Modular forms and Dirichlet series (Mathematics lecture note series)

MODULAR FORMS AND DIRICHLET SERIES

MATH-STAT,

Copyright© 1969 by W. A. Benjamin, Inc. All rights reserved Library of Congress Catalog Card Number 69-17031 Manufactured In the United States of America 12345K21098

The manuscript was pur mro productiOn November, 1968; this volume was published on January 2, 1969

W. A. BENJAMIN, INC. New York, New York 10016

A Note from the Publisher This volume Willi prinlal <.lin:crly fnHoalypcocripl prepared by lhc aulhoc, who lakes full rcspoiWbility for its contenl alld appearance. The Publisher has no! performed his ~ual fur\Ciions of reviewing, ediling. typesetting, and proofreading the marcri.d prior to publk:alion. The Publi•hcr fully cndonc:s rhU. informal and quick mclhlld of publishing lecture notes at a modtratc price, and he wishes lo lhank lhe author for preparing lhe material for publication.

MODULAR FORMS AND DIRICHLET SERIES ANDREW OGG Un1venity of CG!ijorniG,ller~ley

0 W A BENJAMIN. INC. New York

1969

Amsterdam

CONTENTS

INTRODUCTION

ix

CHAPTER I - DIRICHLET SERIES WITH FUNCTIONAL EQUATION

1-1

CHAPTER 11- HECKE OPERATORS FOR THE FULL MODULAR GROUP

11-1

CHAPTER 111 - THE PETERSSON INNER PRODUCT

Ul-1

CHAPTER IV - CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

IV-1

CHAPTER V-A THEOREM OF WElL

V-1

CHAPTER VI - QUADRATIC FORMS

Vl-1

Vli

PRE!<'ACE

These are the official notes for a course given at Berkeley during the fall and winter quarters of 1967-68 on Heeke's theory of modular forms and U1r1chlet series. with Heeke's

~will

The reader who ts conversant find nothing new here, ex-

cept 1 have taken the liberty of including a recent paper of W«ll, whlch st..lmulult"l

interest in this

wy

field. The prerequisites for reading these notes are the theory of analytic ftmctlons of one complex variable and some number theory.

llnattrtbuted

theorems arc generally due to Heeke. A- P. Oee

Uerkeley, California March, 1968

INTRODUCTION

The simplest and most famous Dirichlet seri• is the Rlemaon zata-Cunct!on Re(s) > 1

'(s),

defined for

by

the product being over all primes

p;

the equal11

or the two expressions ls Just an analytic statem• or the

fund~mental

theorem or arithmetic.

[9] proved to 1859 that

'(s)

tlnua t1on to the whole

s-pl~ne

pole of residue

s - l

1

at

functional equation:

lx

Riemaru

has an analytic

COl

except for a simp: and satisfies the

INTRODUCTION

X

is invariant under

s

~

l - s.

In fact, this

functional equation almost characterizes

({s),

for Hamburger [4] showed in 1921 that any Dirichlet series satisfying this functional equation and suitable regularity conditions ts necessarily a constant multiple of

However, t.he

'(s).

"ilu~ttion

did not become clear until greatly generalized by Heeke in his paper "Uber die Bestimmung Dirichletscher Reihen durch ihre Funkt1onalgle1chung", published

in 1936 (paper numbered 33 in [5].) Let us sketch that proof of the functional equation for

({s)

generalization.

which leads naturally to Heeke's

Starting from

r(s) = J~ts-l e-t dt

(Re(s)

>

0)

0

va rtnd JT

-s r(s) l::(2s) = "" ts-1 e -t dt l:n~l f'' (trn2) -s 0

::

where

(Re(s) > l) , .._ l'·ts-1 e -rrn 2 t dt n-1 0 J~ts-l(6(tt) - t>dt 0 e+rrin 2• (lm • > 0)

~(•) = ~ E~=-""

xi

INTRODUCTION

is the basic theta-function.

Now

is holomof

"(-r)

phtc on the upper half plane, and satisfies "''t +

2)

= "(-r)

oH-1/·') "' (I.,l/2 i

,,,,, Re(z) > 0

to

These two equations

sa~

where the square root is defined on be real on the real axis. that

"('t)

is a modular form of dimension

for the group •

4

-1/~,

generated by

0(2)

and

"

Js

't ~ •

-~

+ 2,

up to a constant multiple

the only solution of these equations.

(These fact

have been known for ages, and will be proved later In these notes.)

C(s)

The funr.t1onal equation for

ts now a consequence of that for TT-S

r(s) (:(2s)

"(~):

INTRODUCTION

-~-_l_

2s

1 - 2s '

s ~

visibly invariant under

t- s;

the integral is entire, since c > 0.

for some

furthermore,

~(it) - ~

= O(e-ct),

On the other hand, by Mellin

inversion we have

c >

for sufficiently large

o,

and by similar rea-

soning (carried out in detail below in a more general situation) the functional equation for can be derived from that for proof that

'(s)

'(s);

8(<)

this is Heeke's

is determined by its functional

equation. The above proof follows.

~

directly, as

Given a sequence of complex numbers

a 0 ,a1 ,a 2 , •.• ,an given

generali~es

O(nc)

> o, k > O, C

~(s)

<4f)-

5

for some

= ~1,

r(s)

~(s)

form

c > 01

and

INTRODUC'riON L

r.""

f (;:) (The

""' n=O an

e2Trtn;:/)..

0-conditlon ensures that

somewhere, and

f(T)

f(s)

converges

is holomorphic in the upper

ha u· plane. ) TIIEOnEM.

The following two conditions are equiva-

lent: cl>(s) +

(A)

~ s

+

~ k-s

is entire and bounded in

every vertical strip (henceforth abbreviated to EBV) ann satisfies· cl>(k-

s)

=

C~(s);

k

(B)

c r<•>·

f(-1/T) k

((I)

e

1

k lug

f

where

log

is real or the real

axis.) Note for hnve

C

= 1,

)..

~(s)

= ((2s), f(T)

"(;:),

we

2 t k = l2"

Generally, let

G()..)

be the group of substi

tutions of the upper half plane generated by 't" ... T

+ )..,

't"-.

-1/T.

A

modular fm 2!,

~

~multiplier

C for

function

on the upper half plane satisfying

f(;:)

G()..)

is a holomorph1c

xiv

INTRODUCTION

= f(·t)

1)

f('t + >..)

2)

f(-1/~l = c

k

e27Ti~/>..

series in

"holomorphic at f

by

(from 1)

f<~> = r~=o an e

terms:

f(~)

the expansion of

3)

""''

m<>..,k,C).

in a Laurent

has no negative

2YTin~/>..

'

i.e.

f

is

We denote the space of such

We also denote by

the subspace of m<>..,k,C)

~(X,k,C)

consisting of those

f

which satisfy the additional condition that the for

Fourier coefficients some

c > 0.

The theorem then says there is a one-

one correspondence between elements of

~(>..,k,C)

and Dirichlet series satlsfylng (A); note that '(s)

is regular at

t.e.

f(~)

signature ~.

"vanishes at (>..,k,C)

If'

s = k

'(s)

"""

We say

~(s)

a0 hns

tf (A) holds. 5

C(K 1 s)

of an algebraic number field equation is that

if end only if

is the zeta-function

K,

its functional

01

INTRODUCTION

XV

is invariant tmder discriminant of

s

~

1 - s,

K and

where

r 1 rasp. r 2

of real rasp. complex primes of

K.

d

is the

is the number Note this falls

within the scope or the theorem only when there is only one

r-rtmction, i.e.

nary quadratic. ~(s)

then

If

K is rational or imagi-

K is imaginary quadratic, (~ 1 k,C)

has signature ~(s)

1t turns out that

1s determined by 1ts sig-

d = -3,-4

nature when

= CJTdT,l,l);

but not for

d < -4.

The other part of Heeke's theory concerns the question of whether i.e.

<¥(s)

c

II 'ip (s),

~(s)

has an

where

q>p (s)

~

product,

is a power

p p -s •

series in ~

= 1,

so

G(A)

turns out that space is

0

'

for

q>(s),

is the

mCl,k,C)

unless

= i k,

and

Suppose for concreteness that

=r k

= ~(l,k,C),

'i(s)

It

and this

is an even integer

~

4,

and the only possible Euler product of signature

(l,k,tk),

is

11(1 p

<ji(s)

and

modular~·

has this Euler product if and only if

the associated modular form

f

is an eigenfunction

xvi

INTRODUCTION

for a certain ring of operators on ~

ooerators.

mCl,k,ik),

the

The question of the existence

of Euler products is of course

fundament~l

ber theory, since in practice the numbers

for numan

will

be the number of solutions of some number-theoretic problem and the knowledge of an Euler product reduces knowledge of all primes

p.

an

to knowledge of the

ap

for

REFERENCES 1.

L. Ahlfors, Complex analysis, McGraw-Hill, New York, 1953.

2.

D. Der1ow1tz, "Extensions of a theorem of Hardy", forthcoming.

3.

R. Gunning, Lectures

modular .[Qm, AnAAls

Qn

of Mathematics Studies No. 48, Princeton University Press, Princeton, 1962.

4.

H. Hamburger, "tlber die Rlemannsche Funktlona1gle1chung der

C-Funktlon" I, IT,

III, Math. Zeitschr. 10(1921), 11(1922),

13(1922). ~.

E. Heeke, Mathematlsche

6.

E. Heeke, Vorlesuog

~.

Vandenhoeck

und Ruprecht, G6tt1ngen, 1959.

algebratschan

~ ~

~.

Theorie

~

Chelsea, New York,

1'}48.

7.

L. Mordell, "On Mr. Ramanujan's empirical expansions of modular

function::~",

Proc.

Camb. Phil. Soc. 19(1917). 8.

H. Petersson, "Konstruktlon der sllmtlichon WsungPn etner Rlemannschen Funktionalglcichung durch D1rtchlet-Rethen rntt

MODULAR FORMS AND DIRICHLET SERIES

R-2

Eu1erscher Produktentwtcklung" I, II, III, Math. Ann. 116(1939), 117(1940). 9.

B. Riemann, "Uber die Anzahl dar Prtmzahlen unter einer gegebenen Grllsse", Monetsber. dar Berliner Akad.,

10.

W. Rudin,

E&al

~

lB~y.

comolex analvsls, Mc-Graw-

Hill, 1966. 11.

B. Schoeneberg, "Dns Verhn1ton von mohrfachon

Thetareihen bet Modulsubstitutlonen", Math. Ann. 1160939) • 12.

G. Shimura, "The zeta-function of an algebraic

variety and automorphic functions", AMS conference on algebraic geometry, Woods Hole, 1964.

13·

C. Siegel, "A simple proof of '1(-1/1;)

lit.

= 'IC'tlfi?f",

Mathflmatika l(l9Sio).

A. We11, "Uber die Bestimmung D1richletscher Reihen durch Funkt1onelgle1chungen", Math. Ann. 168(1967).

CHAPTER I

DIRICHLET SERIES WITH FUNCTIONAL EQUATION

All that we need about Dirichlet series is then converges absolutely and uniformly in c + 1 + E, uniformly by

Re(s)

~

since it 1s dominated term-by-term '; -1-E .. n < "'• and hence <jJ(S) n=l

defines a holomorphic function in Cat least) the lle(s) > c + 1.

hall' plane converges at

" 0 = a0 • 1 t 0 ,

s1nce the general term .,(s) !.f

= I~

Conversely, i f

1 ann-s

ann

-s

a

ve :see

"n

tends to

an = O(nc).

r(s)

a

o.

O(n o) Thus

converges somewhere it' and only

As to the gamma-function, we need: a)

ljl(S)

= J~ts-le-tdt,

for

0 1

Re(s) > 0

MODULAR FORMS AND DIRICHLET SERIES

I-2 b)

res + 1)= sr(s)'

c)

r(s)

ls never

O,

n!

d)

'

= .fo •

and ls entire except

for simple poles at

~

nt>

r(l) = 1,

s

-n

~

of residue

n = 0, 1, 2, ... ,

Stirling's formula (cr., e.g., Ah1fors [l,p.l66].) i)

where ~(s) ~

0

as

half plane

~ ~,

lsi

c0 >

a~

uniformly ln a

u,

wnere

s = a + it. 11)

l(s) -

J2U

_.!!:It I

c-l. t

uniformly in

2 e 2

c1

~

lows from i) when

a

~

a1

as

,

~ ~,

t

a2 •

(This fol-

> 0,

and then in

general from b).) e)

Mellin 1nvorpion formulq.

e-x

= ...l... 2Tri

I

o~c>O

x- 9 I'(s)ds,

for

x

> 0,

the

integral taken upwards on a vertical line.

In fact, by the calculus of residues, the right side is E res x-s r(s) n=O s=-n

~

tlt:--

n=o n!

e

-x

•

This being sRid, let us begin the baste theorem.

DIRICHLET SERIES WITH FUNCTIONAL EQUATION

A > 0.

fixed

We have already noted that the growth

cond1t1on on the some~here;

for

grows only

AS

I-3

an f(~)

means that

it means that

pnwPr nf

A

on the growth of

f(r)

~(sl

y as

f(x + 1y)

y- D,

as ~

converges a conditiQn

approaches the real

ax1s, the boundary of the upper half plane.

More

precisely:

PROPOSITION 1.

Given

with the series converging in the upper half plane. a)

b)

If

an~ O(ncl, ~

0,

uniformly in all real

as

y

If

f(x + iy) ~ O(y-c)

uniformly in fl:.2Qf..

f(x + iy) ~ O(y-c-ll

then

Dy Stlrllug',;

x,

then

rurmula

we see that

(-l)n(-~-1 )=

(c+ll ·n·!· (c+u) J'(c+ru rCc+l) (n+ll

- (const.) nc,

as

x.

y ~ 0,

an= O(nc).

rCxl -

x-l J2ii x 2 e-x,

MODULAR FORMS AND DIRICHLET SERIES

l-It

so if

an= O(nc),

then

f(x + iy)

is dominated

term-by-term by

Conversely, if

lfCx + iy)( ~ Hy-c,

tnen

1 -2TTln(x~)/'l-.

1

II f(x

+ n)e

0

I

For future convenience, we state our theorem for two functions instead of one. given two sequences

a 0 , a 1 , a 2 , •••

of complex numbers,

bo' bl, b2' for some

c > o, and ').. > o,

need not be real.)

.

We form

q>(s) " I:n_l a n-s n f(s)

= r.""n=l

~(s)

<1:'->

f(s)

Thus we are

c.2f>

b n-s n -s l'(s)!f(s) -B

l'(s)t'(s)

k )

o,

and an' bn C I 0.

O(nc) (C

DIRICHLET SERIES WITH FUNCTIONAL EQUATION

rm

g (~) = (<jl,

I-5

b e2n1n~/~ n=O n are analytic in some right half plane;

t

r,

g

nre analytic in the upper halt' plHiltJ 1 wlLh the boundary growth condition of Proposition 1.) THEOREM 1.

The following two conditions are equi-

valent: (A)

(s) +

(B)

rc·~) =

fl:QQ..(.

~ 9

Cb

+ ~

ccf>

-k

is EBV, and

ol>(s)

c

Cf(k-s);

g(-1/'r).

~(s) = ~wn=l Jwa (ZEn)-sts-le-tdt n

0

>..

~m

Jma ts-le-2nnt/>..~t n-1 0 u

for

Re(s)

sufficiently large, the interchange of

integral and summation being justified by absolute convergence. but since some

The integral is improper at both ends,

f(lt) - a 0

c > O,

we see

-

O(e-ct)

as

t ~

m

Jmts-l(f(tt) - a 0 )dt

for con-

1

verges untformly on verticAl strips, and so 1s

EBV.

1-6

MODULAR FORMS AND DIRICIILET SERIES

Now assume (B).

Then

I 1 t 5 - 1 (f(1t) -_a 0 )dt

0

-a s

= ----!2 ...

Cb -

__Q_

k-s

Thus Cb

a

(s) + ~ + ~ I~[t 5 - 1 (f(1t) - a 0 )

1

is EBV, with

(s)

= Cf(k-

s) 1

which is (A).

Slmlla!·ly 1 by Mellin inv
from

e-x

=~ 17

x- 5 r(s)ds,

I

we have

Re(sl=-c>O

f(lx) - a 0 for

x > 0,

where

_L -s ( ) 2 i I x ol> s ~s,

u o=c

o = Re(s),

and

c

ls

ct~s~n

large enough to be in the domain of ubsolute

DIRICHLET SERIES WITH FUNCTIONAL EQUATION ~(~).

convergence of

Asst~ing

1-7

now (A), we can

push the line of integration to the left, past picking up residues of at

s

= 0.

Cb0 x-k

al

s

=k

and

o, -a 0

Then

f(ix) - Cb x-k 0

=~ "

I

a~c
x- 5~(s)ds

r(ix) = cx-kg(.1)

X '

which ill (B).

Theorem 1 was a great step forward,

75

years

after the functional equation for the zeta-function, for it reduces a question about Dirichlet series to one about modular forms, which are easter to work wt th.

f

= g,

Taktng for some time hereafter 'I'

c = !.

"' .;, etc., and hence

have the problem of finding the (1-,k,C),

t.e. the

'l'(s)

an 1'

bn,

we now of signature

r<•> c lllo
We consider tho domain

D(h):

ReC•>(.s. A/2,

I-8

1•1

MODULAR FORMS AND DIRICHLET SERIES ~

11

which will turn out to be a fundamental G(~)

domain for

in certain cases, and has different

k > ?1 k

topological character as PROPOSITION 2.

Every

o·()\)

~· three reflections

of a point 1n

3)

i.e. Then T j 2

~

='

Tl' T2 , T3 :

T3("t)

1, T1T2 (-.) + k,

so

-<'T

+ k).

-ll-r 1 TlTJ("') ' -1/"t+l., G•(k) ;::, G(k),

and in fact

consists of all words of even length in

Tl' T2, T3.

Let

s·(k): I·~~ 1, -~2 i Re("t) ~ 0

be the left half of

B(l.);

the proposition is

equivalent with showing that the of

B(k).

the group generated by the

T1

2)

G(~)

k < 2.

or

= reflection in unit circle, i.e. Tl (T) = •11•1'2 T2 = reflection in y-axis, i.e. T2 (•) = -'T T = reflection in lt = -~2. 3

1)

T2T3(•)

be

1

in the upper hRlf plane

G(k)

ls a translate under Let

•

=2

B•(.).)

G•(~)-translates

cover the upper half plane.

DIRICHLET SERIES WITH FUNCTIONAL EQUATION Given if -.;'

'r

=

X

+

1y,

I-9

-t

we can assume

.S,

X

.S, 0;

1-rl 2 1 we are done, so assume I" I < 1. Then and so "'T (-r) .. ~ is higher, y' - --L- l-rl 2 ' 1 1-r I

a•<>..J

we have only to verlfy that we arrive tn after finitely many steps.

). < 2 or ). > 2,

If

we can cover the arc at the bottom or

n•C>..l,

i.e. the Intersection of

B•().),

unit circle,

the

and the upper half plane, by reflecting tn the sides

•

or

and hence assume

ll (>..),

the result 1s clear.

(

or

0(2)

8

d~•) c

c

is a substitution SL(2,Z)

higher than

y'

I

then

~,

1-.;1

).. = 2,

If

'r

I

.s.

;:

~

if

dl 2 < 1,

1:

since

valent to

T

then

T 1

c, d c

~.

If -.;•

and maximally high, and

c B().)

1

I

1s

and there

nro only Cin1tcly many po:s:sibil!Llus ror

given -r,

and with

+ d

C'f

= --...L~~· lc-r .. dl21 IC'f ...

c < 1,

then an element

ts

c 1 d1

0(2)-oqui1Re('r 1ll .S.

t•


We now dispose or the least interesting case

). > 2, where

~(). 1 k,C)

for every value of

k and

has Infinite dimension C.

Roughly speaking,

tho great number of solutions Is because we can

MODULAR FORMS AND DIRICHLET SERIES

I-10

solve our problem in the upper half plane with arbitrary singularities in the lover half plane, H(~)

since

extends into the lower half plane. ~

Given

of

B·(~)

> 2, let z = g(~) map the interior

one-one conformally on the upper half

planA, SO that

g

~,

i, -i

~

o,

1,

s·(~)

defineS a homeomorphism Of

onto the closed upper half ~.

normalized by

plan~,

For the existence of

g(~),

one can appeal to a strong form of the Riemann mapping theorem (cf.[lO,p. 281]), or a generalized Schwarz-Christoffel transformation.

By the reflec-

tion principle, and Proposition 2, we extend

to

g

a function defined on the upper half plane and invariant under

G(),).

'l'he only corner of

tn the upper half plane ts at ls en 11ngle uf

ll.

2'

~

= t,

:so the extended

B"'(),)

where there g

is

nnalyt.lr

on the upper half plane, single-valued by the monodromy theorem.

g

plane since

ls

g

lt is clear that not equivalent to ~·curter

oxpansion

is bounded on the upper half G(~)-1nvnriant

g(~)

is one-one or

"''·

At.

nnd

g(-1)

n~ar

points

""

= ~

we have the

DIRICHLET SERIES WITH FUNCTIONAL EQUATION with

a1 I 0

since

function of T

=i

dim

fgn

=~

since for

n

tr r I

= 0,

g(~)

~

as a

has a double zero at

= -1.

To show ~(~ 1 k 1 C)

lt suffices to show

is one such function, so is

0

1, ~ ••.• ,

independent since T

is one-one near

and a double pole at T

io(~,k,C)

I o,

g

e 2 "i~/~;

I-ll

gn

and they are linearly

has a pole of order

2n

at

= -1. Thus wo only have to construct one non-zero

function

f

£ ~C~,k,C).

We have already noted

g

ls locally one-one except at points equivalent to or Since

m,

g'(T) I 0

so

g(T)

except at those points.

has a double

fine a function

~

~ero

at

in the upper hRlf plane,

since it is simply connected; since

a 1 e 2 niT/~ + ••• , a 1 I

0,

neRr Since

g(-1~)

= g(~),

WA

"'

havo

Jd-r:

so

'

we get

-1,

so substituting Now let

h(T)

T

~

1

g(T)

~ + ~)

Jg(-ljT)

~

1 +

=1

+

= Ji[rr. = ~ ~;

Jgrr) = ~ ~

the minus sign .is correct, for where

we can de-

1,

J£(-1/T) _

./iftT -

t·

g 1 (1:), g](-l~) - gl (T)

= 1. g' (T) ~(g(-.:) - 1)

analytic

MODULAR FORMS AND DIRICHLET SERIES

I-12 and never (g 1 (~)

0

in the upper half plane and at

has simple zeroes at

and at

i

m, can-

ce111ng zeroes of the denominator; the zero at z = e 2 wi~l~.)

ts measured 1n h(~),

and

g(-l~l

g I (-1/'t)•

-\-o

so

=

= g(~l

Then

g I(~)

gives

~

h(-1~) ~ -~ 2 h(~)



~

Thus

h c

never

01

~(~ 1 2,1).

