both in the proof of the
regularity of the functions G of (1.2) and in the proof

of the Rademacher lemma.

The above results are not all new. Petersson has
obtained them in [11]: however, his method of proof is
entirely different from ours. He uses the Hecke idea
[h, pp. h68-h76] of introducing a convergence factor
‘cz + dl's, s > 0 into (1.2). He then takes the limit as

s -> 0+. Our method of rearranging conditionally conver-

-6-

gent double series was first suggested by Rademacher [13].
In some senses our method is more natural since it follows

the proofs for dimension -r < -2.

2. Preliminaries

In this section we give the definitions, notations
and results which are needed for this investigation. No
attempt has been made to give a reference for every fact
which is stated, but rather, to give references for only
those results which lie deeper in the theory. Almost all
of the results for which no reference is given can be

found in Ford's book [2].

We shall be concerned with infinite groups l-of

linear fractional transformations

where a, b, c and d are rational integers and ad - bc

3 1. These transformations map the upper half plane

'7+ = {z : In 2 > 0} onto itself in a one—to-one manner.
The groups we are considering will have the further prop-
erties: (i) for every point p on the real axis there

is a sequence of different substitutions w = Vnz and a
point 20 such that the sequence wn = Vnzo accumulates
at p: (ii) the same statement does not hold for any point
p 574. Groups for which (11) hold are said to be
discontinuous in l+-. Groups for which both (i) and (ii)

Ihold are called horocyclic groups (they are also called
Fuchsian groups of the first kind and Grenzkreisgruppen

-7-

-8-

in the literature). This terminology is not used
exclusively for the case in which the region of discon-

tinuity is the upper half plane 74 .

Let the letters I, U, S, V, M, AJ denote the
following two-by-two matrices (written in one line with
a vertical bar separating the first row of the matrix

from the second)

1 = (1 0| 0 1), U = (1 1| 0 1),
(2.1) V = (a b \c d), S = (0 -l ‘1 0),
M = («.5 ‘Y 8 ). and AJ = (aJ bJI cJ d3)

where all the matrices given above are real unimodular

matrices. Further, for any real it , we write

(2.2) U“: (1) | o 1).
Also, put
-v = (-a, -b |-c, -d).
With each of the above matrices we can associate a

linear transformation, namely

(2.3) w = v2 3 41.1.1).

Notice that V and -V correspond to the same linear
transformation w = Vz. Thus,to any group T— of two-by-
two matrices there corresponds a group I: of linear
fractional transformations. -1 may or may not belong
tol— . However, since our interest lies in the groups

of linear fractional transformations we may assume that

-9-
—I €T—. If it does not then we merely adjoin it to F’
without affecting I" . It follows that F: F/ {1, -1{ .
There should be no confusion when we let V stand for the
matrix as well as the linear transformation. One uses

the same terminology for r-as we did for F': namely,

r'is discontinuous or horocyclic if and only if F is.

Two points 21 and z2 é‘Ft are said to be congruent
or equivalent with respect tol— if there is a V6 VT such
that V21 = 22. A fundamental Eggigg for T". R(i—), is
a subset of?! which satisfies: (1) R( F) is a non-empty
open set: (2) no two distinct points of R(T') are equi-
valent: (3) each point of ?+ is equivalent to at least
one point of the closure of R(l_). A fundamental region
for r'can be chosen so that it is bounded by circular
arcs and straight lines called giggg. A Eggggx of a
fundamental region is the common and point of two sides.
In our case the fixed point of a parabolic element in F'

lies on the real axis and is called a parabolic vertex

 

 

or parabolic cusp. Linear fractional transformations are
classified as parabolic, elliptic, hyperbolic or
loxodromic. We give the same classification to their

matrices.

We shall assume that a fundamental region R(l_) has
a finite number of sides. This is a restriction on F'.
A consequence of this assumption is that the groups we

are considering are finitely generated. There is a

-10-

fundamental region R([_) in which each parabolic cycle
consists of a single vertex: we shall always choose this
fundamental region. Since there are a finite number of
sides there are a finite number of inequivalent parabolic

cusps. Let this number be c (l’).

The modular group, F'(l), consists of all matrices
V = (a b [c d), a,b,c and d rational integers and
ad - bc = l. The modular group is a finitely generated
zonal horocyclic group discontinuous on ?+ . A discon-
tinuous group of real matrices is said to be zonal if it
contains a parabolic element fixing 00. The substitutions
S and U generate F-(l), and U is the parabolic element
fixing 00.

A fundamental region for the modular group is the
set of all z = x+ iy 67+, y) 0, such that
-l/2 < x < 1/2 and lzl > 1. Denote this fundamental
region by R(1). R(l) has a finite number of sides.
Rankin [1h] proves that if ['* is a subgroup of finite
index in.r then T" is horocyclic if and only if F is
horocyclic. Further, he proves that a fundamental region
for T'* has a finite number of sides if and only if a
fundamental region for F'has a finite number of sides.
As we stated at the outset we are interested only in sub-
groups of the modular group. If we add the condition that
the subgroup should be of finite index, then we will know
that it is horocyclic and has a fundamental region with a

-11-

finite number of sides.

We reiterate, r-is a subgroup of finite index in the
modular group. R( F) is a fundamental region for Fwith
a finite number of inequivalent parabolic cusps
p3, J = 1, ..., a (l’). We may choose A365 r (1) so that

-l
J
modulo F'. A coset decomposition for r‘(1) modulo F’

will be

A 00 = p3. Then the AJ do not belong to the same coset

G" 7‘3
r’(1)=U U 1310‘“!-
J=1 k=1

where, of course, the integers 7\J depend upon r..

'AJ is sometimes called the gigth of R([-) at pa. ‘1:

is determined as the smallest non-negative integer so

1
that P3 = A31 U 3 AJ E r'. PJ is parabolic and fixes

p3. P3 generates the cyclic group of all transformations

in l" which fix p1. We now remark that if v is any real
two-by-two unimodular matrix and r'is a horocyclic group,
than V r‘ v'1 is also a horocyclic group. If, in addition,

1on then

r possesses a parabolic element which fixes V-
V F'V'l is a zonal horocyclic group. In particular for
our choice of f_<: r'(1), we know that Ail. A31 is a zonal

horocyclic group for j 8 l, ..., a (r').

A very special class of subgroups of the modular
group are the principal congruence subgroups of level X
consisting of all those elements in F'(l) for which

-12-

V -,3 I (mod N), where the symbol 5 denotes element-wise
congruence. A congruence subgroup G of level N is a
subgroup of [—(1) such that F'(N)CZG<Z F"(l) and there is
no smaller N for which the same statement holds. Princi-
pal congruence subgroups are normal subgroups of finite

index in [—(l).

A 1225; uniformizing variable (hereafter lgggl
ygrigplg) at z0 is an analytic function t(z) of z such
that a complete set of incongruent points of a neighbor-
hood of 20 is mapped one-to-one onto a complete neigh-

' borhood of t = 0. At a parabolic cusp pJ 8 A3100,

t(z) = e(AJz/RJ) maps a parabolic sector onto a complete
neighborhood of t = O. A parabolic sector in our case is
the intersection of a suitable circle orthogonal to the

sides of the fundamental region at p3 and the fundamental

region.

In the case of even integral dimension -r, a multi-
plier system(v‘belonging to F and -r is simply a character
on F'. That is, for M1 and M2 in f'

and VU(M)l = l for M6 l’. We shall need the following
properties of a multiplier system 4J=1J(F',-2) for F and
the dimension -2. If vs F

(2.5) rm) = ”IN-V). mv‘l) = 1/ro(v> = 3301).

-13..
where F007) is the complex conjugate of MW), and,
further.)

(2.6) AMI) = rU(-I) = l.

The multiplier system (V (F ,-2) induces a multiplier
ar(v r v'l, -2). defined by

system on V F' v'l, (U.
' -1 -1
(2.7) (U (M') = flf(M), M' = VMV 63 V F'V .

We will be particularly interested in the case in which
v = A1. In this case we write (113 a (UTAJ 1‘ A31, -2).

Let p3, J = l, ..., a (F) be a cusp of R(l"); we shall

assume that p:l = 00 and A1 = I. 7\J was defined to be the

a
smallest positive integer so that P3 = 11le 3 113 (5r.

Write “AJ = 7. (13,1’ ), then this function satisfies

(2.8) )3 = Na 1‘) = a (1. aJr 1'1).

3' J

In particular 7\= X1 7\ (I,l—). Since [(WPJ)! = l, we

choose K1 so that

Define ((3 = K (13m), then
- - -1
(2.10) «3.. «(13.0- «(1, AJFAJ ).

In particular let K = K1 = K (1,1’).

An automorphic form F(z) on r of dimension -r = -2

belonging to the multiplier system 41’ is a meromorphic

-1h-

function on? which satisfies the transformation equation
(2.11) rm) = (U(V) (cz + (1)2 F(z),

for each V a (a b | c d) e I— . One measures the behavior
of F(z) by its local variable expansion. Furthermore,
F(z) must be meromorphic at each cusp pJ of the funda-
mental region R(I_). A local variable for the cusp

‘1

P3 = A: 00 19 8170a by 8(AJa/’AJ). F(z) is meromorphic

at pJ if it has an expansion

(2.12) F(z) = (ch 4- d3)"2 2? an(F,A3, r)e((n+I(J)AJz/7(J)

n= 3

where s is a finite integer. The set of everywhere
regular automorphic forms of dimension -2 for r— and/U'
forms a complex vector space. By everywhere regular we
mean regular on 7+ 4' 8 ‘H U 0), where 6): {A’loorAC—Z F(l)
We denote this space by' £’(l_,-2,QI). A subspace of this
is the set of all forms which vanish at each cusp of the
fundamental region. This is the space of 2232.22523 and

is denoted by §+( F ,-2, ’0').

To prove that F(z) is meromorphic at a cusp pJ it is
sufficient to show that F(z) approaches a definite limit
as z -> pj through values lying entirely within the

fundamental region.*

——_

*
Lehner, The Fourier coefficients...III, Mich. Math. J.,
p067.

-15-
The 1 transform of F(z) is defined to be

(2.13) Fv(z) = F(z)| v'1 = (-cz + a)"2 F(v'lz),

1

where V- = (d,-b |-c,a). It follows that Fv(z) belongs

to<§7(Vr‘V'1,-2,KU3 where ru' is the multiplier system
on VFV'l induced by-v’onl’. Furthermore,
(2.1%) Fv(z)l V = F(z).

and if A3100 = pJ is a cusp for R(l_), then

(2.15) FA (2) = if? an(F,AJ,F')e((n + «3)2/23)

J n=s

where an(F,AJ, F) is defined in (2.12).

We define the following sets of integers

:(Ajsr) ={C33V6AJF9 v=(oe\Co)} a

oO(c,A,r)={d:3V€Al—, V=(..‘cd)7] s
(2.16) J J

€(Arl‘) = {dc J3(c,Aj,l—) : d6 [0,c7\]}s

QCMJJ‘) = {a :BAVEAJF, v = (a . c J.

 

ae [MN] 3.

where [0,c7\] is the closed interval between 0 andich
(note that c may be negative), [0,c'hJ] is defined in a

similar manner. The following relation is valid
00

(2.17) oo(c.a .r) = U {d +clq : de 00(1) .mi .
J gas-m c J

-16-

 

which follows from Vc’d = (. . c d)Ei Ajr‘, ce:C’(AJ,F'),

dfi d2(Aj.l_) then so is Vc’qu = (. .

 

c,d + cq7\)é AJF.

In the course of our investigation Kloosterman sums

(2018) Wc(n + K’AJ, [1 + K3) 3 Wc(n,p)

(+ K)
= E (U(A-1V )e(12:;£lg + _£___1_: )
d6 000(A‘11r)

with vc,d = (a b |c d)<E Ajf' will arise. When r =I’(l).
A3 = I and.tf= 1, these sums are the classical Kloosterman
sums. We shall assume for the present that

(2.19) ugh...) = o< tell/2 *5 ), 5 > o

for fixed p . In sectinn 8 we shall discuss situations

in which we can make such an estimate.

We include finally the Lipschitz formula [1,p.206].
If t is a complex variable for which Re(t) > 0 and
either 0 < u < l, g > 0 or u = l, g > 1, then

00
8 - -
(2.20) $31)... > (m+u)g 1 e(it (m+u)) = E e(nu)(t+ni) 8

(3) m=0 n=-m

‘where i"(g) is the gamma function.

Any further introductory material will be dealt with

in the course of the text.

3. Convergence and Regularity

In this section we prove the uniform convergence of
the series introduced in (1.2) on compact subsets of 7% .
Thus, these functions are regular in ¥4 . The method

follows ($3 of Lehner's paper.

Consider the series

 

e((p+ /< )V 2/7\ )
um i I: -r‘ ”—4-
c=-m :-m (LE-(A1 Vc,d)(cz+d)

Cۤ(A3,l_) d676(c,Ajsr)

where Vc,d = (a b l c d) 6 A31”)- ATM”? and the sets

é?(AJ,T‘) and 1I(c,AJ,F') are defined in section 2. We
assume u is a non-zero integer. The double series (3.1)
does not converge absolutely and for this reason we must
define in what order the summation is to be carried out.
First, however, we show that while Vc,d is not uniquely
determined by the conditions cc mix-1m), dc- flc,AJ,l—),
that is, by its lower line 3c,d} , the terms of (3.1) are
determined uniquely by these conditions.

Let Vé’d be another matrix in Ajl’ with lower line

it, 3 , then vc,d = Um v; 6. However, vc,d = 13M, vé'd

= AjM', with M, M'ezr‘. Thus 13M = UmAJM', and conse-

quently A31 UmAJ = MM"1€ST'. This is a transformation

fixing p1, and hence, it must equal P?. In other words
-17-

-18-

k7kj
Vé d' Substituting
9

m = k‘hj. This proves that vc,d =

this into the term of (3.1) determined by fc,d( , we find

It}
8“)” ’3’ch 2/7xj) _ e((..+ «3m 31);“, z/iL)

rv(A31 Vc,d)(cz+d)2

 

 

- -l 7M’k t 2
(VIA; U c vc,d)(CZ+d)

e((u+ KJ)(V;,d 24»): 21V“)

 

5113):. -1' 2
’V(A1 U Agrv(A6 Vc’d)(cz+d)
k‘x
since U Jw = w + k'kj. From (2.9) we see that

k7x
mun U A3) - UNP’J‘) - out an).

