The Theory Of Jacobi Forms (Prodress in Mathematics 55)

в

Progress in Mathematics Vol. 55 Edited by J. Coates and S. Helgason

Birkhäuser Boston • Basel • Stuttgart

Martin Eichler Don Zagier

The Theory of Jacobi Forms

1985

Birkhäuser Boston • Basel • Stuttgart

Authors M a r t i n Eichler im L e e 27 CH-4144 Arlesheim (Switzerland)

Library

of Congress

D o n Zagier D e p a r t m e n t of Mathematics U n i v e r s i t y of Maryland College P a r k , M D 20742 (USA)

Cataloging

in Publication

Data

Eichler, M . (Martin) T h e t h e o r y of J a c o b i f o r m s . ( P r o g r e s s in m a t h e m a t i c s ; v. 55) Bibliography: p. 1. J a c o b i f o r m s . 2. Forms, Modular. I. Z a g i e r , Don, 1951. II. Title. I I I . S e r i e s : P r o g r e s s in m a t h e m a t i c s ( B o s t o n , M a s s . ) ; v, 5 5 . QA243.E36 1985 512.9'44 84-28250 ISBN 0-8176-3180-1

CIP-Kurztitelaufnahme Eichler,

der Deutschen

Bibliothek

Martin:

T h e t h e o r y of J a c o b i f o r m s / M a r t i n E i c h l e r ; D o n Z a g i e r . Boston ; Basel ; Stuttgart : Birkhäuser, 1985. ( P r o g r e s s in m a t h e m a t i c s ; V o l . 55) I S B N 3 - 7 6 4 3 - 3 1 8 0 - 1 (Basel . . . ) I S B N 0-8176-3180-1 (Boston) N E : Zagier, D o n B.:; G T

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-

V

TABLE

I.

1

Notations

7

Basic Properties

8

1.

Jacobi forms and the Jacobi group

2.

Eisenstein series and cusp forms

17

3.

Taylor expansions of Jacobi forms

28

Application:

37

Jacobi forms of index one

Hecke operators

6.

7.

8

41

Relations with Other Types of Modular Forms 5.

III.

CONTENTS

Introduction

4. II.

OF

Jacobi forms and modular forms of halfintegral weight

57

57

Fourier-Jacobi expansions of Siegel modular forms and the Saito-Kurokawa conjecture

72

Jacobi theta series and a theorem of Waldspurger .

81

The Ring of Jacobi Forms

89

8.

Basic structure theorems

89

9.

Explicit description of the space of Jacobi forms. 100 Examples of Jacobi forms of index greater than 1 . 113

10.

Discussion of the formula for

11.

Zeros of Jacobi forms

dim J, k,m

121 133

Tables

141

Bibliography

146

INTRODUCTION

The functions studied in this monograph are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jaoobi form on SL2(Z) to be a holomorphic function K x & -> <E

(3C = upper half-plane)

satisfying the two transformation equations 2TTimcz^ (1) * ( £ 3 ) - k e CT +d" • (t 9 e sv«)> /o\ . -\ . \ -2Trim(X2T +2Xz) ,f x , * — _2X (2)

as Jacobi functions, a word

which we will reserve for (meromorphic) quotients of Jacobi forms of the same weight and index, in accordance with the usual terminology for modular forms and functions. 3. The Weierstrass ff—function.

(6)

|»(T,z)

=

z"

2

+

Yl

The function

2

2

( ( z + 0 J) " ­ Ü J~ )

(0€Z +ZX is a meromorphic Jacobi form of weight 2 and index 0 ;

we will see

-3later how to express it as a quotient of holomorphic Jacobi forms (of index 1 and weights 12 and 1 0 ) . Despite the importance of these examples, however, no systematic theory of Jacobi forms along the lines of Hecke*s theory of modular forms seems to have been attempted previously.*

The authors* interest

in constructing such a theory arose from their attempts to understand and extend Maass

1

beautiful work on the "Saito-Kurokawa conjecture".

This conjecture, formulated independently by Saito and by Kurokawa

[15]

on the basis of numerical calculations of eigenvalues of Hecke operators for the (full) Siegel modular group, asserted the existence of a "lifting" from ordinary modular forms of weight Siegel modular forms of weight

k

2k-2

(and level one) to

(and also level o n e ) ; in a more

precise version, it said that this lifting should land in a specific subspace of the space of Siegel modular forms (the so-called Maass "Spezialschar", defined by certain identities among Fourier coefficients) and should in fact be an isomorphism from

^2k-2^ 2^^ L

onto this space,

mapping Eisenstein series to Eisenstein series, cusp forms to cusp forms, and Hecke eigenforms to Hecke eigenforms.

Most of this conjecture was

proved by Maass [21,22,23], another part by Andrianov remaining part by one of the authors

[40].

[2], and the

It turns out that the

* Shimura [31,32] has studied the same functions and also their higherdimensional generalizations. By multiplication by appropriate elementary factors they become modular functions in x and elliptic (resp. Abelian) functions in z, although non-analytic ones. Shimura used them for a new foundation of complex multiplication of Abelian functions. Because of the different aims Shimura s work does not overlap with ours. We also mention the work of R.Berndt [3,4], who studied the quotient field (field of Jacobi functions) from both an algebraic-geometrical and arithmetical point of view. Here, too, the overlap is slight since the field of Jacobi functions for S L ( Z ) is easily determined (it is generated over C by the modular invariant j (x) and the Weierstrass p-function ^ ( x , z ) ) ; Berndt s papers concern Jacobi functions of higher level. Finally, the very recent paper of Feingold and Frenkel [Math. Ann. 263, 1983] on Kac-Moody algebras uses functions equivalent to our Jacobi forms, though with a very different motivation; here there is some overlap of their results and our §9 (in particular, our Theorem 9.3 seems to be equivalent to their Corollary 7.11). f

2

1

-4-

conjectured correspondence is the composition of three isomorphisms

Maass "Spezialschar" C

M^(Sp^(Z))

\l v Jacobi forms of weight

k and index

1

\l

(7) Kohnen s

(r

~F~" space ([11]) C

?

(4))

l

l

M _ (SL (20) 2 k

;

2

2

the first map associates to each

F

the function

(|> defined by (5) , the i

second is given by

Z n>0

,

\

c(n) e

2nrinT

^

—>

V

L

V

2-,

//

2N

c(4n-r ) e

2TTi(nx + rz)

,

n^O r < 4 n 2

and the third is the Shimura correspondence [29,30] between modular forms of integral and haIf-integral weight, as sharpened by Kohnen

[11]

for the case of forms of level 1. One of the main purposes of this work will be to explain diagram (7) in more detail and to discuss the extent to which it generalizes to Jacobi forms of higher index.

This will be carried out in Chapters I

and II, in which other basic elements of the theory (Eisenstein series, Heeke operators, ...) are also developed.

In Chapter III we will study

the bigraded ring of all Jacobi forms on SL (7Z) Z

.

This is much more

complicated than the usual situation because, in contrast with the classical isomorphism

M (SL„(S)) = C [ E , , E ] , C

(J^ ^ = Jacobi forms of weight

k

and index m)

the ring

J

=

©

J,

is not finitely generated.

Nevertheless, we will be able to obtain considerable information about the structure of bounds for

dim J^ ^

will prove that r

ring

M (SL (S)), i

J^ ^.

?

J

*,m

In particular, we will find upper and lower

which agree for k = © J, k k,m

m

sufficiently large

is a free module of rank

(k^m),

2m over the

and will describe explicit algorithms for finding

-5bases of

J. k,m

M (SL ( Z ) ) . >V

as a vector space over

J *,m

as a module over

The dimension formula obtained has the form

2

(8)

dim J

k

m

=

' for

(C and of

r

k

E

dim \

+ 2

r " N(m

r=0

even (and sufficiently large), where

m j- 2 -) £ 7~ r=0 » »

N(m)

4

We will show that

)

N(m)

is given by

( fx ] = smallest integer > x)

m

N ( m ) can be expressed in terms of class numbers of

imaginary quadratic fields and that (8) is equivalent to the formula

(9)

dim J

where

^2^2

= dim M _ ( r ( m ) ) 2 k

k j m

2

0

~*"S t*ie sPace °^ new f°rms °f weight

which are invariant under the Atkin-Lehner r/ \

^

f(T) — > m

"new"

-k+1

T

-2k+2

_,

N

f(-l/mT)

,

and

_new m

2k-2

on

F0(m)

(or Fricke) involution .

,-

, _.

,

_

a s u i t a b l y d e f i n e d s p a c e of

Jacobi forms. Chapter IV, which will be published as a separate work, goes more

deeply into the Hecke theory of Jacobi forms.

In particular, it is

shown with the aid of a trace formula that the equality of dimensions (9) actually comes from an isomorphism of the corresponding spaces as modules over the ring of Hecke operators. Another topic which will be treated in a later paper (by B.Gross, W.Kohnen and the second author) is the relationship of Jacobi forms to Heegner points.

These are specific points on the modular curve

X (m) = 3C/T (m) U {cusps} Q

o

(namely, those satisfying a quadratic equa-

tion with leading coefficient divisible by each

n

and

r with

r

2

P(n,r) £ Jac(X (m)) (C|) 0

< 4nm

m) .

It turns out that for

one can define in a natural way a class

as a combination of Heegner points and cusps and

-6that the sum

V

p

(n,r) q

£

n

is an element of

r

nr

Jac(X (m))(Q) ®

One final remark.

J

9

^

0

, m

Since this is the first work on the theory of

Jacobi forms, we have tried to give as elementary and understandable an exposition as possible.

This means in particular that we have always

preferred a more classical to a more modern approach (for instance, Jacobi forms are defined by transformation equations in 3Cx €

rather

than as sections of line bundles over a surface or in terms of the representation theory of W e i l s metaplectic group), that we have often 1

given two proofs of the same result if the shorter one seemed to be too uninformative or to depend too heavily on special properties of the full modular group, and that we have included a good many numerical examples. Presumably the theory will be developed at a later time from a more sophisticated point of view.

*

*

* &

This work originated from a much shorter paper by the first author, submitted for publication early in 1980.

In this the Saito-Kurokawa

conjecture was proved for modular (Siegel and elliptic) forms on P ( N ) Q

with arbitrary level

N.

However, the exact level of the forms in the

bottom of diagram (7) was left open. as here in §§4-6.

The procedure was about the same

The second author persuaded the first to withdraw his

paper and undertake a joint study in a much broader frame.

Sections 2

and 8-10 are principally due to the second author, while sections 1, 3-7 and 11 are joint work. The authors would like to thank G. van der Geer for his critical reading of the manuscript.

-7Notations We use

IN to denote the set of natural numbers,

We use Knuth's notation

[xj

greatest-integer function

(rather than the usual

max{nG z|n ^ x}

fx] = min{nGZ|n^>x} = -L"" J* By

d||n

we mean

d|n

and

(d,-j) = 1.

0

for

TN U {0}.

[x] ) for the

and similarly

The symbol •

X

]N

denotes any square number.

In sums of the form

^

£ d

l

or n

/ it is understood that the summation is over positive divisors only. ad=£ The function £ d (d G IN) is denoted O (n) . d|n The symbol e(x) denotes e , while e (x) and e (x) (mG]N) m t

V

m

denote

e(mx) and

e(x/m),

complex variable, but in e (ab

)

1

m

means

We use

e m

M

respectively.

e m

( n ) with and

t

I

(x)

2

bn = a (mod m) , and not

m

is a

Z/mZ ; thus

e(a/bm) .

The symbol

[a,b,c] denotes the quadratic

2

denotes the upper half-plane {xG (C|lm(x) > 0 } .

x = u + iv,

z = x + iy,

often be denoted by weight

k

E^ G

E

e (x) , x

for the transpose of a matrix and for the n x n

z will always be reserved for variables in K

with

has

e(x) and

a x + bxy + c y . K

and

In

it is to be taken in

identity matrix, respectively. form

x

on

T

i

F

x

by

q = e(x),

£ = e(z) .

The letters

and

X

4 even) are defined in the usual way; in particular one K

A

:= ® M

= C[E ,E- ]

k

If

6

with

E^ = 1 + 240 X a ( n ) q 3

n

,

= 1 - 504 2 0 ( n ) q . n

5

The symbol

" := "

means that the expression on the right is the

definition of that on the left.

Chapter I

BASIC PROPERTIES §1.

Jacob! Forms and the Jacob! Group The definition of Jacobi forms for the full modular group

r

= SL (JZ)

was already given in the Introduction.

2

F C r

subgroups

with more than one cusp, we have to rewrite the

definition in terms of an action of the groups k)m

(HE

r,)

,

*l x=

One easily checks the relations

-8-

(xez ) , 2

m

where we have dropped the square brackets around the notation.

z

M

or

X

to lighten

-9-

M) I

(cj>L

(4)

V Y ,

k,m

(

(MM)'

<j>L

'k,m

(<j>| x)

,

'm

| M ,

((j)| x)

<j)| (x + x )

=

(MM , G

,

1

>m T

K)IN

m

I x*

y

'm

x,x'

r , 1

z ) 2

G

They show that (2) and (3) jointly define an action of the semi-direct :=

product

r

and group law

Kl

x

(= set of products (M,X) with

2

(M,X) (M ,X ) = (MM , XM + X ) ; f

f

1

r

our vectors as row vectors, so Jaoobi group.

1

f

M G ^ ,

X G Z

2

notice that we are writing

acts on the right), the (full)

We will discuss this action in more detail at the end

of this section. We can now give the general definition of Jacobi forms.

Definition. a subgroup

A Jaoobi form of weight

r C r

k

and index

m

(k,m G M)

of finite index is a holomorphic function

on

<J>: Jf x C -> I

satisfying $1.

i) ii)

M = 4>

X 'm

4>|

iii)

(M G T ) ;

(j)

=

for each

(X G

M G F,

2

(t) | ^ ^M

1

form S c ( n , r ) q V unless

% ) ;

(q = e ( x ) , £ = e(z)) with

n > r / 4 m . (If 2

c(n,r) 4 0

n>r /4m, 2

r

J

Remarks.

J, k,m

for

it is called a cusp form.) T

d) is denoted

J, (F) ; k,m

if

J, (F ) . k,m i

1. The numbers n,r

in iii) are in general in {f^}^

Then the

for some

Is injective by Theorem 1.2.

< £ dim M ^ X F . ) ; this proves Theorem 1.1 and i is 0 for k < 0 unless k = m = 0, in which case it

reduces to the constants. To prove Theorem 1.3, we would like to apply (3) to (A,y) G (Q) . 2

However, we find that formula (3) no longer defines a group action if we allow non-integral

A

and

(cj)| [A y ] ) | [A

1

'm

and

y,

since

y ])(x,z)

=

?

'm

=

e ( A x + 2 A z + A x + 2A(z + A x + y ) ) ( { ) ( x , z + A ' x + y + Az + y)

=

e(2mAy )((x,z)

2

by this group

3R;

i.e.

by the Heisenberg group

H

M

[(À

:=

{[(A U ) , K ]

n),K][a'

1

1

y )^ ]

I

2

(A,y) G 1R ,

=

[(A+A

f

KGI}

f

f

y+y ), K + K'+Ay* ­ A y ]

(This group is isomorphic to the group of upper triangular unipotent 3 x 3 matrices via

-13-

/1 [(A y),к]

JR

/C

2

JK

h(K+\\±)

= ]R .