[

k

(!.) h(~)k/2

we see

't

t2 =

l.

to obtain the (g(T) -

f

t 1 h(~)k1 2 ,

= "" • we see

l)n,

£

t ~(~,k,l),

n

and at

IJl(>..,k,l).

and to show

which is true since Invariant under

at

~),

B(~)

0(~)

= 1,

~

1-'inally,

f(~)

= h(~)k/ 2 .

is an integer ) k/2; f

satisfies the

condition, 1t suffices to show that

tton of

evnlua-

[1' E21

0-condition, consider where

we

~;

h(-l~)k/2

1,

[1 hk/2

ThUS

is

=

for some constants

2 1

ting at

=

h<~>.

k > 0. kSince h k/2 2log h(7:) h(~) e

we can define

h(~

have

2

Now let

HnHlytic in the upper half plane and at + ~)k/ 2

m

h(~ + ~)

lyg

1

0-

g 1 (x+tvl Jg(x + ly)

(x + ly) + ty)

Jg(x

then

Its

and bounded in the tntersec-

and the upper half plane (vanishes

and so bounded in thA upper half plane.

I.HRICHLEI' SERIES WITH t'UNCTIONAL ~UATION

Thus

f

E ~(A,k,l),

and

fJi

I-13

~(A,k,-1),

£

which

proves: THEOREM 2. ntte

A > 2,

If

k > 01

for every

~tmenston

~(A 1 k 1 C)

then

We con:dder next the case

C 1..

has lnfi-

=~

<

1. This time

2.

we work in the upper half plane only, so let us

B(A),

change the notation so

B•(A)

are the inter-

sections of thP. previoug domains with tho upper half plane.

Let

be the lower left corner of

~0

-~2, ~ ~ -1/~+A,

a fixed point or Let

wu

cos wa

be the angle of

= ~~

~ ~ -1/~;

0 < u < ~·

cal sldes under Let

at

~0 ,

f

£

~

G(A).

is a fixed polnt of

-c ... -1/~,

B(A)

11nd thA two verti-

... • + A.

mCI..,k,C),

~eroes

is

i.e.

the two halves of the bottom of

are equivalent under

number of

which is in

li(A) i

~0

and

of

plicltles, except at

f

I 0.

in

f

'o• 1,

1dent1flcat1ons, e.g. a

~ero

B(A), ~,

Let

N be the

counting multiwith appropriate

on n side of

should be counted on only one of the two

B(A) ~ides.

MODULAR fORMS AND DIRICHLET SERIES

I -lit

Let ~0,

n0 1,

ni,

,

LEMMA 1.

of

measured in

m

n

N+ ~

b)

dim ~(h,k,C) ~ 1 + [~(~- u)].

+

~

at

z

a}

Let

n 0 a ~ ~(~- a}

+

C be a contour enclosing the zeroes

in the interior of

f

r

be tho order of zero of

the zero at

~,

frgQ!.

~

B(h); ~0

follows the unit circle from right side follows

=T

follows

y

follows

x = -~

to

x

=~

-h/2

+

the bottom of

to tT,

back down to

~0

to

~ + iT,

+ A,

C the

the top

and the left side

~0 ,

except that

we must detour around Slllall circular arcs to avoid any zeroes on the bolm
13( ~).

Th.,n

N - _l_ I d log r(~) - 2Tri c

and the integrals over the arcs about approach

~ ~ n0 u -2,-2,-2,

the top approaches

-nw,

the vertical sides cancel. gral on the bottom to be

1,

~0 , ~0

+ h

while the integral over and the two integrals on Thus we want the inte-

~
Now

f(-1/t) =

DIRICHLET SERIES WITH FUNCTIONAL EQUATION k

C(I) f(~) 1

so

d log f(-1/T)

k~

c

+

I-15

d log f(T) 1

so this last integral is _l_ 2 rri

I

T

0

+),

d log f(T) - d log f(-1/T)

I

I

T +),

= .=k. arg -~ 1o 2rr

: ..,k (li ... ,. 2

as desired.

.l-'inally 1 1f

r 11 ••• 1 fm

Knd

t ~(>.. 1 k 1 C) 1

1TU) I

m

>1

ze1·o of order 2 m - 1 > n,., at

EXAMPLE.

t(s)

l

1

The zeta function of

and

- a) 1

> 1\..1

r 1 1 • • • ,rm "" 1

has a

and so vanishes 1

proves~).

and ts known to have signature u =

[~(~

then a suitable non-

trivial linear combination of

by al; this

+

dim "11.(./J 1 1 1 1) .S: 1

Q(j:])

is

(./J 1 1 1 ll.

Here

by the above.

is determined by its functional equation.

lienee

MODULAR FORMS AND DIRICHLET SERIES

1-111

At this point we digress briefly for some general considerations.

~ : ~. where ( ~ ~)

A substitution

1s ellintic if

c SL(2 ,i\)

lt llas two non-real fixed points Im ' l

> 0.

L(•) =

T1 ,

where

-.: 1 ,

cT 1 2 + (d - a>• 1 - b = 0,

Since

L

is elliptic ~ (d - a) 2 < -~be ~ (d + a) 2

< 4 <=> the eigenvalues of L, i.e. the roots or x2 - (a+ d)x

+

= 0,

1

are non-real.

We now prove

a very useful result:

PROPOSITION ). point

'1

Let

L be elliptic, with fixed

in the upper llalf plane.

ts a non-zero holomorphic function Im • > 0

f(T)

on

such that

a)

f('t .,.

b)

f(-ll~)

k) =

for some constants L is periodic,

~.

Suppose there

r<•> k

£(~) f(T) ).

) o,

o, and

k )

Then

£.

i.e. Its eigenvalues are roots or

In the variable

t

= T - ~l, T

-

L is a

T}

(complex) linear fractional transformation fixing 0

and

""

'

i.e.

L(t) = P•t,

and we want

p

to

1.

DIRICHLET SERIES WITH FUNCTIONAL EQUATION be a root of Let

I-17

1.

g(-r)

= (-r

k

- T1 l ,

for

Im(-r) > 0.

Then

g(L(d)

~-k

( for some constant

i

g(-r)'l

evaluating at

'1;

Now let

we see Then

l

h(L(T))

en I 0

= t'lh(•l,

h(-r)

and writing

for two distinct values of

f(T +A)= f(-r), hence

p

and

pn

is a root of

the order of zero of

-r 1 = L(-r 1 l 1

K

at

n,

since en~ 0;

whPn

-r 1 ,

f(-r)g(-r).

h(tl

Furth~rmore,

1.

f

~£

~

if

n1

is

we have the for-

mula As an incidental result, if then applying this last formula to • 1 = 1,

= C:

we have cleRrly

p = -1,

0

If

t

L(•l 50

(-1)

llt(A,k,C),

-1/T, n1 =

E

MODULAR FORMS AND DlRICIILm' SERIES

I-18 PROPOSITION~.

f

at

i,

~.

For arbitrary

rn<~,k,C),

0 ;If£

then

and C

ni

= (-1)

n

if

is the order of zero or i.

Returning to our development of the case ~

< 21

LEMMA 2.

If

l!l(~ 1 k,C)

;I O,

then

and

a

k

are

rational.

fiQQr.

is a fixed potnt of

T0

x2 - Xx + 1,

whose eigenvalues are the roots or t.e.

p, P,

Where

p : T

0

+ ~

= ~.

L(T)

= e Jrta

Then

is rational, by Proposition 3, as is then

a

k,

by

Lemma 1. EXAMPLE.

~

=l

0 I r E mCl,k,C),

(modular group), n

then ~ +

n

and then ~~~(mod 1). integer, and so

k

q

:f ~ ~

Hence

If

C = C-l)ni - C-l)k/2.

,J;(~ 1 k,C)

is an integer 2 3

I

O,

If

u

then

(and hence

Also

a ~

(mod 1),

ic un even

k

2 ~. and dim l!l(l,k,(-l)k/ 2 ) ~ 1

LEMMA 3.

.. =~3"

i.e.

n

+

...lL 12 2

1

3'

[i\1.

= ~'l' 2 2 cos

where 'IT 3-

1.)

DIRICHLET SERIES WITH FUNCTIONAL EQUATION £!2Q!.

Assume

prtrne and G(X)

p

a

= p/q

2;

~

where

1

p, q

1-19

are relatively

we will then find an element of

which is elliptic but not periodic, contrary

to Proposition 3·

In fact, one computes that sin (n+ll Q'll' s =

2---~­

sin .R!!: q

choosing lsi

so

(n

+ l)p: 1 (mod q),

< 2, so L ts elliptic.

gates L

n

o1

with

js'l > 2 1

But

s

wA have has conju-

so the eigenvalues of

are Imaginary but not roots of

which is the

11

desired contradiction. ~-

We shall see there are forms when

a ~ ~ - G(X) q

Note that

is discrete exactly in that case.

X

=2

cos

q'IT =(

+ '

-1

is

,

conjugate to (p,q)

= 1,

so

G(X)

and

are isomorphic as

G(),')

abstract groups, but only

G(X)

is of intere9t for

modular forms. tl~t

So far we have proved X < 2,

if

0 If

then 1)

2 cos

rr q'

q

£

£,

q

~

),

£

~(A 1 k,C) 1

MODULAR FORMS AND DIRICHLET SERIES

I-20

2)

N + n.,. +

3)

c = (-1)

LEMMA 4.

Given r~ E

ro, ri'

A

n

~

n

n

+ -: = k

(

\~

2

)

i

=2

mCA,k,C),

Tr

Then there exist

cos Q'•

for a suitable

in each case, with simple zeroes at respectively, and no other zoroes•

c = +1

a)

and

k

r 0 , i, Thuo:

b)

for

f1'

c

-1'

c)

for

r..,, c

+1,

k

As in the case

A

...

k;

..2!L. q - 2'

~

q - 2"

> 2, by the mapping

theorem there exists a homeomorphism B•(A): -~ ~ Re(r) ~ 0 1

the

q - 2'

smallest possible value for

~·

~,

k = j_

k =

c

and

j'l:j 21,

Im

g 1:

of

>0

onto the

closed upper half plnne, mapping the interior of

s•(A)

conformally onto the open upper half plane,

normalized by r 0 g('l:)

1,

w ~

0 1 1,

~.

We continue

to the upper hAlf plane by repeated reflections

in the sides of tion

,

g(T)

D•(A),

obtaining an analytic func-

on the upper half plane by Proposition 2,

the simple connectivity of the upper half plane, and

SERIES WITH FUNCTIONAL EQUATION

DIRICHL~~

I-21

s•ch>

tho fact that the angles at the corners of .!!: q

are of

T

at

the corners.

i,

at

The

B(h).

lent under

g(T)

Thus

G(A),

invariant under

made

2

an integral fraction

which makes the extended

"'•

or

JT

and

0

T

is analytic for

> o,

lm T

and one-one on the interior

oppo~ite

~

analytic at

g

sides or

T + h and

T

~

H/G().) = H U ["'}/G(A),

~

are equiva-

B(A)

-1/T.

Thus we have

II: Im T > o,

where

Into a flielllllnn surr.. ce or t;t!UUS z;ero, ln ract so

~-" g: H/G().) -> ~;

= Riemann

fundamental~

is

D().)

sphere.

G().).

for

J

g

called the elliotlc 1/lU!ll!l!.U: 1Dl1ill:li1Dl<•l Since

doubly

.. t

1'

11/G{).),

is one-one on

g

has a zero of order

at

q

To'

= e 2 "lT/)., -a

a simple pole at

= g' (T),

~.

desired

f0

,

f 1,

f~

are:

Z!!..U. 11 ).

)

1

'

Note

+ •••

also

We also have

g 1 (-1/<)

paring zeroes and poles at

g

l.e.

~

a_ 1 I 0.

= . .21 ·( g' (< + ).)

we see

takes the value

and 11as a simple pole at z

h~s

is a

= 1,

>.

(If

= • 2g 1 (-r). T0 ,

1,

~,

By

com-

we check tl1e

MODULAR FORMS AND DIRICHLET SERIES

I-22

f0

(g'l2 (g(g - 1})

fw :

(

1/(q-2)

)2Q . ) g2q-2(g - l)Q (I?

1/(q-2)

1

l/(q-2)

(

fi

(g'lq gq-lc g -

) 1)

It is now easy to compute the dimension of ~(~,k,C).

Let

f

~(~,k,+l),

c

the above notation, is an integer,

n1

i.e.

minimal value of

k

k,

Since

= r - a0 rm 0 m > q;

r1

f~

t:

r -

110'

~O.,k,•l)

r,,,. fl'

m = k(g4- 2 l

is even and

= mk0 ,

k0

Then, in

4--= --Q - 2

corresponding to

for a unique constant ~

o.

I

f

0

0

ro.)

is constant if

vanishes at

£ ~(~,k-qkn,+l)

m ~ q),

f

0 ,

r~,

(lf

Continuing,

we see that

ts a polynomial tn

Then

also, we have

rl

where

a fm

(the

in a unique way.

DIRICHLET SERIES WITH

Similarly, if is odd end k - ~ q -

2

mk

+

0

f/fi &

£

0

If

(m - llk 01

2a- It ~ -q-:-2

mk

0

+ 2;

~(A 1 ~ 1 +1), so

rf

u,~

I-23

I

ni

Thus

dim ll!(>..,k,-1)

+ [~).

Finally,

uf~ + ~f~, with

(look at the zeroes).

0

then

m is an integer~ 1,

where

=l

£

E ~(A,k,-1),

~(A,k-~ 1 +1).

dim ~(A 1 (m-l)k0 ,+l)

rf

EQUATION

dim ~(A 1 mk 01 +ll = 1 + [:].

Thus

·k

~JNCTIONAL

Thus

f~,

hence any

up: THEOREM j.

0 < A < 2.

Given

except in the case tntcgor 2:. 3 1 tllm nt(>..,k,C)

or

=1

h = 2 cos k -

and

+

Then 11J.(A 1 k 1 C) = 0 1T

...!t.aL ..

l

q-2

where

q'

-

c;

q

is an

tn this case

m+~ [

q

?

] •

course, from the point or view or Theorem 1 1

Theorem 3 gives the nnswer to the wrong question. Let us

~all

f

£

11J.(A,k,C)

-k

and multiplier

at

and 1At

C for

g(>,.,k,C)

n

~

G(A))

IQrm (of dimension if it vanishes

be the space of all such

MODULAR FORMS AND DIRICHLET SERIES

I-24 cusp forms. dim

Thus

~(A 1 k,C)

Givan

as ln Theorem 3· a)

A

1.

cos ~q'

=2

and

=~ q-2 + 1

k

- C

Then: ~(A 1 k,C);

&(A,k,C) c f(x + ly)

in fact,

=O(y-k/2 )

dlm 8(A 1 k,C)

b)

~

- dim &(A,k,C)

THEOREM 3'·

~(A,k 1 C),

aCA,k,C) c

[

if

f

m+~ q 2 1,

t

8(A 1 k,C).

the dimension

of the space of Dirichlet series of signature

ll(A) +dim 8(A 1 k 1 C),

dim lllo(A 1 k 1 C)

c)

which are regular at

= k.

s

~.

(A 1 k,C)

where

ll(A)

or

and

k

0

or

l

is independent

c.

b) follows from the proof of Theorem 3,

since we did construct modular forms not vanisntng at

w,

so

dim m(A 1 k 1 C)

=1

yk12 1r<x + iy)l

F(x + iy) (vanishes at

w)

and

+dim I(A,k,C). is bounded on

For a), D(A)

G(A)-1nvariant and so bounded.

c) ,is equivalent with the following statement: there exists one function f(m)

I o,

then every

h

f t

t

~{A 1 k 1 C)

mCA,k' 1 C')

if

with

also satisfies

DIRICHLET SERIES WITH FUNCTIONAL EQUATION the

0-condttton.

In fact, stnca

I-2'

k, k'

are ra-

tlonal, we can choose positive integers a constant

u

so that

hence satisfies the also satisfy the

O-cond1t1on; then

ll()..)

appears to be open in general. functions not vanishing at by the

LEMMA.

If

h

and

must

O-cona1t1on.

The question of whether

~ttly

n, m and

urn + hm e aC>..,k" ,+1)

For

r•

thon

or

=1,

)..

1

the

are provided expli-

w

Eisenstein~.

k > 2,

=0

as follows. lnT + mj-k

m,n~

converges in sets.

Im

~

> 0,

uniformly on compact sub-

(The prime on the summation symbol means

the term for For

~.

(m,n) ~

= (0,0)

in a compact set, there exists

B >0

with

x, y.

Stnce there are only

lnl + lml

1~ +

= r,

ls omitted.)

Yl

~

B(lxl

+

4r

IYP

fo1· till red

pairs

which is finite for for

with

our series is dominated term-by-

term by

Hen~::e

(n,m)

k

k > 2.

= 4,

6, B,

... ,

the

MODULAR FOHMS AND DIRICIILET SERIES

I-26 Eisenstein

~

r•

(nr + m)-k

n,mcL is holomorphic on

Gk(~~

:g) = (cT

(The

g2

and

g 2 ~ 6~,

g3

PROPOSITION

~.

Im -r

d)k Gk(T)

+

= 140G6 .)

a

v

Clearly 2

Now ~ ;;1n ,,..

for

(~ ~)

£

SL(2,Z).

The Fourier expansion is:

(n) "'

Gk(-r) = 2C(k) + E~=l

= mtL 1: (m

entire, or period

and satisfies

of Weierstrass theory are

g3

where

~.

> 0,

+

1,

-rl- 2 ,

1: (m + n-r)-k. mel:

for the rltfference is

even, and-+ 0

as

Thus: (2nll 2 z 0-z) 2

Im r -•

~.

DIRICIILET SERIES WITH FUNCTTONAL EQUATION

~c~

D1fferent1at1ng wlth respect to

I-27

= 2wiz):

~ (m ~ ~)-k ~ C-2n1~k ~~ nk-1 n '

mtl:

~n=l

Ck-1 !

z •

Thus q.e.d. Ok

Thus

k = 4, 6, ••• , b(l)

=1

lws signature

(l,k,C-l)k/2 )

and satisfies the

for

0-condltlon, so

tn the notation of Theorum ]

1•

The cor-

responding Dirichlet series is

which satisfies t.h .. functional equation

where

~(:;) - (2rr)- 5 r(s)"k(o),

which ur course can

also be derived from the functional equation for

t(s),

which we have not yet proved. C-l)k12a 0

Note

ress=k~k(s) (2JT) -kl'(k).

in

n~rcement

with the aDOve,

assumtr~

known that

MODULAR FORMS AND DIRICHLET SERIES

1-28

dim m(l 1 k,(-l)k/2 )

Actually, since

k

=4

61

1

a4 , a6

we see that

are the

=1 f0

for

r1

,

of lemma 4 (up to a constant multiple) and in par-

~0

ticular B(l).

r

= e 2 n1/3

ig the only zero of

It follows that the

~ SL(2 1 ~)/~I,

transformations + 11

~ ~ ~

form for

G4

in

modular~

the group of all linear fractional fro~

is generated by

SL(2,~),

~ ~ -1/~,

since

a4 is a modular

r but has only one zero in the fundamen-

tal domain for the subgroup. The normalized E1 sens'.§ln llW.l!_ are the series

for

k

~

4, 6, ••. ,

positive real number. (Ak: 2kiB 2k' follows. we get

B2k

= ktb

Starting from

Actually

Ak

is rational

Bernoulli number), as sins:

s~ (1- 0~ 2 ?), n=l

.,.-

DIRICHLET SERIES WITH FUNCTIONAL EQUATION

s cot s

1-29

~ log sin s

=s

=1 -

=1 1

which shows

Ak

power series in One finds

A4

~

is rational since s

s cot s

is a

with rational coefficients.

240,

A6 = 504.

Recalling that the first cusp form occurs

= 1~

k

when

(since the formula of Theorem 3' gives

dim l(l,k,(-l)k/ 2 ) - 1 ~

Writing

lf

= [~]

if

k - 2 (mod 12))

~ (l" )3 - E6 ('I: )2 -. 17?8

k

* = 2 (mod 12)

let us define

Then 6 . 8(1,12,1).

MODULAR FORMS AND DIRICHLEr SERIES

I-30

one finds

3•240 + 2·504 1728

a1

Actually, the

an

=1

•

and all

an

~

Q.

are integers; to see this, we

need, in an obvious notation, (1 + 240U)3 ~ (1- S04V) 2 (mod 123),

i.e.

3•240U ~ -2•504V (mod 123),

that

for whlch 1 t 11Uf.fices

f.e. d3

aJ(n)

=ds

12),

(mod

~

a1

~.

e 8(1 112,1),

= 1.

~(T)

.

t >t zn, n=l n

whuru

By the formula for the number of

zeroes, we see M·d I o

= 1);

(mod 1.?),

which is true since

(mod 121.

Thus an c

= os(n)

U _ V

rnr

Im r ) 0

c...!L 12

=n...

in the Weierstrass theory of elliptic rune-

tions ~(t)

= (2w)-l?(g23

- 2'i'g3 2)

1s the discri-

mtnant.

Now take the quotient or two forms of di-

mension

-12

where

b0 c L,

cations;

j(Tl

to

go:t

the elliptic modular lnyurianL

a crucial fact tn arithmetic appl1-

= j(t

1)

11' and only if

T 1

is

DIRICIILET SERIES WITH FUNCTIONAL I>QUATION equivalent to

t:.

~·

•

under the modular group.

has the product expansion

cr. Siegel [13] for a short proof. follows Theorem

~

Another proof

In these notes.

Thus we have shown the case

I-Jl

=~(A,k,C)

~(A,k,C)

A = 1 by explicit construction (the

Eisenstein series) of forms not vanishing at which do satisfy the result for A=

t

0-conditlon.

"s follows.

2, ], .•• ,

c rrtCJt,k,C).

We can use this

Let

f

€

"t(l,k,Cl.

.p;

= 2,

t

3,

so

= ~(A,k,C)

mtA,k,C)

is a subgroup of

talus three copies of

r,

~

and

G(l),

R(l)

we ace that

we

k

=./2,

• 2;

note

B(2)

con-

Since every point of

G(2)-equivalent to a point of

Proposition 2, and for

G(l) B(l l.