Thus the term of (3.1) determined by Ic,di becomes upon
simplification

9((l‘+ K3)Vé,d Z/RJ)

 

AI(A31V;’d)(cz+d)2

Now we introduce for the c #’0 the auxiliary series

e(( + K )V z/'A )
(302) H(c,z) = i +1 C,d 1

(1,500 was Vc,d)(cz+d)2
65-00(09A31r)

 

which for each c e : (A3, F) is nothing more than the
inner sum of (3.1). In the course of our exposition we

shall prove that this series converges uniformly on com-

pact subsets of 74 .

Let the series in (3.1) be understood in the sense of

-19-

(3.3) 5A3,Ie((“+K)z/7\)+ lim :H(c.z) + 11m H(c,z)

K->°°c=-K - K?-->oo¢:_,1
cedf’éajJ) C(CZCAJ.F)
where 2 if A A I
(30"?) SAJsI 3 { , J l ,
0, otherwise.

If both of the limits in (3.3) exist simultaneously, we
shall define the expression (3.3) to be G(z,rU,AJJ_,p).
The objective of this section is to prove that both limits
do exist uniformly for z belonging to compact subsets

of 7+ .

If AJ = I the terms of (3.1) corresponding to c = 0
arise from d =.: 1 because c and d are relatively prime.
Thus, there are Just two terms and we can choose
Voti:1 =,: I. This will account for the first term of
(3.3) when we show that 065013, F) if and only if A1 = 1.
Suppose 0 €C(AJ, F); then there is a V (5 A1 F with V =
(a b| o d), which implies, a = d = 3; 1. That is, Ube AJF.

We can write 1'51 = M'lu'b, M e F and so A3100 2 M’loo.

-1 ’1

Then pJ = AJ 00 is equivalent to 00, because MAJ oo =

Mid—loo = 00. Because of the way we chose the Ak’ this

can happen only if A1 = I. The converse is clear.

-20-

The following estimate of \cz + dl is essential.

(3.5) \cz + dl _>_ (di sin 5 , o < S: arg. z < 17.

Kli

 

 

 

Fig. 3.1
In Fig. 3.1 we see that lcz + dl is the length of one of

the sides of the parallelogram PQRS with vertices + ldl9

: [cl z. For, \cz + d\ = \cz — (-d)\ is the distance from
cz to -d. Thus \cz + d\ is the shorter side of PQRS or
longer than the shorter side. \d\ sin8| is a leg of a
right triangle which has the shorter side of PQRS as
hypotenuse. In case 2 lies in the second quadrant we

replace 8 by U -8.. The degenerate case is excluded

since z 6 H . Also,
. 1/2
(3.6) lcz + d5: ((cx + d)2 + c2y2§ ,2(CIY. z = x + 1y.
We split H(c,z) into the sum of two series. For
c X 0 we have

Vz = az + b

 

-21-
Let us define for c€Q(Aj,F), c y! 0,

(( 4-K) A ) [-( 4-K )/ C(cz+d)]-1
(3.7) H1(c,z)=ie u 3a/c 3 {e n 3 7(3 }

as-..) mifvc d)(cz + d)

and

6((1H' )8/0 7\ )
(3.8) 112(c,z) = i «1 J

-l
d=-oorU1Aj Vc,d)(c z + d)2

d e 0(3(c,AJ,F)

where vc,d = (a b\ c d). Note that formally R(c,z)
= 81(c,z) + 32(c,z). Once convergence of H1 and H2 has
been obtained, we will have this result. We obtain
estimates of these series which involve c. Expand the

second exponential in (3.7) to obtain

e(( m )a/c a )(-21r1)m( +« )m
(309) H1(c92) = i : +L—J p ...g— .
=-m Ina-1’1“!) vc,d)( 7‘30)mm! (czit-d)m
d ed)(C.AJ.F)

This double series is dominated by

f 1217)!” (pi-1| m
m mm

=-oo mal lei J m! (cz + d! ”+2
de 00’(c.AJ.‘—)

 

Using the estimates (3.5) and (3.6) we see for d i 0
|cz+dl -(m+2)$ lc[-(m+2)/2y-(m+2)/2‘d\-(m+2)/2(81n8 )-(m+2)/2’

and if d = O

-22-

-(m+2) -(m+2) y-(ms-2).

[oz + dl s Icl

With these results we obtain for our dominating series

if gay)" (pi-ll "
“:1 m: c2m+2 y2m+2

 

 

m=1 d=l

+ 2: 2: (2n) UNI)"
m! kf3m+2572y(m+2)72 d(m+2)72r(31n “mi-272

‘- ‘°”'2 HAL") {am 2‘? ”‘7'

m=O ysin 8
f—r‘
d32
d=1

Therefore,

(3.1a) Ipnl(c.z)l 3 (cy)‘2 exp(21r mil/y)

-5/2 ' -1 21r| 4-1)]
+ C lcl (y sin 3) exp[f___I__—y—8Lr;g

where C is a sufficiently large constant independent of z.
When we say that a series converges absdlutely uniformly
we mean that the series of absolute values converges
uniformly. We have proved that H1(c,z) converges
absolutely uniformly on compact subsets of 7+ , actually
for y,2 yo 5 0 and O < ix] 5 x0. Thus, H1(c,z) represents

a regular function in74.

-23-

Note that H2(c,z) of (3.8) correSponds to the missing
term m = 0 in (3.9). The dominating series for H2(c,s)

 

is
i 1 .
da-oo \cz + dli
d608'(c,AJ,T')

Using the results (3.5) and (3.6) we find if d X 0

lcz + cal"2 5 d-2(sin 5 Y2,

and if d = 0
lcz + dl'2 5 c.2 y-Z.
Thus,
132‘ 3 (cy)'2 + 2 (a 51:13)“? .
d=1

(3011) |H2(C,Z)‘ \< (cy)-2 + unul-z—T o
3 sin 8

We proved that for \x‘ $_xo, y‘z yO > 0 the series H2(c,z)

converges absolutely uniformly. H2(c,z) is a regular
function in ’H . Now that we have established the conver-

gence of H1 and H2 we can assert (see lines following (3.8))
(3012) H(C,Z) = H1(C,Z) + 32(csz)o

The estimate (3.11) is not good enough for our
purposes since we will want to sum on c. One first proves
that the terms of H1 and H2 are uniquely determined by the
lower line gc,dfl of vc,d' ‘The proof is almost identical

-2u-

to the case considered at the outset of this section. We

make the choice of the a of Vc d = (a bl c d) unique. We
'

k1

saw that V = U 3v' , where V. is another matrix
c,d c d

ng 9

with lower row and} . That is,

a b) (1 k)1>( a' b') _ (a' + ck); at)
V = 8 - ,
c,d c d O l c d c d
By choosing k properly we obtain ae [0,c )3], i.e.
aeabuyl‘). Now dividing d by c7\ ,d = QC) + d1

where dle (Denim). Then

_ ab 3 a * lq7\ ..

with ce Earl—L dlewcuyl‘) and ae acmyr).

q7\

Using these results and (2.17)

e((,u+ Kin/c 2(3)

 

H2(c,z) = Z i -1

7i (1
d1€ dZULJJ’) q=-oo MAJ vc,d1U )(cz+d1+c7\q)

e((u+KJ)a/c 15) >4: eC-qk)
(cz+d +c7\q)2

2

 

-1
d eagqu) “I”; Vc,d ) q=-m

where we have used 1//U(U(fl ) = e(-q/~1). _The order of
summation is immaterial because H2 is absolutely conver-
gent.

Applying the Lipschitz formula (2.20) to the inner
sum of the above series, valid since g = 2, we find,

whether K>Ooru= K =0:

-25-

 

 

i e(-QI<) 42-: {'21}: 0° e(QK)
q=-oo(cz+d +c 3\ Q) c 7\ q=_m(-1(z/Q\ +d/c>\ )+q1)2

2
if? (n+ K )e((n‘“\ )(z/X ‘ d/c>\ ))
n=0

where in (2.20) we identify t = -i(z/7\ + d/cA ). Hence,

6((“4' Kj)a/c x3)

 

_ 2
H2(c,z)={ _1 . 32—23%} .
deVOc(AJs r)(U(AJ vC,d) c 7‘

00
{Z (m x >e(<n+ K.- )(fl- + c‘; n}
n:

0

 

.-. ('2V1)2{ (m K )e((n+ K \z/’>\ ) } .

e[ (11+ ((1)8 (hi-K )d

1
-1
{d6 00303. F) ”I”: vc,d)

2
(3.13) = (fig-7:?) (n+ K ‘ e({n+ K 12/ A )' Wc(n,p)

n:

0

where we have interchanged the order of summation of the
finite sum and the infinite sum and have introduced the

Kloosterman sum (2.18). The above interchange of

~26-

summation is valid because of the absolute convergence of
the double series involved (recall that cacuj, F) is
finite for fixed c and 3).

At this point we derive an estimate for 32(c,z).
The series (3.13) is dominated by

2 .
35.)? E (n+ K) exp(-21r(n+ K)y/7\) |Wc(n,l*)‘.
c

n=0

Now we use the estimate (2.19) for the Kloosterman sum

and the fact that 0 S K.< l to obtain a dominating series

2
.3121. : (n+1) exp(-21r ny/m) CE MIL/2+2 .
c

n=0
Thus, C
E
(3.1%) Pawns). S. W i (n+1) exp(-21rny/7\)

n=0

-2

.-. cE lcl 3/2 *5 (l - e'ZW/M .
This is our desired estimate involving c.

It is now easy to see using the estimates (3.10)

and (3.1%) that the series

: H1(c,z) and i Hz(c,z)

cs-oo ca-oo
¢€§(A , r) c€€(AJsr-)
egoJ ch

converge absolutely uniformly for y‘z yo > 0, ’x“$,xo;

we proved that H1 and H2 are regular 1:134 . Hence

-27-

the above sums are regular in IL . We have, moreover,

from (3012)

:1 |H(c,z)l g : ‘H1(c,z)‘+ ZR: |H2(c,z)

c=l

 

and¢€€(AJsr—) c..10(AJ,r-) C€G(Ajsr)
mix (H(c,z)[$ Z]: I31(c,z)l + : IH2(C.2)| 3
c=-K =-K

c;.C(AJDr-) C€;(Ajgr—) céé(AJD[—)

therefore, the limits in (3.3) exist uniformly for |x{ 3 x0,
y'z yo > 0. This completes the proof of the lemma.

Lemma 1: The functions G(z,n;,AJ,T-,u) defined in (3.3)

are regular in )4 . Furthermore, we have the expressions

(3015) Gl(zs/U9A}srsfl) 3 8A 19((IH' K. )z/Z )+: R(csz)s
J,

cs-oo
06%(A‘1: [— )scflo

and

(3e16) G(z,/\I,AJ, r3“): SAJ,IC((.L+ K. )a/%) + i 31(C,Z)

c=-oo
+ f H2(c,z) e

c€g(AJ,F),c,¥o
=-oo

Ce £(AJ, r )scylo

The three infinite series appearing in (3.15) and (3.16)
are absolutely and uniformly convergent in each compact

subset of 34 . H(c,z) is defined in (3.2), H1(¢.8)

~28-
in (3.7) and 82(c,z) is defined in (3.8).

For later considerations we discuss 31(2) defined

by the series

H1“) = i . i e((|r"/<J)a/c XJ){ET§%1+E]1}

 

c=-oo =-oo 4f(A31 Vc d)(cs + d)2
(3’17) cEHAUF‘) d6? KC,AJ,T')
c #'0 “

Expanding the second eXponential we get a triple series
(3.178)

: :: e‘((u+ KJ)a/c )jl(-21ri(u+ «Inn
s-oo 11:81

-1 m m+2
c=-oo )4;(A V ) m! (c’X ) (cz+d) '
c65(AJ,F)dc-08(c, Afr) J c,d J
c#b

 

Using an estimation similar to the one following (3.9)

we deduce easily that this triple series converges
absolutely uniformly for y‘z yo > 0,lx‘ 5 x0; moreover, we
get for the sum of the series, 31(2), the estimate

(3. 13) I31“)! 5 Cy 2exp(21rlu.4-1| /y) + C(ysin$)1exp[w]
Mysingj

 

where the constants are independent of z.

h. The Rademacher Lemma

We now come to the main tool of this paper, a
Rademacher type lemma, which allows us to rearrange
certain conditionally convergent double series. In
part we follow the method of Lehner [6, § k]. Some
preliminary investigations will be required before we

can state the lemma.

By a lattice point for AJT"we shall mean an ordered
pair (c,d) of integers obtained from the lower row of
amatrixv =(ab|cd)€A F. Leth.T')be the

ed .1 1
set of all lattice points for AJF':

(k1) (NAVY) {(c.d): 3 V = (a b \c d)6 Air) .

Let ”(K = {w = u + iv: \u\< K, \v) < K) and

(M2) Oxurr‘) = 0(AJ,F)fl Xx .
Let )n consist of the following matrices M =

(on at Y we I' :

(h.3) M = 1;

(MA) Y=1, ogoi,8 <7\ and 084' 82>O;

(M5) Y>1, o<o<,g<m.

If M = (s pix 8 )e r and y 2 1 then the conditions in

(M) and (#5) imply that again») and Se m1, r ).

If Y = 1 then from the conditions in (h.h) we see that

-29-

-30-
o! and S are not simultaneously 0: thus, S = (0 -l \ l 0)
is never inM . Furthermore, in the case Y) 1, B > 0
as can be seen from the conditions in (H.5) along with
0&5 - {3! = 1.