\

i

0

1

y

J

\

0

0

1

/

:= {[(0 0) , K ] , K 6 l j

The subgroup H

i—»

X

.)

is the center of

and

We can now combine (5) and (7) into an action of

H JK

by setting

(<j)|[(X U ) , K ] ) ( T , Z )

=

m

2

e ( X x + 2Xz + Xy + K)(j)(T

M

in (6) is compensated by the twisted group law in H T

Xy - X ' y = det

by S L , the group SL ( TR.) 2

[X,K]M

. B ecause this

on the right by

[XM,K]

=

F

T

and the determinant is preserved

acts on

2

,

e (X y~AU )

and this now is a group action because the extra factor

twist involves

z + X T + u)

5

( X G 1

2

,

1,

K£

M e

SL (IR)) ; 2

the above calculations then show that all three identities (4) remain M , M G SL (]R)

true if we now take

T

0

and

X,X

T

G H

z

and hence that JR

equations ( 2 ) , (5) and (7) together define an action of the semidirect product

SL.(JR) ix H _ . 2

JR

In the situation of usual modular forms, we write where

G = SL (]R)

finite and

2

contains

K = SO(2)

like to do the same.

T

K

as

as a discrete subgroup with

is a maximal compact subgroup of However, the group

G,

S L (JR) tx H

G/K,

Vol(F\G) Here we would

contains

9

JK

F

J

2 = F x 7L with infinite covolume (because of the extra

JR in H

and its quotient by the maximal compact subgroup S0(2) is 5 C x rather than Jfx % .

) JK

C x i

To correct this, we observe that the subgroup

acts trivially in ( 7 ) , so that ( 2 ) , (5) and (7) actually define an action of the quotient group G

J

SL (K) K H / C 2

R

Z

.

ICE

-14Here it does not matter on which side H _

we write C

C

is

is a central extension of JR

by

JK

central in H ;

the quotient

H__/C JK

S

k]=l}

=

1

Now

, since Uu

2

£Ut

( C = e(K>)

and will also be denoted

is a discrete subgroup of

G with

Vol(F^\G^)
|x) ( T , 0 ) ,

<j>CO := x

where

X=(Ay)GQ

<J>|X is defined by (5) (from now on we often omit the indices

on the sign

I ).

For

X

f

= (A

h+X

l(T)

by (6) , so <J>

1

y ) G TL

we have

e (Ay

-A y)(f) (x)

f

=

1

m

T

X

(mod N Z ) if

x

X G N

2

k,m

f

depends up to a scalar factor only on

itself depends only on

Z .

_ 1

2

X For

(mod 7L ) and <j> l

M = ^

d)

G

we have

(cx +

so (j)

D ) -

K

4>(f£}) x

=

W|X|M)(T,0)

=

(4>|M|(XM))(T,O)

=

(<J>|(XM))(T,0)

behaves like a modular form with respect to the congruence

subgroup

{MGr|XM=X

of

F

(mods2), m.det|

X x

M

|G

(this group can be written explicitly

|(c

d )

G

r

l ( -D a

X +

2

b A + (d-l)y, m ( c y + ( d - a ) A y - b A ) 2

2

Gz|

F

-16-

T

element of

=

x

ing

(X

U ) = XM,

x

q C n

N

if NX £ Z

( r ,

r

\ J

2

then

1

(l M)(T) k where

/

F n T

and hence contains

has a Fourier development contain-

this contains only nonnegative powers of

2

e(x) by the same calculation as given for M = Id

after the statement

of 1.3. We end with one other simple, but basic, property of Jacobi forms.

THEOREM 1.5.

Proof. weight k

ring.

That the product of two Jacobi forms (j> and of i

and

2

The Jacobi forms form a bigraded

k

?

and index m

and m ,

x

a Jacobi form of weight

2

k = kj+

k

respectively, transforms like

and index

2

2

we have to check the condition at infinity.

m = m 4-m 1

is clear;

2

One way to see this is to

use the converse of Theorem 1.3, i.e. to observe that the condition at infinity for a Jacobi form (J)(T,Z) of index condition that

f(x) = e (X x)(f>(Xx + y) m

usual sense) for all

m

is equivalent

be holomorphic at

2

0 0

to the

(in the

X,]i £ q ; this condition is clearly satisfied for

(j)(x,z) = <j> (T,Z)<J> (T,Z).

with

f(x) = f (x)f (x). 1

A more direct proof

2

is to write the (n,r)-Fourier coefficient of as

c(n,r)

=

^ n +n =n 1

r

where the c^ since

n. £ n,

l

C l

(n,>r )c (n ,r ) 1

2

2

?

2

+ R

2=

R

are the Fourier coefficients of

(f^

(the sum is finite

r < 4 n . m . ) and deduce the inequality

r

2

< 4nm

2

from

the identity ( ^ ! n

i

+

N

2 "

4(m

+

R

1 +

2 >

m ) 2

/

2

~

\

/

A\ n

i

Anij /

+

V""

_

4_\

4m / 2

(

+

M

1

R

2 -

M

2

r

i

im^im^m^

)

-17-

This identity also shows that (as for modular forms) the product 4 f ) (

>

1

is a cusp form whenever $ for modular forms) (^^ l

The ring

J

or (f>

2

is one but that (unlike the situation

can be a cusp form even if neither $

2

*>*

l

=

© J, k,m > k

m

2

l

nor <j) is. 2

of Jacobi forms will be the object

of study of Chapter III.

§2.

Eisenstein Series and Cusp Forms As in the usual theory of modular forms, we will obtain our

first examples of Jacobi forms by constructing Eisenstein series.

In

the modular case one sets (for k > 2)

where

1^ = {± Q

1L =1, 'k

where

^

|nG 7L}

is the subgroup of 1^ of elements y

1 denotes the constant function.

Similarly, here we

define

(1)

where

n , l i G Z}

Explicitly, this is

(2)

with

.

-18-

where

a ,b

a re chosen so that ^

€ T .

^

As in the case of modular

l

forms, the series converges a bsolutely for k > 4; it is zero if k odd (repla ce

c,d

by - c , - d ) .

The invariance of

J

T

under

m

clear from the definition a nd the absolute convergence.

is

is

To check the

cusp condition, and in order to have an explicit exa mple of a form in J

k

m' w e must ca lcula te the Fourier development of

^ , which we now

proceed to do. As with as

c

is

E ^ , w e split the sum over

0 or not.

E XGZ

(3)

(q = e

If

2 7 T i T

4nm = r

2

,

£ = e

2 7 r i z

c=0, then

d = ±1;

2

m

c,d

e (X T+2Xz)

=

into two parts, a ccording

these terms give a contribution

E q XGZ

m

X

%

2

m

X

n

).

This is a linea r combina tion of q £

r

with

a nd corresponds to the constant term of the usual Eisenstein

series.

If c ^ 0,

we can assume

c>0

(since k

is e v e n ) ; using the

identity

allk

+

2 X

cT+d

_z

J^L

CT+d

=

- c(z-X/c)l

CT+d

cT+d

^

+

c

we ca n write these terms as

c=l

Note tha t

d

d€l XGZ (d,c)=l d+c

a nd

x

X ->• X+c

correspond to

so this pa rt equa ls

(4) d (mod с) (d,c)=l

with

e

a s in "Notations" and с

A(mod с)

z -> z+1

a nd

T -* T + l ,

-19-

F ,.(T..)

==

k

(

E

T

+

p , ^ e - ( - i ^ i )

p,qG IL the function

F^

m

r

is periodic in T

and z,

so (4) makes sense.

Now

the usual Poisson summation formula gives

\, - E

Y(n,D

m

q V

with

y(n,r)

j

=

T k e(-nT)

Im(T)=C

(C >0, 1

C

2

arbitrary).

(x/2im)^ e ( r T / 4 m ) . 2

Y(n,r)

|

e(-mz /i - rz) dz dx 2

Im(z)=C

1

2

The inner integral is standard and equals

Hence

=

}

T

-k

( T

/21m)

, 5

e(^fflT)dT

ImCT)^! 0

if

r > 4nm 2

! 2.k- /2

1-k,,

a^m

3

(4nm - r )

2 /

.

r

if

,

r < 4nm

with

.

c-D

'

2

s

k

k / 2

k _ 2

^

r(k-^)

2

r > 4nm, we can deform the path of integration to +i°°, so y = 0 ;

(if if

r < 4nm, 2

we deform it to a path from -i°° to

-i

0 0

circling

0

once

in a clockwise direction and obtain a standard integral representation of

1/r(s) ) .

Substituting the Fourier development of F^ ^

gives the expression

E w*'> r with

2

r

into (4)

­20­

3

k­ /

OL

(5)

e

k

m

(n,r)

2

dX

1

^ c

(4nm­ r ) m

(for d

~

2

=

/

1

X

2

- rX + nd)

X,d(mod c) (d,e)=l

c=l

, see "Notations").

^(md

To calculate this, we first replace

in the inner double sum

X

by

(since (d,c) = 1, this simply permutes the

summands); then the summand becomes

e^(dQ(X) ) with

2

Q(X) := mX + rX + n.

We now use the well­known identity

e

£ c d(mod c) (d,c)=l where y

(

d

N

)

=

y

(

£ t > a | (c,N)

a

•

is the Möbius function (so­called Ramanujan sum; see Hardy­

Wright or most other number theory texts); then the inner double sum in (5) becomes

'

X(mod с) Q ( X ) = 0(mod a)

Q(X) = 0(mod a)

Now the condition inner sum is

—

depends only on

X (mod a ) , so the

times N ( Q ) , where a

N (Q) a

:=

#{X(mod a) | Q(X)= 0(mod a)}

.

Hence the triple sum in (5) simplifies to 0 0

c=l

i

i

'

a|c

(the last equality follows by writing

N_(Q)

0 0

a=l

c = ab

a

and using S y ( b ) b

To calculate the Dirichlet series, we first calculate

N (Q) a

S

1

= £(s) ) .

for (a,m)= 1;

this will suffice completely if m = 1 and (using the obvious multiplica­ tivity of N ) a

will give the Dirichlet series up to a finite Euler

product involving the prime divisors of m

in general.

If (a,m) = 1 ,

-21then N (Q) a

=

#{X(mod a) | m X + r X + n = O(mod a)}

=

#{X(mod a) | ( 2 m X + r )

=

N (r

2

- 4nm)

2

a

= r - 4 n m (mod 4a)}

2

2

,

where

N (D)

:=

a

#{x(mod 2 a ) | x = D(mod 4a)} . 2

It is a classical fact that

(6)

£ N (D)aa=l

if D = 1 or if D

is the discriminant of a real quadratic field,

S

a

where

L^(s) = L ( s , ~ ) )

=

L ( ) D

,

S

is the Dirichlet L-series associated to D.

It was shown in [39, p.130] that the same formula holds for all D ^ Z if L^(s) is defined by

S

L

O

if D ^ 0 , 1 (mod 4) ,

C(2s-1)

if D = 0 ,

D

(s) • £

y(d)(-^)d~

S

2

0 _ (f/d) ±

if

2s

where in the lasto lined| Df has been written as D .-f S

D

Q

D = 0 , 1 (mod 4 ) , D ^ O

with

f G I and

= discriminant of Q ( / D ) (the finite sum in this case can also be

written as a finite Euler product over the prime divisors of f) . Inserting (6) into the preceding equations, we find that we have proved e

k j l

(n,r)

=

a |D| k

k _ 3 / 2

C(2k-2)

_1

L (k-l) D

if m=l and D = r - 4n < 0, while for m arbitrary there is a similar 2

formula (now with

D = r -4nm) 2

but multiplied by an Euler factor

-22involving the prime divisors of L^(s)

and

m.

Using the functional equations of

C(s) we can rewrite this formula in the simpler form e

(n,r)

fc

=

L (2-k)/?(3-2k) D

where now all numerical factors have disappeared. (D < 0 ,

k

,

The values

L^(2-k)

even) are well-known to be rational and non-zero; they have

been studied extensively by Cohen [ 6 ] , who denoted them

H(k-1, | D | ) .

Summarizing, we have proved

THEOREM 2.1. a non-zero

The series

element of

E

J, . k,m

J

fc

^

(k > 4 even) converges

The Fourier

development

of ^

r

and

defines

E, k,m

is

given by

E

k,m > ( T

-

z )

£

e

4nm > r where

e^ ( n , r )

for

m

wise , while for

4nm = r

4nm > r

(n

r

2

1 if

r = 0(mod 2m) and

0

other-

we have

2

x

( e

equals

2

k,m ' >

k,l

U

,

r

H(k-1, 4 n - r ) C(3-2k) 2

=

;

(H(k-1,N) = L_ (2-k) = Cohen s function) and f

N

e

km^

n

' ^ r

=

~

^rk'^r

T

^ * TT

(elementary p-factor) .

p|m

In particular,

e, (n,r) 6 k,m

r

One can in fact complete the calculation of little extra work; the result for m

0"

7

% ^> m

(m)

e^

m

in general with

square-free is

^

=!TF2krL

d -H(k-l.^zr_)

d |(n,r,m)

K 1

•

-23However, we do not bother to give the calculation since this result will follow from the properties of Hecke-type operators introduced in §4 (Theorem 4 . 3 ) . For of

m = 1

H(k-1,N)

E

E

g

i

8

i

k

we find, using the tables

given in [6] , the expansions

=

1 + (£ + 56C + 126+ 56C" + C~ )q

+

(126C + 576C + 756 + 576?"" + 1 2 6 C ~ ) q

+

(56? + 7 5 6 ? + 1512^+ 2072 4-1512C" + 756C~ + 56?~ )q

2

1

2

2

1

3

=

2

2

2

1

1 + ( Ç - 88Ç - 330 - 8 8 Ç 2

+

E

and the first few values of

_ 1

2

3

2

2

1 + (Ç

2

+ ... ,

+ Ç )q

(-330Ç - 4 2 2 4 Ç - 7 5 2 4 - 4224c" - 330Ç~* )q + ...

=

3

1

+ 56Ç 4- 366 4- 56Ç

1

2

+ Ç )q 2

,

2

4- . . .

2

Further coefficients of these and other Jacobi forms of index

1

are

given in the tables on pp.141-143.

In the formula for the Fourier coefficients of striking that

e^ ^(n,r)

depends only on 4 n - r . 2

E^ ^,

it is

We now show that this

is true for any Jacobi form of index 1; more generally, we have

THEOREM 2.2.

Let

§

development

2 c(n,r)q £ .

r(mod 2 m ) .

If

n

k

only on 4nm - r . 2

Proof.

r

be a Jacobi form of index Then

c(n,r)

is even and

m=l

If

k

m = l and

or

depends m

only on

is prime,

is odd, then

m

then

with

Fourier

4nm - r

2

c(n,r)

(j) is identically

and

on'

depends zero.

This is essentially a restatement of the second transfor-

mation law of Jacobi forms: we have

­24­

n

S c(n,r)q Ç

r

=

<j>(T,z)

т

е ( Л Т + 2Хг)ф(т, + Хт + и)

=

2

2

2

=

m X 2mA п . Х г q Ç Sc(n,r)q (Çq )

=

« z V n + rX + m X S c(n,r) q

v

f

ч

ч

2

^r+2mA Ç

and hence c(n,r)

T

i.e.

r

c(n,r)= c ( n , r ) whenever

as stated in the theorem. c(n,­r) = c(n,r)

=

r

f

2

c(n+rX + mX , r + 2 m X )

== r(mod 2m)

T

and

4n m­r

,

t 2

2

= 4nm ­ r ,

If k is even, then we also have

(because applying the first transformation law of k

Jacobi forms to ­ I £ r

gives J() ( x , - z ) = (­1) 0 ( x z ) ) , ?