Jt < 2, for

Finally, we consider the case

H is

If

g(•l ~ fCJt•> + t-k12 f(~l

then

Taking

conclude that

G(2)

m

A = 1 to prove the same thing for

./2, J'S,

= 1,

in

B(2),

by

is a fundamental domain (l';G(2J)

.S

j.

Defining

Jj.

MODULAR FORMS AND DIRICHLET SERIES

I-32 r(2),

the principal congruence subgroup of

level

2,

o~ where

f

is reduction modulo

r,

has

f

Since clearly

G(2) => !'(2)

and hence that

is a fundamental domain for B(2)

one checks

we conclude

3 in

has index

2,

= 6.

(1'• 1(2))

(G(2)r(2) :r(2)) = 2,

Now

of

rc2> ~ r ! sLC2,~1~>.

1s onto and so

0(2)

r

by

and

B(2)

G(2),

two~

(points where it

meets the boundary of the upper hnlf plane), and

-1;

+1

is equivalent to

and so 1s not counted. 0

at each cusp. 0(2)

modulo

Note

-1 B(2)

~

WA make

H/G(::>)

t =

t

3)

t

4)

=H U {

00

c!......=-1> T + i c

t

2

at

+ 2

,-lJ 0)

as follows:

at points not equivalent to

T

"'• -1

2)

T ~ T

has an angle of

into a Riemann surface (of genus

by assigning local parameters 1)

under

i'

1

em•

at t ,. e-2'111./T+l

Elt

-1.

The reason for these choices ls as follows, at the three corners

i, "'• -1,

Except

a neighborhood of

DIRICHLET

WITH FUNCTIONAL EQUATION

SERI&~

1-33

contains no equivalent point, whence 1).

T

At

those three points, one computes the stability groups tn by

T

~

G(2)

which has order

1

at

-1/T)

(generated

2

an elliptic !1xad nQint, and

i,

infinite cyclic (generated by the "least translation" T ~ T + 2)

~

-1

a parMl!oll c LI.Al:..ll.

""•

whence 2), 3).

~. T

at

-l/T+2 on

(in

-1

Now

G(2)),

and

w, and changes

T ~

~-=1..+

T+2

Is a fixed point or T 1

= -l/T+1

-1/T+2

throws

into

:~:T 1 -1 1•

-1

I

T

Jl.2.1nl, or

T +

1

1

ls the appropriate

thus local variable Rt

-1.

(Gonurally

sp~aklng,

to

treat questions of analyticity at any rational cusp, the procedure is to send It to

w

by a linear

fractional transformation and proceed as before.) We now investigate the meaning or the

0-con-

dftlon: LEMMA 1.

If

regular" Ht

f

'llv(.?,k,C),

£

T = -1 1

k

r

In the VHrtable

tn the sense that rc~ >c.I.....±.J.J i

then

tnh(t)

ls "quasit _ -2rrl/T+l

- e

'

MODULAR FORMS AND DIRICHLET SERIES

I-34

where and

h(t) n

2 0; tn = e- 2 n1n/T+l.

tn general, and ts called the f(T)

at

~· T

.. o

f(T)

(n

= 0,

t

is fractional

~

2t

~

of

= -1.)

Let

is

1

'1:

I 0 at

is holomorphic and

= -1/T+l,

T 1

-~o-r'-l

c!.fllk

as above, so

r(-r)(~)k

Now

•

T'

f(T)(l)

-k _

-

E(

T

~

-l/r+2

= cr(-l/T+2},

~

t

)

-k

..=.1..

f(t+ 2 ),

for

some constant E of absolute value 1. Write e 2 wtp, where p is real, in general 1rratlonal.

E

=

Then

ts invariant under

T ~

-l/T+2

and so has a Laurent expansion

r C·• >c.!..!l.) 1

1:

0

r

Take

T 1 -1)

E a e2w1nr'. n--"" n •

if

n + p

< o. We have

'+1 I r ("t) (~) k e -2nt 0

~

k

an = 0

We now show

(i.e. T'

'

•' = u • ib,

(n+p )-.:I d"t I

1 1 ~ u ~ 2,

h

lArgo.

The

term

l-35

DIRICIILI:."'l' SERIES WITH FUNCTIONAL EQUATION

-k

t

<1->

con be ignored; if

f(x + iy) then

f(~) so for

~) ..,

= f(-

O(e 2ub(n+p)bc)

an

n + p

~1 (2,k,C)

consisting of those

regular at Remark 1.

-r

If

= -I; k

thus

Then

Let

n_ 1

f(-r)(~)

f

so

of zero at

=0

f

which are quasi~l

f

£ ~(l,k,(-1)

at

f

satisfies

-1

k/2 ) c:

= Cf(-1/-r+l),

so

n_ 1

~(2,k,(-l)

-1

and

k/2 )

In fact 1

= n_,

as previously defined.

tt :;hould be, since the cusps

(2,k,C).

~~ (2,k,C),

£

ls an Integer, by Remark 1. k

an

the subspace of

'\,C2,k,C) c:

is an integer,

then the order of zero of

Remark 2.

b,

< O, q.e.d.

Let us denote by ~(2,k 1 C)

for large

the order This Is as "'

are equi-

I-36

MODULAR FORMS AND DIRICHLET SERIES

r

valent under and

k

= p/q

= G(l).

N0te that if

f

~

m1 C2

h = f 12 q/~P

is rational, then

a quotient of elements of

rrtC2,12p,+l)

a meromorph1c function on

HJo(2,.

1

k 1 C)

is

and is then

(By remark 1,

the numerator and denominator are regular, not just 11

qUAs1-regu1Ar 11 , et the cusps.)

Thus

h

has as

many zeroes as poles on the Riemann surface. ~

has one zero on

f

has

D(l),

3 on

hence

....llL =4 k. zeroes on 12q

B(2)

1

B(2)

Now 1

so

measured in

local parameters on the Riemann surface.

This is

actually true generally:

LEMMA 2.

The number or zeroes or

in

1s

B(2) N +

would be called

others are adjusted. as a limiting case, as mula for ~-

~

c

m1 C2,k,C)

k. + ~ 2 + n_l = 4"

n..

n

(~

f

ni

1n local variables; the

This lemma can be considered u-+

o, of the similar for-

< 2.)

This is proved exactly as the analogous

result for

~

< 2; we only have to check that

DIRICHLET SERIES WITH FUNCTIONAL EQUATION

~I d log

f(~)

-n-1 2

tends to

y

arc Now

y

about

~

= -1

I d logCL.f-ll r

k

in

... o,

B(2)

as a little shrinks to zero.

so we want k

~I d log r<~> r

-

-n

~.

which follows from the substitution wntch carries

J•J

1

on

Re(r)

Re(•'l : -

We also have dim

= -1

m1 C2,k,l)

C

I-)7

on

Re(T 1 )

~· ~

= • ~1 1 , 0

and

!·

= (-1)

n

1,

and

~I+(~],

dim ml(2,k,-l) ~ 1 + [~]. as before. ture

Hence

'(2s),

if known to have signa-

(2,t 1 !), is determined by its functional

equation, as Is the ?.eta-function of q.(s)

=~

2

E' Cn 2 + m ) n,m£l:

which has signature

(2,1,1).

-5

,

~(1),

MODULAR FORMS AND DIRICIILET SERIES

I-38

2

00

LEMMA J.

~(2,~,1),

belongs to

!

is one of order ~·

satisfies

.::1

"(~)

at

e mn -r 2n=-""

l

r

and Its only zero in -r

= "(-r),

"(-r + 2)

ls

0-condition.

-r l/2

=

"(-r).

Poisson sumroatton formula:

B(2)

= -1.

is clearly holomorphic tn

"(-r)

and satisfies the show

" ( T) -

The theta-function

Im

-r

> 0,

holomorphtc at

""•

We want now to

For this we apply the if

t f(x + n) n=-co

converges absolutely, uniformly on compact subsets, to a continuously differentiable function where

F(x),

x is a real variable, then F(x)

= ~

9 2Tr1nx

o.

n=-"" n

is represented by its Fourier series;

Applying this to parameter with

f(x)

=

Im -r > O:

where

Is a

DTRICIILET SERIES WITH FUNCTIONAL

I-39

E~UATION

J?i2 - n::.)2 dul

2nu +

T

n=-Q)

-11.,;

_...

n:;::-0/j

Cf>

We cla fm the integral is r =- iy,

take

-1/2

;

it surrtces to

y > 0:

2 "" -wy(u+1n) y du

re

-"'

by Cauchy's theorem,

r..e -u-2 du = y -l/2 •

( wy ) -l/2

-~

"C• ,x) Ct> l/:1 =

Thu:o

X : 0

1

we get

t

Taking

To find

n_ 1 - urder or zero or in

e rrin'?/-r: +- 2 TT1nx

1:

n=-""

,J(r)(.!....f-l)l/2,

= e-2rri/r+l:

"c~.t> I

c

-1ri Cn 2 +n+t>l•+l I

measured

MODULAR FORMS AND DIRICHLET SERIES

1-40

8

-ntflt(Hl )h(t)

t 118hCt), where

h(O) 1 0.

This proves the lemma.

We can now prove whence

~(•)

~ ~1 (2,k,l),

~(2 1 k,l)

1 + [~].

Since

~(•) I 0

Im • > 0, define, for fying

~ 2 k<• + 2)

~

1 + [~],

of dlmen3lon lm • > 0, em•

z

dofin~

= ~ 2 kC•>,

log

and

with non-zero in

~(~)

Thus

~2k

holomorphic at

w,

0-condltion; also

for some constant

= 1.

2

still a power series ln z, We then k > o, ~ 2 k<•) = e 2k log~<~>, satis-

and satisfying the

at

for

is a power series in

constant term, we can

t

dim ~(2,k,l)

•

E• £

substituting

~(~,k,l),

•

~

we see

with its only zero

-1.

The Eisenstein series

~(>)

~(2,4,1),

with its only zero at

Now if

f c

~ 1 (2,k,l),

at

for " unique constant

then

r-

c

~(1,4,1)

c

• 0 = e2n1/3 • a 0 " 2k

so

vanishes f - " 0 ,2k

1-~1

DIRICHLET SERIES WITH FUNCTIONAL EQUATION

= E4.

f1,

if

= 4,

k

where

fl

E

mC2,k-4,l).

1f k < 4.)

0

(fl

1s constant

Continuing, any such

f

is uniquely of the form 1: a ,2k-41 E.i i,ik/4 1 -'t

which proves l

= ~(2,k,l),

ml(2,k,l)

+ [~].

For g = ~8 -

C

= -1,

aE4

chooge

n

I 0

u

vanishes at

not have a zero at

1.

= -1,

•

zero at

= Jg,

Lhtm f1

t

= i n

-1 ~ (-1) 1 ,

E ~ 1 (2,k-2,+1).

nomtal in

~

=1 and

r,

If so

g

Then

dim

= u,

f =

Note

h,

f

mo<2,k,C)

o, where

m1 (2,k,-l)

+ [~]. so any

f 1

h · fl'

E

E4

is a poly-

m1 C2,k,C)

We have proved: THEOREM 4.

does

has a double

m1 c2,k,-ll,

1"(1)

Thus

g

so the formula

and no other zeroes.

,,2,2,-Il.

dim ~(2,k,-l)

so that

Clearly

1 = ~ = N + ~ + ~ shows that

h

of dimension

= m1 (2,k,C), and

is of

MODULAR FORMS AND DIRICHLET SERIES

1-42

whuru

ar(v)

as a sum or

is thu numbur of ways of writing r

(2~(T))r c m<2,~ 1 l).

squarus;

onu knows an uxplicit basts for "' ( n)e 'ITin't 1 + En=lar

writu

mC2,~ 1 l),

v If

one can

in terms or this basis

(which involves only knowing the first few coefficients) and thus get an explicit formula for for all 1)

n.

ar(n)

For examplAs

r~=la2(nln-s = 4'(~(i),s)

~

=4

n=l

n-s E (- 4 ). din d

E (~), din d

=4

Thus

a 2 Cnl

as is well known.

2)

E~"' 1 a 4 Cnln-s

C 2- 8 !;:(s),(s-1.) (2s-2 2-s)

3)

E~=la8(n)n-s

2- 5 !;:(s)C(s-3)(C 1+c 2 (2 8 +2 4 -sJ)

1tl

I~= 1 a 12 Cnln-s = C12- 5 r;;(s)!;:(s-5) (2 5 -26-s)

DIRICHLET SERIES WITH FUNCTIONAL EQUATION where

~(s)

1-43

J[rtl.

is associated to

(The first three formulas are classical.)

The

general principle has vast applicab111ty--if you know a basis of a space of modular forms, then you get artthmetlc identities. modular group

For example, for the

we have

I' I

~ = Ea,

~ ~E6,

ElO

etc. ~-

for

Now that we have the functional equation we can give a very natural proof, aue

t;;Cs),

to Weil, or the product expansion

~ = e 27T1• •.

where

A(~),

nition of ~1 2 A(T) 0 ~(~)

for

~(~) = ~~ (l-zn) 24 , n=l

Taking this product as the defiA(-1/~l

it suffices to show

~1nco clearly

~(1:

+

1) = ~(T)

is the unique cusp form of dimension

r.

Extracting the

-12

24th root, we have

Il!:!.l!:~1n!.l':~ [l.lll~t1QDo

'l(,;)

"

,.m•/24IT 0 n=l

and it suffices to prove Now let

f(·t)

Iff-

- zn) : (11/2lf(T)

~(-1/r)

log '1(1:)

r



=

and so

1/2

IJCrl.

MODULAR FORMS AND DIRICHLET SERIES which we associate as usual the Dirichlet series ~(s)

m- 1 Cmn)-s

t

C(s)t(s + 1).

m,n=l Thus

=k

f(-.:)

o1

where

I Cf> o=ol

> 1, and

-s

the

P-function)

= (-s).

= (2rr)- 9 f'(s)f(s).

~(s)

C(sl

functional equation for

(s)

(s)ds

gl~e

(and identities for

the functional equation

Furthermore,

for simple poles at

The

s

and a double pole at

=~

(s) 1

s = 0

is entire except

of residue

~ ~

~(s) + ~

with

2s

regular at poles,

exc1ud1ng a neighbOrhood of the

0;

~(s)

is bounded on vertical strips.

The

method of Theorem 1 is therefore applicable--shifling the line of integration to the left of noting that

.l. -r 2 lug I• ¢(s)

-m 121

= (-s),

-r


-s

at

fs)ds

s =-

1 ·,

1

has res dues

o,

-1,

-1,

..!J..l... 12 1:'

and applying

we get

f(T) : ~ 12t + .l, 2 log ~ 1 - lnT'f + f(-1/T)

DIRICHLET SERIES WTTH FUNCTIONAL EQUATION i.e.

= log

log 'l(-1/"t")

'l(-r) +

I-~5

t log f·

Quite stmtlarly, we can derive tho product expansion for the theta-function:

where

z

= errt•;

as above, i t suffices to take the

product as a definition of

~(•)

and prove

' 1/2 = Cyl ,,(•l.

"C-1/'t)

Here log "("t")

is associated to the Dirichlet series

.,Csl

'l'hus

where

log

; -C2mol-s + 2{-l}m-lm- 5 (2n-ll-s m m n,m-1

MT}

_l_ 2 rr1

o1 > 1 and

I

o=c1

~(s)

-r -s r(slCs )ds,

MODULAR FORMS AND DIRICHLET SERIES

I-46

at

s =!. 1,

entire.

~(s) - ~ is

we have this time

2s

By the same method as before, we get

log ~(T) + l log~= log ~(-1/T). 2 i This completes our general program of determining all solutions to the functional equation of Theorem 1, except for the ambiguity on the d1t1on of Theorem 1.

0-con-

Defore going on to the theory

of the Euler product, we close this chapter with a theorem on the zeroes of a Dirichlet series with functional equation! this theorem, due to Hardy and Heeke, is a good illustration of the technique of passing back and forth between D1richlet series and associated Fourier series, ~ la Theorem 1. First we need: DEJ.'INITION. domain, has lsi

~ ~

A function ~

c

f(s),

if

f(s)

in that domain, for all

PlffiAGMEN-LINDELOF THEOREM.

Let

and of finite order ln the strip ~

t,

where

s = o + 1t, k

f(
it)

O(t J)

regular t n some O(e I sl [

as

be regular ~

a

and suppose for

)

> 0.

f(s) o1

c+E

1, 2.

~

a2 ,

I-~7

DIRICHLET SERIES WITH FUNCTIONAL EQUATION Then o1

~a~

with

= O(tk(ol),

f(a +it) o2 ,

k(oj)

~·

where

= kj'

Say

= o1 ,

~ Aet 0 •

= k2 = 0, and t = 1.

k1

a= o2 ,

ger > c, whore

is the linear function

= 1, 2.

I f(o + it) I

first the case ~

k(o)

m: 2 (mod~).

p(g 1 t)

.S. BetD

t,

t on

> 0,

let

a~ "1'

~

Hence

lg(s)l ~BaED

t -• 0

Lot

A... tc+t(-tm+atm-

2)

= -tm

Re(~)

= ets

g(s)

a= a2 , ~

s,

t

we get

Note ,

f(s). t

Then

= 11

and

2 T(t).

by the maximum

lf(s)l

For the general case, let

or

Ill

and

B for

+ p(o,t),

t2

and

a

(some constant).

for all

s),

B on

m be an inte-

lf(s)J .S. Bet(D+Islm>.

(for fixed

S

Jp(o,t)j ~ atm- 2 •

so

lg(sll

principle, so

lfC~ll

so

Then

ReCsm) ~ -tm + atm- 2 ~ D

Jg(s)

Let us consider

is a polynomial ln

degree< m in

For any

uniformly in

Letting ~B.

h(s) =- e

k(s) log

Now Re(k(s) log

f>

= k(a)loclsl

-at arg(t - ta)

f

MOUULAR FORMS AND DIRICHLET SERIES

= k(a)1og So

= tk(v)eO(ll.

h(s)

t + 0(1)

Th en

~ hTST

sa ti s fi es t'•le

conditions of the special case, so

= O(tk(a)),

f(s) = O(h(s)) COROLLARY. in

a1

Let

~ a~

f(s)

a2 ,

Let

IL(a)

be

o1 !,. a!,. o2 , numbers Then

and suppose

c(a),

i.e.

IL(a)

so that

a£ [o1 ,a2 l.

for each

the~

c(a)

IL(O)

be regular and of finite order

1 !,. t,

for some constant

q.e.d.

functlon or

f,

is the infimum of the f(o + tt)

= O(tc(n)).

is a convex function.

PROPOSITION 6.

Given

~(s)

Uirichlet series converging somewhere, and C =

k > 01 (~,k,C),

~

1.

Then

~(s)

~

> 0,

has signature

i.e. satisfies the condition (A) of

Theorem 1, if and only if

~(s)

satisfies the

followlng condition: (A') (k-

s)~(s)

is entire or finite order,

DIRICHLET SERIES WITH FUNCTIONAL

~UATION

1-~9

and satisfies the functional equation

~(s); C~(kft22r·

~(s)

Let

has order

r(s)

s),

~

in

h~ts

order

> 0,

a0

k - a 1 ~a

boun
l.

there since Thus if

.f o1

"''")

for 0 l 01

(by the functional equation). in any strip

a

~ a~

by

f' +

Ca 0 k-s

~(s)

is

is

b,

0

"aLbofl"" (A')'

so also

l 01' and

0

H~nce

1 .f t,

~

k - 01

~(o

+ it)

= ocul

by the Phragmen-

L1ndelof theorem, which proves (A). ~·

the

satisfies (A), then it has

Conversely, 1C

for

is

remains;

<%>(s) +

't'(sl

(!>(a+ it) ,.. 0(1) + it) = O(l)

IJ>(S)

a ,i k - a 1

a

of order

in

~(s) = c1r>"~f:l·

functional equation, as doos Only the strip

- s)

~(s) ~ (~)-sl(s)y(s)

so

there and al:so in

1

Now

by Stirling's

By absolute convergence,

a 2 a 1 > k,

bounded in

2

r(;) = ~1(1

formula, as does then

a .f -a0 < 0.

(~,k,C).

have signature 1

= Cifl- 8 r(s)~(s).l

~(s)

where

You can state Proposition 6 for two

MODULAR FOHMS AND DIRICHLET SERIES

I-5'0

functions, as in Theorem 1.

(A') is actually the

condition stated in Heeke's papers.

THEORF>I 5'.

Let

be a non-zero

E

'i'(S)

n=l

.t.l!.lll. Dirichlet series (Le. an ().,k,C)

the associAted Fix

0 < a0 < k,

~(c0

+it),

I(t)

Im

~(c0

+it),

~(s)

us

~ -s = ().) l'(s)q>(s).

y ~ 0

or

p, 0 .S. p < c0

I(t)

and let

Suppose

through positive values, Cor +

t•

Then either

changes sign infinitely muny times.

(Iu ..·~rtl~ulur, tf

= 1,

,

Re

some constant

C

a0

R(t)

O(y-P)

R(t)

of signature

R)

and

modular form.

vhere

E

a0 ~ k 2'

~ck 2

or purely imaginary, 1f

+

ttl

C

= -1,

ta real, if

and so

has infinitely many zeroes on the middle line

ljl( s)

c=k

provided

~·

Since

2'

p < ~.)

~(s)

1s real on the real axis,

DIRJCHLF:I' SERIES WITII FUNCTIONAL E(,JUATION =-

1-'il

.l((a + it) + (ao - 1tll 2 0

= .l((a 2 0

+

ttl +

C~(k

- "o + it);

similarly, l(t)

= ..Lc<:>Ca 2i 0

+ it) - C(k -

a0

Note that replacing

by

k - a0

a0 + it)). does not affect

the statement to be proved, so we may assume

For any

a1 ,

0 < a 1 < k,

we have as usual:

~

= 21Ti y > o, side above 1s then

Hence

so

J (y)

a=a 1

y = e iCy-~)

-s

(s)ds.

The right

MODULAR FORMS AND DIRICHLET SERIES

I-52

Since t(y-!l.) 2 ~Cu1 + lt)dt

00

I e 0

0
say for

f,

= 0(1),

we get (change

t

4

-t,

and

conjugate): 00

I e 0

Hence

-t(y-!l.) 2 ~Cu1 + 1t)dt

G(t)

= R(t)

or

.. -t(y...!!) 2 G(t)dt I e 0

Now assume t >> 0.

O(t)

= O(y-P).