Each V 6 I" has one of the following representations:

(k6) v = _+_ U“ MUn)‘ with MEWZ,
or, possibly, if S = (0 ~11 l 0) e F
0+.?) v = 1 UM“A so"X .

The representation for each V e l- is essentially unique.

To prove this result let V = (a bl c d) € V be an
arbitrary element. We may suppose c 2 O, for otherwise
-V has -c > 0. Now consider U-mX VU‘nx:
(a-mc'A *1 c, d-nck ). If c f 0 there are unique choices
of m and n so that 0 g a-mc7x < c7\ and 0 g d-nc'A < c7\ .

-m7\ vu'n7‘ . If c = r > 1, then o<= a-mc7\ and

Let M = U
S: d-nc7x satisfy the inequalities of (#35) (neither

at or 8 is 0 since (on?!) = (Y,5) = 1): thus, 145W). If
c = Y = 1, either both a and d are integral multiples of?
or this is not the case. In the first instance a - mc7\ = 0
and d - nc?\ = 0, hence, U'mxv U'n)‘= S = (0 - l \ l 0).
This case leads to the representation (h.7). 0n the other
hand, if a and d are not both multiples of 7\ , '
M a U-nfl VU'nzem since the inequalities of (MM) are

satisfied. Finally, if c = 0 then a = d = _4; 1, hence,
v = 3 0M .

-31-

We can think of M<?Yn as acting on the u,v plane
not as a linear fractional transformation, but as an

affine transformation given by
(k8) (u,v)(o; {53) = (u'.v'). u' = Mu + 7v, V' = Bu + 5v.

IfAf is a set in the u,v plane let A/M = {(u,v)M: (u,v)
enf}. In particular,

(M9) fixuyr )M = {( act Yd,$c+ 3d): (c,d)E fluff )1 .

The square xf’x, with sides u = 3 x, v = 1 x is mapped by

M onto a parallelogram whose sides have the equations
('+.10)0Lv-8u=1K and Xv-5u=_tK.

As an example, suppose M 6W, B -0( < 0 and 5- B < 0,then
the image of 4/K, A/K M, is given in Fig. h.l.

(01+7)\<,(3+fiQPQ

 

 

3;

L8
x¥ K
Z rfiVijfili—i'wi C-(Y-d)l<\—( 839K}

(" Ht 71") K;‘( S+E.)K)

 

F180 1+0].

-32-

We see that
(tun) 6*(AJ, Fm = on”)

for each M e F. For, if v = (a b \o d) 6 if then so is

VM€AJF3 thus, the lower line of vs is in amyl— ).
This proves O’(AJ,F) 301113,? )M. Next, 1et (c,d) e

(NAVY— )M: in order to show that (c,d) 60(AJ,T_) we
must show there is a V6 A3!“ with lower row (c,d). Since

(csd)€ (5(Ajsl— )Ms (cad) = (C'sd' )M where (C'sd')50(AJs|—)o

Let vv = (a’b' ! c'd')€ A 1‘. mil: (a*b*\ c d)6 AJF.

.1
If/ andy are any two subsets of the plane, we

have (J03 )M =4/Mfl 27M. Thus,
(thiz) otxujm' )M= Xxx/i 61AM”).

In order to make the notation more concise we will

write g 3 S in place of the more complicated
J
c=-M d=-oo

summation conditions 2
=-M ds-oo
¢€é (Ajsr) d6 mcsAJsr)
If a prime (') appears on a summation symbol it means
the summation variable c does not take the value 0. K

and M may take on any values so long as the resulting

sum has meaning.

-33-

We are now ready to state the Rademacher lemma.
Lemma 2: Let M be in the class 7H.defined in (h.3),
(k.k) and (h.5), then

8((p+ K )a/c'k)
lim 2:3 i 3 L.
K->oo

 

 

J 41(A? wd)(cz+d)
d=-oo
ea,» «3)a/c ’AJ)
K“>°° 4f(A3 Vc,d)(cz+d)

(csd)€O-K(Ajs r )M

where vc,d = (a b \c d)<E Ajf' and C9k(AJ,f—)M is defined
in (“09).

In terms of the notation of section 3(see (3.8))

this lemma becomes

M 1+ it Z. e((u+'< )a/cl )
( .l ) H (c,z) = lim ----1---1---
i 2 K

">00 ’U(A31Vc’d)(cz+d)2
(c,d) 6 0K”), r’ )M

63.00

We have shown that the series in the left member
of (h.lk) is absolutely uniformly convergent for yIz
y0 > 0. Further, for fixed M = (a pl 1 S) 6m the

2:: J H2(c,z)

lcl >(d + Y)K

series

can be made arbitrarily small by choosing K sufficiently
large. This is because 0! + Y is definitely positive by

our choice of )n , (k.3), (h.h) and (h.§). Moreover,

-3h-

 

the sum
(Ot+7l)K:'1 c=(01 4- 1)K-1 ‘
(this) Z H2(c.z) = Z, i a“: “we 73)2
j .1 ' .1 ”(83 Vc d)(°2+d)
=-(o<+1)x+1 6"” '

c=-(0< +Y)K-l
is an absolutely convergent double series for fixed M
é7n,and K. This can be seen by using the estimate (3.11)
obtained by taking the series of absolute values of terms
of H2(c,z) and then performing the finite sum on c. We
shall want to rearrange this series, but first some new

notation will be introduced.

For fixed M<:)n_define the regions-Q1 = 111(u,v)
5331 = $331(usV), i = 1,2, as follows:

£1 = {(u,v): (If-00K 5 u < (THOUK, otv - Bu 2, K),
512 = {(u,v): -(°‘+V)K < u < ($00K, 3"? " 5“ 2 K? s
.031 = {(u,v): «wax < u g -(x-um, ow - Bu 3 «L
_fiz = {(u,v): -(V-d)K < u < 01+X)K, 1v-- Eu 3 -K} ,

The cases where d = O and M = I merit special attention.
Iro<=0setfllan'1=¢. ITMaI,setIl2=—Qz=¢.
The regions are given in Fig. k.2, we have used the same

choice of M as in Fig. h.l.

-35..

“(1' ((stT)Kv(<+%)K)
,/7 '
[/4 //,
4”?“ *‘
Ja/
a __ w - u

 

k” _{11

(~(NK)K,—(8+§)K)

 

 

Fig. h.2
Define
(this) wim = 319-415 , ”1“” = 2: $3315 .
(cz+d) (cz+d)
111(c,d) 431(c,d)

i = 1,2, where

(h.l7) g(C.d) =FU(A;1VC d)e((u+ K3)a/%'Zj)

for (c,d)e C9(AJ,r—). The summation conditions indicated

in (h.16) mean summation is performed over all lattice
points for AJT' which lie in the regionfl.1 or .Il'i,

If 0t=OsetW1=Wi=0. IncaseMzIsetw2=w§=o.
One could check directly, using the methods of section 3,
that these series are absolutely uniformly convergent for
y'z,y0 > 0. Their convergence will come out in our

development. Notice that the regions .01 and—EL'1(i=1,2)

-36-

are symmetric with respect to the origin. This implies
(ma) W100 = w;<x) (i = 1.2).

Indeed, if (c,d)é C}(AJ,r-) appears in the summation
W1(K) (i = 1,2), then (-c,-d)€ C?(AJ,T‘) since - 16 r',
and because of the symmetry (-c,-d) will be a lattice
point for AJT'appearing in the sum W£(K) (i = 1,2).
Furthermore, because the terms of the series depend only
upon the lower row of the matrices involved we can

assume that V = -V . Recall that ”vi-I) = QI(I)
-c,"d c,d

 

 

= 1. Thus,
mini; 2 e“'*”a’<‘a/'°“a’ g g(-c,-d)
(cz+d) IU(A31V__C’_d(-I))(cz+d)2 (-cz ad)‘2

The parallelogram of Fig. %.2 is simply the boundary
of’4(K M. Thus, because of (h.12) we can write

 

(at+7)’K-1

Z3 H2(c,z) == 2W1(K) + 2W2(K)

c=-(o¢+X)K-l '
as, A; “KW “ii—5

’V(AJ Vc,d)(cz+d)
(c,d) e 0K”? F )M
Now write
c=o; -
Z 332(c,z) = Z3 H2(c,z) + Z3 H2(C.z)3
°='°° \cl <(Y+ot)K \cl 2(a+V)x

therefore,using (k.l9) we get

-37..

a) o
/ e((p+K )a/c A )
J /v(A11Vc’d)(cz+d)

=cm

 

(c,d)E @KMJ, 1' )M

(too) = 21.4100 + 2w2m + :3 H2(c,z)
lclz(o<+Y)K
Therefore, in order to prove the lemma we must prove that
the three terms on the right side of (h.20) can be made
small by choosing K sufficiently large. As we have al-
ready remarked this can be done for the last term. Thus,
it remain; to 25232 that W1(K) ggQ‘W2(K) £32 23 Eggs
arbitrarily small by choosing g sufficiently large.

This task will occupy us for most of this section.

Following Lehner [6, p. 77] we extend the definition
of g(c,d) by setting

(l+.l7a) g(c,d) = o for (c,d) ¢0(A ,r‘),

J
c and d rational integers. For (c,d)E‘Ct(AJ,V')

g(c,d+c)\) zli(A31vc,d+c%.) e((p+ AJ)a/b 23),

X

but we may choose vc,d+c = vc,dU , therefore,

_ - _. 2
mullvmmw ) e((p* KJ)a/cZJ)

g(csd+C)~)

e(- K)g(c,d).

We used the fact that vc,d = (a bl c d) and vc,d+c =

(a *l c *). We now define a periodic function

-38-

Let
(H.21) @(cm) = e(dK/c?\) g(c,d)

for c and d rational integers. Since
Cp(c,d+c’l) = e(K)e(d«/c >i )g(c,d+c?t) = e(dIa/c?t )g(c,d).
97(c,d) is periodic in d with period C7\. . Therefore,

it has a finite Fourier expansion

¢(c,d) = Z: Bke(kd/cl ), Bk = llcTA— :43(c,d)e(-kd/c'>\).
k

c7\)
(b.22) “cm

where in each case the sums are extended over a complete
residue system modulo c'A, say 0, 1, ..., lcl7\ - 1.
From the first equation of (k.22) and the definition

of Cf (c,d), (ll-.21), we see

(M23) g(c,d) = Z Bk euk- mam).

k(c7\)
From the second equation of (h.22) and the definitions
of (70(c,d), (h.21), and g(c,d), (ml?) and (1+.l7a), we
obtain

\CM -1
(than) Bk - \cz-Ill: d2 J’VUJ Vc’d)e((p.+ KJ)a/c ’AJH-ki-IOd/cM
=0

The summation conditions mean d 6 flcm, l.) and 0 g d <
lcl X . However, the above sum is periodic in d with
period loll , thus, the finite sum in (M210) is nothing

more than the Kloosterman sum of (2.18).

(”025) Bk = -l-:ll- wc(-k+ ll sAJsll+ K3).

-39..

If we use the extended definition for g(c,d) in
(h.16), we can drop the conditions on the summation
variables symbolized by the J on the summation sign: that
is, we no longer require c Q g(AJJ’) and d6 05(c,AJ,r ).
Now insert for g(c,d) in (k.l6) the finite Fourier expan-

sion (h.23) to obtain
(0‘ ”OK-l lo! 7\ -l

- 9((k- Ac )d/9_7‘)
(ll-.26) W109 "' Z Z Bk Z (cz+d)2
c==()(-0()K k=o

 

atd-Bczx

(LOOK-1 (CM --1
e( (k- KBd/c A)

(“027) W (K) = B
2 c=§(:r+ot)K-l ;0 k Z (CZ+d)2
YdPXcZK

The dependence of the Bk on c has been suppressed by the
notation: however, it is clearly present as is shown

in (k.2h).. It is in the above form that we will make

our estimate on W1(K).

In making our estimate on the inner sums of (h.26)
and (h.27), we will want to identify two cases. First,
we shall suppose d may take small values and c is
bounded away from 0 by a multiple of K, and secondly,

c may take on small values and d is bounded away from

O by a multiple of K. We are excluding the case that
‘both c and d can take small values. That these two
situations, and only these two situations, are realized
is a property of the class'MW. The proof of this state-

ment is deferred until later.

-ho-

Define

(”028) T(k,C,K) = i 9((k-K)d/C7\.)
d=QK (C2 + d)

 

where Q = Q(c) is defined so that QK is the lower limit
of summation on the inner most sum (on d) in the equa-
tions (h.26) and (b.27) for W1 and W2 respectively.
Notice that Q(c) depends upon M. We assume that either:

(I) -oo< Q(c) < + 00 with ch> BK,
(h.29) R a positive constant: or
(II) 0 < Q(c) < + 00.

These two situations are not mutually exclusive. In

the proof of this lemma we always exclude c = 0; this

comes from the fact that the summation variable c in

the left member of (%.l3) is not 0. Notice further,

that in (I) d #’o since Icl > 1 (for sufficiently large K).
and (c,d) = 1.

We intend to make an upper estimate for T(c,k,K).
This estimate will be carried out in four stages listed
below:
(I)1 situation (I) with k = 0,
(1)2 situation (I) with 1 5 k g ld7\ - 1;
(II)1 situation (II) with k = 0,
(II)2 situation (II) with 1.3 klg lclh - l.

(n.30)

In order to carry out this estimation we introduce

some preliminary material. Let Sd = E e(o(k-K)t/c7\ )
t=0

-hl-

witha=_4_-l. ThenSO=landford21

(10.31) 8d = : e(a(k- K)t/c7\) g 1'e(°(k-’t)(d+l)/c7\)
t=0 1 - 9(a(k-K)/c 1)

provided It - K i 0. Using the inequality
sin n x 2 min. l2x, 2 -2x l for 0 < x < l,

we find that for k and K in their ranges (see (h.30))
and k - x i o

lSdl _<_ {sin “gull-15 {min[2 l(k-k)/c')\l ,2-2l(k-K)/c7tl]l-1.
Thus,
$.32) lSdl S loll /2 llk-Kl'l + [M1 - lk- Kl I’ll .