2

so if m

is 1

or a prime, then f

4n m­r'

2

=

4nm­r

Finally, if m = l and k 2

but

4 n m ­ (­r) = 4nm ­ r

Remark:

2

r

T

f

is odd then <J) = 0 because 2

and

f

= ±r(mod 2m) => c(n,r) = c ( n , r ) .

­r = r(mod 2m)

c(n,­r) = ­c(n,r)

in this case.

Theorem 2.2 is the basis of the relationship between

Jacobi forms and modular forms of half­integral weight (cf. § 5 ) .

In the definition of Jacobi cusp forms, there were apparently infinitely many conditions to check, namely 2

4nm = r .

c(n,r) = 0

residue classes

r(mod 2m) with

largest square dividing 1

2

2ab I r) . 1

m

r

2

T

è G J k,m

2

with

a

2

is the

square­free, then

we have

2

c (as , 2abs) =

c u s

in particular, the codimension of j P k,m 5

The number of

= 0 (mod 4m) is b , where b

(namely if m = a b

Thus for

<J> a cusp form

1

with

Theorem 2.2 tells us in particular that we in fact need only

check this for a set of representatives of r (mod 2 m ) .

4m ! r

for all n,r

±

n

0 for s = 0,1,. . . ,b­l

j k,m

is t most b. a

;

Using

-25k c(n,-r) = (-1) condition

c(ti,r)

if

THEOREM 2.3. ———

largest integer

k

s = 0,1,

is odd.

(vesp.

of

jf k,m

i

is even and

k

f

in

U S p

J

|_"^2^J if

such that

>["jJ

Hence we have

The oodimension

k is even

+

for

2

s = 1,2,.. ., |^~—J

["2"] l

we see that in fact it suffices to check the

c (as , 2abs) = 0

k

J, k,m

is at most

is odd), where

b

is the

b | m. 2

On the other hand, if k > 2

then for each integer

s

we can

construct an Eisenstein series

E, (T,Z) :k,m,s

(8)

V /

e r 1

(m=ab of

2

q

j

a s 2

S

2 a b s

|Y

J\ J oo r

as above), where the summation is the same as in the definition

E = E, ~. k,m k,m,0 T

Then repeating the beginning of the proof of

Theorem 2.1 we find that

(9) \ , , m

-

s

*

£

r 2 / 4 m q

2 , then J

is the space of cusp forms in

m

J, k,m

= jf k,m

and

J^ ^

u s p

1

© jf k,m

1 8

,

where

J? K,m

U S P

is the space spanned by

-26-

the functions

E. . k,m,s

J

or

The functions

E, k,m,s

J

0 < s < 77 (k odd) form a basis

J?

2

i s

with

0 < s < ~ (k even) = 2

.

k,m

We will not give the entire calculation of the Fourier developments of the functions the result.

E. k,m,s

here, since it is tedious and we do not need

However, we make some remarks.

certain operators higher index.

and

In §4 we will introduce

which map Jacobi forms to Jacobi forms of

These will act in a simple way on Fourier developments

and will send Eisenstein series to Eisenstein series. combinations of the E, k,m,s

("old

Hence certain

forms") have Fourier coefficients which

can be given in a simple way in terms of the Fourier coefficients of Eisenstein series of lower index (compare equation ( 7 ) , where the coefficients of

E

K. , m

are simple linear combinations of those of

and we need only consider the remaining, "new", forms.

E

),

K. , x

A convenient

basis for these is the set of forms

(10)

E

( X )

k ,m 1

:=

Z

X ( S )

Vm.s

(

m

=

f

2

)

s(mod f) of index

f , where

X

is a primitive Dirichlet character (mod f) with

k X(-l) = (-1)

.

Then a calculation analogous to the proof of Theorem 2.1

for the case m = l shows that the coefficient (11)

e

k*m

if (r,f) = 1 , where and

e(X)

( n

'

r )

L^(s,X)

=

e

(

X

)

X

(

r

)

V_

4 n m

q Ç n

r

in

is given by

(2-k,X)

is the convolution of L^(s)

and

L(s,X)

a simple constant (essentially a quotient of Gauss sums 2

attached to X

and

X

—2

divided by L(3-2k, X

)) ; in particular, the

coefficients are algebraic (in Q(X)) and non-zero.

If

(r,f) > 1 ,

then

(X) e

k m^n'r^ "*"S given by a formula like (11) with the right-hand side

multiplied by a finite Euler product extending over the common prime

-27-

factors of

r

and

f.

If k = 2, then the Eisenstein series fail to converge; however, by the same type of methods as are used for ordinary modular forms ("Hecke's convergence trick") one can show that for X Eisenstein series

non-principal there is an

E» G j having a Fourier development given by ¿-, m, x ^>, m k

the same formula as for k > 2.

Since X

must be even (X(-l) = (-1) ) and

since there exists an even non-principal character or b 2 7, such series exist only for

m

(mod b ) only if b = 5

divisible by 25, 49, 6 4 , . .. .

There is one more topic from the theory of cusp forms in the classical case which we want to generalize, namely the characterization of cusp forms in terms of the Petersson scalar product.

T

=

u + iv

and define a volume element

(12)

dV

:=

(v > 0)

dV

on K

,

t

x

We write

z = x+iy

by

v ~ d x dy du dv 3

It is easily checked that this is invariant under the action of JCx C defined in §1

on

and is the unique G^-invariant measure up to a —2

constant.

(The form v

on 5C; the form normalized

du dv

v ~ dx dy 1

is the usual S L (IR)-invariant volume form 2

is the translation-invariant volume form on

so that the fibre

(C/ZT + Z

transform like Jacobi forms of weight

has volume k

and index

1.) m,

If

(J) and \¡)

then the

expression

k -4™,y /v ^ 2

v

e

( T ) Z )

is easily checked to be invariant under Petersson

(13)

scalar product

(4),*)

of

$

:=

and

J

F \JCx(D J

, so we can define the

ij> by

v

k

e

"

4 T r m y 2 / v

around z=0 ;

by forming certain linear combinations of the coefficients one obtains (D^

a series of modular forms with

2) $ Q

= a modular form of weight k + V.

The precise

For

polynomial

result is

THEOREM 3.1. p

2v ^

(1)

°^

variables

Ç2v)Hk-lj\

Then for

p

j G j^

y c(n,r)q ç n,r n

r

2v"

v£3N , 0

k6 I

define

a homogeneous

by

1 ) ( r

'

n )

(r)

a Jacobi

, the

function

=

coefficient of

form with Fourier

2 t

V

i n (1 - rt + n t

development

2

) " ^

-29-

(2)

i>

2

is a modular

*

:=

I ( I p^ (r,nm)c(n,r))q n=0 r 1 )

form of weight

k + 2 v on T,

n

If v > 0 , it is a cusp

form.

Explicitly, one has

3)^

=

I(Ic(n,r))q n r

=

1(1 ( k r - 2 n m ) c ( n , r ) ) q n r

=

l h ((k+1) (k+2)r n * r

2

(3)

(k-1) p ^

p - ( r n) P U,n; ( k

2

r is finite since

2

2

n

c (n,r) ^ 0 =» r < 4nm.

is given explicitly by

•= .

1 }

2 v

,

11

- 12(k4-l)r nm+ 1 2 n m ) c ( n , r ) ) q

k

Notice that the summation over The polynomial

,

n

T ^

(-if l i)

(

2

y ! ( 2 v

V

_

)

S

2 l i )

,

(k+2V-y-2) (k + v ~ 2 ) !

r

2v-2

y n

y

r

and is, up to a change of notation and normalization, the so-called Gegenbauer or "ultraspherical" polynomial, studied in any text on ortho(k-1) gonal polynomials;

w e have chosen the normalization so as to make p

2

a polynomial with integral coefficients in k,r,n in a minimal way (actually, ~ - times

p ^ ^

a function of r and n of the polynomial where

Q

p ^

would still have integral coefficients as

for fixed

k G I ) . The characteristic property

is that the function

p ^ "^(B(x,y) ,Q(x)Q(y)) ,

is a quadratic form in 2k variables and B the associated

bilinear form, is a spherical function of x and y with respect to Q (Theorem 7 . 2 ) . There is a similar result involving odd polynomials and giving modular forms 2)^,1)^ and replace

(k+v-2)!

, . . . of weight by

k+l,k+3, . . . (simply take

V G % +1^

(k+v- / ) ! in (1) and (3)), but, as we shall see, 3

2

-30-

this can be reduced to the even case in a trivial way, so we content ourselves with stating the latter case. As an example of Theorem 3.1 we apply it to the function

^

studied in the last section; using the formula given there for the Fourier coefficients of

E. - we obtain k,l

COROLLARY (Cohen [ 6 , Th. 6.2]) . be Cohen s function

Let

k

be even and

H(k-1,N) ( N G I ) 0

-

1

L_ (2-k)

if

N > 0, N = 0

C(3-2k)

if

N=0,

if

N = 1 or

N

or 3(mod 4) ,

! Then for each

v £ ]N

0

0 the

2(mod 4) .

function

r 1 4n is a modular form of weight k + 2y on the full modular group T . If x

it is a cusp

v > 0,

form.

Cohen's proof of this result used modular forms of half-integral weight; the relation of this to Theorem 3.1 will be discussed in §5. Yet another proof was given in [39], where it was shown that

C^ ^ V

has

the property that its scalar product with a Hecke eigenform f = Sa(n)q

n

G S^ 2 +

V

^

s

equal, up to a simple numerical factor, 2

to the value of the Rankin series

—S

2 a(n) n

This property characterizes the form

c£ ^ V

at

s = 2k + 2v - 2.

and also shows (since the

value of the Rankin series is non-zero) that it generates S^ 2-^ (resp. +

M^. if V = 0 ) as a module over the Hecke algebra; an application of this will be mentioned in §7. To prove Theorem 3.1, we first develop expansion around z = 0:

(j) (x ,z)

in a Taylor

-31-

(4)

(T,Z)

=

E

X

(T)Z

V

v=0

and then apply the transformation equation

to get

2 f£\ ( 6 )

X

/ax+b\ v ( ^ J

i.e. X

=

v

, ^.jxk+v/v , . , 2irimc \ V v-2

/^\ , 1 / 27rimc\ 27 v-4

v

(

c

T

+

d

)

T

)

+

X

( T )

+

X

V like a modular form of weight

transforms under

corrections coming from previous coefficients.

( T )

. •• ) '

+

\

k+V modulo

The first three cases

of ( 6 ) are

* (tSr) = 0

^(iff)

X(f^±|) 2

C« d) x (T) k

+

B

d)

=

< «

=

(cx + d )

+

X

k + 1

k

+

l (

T)

X (T) +

2

2

2irimc(cT+d)

k+1

X„(T) .

Differentiating the first of these equations gives X

o (fSJ)

=

kc(cx+d)

k + 1

X (T) + ( c x + d ) o

k + 2

x;(T)

,

and subtracting a multiple of this from the third equation gives

?

2

:

=

x . 2

*g5L ; x

e

M

Proceeding in this way, we find that for each

(7) U

;

E (T) V T

;

•= •

V* /

k

+

V

2

(D

the function

(-2^im) (k+v-y-2)! (k+V-2)! Y!

transforms like a modular form of weight

y

k-fv

.

on T.

(Y) \-2\i (

}

KT>

The algebraic

manipulations required to obtain the appropriate coefficients in (7) directly (i.e. like what we just did for v = 2 ) are not very difficult

-32-

and can be made quite simple by a judicious use of generating series, but w e will in fact prove the result in a slightly different way in a moment.

If <j) is periodic in z and has a Fourier development

£ c(n,r)qV, n,r

then

X

= ~ r l ( l (27Tir) c(n,r))q n r (8) V

V

n

and hence

so

(9)

^ . ( T )

-

( ,IR

2V (

k

2

^ - ; f

)

! ? 2 v

(x)

Thus Theorem 3.1 follows from the following more general result:

THEOREM 3.2. with coefficients holomorphic ?

v

defined

Let (J>(x,z) be a formal power series in z as in (4) X

which satisfy

v

everywhere

(including

by (7) is a modular

Let M^.

Proof.

(O

(6)

aZ-Z ^

d )

the cusps of T).

form of weight

G

^

(Note that M

"

a

p

e

.7%en tTze function

k + v on T.

denote the set of all functions

the conditions of the theorem.

an(2

1

(J> satisfying

(T) is isomorphic to

k,,m M

k 1 ^ via z t—> y/m z .)

we can split up M ^ ^(r) M

k m ^

Since

y

involves only

X^» with

v =v(mod 2 ) , !

into odd and even power series, say,

**k m ^ ® k m ^

=

£

and look at the two parts separately

M

(this corresponds to adjoining

-I

2

to F and looking at the action of (_l) M. (F) = M (F) ) . K.,m ic,m k

-I If

on <j); if T already contains - I . , then ^ ^ d) £ ML (T) , then Tc.m T

^ V ' ^ V

k

d> = zd) i

r

^

O

R

^

A

N

C

*

with d>, G M' , ( T ) i k+l,m

T

Y

and the functions

are the same except for the shift

k + l . Hence it suffices to look at

(F) .

differential operators . 3 3 87Tim -r— - — 7 3Z 2

L

:=

Q

8

T

2

V

->

V-l,

We now introduce the

-33(the heat operator)

and L

_ zkzi f

L

k The operator

L

z

.

8z

is natural in the context of Jacob! forms because it

acts on monomials

q £ n

by multiplication by (27Ti) ( 4 n m - r ) and hence,

r

2

in view of Theorem 2.2, preserves the second transformation law of Jacob! forms; this can also be seen directly by checking that

L(<j>| X)

(10)

=

'm If

L

(X e

(L(J))| X

IR ) 2

,

'm

satisfied a similar equation with respect to the operation of

SL (M),

then it would map Jacobi forms to Jacobi forms.

2

Unfortunately,

this is not quite true; when we compute the difference between L(<j>|.

K,m

M)

and

(L(f))L

M

0

KiZjii

we find that most of the terms cancel but ($1 ^ M ) (T ,z) ,

there is one term, 4Trim(2k-l) ^ ^ c

k = weight

in which case %

L

+

m

left over (unless

really does map Jacobi forms to Jacobi forms of

and the same index m ;

which are annihilated by

L ) .

examples are the Jacobi theta-series, To correct this we replace

L

by L^,,

which no longer satisfies (10) but does satisfy

(11)

L

k *lk,m (

M )

=

( L

k*>lk+2,m

as one checks by direct computation. nator,

CMeSL 0

K+z,m

1

A

Iterating this formula composite map

(D .

£ (8iTimX^ - 4 ( X + l ) ( X + k ) X , , ) z X>0

X

k

to

K,m

k,m Explicitly, we have

V

(D

A

V

A

+

times, we find by induction on v

2X

.

1

that the

­34­

^ v 2A > X^z X v

maps


2\) ^ ^ k + 2 v *

^^­s Proves Theorem 3.2 and hence also Theorem

3.1 except for the assertion about cusp forms.