I(t)

satisftes

= O(y-P). has constant sign for

Then the integral converges absolutely:

If the the:u•·.,m is false, th1s ls true for both

R(t)

and I~Ca0

I(t)

and hence for

+ 1t)l

~

jR(t)j + ji(t)j:

DIRICHLET SERIES WITH FUNCTIONAL EQUATION Now

r(o0 + 1t) ~

J2W

-.Jk

e 2

t

0' 0

_l 2

(t ~ +~),

I-53 so we

get

where

= o0

a

we have

~

-

~.

so

0

I

Setting

y

=f

fortiori

0' +1T

Then

> p.

u + 1

R

Q

Is <j>(s) lids I

= O(Tt'),

so

0'0+1

for any natural number

where

13 C a+ 1,

m,

and so

~

whlch leads to a contradiction,

via Phragmen-Llndelllf, as follows. Choose

m so

fortiori

am I 0

and let

MODULAR FORMS AND DIRICHLET SERIES

I-51t

for

a > 0,

(of order

t

~(z)

actually), we soc

1,

z(a0

order,

2 1. Stnce

+ it) : O(tP). ~(o)

growth function

for

is of finite order z(o)

has finite

We will now show the Z(s)

is

a + 1

for

a

oufficiently large (in the domain of absolute con-

vergence of ~(a).

~(s)),

contrary to the convexity of

In fact, for large

01

Z(o + it)

+ 0(1) =

( 9 + it) a+l 8 m a + 1

+ O(ta) + 0(1)

since

o+lt a z I z (~) dz O+i

~.

O+it aza-l(.l!l)z I ----'0~-ddz

o+t

log m

Looking at other lines In the critical

n

DIRICHLET SERIES WITH

~~NCTIONAL

EQUATION

I-~~

strip besides the middle line was suggested by Berlowitz [2].

EXAMPLES. 1)

~

If

< 2,

then one can take

P

and

= ~

a0

~(s)

l.u.

o0

we can only k -

- t(s)t(s • 1 - k) 1

for

= Gk(T : 1 1 > I 0 1 m as take ~ = k; the theorem

t < 00 < k.

£

8(~ 1 k 1 Gl,

•

corresponds to the Eisenstein series Gk(T)(T - l)k

r

3'· Hence

by Theorem

the theorem applies for any 2)

o,

z

k- 4, 6, ••• ,

Gk(T). T

~ 1.

Now Thus

applies for

CHAPTER II IIECKE OPERATORS FOR THE FULL MODULAR GROUP

Let

k

be a fixed positive integer (unlike

the first chapter, where be real.)

If

L

= (~ ~

k

was only required to

is a renl mntrix w1 th

positive determinant, and

f(r)

is a holomorphic

function on the upper half plano, let

fjL

be the

holomorphic function on the upper ha u· plane defined by

note

(o ~) • f

f1

ror

a >

o.

Passing to the

corresponding homogeneous function F(w1 ,w2 )

= w-kf(w /w ) 2 1 2

Con the space of two variables

Cw1 ,w2 l

with

MODULAR FORMS AND DIRICHLET SERIES

II-2

we get (~1 • ~)

-k

awl • ~2 + dw )

f(~

1

2

(ad- bc)-k 12w2kCfiL)(w1/w2 ).

r

Thus

~

F

inducua

fiL ~ Cad - bc)k/2r • L

= ILif • L,

which proves the rule

= fi(LM).

CfiLliM f

will be a modular

for a subgroup

G of

£

G;

G.

(1 1\

SL(2,Z),

o

v·

modular~

~

T+l 1 T

just that ~(r,k)

k

-k

If, besides cer-

fiL = f

for all

note it suffices to chock this for a set of

generators of

T

of dimension

GL.(2JR)

tain regulartt.y r.ondttions, L

~

~

is

For the homogeneous modular

(0 -1\ (-1 0\ 1 oJ • o -v r

-1/T);

f(T)

= SL(2,Z)/~I,

~

will do (the generated by

the regularity condition is

is holomorphic at

be the space of

~.

We let

n11 such, where we now take

to be an even integer;

HECKE OPERATORS FOR TilE FULL MODULAR GROUP

a
in the earlier notation.

Similarly,

the space of

vanishing at

~ ~

f

r

for

mer ,kl

lllU',k)

£

or dimension

C:·~ 6)

-k.

Il-3

is

~,

the

We know

&(l,k).

We now give a "geometric" definition of the llecke operators. L

1n

each

C:;

AmCro1 1ro2 l > 0,

SL(2,Z).

X be the set of lattices

Let L

has a basis

unique up to the action of divisor~

D be the

Let

wl th

w 11 w 2

i.e. the free abelian group on

t

formal finite sums

X,

nLL =A,

i.e. the set or with

L~:X

A

the

is nos1t1ye if all ~

~

A is deg A

of

correspondence on

X

mapping" of

n1.

written

0,

=t

L

X or degree

nL. n

x,

or

n1

£ ~;

A~

0;

An abstract is a "one-to-n

into itself, i.e. a homomorphism

D ~ D which carries each positive divisor or degree for

n

1

to a positive divisor or degree

= 1,2,3, •••

n.

we define three types of cor-

respondences: 1)

T(n)

associates to a lattice

sublatticos

L'

of index

n.

L all

MODULAR FORMS AND DIRICHLET SERIES

II-4 2)

TCl,nl

lld1111t1ll!l sublattices or index

L/L'

Le. such that J)

T(n,nl

LEMMA. any

Any

L all

associates to a lattice

associates

T(a 1 a)

n,

is cyclic of order

nL

L.

to

commutes with any

T(n)

and

T(n,n).

frQQ!.

Reading from left to right, say,

aosoo1ntes to a lattice

in

aL;

note

L'

= aL"

for a unique

THEOREM 6. t1cular, T(n)T(m)

(L:L')

T(n)T(m) T(n)

~

T(mn)

L

all

T(a,a)T(n)

L'

of index

= a 2n.

Since

L' caL,

L"

of index

n

in

E dT(d,d)T(~); dln,m d

n

L.

in par-

is multiplicative, i.e. if

(m 1 n) - 1.

Hence tho

T(n)

generate a commutative ring of correspondences on

x. ~·

These correspondences were known to

llurwitz, but he did not know their commutativity.

fr2Qi. n

=p

The first and key step is the special case is prime,

m

= ps,

so we want:

n.

HECKE OPERATORS FOR THE FULL MODULAR GROUP

The left side takes a lattice sublattices

Li•···•Lp+l

Li

thon takes each Thus

L"

uf'

s+l p

has index

p + 1

L of index

L"

the

to

L to the

of

in

index

L,

11-5

p,

and

p"

ln

and each

L"

certainly occurs; what duplication takes place?

If

L"

comes from

Ll n Lj = pL,

L" c

L"

and then

Li;

then

is contained in all

(p + 1)

of the

thus we get the right side. By an induction, left to the reader, we get

T(pr)T(ps) ~

E

pvT(pv,pv)T(pr+s-?vl.

v~r,s

Finally, lf' for if

L"

(n,m) = 1, has index

for a untque

L'

then nm

of tndex

T (n )T (m) = T (nm) ,

in

L,

then

n

in

L;

L ;:, L' ;:, L" this proves

the theorem. ~.

(C.T.C. Wall)

T(n)T(m)

=T(m)T(n)

A facetious proof that

MODULAR FORMS AND DIRICHLET SeRIES

II-6

ror arbitrary

m, n

of index

on the one side we wish to know the

nm,

is to observe that given

number of subgroups of order

n

or

L/L"

L"

and on

the other the number of factor groups of order

n,

and these two numbers are the same, by Pontrjagln duality. Now let

FCw 1 .w2 l,

for

ImCw1!w2 l > O,

homogeneOUS functiOn Of dimension invariant under

SL(2 1Z).

Then

-k

be a

Which is

FCw1 ,w2 )

= F(L)

may be regardad as a function of the lattice L

= Zw1 + zoo2 •

Define an operator

T(n)

on such

functions by F·T(n)(L)

where in

L.

L'

runs over the sublattices of index

(The factor

nk-l

Simllarly,

so the 1dent1 ty of Theorem 6 becomes T(n)T(m)

n

turns out to be convenient.)

HEeKE OPERATORS FOR THE FULL MODULAR GROUP operating on

ll-7

F.

Another way to do all this is as follows. Let

M(n)

2

be the set of all

mntrtces of determinant

n,

K

and

integer

?

M•(n)

the pr1-

m1t1ve ones.

u

LEMMA.

p(B0 db)

ad=n

I

d>O b moll d

(a,b,d)=l

r• = SL(2,Z),

where

and the union is disjoint!

the index (number of cosets) is (M.(n):f 1 )

= tCnl = nTI (1 pjn

+ .l.)

P •

Similarly, M(n)

=

I Jr a¥n

1 ("

o' r

o d)

1

r•(a b)

_

0 d

d>O

•

d

aid

frQQr.

The decompositions fol:ow from the fact

that left multiplications from

r

1

are row opera-

lions, right multiplications are column operations. For the index formula, note first that the index is multiplicative, for if

(n,m) ' 1,

M•(n)

=Ui·•a 1 ,

11-8

MODULAR FORMS AND DIRICHLET SERIES

the union ls diojoint, for if

19 n matrix with coerrt~1Rnts 1n

and n

~.

= pr,

pr-1 - p

z,

hence in we get

r-2

pr

For

(~ p~)•

(P0 pr-lb) , ... ,p-1 aml ( 0Pr o) 1

representatives

Now given a lattice n,

oL

aL

I

L,

so

u

or

is a sublattlce or index

n,

prlmlt1ve lf and only lf furthermore,

1 = 1'.

representatives

( r-1 b) representatives\~ p ,

determinant

= j',

so

~ n

a

and a matrix

ls a primitive matrix;

depends only on the coset

r•u,

and we get a one-one correspondence between coscts r•a

r

£

and sublattlces ~ { r,k),

and

uL

of

L.

Hence if

F{w1 ,w 2 > -- w2-k f{ ~) w2

is the

HECKE OPERATORS FOR THE FULL HOUULAR GROUP

Il-9

corresponding homogeneous function, we define the n!h ~ OPerator by

.1!.._1

f!T(n)

where

M(n)

= n2

r flu a

~v1·•u,

which 1s the inhomogeneous

function corresponding to fiT(n)IA

~

CIT(n)

for

F·T(n).

A£ r•,

still formally a modular form for sion

Then fiT(n)

su

r,

-k; we will verify shortly that

holomorphic at

~.

T(n)T(m)

mer ,k).

PROPOSITION 7.

Then fiT(n)(r)

Let

of dimenf!T(n)

is

We still have the same identi-

ties:

on

is

f

£

111(1 ,k),

say

MODULAR FORMS AND DIRICHLET SERIES

II-10

where a (n) v

Henco

r(T(n) c

~(r,k),

and is a cusp form if

f

is. ~·

~ a

t

f(TCn)(T)

ad=n v=O

8 2n1avr/d

v

q.e.d.

(We used the fact that the character sum e 2 "lbv/d

t

is

d

1f djv

and

0

otherwise.)

b mod d

We now form the formal Dirichlet series D(s) = I T(n)n- 5 1 n=l

operators on space.

whose coefficients

~Cr,kl,

T(n)

are

a finite dimensional vector

The Identities among the Hacke operators

are most neatly expressed

8SZ

ijECKE OPERATORS FOR THE FULL MODULAR GROUP THEOREM 6'.

= "'I:

D(s}

II-ll

T(n)n-s

n=l

=TT CI

_ T(p)p-s + Pk-l-2sn-l,

p

where

is the identity matrix.

I

~·

In view of the multiplicativity, we have

only to check that I

"'I: TCv">r-vscr- T(p)p-s + Pk-l-2sr>, v=O

whtch follows from the previous 1dent1 ties (and vice versa}.

COROLLARY.

Let

and suppose

1'

~;

fjT{n)

= c.n r.

so normalizing to have en an'

of'

f(T)

"

E

e21riv"<

\1=0 a"

1s an eiscnfunction f'o•· the llecke

f

algebra, say

1!\CI.,k),

a1

Then

= 1,

en· al

= an'

the eigenvalue

ts the same as Fourier coefficient

T(n)

and the associated Dirichlet series has the

l!:ulRr product <j!{s) =

I:

n=l

ann-s = ITCl - app-s + Pk-l-2s) -1 p

'

MODULAR FORMS AND DIRICHLET SERIES

II -1:.>

rest ts clear. We now prove the converse or the corollary ln a strang fnrm--t.he only possible Euler product for

tCs)

is when

T(n).

is an eigenfunction for the

f(-r)

First we prove a useful preliminary result;

PROPOSITION 8.

Let

f

1)

If

n > 1,

then

2)

Let

p

be

a)

If

fla

t

mcr,k)

If

f.l:QQ!.. Q

£

f =

l)

f apn

Then

fl (6 ~H6 n(g ~)

w1ere I

L, Mt r ' ,

a

£

M00 (n),

r "' o.

= 0. =0

for all

so

But then

m with

for all

We lllliY as well take

o)r• •

r•(l 0 n

for some

a fixed prime.

a m "'0

then b)

"lCI',k)'

E

(I

n, =

then

(l0 0)n '

fl(b ~)(~ i)= f1(6 ~). = fl(g ~)· Now (~ ~ fl(ol no2) ,-, r, f

0

p .( m,

f

o.

since so

= L(~ ~)M,

i.e. f(-r)

(look at the Fourier

r&

HEeKE OPERATORS FOR THE FULL MODULAR GROUP

II-13

series). 2)

fC·rl 8

f

pn

1

0

:.

for

i.e. f

= fj (g

then

fiTCpl

I

p ( m1

;) ,

so

f

= p~- 1 r1(~

then

~

0.

f(~

+

i>

If all

~) ~Cr 1 kl 1

so

£

= o. Now let

f

=0

am

If

£

~(I ,k).

We say

re 1 a t1 v e 1.2. .tla

i.e. a

mp"

THEOREM 7. ~ay

haJi.

correspond to

OUl ~

product

for

p ( m1

v ~ 0.

Note

by 2a) of Proposition 8.

'l-Cs)

if and only if fjT(p)

p(s)

ann-s I 0

JU:1.Inl2. p 1f

= Rmc(pv)

c(l) - 1 1

p

= E~..1

~(s)

has an Euler product relative to

f

= c·f.

is an eigenrunction for If

~o 1

then the

T(p)

p-raetor is

necossRrily

~.

Assume first

relative to

p1

~(s)

has an Euler product

II -14

v

~

MODULAR FORMS AND DIRiaiiLET SERIES

0.

Now

fiTCp><~> ~ pk- 1 r
~

+

r

Pt mod p

p~-lf(p~) + E a

pn

f(~) P

zn,

so fiT(p)(~) - c(p)f(~) = pk-lf(pT) +

r(apn - c(p)an)zn

=a power series in

0, by Proposition B.

so is

Conversely, suppose any

zP,

fiT(p)

C•

f.

Then, for

n,

an

(p)

=

{anp nnp +

if p

p

k-1

.f n

an p

by Proposition 7•

Hence

if

Pin,

HECKE OPERATORS FOR THE FULL MODULAR GROUP I

p(m

amm

fj)(s)

Thus

has

f(•)

the associated Heeke operators. series

Gk

-s

q.e.d.

an Euler product if and only if is an eigenfunction for the

For example, we know the Eisenstein

corresponds to (a constant limes)

q>(s) "'l;;(s)t;;(s + 1- k), duct, so

11-15

Gk

which has an Euler pro-

is an eigenfunction.

Thus the do-

composition

ts preserved by all Hacke operators, and to show

mcr,k) to show

haS

II

basis of eigenvectors it SUffices does, which is the goal of tho

8(1' 1 k)

next chapter.

There we will show R(r,k)

an inner product in which the npE>rators, and so can

l!u

T(n)

has

are Ilermitinn

simultaneously d1agonal1zed

by standard linear algebra. ts one-dimensional, generated by

8Cl',l2)

z e 211 1L;

n 0 - z1124 )

n=l

hence the corresponding Dirichlet series

MODULAR FORMS AND Dli:Ul:IILET SERIES

II-16

r

q.(s)

n=l

a n-s n

""Tin p

q>(s)

has the Euler product

- app

-s + Pll-2s) -1 I

first proved by Morde11 [7] tn 1?17. conlecture says that the polynomial has conjugate roots, i.e. lapl

~

2p

conlecturr io thnt any dgenvalU('

ltal!!ilnulan's 1 -

8

11/2

•

np

or

11 2 pt + p t

Peterson's T(p)

k=.l. on

acr,k)

prime

satisfies

lapl

~

2p

2

,

for any

p. As a final remark, the Eisenstein series

Gk

ia characterlzAd as the eigenfunction not vanishtng at n, f

r

where

= a·Gk

form, so

In facl, if

w,

+

g,

fiT(n) = An f,

for all

is not a cusp form, we write wherH

g

An- ~- 1 (n),

is a cusp form; then

f

=constant ttmes

r.k.

CHAP'f~

III

THE Pf:I'ERSSON INNER PRODUCT

Although we are mainly interested in the full modular group

r,

it is convenient at this point

to develop some machinery relative to a subgroup

G of

r of finite index

want to make

Ji/(}

the natural map

~·

In particular, we

into a R1ems.nn !lllrface, so that

tVG- H/I'-

is holomorphtc; there

ts very little to do, since the fundamental domain D(G)

will be a union of

mental domain Any

~

copies of the funda-

ocr).

element or

,. carries

onto

..,

o,

say,

ra tlonn.l number; in thl s way we get tnequlvalent (under

G)

points

P1 '- .. , + l'

T

so the stability gr·oup

r(p 1 )

or a

of

P1

in

r

is

MODULAR FORMS AND DIRICHLET SERIES

III-2

infinite cyclic, genetnted by

gemu·ated

el

u

by

•

local parameter at near

G is infinite cyclic,

in

We tnke pl

Lj t

r,

zl = 8 2Trh/e1

(it is one-one on

the stability group

tnr1n1te cyclic on

is tho

then the

At one of the other cusps

Pl) •

where

pl

e1

If

U81 & G,

least positive integer with stability group of

U.

PJ r(p j)

as H/G 1

lt•

ts

uJ and

appropriate local parameter.

is a fundamental domain for ~

H/G with

= H lJ

{P 1 , ... ,P 0 }/G

fOG .. H/r

cation index of

Then

G,

and we have made

into a Riemann surface,

holomorphic (and PJ

over

~,

eJ

the ram1f1-

i.e. the number of

sheets in the covering which stick together there) except we still have to treat the elliptic fixed points.

But this is trivial, for

H

~

11/l'

III-3

THE PETERSSON INNER PHODUCT

ramifies only over

i

and

T0

2

stability groups of order

= e 2n1 13, 3,

and

with

so if

P

1s

an elliptic fixed point, we take the same local

then

T0 ,

only if if

Note if

P

~

i

or

is an elliptic fixed point if and

P P

r.

G as for

parameter for

t" 1mramlried over

11 1 rasp. ll-o

points of order

or

1

-r 0 •

Hence

is the number of elliptic fixed 2 resp. 3,

then ~ resp. ~ 2 3

are the number of ramtfled points over

1 resp.

Now the Riemann-Hurwitz formula for the genus of

H/G

2p(G) - 2

p t

where of

p

over t P~1

11/1-

ls (since

(ep

has genus

= J•(-2) + t Cep -

H/G,

10.

p

and

ep

p(G)

0):

1)

ls the ramification index

By tho above,

- 1) :~ 2

T0 •

MODULAR FORMS AND DIRICI!Lm' SERIES

Ill-It

Thus p(G)

proving:

PROPOSITION 9.

~·

index

If

G

19

a subgroup of

of finite

l'

~ is

the genus of ~

~

p(G) = 1 + ~- ·4·- ~

whore

~i

resp.

~0

ts the number of elliptic

fixed points of order

2 resp.

3, and

a is the

number of cu:sps. To define modular forms, it is convenient to use homogeneous notation. of

1"'

= SL(2,Z),

Let

G'

be a subgroup

of finite index, and

the

G

group of linear fractional transformations defined by

thus

G

G'/!:.1

and otherwise

G

G'.

G';

lr

I= ( 01

_v

£

G',

We have already defined

(ad - be) k/? ( c-c

+

d) -k f ( ~) c-c + d •

TilE PEI'ERSSON INNER PRODUCT

f2Lm

A modular f(~)

function 1)

fiL

2)

f

on

for

III-5'

G'

is a h9lomorphic

H such that

=f

for all

o•,

L t

is holomorphic at the cusps.

The meRning of 2) is as follows. we can wrl t..

f ( •) -

E

m,

At the cusp

nnzl n

as a Laurent

n=-""'

series in that of Lj

£

=0

Rn f

z1

at I''

I

and the condition is n < 0;

for

a0

At a cusp

~

we throw

ceed as before.

pj to

PJ

is onlled the

..

= Lj(oo),

where

L-1

and pro-

fiLj

satisfies

by

More precisely,

j

L-l 'L j G J'

condition ll for the cunJugate group and condition 2) at holomorphlc at will be called

PJ

is just that

the value of

""''

the~

of

f

at

l!l(G' ,k)

the space of all sur.h

11"

-I

then

£

G'

k

f.

a~

fQrm If

f

We denote Actually, "l(G',kl I Ol

must be even (1f

l!l(G,k) "' I!I(G 1 1 k).

G and f

Is

at

Pj.

and there Is no danger In confusing so let us write

fiLJ

fiLj

by

~

£

G',

rrt(G,k)

vanishes at all cusps;

Is

aCG,k)

MODULAR FORMS ANI.l DIRICHLET SERIES

Ill-6

denotes the space of cusp forms. If

on

~.

~or

f

£

mCG,k) 1

then

r

is not a function

but we can still speak of the

r at P

~.

£

~

Q!

measured in local para-

meters on the Riemann surface.

PROPOSITION 10.

Let

f

£

r,

the total number uf zerueH of

~·

tiplicit1es 1 is

Hence

Then

counting mul-

dim ~(G,k) ~ 1 + [~]. 12

so

.E..I:QQf.. A

meromorphic function on

f I 0.

mCG,k),

11/G,

h

= CAk

is a

kl'

poles

with

and hence an equal number of zeroes. l!§.mlu:k.

This Is not a good bound.

an exact formula for

dim mCG,k)

For

k

~

:?,

follows from

the Riemann-Roch theorem; cf. Gunning [3].

~·

Take

k

2.

(~ ~)

Since for

d(~) = dT c-r+d (cT+d)2'

we see

d1tion 1) if and only if

f

f(-r)d-r

£

satisfies conis

G-invurlant,

i.e. cnn be regarded as a differential on If

f

£

mCG,2),

then

f(T)d-r

SL(:?,R),

H/G.

is certainly holo-

111-7

THE PETERSSON INNER PRODUCT A

morphic on 11/G

except at the cusps.