This inequality is valid for dlz 1. Under the conditions
on k and k the right member of (h.32) exceeds 1, there-
fore, this inequality still holds if d = 0.

Recall the estimates (3.5) and (3.6) for Icz + dl .
Let w=w(z) = min. lsin 5', yl where 0 < S. = arg. z

< w . Combining these estimates

(1533) lcz+dl 2 ldl sins '2 ldlu) and lcz+dl 2 lcl yz lclm,
we conclude that for each 11 with 0 5 7L 5 1

any.) lcz 4- dl 2 Id1 "1 lat) .

We continue with the estimation on T(k,c,K).
(I)l Suppose we are in case (1)1. Then with 92 = 5/8

we get

-hg-

(M35) lT(0,c,K)l g i lcz+dl'25_ L52 ldl-b'A lcl'3/1+
d=QK d=QK

s :w'2 ldl‘5fl‘lcl'3/l‘ < c lcl'3/1”,

=-m

C = C(z) is a general positive constant depending on

the parameters indicated.

(I)2 Suppose we are in case (1)2. In order to use the
preliminary material uniformly we decompose the sum
T(k,c,K) into two series so that in each case the sum-

mation index (1 will be positive. If Q(c) < 0, write

T(k,c,K) = T1(k,c,K) + T2(k,c,K)
with

- T1(k,c,K) = (cz+d)”2 e((k-K)J/c7\)
d=QK
- K

= (oz-d)“2 e(-(k-k)d/c 7x)
d=l

*‘3
H
I

and
CO

T2(k,c,K) = Z (cz+d)-2 8((k'K)d/C7\.)
d=QK
d‘Z].

T2

In case Q(c) > 0 define T1 = 0. In T1 and T2 we replace
the exponential by Sd - Sd-l’ Then
-QK

T1 = Z (Sd - Sd_1)(cz - d)-2
d=l

with Sd defined in (1+.3l) for a = -l.

-h3-

Thus, _ -I
T = S [(cz-d)-2-(cz-d-l)'2] + s (cz-QK)'2
1 d -QK
d=l _2
-So(cz-l)
Therefore,
-QK-l
lTll 5.2:: \Sdll'lczwdllcz--d--lll"1llcz-dl"1 +lcz-d-ll'll
d=l

+ lS-QKllcz’QKl-2 * lSollcz - ll-2
Using the estimates (n.32) and (“.3“) with 72.: 1/)+,o,1,
lTll S llCIA/b ( lk-«Fl + {Icy} - lk- KlJ-1)l'

-QK-l

l}: (2 w3 d(d+1)1/‘* lcl7A)-1 + 2w'2 52 l,
d=l

hence,

(1.36) ”1! g llk-Al-l + [ (c1) - \k- ml]'1l' c: Icl’3/‘*.

By similar methods an upper estimate of T2 is made.
Let Q' = max qu, ll , then with o = 1 in the definition
of S

d
T2 3 : Sd[(¢2+d)-2 - (cz+d+l)'2]
d=Q'
-Sq._1 (cz+Q'+1)‘2,
We see that

lT2l S {OCH /2)( lk- Kl-l +[ loll - lk- Kl]'1) }

a)
l: [ km“ l “*‘ml VII l°z+d|-1+|°z+d+ll "11+ l¢z+Q'+1l-2l.
d=Q'

-hh-

hence,

(M37) lT2l < lit-(r1 +[ let) 'lk'Kl1-1}C°lcl-3fl+ .
Combining the results of (n.36) and (n.37),

(M38) lT(k,c,K)l < c lcl‘3/‘+l “Ml-1 . [ lei) -lk- m-il
for the situation (1)2.

Now we turn to (II), namely 0 < Q < + co.
(II)1 Let the conditions of (II)1 prevail. Then as in
(“-35).

lT(0.c.K)l < 2:: d'S/u lcl-3/h (n72.
d=QK '

Using an integral estimation for the series on d, we find
(v.39) lT(o,c,xl < clcl’3/h K-l/h.

(II)2 Now turn to case (II)2. Then as before

T(k,c,K) = <¥L

z;_ Sd [(c2+d)-2 - (cz+d+l)'zj
d=QK

-sQK_1 (cz + QK)”2

with o = l in the definition of Sd‘ Making the usual

upper estimate,

q

”(hunk lid/A /2 (ht-«1’1 + I M) - lk-Kl]-1)l

l3: n-z d'5/‘+1c(‘7/‘* W-z (an-w a M}.
d=QK

Thus,

(‘+.‘+0) lT<k.C.K)l < C lcl'3A K'l/t‘l l k-d’1+[ [clk- k-ItU'll.

_hg-

We see loll-l
(Min) winc) = Z Z Bk T(k.c.x) (i=1.2)
c k=0

provided 0 is summed over the proper set of integers -
as dictated by (h.26) and (h.27) according as i = l or
2 respectively. We have essentially two estimates for

T(k,c,K) correSponding to (I) and (11). Let

\cH\-l
(Muz) T(c,K) = B0T(O,c,K) + Z Bk T(k,c,K).
k=1

An estimate for Bk is obtained from the relation
(n.25) and assumption (2.20), the estimate for the

Kloosterman sums, We get

(Mk3) 13k = 0(lcl’1/2 +E ), k = o, 1, ..., loll -1

where the constant in the (9-symbol depends upon 9 and E .
From (h.h3), (h.35) and (v.37). we find for (1)
(MM) lT(c,K)l < cE lcl‘5/‘* *5
MA -1 g

+ log lay-5A +9 2: (ls-«r1 4 (cm ~lk-AU‘1l.
k=1

In (II), using the estimates (v.39) and (h.h0) in conjunc-
tion with (h.h3), we get

(Mus) lT(c,K)l < C: lcl"5/‘+ *E- 161/“

lclk -l -
r e, lo! -5/‘+ “rm )3 «taunt-(11')

k=1
(:2. = C(E,z) is a general positive constant depending upon

-h6-

the parameters indicated. Recall c and K are integers
cflo. Thus {cllzL If lcl')‘=lthe sum onkin
(h.hh) and (h.h5) is empty and hence 0. Otherwise using
an integral to estimate this finite sum,

lch-l

(hr-Kl"l + [ lo?) -ik- «(r-1) = 0(1og lcl) = 0( Isle ).
k=1

Using the above result in (H.M#) and (h.h5) we obtain:

(Mh6) in (I) T(c,x) out-1'57“ ” 28);

O( lcl"5/‘+ * 2“ 161/”).

(h.h7) in (II) T(CsK)

From the definition of T(c,K), (%.h2), and from

(h.hl), we see‘W1 = E T(c,K) , i = l, 2. Let us
c

assume that we can partition the set over which c is
summed into two disjoint sets, one in which (1) holds,
and in the other (II) holds for the same choice of R

(see (h.29)). This decomposition, if it can be effected,
need not be unique. Write symbolically

(Mi-+8) w (K) = T(c,K) + T(c,K).
1 2:(1) >;—-(n)

C

'where E represents a summation over those c in our
-I
c

decomposition for which (I) holds. E ( ) is defined
, II
'by analogy. Then c

-h7-

-5/‘+ + 26 _ E _
lw1CK)l S :(I)C£lcl + g(II)CE kl 5A*2 K 1A

5 CEZlcl-SA +2E+C£K-1/h if lcl-S/h + 2?;
lcl>RK ='°°

We place the restriction that o < 2 a < l/h, then,

vim = 0061/“ +26) + 0061/”) = 0061”+ ”'25).

The constant involved in the (9-symbol involves only
p, 5 and z. This shows that W1(K) -> 0 with K ->oo,

as promised.

To complete the proof we must show how the decom-
position (#.h8) can be effected. This will, of course,
depend upon the particular MEEXK. Certain cases must
be identified and handled separately. Recall the def-
inition of TN given in (h.3), (h.h) and (k.5). We
assumed that for M = (d Ellis ) all the entries are
positive, further, a.and S are not simultaneously 0. We
identify six cases:

1) M = I,

2) M

3)M
Let M = («.8l‘38 ). The remaining cases have positive

(OH-lll 0) with (X > o,
(o 4118) with 8> 0.

entries and we identify them:
1,) r-oL>o, <§-£s>o,
5) X—d<o, 5-B>o
and, finally,
6) 6-q<o, 6-B<o.

-h8-

The only other possibility Z’-ci > O and 5.- B < 0
is excluded by the fact that «8 - sb’ = 1. We determine
the image of ’J7K under M in each of the six cases. From

the geometry it will be clear how we can effect the
decomposition (%.h8).

We refer the reader to either Fig. h.1 or Fig. h.2.
1) Let M = I.

 

 

 

’1.
~35}
(— K’L..._._a_.uaaw - --__.. ,_____. ( K» K)
J I
% K
___._- puma--- hi...”
I i
(~ch) (m—K)

 

Fig. h.3

In this case W2(K) = 0. Further, in the sum for W1(K),
(II) always holds with Q(c) = 1.

-hg-

2) SupposeM=(oL,-l\lo)€7nwith o(>0. The

transformed region is given below.

 

 

V
l 111 f
(-( \+<><)K)KL_ (Cl-00 K,K)F
\ _(1‘
”ix” xi? ‘3)
\ U
‘ ~‘i((d+0b(;K)
Fig. link

We see that for W2(K) situation (II) prevails with
Q(c) = 1. We handle W1(K) in the following way

(1+0L)K-1 . [x 2]
W1(K) = Z T(c,K) + i T(c,K)

c=[K/2]+l c=(1-d)K-l
to obtain the decomposition (h.k8). In the first sum
above (I) holds with R > l/h; in the second sum (II) is
satisfied with Q(c) 2 1/2oc .

-50-

3) LetM=(o-1(1£)emwith (>0.

i V /(K\(§+\)\"\)
. , L
~Q .K'/

I.

6 :1)“//// (Kfikflkv
//

1

T u
/////f //:;14
(hum—M) A “

(9 l-<,-Lg+ 3‘; i0;

 

334$

 

 

Fig. h.5
For this choice of M, W1(K) = 0. We write

[-K/28] K-l,
wzm = Z mm + Z T(c.K)
=-K+1 c=[-K/28]+l

to obtain the decomposition (h.h8). In this case for
(I) R = 1/28 and for (II) Q(c) > 1/2.

-51-

u)LetM=((xs|zg )cm with x-a >0 and
S - B > O. The configuration of regions is given in

Fig. h.6.
i V :
‘ «a+r)K,¢d€)K)

  

 

U

_ --_—.—_.—- ._._(_

 

 

(~((+d)K‘—(6+S)K)

L

Fig. h.6.
For W1(K) we always have (I) with R = ()’-0<). However,
we must decompose W2(K). Write
[-K/gg] (IrdeK-l

W2(K) = z T(C:K) 4' Z T(C,K)
C=-(Y+d)K+l c=[-K/é$]+1

to Obtain (“01*8).

The cases 5) and 6) are handled in a manner similar
to h). We give their configuration of regions in Fig.
h.7 and h.8 respectively. The reader can see how these

situations are handled.

-52-

5) M=(0(Bl‘(8)67)1with J-(X<Oand 8-B>o.

52‘ ((Y+o<)K,(£+fs)K)

 

 
    

a)K€(S-@)KQ

 

(«(1400 K , -(S+g)m

 

Fig. h.7
6) M: (ka]h’8)éWlwith y-ot <0and 5-p<o.

(em) K)(8+@)K)

 

 

 

(-(o<4ar)f<c(5 -- a) K)

Fig. 1‘08

-53-
A useful consequence of lemma 2 is the following.

Lemma 3: Let M 6M then

 

e((p.+&)V z/K )
(“01.9) C(zs'm j, 1'9“) = 11111 T- ‘—"""'J c,d L 0

K '>°° [— miglvmxcmfi

(c,d)E 036A?“ )M
Proof: In lemma 1 we proved that

(D.

G(z.rv.AJ,I’ .n) = 3AM e((p+K)z/1) + :3 H1(C.z)

c=-oo
2:“
+ H (c,z).
J 2

=om

Now, lemma 2 gives for M 5 m

u e((p+KJ)a/c 7&3)

 

(MSG) H2(c, z) = lim Z 2
K ‘>°° ’\J’(A31Vc,d)(czifd)

(c,d)ee’KuyFm

Furthermore, we proved that the double series H1(z) of
(3.17) which sums to EC, J111(c, z) is absolutely conver-

gent. Thus, we may write

-(p.+K)
-1
I e((m/cj)a/C’KJ) §e[+)c 7‘ cud] 3

(#051) H1(2) = 11111 #L———

K ->oo mAilvc’d)(cz+-d)2
(c,d) 591((AJ9 r)”

 

Both the limits in (#.50) and (h.Sl) exist, therefore,
we may add them. Using the fact that for c y! o,

-51}-

V2 == (az+b)/(cz+d) : a/c - l/c(cz + d), we see that

 

“(Mk )V z/7\ )
G(Z,U,Aj,r,p) = lim Z J cod 1

K ->oo 47(A31Vc,d)(cz + d)2
(c,d) fix”? )M

where we have incorporated the term correSponding to

c O in with the limit.

5. The Functional Equation

In this section we prove that the functions
G(z,m;AJ,F',p) satisfy the functional equation (2.11).
That is, for every V = (a b lc d)

(5.1) G(VZ.UrAjrr,u) = rm) (cz + d)2 G(2.’lf,AJ,f—,p).

We use the results of the preceeding section; namely,

lemma 3 and the representations (h.6) and (H.7) for VEET'.
The result (5.1) follows from the special cases:

(502) G(U7\ 2945-1339 Fall) 3 “(01) (“Zn/3A..” rail);

and forM= (oKBlTS)e'm,

(5.3) mummy 1",») mum (6’2 +8 r2 G(z.4r.AJ. It»);

finally if, in particular, 3: (o - 1| 1 o)€ I“ , then

(5.1+) G(Sz/U‘.AJ.F .u) = ms) 2.2 c<z./U.AJ.F’.M.

Indeed, suppose first V = Um“ MU“) with M = (a BU 5 MM.