B ut the latter is (k­1)

clearly true, because the constant term of (2) is p £

V

(0,0)c(0,0),

which is 0 for V > 0, and the expansion of jD^jfy at the other cusps is given by a similar formula applied to (j> | ^ ^M,

M £ Tj.

By mapping an even (resp. odd) function

({> €

m

( F ) to

(^ ,? ,^,...) (resp. to (? , C , . . . ) ) , we obtain maps 0

2

x

3

It is clear that these maps are isomorphisms: terms of £

one can express X

( 1 2 )

X

Y (T) v " (

i

)

Ц

V (2тг1ш) (^У-2и­1)! Л и ) Z­rf (k+v-u­1)! y! Ч ­ 2 у Oiy

-

v

?

is injective.

0

^

,

J k

(

r

~* V

)

m

r

)

.

\

0

Note that half of the spaces

in particular, for

V =T

or

<J> = 0 ( z

2 m + 1

),

+

i

(

r

)

m

•••

(r)

W

e

r

)

are 0 if - I ^ F ;

we have

l

1

= 0

0

zm

5dimM +dim Sk+2 4- . . . 4-dim S k

dimS,

-4-dxmS.

0

4...4dim

k

+

2

(k

m

(k

S. „ L

even) ,

odd) .

Here the second estimate cank+1 even be strengthened to k+2m-l k+3 (16)

dim

J

K ( F F I


*

=

© J. k,m > k

as given in Chapter III, to which the reader may now m

skip if he so desires (the results of §§4-7 are not used there).

As an

-38-

example, we now treat the ease m = 1, which is particularly easy and will be used in Chapter II.

0

s

Equations (15) and (16) (or Theorem 2.2) give

(k

odd) ,

dim

^ ^

dim M ^ + dim

On the other hand, the Fourier developments of

E^

and

l

^

E

g

even^ *

as given

1

after Theorem 2.1, show that the quotient

6 i '

E

( T

-i I " (144C + 456 + 144? ')q + ...

z ) =

E

( T

z )

>i depends on

z and hence is not a quotient of two modular forms, so the map

\ - 4

e

M

k - 6

~*

(F,G)

is injective.

J

k,l

F(T)E

Since

+ g(T)E

( T , Z )

dim M^_^ + dim M^_^ = dim M ^ + d i m ' S ^ ^

(this follows from the well-known formula for

THEOREM 3.5. free module

(T,Z) b, 1

(2) ,2) Q

E

#2

:

J k

)

l

—

\

3.1) is an

E

= E, • E

H

8,1

&

and 4

+

»

i

E 6

,

.

The map

i

S k

+

2

isomorphism.

In particular, we find that the space with generator

k

2

as in Theorem

2

^

d i m M ^ ) , we deduce

* +

r a

The space of Jacobi forms of index 1 on S L ( Z ) is a

of rank 2 over M , with generators

*>*

f°

,

(E

h,l

J

8, 1

is one-dimensional.

= 1 + 240q + . . .

U

the Eisenstein

n

series in M ^ ) , while the first cusp forms of index 1 are the forms

(17)

*IO,I

*

m

6

=

T77144

Y

i2,i

(

S \ I

E

"

(E E 2

v

k

k,i

V E , I >

- E

6

E

>

6,i

y

)

of weight 10 and 12, respectively (the factor 144 has been inserted to make the coefficients of

<j)

and

(J>

integral and coprime).

We

-39have tabulated the first coefficients of

<J>,

e

v

of

1

E,

(k= 4,6,8) and

1

c

1

(k = 10,12) in Table 1; notice that it suffices to give a single

K, I 1

sequence of coefficients

c(N)

(N > 0, N — 0,3(mod 4)) since by 2.2 any

Jacobi form of index 1 has Fourier coefficients of the form c(n,r) = c(4n - r )

for some

2

{c(N)}.

To compute the

use either assertion of Theorem 3.5, e.g. for

<j> , iQ i

c(N) , we can <J>

^ we can

12

either use (17) and the known Fourier expansions of

and

E^ ^

or

else (what is quicker) use the expansions

(18)

00

(A

=

00

q TT ( l - q ) n=l n

S c 1 rl < 2/n

2 i f

= £ T(n)q ) n=l n

(4n-r ) = 0 ' 2

i n

1

Yl c ]rl z

times the Weierstvass

C - 2 +

c""

1

p.-function

J O ( T , Z ) .

-40-

Indeed, since nowhere else in

(f)

vanishes doubly at z = 0 and (by Theorem 1.2)

10 l

C/Zx+Z, 4>

10

and since by (18)

=

1

(2TT1) A(x)z 2

2

+

0(z ) k

(19) (j)

12>1

=

12A(x) + 0 ( z ) 2

the quotient in question is a doubly periodic function of 12 double pole with principal part

—

z with a

-2

r- z

at z - 0 and no other poles

(2,i) in a period parallelogram, so must equal

^

2

r-^2.(x,z). (2iri) Finally, we note that, just as the two Eisenstein series 2

E,. , form a free basis of 6,i

ct>

form a basis of

JCUS^

J

over over

( f , g )

*»

the two cusp forms

d>

^

T

and . and

10,l

M , i.e. we have an isomorphism

*,1

'12,1

M, ,

E^

* '

«—>

r

f(T) (

(2)

: JC* C

I™

+d)_ke

(*5)e \M (D ad-bc = J, r i

- * ~

( 3 )

k

2



are well defined (i.e. on

^ and map

m

to

^ope^vely.

The well-def inedness and the fact that

$1 £> '$1V^, '((> JT^ V

transform correctly follow by straightforward calculations from the properties of the Jacobi group given in §1; the conditions at infinity will follow from the explicit Fourier expansions given below. Before proceeding to give the properties of the operators we explain the motivation for the definitions given.

U, V, T,

The operator

is an obvious one to introduce, corresponding to the endomorphism "multiplication by £" on the elliptic curve two,

C/ZT+22.

A S to the other

we would like to define Hecke-operators as in the theory of modular

forms by replacing

<j) by 4> | M,

where

M

runs over a system of represen-

-42-

tatives of matrices of determinant T .

elements of

l

£

modulo left multiplication by

Doing this would produce a new function transforming

like a Jacobi form with respect to

F .

However, since

l

<j)| [M]

is a

multiple of ,/aT + b

/1 \

/1

z

H^FTd ' ^FTd)

( 4 )

(M = ^

d)' ^

=

det ^ »

( | ) |[M] transforms in with

z

anc* ^

is in general irrational, the function

for translations by a lattice incommensurable

Zx +2£, and there is no way to make a Jacobi form of the same index

out of it.

However, by replacing

/£ by

ity; since this is formally the operator index by by

\

£.

£ , 2

£

in (4) restores the rational-

Uy^,

and

multiplies the

we obtain in this way an operator which multiplies indices

This explains the definition (2).

Finally, if

£

is a square

then the function (4) transforms like a Jacobi form with respect to translations in the sublattice

/£~(ZT + Z ) of

ZT + TL,

so we get a

function with the right translation properties by averaging over the quotient lattice.

This explains (3) except for the condition on g.c.d.(M), o

which was introduced for later purposes:

If we define

formula as (3) but with the condition " M

primitive" (i.e.

then

T^

and

T^

T^

by the same g.c.d.(M) = 1),

are related by

Eventually we want to show that the Jacobi-Hecke operators to the usual Hecke operators

T^

T^

correspond

on modular forms of weight 2k-2, and

equation (5) is precisely the relation between these Hecke operators and the corresponding operators defined with primitive matrices. Our main goal is to describe the action of our operators on Fourier coefficients and to give their commutation relations. with

and

since they are much easier to treat.

We start

-43-

THEOREM 4.2.

i)

(6)

|U

§ = 2 c(n,r)q ç . n

km

=

£

(with the convention

l k

Since

(j) | V

Q

V l

(

n

)

q

I

1

]

should belong to

Q=M^,

we

2k c, = - - — so that d> V is a multiple of the Eisenstein series 2k of weight k. This definition will be used later. take

k

Proof.

B

0

Equation (6) is obvious.

For (7) take the standard set

of representatives

(11)

^

for the matrices in (2).

,

Then

a,d>0

,

b(mod d) ,

ad = I

-44W|V )(T,z)

=

A

Y, ad=£

~

A

K

b(mod d) an

=

A*"

1

J

d"

]T

k

ad=£

b(mod d)

k-l V £ 2j

,1-k

0

=

ad=£

E

,

c(n,r) q

Ç

d

a r

e (nb) d

r

\ ^ /

.

v

c(n,r) q

an d ar Ç r

n,r n = 0(mod d)

k-1

^

a|£

/ n£

n,r

which is equivalent to (7).

£ n

V

\

an^ar

'

a

Equation (8) is obvious.

By (6) and ( 7 ) ,

we have

coefficient of

q £ n

r

in (4> I U ) | V^,

=

£

^

^

a

k

c^—j ,

1

a| (n,r,£ ) f

£lH a

0

!

if

£f r

a|(n,|,r) =

coefficient of

q C n

in

r

(0|V^ )|. t

Finally, using (7) we find coefficient of

q £ n

in (<j) | V ) | V^,

r

?

-E

E

a|(n,r,n

-

where

N(e)

I N(e) e ^

b

. f )

,

is the number of ways of writing

e

preceding sum.

1

c

|(^l,f,*)

(

^

as

If such a decomposition exists then

a*b

in the

e ] la

and hence

-45-

a =

v 6 for some integer (e, 36; b = e / a we find the formula

6; writing down the conditions on

e

(

0

6

N(e) = number of divisors

(=0 unless

e |(n£, n £ , £ £ ) , T

coefficient of

q ^ n

r

^ n , £ , £ , e,

2

£

oV

,

!

e |n££ ).

f

calculation just given for U

of

,

a

and

,

On the other hand, using the

f

we find

£ T

in

d

k

((|) | U ) | V

^

1

d

£ £ T

2

d| (£,£')

V*

,,k-l ä\a,V)

where now

?

elnd

we find

1

d

as

,

e

v6

\Ti5 c)

"l

< nM

rN

1

a | ( n , f , ^ )

N (e) counts the decompositions of

the conditions in the sum; from writing

k

we obtain for

N (e) T

e

as a d #

,

e

,

satisfying

divides

d,

and

(n,e) the same formula as for N (e).

This proves (10) (another proof could be obtained by combining the Corollary below with the multiplicative properties of ordinary Hecke operators) and completes the proof of Theorem 4.2. As a consequence of the formula for the action of V

£

on Fourier

coefficients we have the following COROLLARY.

where

T

£

For



(cx+d)

e

— 7

X

C T + d

L

zL /a b \

+ 2X

mz

mcz 7

C T + d

2

7 c T + d

»

J

Xez

d

\c ; where now

^

^

square­free.

r

(c,d) = 6

Then

It follows that of

runs over

^

a

^

divides

m

Now suppose

m

6' = |r is prime to

and

is 6.

can be multiplied on the left by an element a = b = 0(mod 6 ) ,

so as to make

r

I^XJMldet M = m } .

so

-\((" "„)• ««=•-.

- (S

W••«•­••=•>M».

and hence

E. J V k, 1 m 1

k _ 1

= m

V ô" /1,

k

6|m

v

V** e

Replacing

^

by

l

x

/

^ ^

(cx+d)~

k

(:SKi?)w« ^'

ax+b , ,

T~

cT+d

l)(c

0

d)

+

2 A

f

2

6 z cx+d

c z m ~| '

cx+d J

*

multiplies the final exponential

­48­

2

e(X 6'/6),

square­free).

„ К

f

and this is 1 only if 61X

( since (6,6 ) = 1

Therefore the terms with

6fX

, k­1 V ol­k V , ^ ,.­k "ST mfX v = m > о > (cT+d) > e — k,l' m Z­, L¿ L ô 6 Im Г00 \ Т 1. * XeôZ

and

5 is

can be omitted, so

2

2

ат+b , X z cz ­—- + 2 — 7 ­ 7 cT+d 6 CT+d ст+d

TT

0

1

= т У к_1

Ô

Z-j

1 ­ k

E, ( T , Z ) = k,m

a, (m)E, (T,Z) k­1 k,m

.

1

ôlm

This proves (12) ; the formula for the Fourier coefficients of E k,m

now

1

follows directly from Theorem 4.2. For m not square­free we find after a similar calculation the more general result

(13)

E

= m­

k m

TT ( l + P

k + 1

_

k

+

1

)

_

1

VW

£

)

\

U

5

ll d° J!L V

Eis yet more generally, we see that the space is mapped by U

£

and V

^

defined in Theorem 2.4

to the corresponding spaces with index m £

£

m£ , respectively, the precise formulas for the images of E^. m

U

£

and V

(terms

g

2

and

under

being easily ascertained by looking at the "constant term"

£

n

q ?

r

2

with

4nm=r ).

One then sees by induction that all

Jacobi Eisenstein series can be written uniquely as a linear combination of E ; |u O (£,£'e]N, X a primitive Dirichlet character modulo f K, m Xj Xj A ;

0

with

V ( M

k

X(­l) = (­l) ,

m=f

THEOREM 4.4.

Le t

2

E ^ as in (10) of § 2 ) . Combining these k,m remarks with the commutation relations of Theorem 4.2, we obtain

l d

J ° k,m 1

and similarly J

J

=

for jcusP>°­^ k,m

E £ , . 2; d m d>l

I

0

J, m I U o V ! £ d d —3­

£ I d k,­

0

T

ancf jEis,old ^ k,m

36

De fine

jcusP>new k,m

as ­^e

-49-

orthogonal

complément

of

scalar product)

and

(X

Dirichlet

a primitive

J^ ^ 1

J^^'

1 1

^

j£ ^ u

in

p

(with respect

as the span of the

character

to the

Petersson

functions

(mod f ) ) if

m = f

2

and

0

otherwise.

Then Eis

J

_

k,m

Eis,new

m

"

£

®

,

\,m/£'£ J £ £' 2

U

'

V

£ £'|m 2

ana

J

cusp k,m

_

cusp,new , Z-r k,m/£'£ £ £ £,£' £ £'|m J

2 1

U

V

?

2

Eventually, we will show that the latter decomposition is also a direct sum, but this will be harder and will depend on using a trace formula. It remains to describe the action of the Hecke operators the Fourier coefficients of Jacobi forms.

T

on

£

We do this only for (£,m) = 1.

The formula obtained involves the Legendre symbol

( j - ) if

£

is an

odd prime; in the general case it involves a slightly generalized symbol £^(n) , (D,n £ 7L) , which we now define. If

D

is congruent to 2 or 3 (mod 4) we set

£ (n) = 0 . D

For D = 0

we set

{ Otherwise

D

q

if

n= r ,

0

if

n^d

corresponding to

Then let Q(/D),

X

(p odd) ,

f^ 1

and

D

Q

is the

i.e. the multiplicative function with

X(2) =

1

0

2

can be written uniquely as D f

discriminant of (mod D )

r

0

B D D

0

0

S

1(8)

= 5(8) ,

o -

°< ) 4

X ( - D = sign D

-50-

e (n) D

The function

-

g|f,

if

n=n g ,

if

(n,f ) i •

2

o

n)

(|,

=1

0

,

o

.