At

.!k=.~ z e '

then so

f(T)dr

= (~)a 0

if and only if

f

~

+ •••

sideration holds at the other there is an isomorphism of ot"

holomorphic at

1

vanishes at

""•

A similar con-

~. cus~s,

g(G,2)

so we see that onto the space

holomoqJ!olc dlfTerent1als (and hence

dim S(G,?)

= p(G).)

The holomorphic dift"orentials on a c0111pact Riemann surface

(..,,..,•) =I

x

The

Peteq~un

X have a natural inner product

Wll;;;-;-

.1llnlU: pj'oduct 15 the natural genera-

lizatlon of this inner product to arbitrary

S(G,k)

for

k.

Given holomorphic runcllons

r

and

g

on tho

upper lmlf plano, consider .the double differential b(f,g)

!r(T)gr.}{Im T)k- 2dT II dl" f(x + iy)g{x • 1y)yk-2dxl\dy

PROPOSITION 11.

For any real matrix

L

MODULAR FORMS AND DIRICHLET SERIES

III-8

of positive determinant, we have ll(fjL,gjL) " ll(f,g) • L • .E.r.2Q!.

The right side means replace

L('t) " ~~ : ~

1:

by

throughout anrt so depends only on

the llnear fractional transformation

L(r),

as is

also the case with the left side; hence we can assume

det(L)

f(L(•)) dL(•)

= 1.

= (C't

= Cc•

Im L(•)

Then +

d)kfjL(<)

+ dl-2d•

= jc•

+ dj-?Im •,

and the proposition follows. In view of Proposition 11, if

f,g c

mCG,k),

then Cf,g) "' (f,g) 0

! ll(f,g) D(G)

ls well-defined, l.e. independent of the choice of fundamental domain

D(G),

provided the lnte-

gral converges.

L~.

(f,g)

exists i f

f

or

g

ls a cusp form.

III-'}

TilE PETERSSON fNNER PRODUCT

Clearly the integral converges if we ex-

~·

clude a neighborhood of the cusps. f(~)~

= 0(e-cy)

for some

1

C

> 01

so the in-

7e-cyyk- dy < ""• 2

togrol is domin"t"d by

m,

At the cusp

The

yl other cusps go the same way, using Proposition 11. The inner product satisfies the rules1 (f,g)

1)

Jn

is linear in

r,

2)

(g,f) = ~

J)

(f,fl

4)

If

H is a subgroup of

v1

then

~

o,

and

II(G,k)

(f,f)

=0

only if

contains

CD11

Since the

v

D0 l. 1:l

a

Cinitt~-
ktllbert space.

We wish to show the Heeke operators Hermitian on

r =0

G of finite indox

(f ,gl 11 = v(f ,g) 0 •

copies of Thus

conjugate linear

g

3(G,k),

T(n)

i.e. (fJT(n),g)

T(n)

are

= (f 1 gJT(n)).

are polynomials in tho

T(p)

with

real coefficients, it suffices to treat tho case n -

p

is prime.

LEMMA 1.

There exists a set

{a}

sentatives for tho left ond rlght

of common repreco~Hts

of

M(p)

MODULAR FORMS AND DIRICHLET SERIES

III-10

r• = SL(2,.Z),

modulo

=UI''a

M(p) ~·

Since

i.e.

= Uol''.

M(p)

= r•ur•

ts a single double

coset, every right coset meets every left coset, the lemma.

which proves

L~A

2.

Let

(mod nl},

f'(n)

and

= l(~ ~)

l'(n)

£

r•:(~ ~) "'(~ ~)

th .. corraspondlng group nf

linear fractional transformations.

£..1:2!1!.. Then

S11y

Then

= (~ ~)

y

yuny -1 = ("c

(~ ~)

Let

Now

Let

(

-b)

a

t:

M(n).

1

r(mn).

f(mn) c y- 1 r(m)y.

Than

1

M(p) = Uf'a T(p)

~-1

=p

aqulvalently

a '

b} {-cd -b}a ·(n0 0)n (mod mn).

d

= (-~ -~) =

=uar•, ~-1

tu = p 2

(ad -

so

M(p)

ta',

= (f,g(TCpllp

(f(TCpl,glr

=1( d n -c

Y-1

I

for

bel(~ ~)-1.

= M(pl'

so to show f,g

£

3Cl',k),

(f(T(p) ,g)I.(p) = (f,g(T(p))r(p)

(by rule~)

above), it suffices to prove

(f(a,gl 1 (p)

= (f,g(u')l'Cp)'

f(u,c(u'

£

"Ul''••'·

s(r(pl,kl.

since

by lelll!llll 2,

or

THE PETERSSON INNER PRODUCT Let so g

D(p)

a- 1oCpl

III-11

be a fundamental d9main for a- 1 r(p)a;

is one for

are forms for both

have the same index.

f(p)

fla

a- 1 r(p)a,

1

and which

lienee:

Cfla,g)

(flu,g)r(p)

and

then

f(p)

1

a- f(p)u

I b(fla,g) a- 1D(p)

I b(flu,g) • u- 1

D(pl

"

I b(f,gla- 1 ), D(p)

by Proposition 11,

I b(f,glu') = (f,gla'lr(p)

D(p)

This proves:

Tlli::OREM 8 (Petersson). are Hermitian on

The Heeke opera tors

g(f,k),

T (n)

i.e.

(fiT(n),g) = (f,giTCn)l.

COROLLARY.

The eigenvalues of

T(n)

are totally

real algebraic numbers. fiQQ!.

The eigenvalues are real by the theorem.

Now any element of

~(f,k)

1s a polynomial in

Eq

MODULAR FORMS AND DIRICHLET SERIES

111-12

elements Hence

with rational Fourier coefficients.

T(n)

is represented by

B

r&tion&l matrix,

so its characteristic polynomt&l det(xl - T(n))

has

rat tonal coefficients; hence

= >.. (1) n

>-n

is

algebraic and all its conjugates are real. If

fC~>

=

n r~=l anz , z = e2n1t T(n),

eigenfunction for all

and if we normalize to have

1

is an

then we know a1 " 1 1

a 1 I O,

then the

·Fourier coefficients are the eigenvalues, i.e. fiTCnl =an f

for all

PROPOSITION 12. functions. Say

f<~>

el = bl

= L

If

so

f,g

Then either

fr.Q.Q.(.

"n(f,g)

Let

(Corollary to Theorem 6').

n

be normalized eigenf = g

or

"r~=l anzn, Cr,gl I

g(1:) ~

o.

rb zn n'-1 n '

o,

then for all

f

g.

= CfiTCnl,gl Hence

(f,g)

n,

III-13

THE PETERSSON INNER PRODUCT

It follows that if

f 1' .•. , f r

are a maximal

set of normalized eigenfunctions, then they are linearly

ind~pendent

and

r

~dim

g(r,k).

That

actually there exists a basis of eigenfunctions follows from Theorem 8 and linear algebra:

LEMMA.

Let

be a commutative ring of Hermitian

R

operators on a finite dimensional Hilbert space Then

V has an orthogonal basis

eigenvectors of ~.

being a matrix ring). first

r 1 , ••• ,fn of

R.

s 1 , ... ,sm

Let

span Asstme

V contains one

R (finite-dimensional,

VI o.

~1genvector

s 1 , ••• ,sm' by Induction on m. Let elgenvalllM nf

S)'

and

vector thesis. (C·f)L

s 3s1

= s 1s 3 ,

f 1 of Then

= [g

We show

f1

of ~~

be an

Vl

the corresponding eigenspace. stnce

and so

s 2 , .•• ,sm

Then

SJVl c

v1

v1 contains an eigenby tho Inductive hypo-

V " (C:.f) 6l (C:·fY'",

c V: (f,g) =OJ

wher"

is Invariant under

s1 , .•• ,sm since the s 3 are Hermitian, and then hA<~

V.

on orthogonal basis

r 2 , ••• ,r11 ,

by induction

MODULAR ··ORMS AND DIRICHLET SERIES

III-14

on

n. lienee

f 1 , •.. ,rr

a(f',k)

has an orthogonal basts

of normalized eigenfunctions of the

Heeke operators.

The set

(r 1 , ••• ,rr)

is the set

of all normalized eigenftmctions, by Proposition 12. The rlng gonal

R or l!ecke operators is a rlnK of matrices, and so has rank i r;

r x r

actually the rank is

r,

since the

r x

~

of Fourier coefficient!l (eigenvalues) of has rank

r.

THEOREM9.


matrix

r 1 , ... ,fr

We have proved:

a(f',k)

hasabasis

r 1 , ... ,fr

of

normalized eigenfunctions of the ring of Heeke operators

R,

and

[r1 , ••• ,rr)

normalized elgenfunctlons or rank

is the set of all

R.

R has maximal

r =dim acr,k). In terms of Dirichlet series, there exist

exnctly series

r

= dim

~ 1 Csl,

have signature 9

= k.

a(r,k)

•••• ~r(s)

normalized Dirichlet with Euler product, which

(l,k,(-l)k/2 )

and are regular lit

They have real coefficients and so have

infinitely many zeroes on the line

o

= k/2,

by

THE PETERSSON INNER PRODUCT

Theorem If

III-15

5. TCn 1 l, ••• ,T(nrl

are a basis of

R,

their

eigenvalues lie in a totally real number field then

F

contains all eigenvalues of all the

For example, if k

I 26, 36,

some

T(nl,

dim g(l,k) : 2, R

Is

T(n).

i.e. 24 i k i

spanned by

I

= T(l)

)~,

and

diagonalized over a real field

11 > o. = 14416?, a prime.

F(k) = Q(J{l).

d

then

F;

When

k - 24,

llecke found

CHAPTER IV CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

If

N is an integer

~

1,

the homogeneous

pr1nr1pal congruence subgrouP of level

r•

= SL(2,Z:l

ls

l"(N) =

1(~ ~)

PROPOSITION 1).

Hence

Cl':i·•cNJ)

We have Hn exMct sequence

= N3Tio-

~,b,c,d t

we have to adjust

a, .•• ,d 1

p

r•(N)

wlth

Z

~l.

SL(2,Z)

We have only to show

is onto, since the kernel is i.e. given

(6 ~)<mod Nl}.

(~ ~)

c 1'':

riN

fiQQ!.

N of

~

SL(2,Z/NZ)

by definition,

ad - be

~

1 (mod N),

by a multiple of

N to

IV-2

MODULAR FORMS AND DJRICHLE'l' SERIES

get determinant (c,d,N) • 1 1

so

ed - fc

ad - be

(c,d + tN)

b +dfNI = 1 1

+c eN

i.e.

Now

(c,d) = 1.

we can assume

18

1.

=1

be so chosen since

-

1

=1

PIN

so assume

and

1 =ad- be+ (ed- fclN,

e

and

f

can

1.

The index formula is the same as

~(N)N3,-rcl- ~).

t

for some

ag +be ~ -k;

GL(2,Z/NZ) = Aut((~/NZl 2 )

so

Thus we want

i.e.

(c,d)

+ kN,

thRt

~hnwln~

has order

The question is multiplicative,

p

N : pr

is a prime power.

The order is

then the number of ways of choosing a basis of

Z/NZ

$ ~/NZ;

N2 (1 - ~)

if

a,b

is a fixed basis, there are

choices for

o(al

(o

an nutomorplllsml 1

p

and, given

u(al 1

N·q,(N)

choices for

for

<J(b),

this is the order of the union of the cosets of

{oCal} which are primitive in l./NZ. The principal congruence

level

of

of

N is group of linear fractional transfor-

mations determined by N > 2.

~~QYL

r(N)

I"'(N).

l(N)

= I"'(N)

is a normal subgroup of

if

of Index

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

={ 1

(1:1(N))

if

IV-3

N

6 if N

~3Tic1

otherwise.

piN

A congruence subgrouo of level diate group

G,

r.

1(N) c G c

A modulai' !.2l:m 2!. lJw!l. l>~

'"' tlltlmtmL uf

N is an interme-

N

~

dimension

-k

1'1\(1(N),k).

MOdular forms of higher level arise from modular forms of level f o: 1'1\[l·,k),

and

1 as follows.

L o:'l!l(N),

then

If

fiL

is a form

for the group

an~

GL

contains

In particular,

1(N)

(Lemma 2 before Theorem 8).

(~ ~)

L

gives

f(~ ~) Any other nrlmltlve the double coset where

A,B

£

I',

'

1':

Nlc] •

L of rlAtermlnant

rL0 1,

givtng

L0

= (~ ~).

N is in say

L

= AL0 D,

MODULAR FORMS AND DIRICHLET SERIES

IV-6 Let

G

= GL0 = r o (N);

= e2nt/3,

~o

stability group

Let

A

H- 1AMP 0 A c

= P0

~.

or

G(P)

= MP0 ,

and

M c L

where The

is determined as follows.

=P

A • G(P) +--+ AP

Thon

G.

c

P

let

L0 M = BL,

Now write

G(P) ~A c MI'(P0 )M-l ~A

B

~

c

r.

= MEM-l,

Then where

1 .e. E c M-lL-o 1 ILoM

For

P

= ~,

the ramification index

the least positive integer such that

./1 e)L-1 ., {a0 "\O 1

b){l e)(a b)-l =

d

0 1

ea: 0 (mod d),

0 d

Le.

or

b

for each

to

d,

whence

ve see if

I'

o

=

P0 = i,

For P

so

d,

i.e.

Thus

~(t)

(~

e

= l1t"

values

t·ep(t) = eep(t)

t ~(t) diN

since

t>·

is

contains

(10 "llf) 1 •

e ; 0 (mod

We noted above there are

r

e

cusps corresponding

=

t ~((d,~J). diN

-5)

generates

r(p 0

),

is an elliptic fixed point if and only

contains

(a

b){O1

0 d

-l)(a 0 0

b)-l

d

=( R n

l1 11

~) N ' R a

TV-7

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP i.e.

a "' 1,

b2 • 1: 0 (mod N).

number of solutions (mod N)

is the

Thus

x2 ~ 1: 0 (mod N) 1

of

c1;>

whirh is given by the formula stated, where is the Legendre symbol.

P0

For

=T 0 ,

the question Is whether

1 =(~ "b){o-l){ab)( Od 1 l Od d

cont~tlns

_b-qra)

a

i.e.

= 1,

a

b2 - h + 1

f ;>

2

1-.h

I

a

0 (mod N);

this gives

the formula for If

~·

Q

is a subgroup of the modular group,

of f1n1 te index, then the defini Uon of a modular form for

G can be stated as

l)

fiL

" r

for

L

2)

for all

Ac

I,

(resp. vanishes at fjA(d

, i no.D

n e2»1n~/N n

£

fiA

sm~tller

group, then

is holomorphic at

for cusp forms), I.e. (rasp. also

for some positive Integer by a

G

N. f

B

0

= 0)

1

If we replace

G

is a form for that

group, for condition ;>) is independent of what group we cular, tf

reg~trd

f

as being a form for.

G ts a congruence subgroup,

In parf

can

IV-8

MODULAR FORMS AND DIRICHLET SERIES

be regarded as a form of various levels, and the question of whether

is a cusp form is indepen-

f

dent of the level. We now define the Heeke operators In a reasonably general setting. of

Let

G be a subgroup

r• = SL(2,.Z.) of finite·tndex and

t:. c GL+(2,R)

a set of real matrices of positive determinant, closed wule•· a £

t:.,

and such

multlpllc~tlon,

the double coset

(u)

= GaG

thnt for

finitely many right and left cosets with to

G.

each

contatns only

(Chapter II treated the case

re~pect

G-

r•,

t:. =integer matrices of positive determinant.) Let

R

= R(G,A)

be the free

Z-module (or

module) on the double cosets

= GuG,

for

R is a ring under

a c t:..

(a) •

where if then

(a)

~­

r cY ". (y) (y)

a,.-

(a) =UG"'t•

cY

u,~


~

u GP j

is the number of pairs y

Ga 1 p3

Gy;

co sets

(u), (~)' (y).

ca,~

(disjoint), ( 1 'j)

wllh

depends only on the double We leave the verification

of the associative law to the reader.

CONGRUENCE SUBGROUPS 0¥ THE MODULAR GROUP Now take

G

= r• =

~

SL(2,Z),

= r•ar• = LJI"u~

= integer A double coset

matrices with positive determinant. (a)

IV-9

operates on the group

of

11

1

divisors (i.e. the free abelian group on the set X of lattices) by

L-+ l:ai(L).

T: R(I" ,A) -• R(X)

of

R

= End(C)

into the ring of correspondences which is

linear (by def1n1tton) T((u) •(jl))L

=

1:

1:

For f1xerl

J.,

~nd

1:

uljlJL

may be written

1:

ad=n

ll

J

a jl L =r•y, 1 j k

= T(a)(T(~)·L)

aL

r•u

of

correspond

L,

so

T: R

Furthermore, any double coset (a)

and we write

T(n) "'

1:

i

the loft cosets

one-one to sublattlces is injective.

mult1plicnt1vo:

(y) k r•u

i,J

ajd,

Thus a mapping

T(a,d).

= r·(~ ~)~··. TCa,d) Then

~

T

whore

«~ ~]

R(I',A)

,

~

R(X) (a)

d > o, and

ts the ring of

aid

Heeke operators defined 1n Chapter 2, and we have

MODULAR FORMS AND DIRICHLET SIDtiES

IV-10

the baste identity

=

T(n)T(m)

E dT(d,d)T(~). dJn,m d

In general, we get a representation of on

( (u)

V

c

=~(G,k)

or

a(G,k)

R(G,6)

by

l.)Jni).

Now let

G

= f'(N).

Let

6'(N)

be the set

or integer matrices of positive determinant such that

(n,N)

at A1 (N)

with

1,

and

6(N)

a=(~ ~)(mod N).

the set

n

or

We have a

natural map 111CN): R(f'(N),6(N))-+ R(I'',6'(N)) by

I''(N)al''(N)-+ f'af'.

is

isomorphism.

LEMMA.

q>(tl)

~-

We know the set of pr1mitlve matrices of

determinant

nn

m is

CONGRUENCE SUUGROUPS OF THE MODULAR GROUP M•

JV-11

r• (~ ~)r• =ur·(~ ~)

m

ad=m,d~ b mod d

(a,b,d)~l

We claim that the set of determinant

(m,N)

m and

M;(N)

-

of primitive matrtcoo

(~~)<mod

(for

N)

is similarly

1)

=

ur• CNlRa (ao

bN) d

ad=m

b mod d

(a,b,d)=l

(~-l ~)<mod (Ra ts.)

is not well defined, but the coset

l'(N)•

Jt is clenr (from the decomposition of

we have distinct cosets adding up to

N).

M:(N),

Ra

M;) and

so only have to cherk there is only a single double cos.,L

1110d I' (N).

Thus we have to solve

wlth

(~ ~)

£

j

I I

(~ ~)

MODULAR FORMS AND DIRICHLET SERIES

IV-12

(ya e)

(ao bNd)

u

30

= 1,

i.e.

we can choose Then

& ~ bN~ = ax ~ a (mod N)

y = dN,

take

(y,b)

( a.Y • db+bNy • ) '

x,

(mod N)

hence and

y,

b,

and

(dN,ax-bN 2 )=1;

since

(a,b,d)

= 1.

-= ( 0• 80 } u ( uy ~~} __ ( ry s~N}

The lemma now follows.

Let TN(a,d), R(f''

=1

syN ~ 1 ,

r& -

wtll do.

x

vlth

(N),A(N))

TN(n)

be the elements of

mapped on T(a,d),

The basic identity in

R'(Nl

T(n)

by

cp(N),

= R(f'(N),A(N))

is

then

for

(nm,Nl

=L

However, when we operate on "'.(fCNl ,k), k_l TN(d,d) operates as (d 2 l 2 H (d 0 } = dk- 2 R d 0 d

since

Rd(g

~}

(~

d02 }<mod Nl.

the basic identity is

Thus, on

d'

"l(r(N),k),

cONGRUENCE SUBGROUPS OF THE MODULAR GROUP for

IV-1]

= l.

(nm,N)

This suggests we should IDake the following V = mC&(N),k)

decomposition of d

~

Rd

g{l(Nl,kl.

is a representation of the abelian group

(Z/NL)•

on

V,

and so the ineducible subspaces

are one-dlmenslonal. dE (L/NZ)•, CZ!Nt l •

V

or

If £(~)

then

fiRd

= t(d)•f

ts n character

for all of

Thus "'C!l

VCt:l,

E

summed over all characters of operates as

LEl-IMA.

R n

Hence

V(E)

£(d)

and

on

CZ/nE)•,

where

Rd

V(E).

TN(m)

COIIUIIUte

is invariant under

((nm,Nl

1).

TN (m).

lt_l

l:J.:QQ£.

TN (m) ~. m2

l: L, L

trices or determinent

where the set of all

m,

'

(~~)(mod

Nl

is

~(N) =IJf'(N)•L

Uut

M111 (Nl

R~ 1 M111 (N)R 11

=U

I"' (N)R- 1LR

n

n'

so

IDS-

MODULAR FORMS AND DIRICHLET SERIES

IV-14

TN(m) = R~~(m)Rn' Thus

TN(n)

q.e.d.

operates on

V(E) 1

with baste

identity

for

(nm,N)

= 1,

or equivalently: the Dirichlet TN(n)n-s

I!

(whose cooff1-

(n,N)=1 cients are operatoro on

V(&))

h3s the Euler

product

n p.fN Thus the Heeke operators

TN(n)

for

(n,Nl = 1

behave much as in level 1, except for the 1ntroduction of the characters

they are

for

f,g

c(n).

"£-Hermitian",

£

v

then

fjUN

i.e.

l(l"(N),k)(E).

In order to define spll. t

Wo will prove lDter

up further.

=f

nnd

TN(n) Let

u

for

= (b

we must

niN

0.

If

f

1:

v,

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

IV-15

(z

Let An

tiN·

Let us say (n,N) = t.

In~

divisor

t

Now

f

has dlvlsor

Tho sot of all

is then a subspace v

t

f c V

of

V(t). f ~ rjuv;

Z/NZ operates on V by

£

if

the group is abelian we can again decompose

sfn~e

v according to characters, say V

=$

VX,

where

X f

£

=xCvl·f,

VX~ fiUv

ur:.r..