If V = (a bl c d), then c = Y and d = 6 + nc7\ . Apply-

ing (5.2) and (5.3):

n}

G(Vz.ar.AJ. F...) mu“) emu

zquAJrr 9P)

IU(U‘”)‘M) (I'Unlz ’rzi)‘2 C(Unxzn/‘MVRM
=/U'(Um’>\M) IUUJn ) (Yz +7n>x + d)2 G(z,’1r,AJ.r 9P)

= 4,7(V) (C2 + d)2 G(29V9A FDP)°

j,
-55..

-56-

If, instead, V = Um SUdk: (a b\!c G) than c = 1 and

d = n7\. As above,

ax

=4;(u“‘7‘5;) (11“7‘2)2 ow z,’V,AJ.r .u)

=4)‘(v) (z 4- 11702 G(z,’U’,AJ, rm).

There was no loss in generality in assuming c 2,0 since
we were dealing with linear fractional transformations

rather than matrices.

We defer the proof of (5.2) until the next section
where we obtain the Fourier expansion of G(z,®CAJ,Y.,p).

To deal with (5.3), we have from lemma 3,

11m : e((u+ "1)Vc d z/kj)
K “>00 V(A31Vc’d)(cz + d)2
(c,d)c [figuym

G(29/U9A39 r 9P) 3

where in that lemma we have taken M = I. Now for any

M C- 'm we see upon substitution

 

e((u+’<) dMZ/X )
MHz/1518.1. F...) = lim :0)! ——-Jv fil—

K->oo IVc’ndMz-td)2
(c,d)e (9::(115, F)

 

e((u*< J)V Mz/ij)
lim mmofmg)? Z ° 6
K'>°° 4/‘(A 'lv: drmc' z+d' )

(c,d)éCKMJJ')

2

-57-

where vc,dM = (* * \ c' d') = (* * tac+ Yd, Bo +Sd).
If (c,d) runs over C}K(AJ,[-) then (c,d') = (c,d) M runs
over K(AJ.f—)M. The terms of the series depend only

on the lower row of vc,dM‘ thus,

d((l5+a’f _____J)vc ,dz/AL,
,fliAJ 1vc d)(cz+d)2

 

con/minim) =U(M)(Xz+5)211m Z—-——1

K->oo

but by lemma 3
7i

e((u+l( )V 25/
Cami, ..n 2.1—: cd 4..
K">°° «MAJ Vc d)(<:z+d)2

rm

therefore,

G(Mz,/U,A3.r,p) = «mum +5 )2 C(Z,’U’,A3,r,p).

Suppose, finally, that S = (0 -l\ l 0)€Ir . Then
from lemma 3 with M = I,

e((p+fi. )V Sz/A )
11““ Z ‘4 M 51‘
K->oo an Aglvc , d) (cSz+d )

(c,d)eoxu r)

C(3294LAJ, rill)
J,

2 e((u+1<J)Vc’dSz/7\J)

lim/U(S) 2
K ->oo

 

/U'(A31Vc dS)(dz-c)2

(c,d)60 KMJJ‘)

-58-

As (c,d) runs over O'K(AJ,F ), (-d,c) = (c,d) 5 runs

over fixujf )3. But O'K(AJ.F)S = @(AJI ). Thus,

 

Z 3((P+KJ )Vc,dz/AJ)
Qf(A31Vc,d)(cz+d)2 '
(Cod)€ §K(A31r )

G(SZ{U3AJ,f—,p) =’UKS) 22 11m
K ->oo

=’U(S) 22 G(z,/U',A I'M,

J,

6. The Fourier Expansion
As mentioned in section 5 we must yet prove that
’A 1
(6.1) G(U z,/v,AJ,I”,p) =rU(U )G(z,’lf,AJ,r .p).

This is accomplished by expanding G(z,V}AJ,V_,p) in a
Fourier series, which, furthermore, will give the behav-

ior of G(z,v}AJ,Y',p) at the infinite cusp.

We begin with the series

I
(6.2) H2(z) = ijfizhn)

03'00

’ 2 2 2
= 3 (2n) /c A (n+x<>wc<n.u)e<<n+L:)z/7\)
n=0

c=-oo
where we have used (3.13). The above series was shown
to be absolutely uniformly convergent for y 2,yo > 0.
Each of the functions H2(c,z) is regular in y 2 y0 > 0.

Thus, by the Weierstrass double series theorem

I
(6.3) H2<C.z) = f “(man/fl): 3{-21ri/c7\)2\‘!c(n.u)(MM)

=0
n c=-oo

-59-

-60-
Following lemma 1 we proved the double series H1(z)

oo
(3.17), whose sum is :3

S-m

H1(c,z), converges absolutely

uniformly as a double series for y.2 y0 > 0. Thus we may

rearrange the order of summation in the manner of H2(z).

Proceeding from (3.17a)

(- '21T1(p.+K))m e((p+KJ)a/c >13)
H1(Z)= cm 'xm m: J '1 n+2
2-00 d=-oo «NAJ Vc'd)(cz+d)

(6.h)

 

 

Now, as in H2(z), we can write the inner most sum (on d;
see section 3, the development following (3.12))

i——-1———L-— e((p+‘< )a/c R ) { e((p+xj)a/c ’AJ)
= Z: -1

=_égv(A31Vc d)(cz+d)n+2 (UMJ vc,d)
deficurr)

{c-nmz i saw)
(chmz q=_°o(-i(z/7( +d/c7\ Mu)”2

 

 

 

 

 

g 9((p+«3)a/b‘xj) (_ 2w1)m:2
- Z ’U(A31V (c Mm+2 NM?) 3 .
d6 003113.!” )

c,d)

00
a: (Iv-K)!"+1 e((n+K)z/>\ + d/cl D}
n=0

where we have used the Lipschitz formula (2.20). Inter-

-61-

changing orders of summation, valid because of absolute
convergence, and introducing the Kloosterman sums, we

obtain finally

0° (-2vi)m*2 (n+K)m+1
co. = : e((n+K)z/)\) wc(ngll)o
n=0

3 (c70m+2 (m+l):
=-oa

 

Introduce the above expression for the inner sum in (6.h):

(6.5) 111(2) = i e((n+&)2/?\)

n=O

 

m ' (-2"1)2m+2(n+K)m+l(p+KJ)m
i wc(nrll)

0n comparing (6.3) and (6.5) one sees that (6.3) corres-
ponds to the missing term m = O in (6.5). Thus, on adding
(6.3) and (6.5), we obtain

cum“. F») = SAJ'IeUp 4402/1)

 

(6.6)
t i cn e(n+K)z/7L )
n=0
n+K>O
where ‘
i’ i ('2V1)2m+2(n+K)m+1(p+Kj)m
C = W (n, )
n c=-ooc p m=0 c2” AW A?(m+1):m:
CG é (A3, r)
(6.7)

3 Cn(/U'9AJ: r- all)

-62-

We note in (6.3) and (6.5) that if K.‘ 0 then the coef-
ficient co = 0. This is the reason we restrict our sum

in (6.6) so that n +I< > O.

A simplification of the coefficients c can be

 

n
effected if one used the Bessel function [18, p.358]
(6.8) J (z) = i "'1’" ‘z/Zfim .

1 m=o m: (M1):

The sum on m in (6.7) can be written in the form

') m 2m*1‘¢""' "" 2m+l
_ 217 ng-K X1 ('1) (217) ( “:51 “mi-K)

ch 44 «3 J? M m:(m+1):(./X;fx’)2m+1 Mam“

- 2 _
(6.9) = - 2" (4‘1“) (2:5)12‘J(21r_f‘“*“1"““’)
Icl7x M “A; 1 Icl M A

We assumed that p #'0, therefore, p + K3 #'0. Using

 

 

this expression, we may write the coefficients on in the

following form

1/2 1/2 I
u+K - un'(u+K')(n+K)
.211 3 2:13 2': , _. L
cn ’>\ < 7‘ ) (X ) =-o[:' 1Wc(n NHL 7‘3 k )

J

 

 

 

 

When u + H1 < O‘we can replace p +I(J by (p + Kit and

then we must also replace J1(z) by 11(2) = 1-131(12) [18,
372], the Bessel function with purely imaginary argument.

-63-

We now derive (6.1). Replace a by U z = z +?\ in

(6.6) cw‘zwmjfim) = 5%,: emu-u )(z+7t m)

+ i on e((n+K)(z+ 70/1)

n30
mm )0

= e(/i)G(z.’U.AJ.r.M =U(U1)G(zr’1fsAjrr-— 99')

As 2 ->ioo within the fundamental region R(r),
G(z,v,AJ,F ,p) tends to a definite limit, finite or
infinite. We see that it p + x > 0 or A: f I this limit
is o. If A3 = I and p. 4» I4 < 0 then G(z,’U’,AJ,[—,p) has

a pole at 1 oo.

7. The Behavior at the Cusps

In order to show that G(z,v3AJ,Y-,p) has the correct
behavior at each of the parabolic fixed points of r-, we
establish the formula

(7.1) G(stsAJsr—sfl)’ B 3 G(Z,’\J!,AJB.B-1r Bop)

for each Bti T'(l)(see (2.13) for the definition of IE).
a}! =/U(B'1F'B,-2) is the multiplier induced on B’ll' B by
4fa(U(I—,-2). This formula is of interest in its own
right.

Sincel— is a subgroup of the modular group and B
€1"(l), then B'li' B is a subgroup of the modular group.
If R( r) is a fundamental region for F then B'1R( r) is a
fundamental region for B'lr B. Write R(B’1 r B) =
_B'IRH' ). If Mr) is bounded by a finite number of
sides consisting of straight lines and circular arcs
then so is R(B-1I‘ B). Furthermore, if the parabolic
cusps of R(T') are inequivalent, so are the parabolic
cusps of R(B’lf' B). If Asloois a cusp of R(T') then

(AJB)'loo= B.1 3100 is a cusp of R(B'lr' B). Thus, all

the developments of sections 2,3,k,5 and 6 hold for the

Poincare series in the right member of (7.1).

We shall assume for the moment that (7.1) holds and
show that G(z;U;AJ,f—,p) has the proper behavior at the

~6h-

~65-
cusps. In section 6 we showed that G(z,’U’,AJ,\— .9) has
the proper behavior at z = 1 00. We consider first the
remaining cusps of R( 1’). Let Ail 00: pk for 2 g k g
a(l-). Then from (7.1) applied to G(z,Nk,AJA;1,rk,p)
with B = Ak and FR = AK )- Ail we get

(702) G(Z,’Vk,AJA;19 rksfl) I AR 3 G(zrvsAJor—DF)0

"i is the multiplier system on r}: induced by "If on r .
ark induces ’Von F . Rewriting (7.2)

(Ckz *' dk)-26(AkzsrvkoAJA;1sT—ksl5) = C(zsvsAjsr a“)

where Ak = (ak bk, ck dk). As z'-> pk = -dk/ck, Akz

--> 1 00. Since G(Akz,’lfk,AjAl;1,r-ksu) = O(‘e(kakz/?\k)( )
with wk=l<k if Kk>0sndk;13, wk=1ir Kk=0and
3 )1 k, and if k =1 then wk = u + k3, it follows that

G(z,ir,AJ,T-,u) tends exponentially to a definite limit
as 2 -> pk. Indeed, if k i 1 each of the terms of the
Fourier expansion of G(Akz,’UL,AJA;1, rkm) tends expo-
nentially to O as 2 -> pk.

Now let p be a parabolic fixed point of T— . Then
there is a V = (a bl c d) such that v-1p a pm is one of
the cusps of R(f'). The functional equation (5.1) gives
G(Vz,’lf,AJ.r,p) =4)'(V)(cz 4- d)2 G(z,’U',AJ,r,u). As a -->
pm in a parabolic sector in R(T' ), Vz --> p in a parabolic
sector at p. G(z,’U’,AJ, 7,») tends exponentially to a

-66-

definite limit, therefore, 6(Vzgr,AJ,V'.u) does also.
This completes the proof that G(z,¢5AJ,i-,p) has the

proper behavior.

It remains to establish the formula (7.1). We do
this by induction after we have established

(703) 6(29’VoA19 Fall), U G(Z,’\J!,AJU,U-1rU,|L),

and

(70"?) G(z,’U',AJ,r gun) I S 6(294}'9AJS:S-1[—S,|5)

for the two generators U = (1 1| 0 l) and S = (0 -ll 1 O)
of f'(1). a? is the character on U'll_'U induced by «F on
I'. a; is similarly defined.

Proof of (7.3): In (3.2) we defined H(c,z); and

G(U flu-9A3: Y‘all») 2' iJH(CsUZ)
c=’m

where

H(c,Uz) = iJIi-r(A31Vc,d)(cUz4-d)-2e((pi-KJWc’dUz/KJ).

=-oo
Now
gnarl”) = g(ljum‘lr' U)

and

00(c,AJU,U'1|" U) = ic+d2d600'(c,Aj,\—) i.

-67-

The proof of these facts is almost immediate. Further-
more, if IU’ is the multiplier system induced on U'lY’ U

’k

c,d+c

4f(A;lvc,d) = a}(u‘1A31vdU) =ar'usju)’ 1v; Md).