2

e^, which reduces to X

if

D =D ,

occurs naturally

Q

whenever one studies non-fundamental discriminants; it has the properties e ( n + D)

=

D

e (n) D

and (as shown in [13], p.188), »

£ (n)

£ ^ n=l where

L (s)

= vs) •

n

is the L-series already introduced in connection with

Jacobi-Eisenstein series (eq. (6) of

§2 and the following formula).

We can now state the formula for the Fourier coefficients of

THEOREM 4.5. and index 4>|T

£

=

m

and

Let

£

é = X c(n,r)q ? n

a positive

2c*(n,r)qY

one checks that 2, a

only on

n

4nm - r

i.e. such 2

and on

N = (a,£) works.) in ip

r

r

Letting

(so that C(r,A)

depends

r € Z/2mNZ ) , we easily deduce that the coefficient C*(r, r - 4 m n ) with 2

^2 ' C(R

(19)

,

2

C*(r,A)

=

|

A)

R (mod 2Nm) r (mod 2m)

R=

(which depends, as it should, only on A and on this to

IL» =

^

r (mod 2 m ) ) .

we find that the coefficient of

q Ç

in

Applying <j> | T

£

is

given by

(20)

c*(n,r)

=

a ad=£

with n

C*

k

C*(r, r - 4 n m ) a

1

2

2

related as in (19) to the coefficient

(R -A)/4mçR

i

.

n

^ 2 ,a

C (R,A)

of

a

From eq. (18) we see that this coefficient

is given by „

/

A N

C (R,A) a

«

A / / n A((a,d),

A (a, 3 ) and

with the convention that

R -A 2

W

(

a

?

d

c (n,r)

)

j

are

\

/, R - A c(d ^ r r - , 2

0

T

dR\ ) ,

unless a, 3 , n

and

2

are integral.

Set a = (a,d);

then

ad = £

implies that N

a = xA~ a ,

for some

x,y€ W

with

d -= y„ 2 a ,

A

(x,y) = 1.

£ = xya ,

Then

(j¿,a)

(Ì y

'

(Ä,d)

X

r

-55-

x|R ,

Conversely, if

Hence (taking

x |A , - \ = ( | ) 2

A = x A ,

R = xr ,

A = r

Q

Q

N = (a,£) = xa

Q

Q

2

(mod 4m) .

(mod 4 m ) ,

then

in (19))

(21) r (mod 2am) xr = r (mod 2m) r = A (mod 4m) Q

Q

2

Q

if

A = x A 2

Q

and

C*(r,A) = 0

if

x fA 2

Note that, since m is prime to £ = a d 2

r

2

= A (mod 4m) and

The fact that

m

r' := r y = r y x Q

/ number

?

number

2

by hypothesis, the conditions r

2

Q

2

= A

Q

(mod 4m) .

is prime to £ also implies that the two numbers - 1

= r£/d

r -A

r and

determine one another (modulo 2m) and that the \

— 4 m ~ " ' o ^ / in the last formula is the same as the

c(n ,r') f

x ~ A = 2 or 3 (mod 4 ) ) .

r = A (mod 4) already assure that

2

cly

(or if

r

4'^)

in (14) (with

A = r -4nm). 2

]T

(22)

Moreover, we have =

£

A /

a

)

>

r (mod 2am) x r = r (mod 2m) r = A (mod 4) Q

Q

2

0

as we will show in a moment, and substituting this into the last equation and into (20) gives the desired formula (14), if we observe that

^ x

a

e

(a) = - — - e (ax ) o x a o 2

A

A x 2

= ~ e (a) . A

It remains to prove equation (22).

From the definition of A(a,$)

and the fact that a and m are relatively prime, we find that the left-hand side of (22) equals

-56-

Y | ot

r xr r = o

0 E

Q

2

(mod 2am) r (mod 2m) A (mod 4 Y ) o

ot

y N^(A )

The inner sum equals

ing equation (6) of §2.

^ Ç

K

N^(A )

with

q

as in the formula preced-

Q

That equation gives

oo

oo

E N

S

y=l

)

( A

O

) Y "

A

y

in

s

e (a)a

s

à

a=l

0

or (since the coefficient of n ~

E

=

L (s)

=

S

o

Ç(2s)/Ç(s)

is X (n) )

This completes the proof.

We remark that the hypothesis

(£,m) = 1 was not used in the

derivation of equations (20) and (21), so that we have in fact obtained a formula valid for arbitrary

*,

NT

X

a

k

£ and

X

^

m:

^

*( a

(23) *(n,r) = 2 Tz^y

V'"~~^"/ l

i_

a x = —,—jrr- , (a,£)

in (21).

J

y =

£ . (a,£) /

r+2mX\

2

~T~' Ti'

y

y

2

, _ . as before and we have set

r

o

ft

=

r + 2mX x

In particular, we find :

£ prime, £ | m

(24)

n+rX+mX

2

X(mod(a,£)) r + 2mX = 0 (mod x) n 4- rX + mX = 0 (mod x )

2

2

where

/

2

C

c

a|£

n+rX+mX \

S

=>

coefficient of

q Ç n

in <j) | T

r

c(£ n,£r) 2

c(£ n,£r) - £ 2

k _ 2

c(n,r)

c(£ n,£r) + £ - (£-l)c(n,r) + £ 2

k

2

2 k

-

3

^

£

if

£fr

,

if

£.|r, £ f n , (Bll2^1

c

,

X (mod £) | + | X + ^X =0(mod£) if £| r,n . 2

^f^)

CHAPTER II RELATIONS WITH OTHER TYPES OF MODULAR FORMS

§5.

Jacobi Forms and Modular Forms of Half-Integral Weight In §2 we showed that the coefficients

index

m

depend only on the "discriminant"

c(n,r) of a Jacobi form of r - 4nm and on the value of 2

r(mod 2 m ) , i.e.

(1)

c(n,r) = c ( 4 n m - r )

,

2

r

c ,(N) = c ( N ) r

r

for

r = r (mod 2m) . 1

From this it follows very easily, as we will now see, that the space of Jacobi forms of weight

k and index m

is isomorphic to a certain space

of (vector-valued) modular forms of weight

k-^

in one variable; the

rest of this section will then be devoted to identifying this space with more familiar spaces of modular forms of half-integral weight and studying the correspondence more closely. Equation (1) gives us coefficients all integers

0 satisfying

well-defined modulo 4m if

(2)

c (N)

(since y

:=

N = -y (mod 4m) 2

(notice that

y

2

is

c (^J^-

> )

(any

R

r€E Z ,

r = y(mod

2m))

re y

r = y(mod 2m) ; we permit ourselves the slight abuse of We extend the definition to all

N

by setting

c^(N) = 0

N ^ -y (mod 4 m ) , and set 2

oo

(3)

and

is given modulo 2 m ) , namely

is a residue class, one should more properly write

rather than notation).

y

c,^(N) for all y e Z/2mZ

h () T

M

:=

Z c (N) q N=0 u

-57-

N / 4 m

(y € Z/2mZ)

if

­58­

and z/ \ (4)

r% /_ \ V/ > > T

Z

2

< r /4m?

:=

r

r = y (mod 2m) (The

8^

are independent of the function

)(T

'

Z)

=

E

E

E

2

c (4nm­r )q.V y

rGZ n^r /4m r = y(2m) N+r

¿2 E

=

Then

2

y (mod 2m)

(5)

(j).)

li (mod 2m) r = y(2m)

L

h..(T) e

У

N

V> *

I

ro

m,y

2

4M

?

N2 0

(T,Z)

.

y(mod 2m)

Thus knowing the (2m)­tuple

(h ) , . _ of functions of one variable y y(mod zm; x

is equivalent to knowing that given any functions

Reversing the above calculation, we see h ^ as in (.3) with

equation (5) defines a function

c^(N) = 0

e

2

N ^ ­y (mod 4m),

z + X x + y

which transforms like a Jacobi form with respect to (X,y

for

Z ) and satisfies the right conditions at infinity.

In order for

(j) to be a Jacobi form, we still need a transformation law with respect to S L ( Z ) . 2

while

Since the theta­series (4) have weight

<j> has weight

k

and index

be modular forms of weight

h

k­%.

6

^

For the first we have

e

m,y

( T + 1

'

Z )

=

e

*n

( y 2 )

9

m,y

( T

>

z )

and (7)

H (T l) Y

+

=

must

To specify their precise transforma­

1



m,

m, we see from (5) that the hy

tion law, it suffices to consider the generators ^ of T .

and index

% (V) M

1> (T) u

,

and

^

-59as one sees either from the invariance of the sum (5) under or from the congruence

N = -y (mod 4m)

in (3).

2

x «—>x+l

For the second we have

as an easy consequence of the Poisson summation formula the identity

(8)

e J-I,f)

= V W M e ^ ^

m

J

e (- v)e 2m

y

m(V

(x, ) , Z

V (mod 2m) so (5) and the transformation law of

(9)

h (-1)

(f> under

-

(x»z) J—> ^- ~ ; , y ^

J e (yv) h (x) v(mod 2m) 2m

V 2 m T / l

give

.

v

We have proved

THEOREM 5.1.

Equation

(5) gives an isomorphism

the space of vector valued modular satisfying

the transformation

forms

( ) y( n

u

moc

between

[ 2m)

o

n

S L

laws (7) and (9) and bounded

J

fc

and

m

2^)

as Im(x)

°°.

When we spak of "vector-valued" forms in Theorem 5.1, we mean that the vector

h(x) = (h ) , , _ satisfies ' V r y (mod 2m) x

(10)

where

h(Mx)

U(M) = (U

=

( c x + d ) ^ U(M)h(x) k

(

M

(M) ) is a certain 2m x 2m matrix (the map

=

(c

d)

G

r

i

}

U: Y ^ -> G L ^ C )

is not quite a homomorphism because of the ambiguities arising from the choice of square-root in (10); to get a homomorphism one must replace r by a double cover). identify weight

k- h

in the cases

The result 5.1 would be more pleasing if we could

^ with a space of ordinary (i.e. scalar) modular forms of on some congruence subgroup of m = 1 and

case a little. of Theorem 5.1.

m prime, k

r .

We will do this below

even, and also discuss the general

First, however, we look at some immediate consequences

-60-

First of all, by combining (5) with the equations

0^

(T,Z) =

k

0

^ (T , - Z )

m

$ (T ,-z) = (-1) (j) (T , z)

and

(11)

h_^

=

(-l) h

we deduce the symmetry property

(y G 2/2mZ)

k

y

(this can also be proved by applying (9) twice), so that in fact (h ) reduces to an (m+1)-tuple of forms to an (m-l)-tuple

(h - h ) _ y -y 0 f (4x)h(x)

Tp:

M

^ -> M^_^

, := © M. ,

(h G M

We can now state:

l 9

*—2

h and

"

h G M^._^

h under

Tp

(k even) is given by

is a module over

f G M, ) .

then

over the ring of Hecke operators,

i.e. there is a 1-1 correspondence between eigenforms h G 2k-2

for

M,

by

and

agree for

-64-

THEOREM 5.4.

The

correspondence

n r

.2n

(15)-

gives an isomorphism is compatible

M^_^ and

between

with the Petersson

Hecke operators

9

r

^ (k even).

wtt/z t/ze

scalar products,

and with the structures

of

M

x

№£s

J

and

actions

of

. a s modules

L"k~^2

oyer

isomorphism

#,1

M .

Proof. (15) by

h(x)

D enote the functions defined by the two Fourier series in h (x) = 2 c^(N)q

0r,z) and set

and

N/4

with

y

2

N = -y (mod 4)

c(N) c (N) '

c(N) = c (N) + c (N)

so that

Q

By Theorem 5.1,

(V N)

l

0

otherwise

and

h(x) = h (4T) + h (4T) . Q

<j) is a Jacob! form if and only if

h

Q

and

h

satisfy

the transformation laws (7) and (9), which now become

(16)

These easily imply

(for the three-line calculation, see [40], p.385), and since the matrices ^

^

and

and hence

^ h G M^_^.

h e M^^,

Conversely, if

same calculation shows that Theorem 5.1 that

^ ( 4 ) , it follows that

generate

h

0

<j>(x,z) is in

and ^.

h

T

h G

^ ( ^ ( 4 ) )

then reversing the

satisfy (16) and hence by This establishes the isomorphism

­65­ claimed.

To see that it preserves the Petersson scalar product

(up to a constant), we combine Theorem 5.3 with the fact that (h,h) = const. [ ( h , h ) + ( h , h ) J Q

0

1

1

for

h

and

h^

related as above.

This fact follows easily from Rankin's method, which expresses (h,h) as ( ^ | c ( N ) | N J and similarly for h and h ~ ^N>0 (for more explicit formulas, look at the proof of Corollary 5 in [13],

a multiple of

2

Res

S

Q

1

S

2

l

/

pp. 189­191, and take residues at

s=l

in the identities proved there).

The compatibility with Hecke operators is clear from (14) and Theorem 4.5. The compatibility with the structures of v

over

J

M , , 2*-h

and

0

J

. as modules *,1

is also clear. Theorem 5.4 tells us that the spaces

J

_. and

M„ , , are in

*,1 some sense -identical:

l*~H

they are related by a canonical isomorphism

preserving their Hilbert space structures, their structures as modules over the Hecke algebra and their structures as modules over the ring of modular forms of integral weight, and such that the Fourier coefficients of corresponding forms are the same (up to renumbering).

Combining this

with results on Jacobi forms proved earlier, we can obtain various (previously known) results about forms of half­integral weight, in particular the following two: COROLLARY 1. Theorem

2.1.

lies in

М^_^ •

Le t

The n the

k^4

be

e ve n and

H(k­1,N) = L_ (2­k) N

as in

function

This was proved by H.Cohen for both odd and even

k

in [6]; it was

Cohen's discovery of the existence of Eisenstein series, half of whose Fourier coefficients vanish, which led to Kohnen's definition of the space

M^_^(4).

Cohen's calculation of the coefficients of the two

-66Eisenstein series in M ^ _ ^ ( r ( 4 ) )

is of the same order of complexity as

Q

the calculation leading to Theorem 2.1, but our proof makes it clearer where the condition

-N = • (mod 4)

comes from.

does not apply at all to the case of odd

COROLLARY 2. generators

3 f and

EL K

5

.

On the other hand, it

k.

is a free module of rank 2 over M , with t

.

This was proved by Kohnen [11, Prop. 1 ] , who also gave the corresponding result for M

,

(it is also free, with generators

9

and 7C ) ; our proof, which is a restatement of Theorem 3.5, works only 2

for even

k.

Conversely, by combining Theorem 5.4 with Kohnen s results as 1

quoted above, we obtain

COROLLARY 3. operators

If

<j) E

1

is an eigenfunction

T^, then there is a Hecke eigenform

same eigenvalues.

{eigenforms in

is bijective* there is a map

The

in 2 k - 2

More precisely,

> {normalized eigenforms in ^2k-2^

for fixed

: J, -, ->- M

n

Q

and r

Q

with

All of these maps are compatible of them is an

N = 4n -r Q

2

> 0,

given jby

(n'.r'e Z ,

combination

the

M

correspondence

^}/scalars

M

of all Hecke

with Hecke operators,

4n -r f

f 2

= il N/d ). 2

2

and some

isomorphism.