If the exact period of I; ~ primitive

r;.f,

flU '

e 2 nln!N, VA c V(t). V

X of

for each character X

H

is

i.e.

t•

X-th root of

and then

(n,N)

then

1,

t

= t;

thus

This proves that Gl

V(t) •

tiN Nr)te

f

£

V(t)

~ f'

=E

r 1,

whore

f

1 ju

r,; 1. f1 ,

r; 1 = primitive ~-th root of 1. LEMMA. for

V(t)

is lnva~lant under

(u, N) = 1.

Rn

and

TN(n),

IV-16

MODULAR FORMS AND DIRICHLET SERIES Run= UR (mod N); n n

.E..r.22!· f(RnUn

= (f(Rn'

which shows n2

into itself, since if

so

Mn(N),

L c

u-lwn

t:

TN(n)

carries

Mn(Nl;

Thus

then

f(U "' (•U,

Rn

carries

generates

Z/Nl.

V(t) Similarly,

L _

and

hence V(t)

V=

1f

Into itself.

V(E,t),

~

where any element

r

E 1t

of

V(t,t)

has tt1v1sor

(n,N) = 1.

antt

t

Note that 1f

V =-

fiRn

'!l(r(tl)

~

£(nl·r, then

,k),

V(1,N) = m(10 (N),k). We now define Heeke operators for

PIN

on

V(t.);

Tt(p) ·- TN't(p)

as the notation suggests,

they will be different for the various

LEMMA.

On

V(t),

f!:Q2!.

If

f

fl

~)

( 01 m

Now if m

c

rllvlcles

depends only on

V(t),

fl

t.

th.,n

f(UN/t

= f,

(1 (10 !1.)~ (10 tf.l.) mt = f( 0

b mod m. so

(b+mm)~)

m ts such thnt every prime factor of N,

we define

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

on

V(t).

We will check shortly that

to show the divisor is still

t,

IV-17

fjTt(m) c V;

we look at the

Fourier expansion:

rC·t)

=

(rhnnea nf not .. tt.>n).

Thu:~

Tt(o,)

anm'

whence:

Then

hus t.ln• effect of replacing

l)

rjTt(m)

2)

the

3)

Tt(m)

still has divisor

Tt(m)

=0

To check that

an

by

t,

commute with each other,

if

(m,fl

fiTt(m)

I 0.

Is still a form for

MODULAR FORMS AND DIRICHLET SERIES

IV-18 r(N),

m =p

we can assume

by the remarks above.

A' :

(~

fiTt(p)

P;) (mod N), E

V(t)

r

Finally, the and the T(n)

on V(t). prime.

Cror

Tt(m).

plm --+piN)

(n,N)

To see this, we can take

Now T(p)

Tt(p)

hence Tt(p)

and hence also (wher"

N).

This proves

and thet

Cn,N) = 1;

Tt.Cm)

= TN(n)

(mod~).

0

V(t) 1

E

for

operates on V(t,c),

N t'

r•, A~(~ ~)(mod (1 bb2H)-l (10 ~) Pt A 0 P t c r•,

p•:

tr

commutes wtth Rn

p (

Let A c

A' =

One computes that

is prime, and

= l) n

commute

=p

to be

is the sum or the operator

~1 p

2 RP(g ~)

coefficient

And the one which replaces the Fourier

a0

by

anp;

this second operator

certainly commutes with Tt(m), above, and we Just vroved that remains to prove that Tt(m)

by the remarks Rp

But ( 01 bp) bl

which proves it.

does, so lt

(g ~)

and

•

commute.

CONGRUENCE SUBGROUPS OF TilE MODULAR GROUP Finally, we define V(t),

where

Tt(nm)

= T(n)Tt(m)

Cn,N)

1

and

pjm

E(n)

0

ror

(n,N) I 1

we agree that

IV-19

~

on

pjN.

If

by the

usual convention, then ve have defined Heeke operators

Tt(n)

on

V(t,E)

for all

n 2 1,

satis-

fying

(for all

n,m

2

1).

Equivalently, we have an

Identity of formal Dirichlet series

note

dp)

PROPOSITION

0

if

pjN

15. If

fjTt(n)(t)

=r

c V(t 1 £)

f

f(r) ~ r a zn, n n

sion

Tt.(p) = 0

and

z

tr

PI~·

has Fourier expan-

= e 2 " 1 r·t/N,

thon

Rv(n)zv,

v

where

a (n)

" Proof.

We r.an a!l!lUDie

(n,N)

=

1,

by Lhe remarks

IV-20

MODULAR FORMS ANU DIRICHLET SKRIES

above; in that case the proof is the same as that of Proposition 7 in Chapter II.

THEOREM 10.

Let

T(n)

= Tt(n)

tor on 1!\(r(Nl,k,t,t:l, ...

T(n)n-s

I:

l)

n=l 2)

If

n .2. 1.

ror

·TI (1

be the Heeke opera-

-

T(plp-s + dplpk-l-2sl

p

is on eigenfunction for nll tho

f

T(nl,

normalized to have

a1

= 1,

then

the associate Dirichlet series has the Euler product

Next we generalize Theorem 7 on the uniqueness of the

p-factor in the Euler product (for

p ( N).

We need first the corresponding generalization of Proposition 81

THEOREM 11.

Let

f, flo

fll(l"(N),Jd,

&

where

a prlmttive integer matrix of determinant

n > 1,

(n,N) = 1.

£.l.:.Q2I..

Since

Q

&

Then

f

n,

u

= O.

ro(l0 no}t·• '

assume

Q

ts

with

= (~ ~)·

-1

CONGRUENCE SUBGROUPS OF THE MODIILAR GROUP

Since

r1(~ ~)

IV-21

(6 ~)

The powers of

(~ ~) are not all primitive, so

replace 1t by

V = A (~ ~) ,

n:)(wud

A '=

nl.

Tllen

i

Are

and all powora

A c r(N),

where

(g ~)

V -

pdmltlve.

Vt ;;

1

Slnce

(~ :i) (mod n) J Vt

2t primitive of determinant n , write Vf. "'

B(~ n~t)

where

C,

V -(~~)<mod N), n

f

l(mod N),

fjB = fiC- 1 • f

so

G

we have

fiD & -

~(I(N) 1 k) t

g ( ~)·n n~

g

0,

r•.

BC

£

r(N),

Finally,

n~

)

£

so if we choose

= fjVt = f1B( 01 ~t)

g(T

so

B1 C

f

= o.

-k~

C,

sat1sf1es

Now {

so that so

is

MODULAR FORMS AND DIRICHLET SERIES

IV-22 COROLLARY. f

p be a prime, p f N; t 0 e2wtn~IN.

Let

f(t)

c mcrCNl,kl, 1)

~.

u

If

for all

=0

apn

for all

ll: Then f(T) = f(T

= (g

:).

we get

has level

m wlth

N;

= 0.

f

taking

+

then

n,

g>;

produ.,t. rolatlvo tn

- (

E

pfm

a

mp 'II

and

=a

p .( N,

THEOREM 12. assoclat.a! to Let

p .( N.

2): Then

a= Rp(g

?),

we get

f

= O.

has an Euler

1r

p

a m- 5 )( "' E c(pv)p-vsl m v=O

c(p"~~)

m

then

Let f(-r)

Then

r = o.

taking

We recall that cp (sl " t ann-s

~(s)

p ( m,

r "' o.

then 2)

n

am = 0

If

let

for

p

c(l) ~ 1,

~:~~Cs)

f

~:~~Csl

; o,

by 1) of the corollRry.

! 0 be the Dirichlet series

= E a n e 2 trlnT/N cp( s)

If

'II•

£

III.(!'(N) 1 k).

has 11n Euler product

CONGRUENCE SUBGROUPS OF TliE MODULAR GROUP

relative to

p

function fo1' flfip

= E·f,

if and only if

Rp

and

T(p)

fiT(p) • C•fo

f

is an eigen-

= TN(p), Jf

S0 1

IV-:>3

say

the

p-factor

is necessarily of the form

!'or "only if", assume

fl:Q.Q£.

for

p ( m,

v

~

o.

Then

fiT(p) - c(p)•f

.k_l p2

fiRp(b

~)

+ ~ (a

power series in - 0,

n

pn

- c(p)an)e2"1n~/N

e 2 " 1pr/N

by 1) in the corollary.

Stm1 hrly,

~-1

p ·-

0 fiR p (P 0 1)

fiTCp) - E apnzn L

whtH'A

z

e:>rr! r/N.

(c(p)an - apn)zn

Then

MODULAR FORMS AND DIRICHLET SERIES

IV-2lt pk-lfiRp

=t

(c(p)a n - a 2 lzn p

P n

CcCpl 2 - cCp 2 ))f power series 1n

+

zP

(c(p) 2 - c(p?))f.

= c•f, = pk-lt:f(pT) +

Conversely, suppose fjRP

= t:•f.

Then

so

fjT(p)

c•f(-r)

1f

t apnzn,

p.(n

k-1

P

pjn

if

hn/p

Thus

E snn-s • t

p.(m

~P

E anpk-l(np2)

£

k-1

q.e.d. '

THEOREM 13 (Petersson). for

I'(N) ,

for

(n,N)

Let

of character

CfjT(n),g)

= 1.

- E "pn(pn)-s

2 -s r an(np )

a m-s m

-s

£.

f,g Then

= t(n)(f,gjT(n)l, (T(n)

= TN(n).)

be cusp forms

IV-25

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP It suffices to prove this for

f.1:.2Q..t.:.

n = p,

power, and then for result for

n

T(p")T(p)

= p,p

2

a prime

for 1 f we have the

\1

then from

, ••• ,p '

= TCp"+ll

n

+ ECplpk-Lrcp"-1>,

we get CfiTCp"+ 1 ),gl CfiTCp")T(pl,g) - E(p)pk-l(fiTCp"- 1 l,g) - c(pv+llCCr,~jT(p")T(pll - ETPJpk-lcf,giTCp"-1>> E(pv+l)(f,giTCp"+lll, using the fact that g.

Thus we assume

(f,g)

=p

n

is conjugate-linear in is prime,

p ( N.

Now the set of Integer matrices of determinant p

~

and

M•(Nl

p

(6 ;)cmod Nl =

r• (N)(01

is a single double coset

O\f" (N)

p}

which shows Cas in the case

'

N

= 1,

Theorem 8)

that every left coset meets every right coset, and so there exists a set rep1·osenta ti ves:

(a}

of left and right

MODULAR FORMS AND DIRICIIL.E:l' SERIES

IV-26 M. (N)

p

Letting

=u !'' (N) = u or Q

b)'

( ac d

= (-c d -b) a

I

(N) •

we see that

'

Hence

so we are to prove

(E fju,g)&(N) u

= Cf,E

gju'lr(N) •

For this it suffices to prove that

(fja

and

gju'

are forms for

r(pN).)

This is

proved as before; in the notation of Chapter III, we have

(f'ja,g)

1

a- r(pN)a

I

l>Crju,gl

a- 1 D(pN)

I ll(fja,g) • u-l D(pN) I b(f,gja- 1 > D(pN)

IV-27

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

"I b(f,gla') D{pN)

= (f,glo'l 1 (pNl, where

D(pN)

ls a fundamental domain for

a- 1D(pN)

and so

is one for

a- 11"(pN)a,

~

Note the eigenvalues

r(pN),

uC

T(n)

q.e.d. {(n,N)

= 1)

are not necessarily real this time; the rule is that f:(n)Xn

),.n

.J.

l.e.

XnECn) 2

real if

is real.

tCnl = I,

In particular,

purely imaginary if

),.n E(n)

It follows as before that the space or cusp forms of

d1m~nsion

r(ll)

-k

divisor

c;

r 1 , ... ,fr

of eigenfunctions for ell

Cr,r 3 l I 0,

If

f

50Y 1

Xn(r,r 3 >

=-1.

aCN,k,t,F) t,

and

has an orthogonal basis

character

Cn,N) ; 1.

t'o,·

ts

T(n)

with

1s any eigenfunction, and Lhen


ECnl~J>cr,r.1 >

=

~J>cr,rJl

HODULAR FORMS AND DIRICHLET SERIES

IV-28 and so

f

and a suitable constant times

the same Fourier coefficients

a0

for

fj

have

(n,N)

1.

This is all you can say in general; however, if pit_,.. PI~. f

then

T(n)

is a constant times

=0

for

(n,N) I 1,

and

fj,

We have the following general estimate on the Fourier coefficients of cusp forms:

PROPOSITION 16.

Let

finite index, and f(T)

= O(y-k/2)

G be a subgroup of

f ' &(G,k). as

~

y

0,

r

of

Then

uniformly in

hence the Fourier coefficients of

r

x,

and

satisfy

an "' O(nk/2). .fl:.Q..Q.f..

Write

I'

=U

Gl.

(disjoint).

Then

L

is invariant lUlder mental domain ThUS an

D

r, of

h(•) - O(y-k), O(nk/2)'

and bounded on the funda-

r so

since it van1she" at f('t")

by Proposition 1.

O(y-k/2).

~

Then

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP Now 11' for

T(p),

f c aCr(N) ,k,E) where

p ( N,

IV-29

is an eigenfunction

then the

p-raetor of

the associated Dirichlet series is

J.et real.

satisfy

'l

fjq

c

l,

is

90

Petersson's coolecture states that 1 - Bp~t + p

k-1 2 t 1

which has real coefficients, hRs conjugate roots, i.e.

.lcl I~ I ! 2p 2 ,

which would or course be much

strorrger than the general est111111te of Proposition

16. Thus we see that the theory of the Heeke operators

T(n)

in level

parallels thHt in level

N, 1,

for cusp forms, at least for

(n,N)

'lo treat the non-cusp rorms, we aga ln need the

explicit construc-tion of Etsenslein Let

k ~

3 and

c, d c L,

and consider the

Eisenstein~

Gk(~;c,d;Nl =

E' (mT + n)-k; m-e (mod N) n-'d (mod N)

~·

1.

MODULAR FORMS AND DIRICHLET SERIES

IV-30 k l 3,

since

this is an absolutely converging Im(•) > 0.

double series, for

Eisenstein series for

(There are also a modification

k = 1,2;

to ensure convergence is necessary -- cf. Heeke's 2~ paper [5),)

GkiL for

L

has the term for

E

r• =SL(2,L),

(m,n)

then

replaced by that

(m,n)L:

Since clearly and

If

d

Gk(r;c,d;N)

modulo N,

depends only on

we see tt is a form for

provided it is holomorphic at the cusps.

f(N), To show

this, we determine the Fourier expansion:

PROPOSITION 17.

.... E

Gk(~;c,d;N)

},.-0

"'

H,Z ' "

z

dnd for

t

if

c

tf

c

E

(sgn

0

(mod

0

(mod N)

N)

>. l 1, (-211t)k

Nkf'Ck) m""'"' m::::c(N)

v )v

c

k-l e 2TT1 vd/N

CONGRUENCE SUBGROUPS Of THE MODULAR GROUP

IV-31

5,

we start

f£Qu[.

As in the proof of Proposition

from

Clearly

E r (mT necn

+

nN

+ d)-k

m/0

u ..,

ami this proves

Thus

Gk(r;c,d;N)

Gk(~;c,d;N)

tf

Cc 1 d,Nl

I'I'"Uj)OS1 t1on. £

mCI (N),k).

is called primitive tr

= t > 1,

then

Is "' primitive Eisenstein series of level e(Nl - e(N,k)

(c,d,N)

~·

Let

be the 5pacu generated by all prl-

m1tive Eisenstein series of level

N.

1·

•

MODULAR FORMS AND DIRICHLET SERIES

IV-32 Now ~I

since

N fundamental domains for r(N)

r

is a normal subgroup or

same is true at any other cusp.

meet at the

1,

The number of

cusps is then

o(N)

(r:r(fill N

{

~rru pjN

- ..l...) p2

if

N

1

tr

N

2

if

N>2

We have an obvious map eCNl -. C:o(N) by evaluating at the cusps, and we want to show

this is an isomorphism.

Now the number of pr1m1-

tivepo1rs

is

(c,d)modN

N 2 TIC1-~) 1 pjN P

and

cleRrly

so

dtm eCN)

~

a(N)

and so it suffices to prove

the map ts onto. For this, lt is convenient to consider the

IV-33

CONGRUENCE SUHGROUPS OF THE MODULAR GROUP rostrlcted Eisenstein G;(l:;c,d;Nl =

again a form for

for

L

£

r•.

~

t (11!1: + nl-k, m=cCNl n.od(Nl (m,nl=1

r(N),

with

To connect the two kinds of Eisenstein

serioc, wo use the~ function

..,.
the

mu1tlpl1cative function of positive integers with

..,.C1) = 1,

..,.Cnl = 0

if n

..,.Cp)

= -1

for

p

has a square factor.

The MObius

function sa t1sf1es t

djn

..,.Cdl

(d>O) Thug

G:(T;c,d;Nl

=

1:'

m=c

(m~ +

n)-k

n

prime, and

t

j.L(a)

a! Cm,n)

n~d

; ..,.Cala-k t• Curt" + n)-k • a=l ma_oc na:;d

MODULAR FOHMS AND DIRICHLET SERIES

IV- 3lt

Now assume

(c,d,N)

•

okc~;c,d;N) c

0).

assume

= 1.

with

(a,N)

11Cala -k.

E

choose

a'

•

Gk(~;c,d;NJ

Thus

.

The value of

a,

Then

at:::l(NJ a>O

e(NJ.

o1h

(otherwise

For such an

aa' : 1 (mod N).

where

€

=1

Then In the sum above we can

Gk(T;c,d;N)

at

~,

i.e. Its

Fourier coefficient, is visibly if

(c,dl

((),1)

(mod N)

otherwise

•

Gk {T 10,1 ;N)

and

0

takos the value

1

at the cr1sp

at the other cusps; similarly,

takes the value all other cusps.

1

at the cusp

This proves:

-d/c

-

a:(·t ;c,d;N) and

0

at

IV-35'

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP PROPOSITION 18.

The 111ap

isomorphism, and

t(N,kl, • c:"(N)

t(N,k)

is an

is generated by the re-

stricted Eisenstein series.

PROPOSITION 19. Hence

e(N)

of level

If

N'jN,

then

t(N',kl c eCN,k),

is the space of all Eisenstein ser!As

N,

primitive or not.

f.l:QQ!. G.(T·c' d'·N' l

k

'. '

G~C~;c,d;Nl,

r

'

C=C' (N)

d:=d I (N) c,d mod N which proves the first statement, in view of Proposition 18; we have already observed that an 1mprim1tive series of level

nt'

series of lower level

N is a primitive which proves the second

statement.

PROPOSITION m~trix

fiL

t:

~0.

If

of determinrlnt e.CrnNJ.

Also,

L

ls a

prlmttlve integer

m l l,

and

f e eCNl,

tCmN,k) () ll!(rCNl,k)

f£22£.

We have proved this for

L e f',

so we may as well take

m = 1, L

then

= e(N,k). i.e.

(6 ~) ;

then

MODULAR FORMS AND DIRICHLET SERIES

IV-36

For the second statement, note that we have as a result of Proposition 18 a direct sum decomposition. ~(r(N) 1 k)

~

e(N,k)

~

a(N,k).

f c eCmN,k)

n ~(r(N),k),

write

E ~ eCN,kl,

g c a(N,k).

Then

Proposition 19, and

i

If f

=E

+

g,

where

g , eCmN,k),

is a cusp form,

~o

by g

=0

by Proposition 18. In vlew of these propositions, let us call any element of

eCN,k)

Since e (N ,k)

an Eisenstein series.

is invariant under all modular

transformations, in particular by (n,N)

&

eCN,k)

1

and

(6 }),

U

for

we can decompose

according to divisors

characters

Rn

t

of

N and

e of (Z/NZ)•, getting

~(r(N) 1 k,L 1 t)

-

eCN,k,t,t) W &(N,k 1 £ 1 t).

Furthermore, this decomposition is respected by all Heeke operators

T(n)

= Tt(n),

n ~ 1,

by

Propos! tion 20. A way to construct modular forms (of higher level) from given ones, using chnracters, is given

IV-37

CONGRUENCE SUBGROUPS OF THE MODULAR GROUP

bY the following theorem; this technique 1s aha emphasized in the following chapter. m 2 l,

integer

a character modulo

X on

character

(L/ml)•; xCnl = 0

convention that

m is a

X to a

we extend

function of all positive Integers

n

by the usual

(n,m) > 1.

if

that we do not in general require ~

Given an

Note

X to be Rtlml-

(not definecl modulo a proper divisor of

in particular, even the identity character

m satisfies the convention xCnl

modulo

ml;

x =1 = 0 for

(n,m) > 1.

THEOREM z

Let

1~.

= e 2wiT/N

rx

Then

- 1:

n

m·¥·

and let

whore

(m,Nl

[i, tm2 ),

form (rasp. Eisenstein series) if E.l:QQ£.

Let

LX - 1

X

= 1.

be a Let

xCnla n znt .

l!l(r(m2 Nl ,k,

f:

r znt an ' (n,~)=l

c l!l(r(N),k,t,t),

<'har.. cter modulo

r x
f(~)

and f

rx

1s a cusp

is.

and constder the operator T

xCx)c-2wlxy/H (m y) ; 0 m

n x,y mod M

MODULAR FOHMS AND DIRICHLET SERIES

IV-38 then

fiLx

~(1Cm 2Nl,k).

£

ane 2 n1n•t/N

nnznt

l

E

Applying

LX

to

contributes a factor of

xCxle2n1(n-xlyiM,

M x,y mod M

which is X

In.

x(nl, Thus

since the sum over

rx = fiLx

and divisor

m2 t,

series) if

r is;

character is

{~-l ~)

Rn:

£l.

is

0

ror

m2 N

and a cusp form (rasp. Eisenstein 1t remains to check thllt thA

Let

(mod m2Nl.

(~ ~fn -: Rn{~ Y~2)

y

is a forD! of level

Rn

£

r•

'

Then

(mod N),

so

2 2 fXIR = !Jnl r x(xn2)e-2n1xn y/Mfl(m yn) 0 ·n M x,y mod M m

For example, we know the Eisenstein series of level

1

is associated to the Dirichlet series

C(s),(s + 1- kl

lf

=

E ~k-l(n)n- 5 ; n=l

X ls a character modulo m,

then

CONGRUENCE SUBGROUPS OF THR MODULAR GROUP

IV-39

LCx,s>LCx,s-H-k)

= nr xCn 1 ln 1 -s 1

=

...

r xCn)ok_ 1 Cnln-s

n=l

assodatell to an ~tsenstein series of level m2 , divisor m2 , and character x2 , and is an eigenIs

function for the TIIEOREM 15'. ciated to

T(n),

= 1.

(n,m)

More generally:

The space of Dirichlet series asso~CN,k)

is generated by the series of

form

where

h

a character

J.(l(,,.)