Also,
7\. = R(AJUsU-1r-U)=7\(AJ.F)=13.

and

It = R(AJUN'IF U) = K(AJ. F) = «3.
Substituting these results into H(c,Uz),

R(C.Uz) = Z-_:71"J"((AJU)1Vc dfic)(cz+d+C)'2e((u+KJ)V;,d+c2/7\J)
d€£(C,A Ajsr )

Z: 7m '(u U)‘1vd)(cz+d)'2e((p+ «ch’dz/AJ)
d6 flC.AJU.U'1I—U)

g(cm)

since as d runs over ¢7(c,AJ,F') d' = d+c runs over

00(c,AJU,U'1 T' U). Thus,

if H(c,Uz) = {g(cm) = (}(z,/\/!,AJU,U"1 \— U,p.).
3 -l
c="m C€;(AJU, U rU)

Proof of (7.k). From the definition of the operator

-68-

S and from lemma 3 with M = I, we see

G(z.¢r.AJ,T'.p)|s

. 5:13:00: ri'xi'lv d)(cSz+d) 2e((u.+ xjwc dSZ/A )
(c, d)€€9K(AJ,r )

. = 3 - - ' =
Let vdrc vc'ds (b, a|d, c). vd,-c€ AJVS

AJS s’1r 3. Thus, (d,-c)eO(AJ.F) s and

- -1

Moreover,

M(A31Vc,d) = q}'(s'1A31vc c, S) = 13((AJS)-1V;'_ )
and

R(AJSfi’lr‘ s) = 13 ., &(As,s'1r S) = «3.
Therefore,
2’2 Lim ZfiAslvc,d)(cSz+d)'2 e((u+IiJ)Vc’dSz/3\J)

K ->oo
(C,d)€0 (Ajtr)

11m Zrufiszrlv d)(cz+d)‘ Zeuw «ch d2M)

K->OO

(c,d)eO (A s,s‘1r‘ S)

J

N .-
G(Z.€U’,AJS,S 1 r‘ 3.»).
This last equality follows from lemma 3 with M = I.

Since 8 and U generate F'(l), we can write any B
b
1

a a b
em) in the form B = s 10 s ”U “, a1 2 Leo 2 0.

-69-
b1 2 l and bn‘Z 0. Now we need to know that
F(z) v1v2 = (F(z)‘ vl)\v2.
Let N' be the multiplier system on V? 1' v1 induced by
”VOn F , «P be the multiplier system on (V1V2)-1Y' V1V2
induced by ’V’ on Vilr’vl, and let nffl'be the multiplier
system on (v1v2)'1r' Vlv2 induced by’U'on F'. Then

4f =/U' (7.1) is now proved by induction from (7.3).

(7.h) and the above mentioned facts.

8. Kloosterman Sums, Main Theorem, and Examples

We introduced Kloosterman sums associated with our
modular forms of dimension -2 belonging to r-and the
multiplier system /lr=’lF(F,-2) in (2.18). At the same
time we assumed that for fixed 9 {'0 these sums could

l1/2 +E‘)

be estimated as O( \c . With this assumption

we have proved the following theorem.

THEOREM 1: Let [_be a subgroup of finite index in the

modular group (“(1) and let Qr=flf()",-2) be a multiplier

system for(— and the dimension -2. Let A3100 = p: be a

cusp of the fundamental region of F (A3100 3 00 if and
only if A3 = I). Then

e( (“+KJ )vc’dz/RJ)

(8.1) G(z,W,AJ,r oil) = i
c=-oo =-oo /U(A31Vc,d)(02+d)2

“(3”)“? ) dEo<9(c.AJ.l’)

 

is a modular form of dimension -2 for r and the multiplier

system «1T=Qf(f-.-2)s provided the Kloosterman sum

(8.2) Wc(n+K.AJ,p+’£J) = Eluslvmyd(n+K)/c?«+(p+KJVol.1)
dEfié(AJOI—)

can be estimated as O ( (ell/2 +E ) for fixed p. ;{ O. The
series in (8.1) is not aboslutely convergent; we under-

stand that the summation is to be performed in the order
-70-

-71-

f a) -1 oo
11m 2 °'° + 11111 E 00.
K ->oo K'->ool-—-

c=0 d=-oo c=-K d=-oo
effgflAJ’I-fl) deflcrA1sr) (366(Ajrr ) dfflcsAJsr)

The sets 5 (A1,? ),0‘~9(c,AJ,r ) and ”6::(A3’r) are defined
in (2.16). Furthermore, G(z{U;AJ,[-,p) has the Fourier

expansion

(8.3) SAJ'I 9((IH’K)Z/7\) + Z cn(V.AJ.r.p)e((n+K)z/7t )

 

 

 

n+K>0
where
' w ‘13 n+Kg ”C(nrfl) kn “+K3‘nIX
‘8'”) “““(ALWTJ Z TJ1(T°-F{XJ T)
n+K>0

where we use the positive square root. J1(z) is the

Bessel function defined in (6.8).

In regard to the estimates of the Kloosterman sums

Petersson[11] has proved the following theorem.

Theorem (Peterson): Let F'be a congruence subgroup of

the modular group, wran.abelian character for F‘, and
suppose there is a principal congruence subgroup T'(N)
such that Hmc F and 1f=4f( r‘,-2) is identically l

on Y'(N). Then the Kloosterman sum (8.2) has the estimate

8.
(9(lcl1/2 + ) for fixed u #'0. The constant in the
C7-symbol depends upon 4F,r—,p and E.but is independent

of n.

-72-

We shall give Petersson's proof of this result. The
material given in thelnext few pages is an elaboration of
pages l6, l7 and 18 of [11]. The proof proceeds by
showing how to reduce the sum (8.2) to a sum of original
Kloosterman sums. Then the results of Salie [l5] and Weil

[17] for these sums are applied to give the final result.

("(N) is normal in F‘. Suppose V has the coset

decomposition

(8.5) =U Ks I‘m), K e (—

s=1 s
where 1!: [F’:)—(N)]. Then

V
(8.6) A r = U A

j 831 JKs rm).

We state a lemma.

Lemma 1+: 0(AK ,l’m» and 0(A3Ks' (’01)) are either

3K m’

disjoint or identical; they are identical if and only if
-113

Km = PJKSM where n is an integer, PJ = AJ U A:j and

M E F(N). Finally, for ceéu r)

J,

(8.7) flcmjf) = U1 Ole. AK K.Y‘(N)).
s-

n
. z n - J
Proof. Suppose first, Km PJ KSM then AJKm - U AJKSM.

If (c, d)€(7(AJKs M, ( (N))= C9(A3Ks ,('(N)) then there is

a V<E AJKSY‘(N) of the form V = (a b \c d). Furthermore,

-73-

u 3v = (. r) c d)€ ijmr'm). Thus. 6(ij 5. Foo)
26(A3Km, (‘01)). If, on the other hand, (c,d)eOUAK m.[‘(N))

n
then there is a V = (a b\ c d)<E AJKm('(N) = U'73 AJK:('(N).

:
‘39

Thus, V = U V' with V'6 Ast Y’(N), but V‘ has the

lower row (c,d) . Therefore, (c,d)€C9(AJKs,T'(N)).
Now suppose (c,d)€(?(AJKm, F(N))n@(AJK3. l-—(I‘1)).

We shall show that, indeed, the above two sets are equal

and that Km = P’J‘KSM where M6 r (N). By hypothesis, there

are two matrices v and v' such that v = (a bl c d)

E AJKmT'(N) and V' = (a' b'l c' d')€3 AJKs T’(N). Write

V = AJKmM and V' = ijs M' where M,M M'€('(N). Since V and
V' have the same second line, V = UrV'. Thus, AJKmM =

Ur AJKS M' or Ks = A311)" AJM'M'l. we see that AslUrAJE Fand

fixes p3 = A3100. Hence, ASIUrAJEZFE , the cyclic group

3 h
which fixes p3 and is generated by PJ = ASIU jA1. There-
fore, AglUrAJ = P? for some n. By part one of our

proof the sets are equal.

Let K1, K2, ..., Ky, ‘be a complete set of represent-
atives of F}/r’(n) no two of which differ by a power of P3.

We shall prove that for ceéujf )
a:
V

(808) @(csAjsr ) = U fi(cpAJngr(N))
s=1

-7h-

and the sets «9(C,AJKS,[-(N)), s = 1,2,...,v’ are pairwise
disjoint. .This is a restatement of equation (8.7) of
lemma k with only the distinct “9(c,Ast,lfi(N)) chosen.
The proof is immediate from the decomposition (8.6), the
way in which the first Km's were chosen and the first

statement of lemma h.

Before turning to the Kloosterman sums we need some
auxiliary results concerning F andKU‘. Ujk is the least
translation in Fiend UN GT_(N)C.r fixes 00; therefore,
g‘x = N for some integer g. Similarly gJ‘XJ = N. Now
N )

consider 4f(UN) = e(gk) = l, and /V(A31U AJ = e(gJ/(J)=l.

Hence, g)« and g: K3 are integers.

Consider the sum (8.2)

wc(n+K,Aj,p+/\J) = Z’Tf(A31V'c,d)e((n+K)d/c1 +(p+KJ)a/c 1‘1).

deoégujx)

The summation conditions mean that the sum is extended
over those matrices V in A1?" with V = (a b\ c d) and

d6 [0,c3~]. The terms of this sum are periodic in a with
period c”J and periodic in d with period c} (see material)
following (h.l7a). One could make a unique by requiring

0 < a g \c\>\J. Therefore, we can write

-75..

loll
EWCCn.,'») = g Z
d:
6681::ij )
g\¢Db_ 1
(8.9) "' (H (“A A) Va, 8’ “(WWW/ck ((MKJVc a3)
- J 00% 84.1")
N (cl

= Waglvm) e([g(n+/( )d+gj(u+/(J)]/cN)

d=1 .
d6 @(C, A37r)

where we have used the periodicity in d and that g?t = N
and gJ 8:] = N. Now using the decomposition 16(c,AJ,T )

gt
= U 08(c,A K F(N)) and the fact that these latter sets
s=l

38'

are pairwise disjoint,

u" N|c(
8Wc(nw)= 2; dzg’vmglvcm) e([g(n+/()d+gj(u+/<J)aJ/CN),
5:: =
(8.10) d€45(¢sAJKS,1—(N))

Now v = (a bl c d)€ A K F(N). Let p(C) denote the
c,d .1 s

non-empty set of integers s, 1 S s g I}, for which

)0(c,AJKs,l—(N)) ;( 8. Write vcd = AstM(a,c,d) with

ma,e,d)er(m). Then Em‘fv c cl)= (U’(K )‘a‘;(h(a.c. 8))
=7(—I(Ks). Introducing this into (8.10) we obtain

Nlcl
ch (n,u)= Z/UUCS ) Zl e([g(n+/<)d + g3(n+hj)a]/CN).
569w) d=1

(8.11) d6 @(coAJKss F(N))

-76-

Define the integers m = g(n+K) and wj = gJ(p+/(J).
Define

(8.12) ScN(m’AJKs’ F(N),LUJ) = 2:: e((md + 03a)/bN)
d=1
d6 ficfiins. l—(ND

where the summation conditions mean that d runs over a

set of integers with 1 g d g N lcl and with each d there
must be associated an element V = (a bl c d)6 AJKSF(N).

We may suppose I g a g Nlcl , for UrNV = (a+rcN *Ic d)€
AJK8['(N). We see that

_ Y— —-
(8.13) ch(n,p) - C Arms) scN(m,AJKS,F(N).wJ).
$6p(C)
Let us consider the sums ScN more carefully. Suppose

M = (8 8| )8 )6: Tu) and cegom' (N)) then

(8.1%) SCN(m,M,(—(N),oJ) = 2:: e((md+0Ja)/cN).
d6 06cm. F (N))

We can state the summation conditions in the following

form:
d mod cN; (d,c) = 1;
(8.15) .
d58,a:'-.o(modN, ad§1+cBmoch.
Now let M' = Uk*MUk = (a‘ B'\¥' 8'). Then
(8.16) o? B'\ = (8%! B+kot +k"(8 +kY )j
\8‘ 51/ v (+1.1 .

-77..

The object of introducing M' is to replace the last

congruence of (8.15) by ad - 1 mod cN. This will require

an appropriate choice of k and k*. We have

(8.17) scNonm; F(N).w) = Z e((md' +wa')/cN).
d'é otgm', (‘01))

If de afgm. (“(N)) and v = (a b | c d)€ if (N) is the
matrix determined by this choice of d, then by the normal-
ity of F(N) it follows that Ukmvuk = We 14' F(N) and
d' = d+ck€C£Kc,M',r-(N)). Furthermore, as d runs over
d%(M,(_(N)), d' = d + ck runs over a complete set of

integers congruent modulo cN to afié(M'.l_(N)). Thus,

(8.18) ScN(m,M', F(N),w ) '= Z e([(d+kc)m+w(a+£)c)J/ch)
(168304. F(N)

= e((km+K"co)/N) ScN(m’M’ F(N).w).

We return to the choice of k and k‘. Since ( 8,7/) =
1 there are infinitely many primes in the sequence '8+k7';
choose one which is not a prime factor of N. This fixes k.
Then we can choose k” so that B' 5 0 mod N (see (8.16)).
Now the summation conditions d€d2(M',Y-(N)) for
ScN(m,M',r—(N),aJ) become (see(8.15»:

d mod cN; (d,c) = l;

d 8' mod N, ad a 1 mod cN.

However, ad 5 1 mod cN implies (a,cN) = (d,cN) = 1, and

-78-

a 5(x' mod N and d a 8' mod N are equivalent. To see
this last remark, note that ad - bc = 1 =C*g - 83’ ,
I

B E b a 0 mod N; thUS. ad 2(1' 8' mod N. Now use (a,N)

= (d,N) = 1. Therefore, we are permitted to write

(8.19) SCN(m,M’, F(N),w) = Z; d((md +wa)/ch)
d

* cN)
ad a 1(cN)
a E‘XI(N)

where the asterisk indicates that d runs over a reduced

residue system modulo cN.

We are now in a position to reduce the above to a

sum of Kloosterman sums. We use the fact that

l/N Er: “(POUR/N) -{1 when r :3“! mod N,

0 otherwise.

t=1
Therefore,
SCN(m,M', F(N),w) = : e((md+wa)/CN) l/Ni e((a-o(')t/N)
d*(cN) t=l
ad§l(cN)
= l/N : e(-o('t/N) Z; e([md+(w+tc)a]/cN)
t=l d* cN)
ad§1(cN)

(8.20)

l/N : e(-or't/N) S(m,w+tc;cN)

t=1

where

-79..

(8.21) S(u.vzq) = E e((ud + va)/h)

d*(q)

ad51(q)
is the classical Kloosterman sum. We remark that if
either u or v 5 0 mod q, this sum reduces to the

Ramanujan sum. We shall not be required, however, to

make a distinction.