The second statement follows from the explicit description of the

-67-

correspondence

^ -> 2 k - 2

and from the corresponding result for

M

half-integral weight, namely that the map

sends M ^ _ ^ to M ^ _ 2

the g

for all N > 0 and that some linear combination of

2

is an isomorphism ([11], Theorem 1, iii; Kohnen states the

result only for

-N a fundamental discriminant).

However, it also

follows from the first statement of the Corollary and from Theorem 4.5, by noting that the map M

is simply

n

o» o

Finally, by combining Theorem 5.4 with the main theorem of [13], which is a refinement for modular forms of level one of a theorem of Waldspurger [36,37] one gets:

COROLLARY 4. f G 2k-2 S

(f =

§ G j£ ^ U

oorresponding

Then for all

where

Let

n,r with

be an eigenform

P

normalized

of all

T^ and

eigenform as in CoroIlary 3.

r < 4n, 2

L(f, e^, s) denotes the twisted L-series

Xe (n)a(n)n~"

s

D

Sa(n)q ). n

We mention one more result about the correspondences

THEOREM 5.5.

Let

<j> G

to one another as in Theorem

1

and

5.4 and

h G M _^ k

v^O

be forms

an integer.

J^_^ ~>

corresponding

Then

-68D

where 2)^

V

W

=

F (e,h)|u v

is the Taylor development

,

1 (

operator

of §3,

F

is Cohen's

v

2

operator as defined in the Corollary and

is the operator

to Theorem

3.3,

6(x) = 2 q

£ c(n)q »—» 2 c ( 4 n ) q . n

n

This can be checked easily by direct computation. case

(j) =

^,

h=5C^ ^

to Theorem 3.1. defining it as

gives Cohen's function

(Indeed, Cohen proved that F (0,7C _ )|U v

,

k

1

The special as in the Corollary

C ^ ^ belongs to M ^ ^ y V

^

.)

I+

We now turn to the case of general m, where, however, we will not be able to give such precise results as for m = l .

By Theorem 5.2 we have

a splitting

(17) where

e runs over all characters of the group

E = {£ (mod 2m) I £ = 1 (mod 4m)} = (m/2xf 2

with

£ (-1) = (-1)

k

and

£ J. k,m

± . . . ± J. ' * * * ' k,m

(which was denoted

in the remark

following the Corollary to Theorem 5.3) is the corresponding eigenspace, i.e. if to

(h,,) , , s are the modular forms of weight M "M (mod 2m) 0

<J) € J^.

m

then

tiQj = £ ( O h ^

for all ^ € H,

k-%

associated

In particular, the map

(18)

(which generalizes the map

<j>

Theorem 5.4) annihilates all

h ( 4 x ) + h (4x) Q

k,m prove the analogue of Theorem 5.4:

i

with £ ^ 1.

used in the proof of

For this map we can

-69THEOREM 5.6. If

The function

m is prime and

between

m

^th p i ouv

k even, then the map (18) defines an

and the space coefficient

eT

h(f) defined by (18) lies in

isomorphism

M^_^(m) of modular forms in

is zero for all

N with

M^^Cm).

M^^On)

whose

("—") = -!.

More generally, the image of (18) is contained in the subspace M^_^

(m)

whose N

consisting of all modular forms of weight

th

Fourier coefficient vanishes unless

and for m and

r (4m)

on

Q

-N is a square modulo 4m,

square-free the map (18) gives an isomorphism between

M, V * (m) K-h

spaces

k- ^

M^_^

(note that

J, = J. k,m k,m

for m prime and

of modular forms

2 c(N)q^

were studied (for m

k even). '

F

(in), and more generally the spaces M^/^.

^

(m)

The

consisting

with prescribed values of ( )

for

p|m,

odd and square-free) by Kohnen [12], who showed

that M _^,(m) decomposes in a Heeke-invariant way into the sum of these k

spaces.

It is tempting to assume that this decomposition corresponds to

the splitting (17) of

but this is not the case.

m >

We discuss this

in more detail below after proving Theorem 5.6.

Proof (sketch):

It follows from equation (9) that

is a multiple of h ^ ( T ) and hence invariant under the invariance of

h

h G M

We have

k—2"

(F (4m)). o

itself under

h = 2 c(N)q

c^(N) is non-zero only for in M | for m

* . +

k~^2

square-free it is infective on

this shows that

c ^ ( N ) = c^(N)

J,

1

.

jf

K,m

for all

since

h

lies

with £ # 1 , but 6 €E J

Indeed, for

k,m

J

implies that of

li = -N (mod 4 m ) , 2

this and

c(N) = 2 c ( N ) ; y M

with

N

Tf—> T+l;

suffice to show that

The map (18) is clearly zero on all

H

we have

T I—> T+l

T

1

k,m

Y

^ G H , and for m square-free this

Cy(N) is independent of the choice of the square-root

-N (mod 4m);

hence

c(N) = 2 c^(N) t

and therefore

y

h = 0 =>(j> = 0.

Finally, the surjectivity claimed in the theorem results from the formula (19)

dim Jk m

=

dim M2k-2^r*^m^

^m Prime>

k

even)

-70which will be proved in §§9-10 and the isomorphism r*(m)

proved by Kohnen [12], where SL (R).

M^_^(m) =

large, but the inequality with

T (m)

denotes the normalizer of

(Actually, (19) will be proved in §§9-10 only for

2

(m))

= replaced by

k

in

Q

sufficiently

£ will be proved for all

k,

and that is sufficient for the application here; it then actually follows from Theorem 5.6 and Kohnen s work that one has equality in (19) for T

all

k.)

This completes the proof.

Theorem 5.6 and the remarks following it describe the relation between

^ and forms of half-integral weight.

As we said, the

£

situation for the other eigenspaces

^

is more complicated; the

authors are indebted to N.Skoruppa for the following remarks which clarify it somewhat. Suppose that

m

character as above.

{C (mod p) | £

is odd and square-free and let Write

= 1 (mod p)} =

£ = TT P|

£

with

£

£: H •> {±l}

a character on

m

and let

TLllTL

We extend

f

be the product of those

for which

£p is non-trivial.

by taking

£ to be the product of arbitrary odd characters

for

be a

£ to a character

£ (mod f) £p (mod p)

p I f and define

(thus

h~ is our old i)

h

h

for £ = 1 ) .

Then

lies in M ^ C r ^ m O . X ) ,

g

where

X = (^) £

;.

k

£ on all J, k,m f

ii)

the map r

Y

h~ £

is infective on J

iii)

is

0

such that the representation of 2

and

£ J. ; k,m

the image of this map is the set of

SL (Z)) on

for £ ^ £ T

^ Y E SL CZ)

* lY

c

h

h

in

M^_^(r (4mf),X) Q

S L ( Z ) (= double cover of 2

is isomorphic to

C

£

(or to 0 ) ,

2

where

C

£

is a certain explicitly known irreducible

representation of

SL (2£)/Y (4m) . 2

p

-71This set can be characterized in terms of Fourier expansions and its dimension (resp. traces of Hecke operators) explicitly computed; it is contained in but in general not equal to the space of modular forms in M, ^ ( F (4mf) ,X)

2-/ c (N)q^. -N = D(mod4m) Notice that there is a choice involved in extending £ to £ but 0

2

with Fourier expansions of the form

that the different functions twists of one another. arbitrary (if m

h~ which are obtained in this way are

Skoruppa also has corresponding results for

m

is not square-free, then one must restrict to the U-new

part, i.e. the complement of the space / -4 \ then we must take

e

2

to be ( — )

© C

/d l^d »

if m

2

m

is even,

^ ^ >

of conductor 4 rather than 2 and thus

to be twice as large as above; the level of

hg

in general is four times

2

the smallest common multiple of m and f ) and has given a formula for dim jf for arbitrary m and £ [34]. k,m J

We mention one other result of Skoruppa, also based on the correspondence between Jacobi forms and modular forms of half-integral weight:

THEOREM 5.7 ([34]):

J, = {0} for all 1 ,m

m.

The proof uses Theorem 5.1 and a result of Serre and Stark on modular forms of weight 1/2. Finally, the reader might want to see some numerical examples illustrating the correspondence between Jacobi forms and modular forms of half-integral weight. at the end of §3.

Examples of Jacobi forms of index 1 were given

According to Theorem 5.4, these should correspond to

modular forms in M

, , so that, for instance, the coefficients given

in Table 1 should be the Fourier coefficients of the Eisenstein series in

(k = 4,6,8)

and cusp forms in M^_^

(k = 10,12);

one can check

that this is indeed the case by comparing these coefficients with the table in Cohen [7]. Numerical examples for higher

m

illustrating

Theorem 5.6 and the following discussion can be found at the end of §9.

f

-72-

§6.

Fourier-Jacobi Expansions of Siegel Modular Forms and the Saito-Kurokawa Conjecture We recall the definition of Siegel modular forms.

upper half-space of degree symmetric n x n matrices

n

is defined as the set JC

n

The Siegel of complex

Z with positive-definite imaginary part.

The

group

sp

2 n

(i)=

{ M ^ M ^ W l M j / ^ g

= j(£

^

,

J

2 n

=

I A,B,C,D G M ( m ) » A B = B A , C D ^ D C , A D - B C = I | t

t

1 1

t

t

n

n

acts on 3 C by n J

D)

(c

#

Z

(AZ + B M C Z + D ) "

=

a Siegel modular form of degree full Siegel modular group F: JC — > n

€

and weight

= Sp ^(S)

n

k with respect to the

is a holomorphic function

2

satisfying

F(M«Z)

(1)

for all

F

n

;

1

Z G 3C

n

and

=

det(CZ + D )

M = ^

e V^.

^

F(Z)

k

If n > 1, such a function will

automatically possess a Fourier development of the form

(2)

where the summation is over positive semidefinite semi-integral (i.e. 2t.., t. . G z) n x n matrices lj' n

T; if n = 1, of course, this must be posed

as an extra requirement. If n = 2 we can write Im(z)

2

< Im(T)Im(T )

we have

r

Q

and we write

f

T = f \r/z

Z as

^ ) with m / 2

^

with

F(t,z,T )

n,r,m 6 2 ,

?

T,T' G 5 C ,

instead of

n,m 2 0,

r

2

z G C,

F(Z);

< 4nm

similarly

and we write

-73-

A(n,r,m)

for

(3)

A ( T ) , so the Fourier development of

F(T,z,T' )

=

^

^

F

becomes

A(n,r,m) e(nT + r z + m T ) f

n,r,i€I n,m,4nm-r >. 0 2

The relation to Jacob! forms is given by the following result, which, as mentioned in the Introduction, is contained in Piatetski-Shapiro s 1

work

[26].

THEOREM 6 . 1 . degree

2 and write

Let

F be a Siegel modular

the Fourier

development

form of weight

of

k

and

F in the form

oo

(4)

Then

F(T,z,x )

=

f

£ m=0

4> ( > > ^ ' ) T

<j> (T,z) is a Jaoobi form of weight m

Proof.

For

(

ca

" ) G

and

z

e

'

m T

m

k

and index

(X y) G l

2

m.

the matrices

and

(5)

belong to

F

£

and act on 3C

2

(T,Z,T')

respectively.

Applying

by

*—>

(1),

(T, Z + À T + y, T

Fourier-Jacobi

expansion

+2XZ +À T)

<j>; the condition at infinity m

Following Piatetski-Shapiro, we call (4) the

of the Siegel modular form

F.

Note that the proof of Theorem 6.1 still applies if by a congruence subgroup.

,

2

we deduce the two transformation laws of

Jacobi forms for the Fourier coefficients follows directly from (3).

!

F

2

is replaced

Note, too, that the first collection of

-74-

matrices in (5) form a group (isomorphic to

S L Z ) but that the second 2

do not; the group they generate has the form

2

so that we again see the necessity of replacing group 3C_. The complete embedding

of

%

by the Heisenberg

SL_ (3R) H 3C

iL

into

TO

IK

Sp (1R) k

is given by

where

(A

1

y ) = (A y) ^

^ ^.

f

However, we shall make no further use

of this. The real interest of the relation between Jacobi and Siegel modular forms, and the way to the proof of the Saito-Kurokawa conjecture, is the following result, due essentially to Maass [21].

THEOREM 6.2. Then the functions coefficients

Proof.

Let §|V

§ be a Jacobi form of weight m

(m^O)

of a Siegel modular

form

V§

in §4 are the of weight

and index

1.

Fourier-Jacobi

k and degree 2.

Reversing the proof of Theorem 6.1, we see that the

function defined by ( 4 ) , where k

defined

k

cj> (m^ 0) are any m

Jacobi forms of weight

and index m, transforms like a Siegel modular form under the action

of the matrices (5).

In particular, this holds for the function

On the other hand, Theorem 4.2 gives the formula

(6)

­75­ for the Fourier coefficients (defined as in (3)) of V§ n

^ ^ c(n,r)q Ç n ^ r /4m

r

is the Fourier expansion of

, where



2

J

x j

k ; 0

k

j

l

x J

k ; 2

x ...

and a map

v--

J k

(

i

№

— »

such that the composite

is the identity. of

F

with

Thus V

is injective and its image is exactly the set

F = #/( =

(V m ) .

m

that the Fourier coefficients

A(n,r,m)

(defined by (3)) are given by

the formula (6), where ( A(n,0,l) (8)

c(N) =

This means

if

N = 4n

if

N = 4n­l

I ( A(n,l,l)

^

-76(The last statement could be omitted, since the validity of (6) for any sequence of numbers je(N)} forces the

c(N)

to be given by this formula.)

Equivalently, we can characterize these functions by the Fourier coefficient identity

(9)

M^(T)

The space

2

of functions satisfying (9) was studied by Maass

([20], [21]; see also [17]) after the existence of Siegel forms satisfying these identities had been discovered experimentally by Resnikoff and Saldana [27] (whose Tables IV and V should be compared to our Tabled 1 and 2 ) ; he called it the "Spezialschar".

We thus have established

inverse isomorphisms

combining these with the isomorphism

COROLLARY 1.

If

h(x) =

^-^-~» M^_^

^

c(N)q

given in §5 we obtain

is a modular

form in

N = 0,3(mod 4) Kohnen s "plus-space" 1

equation

(6) are the coefficients

"Spezialschar"

M£(T ), 2

then the numbers

M^(F ). 2

The map

A(n,r,m) defined by

of a modular

form

F in Maass'

h »-> F is an isomorphism

the inverse map being given in terms of Fourier

by equation

from M^_^ to

coefficients

(8).

From Theorem 3.5 (or Corollary 2 to Theorem 5.4) we also deduce:

COROLLARY 2.

The "Spezialschar"

M*(r

over

M (SL.(Z)) on two generators,

© M*(r k even 4 and 6 .

) = 2

of weights

K

2

)

is free

-77This result was proved by Maass [21], [22]. Finally, we must check that the map V action of Hecke operators in I: T

algebra map of weight weight

•>

g

and index

1 such that

W)|T

=

the chapter we write and

TTj) .

T (£)

T g ( p ) with 2

and

g

g

maps Hecke eigenforms to Hecke

It is well known (cf, Andrianov Tg(p) and

V T G TT .

M<J>|l(T))

This will imply in particular that V

by the operators

i.e. that there is an

2 to the Hecke algebra for Jacobi forms of

(10)

eigenforms.