=

.. l:

.1i.. t ' J

mnrlulo

xCn)n-s.

Th"

n'"l

x1 .x. .

E

cOl'respnndlng El""'''atein :serlo:;

Is an o1genfunct:ton of the ~·

.IJy Proposition li',

T(n), the

r

has

(n,N) = 1.

Dir1rhlet series

IV-40

MODULAR FORMS AND DIRICHLET SERIES

times)

""t n-s t sgn(v)vk-le2n1vd/N. n=l mv=n m=c(N) Let

Cc'd(s) of

=lN a

Cc'd(s)

t e- 2 n1ad/NC (s); mod N c,a

generate the same space.

n- 9

in

1

I:

Cc'd(s)

the

The coefficient

is

t mv=n m::c(N)

8 -2rrlad/N

N a mod N

t sgn(vhk-l m,=n m::c ,:;d

r "k-1 + C-llk m,=n m:c

,:;d,,>O

m-s

t

m c(N) Now fix

nk-1-s.

t

nod(N)

t 1 ,t2 jN.

Cc'd(s)

r "k-l mv=n m--e ,=-d,,>O

with

Then the space genurated (c,NJ

= t1

by

the

and

1~

the SRme as that generated by the

(d,NJ

= t2

CONGRUENCE

OF THE MODULAR GROUP

is a character modulo Jl t .

x1

where

&~BGROUPS

JV-41

Now

1

r 1!_11?2 X Cb >x Cb >t b 1 mod t

b

1

t

,b t

1 2 2 csl

1

hence

Lxl.x2Csl

= t~- 1 CI+C-llk<x 1 x 2 >C-lllCt 1 t 2 l- 5 LCx 1 ,slLCx 2 ,sl. This proves the theorem.

COROLLARY.

"l.(I'CN),k,t:,t)

functions for all £LQQ!.

T(n),

has a basts of eigen(n,N)

= 1.

In view of the decomposition

~(r(Nl,k,c 1 t)

~ e(N 1 k 1 ~ 1 t)

W 8(N 1 k 1 E 1 t)

1

IV-42

MODULAR FORMS AND DIRICHLET SERIES

and the fact we have already diagonal1zed the T(n) the

on the cusp forms, we only have to dlagonalize T(n)

on the Eisenstein series; this is done

by Theorem 15, in view of Theorem 12. That one cannot in general diagonal1ze the T(p)

for

Let

q

and

xl'

pjN

is shown by the following example. N = q3,

be an odd prime,

t1

= t 2 = q2 ,

x2 characters modulo q with

Ct 1t 2 lC-l)

= C-llk.

(ThAy exist.)

The theorem

gives an Eisenstein series

t ~ N ~ q3.

with divisor the

n1h Fourier

Since

coefficient by the

T(q)

replaces

(nq)!h, we

have h(-t)

h(t) fTCq)

fjT(q)

= o.

(1:)

Thus

= ao T(q)

on the space spanned by

+

r a ti2mnt q.(n n '

hBS

f, h,

matrix

(g ~)

anrl so

T(q}

1s

CONGIIUENCE SUBGROUPS OF THE MODULAR CROUP

lV-43

not diagonalizable. ~·

We have shown that nny form

N has an R

n

associ~ted

f

of level

Dirichlet series,

= O(nconst.);

by Theorem 15 if

Eisenstein series and by Proposition 16 if a r.usp

form.

f

is an f

is

CHAPTER V

A THEOREM OF WElL

From now on we

only wlth forms

~eal

N and maximal divisor

level

f(~ +

I.e, modulo

N,

1) ~ f(T). let

r.(N,k,t.).

(~ ~) f

r~(N),

c

=0

unless

fj (-01

~-;) Let

~

r

Thus tf

t.e.

d-ll

= C-llkf.

(~ ~)

r~(Nl,

and

r1(~ ~) ~ c<~>·r;

then

since

l!l(r0 (Nl 1 k)

Note that

-c/N) a

and so:

and

8(N,k,t),

l!l(N,k,El,

Note

-1)

c

3)

~

= (-l)k,

b) H -1 ~ ( d 11 N c d N -bN tf

k

NjC',

0 O • HN = ( N

(a

is a character

= l!l(I'(N),k,t:,Nl,

rnCN,k,d

similarly (at least for

of

t = N,

E

lf

f

t

1 , (N)

o

= II!CN,k,ll.

MODULAR FORMS AND DIRICHLET SERIES

V-2

(of substitutions of the upper half plane), containing subgroup of index 2)

f

~fiHN

r 0 (N)

as a (normnll N > 1 l.

(for

;>

defines an isomorphism

1!\(N,k,d ~> 1!\W,k,e).

3)

In particular, if

=E

E

f~

fiHN

and

fjH~ ~

diagonali~e

(f

f+

f-,

f-

f - i-kfjHN).

+

we will say or that group

f

(n,N)

where

f

= 1,

=f

f+ If

f

£

+ 1-kfjHN,

~(N,k,E),

has a funct.lonal egu,Jtlon,

ts a form for the extended

r.(N),

multiplier

21

+ llr(N,k,El-==> fjHN = ±1 k f.

~

f

fl(ol _01) : (-l)kf.

this representation

of the group of order

where

then

is an automorphism of

mCN,k,E), We can

is real

~1),

(1.e. 1ts values are

C

of character

= ~1.

The

operate on

E and

T(nl,

for

~(N,k,El,

a5

V-3

A THEOREM OF WElL follows from:

LEMMA.

HNT(n) "' ETriTT Cn )HN,

~-

T(n) "' nk/2- IE L,

matrices of determinant Now U r•Hr;LIIN-l and

n,

III(N,k,E).

where U l'L and

L:

is all

(6 ~)(mod N).

is still disjoint, as one checks,

(g ;)cmod Nl,

Hr;LHN-l

on

RniiNT(nlH11 -l- T(n),

so

q.e.d.

The theorem of Wail [14) we are about to prove is in the spirit of Heeke's basic Theorem 1. When

N : 1, 111 =

ments for

r

r

(~

~ 1·.(1)

-01) ,

U

1s generated by tvo ele( ~ {) ,

so modular forms

are defined by two functional equations

(plu3 regulal"lLyl; periodlclty (functional eqll8t1on for

U)

~(s),

gives an associated Dirichlet series

and tho functtonal equation for tional equstlon for

~(s),

Rnrl

11 1

gives a func-

v1co versu.

result is to characterize forms for

r.(N),

Well's which

hRs l n gene1·a I more than two genera tors, by functional equations for many essociRted Dirichlet series. R,stde<:

Theorem 1, we need Lhe notlon of

MODULAR FORMS AND DIRICHLET SERIES

V-4 Gauss sums.

m,

Let

x

i.e.

X be a character of

ls a prlmitiyc character modulo

m,

X is a character modulo m and not modulo

t.e.

any proper divisor of have

condu~tor

m.

For

n

£ ~/mZ,

we

the~~

~

xCx>e2"txn/m.

x mod m

PROPOSITION 21.

~·

If

as desired. to show

g_<'n)

2)

lgxl = Jm 1,

Cn,m)

J.Cnlgl.(n)

=

Now

gl.(n)

= ~Cnlg.<

1)

then

~ xCnx>e 2 " 1 nx/m x mod m suppose

= 0.

Write

Cn,m)

=t

n0 t

= n,

Then

gx

E

~ x<x>e 2 ~nux/mo x mod m0 t

=g

X'

> 1; we wanL m0 t

= m.

v-;

A THEOREM OF WElL

o,

and this is

I ACyl = 0,

since

because

x

is a non-trivial character on the group of y: 1 (mod m0 l,

modulo

m0 •

since

r

x<x> r xCxy)e2rrtxCI-yl/m

(x,ml-1

= If

y

E r ~(yle2"1x(l-y)/m Cx,ml"l y

Cx,ml > 1,

r

~hsracter

Is not a

A

For 2l,

then

~Cyle2~x(l-y)/m ~

8 2rr1x/mg;rxr

~

o.

y Thus

lsll2 = ~ ~(y)e2rrix(l-y)/m

m,

x,y the sum over

sln~e

y

x

is

0

for

y

I 1, m for

= l. Nolo/ if

and

A a chnracter of conductor

in Theorem 14,

is a Fourier series, m,

we define,

as

MODULAR FORMS AND DIRICHLET SERIES

V-6

By the proof of Theorem 14, this is also ~ X(X)e-2WiXy/mf(~ + ;) m x,y mod m

=1

fi((T)

:.1

r

m y mod

:!x m Now let

gx'-ylf('t + Dl

Dl

ond

a 0 ,a 1 ,a 2 , ...

b0 ,b 1 ,b 2 , •.•

sequences of complex numbers, some

Let

a > 0,

Dl

"iC-y)f(~ + Y). m

~

y mod

Y)

an' bn

end form

..

f('t)

I a 0 2wirn n=O n

g (-r)

q:( s l

I a n -s n=l n

+Cs)

(9)

(217) -s res )cp ( s)

f(sl

C I 0,

A > 0,

k > 0;

.. ..

be two

= O(n°l

I b 0 2Trinr n=o n ~

b n- 9

nal n

C21T) -s rc 9 >.;c s >

recall that EBV means

"entire and bounded ln every vertical strip".

LEMMA l.

Equivalent are 1

I a Cb (All 4>(9) + A-s?(~+ ~l

s

nnd

4>(s)

for

k-s

k_s CA 2 V(k

is lillY, sl

arul

A THEOREM OF WElL

V-7

= CAkI 2 C~l

(Bl) f(~)

Now assume

is a positive Integer, and so

k

1"1 (~ ~)

wo hnvo the notation (

~ ""o~ ,

rjt r. 1L1 Or

= (w

t[GL+(2,Rl]

=t 1:

for

c 1 fjLi,

=R

X

(•)

operates on

f

and

R.

We now ch:tnge the notation.

a 0 ,a 1 ,a 2 , .•• ,

f

Letting

R: fj<" =OJ

ts a right ideal in

of conductor

k.

rond:J

(Bl)

The group ring by

g(it)•

This ts a reformulation of Theorem 1, with

f.I:Q.2r.

IIA

-k

a 11 m,

~

= O(n°),

define X(n)a e;!YTin-r

n=o

n

...

I: ~(n)a n•S

n=l

(~)

n

-s rcslL.c;(s).

Given

and Rny ~hRractor

~

V-8 For

MODU!.AH FORMS AN!J DIRICHLET SERIES

m

= 1,

A= t. 1 .

we write simply

Letting

a(x)

=r 1 ,

f

= (6 ~)

for

x

L

= L1 ,

£

~.

we

have fl..=

Let

~

!4

iC-ylflu(~).

mymodm

m

N be a fixed positive Integer, and

LEMMA 2.

C

= ~1.

Equivalent arc:

(A2) A(s) + N- 512 C80 s

Ca k-s

• -----'1)

is EBV,

and

.l!._s

= CN 2

A(s)

Thls ls lmmed1ate from lemma 1, wtth

~-

f

= g

etc.,

LEMMA].

A= N.

Equtvnl~nt

(A3) Al..(s) (B3) g

.i:r..Q.Qf.

A(k - s),

are (for

is EBV, and

m > 1,

CX / 0):

.l!._s Al..(s) - ChN?. Ai(k-s)

~ ~(y)u(~) -
AgHili immediate from lemma l, with

A " m2 N.

A THEOREM OF WElL ~-

and

If the

lc.
V-9 an

(m,N)

(a,m) = 1,

-

abN

a

b

Then for each

a

b with

8

and so

m yCa, b) ~ {-Na

since

= 1.

there exists

-1 (mod H) I

-:)

f~(N),

belongs to

= ~~ ,

(If (A3) holds.)

Now suppose wlth

~A

are real, than

for some

runs over

n.

(b,m)

does, and writing

One computes

b (mod m)

1,

(-~a

y(b)

-nb)

as

= yCa,b),

we see that (B3) is equivalent wllh

CD3' >

r

:(Cb>Cl b mod m

c,t g

.(

k

.12.

'>{

xC-NlliNy(b))aCm>

- o (mod l.f). (-ubN

l(m) 1

Now let tion

so

.((R)

= :{C-t>N).)

f c mCN,k,L) 1

f · ClkfiHN

character modulo

(C ~ ~1); N.

Then

with functional equahere f(~)

=

£

is a real

::' 2TT1nt ~ a e n=O n

MODULAR FORMS AND DIRICHLET SERIES

V-10

with

an

O(n°)

by the preceding chapter. tru~;

(82) holds by hypothesis, so (A2) is y(b)

= t(m)

t(n)

modulo

Cg iX ( -N )dm)

hence (AJ).

This fJI"OV""'

THEOR~

Let

16.

equation Then

f

also

Thus, taking

we see (BJ') holds, and

~

ex

Ur·

Thus

f

£

"l.(N,k,E)

= CikfiHN' c = ~1

A satisfies (A2),

and

with functional Cso

"x

E Is rAnl). satisfies (AJ),

for every character

x whose r.ondur.tur m is

relatively prime to

N,

the value of

CA

being

taken as

We now turn to the converse, in a stronp, form; Le. assuming the functional equation for LX

for "sufflclently many"

~

r~.J.5,7,ll,

X modulo

LEMMA 4.

..• J;

m & Ill

Let

m

x's.

Let

any non-identity character

is primitive.

£ ~.

with

(m,NI

l;

let

V-11

A TIIEOREM OF WElL c~

! o. Then equivalent are: (A4) for every pr1m1 t1 ve (A3) holds wtth

cJ(

= (b',m)

).(b) ~ (Lilt•)

(1 -

k

l,

•

1~

whenever where

C~HNy(b))a(~);

r X(b)).(b) b mod m primitive

~.

~

m,

= C~x(-N)gli;.

(B4) ).(b) = ).(b' >(mod llr>, (b,m)

modulo

1..

= O(mod

for every

llr)•

X modulo m.

That (A4) is equivalent with (B4') follows

from Lemma 3.

Clearly (84) implies (B4'); con-

versely, given (84'), we have

o

r

x.h

<x-x

r;;Cm)().(b')-Mb">>

where the sum is over all primitive all

The last

LF2-IMA 5. m, n

hence over

1..•

X•

t ~.

Let

an~

y

key lemmn

=( -RN m -b\ nJ

lsi

,

1 ~ (N),

Assume (A4) holds for

C'C' m n = (-l)k,

with

m Rnd

and assume (A2) holds.

n,

Then

with

MODULAR FORMS AND DIRICHLET SERIES

V-12 k

fly=~ m

Let

f.I.:.W:· y

f

= y(b)

Y' = (aNm nb) '

so

0 - Er>a -

= c:•c~-k. m

y(-b)

No~

b, b'

m

-b,

we

whore

Thug

(1)

l - ~y' ~ (1 - ~y)u(~)(mod Qrl·

Now

n b) Y-1 -_ (Na Y,-1 m '

similarly (m

~( n -b) -Na m '

replacert by

so

~e

get

nl:

Thus

since

-Cl

1 - fY'

-

Thus

and (A1ol I

= b,

(1 - fy' )u(=.l!)

1

(A2), hancu

1 ;; CtkHN (mod Qf),

hence (84), holds; taking

F

:

in the notation above.

(B2), holds, so

have

Y1

-

f-1 ('-l)l;y'

-f(l - F-ly-l)u(~)y', (1

Fr'r- 1 uc-:>bly' n

(l

fy)o(~),

(1- fy)(l-

~)

by (:')

by (1).

_ 0 (mud Qf)'

where

V-13

A THEOREM OF WEJL

The eigenvalues of

x2 of

cJL-2lx mn thus

1;

are the roots of

~

1, and are imaginary but not roots

+

is elliptic of Infinite order.

I'

fl

lienee, by Propos! tlon 3,

THEOREM

17. Let m•

(1 -

!;y) = 0,

m

be a subset of

every primitive arithmetic progression with

1,

(a, b) £

pose (A2)

f

m c

m•.

C

= ~1.

Let Sup-

ThAn

f

£

m•,

ll).(l'l,k,£),

satisfies the functional equation

f = Ctkflllw

=k

s

and

a + nb,

X of conductor m c

C.<.= CdmlXC-NigAjl7.

and

at

for

N,

meeting

Is satisfied, and (A3) Is satisfied

for every character With

=1

(m,N)

character modulo

be a

q.o.d.

-

If ~

L(s)

converges absolutely

ror snmP

h

>

O,

then

f

Is a

cusp form. (A2)

f.o:.Q.Q.t:..

Let fly

y

:.

holds, hence (B2l, so

(N~ ~)

= ddl· r.

£

If

l ~ (N). b

~

o,

f = ctkriHw

We are to show tho>n

MODULAR FORMS AND DIRICHLET SERIES

V-14

1 0)

=H(l-c)H-1 N0 1 N

Y = ( Nc 1

Assume now

desired. (a ,Nb)

I

b

= 1,

(d,Nb)

r =

and

o.

fly,

as

Then

so

m = a + Nbs d +

n •

for some

m, n

holds, with C'C' mn

Nbt

t:( d) f,

Thu,;

r

£

z_

Now (A4)

c~ • Ci kE(m),

(-Uk.

(~ ~)

s, t e

and

1'11'

£

Let

f~(N);

so

y' = (Nlt

then

by Lemma 5'.

flY' = E(m)-lf lienee

l:t rorUJally H UIU
wlth character

E,

E(n)f

'o(N).

and with the desired functional

equation; it remains to verify the regularity conditions at the cusps.

LEMMA.

Let

f(-.:)

be holomorphic in the upper

lmlf plane, satisfying al

fjL = f

b)

f

ls

for

L

£

I(N)

holomorphlc at

i.e. nt

0

A THEOREM OF WElL

V-15

in the variable c)

f(t)

is a modular form of level

o < k,

cusp form 1f ~-

r.

z

e27T1 t/N:

13y a),

fiL

t

fjL(-r)

8

and a

Laurent expansion 1n

8

zn

n=-'"' n

if

0

;

N,

be a rational cusp, where

has

and we want to show 80

uniformly

= 0.

f(~)

and

L(m)

xo

Let

L .,

also

y ~ O,

as

x.

in Then

= e~wt•/N

z

f(x + iy) = O(y- 0 )

=0

an

a< k.)

n <0

for

(and

Nov

• +1 0

I fjL(<)z-nd,

"(

0

Let

(~~);note

L

where

b

Then

X +

is a large constant, iy

= L(<),

where

f(L(<)) - O(b 0 ) 1 O(bo-ke2T7llb/N) as =

Hence ll

n for

let

c/0;

n < 0,

and

80

=0

u0

•=u+1b, ~

u

~

u 0 +1.

Im •

y

jet:+ dj2

= o(l.).

r 1L(• > ; O(ba-k)' b

-t

if

GCi a

thus

< k,

an = 0 q.e.d.

b

V-16

MODULAR FORMS AND DIRICHLET SERIES

~.

It is reasonable that condition c),

uniform tn all

x,

should imply regularity at

the cusps; we know ths converse only for congruence subgroups, via the Eisenstein series. Returning to the proof of Theorem 1?, we are gtven

an= O(nc),

and so by Proposition 1 1

f(x + ty) = O(y-c-l), form, by the lemma. verges absolutely

nt

and

f

is then a modular

Finally, suppose o

= o,

a < k.

converges absolutely in the half plane and so

n0 = 0 1

by (A2).

fices to show that

f(x

+

L(s) Then

conL(s)

Re(s) > a

lly the lemma, it sufiy) = O(y- 0 )

as

y ~

Now

~~ Ia I v=l v

~

... ~E Ia lv -a nu v~l v

~ t

janje-2T1Tly

= O(nv-, .

Hence

1r<x

~ 1y)l

~ (1 - e- 2 "Y) r s e- 2 771lY n

q.e.d. (Cf, proof of Proposition 1.)

Wc>ll's theorem

lead~

to a very interesting

o.

A THEOREM OF WEIL

V-17

conjecture on the zeta-function of an elliptic curve

Q.

E defined over

C(st~~,-ll,

be written

Thls zeta-function can

where

L(s)

= n Lp(s), p

the local factor If

Lp(s)

being defined as follows.

E has non-degenl!rate reduction at

p

(which

will be the case for all but a finite number of the p),

then

LP(sl

- ,. p

1 + p

(l - app-s + p1 - 29 )-l,

where

is the number of points on the reduced

curve with coo1•dinates tn the field elements.

If the reduction of

singular, we set

LP(s)

polnt is a cusp,

and

the singular point is

Glo'{p)

E modulo

of

p

p

is

if the singular (1 - app -s)-1 LP(s)

1f

= +l

if

=1

=

node, where

Et

ap

the tangents at the double point arc ratlonal over

GF(p),

and

ap

= -1

otherwise.

One can

n

define an integer

n p P,

N

the conductor of

p

E,

with

tion Rl ~tnd

np

np p,

n

> 2 p-

=2

for

=0 np

if

E has non-degenerate reduc1f

the reduction has a node,

if the reduction has

pI?, 3).

:1

cusp (and

The Hasse-Wcll conJecture

MODULAR FORMS AND DIRICHLET SERIES

V-16

is that the hypotheses of Theorem 17 hold for L(s), E

= 1.

with

N the conductor of

It would then follow that

elated to a form

r

of

~1mengton

with a functional equation. fapf ! 2p 112 ;

E,

k

L(s) -2

2,

and

1s assofor

f 0 (N),

Actually, one knows

this is the "Riemann hypothesis"

for elliptic curves, proved by Hasse in 1934. Hence

L(s)

converges for

will be a cusp form.

Re(s) > 3/2,

so

f

Note the agreement or the

"Riemann hypothesis" and the Petersson conjecture.

A good Introduction to algebra-geometric aspects of this subJect 15 Shlmura [12) (and many other papers of Shimura.)

CHAPTER VI

QUADIIATIC FORMS

The most fruitful method of constructing modular forms of higher level 1s to form thetaseries of positive definite integral quadratic forms.

Thll basic references are Heeke's

"Aru.lytlsr.he Ar1thmetik der posltiven QUlldratischen FormPn" [5',No.ltll, and Schoeneberg [11]; we will give only a very small part of the theory here.

: 1 ~

n

2t,j=l 1,1

(xt = (x 1 , ... ,xr)' be a I.e.

Q(x)

Ca 1 J)

>0 a 11

for is

X

R,

1

J

A= r x r

matrix)

integral 4uadratic form,

posltiv~ d~flnlte

an lnleger, A=

xj E

X X

I

~n

o,

and

is

even integer, i.e.

is an integral symmetric matrix.