Thus, we see from (8.13). (8.18) and (8.20) that

(8.22) Wc(n+/(,p+/(j) = l/gN{:(((-I§Ks) e(-[ksm + k; (831/10;
56p c

{ie(-°£;t/N) S(m,wJ+ct; cN);

t=l
where ks,k;,cxé are integers depending upon Ast‘ m =

g(n+K) and (m3 = gj(p +ch) are integers.

We now concern ourselves with the estimation of the
sum appearing in (8.21). It is known [1.1, p.91] that
S(u,v;q) is multiplicative; if q has the prime power

m1 m m

decomposition p1 p22 ... pr”, then to each integer u and

to each integer v there are integers v1, V2,...,vr

such that

r

m

(8.23) S(u,v;q) = T1” S(u,vt;ptt).
t=1

Salie [15] proves that if q = pm, p a prime number,

-80-

and (u,v,q)‘% pm-1, pm, then

(8.2%) (8(u.v;q)l < 2 5 led”2 «11/2
where
(8.282) d = min ((u.q). (V.q){ .

Weil [17] proves for p a prime, uv‘7 0 mod p and
(u,V,Q) = 1, that

(8.25) (8(u.v;p)l < 2 (nil/2.

In the case in which (u,p) l and v a 0 mod p we find
S(u,v;p) = -l. The case q = pm, m 2 2, p a prime and
(u,v,q) = pm"1 is handled by using (8.25) and the known

result [1.1, p.90] that S(u,v;q) = pm'13(u/pm-1,v/pm'1;p):

(8.26) (S(u,v;q)( < 2 pm-1 J p .

The one final case to consider is q = pm and (u,v,q) = q.
Then,

(8.27) S(u.v;q) = C(q) = pm - pm'1 < q-

We can write the inequalities (8.2h), (8.25), (8.26)
and (8.27) uniformly as

(8.28) |8(u,v;q)( < 2 f3 lqtl/Z 81/2 .
Now from (8.23) and (8.28) we obtain

r
\S(u.v.q)| < (2 J‘z'mqi 1/2 E (ii/2

-81-

mt mt
where dt = min {(u,pt ), (vt,pt )( . Thus,

(8.29) |S(u.v;q)l < (2 f5 )r lqll/2 (u.q)1/2.
Now using the symmetry of S(u,v;q) in u and v,
(8.30) \S(u,v;q)l < (2MP?)r Iq‘1/2(W.q)1/2.
Therefore, combining (8.29) and (8.30)

(8.31) |s(u,v;q)( < (2f? )1. lq(1/2 d1/2.

Let r = r(q) be the number of prime divisors of q.
Then it is known [3.1, p. #6] that

(2./"2‘)"'(‘1)<c:E qg, 00.

therefore,

(8.32) (S(u,v;q)) < CE (qil/é +5; dl/é.

Now using (8.22) and (8.32)

/2 5 1 l
{Wc(n+K.u+/(J)| <(y*/gh9 Ci CN 1 4' min a(m’cN)2(wj+ct,cN)2}
t=l
(8.33)
< (y/SN) CE \(:N(1/2+E :(wJTCt,CN)1/2 .
t=l

Now (203 = gJ(p+/(J) f' 0 sincep ;{ 0. Then the above sum

does not exceed

N1/2 i: (wJ*Ct.C)1/2 s N3/2 (Loyal/2 S 113/2603.
t=l

-82-

Therefore,

(8.3s) \Wc(n+8,p+xj)l < cm ,2...) loll/2 *6

where C(Qf,[_,8,u) is a positive constant depending on

the parameters indicated.

We now give a series of examples in which our method
applies. From theorem 1 we must only show in each case

that the Kloosterman sums have the proper estimate.

EXAMPLE 1: van Lint [7] has shown that the commutator
subgroup, 1"(1), of the full (inhomogeneous) modular
group contains the principal congruence subgroup of level
12, [‘(12). That is,

F(12)c r'(1)c (“(1).
Recall that/U'(-I) = 1. Every character (there are only
six) on ("(1) will be identically l on (-'(l). hence on
F (12). By Petersson's theorem the Kloosterman sums
associated with I‘ = ("(1) and any character’U'on T‘ (1)
will have the proper estimate. These constitute the cases
considered by Lehner [6]. We can make the following easy
extension. Suppose ['13 a congruence subgroup of level
N and Qi=-1f((”,-2) is a character on [_(1) restricted to
1" . Since F(12N)cr(l2), we see that ”V: l on 171211).

Again Petersson's theorem applies.

EXAMPLE 2: We consider a case similar to example 1, in
fact, example 1 suggested this consideration. Let I‘:
('2. the subgroup of the modular group generated by

-83-

US = (1, -l} l 0) and so = (o -l\ 1 l). [g is a normal
subgroup of index 2 in ['(1). Its fundamental region
consists of two replicas of the fundamental region of

R = R( F(l)), namely, ROLJURO = R(Fé). We shall prove

O

THEOREM 2: The commutator subgroup of [é contains the

principal congruence subgroup of level 6:

(xmcré.

The proof of this theorem rests on the following
well known result: If 1”* is a subgroup of‘v,I:/(”' is a
finite group, and every character of ['is identically 1

on T * then (”*CIW'.

('2 is generated by us and SH, and (US)3 = (SU)3 = -1.
If Hf=(U( (2,-2) is a character on (‘2, we find

81((Us)3) = U((SU)3) -U(-I) = 1.
Therefore,

xt{us) = e(K/3). /V(SU)

e(m/3). k,m = 0,1,2.

2 1 2

Let U1 = U = (1 2\ o l) and U2 = s' U s = (l o {-2 1).
We use the Reidemeister-Schreier method for calculating
generators for i-(6) as a subgroup of {—(2), which is
generated by U1 and U2. We omit the details of this

calculation: the method is given in [3] and [9]. If

a b a b
" - 1 1 000 n
vc (”(2) and v .. U1 02 ulna2 then 2 a1, E b1 is

-gh-

called the exponent sum of V in the generator U1, U2,
reSpectively. Using the relations U1 = -Us-SU and
U2 = ~SU°US we can calculate the exponent sums of V in

the generators US and SU of ) 2.

['(6) is a free group. Let V1, 1 = l, 2, ..., 13
be the free generators of ("(6),”7:1 and (01 the exponent
sum of v1L in (11 and (12 respectively, and, finally, E1
’Qi the exponent sum of V1 in US and SU, respectively.

The results of our calculation are contained in Table 8.1.
Now from Table 8.1 we see that S 1 and ’2 i are multiples
of 3. Therefore, if 4f=/U(I'2,-2) is any character

on {—2, then for any generator Vi of r—(6), we see that

mvi) = /U(US)3n/U‘(SU)3n = e(3kn/3)' e(3mn/3) = l

n an integer. Hence,a is identically 1 on ['(6). This

completes the proof of Theorem 2.

In Table 8.1 we have let p1 and 01 be the exponent
sum of V1 in terms of the generators US and S, respective-
ly. Using the relations U1 = ((13-8)2 and U2 = S-US-S-US
we can calculate the exponent sums of V in the generators
of S and US. Then from Table 8.1 we can see that van

Lint's result can be improved to (”(6)CZV'(1).

Now we apply Petersson's theorem to obtain the esti-

mate on the Kloosterman sums for i—2 and’U'.

-85-

Table 8.1 Generators and exponent sums for (‘(6).

 

 

 

 

 

 

Generators Representation in pi a1 ti 0J1 E1
of [(6) lterms of U1 and U2 US S U1 U2 US
T-
! 3
V1 :U1 6 6 3 O 3
v2 gngulugluiz o o -l 1 0
v3 guzuluzzuil o o l -1 o
vI+ :Ulugulugl 6 6 2 1 3
v5 jUIUZU§U52012 o o 1 -l o
2
v6 guiugulugluil 6 6 2 l 3
v7 iUiUZUiUEZ 6 6 u -l 3
U8 U3 6 6 3 o 3
v9 UZUIUZUIIUEIUil o o -l l o
v10 Eulugull 6 6 g o 3 3
v11 :(U1U2)2U11U;1U;2 o o -1 1 l o
v12 (Uiuguiz 6 6 i o 3 3
v13 Uiuzuluzuilugl 6 6 : 2 l 3 1 3
[

 

 

 

 

 

-86..

EXAMPLE 3. Let F: You) where q is a prime of the form
hm + 1. [_o(q) is defined to be the group consisting of
all those elements V = (a b[ c d¥3F(l) for which c 5 0
mod q. Let the multiplier system for r0(q) and the
dimension -2 be defined by

arm =/U'((a b I c d)) = ({3)

where ('3) is the Legendre symbol. Hecke [N3 p. 809]

calls a modular form for this group and multiplier system,
a form of type (~2,q,(§) ) or of real type. We see

f (q)CZFO(q)C r(l); since -1 is a quadratic residue of.
primes of the form hm + 1, we have

’U(V) = (g) .3 fig): 1

for V€W_(q). Petersson's theorem applies.

EXAMPLE h. The principal congruence subgroup Y—(2) of
level 2 is a free group with two generators U1 and 02.
One can obtain a character on §‘(2) by defining its values

on U1 and 02. Say,
fWUl) = e(<></8), Mug.) = e(B/8), o g a, B < 8.

We choose this notation to agree with Mask [8]. Any
element of T‘(2) has a representation.of the form V =

a b a b
1 l n
U1 02 . . . Ul 2 0 Then ’

-87-

Z X b
8
mm = xL“(U1) 1muz) 1 = e(Zaiot/8)e(ZbiB/8).

Maak, in the paper cited above, solved the more difficult
problem of determining Qf(V) in terms of the elements

a, b, c, d of V. We shall not have occasion to use his
results. We use the Reidemeister-Schreier method to
calculate the free generators of Y’(h) and F'(8). In
tables 8.2 and 8.3 are tabulated these generators along
with the exponent sums 1:1 and 031 of these generators in
terms of U1 and U2 respectively. The last column gives

the value of the character.
Table 8.2 Generators and exponent sums for E-(%)

Generators Representation in '31 (”i

 

of (KM) terms of U1 and 02 fif(j:>
U1 U2

T U2 2 0 e(“/“)
1 1
2

T2 U2 0 2 e(B/h)

2 - x
T3 U2U1U2 2 0 e( /h)

 

 

r5 ”2U1U2Uil 2 o e(B/h)

 

 

-88-

We see that the’U’=“*(T‘(2),-2) determined by

1) Cl: 0,

2) (i=0,
(8.35)

3) O(=’+,

1+)0K= h,

B
B
B
B

o
u
0

u

will be identically 1 on ['(h).

From the data given in Table 8.3 we see that, in

addition to the above multiplier systems, the multiplier

systems determined by

5) mi: 0,

(8'36) 7) O(= O,

are identically l on T'(8).

'UD'CD'UD

B

2
2
6
6

Petersson's theorem applies to these cases. We

shall return to these examples at the end of the next

section.

-89-

Table 8.3 Generators and exponent sums for r(8)

 

 

 

 

 

Generators Representation t1 (1)1 00(1)
for F(8) in terms of T1 (01) (U2)

X1 Ti 1+ O eé)

x2 Tzrlrglrll o o 1

x3 Thrlrglril o o 1

XL, : T1T2T1T31 1. t o 961;: )

xs grlrurlrgl u E o 3 e63)

E 1 -1 i :

x6 ETSTITS T1 o i o a 1

x7 :TlTsTngl u g o : e63)

x8 r3 0 1. fig.)

x9 'rhrz'rglrll -2 § 0 ' e(-°f;)

x10 rsrzrglril | -2 g h i e(-‘E'+ 3)

x11 TngTil ' o g h g e(3)

x12 rlrurzrgl 2 g o g e63)

x13 «_Tl'rsrz'rgl ; 2 u 3 e(?;‘ + 3)

x1“ ér3ril % 0 § 0 1

x15 §r1r3 E h 3 0 e63)

x16 Tur3rglri I o g o 1

x17 r5r3rglril o 1 o 1

-90-

Table 8.3 continued

 

 

Generators!Representation 1'1 C01 ’W(X1)
for i"(8) in terms of T1 (01) (U2)
-1 h N
-1 g_
X19 TITMTBTM H O e(2)
-l -1
X20 T1T5T3 T5 0 0 l
-1 -1 a
X21 TzTth T1 -2 0 e(- g)
2
X22 Th 0 0 1
x r r T’lr'l -2 o (- a)
23 s h 2 1 e E
-1 o<
2 -l
X25 TlTth . 0 0 l
-1 g
X26 jTlTSTuTZ g 2 0 6(a)
1 -1 -1 z _ - x E
X27 {TZTSTM T1 E 2 h e( K + 2)
-1 -1 1 g
X28 THT5T2 T1 i -2 0 3(k)
2 9 E
2 -1 3 E
-1 3 N E
-1 g x
X32 TlTkTST2 2 O 1 e(:)

 

 

 

 

;_ -1 -1 g 1
x33 ,T2T3T2 T1 0 5 o 1

 

9. The Inner Product Formula

The set of all cusp forms g? F,-2,1r) is a finite
dimensional vector space. Petersson [10] introduced an
inner product
(9.1) (F(z), 6(2); R(i”)) = ,{i F(z) 5??) dx dy

R(T‘)
on this Space. The integral is a Lebesgue integral,
and R( r) is a fundamental region for F'. The integral

is known to converge and be independent of R([—) [10,
pp. h9h-h96]. The object of this section is to establish

the inner product formula:

111130313143: For .1 _>_ 1, F(z) 5V F,-2,1r), we have

a (F,A ,F )K2
(9.2) (F(z). G(z.’2r.A Y3»); MD) 2 41—4—1—

3, [Hr(p+KJ)

where a“(F,AJ,f—) is the p-th coefficient in expansion

of F(z) at the cusp A3100 = pJ (see(2.12)).

We start with
Lemma 5: If F(z)€é§+(T-,-2{U) then for y 2 yO > O

(9.3) F(z) =O(exp[-2wy/7\])

where 0):.K if H.) 0, otherwise a): 1.