M^CI^),

from the Hecke algebra for Siegel modular forms

k and degree

k

^ and

is compatible with the

Tj(£)

[1]) that

p

TTg

prime (until the end of

for the Hecke operators in TFg

We use the somewhat more convenient generators

Tg(p) = T ( p ) g

2

T (p) g

and

- T (p ). 2

g

THEOREM 6.3.

The map

Pi

-> M ^ ( F )

1

i: TTg -> TTj. defined

on generators

is

2

(in the sense of (10)) with respect

Heoke-equivariant

to the homomorphism

of Heeke

algebras

by

i(T (p))

=

Tj(p) +

, / w l(Tg(p))

=

/ k-1 , k-2s m / n , o 2k-3 , 2k-4 (p + p ) T ( p ) + 2p + p

s

Proof.

is generated

k P

_

1

+ p ~ k

,

2

J

In [2], Andrianov proves that Maass' "Spezialschar" is

invariant under the Hecke algebra by calculating the Fourier coefficients of the two functions F

for

F G M^

x

=

F|T (p) s

,

F

2

=

F|Tg(p)

and checking that they satisfy the identity (9).

Spezialschar is just the image of V,

this says that for

<j) G

Since the ^,

-78-

for some

b >t




2

t

^ ^

1

explicitly given coefficients, so to prove

the theorem we need only check that

Х

=

Ф|(Т (Р) + Р " '

, Ф

=

Ф| ( ( р

Ф

Л

1

К

2

+ Р " )

,

(11) 2

к

"1

. к-2 , ч , 2к-3 ^ 2к-4 + Р )Tj(p) + 2р + р ) Л г г

0

ч

.

The coefficients of a Siegel modular form in the Spezialschar are determined by a single function If

c ( N ) , C j ( N ) , and

way to

(j), (j> and i

c (N) 2

c(N)

(N=0,3(mod 4))

are the coefficients corresponding in this

(f> , then equations (13)- (16) of [2] (with D = - N , t=l, 2

d=0) give

if

p fN,

if

p

2

if

p

2

if

p j N. In a unified notation, this can be written

2

I N,

f N,

as in (6),(8).

and

2I

-79-

with the usual convention

c ^~^r) = 0

from Theorem 4.5 with m = 1, i = p ф I Tj_ (p)

if p f N. 2

On the other hand,

we find that the N ^

coefficient for

is given by

(cf. equation (14) of § 5 ) , and comparing this with the formulas for and

c

2

gives the desired identities (11).

In [1], §1.3, Andrianov associates to any Hecke eigenform F e S (F ) k

where

the Euler product

2

F|T (p) = Y F , s

p

F|T^(p) = YpF.

If

7 =Щ

with

|Tj(p) = Л ф , р

then it follows from Theorem 6.3 that

and hence

By Corollary 3 of Theorem 5.4, there is a 1-1 correspondence between eigenforms in M^^ deduce:

2

and

^,

the eigenvalues being the same.

We

c

1

-80-

COROLLARY 1 (Saito-Kurokawa Conjecture). spanned

by Hecke

normalized

eigenforms.

Hecke eigen forms

The space

S

£(T )

These are in 1-1 correspondence f £

2 » ^he correspondence

is

2

with

being

such

that (12)

Z (s)

=

F

Ç(s-k+l)Ç(s-k+2)L(f,s)

As stated in the Introduction, most of the Saito-Kurokawa was proved by Maass

[21,22,23].

In particular, he found the bijection

between functions in the Spezialschar and pairs of functions satisfying

conjecture

(16) of §5, showed that

dim M^(T^)

= dim 2 k - 2 ' M

(h ,h ) 0

anc* ^2\a-2

established the existence of a lifting from the Spezialschar to satisfying

(12), without, however, being able to show that it was an

isomorphism.

The statement that the Spezialschar is spanned by eigen-

forms was proved by Andrianov

[2] in the paper cited in the proof of

Theorem 6.3, and the bijectivity of M a a s s authors

1

[40].

See also Kojima

[14],

lifting by one of the

1

A detailed exposition of the

proof of the conjecture (more or less along the same lines as the one given here) can be found in [40]. A further consequence of Theorem 6.3 is

COROLLARY 2.

The Fourier

(sum over non-assodated in

M^(T ) 2

coefficients

pairs of coprime

of the Eisenstein

symmetric

matrices

series

C,D e M (2)J 2

are given by

(2) Indeed, since with

A(^|J Q ) ) = 1 ,

is the unique eigenform of all Hecke operators it must equal

j) >

and the result follows.

-81-

The formula for the coefficients

A(T) of Siegel Eisenstein series of

degree 2 was proved by Maass ([18], correction in [19]), but his proof involved much more work, especially in the case of non-primitive

§7.

T.

Jacobi Theta Series and a Theorem of Waldspurger Perhaps the most important modular forms for applications to

arithmetic are the theta series

(i)

where

Q

is a positive-definite rational-valued quadratic form on a

lattice A %rk(A)

of finite rank.

This series is a modular form of weight

and some level; in the simplest case of unimodular

Q

(i.e. Q

is integer-Valued and can be written with respect to some basis of A \ x Ax

with A

t

as

a symmetric matrix with integer entries and determinant 1)

it is a modular form on the full modular group.

As is well known, the

theta-series (1) should really be considered as the restriction to z = 0 ("Thetanullwert") of a function 0 ( T , z ) which satisfies a transformation law for

z i — > z + AT + ]i

and is, in fact, a Jacobi form.

(This fact,

which goes back to Jacobi, is the primary motivation for the definition of Jacobi forms.) modular

We give a precise formulation in the case of uni-

Q.

THEOREM 7.1. form on a lattice with

Let A

Q(x) be a unimodular -positive definite

of rank

Q(x) = h B(x,x).

2k and

B(x,y)

Then for fixed

y£A

the associated the series

quadratic

bilinear

form

-82-

(2)

is a Jacobi form of weight

Proof.

k

and index

m=Q(y)

on

SL (Z) . 2

The transformation law with respect to

is obvious and that for ^ ^

^^

l) ^

0

2

x

The transformation law with respect to

is equally clear:

e (A T + M

( Z )

are particuH »

z i—> Z + A T + U

2

an immediate consequence of the Poisson

summation formula, so that the modular properties of larly easy in this case.

S L

2Az)0

( x , z + A T + p)

xGA

Finally, as pointed out in the Introduction, the conditions at infinity 2

are simply

Q(x)>0,

Q(y)>0,

fact that the restriction of 2 x 4- TLy C A If

Q

is positive

Q

B(x,y)

< 4Q(x)Q(y),

which express the

to the (possibly degenerate) sublattice

(semi-)definite.

is not unimodular, then

0^ Q,x

will have a level and

character which can be determined in a well known manner. We recall two generalizations of the theta series (1) and explain their relation to the Jacobi theta series ( 2 ) . insert a spherical polynomial (recall that "spherical" means with respect to a basis of

P(x)

in front of the exponential in (1)

AP = 0, where

A®3R

for which

homogeneous of degree 2y, then the series (3)

First of all, one can

A Q

is the standard Laplacian is

Sx

2

):

if P

is

-83is a modular form of weight

k + 2v

(2k = rk A ) and the same level as

and a cusp form if V > 0 (see for example Ogg [25]).

THEOREM 7.2. the

Let

A,Q,B,y

be as in Theorem

0^,

We then have

7.1,

v G ]N . Q

Then

polynomial

(4)

(

^

as in Theorem

Proof.

3.1) is a spherical polynomial

2v

and

The final formula is clear from the definition of - ^ v *

and this makes it morally certain that P^ ^ to Q,

of degree

since (3) is never modular unless

is spherical with respect

P

is spherical.

To check that

this is really so, we use eq. (3) of §3 to get

?

v

y (x)

=

(we may assume y ^ 0, (choose a basis so

constant x coefficient of

2 V

in Q ( y - t x )

^

so m ^ 0) , from which the assertion follows easily

Q = Ex.

2

and compute

1

recalling that

t

V

^ ) Q(y - tx) 3x /

,

2

k = % rk A ).

The other generalization of (1) is the Siegel theta series

(5)

where

r^(T)

form T

by

is the number of representations of the binary quadratic Q

(just as the coefficient of

representations of we have

n,

q

11

in (1) is the number of

or of the unary form n x , by 2

Q).

Explicitly,

-84-

Then the following is obvious.

th THEOREM 7.3. Jacobi

Let

coefficient

of

Q be as in Theorem (2) 9^ equals

7.1.

Then the m

Fourier-

Theorems 7.2 and 7.3 show how the Jacobi theta series fit into the theory developed in §3 and §6, respectively. is also easily described: to

0

the modular form of weight

y.

Q

associated

to the orthogonal complement

We illustrate this in the simplest example, with

Then (A,Q) is the

The form

0^

Aut(Q).

and since

J

E

is

4,1

k =4

and

m = 1.

lattice (cf. Serre [ 2 8 , 5 , 1 . 4 . 3 ] ) , i.e.

g

(since

240 choices of y under

k-^

is essentially the theta series associated to the quadratic

r t

form of rank 2k-l obtained by restricting of

Their relation to §5

with

dim

Q(y) = 1,

we can choose

=1);

in particular, there are

but since they are all equivalent

y=(%,...,h).

Then

is one-dimensional this must equal

formula for the coefficient of

q Ç

or, more explicitly

x^

(replacing

n

r

by

in

E

fc

±

h x ),

E

4, l

.

Hence the

(Theorem 2.1) implies

­85­

(6)

G Z

#{x

= 8n, 2 X = 4r}

2

= . .. = x (mod 2) , 2 x

I

g

= ­252 H ( 3 , 4 n ­ r ) . 2

This first seems like an infinite family of identities, parametrized by r E 2£, but in fact the identities depend only on if

r

is even then replacing

x^

x^ 4- \ r

by

of (6) by the same expression with so (6) for any even

n

r (mod 2 ) .

Indeed,

replaces the left­hand side

replaced by

n ­^ r

2

and

r by

0,

r is equivalent to the theta series identity

(7) X G F , X X, =

+

1

...+X

. . . =

X.

=0

8

(mod 2) (= l + 1 2 6 q + 756q

Similar remarks hold for 8

replaces

"x e E " by

this condition into for odd

r

r

2

+ 2072q

3

odd except that now replacing

"xe

8

(Jg,...,%)+ Z

8

" x G I , all

x^

" ; doubling the

odd, x == . . . = x x

Q

+

...)

x^

by

x_^+%r

x

then turns

(mod 4 ) " ,

so (6)

is equivalent to

(8) 8

-хЕЖ

..,+х

,х + х

8

=

О

x.¡_ odd, x = . . .= х (mod 4)

Thus the Jacobi form identity

0_ = E Q,y

modular­form­of­half­integral­weight the integer­valued quadratic form 8

{x G Z | S x = 0, x = . . , = X ±

x

V

is equivalent to the k,i

identity 8

2 x

2

0^,=H , 3

where

Q

on the 7­dimensional

F

is

lattice

(mod 4) }.

We end this section by combining various of the results of Chapters I and II to obtain a proof (related in content but different in presentation from the original one) of a beautiful theorem of

-86-

Waldspurger s

[35], generalizing Siegel's famous theorem on theta-series.

1

Siegel's theorem [33], in the simplest case of forms of level 1, says

(k e 4 Z )

(9)

Here

(A^,Q^)

(1 < i < h)

denote the inequivalent unimodular positive-

definite quadratic forms of rank of

Q^.

Explicitly

2k

and

the number of automorphisms

(9) says

(10)

(n> 0)

-1

w.

(e. :=

ZTTT

)>

i.e. it gives a formula for the average

w, + . . . + w,

1

h

1

number (with appropriate weights) of representations of an integer the forms

Q^.

n

by

The number of representations by a single form, however,

remains mysterious.

Waldspurger s result in this case is 1

(11)

where

C (x) = fc

(H _ (x)6(x))|is k

1

Cohen's function.

Thus one has

explicit evaluations of weighted linear combinations of theta-series CjJ

with variable weights; in fact, since the

T

are known to span

n

(cf. discussion after the Corollary to Theorem 3 . 1 ) , one gets all modular forms in this way. n=0

(with the convention

Equation (9) follows from (11) by taking f|T

n

= 2C(l-k)

-1

a(0)E

k

for

f =

2a(n)q € n

or by computing the constant term of both sides; taking the coefficient of

q

(12)

m

on both sides gives

-87-

Finally, Waldspurger has a generalization of (11) involving theta series with spherical polynomials. 1

To prove all of these results, we start with Siegel s own generalization of (9) to Siegel modular forms of degree 2 :

(13)

(2) Here 0

is the Eisenstein series of degree 2 and weight k and th the theta-series (5). We then compute the m Fourier-Jacobi

i coefficient of both sides of (13); by Theorem 7.3 and (the proof of) Corollary 2 of Theorem 6.3, this gives (14)

an identity of Jacobi forms of weight the development map

k

and index m.

of §3 to both sides of (14).

We now apply By Theorem 7.2,

the left-hand side of the resulting identity is

where

P?

m

is given by

(15)

and is a spherical polynomial of degree 2v with respect to Q . i

By the

Corollary to Theorem 4.2, the right-hand side is

But

j£> (E, , ) 2v k,l

is Cohen's function

C ^ к

as defined in §3 (Corollary to

-88-

Theorem 3 . 1 ) .

Hence we have the following identity, of which ( 1 1 ) is

the special case V = 0 .

THEOREM 7.4 (Waldspurger inequivalent and

PV

quadratic

(v £ 0 ) the polynomials

m

V

where

unimodular

C^ ^

[35]).

is Cohen's function

Let

Q

(i = 1,. . . ,h) be the

±

forms of rank p^j

(15),

2k ( k > 0 , k = 0 (mod 4))

as in (1) of §3.

(as in the Corollary

to Theorem

Then

3,1)

th and

T

m

the m

Hec k e operator

in

№y 2 * i+

Since, as mentioned in § 3 , the

V

C | T

m

are known to span

S^^^*

one obtains

COROLLARY. P \

spherical if

For fixed

of degree

k

and

V,

2v with respect

the functions to

) span

0^ S

j^ 2 +

p

(i = 1,... h, (respectively

V

v= 0).

In particular, the theta-series associated to the E -lattice and g

to spherical polynomials of degree k-4 span

S^. for every

k.

CHAPTER III THE RING OF JACOBI FORMS

§8.

Basic Structure Theorems The object of this and the following section is to obtain as much

information as possible about the algebraic structure of the set of Jacobi^forms, in particular about i)

the dimension of

J^ ^

(k,m

fixed), i.e. the structure

of this space as a vector space over ii)

the additive structure of

J *,m

module over the graded ring

€;

= © J, k,m

(m

fixed) as a

k

= ©

of ordinary

modular forms; ii!)

the multiplicative structure of the bigraded ring J - © J, of all Jacobi forms. *,* k,m k ) t n

We will study only the case of forms on the full Jacobi group (and usually only the case of forms of even weight), but many of the V.

considerations could be extended to arbitrary

The simplest properties of the space of Jacobi forms were already given in Chapter I. for all k

and m

There we showed that

and zero if k

or m

J^

m

is finite-dimensional

is negative (Theorem 1.1 and its

proof) and obtained the explicit dimension estimate m + £ dim V=l

dim

(1)

dim J

k,m

(k

even)

,

ID-1

E V=l

d

i

m

S

k v-i + 2

-89-

(

k

o

d

d

)

'

-90(Theorem 3.4 and the following remarks). reduces to M, E

if m = 0

for m = 1.