The

VI-2

MODULAR FORMS AND DIRICllL£1: SERIES

1D&la-fuoction associated to

Q is

~(r;Q) • ~ e2w1Q(n)~ n

where

nt

Cn 1 , ..• ,nr) c zr,

and

number or tntogral solutions of show eventually that of dimension case where

-k

C > 0,

for some and so

,, (T ;Q)

is evon.

e -2 eycl:n i

y

N,

in the

Q(x) > C

Note

( I: e -2 eycm

nt.lr where

is a modular form

-

i

2

i=l xi,

domina ted term-by-term by

2

I:

We will

Q Is positive definite,

since 1s

is the

Q(x) = v.

of a certain level

= 2k

r

,,(T,Q)

aQ(vl

2

f,

m£1:

= Im

T,

and so

~(t;Q)

is a holomorph1c

functton on the upper half plane. A is the

~of

is the determinant of 6

= (-l)kD

LEMMA.

Let

IDMtrlx

(Ktt

Q,

Q,

its determinant

and (1f

is the discriminant of

A

= (a 1 jl

evcnl.

r

~

2k

D

is Hven)

Q.

be an Integral symmetric If

r

ic odd, then

U = det A

VI-3

QUADRATIC FORMS

is even; if

r

is even, then

~

: 0 1 1 (mod 4).

o of n

ing over all permutations Since

letters.

A is symmetric, the terms ror

o and

o 1 o- 1 ,

are the same, and so cancel modulo 2 tf l.e.

a211.

fixes

~ome

If

letter

r

is odd, any and olnce

i'

D = 0 (mod 2).

we get

a2

o wtth "tt

u- 1

~ 1

is. &Ven,

We leave the second state-

ment, which we do not need, to the reader. Let

even.

A be integral syiiUDOtric, with

Then

DA-l = (A

ij

)

r

= 2k

ts the cofactor matrix,

whtr.h is still integral symmetric, by the lemma. Thus

DA-l

is again the matrix of ~n integral

positive definite quadratic form least positive integer ls the

l..l::£ti (Sture) or

N with Q,

~tDA- 1 x. NA-l = A•

.

=htNA-lx 2

ls the adlolnt

f2.J:m to Q;

integral

and tho corresponding

qu,.dratic form

Q (xl

The

note

NID,

and

VI-4

MOUULAR FORMS AND DIRICHL.t:r SERIES

is prlmitiye, i.e. v > l,

for

i.e. the greatest common divisor of

~;i

the coefficients

particular, if

1.

Q is primitive, then

PROPOSITION 22. Q satisfy

ts

and

Q and its adjoint Q•• ~ Q. In general,

t.a. and

of

1s not integral

Q•

N

~

•

N ,

have the same level,

N•jN,

and

The determinant NIDIN 2k;

The

hence

D and level

N and

D have

the same prime factors, and for a given level und number or variables

~k,

N

N

there are only f1-

ntte1y mAny corresponding discriminants

~.

The basic result, due to Schoeneberg [11], which we prove eventually, is that where

dnl

= (!!) n

d-n) = (-l)kdnl;

.t(T;Q) ~ !Tl(N,k,rl,

(Jacobi symbol) for

n > 0,

the two uses of the word level

then agree. We also need a mod1f1ed thetn-functlon, ustng

VI-5

QUADRATIC FORMS spherical functions.

Let

A be a symmetric posi-

tive definite real matrix of degree a quadratic form variables

y

xtAx.

= Dx,

r,

defining

By a linear change or

we dtagonaltze the quadratic

form: r

2

£ y 1=1 i t.e.

I

= (B-1)

t

AB-1,

or

A

BtB.

(B

is a real

matrix.) Now a function

f(x)

is a lillb!lt1s:al !UDs:t1!2n

with respect to the quadratic form £

a

i.e.

2 = 0,

xtAx

if

0

IIYt

~~=-£. • 1 8Y1 IIYt 1 bjl bk1

..

..

f(x)

so

• = ajk

'

is a spherical function relative to

tf and only if

• ____.d:_ :r a ij IIXtiiXj = o.

There is an inner product on functions X

£

Rr

'

by

I f(x)ilxTdx 1 • • •dxr xt.Ax,S.l

Q

f(x),

MODULAR FORMS AND OTRTCHLET SERIES

VI-6

fs"r /

=-

Y

THEOREM 18. of degree cients.

Let

f(x)

in

v

f(x)gri(Tdy 1 · · ·dy2 •

YD be

a homogeneous polynomial

x 1 , ... ,xr'

with complex coeffi-

Then the following three statements are

equivalent: l)

f(x)

is a spherical function with res-

pect to

f(x)

2)

xtAx.

is orthogonal On the above inner

product) to all homogeneous polynomials of degree <

3)

r

is a linear sum or functions of the

form ~·

V•

(!;;tAx)",

whore

!;;

£

Translating )} into variables

tr,

CtAC - o.

y,

If we '1

ty.

•

thus we can assume wl thout loss of gcncral1 ty that

A= I

is the identity.

If 3l holds, then so docs 1), since .....IL v [ ~~ - t 2 Ct (jxj)

ax 1

vCv-l)(f ~~)(

ax 1

0

1f

l:

(;~

o.

}v- 2

VI-'7

QUAJJltATIC FORMS

Tn

gener~l,

ts homogenebus or degree

f

and so

~.

hence the divergence

theorem gtves

" I fw

(l)

OK

=

I

(1: .dL.x )co

OK 1 exl

1

=I

K

Mdx

where

K•Ex~~l, r-K•Ex~-1,

and

"'

= 1:

(2)

A(f'g)

= (-ll t-ldx1 ..• ~1 ., .dx

"'t

x 1
!if'=:E-4 ox 1 1.,

Uy Stokes' theorem,

I

aK

fw

= I

I:

eK

rx1w1 = I

1:

K

.-lLcrx )dx

ax 1

t

(" .,. r) I r<Jx. K

Thnsl (3)

I Af

\1(\1

+ r)

K

We now prove Not<> that

I f, K

l:lr

~

1)

=->

2) by induction on

o =;, l:l(.II.L) 1

ax

e

Q,

Assuming

"'

lif'

= o,

VJ-8

MODULAR FORMS AND DIRICHLET GERIES

deg(g) < deg(f):

and

= ( )

I fg

I ll(fgl,

by (3)

K

K

> I fllg,

by (2) and induction

K

o.

( l I rll 2 g K

gonal to all

g

to all

cctx)v,

r = o.

Let

)), let

~>

Finally, to chov :?)

be ortho-

f

of lower degree, and orthogonal where

etc

= 0;

= cctx)V;

g(x)

we are to show

then

g

and all of

its partial derivatives satlsfy )l, hence 1) and 2).

Then

= 1 fg

0

K

Now iteration of

avr """6X

'f - ~

V•

- ~ ftX

il

glves 0

iv

an

d

so the Hbove gives simply that etc =

o,

and hence

"(x) = xtx

= E x~,

f(x) say

rCCJ =

o

when

is divisible by rCxJ

~

~,(xle;CxJ.

"I hen,

l;lUADRATIC FORMS

VI-9

from equations (l) and (]):

l I gji(..

I gg "'

aK

K

= ( ) I llgi(u,

since

oK

on

1

b

aK

)I fg '- 0. K

Thus

so

01

g

COROLLARY.

f

- 01

The space

q.e.d.

Hv

of spherical functions

which are homogeneous polynomials of degree

v

has dimension ( r-l+v) -(r-3+v) r - 1 r - 1 !l:2Q.l:..

of degree 11 < v}. f

£

p

v'

p

Let

v

Then

\lo

Now f

~

other hand, t" r: ll" ~ f

be the homogeneous polynomials

pll H v

~

f

v

l

l Pv-2'pv-4•"""•

(f,g) - ( lCf,llgl,

l

p : f'

p

f'or

11

l pv 1f 11+v is odd, so given

thus

P,_ 2 ;

( r-l+v) _ (r-l+v-2) •·-1

Hv - (r .,

r-1

1

so

dim ll

q.e.d.

On the

\1

- dim P

v

- dim P

v

VI-10

MODULAR FOnMG AND DIRICHLET SERIES Given an integral positive definite quadratic Q(x) ~ !xtAx,

form

tion of order

n spherical func-

P(x)

with respect to

v

I:

"(qQ,P)

and

Q,

we have a

P(n)c 2 TT1Q (n)T

nc~r

which we will prove is a modular form with charector (lf

r

&

= 2k

r 0 (N) 1

for

1s even),

The introduction of the

of dimension

-(k+v)

and a cusp form if P(x)

passing from zeta-functions to

v

> 0.

is somewhat like L-series by intro-

ducing characters.

PROPOSITION 23.

Given a positive definite symme-

tric real matrix

A of deg1•ee

define E 6 2w1Q(n+xh

"(r,x)

nczr for a parameter

x c Rr.

Then

r

1

Q(x) = !xtAx,

QUADRATIC FORMS

frQQr.

VI-11

(Cf. the same result for

r

1

in

Chapter T.) periodic function of

x

its Fourier series E

ram

e211tmtx

m£Z

where 1

1

I ···I "C•,x>e

0

-2wimtx

0

dx · ··dx 1 r

Completing the square, I.

T(x - ,-lA- 1m) A(x - ,-lA- 1m>

Hence

a

m

~ e-wi""<

-1 t -1

m A mb m'

wnere

-1 -1 )t ( -1 -1 ) _ I e11l t ( x-• A m A x-• A m dx

b m

Rr t

I e"'i<x Axdx, Rr

by Cauchy's theorem

MODULAR FORMS AND DIRICHLET SERIES

VI-12 y = Bx,

where

dy

= jBjdx,

= ...1.. -r/2, J5 COROI.J.ARV.

Tf

A = BtB

q.e.d.

Q(x)

is integral in

= 2k

r

variables, then

ft22!.

Set

x

0,

and note

Q•

has matrix

NA-l. We now generalize the

~bove

,,('t;Q,P) = r P(n)e 2 '1T1Q(n)'t,

spherical function relative to on,

Q(x)

corollar~

where

P(x)

Q(x).

to is a

From now

will always be an Integral positive

definite quadratic form ln an even nlmber

2k

of variables. We take first a typical spherical functi'>n P(x) = ''tAx)~,

where

Q(') = 0.

The transformed

theta-function w111 involve

p•(x) = P(A- 1 x>

= (~tA- 1 x>~,

satisfies

where

~tA- 1 ,1 = CtAC ~

o,

~ =A' t.e.

~·(q) =

o;

thus

p•(x}

QUADRATIC FORMS

VI-13

is a spherical function relat{ve to Q•(x). We start with ~(•,xl ~ E e 2 wiQ(n+xl• and n

L = E (i ~·

apply

i

LQ(x) - (tAx, (1)

L"•'

L2Q(x)

~

times. (tA(

Note

= o.

Thus:

E (2wi't)"((tA(xn+xll"e 2l71Q(n+xl• n

Dut by the transformation forroula (Proposition 23), we have also {I

nnd hence (2)

proved for the special spherical function and hence (by Theorem 18) for nny spherical function P(x),

TIIEOR~

by comparing (1) and (21:

19. (Schoeneberg)

If

P(x)

is a spherical

MODULAR FORMS AND DIRICHLET SERIES

VJ-14 function for

Q(xl,

adjoint function

and

(R

P•(x) ~ P(A- 1xl

the

spherlrAl function for

Q•(xJ),

than:

E P(n + xle2wiQ(n+x)~ n

This suggests our "k" will be

k+v,

so we

set

selling

x

=0

above, and

HN

= (~

-j)

we have "(qQ,P)

Thus: COROLLAHY 1. ~

"(r ;~,Pl

..

--~ ./5 ,t(r,Q ,r,• l IHI\.

as usunl,

VI-15'

QUADRATIC FORMS Taking

X

=h

ls

N'

lntugral, the rormulo of Thoorom 19 reads•

o>

,c~;Q,P,hl

= N-"

PCnle 2 YTtQ(n)1;/N nO:h(N) I:

=~ r JiSt k+v n

ir

1~ tntecral.

N-"P(m).

m =- liA- 1n.

Am: 0 (mod Nl;

m is integral, and

n = N- 1Am

(der. l

~·(n)P(-vi/~)ntA-ln+2YTlnth/N.

On the right, substitute integral, and

2

Am

m is

on the other hand,

~ 0

Also,

Then

(mod

N),

then

P•(nl • P(A- 1 nl

The right side of (3) is thus:

(4l

Now suppose rlep~nds

Ah

0 (mod N).

only on

m modulo N,

':'hts proves:

Then so (4)

t

e 2 wim AhiN u~cum~s•

2

MODULAR FORMS AND DIRICHLEt SERIES

Vl-16

COROLLAkY 2.

For an integral vector

Ah: 0 (mod Nl, ~(T;Q,P,h)

N-~

k

"(qQ,P,h)

h

with

define

; ~

./5

t P(n)e 2 wlQ(n)T/N n:h(N)

2

2

t

E 0 2w1g Ah/N "(T;Q,P,g)IH 1 g mod N Ag-o(N)

We also have obviously that ;>

"(T+l;Q,P,h) = e 2wtQ(h)/N "(T;Q,P,h). Hence the vector space generated by the

"(qQ,P,h)

is operated on by the full modular group series are clearly regular at at

..,

if

" > 0.

thclr tnvartance under

r(N) N

Suppose

r.

These

and vanishing

lienee we only have to check

modulEtr forms of level ~·

~,

N

= 1,

to know

they are

(cusp fu1·tnS 1 f i.e.

By ChRpter 1, we know

tk = 1,

number of variables is

li&US

D

= 1.

"

> 0).

Then

I.e. ~jk;

divisible by

the 8.

We

VT-17

QUADitATIC FORMS

have

is a modular

~(~,Q)

and level

wtth

1.

of dimension

for~

An example with

~(t,Q) = 1 ~ E:=laQ(~)e 2 F1~t

dim rnCC',4)

= 1,

Similarly, tf aQ(~)

is

Since

we have necessartly

Q has representation numbers

so

=8

r

-k

aQCvl

Q has 16 variables, then

= 480c7 Cvl.

For

there exist two forms with dlscrlmlnant

1

k

= 12,

Q1 , Q2

Siegel proved In 24 variables

anrl rllffarent theta-series;

then

ts a non-zero cusp form of dimension t.e.

~~(T),

c I

o.

-12,

Ramamyan's conjecture can

be thought of as an assertion about the difference (~)

aQ I

(~).

- aQ 2

In general, If

Q(x)

is a form

MODULAR FORMS ANO DIRICHLET

Vl-18

or discriminant

where

g(-r:)

b

v

2k variables

is a cusp form.

coefficients, we

where

in

1

S~Hl~~

(41k>,

then

In terms of Fourier

~ve

= O(vk12 l,

by Proposition 16, so the

theory of modular forms gives asymptotic results about the representation numbers

aQ(v).

Returning to our general development, btisidos the rules above, we also have, for any natural number (6)

c:

"(-r:;Q,P,h)

I: " ( C1: i CQ 1 p 1 g )

g

Let

g=h(N) mod eN

(~~),r•,

~-a....L cr+d cT+d' o)(r;Q,P,h)

with

c>O.

Then

so:

1(~ ~)

.,. ( ____l__ ,, r ) "'' a-cT+d;c ... ,.·,c

(C L+d)-(k+vl

g'h(N)

c

mod eN

QIIAilRATIC FORMS

VI-19

ik(_l)k+v ck./D

2 I e2wial;l(g)/cN " g-h(N) g 010d eN

r e2wtttAg/cN~~(c~+d;ci;I,P,t) t

mod eN

At f)(N)

(The },.,.t by Corollary 21 the dotormlnant of

Js

c 2ko.>

(71

Thus:

( b)

~(T;Q,P,h)l ~ d

~

1-k-2v ckJD

x

~ t(h,~l8(cT;cQ,P,t)

t

mod eN

A' IO(Nl

where

(A)

e2111CaQ(g)+~tAg+dQ(t))/cN 2

f

(:(h,·) g

h(Nl

g mod eN

for

{~ ~)

t

1 ',

c

~ 0.

One

~omputes:

cQ

MODULAR FORMS AND DIRICHLET SERIES

VI-20

This shows that

(10)

t(h,tl

depends only on

t

hence, using (6), we can rewrite (7) as

modulo N;

~(T;Q,P,hll(~ ~) _ 1-k-2v . k E t(h,t)~(T 0 Q,P 1 ~). cJfitmodN At:O(N) In particular, if

d- O(mod N),

then (10)

becomes (11) ~(T;Q,P,h)l ( ac

bd)

t(h.O) E e-2w1hLAt•b/N 2 ~(<;Q,P,t) ik+ 2 vckJfi t mod N At.:O(N) and applying

H1

(~ -~)•

E e2wtttA(g-bh)/N 2

llow

t

lf:~ N

a finite group with D if

g

becomes:

~

bh (mod Nl

we get

ts a character sum on

D elements, so this sum ts and

0

otherwise.

Thus

(l?)

VI-21

QUADRATIC FORMS

for

c >0

c < 0,

and

Njd.

This then holds also for

(~ ~)

since if we replace

this gives a factor of

(1]}, stnce

(-llk+v

~(-h) = (-llv~(h).

by

(:~ :~)•

on both sides of ChRngtng the

notation,

for

(~ ~} e r~(Nl,

(15) tp(h)

where (cf. (8))

I: 8 2111bQ(g)/dl?. h(N) g mod dN g

In this sum, we can wl'i te g 1 mod d, (1 6 ) cp

g

= adh

+ Ng 1 ,

and (15) becomes

(h) - e2'11"labQ(hl/N 2

Thus, (14) becomes

I:

s1

9

211lbQ(nl/d.

mod d

VI-22

MODULAR FOHM:> AI'W lJIIHCHLET SERIES

= d-k

~(b,d)

where

(~ ~} r I'~ (N).

t e 2 nlbQ(g)/d, g mod d

In order to 1 nvestigate the

= 1,

further, let us take

~

~(T;Q,P,hl = ~(T;Ql I

0.

h

1s lp the field or ~

and hence

1

the automorphism

= ~(l,d)


(~ ~') t(d)

c

r~(Nl

for

~(b,d)

b)

( ac <1

modulo N, taking

r·•0 (Nl '

(~ ~)

e

c

l~(Nl,

d modulo N.

where

satisfying

1

ror all Applying

we find

Finally, s1nce

(N~' ~)

lf

to

~(b,d) ~ ~(b+na,d+ncl

is rational.

(Schoeneberg).

c

{b ~}

e2 rrib/d --> e 2 ui/d,

£(d).

l>(b,d l

so we knuw

(d+ncl!.h roots or

depends only on

THEOREM 20.

= o,

Applying

both sides of (17) 1 we find

n

for

We

we see have proved:

We

have

£

is a real riiRracter

t(-1) - (-l)k.

r• CNl,

~(<;Q,P,h)j(~ ~) ~ ~( j~,P,h),

In pRrticulnr

QUADRATIC FORMS

Vl-23

and hence

'(t;Q,P,h)

dimension

-(k+-v),

if

is a modular form of

N,

of level

and a cusp form

v > 0. E(d)

It remains to determine

(we know E(d)

(-l)k),

d-1)

r:

d-k

d >0

for

We have

e2JT1Q(g)/d,

g modd

and

E(d)

Q(x) say

d modulo N.

depends only on

be an odd prlme with

p

d (mod N)

;

can be diagonelh:ed modulo

Q(x)

p

£

z.. Then

dp)

p

p

2

-k 2k

2rriajgj/p

Ee .1"1 g.l mod p II

-k 2k II j~l

where

~

E(lt-(p))e .:.1 mod p

2rr1a Jzj/p

is the Legendre symbol,

(~) p

"0 1 1,-1

as

PIZ

is solvable, or oth .. rwige.

p

wl th integers,

r: a.lx~ (mod p), where aj .1

E(d)

Let

and so that

J'

Note t.h>ot

VI-24

D:

MODULAR l''OKM:; AND DIRICIILET SERIES

2k II (2a 3 ) (mod p); j=l

piD,

PIN,

plri,

hence

p .( a 3

contrary to

r e2"1a3"'lP z 3 mod p

0,

(d,N)

(othervise

= 1).

Thus

and the above becomes:

c(p)

dd)

P-

p-

k 2k

n gX(a 3 J

(Gauss sums)

j=l k 2k

n Cx(aj)gX)'

by Proposition 21

j"l

p

-k 2k ( )

gX X D '

2k

since

U-

n (2aJl (mod

j=l

2 x<-llgx,

E(d)

p

Nvw

p

~ lgxl 2

so 'ole have finally that

= t:(p)

Thus we have proved that prime

p).

E(rl) ·

(~) p

for any

vhirh is sufficiently large and Hatlsfies

p = d C111od N).

Hence

6

is a

11 <.!1scrlm1nant.",

wA

QUADRATIC

VI-25'

FOI\MS

0 1 1 (!nod 4),

dd) " (~)

neressartly have

'·

and

(Jacobi symbol).

(Here we are appealing to basic

facts about quadratic number fields; if then the conductor

or

N'

/j.:

2,3 (mod 4),

is

4 or 8 times some of tho odd Primes dividing

D,

whence

tl'

does not divide

N,

since

Cf., e.g., Heeke's book (6), Chapter VII.)

THEOREM 20+. for

O,l(mod4J,

NID· Thus:

E(d) .. (~) d

and

> o.

d

~·

As sketched earlier for the case

N " 1,

one gets asymptotic results on the representation numbers st.Hin

aQ(")

by writing

,)(-.:,Q)

as an Eisen-

series plu:1 ,. cusp form. finally, the theory of the

T(n)

gives a

way of deriving knowledge of the representatlon numbers

aQ(nJ

from those for primes.

Starting

from

r. a (n)e?.n1nr 1 n=O Q one gets a basts

f 1 , ... ,fr

for the least space

VI-26 V(Q)

MODULAR FORMS ANU UIRICHLET SERIES ~(T,Q)

containing

T(n),

(n,N)

= 1,

by taking

= f 1 (T(nj)

for Note

has integral Fourier coefficients;

hence we

CBll


number field gl' ••• ,gr for the

fj

1 = n 1 < n 2 < ..• < nr'

suitable integers each

nnd closed under all

K.

of

fl''''tfr

T(n),

(n,N) • 1,

Fourier coefficients once the

ap'

p

T(n)

Thus certain

~

an'

N,

will be eigenfunctions and hence their (n,N)

from

Q

by

= 1,

are known (for

Furthermore, one can determine g 1 , ••• ,gr

ovor a certain

K-llnear sums

are known g 1 , ... ,gr).

r 2 , .•. ,fr'

a flnitA process.

For

details and examples, we refer again to Heeke's "Analytische Arlthmetik" [5,No.

411.