Proof: F(z) has the expansion in (2.12) with AJ = I

where s + K > 0 since F(z) is a cusp form. This Fourier

-91-

-92-
series converges absolutely uniformly in y 2 yO > O. The

result follows.

Let R( F) be a fundamental region for I’ which is
connected and lies in the strip E< x < 2 +1 where E is
a cusp Ailoo. This fundamental region is to be bounded
by a finite number of straight lines and circular arcs.
Each parabolic cycle is to consist of a single element.

We begin the proof of (9.2) with A = I; let

3
(9A) J = (cu/151$”), F(z); R).
-1
For each p = A 00, J = 2, 3, ..., a(I‘) let R = R
J J .1 p3

be a parabolic sector of R at p3; We suppose the sectors

are chosen small enough so that, for j #’k, Raf) Rk =,¢.

Let R1 = R - 3C} RJ. We are now able to write

(905) '1‘: (C(zr/UaIrx—tll)’ F(Z); R3): :11} J3
Jfl

where RJ indicates the set over which the functions are

integrated.

We now introduce two results of Petersson [10]. If
G315 an "admissible" region, then for V = (a b \c d)

a real unimodular matrix

(9.6) f, (F(z)\V) (G(z)\V)dxdy = J F(z) 6(2) dx dy;
v 10.3 63

-93...

and for L € V.

(9.7) J, .F(z)'G(z) dx dy = J’F(z) G(z) dx dy.
LCB 8

Suffice it to say that R, R1,... RT are "admissible"

regions.

We now apply (9.6) to Jk, 2 g k g o, with V = Ail.

(9.8) Jk f {mm/11.1.1311) \ Ail} { F(z)‘ affix dy
AkRk

if 0(2'WiC’A121' rkrl-l) F(Z)\ A121 d1 dy
AkRk

The last equality comes from (7.1); (L = Ak F Ail.

We now use the representation
0 ’U’ '1 F -
(90/) 6(29 k’Ak, kg“) " H(C,Z)o
“may. FR)
This converges absolutely uniformly for y 2 yo > O. The
parabolic sector Rk is mapped by Ak onto a strip ’1; =
i2 = x + iy: ék < x (:1: +7\k, y >7Zk > O }. Introduce

(9.9) into (9.8) and interchange summation and integration;
thus,

(9.10) Jk = 2::3 ‘{ HCc,z)‘F(z)\ Ail dx dy .

-9h-

We now justify this interchange of integration and sum-

mation. We see from (3.10), (3.12), (3.1%) and lemma 5
ifcflo

H(c,z)F(z))A£1k< C(Yk)exp[-2wwy/k]{(L/02Y2)exp[2fl(u+l)/V
+(Q/IclS/2 y sing )exp[2n(p+l)/V ysinS ]
+(C5/ [cl 3/2-E)(1 - exp[- 2fiy/13Y2

Recall that 8: erg. 2; since gk < x (ER +>‘~k, y >21: > O,

we see that sin): is bounded away from 0. If c - O

‘ H(o.z) F(z)! A31

 

< C(Qk) exp[-2 (p+lt+UJ)/A.].

Therefore,

00
2:: .[ \H(°’z) F(Z)‘ Ail} dY < + 00,
c ’Lk

and this completes the Justification of interchange of

summation and integration.
H(c,z) is the series

Z’fiflfivcmflcz ” d)‘2e((p *“)Vc,dz/7\ )°
d€fi(C9A-lsr—k)

Recall that K = «(1, F) = M11121, [1) and 1:1(A121, rk).

This series is absolutely uniformly convergent for yEB

yo) 0. Introduce this expression into the terms of

-95..

(9.10) and interchange orders of summation and integration.

 

We obtain
(9.11)
-._ e((p+fl)vc dZ/W) ._
J = y ' F(z)\ -ldxdy.
R Z L. ’lfkhikvc’d)(cz+d)‘2 Ak
ce as “1531;

5(A;1.I"k) acmklxk)

We understand that the summation in (9.11) is to be car-

ried out in the same manner as indicated in Theorem 1.

We now justify the interchange of summation and
integration on d. We see from lemma 5 and (3.5) that
irdgo

“(1”ch dz/X ) exp[-21rwy/1]

flr(z)]ak1 < C(7k) ‘cz + d[ 2

 

Mk(Akv d)(cz+d)2

5 ch)(1/d2) exp[-21rduy/7\ ].

Therefore,

Z [w e((u+K)Vc dZ/W)

QR Vk(Ak1V C d)(cz+d)2

 

             

< CQk) {I + 2:.1/d2} foo exp[-21rw y/A ]dy
7

d k

< + 00.

-96-

In the terms of the series (9.11) we make the change
of variable w = Ach d z. The Jacobian of the transforma-
.9
tion is
, -h

Since F(z)\ AQ1€'C:(rk.-2{Uk)
F(ngdAilw)lA;1 =’“R(ngdAil)(-(cdk+ack)w+(cbk+aak))2F(w)\A;1

On computing one finds that

1

- -1
(ch,d Ak

"4- (1).-2 = {-ckw 4' ak)2(-(Cdk+ack)W+(0bk+aak))-20

Therefore, substituting these results into the terms of

(9.11), we find they become

J, 6((p+KJA£1W/7\)(“Ckw + ak)'2'F(z)l Ail dudv.
Akvc,dAkRk

Once again we make a change of variables. Let z = Ailw

then the above integral becomes

I e((u+K)Z//\) F(z) dx dy .

Vc,dAkRk

We have used the fact that the Jacobian is lckz + dk) -h s

and the identity

(ckz + akrzmikz) \ A121 (ckz+ak)’2(ckakz + dk)-2 F(z)

F(z)

-97-

Thus,

(9.12) Jk = Z Z I e((p+K)z/7\) F(z) dx dy.
c d
vc,dAkRk

c and d are summed over the same sets as in (9.11) and

summation is carried out in the manner of Theorem 1.
-1
We mentioned that vc’de 1’ 1k so that Vc,dAk€ V .

As (c,d) runs over @(Ailfk), (c,d)Ak runs over 0 (I,\'-).
Let 02 be a complete set of matrices in r with different
lower row. We shall make an appropriate choice of the
upper row. Let [Ux'] denote the cyclic group generated

by fil . Then

(9.13) V = "J [U1] v .
e<2

Let

(9.11.) 08:

V902 VRH') .

dfiis a fundamental region for [U?\]. We suppose that
-1
vc,d€ T'Ak is chosen so that Vc,dAk 6 C2 . Define

03k = u
Vc,dAk€Q

R for k = 1,2, .00, O 0 Then,

Vc,d Ak k

by the completed additivity of the integral, we see from
(9.12)

(9.15) Jk = d!- 6((p+k)z/?\)Ԥ(z) dxdy .
k

-98-

O
H-
Furthermore, U @k = W. Thus, from (9.5) we se
k=1

that

(9.16) J = [8((314402/7‘.) F(z) dxdy .
667
ThestripJ ={z=x+iy: §<x<§+l , y>0‘( isa

fundamental region for [U1]. Due to the way we chose
R( F), there is a determination of the V's€ 01 so that
00:1! . We shall use this determination.

Define 00'(y0.y1) £2606 :yO < y < y1% then

J = lim J(yo,yl) = lim f e((u+K)a/7t )F-(z)dxdy.
y1 ">°° y1 "‘>°° (76(y0.y1)
yO ->O yo ->O

By the lemma of this section

I - C(y0)7\1
‘ F(z)e((u+’<)Z/A )(dxdy < 2n<u+K+w)exp[~21r(p.+x+w)yO/7\ ]
(KC) (yosyl)

 

If now y1—>oo, the integral tends to a limit J(yo,oo).

Introduce the Fourier expansion for F(z) and interchange

orders of summation and integration:

 

mom) = Z 3n(F.I.V)[e((n+K)z/a )e((u+K)z/1)dxdy
n+K>O OWYONO)

The interchange of order is justified by the Fourier series
converging absolutely uniformly in y'z yO > 0. Consider

-99-

 

f e( (n+K)z/)\ )e((p+/<)z/?~) dx dy
w‘YODW)

00 $4.)
I exp[-21r(n+K+p.)y/ 7( ]dyf exp[ 21ri(p.-n)x/’A ]dx

 

Yo g
= 2'2 exp[-hw(p+k)yO/z_] S .
hfl(p+K) p,n
5“ n is the Kronecker symbol. Thus,

._ F 2
8“(F:I: )%

 

J(y0.oo) = exp[-hw(h++<)yo/h 1.

hV(p+K)

 

Now let y0->O. Since (F, G: R) = ( G, F: R) the result

(9.2) follows for Aj = I.

Consider

)

J(A (F9 6(29M9A3, r3“); R(r ))o

3

Then by (9.6)

MAJ) = < WEI: t(znmjm .p)lA31: R(I‘Jn
.. -1 .
" ( F‘AJ 3 G(zsqrjrlp[31u)9 R(rj))o
= ‘ = = F; r = ,
Now 13 MAJJ’) MIA’J) and n3 (13, > K(I,rj)

Thus, from (2.15) and the case already proved

-100-

2
afl(F’Aj, r) x3

 

-1 . -
(F A3 : G(Z.VJ.I.FJ.|1), R( rj)) "

This completes the proof of the inner product formula.

The following well known result is an immediate

consequence of the inner product formula.

Theorem: The vector space Spanned by the Poincare series

G(Z,V,A39rgfl)g [L = 1, 2, .0. equals 5+(r, ’2,V)o

We return to the examples we considered at the end

of section 8.

P-

EXAMPLE 1. I = r (1): there are six characters on T-(l).
In only one instance is the dimension of the space

5% r(l),-2,’U‘) positive. Petersson [12, p. 189] gives
a formula for calculating the dimension of £7+. Using
his formula, we find that when Qf(S) = -1 and QT(US) =
e(-l/3) is the character defined on the generators S and
US of R1), the dimension of 5+(1),-2,V) is 1. This
proves that in this instance not all the functions
G(z,U3I,I",p) with p > O are identically 0. However, for
the remaining five characters on ('(1) and for p > O,

we have G(Z,‘U,I,r ,“) = 0.

EXAMPLE 2. T =T"2 = [SU, US]. There are nine characters
on (‘2. As in example 1, there is only one character for
which dim {‘1 (12,-2.1)") is positive. If ”1"(SU) = e(-1/3)
= g(US) then the dimension is l.

-101-

EXAMPLE 3. In the case I‘d—0m), (NV) = ('3): v =

(a b] c d), q a prime of the form h m + l, Hecke [h,
p. 815] gives the dimension of g*(i’0(q),-2,U). The
first prime for which this dimension is positive is

q = 29, in which case the dimension is 2.

EXAMPLE 1+. If I” = F(z) and’U’ is the multiplier system
determined by 4I(U1) = e(l/2), ’U(U2) = e(1/h). then the
dimension of the space é§+( F(2),-2,1I) is 1. In the

other seven cases considered the dimension is O.

BIBLIOGRAPHY

l. L. E. Dickson, Studies 19 the Theory of Numbers, Univ.

 

of Chicago Press, 1930.

1.1. T. Estermann, Beweis eines Sates von Kloosterman,

Hamb. Abh. 7(1930), 82-98.

2. L. R. Ford, Automorphic Functions, Second edition,
New York, 1951.

 

 

2.1. R. C. Gunning, Lectures gn_Modular Forms, Lecture

Notes, Math. Dept. Princeton Univ., 1958.

3. M. Hall, Jr., The Theory of Groups, New York, 1959.

 

3.1. G. Hardy, Ramanujan, Twelve Lectures on Subjects

 

Suggested by His Life and Work, Cambridge, l9h0.

 

h. E. Hecke, Mathematische Werke, GSttingen, 1959.

 

5. M. I. Knopp. Fourier series 9; automorphic forms 9:

 

non-negative dimension, Illinois J. Math. 5(1961),
18"“‘20

 

5.1. , Automorphic forms of non-negative dgmen-

 

sion and_e§ponentia1'sums, Michigan Math. J. 7(1960)

‘- s.“

257-287.

6- J - Lehner: 9.9. 14.99.11.191‘ f. 9.3.28 .0! negative dimensies.

Michigan Math. J. 6(1959), 71-88.

-102-

7.

10.

11.

12.

13.

19.

15.

16.

-103-

J. H. van Lint, On the multiplier system of the

 

Riemann-Dedekind function 71 , Neder. Akad. Wetensch.
Indago Math. 20(1958), 522-5270

W. Maak, Fastperiodische Funktionen aufgdgr Modul-
gruppe, Math. Scand. 3 (1955), hh - #8.

 

W. Magnus, Discrete Groups, N. Y. U. Inst. of Math.

 

Sci.,l952.

H. Petersson, Metrisung der Automorphen Formen und

der Theorie der Poincaréschen‘Re1hen, Math. Ann.

 

- -4. '“

117(1990). lt53 - 537-

 

. fiber Modulfunktionen End Partgtionen'

 

probleme, Abh. D. Akad. Wiss. Berlin, 2, 195k, 1 - 59.

 

 

, Automorphe Formen als metrischei1n-

 

var1anten 1, Math Nachr. 1(l9h8), 158 - 212.

H. Rademacher, The Fourier series and the functional

-..II‘

 

 

equation of the absolute modular invariant J(‘C),

Amer. J. Math. 61 (1939), 237 - 298.

 

R. A. Rankin, On horocyclic groups, Proc. London

Math. Soc. (3) h (195%). 219 - 23%.

/ H
H. Salie, Uber d1e Kloosgermanschen Summon S(u,v;q),

Math. Zelt. 3“ (1932)’ 91 - 1090

E. C. Titchmarsh, The Theory of Functions, Second

 

 

Edition, Oxford Univ. Press, 1952.

-1ou-

17. A. Weil, On some exponential sums, Proc. Nat. Acad.

Sci. U. S. A. 3% (1998), 20% - 207.

18. E. T. Whittaker, and G. N. Watson, Course of Modern

 

Analysis, Fourth Edition, Cambridge, 1958.

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