We also proved that

m

and is free over M , on two generators

The very precise result for

J

6,1

E

,

was obtained by k, 1

comparing the upper bound (1) with the lower bound coming from the linear independence of the two special modular forms E and E 6,1

Similarly, we will get information for higher

m

by combining (1) with

the following result.

THEOREM 8.1. independent

over

Proof.

The forms

E^

and

l

E

g

are

1

algebraically

M^ .

Clearly the theorem is equivalent to the algebraic indepen-

dence over M

of the two cusp forms

d>_ , and ' <J>

. defined in §3, (17).

A

10,1

k

Suppose that these forms are dependent.

12 1 9

Since both have index

1 and any

relation can be assumed to be homogeneous, the relation between them has the form m (2)

E

. M

T

*io i

)

for some m, where the the smallest

j

gj

for which

( T

'

z ) 3

* 1 2 i< > > " T

z

m

=

3

0

are modular forms, not all zero. g_, is not identically zero.

Let

j

q

be

Substituting

into.(2) the Taylor expansions given in §3, (19), we find that the left-hand side of (2) equals

(constant)A(T)

m

g . (x) ' o

2 J Z

° +

0(z

2 3

°

+ 2

)

3

and hence cannot vanish identically.

Since the two functions our analysis of

J

E^

J

This proves the theorem.

and

E

g

i

will play a basic role in

, we introduce the abbreviations

A,B

to denote them.

k , k

Thus

A G J

4, i

M [X,Y] -> J k

,

B G J

6,1

sending k

k

X

and the theorem just proved says that the map to A

and Y

to B

is injective.

The (k,m)-

­91­

graded component of this statement is that the map

^k­4m (f ,f 0

is injective.

^k­4m­2

l S

* * * ^

...,f ) m

X

f ^

^k­6m

^

^k,m

+ f ^ - h

+

...

¡ y

+

This implies

COROLLARY 1. —

dim J, k,m

> =

/ . dim M, . „. *­l Tc­4m­2i

We now show how this estimate can be combined with (1) to obtain algebraic information about the ring of Jacobi forms.

COROLLARY 2.

Fix an inte ge r m > 0.

Jacobi forms of inde x

Proof. over dim

m

The n the

and e ve n we ight is a module

0 0

dim ^ + 0 ( 1 ) = dim ^ 4 ­ 0(1)

and

•

of

of rank m+1 over M .

The linear independence of the monomials

implies that the rank is at least m+1. ­>

space

3

A B

m

(0 < j < m )

Using the two facts

(we do not need the more

k precise formula

dim

­

+0(1))»

we can write Corollary 1 in the

weakened form dim X

If there were m+2

к,m

> ­

(m+l)dim M. + 0 ( 1 ) к

Jacobi forms of index m

00

(k ­>• , к

even) .

linearly independent over

M^,

then the same argument used to prove Corollary 1 would show that the factor m+1

in this inequality could be replaced by m + 2 ,

the upper bound

dim J

coming from (1).

fc

m


G

as a

which are meromorphic

modular

forms).

, then the forms

be linearly dependent over must involve

modular

uniquely

(f) , A , A m

m

"*"B, ... , B

m

must

by Corollary 2, and this linear relation

<J) by the theorem, i.e. we have the formula

(3)

with

f,f. € M. J *

and

f ^ 0 ; dividing by

f

gives the assertion of the

corollary (the uniqueness follows at once from the algebraic independenc of

A

and B ) .

Looking more carefully at the proof just given, we can obtain an estimate of the minimal weight of the form

f

in (3).

Indeed, the

proof of Corollary 2 depended on the fact that the upper and lower bounds we had obtained for

dim J. (m k,m

fixed)

differ by a bounded

amount, i.e.

for some constant for

dim

C

we see that this holds with

for all even k. bounded by J

C

for all k.

M,[A,B1 n J , * k+h,m

using the explicit formulas C =

Hence the codimension of

of the form (3) with

is < C.

depending only on m;

of

f

Now if of weight

d) G J, k,m h,

m

^

^

and with equality

M [A,B] n j

in J.

is

* k,m k,m and there is no relation

then the subspaces

(f> •M^

and

J, . are disjoint and hence the dimension of M, k+h,m n J

Therefore there is a relation of type (3) at latest in weight

h = 12C = 6m(m-l).

(Later we shall obtain a much better bound.)

Corollary 3 says that

,® K ,

where

K

= C(E. ,E_)

is the

-93-

quotient field of M^,

is a free polynomial algebra

In particular, the quotient field of

J, 7F ,

, is

K^[A,B]

over

€ ( E ,E , A , B ) .

7t

K^.

In view

6

*T

of Theorem 3.6, this is equivalent to the statement that the field of Jacobi functions (= meromorphic Jacobi forms of weight for

SL (Z)

is

2

C(j(T) ,p(x,z)),

0

and index

0)

a fact which is more or less obvious

from the definition of Jacobi functions and the fact that every even elliptic function on

C/ZT+Z

is a rational function of

^-(T,Z).

Before proceeding with the theory we would like to discuss the case of forms of index

2

illustrate our results.

in some detail; this will both motivate and Here, of course, we do not need Theorem 8.1, 2

since we can check the linear independence of monomials

A ,...,B m

m

for any fixed

m)

2

A , AB

and

B

(or of the

directly by looking at the first

few terms of their Fourier expansions, as was done in the case m=l in §3. Thus we obtain the lower bound of Corollary 1 "by hand". the upper bound are given for small

k

k dim 1^. + dim S

+ dim S

k + 2

dim M ^ g + dim \ _

1

Q

f e + 4

+ dim \ _

±

2

This bound and

by the table 2

4

0

1

0

0

6 1

8

10

12

2

3

1

2

2 0

1

Thus the upper and lower bounds no longer agree, as they did for m = l , but now they always differ by

1.

in fact always the correct one. J

even

k>4

and all m > l .

We will see that the upper bound is In §2 we showed that

J

k,m

t

4- 0

Hence there exist non-zero forms

for all

X £ J

Y e J

g

^.

, if , 2

=

By Corollary 2, there must be two linear relations over 2

among the five Jacobi forms

X,

Y,

A , AB

2

and

B

of index

them, we could calculate the leading Fourier coefficients of

2. X

To find and

Y

(we didn't give complete formulas for the coefficients of Eisenstein series of index

>1

in §2, but from §4 we know that

E

and

E

are

-94-

proportional to

^| V

and

£

E

g

1

1V ,

from which the Fourier coeffi-

2

cients can be obtained painlessly).

However, we prefer a different

method which illustrates the use of the Taylor development coefficients .0» d) of §3.

Since

S = S = S, = { 0 } ,

V

V = 1,2; (thus

e

6

we normalize X = E^

2

,

X

Y = -E

6

and Y 2

we must have

) .

Q

0(z ) 6

X,Y

= -E

g

are

since a Jacobi form of index 2 is z

convenience we introduce the abbreviations

Q

(just as we already are using

A,B

for the near-Eisenstein series

for the derivatives of P,

Q

Then equation (12) of §3 shows that the

determined by its Taylor development up to

are those of Ramanujan).

for

V

j2? X = E^, !D Y

by assuming

beginnings of the Taylor expansions of

we do not need to go beyond

JZ>,.X = 2) Y = 0 V

10

8

for

E^

1 - 24 2 0

J

1

(by Theorem 1.2).

4

and and

(n)q

11

R E

g

for ^),

E^

and

For E

as well as

(the notations

g

P

P,Q,R

Then one has the well known identities

Q

and

R,

so the above expansions become

Similarly we find the expansions

for the two basic Eisenstein series of index 1.

Hence the five forms

-95-

2

with

X

2

= X,Y,A ,AB,B

have Taylor expansions

+ X z

Q

+ Xz

2

+ 0(z )

h

2

6

k

given by the table

_Jl_v

*

o

2^

X

Q

R - PQ

Y

-R

PR - Q

Q

QR - P Q

A

X

2

2

AB

QR

B

R

2

2^ P Q 2

%(Q + R ) + p Q 3

2

Q

2

2

2

2

- 2PQR

2

-PQ +Q R

2

3

2

^Q(Q + R ) + P R - 2PQ R 3

R-PR

2

If any linear combination of the rows gives

0

the right, then the corresponding form <j> is zero.

2

P QR-PR

2

2

2

- 2PR

2

- P R - Q R + 2PQ

2

% ( Q + R ) - PQR 3

+Q

2

h

2

2

2

in the three columns on 0(z ) 6

and hence identically

Therefore by linear algebra we find the formulas

expressing

X

and

Y

as polynomials in A

and

B

as in Corollary 3.

2

At this point we can discard AB, B

2

and

2

B

from our collection

X, Y, A ,

since 2AB

=

we can also replace

B

RX - QY

A

Z (Z

AB

2

=

Q X - RY - Q A 2

;

2

by the more convenient basis element

=

2 QX - A

is a cusp form and is equal to

R A 2

2

- 2QRAB + Q B 2

Q -R 3

124>

cusp form of index 1 defined in § 3 ) .

10

Then

, X,Y,Z

2

2

where

4>

iQ

l

is the

are Jacobi forms of

index 2 and weights 4, 6 and 8, respectively, and since they are clearly linearly independent over

we can improve our lower bound to

-96-

dim

2

^

m

=

^k_4

+

c

^

m

^k-6

+

^

M

\

8

'

Since the right-hand side of this inequality is equal to the upper bound (1), we deduce:

THEOREM 8.2. even weight

The space

is free over

and

8, respectively.

and

B by

M

2

°f Jacobi forms of index

on three generators

The functions

X, Y

and

0

0

and

X , Y , Z of weight Z are related

There are two striking aspects to this result: J

2

to

4, 6, A

that the module

is free over M. , i.e. that we need no more generators than are

required by Corollary 2, and that the only modular form we need to invert in order to express these generators in terms of A the discriminant function

A =

1 _

3

0 0

1 / 2o

2

(Q - R ).

and

B

is

We now show that these

two results hold in general.

THEOREM 8.3. In other words, of Theorem

The ring

the meromorphic

8.1 are holomorphic

Proof.

E

6

is contained

modular except at

in

forms occurring

in Corollary

Z

infinity.

Using the fact that

A(x) vanishes nowhere

we find with the same proof that the functions

^(TQJZ)

M ^ [ -j- ][A,B].

In the proof of Theorem 8.1 we used the fact that A ( T )

is not identically zero. in Ky

^

E^ ( T , Z ) i

Q

are algebraically independent for each point T G5C. Q

and Indeed,

-97-

replacing the functions

g^(x)

in (2) by complex numbers

c^, we find

that the left-hand side equals

(const.) I

as

z-*0, where j

because A ( T ) ^ 0.

o

is the first

j with

Now suppose that

c. ^ 0,

d> e j

and this cannot vanish is a non-zero Jacobi

k,m

form and let f (T) be a modular form of minimal weight such that f(j) £

[A,B] ,

i.e. such that there is a relation of the form (3)

(we have already proven that such an of A , then

f

exists)

f vanishes at some point T GdfC, Q

all j by the algebraic independence of

E

If

f

and then

is not a power

f (x ) = 0 for Q

(x .z) and

E

4 , 1 °

(x ,z) . 6 , 1 0

f and f. have a common factor (namely E or E if T is iri/3 3 equivalent to e or i and E ^ - j ( T ) A otherwise) , contradicting But then

Q

the minimality of f.

THEOREM 8.4.

Proof.

J. . is free as a module

over

M .

The proof is similar to those of 8.1 and 8.3.

Let us

assume inductively that for some k > 0 we have found Jacobi forms <j> , . . . ,<J> l

M

*

r

of weight

in weights G

< k,

k ,. . . ,k i r

< k

which are a free basis of J & ,m

i.e. such that every form in J, . . k ,m r '

be written uniquely as

T

\ f _^ ( x ) ^ (x, z)

certainly true if k = 1 or k

with

over

for k* < k can

f^ G M ^ ? ^ .

(This is

is the smallest integer with J^ ^ ^ 0 .)

If we can show that the <j)^ are linearly independent over M^ in weight k

also, i.e. that there is no non-trivial linear combination

S f .

for

°^

m

~*isa

direct

sum, and choosing a C-basis

its complement gives us a new collection of Jacobi forms

satisfying the induction hypothesis with k

So assume that we have a relation

Sf.d). = 0 in J, i

' ~]

k.m

replaced by k + 1 . .

Since k. < k, X

'

-98-

weight(f^) = k­k^ > 0 ,

we have

­ 2m

V

R

k,m

z= 0

for all

J

A N D THE SPACE

M , ^ ^ INTRODUCED K,M

k,m

for v = k (mod 2)

vanishing of a Jacobi form at

Z -> 0}

T H E N W E H A V E A FILTRATION

k,m

J5 ~^ k,m

for

V

1

because the order of

has the same parity as the weight,

k

and for V > 2m-3

for

k

odd for

the same reason used to prove (1) in §3 (a Jacobi form has 2m zeros altogether in

C/Z+Zx,

and for

zero 2-division points).

k

odd three of these are at the non-

On the other hand, from the defintion of

Jacobi forms we see that if a Jacobi form of weight m 2 +

k

=

The same argument shows that

m,m+l

_ =>

=>

if V = m

„ d i m M ^

2

dim M ^ ^ ^ >

we use instead >

k+2m-2 - 3 5 -

k+2m ^ 2 (mod 12)

proving the assertion in this case also.

=>

v

>

m ^

dim \

,

+

2

m

>

> f

,

-104-

We now turn to the lower bound; for this we have to construct Jacobi forms.

To do this we begin by enlarging the space of forms under

consideration.

Definition.

A weak Jacobi

form

of weight

k

and index

m

is a

function satisfying the transformation laws of Jacobi forms of this weight and index and having a Fourier expansion of the form

(7)

The space of such forms is denoted

J, k,m

r

The space

J,

is, of course, in general larger than

it is still finite-dimensional.

restrictions to

z = Ax+\x

as in Theorem 1.3 the condition

, but

Indeed, a weak Jacobi form also has 2m C/Z+Zx

zeros in a fundamental domain for transformation law under

J,

z -> z + Ax + ]i (A,11 G Q )

(since it satisfies the same

as a true Jacobi form) , and its

give modular forms in the same way

(the conditions at infinity are satisfied because of

n > 0

in the definition of weak Jacobi forms), so the

proof of Theorem 1.1 carries over unchanged.

Moreover, all of the

content of §3 also still applies, so we get maps

(8) ^

:

~k,m

J

-*

and they are still infective.

\

+

l

9

\

+

3

9

•••

9

\

+

2

m

-3

( k

°

d

d

)

The only point that needs to be checked

in order for, for example, the map

to make sense is that for a given

n

there are only finitely many

r

-105-

with

c(n,r) 4 0.

the coefficients

But this follows from the periodicity condition on c(n,r)

Jacobi forms, for given and

r = ± r (mod 2m) ,

(Theorem 2 . 2 ) , which applies unchanged to weak n

and

r we can choose an

by the condition defining a weak Jacobi form, so r

2

T

with

|r | < m f

and then

f

soon as

r

c(n,r)

vanishes as

- 4nm > m . 2

The basic result on weak Jacobi forms is the following.

THEOREM 9.2.

The map

(8) is an isomorphism

for all

k

and

m.

Notice that the statement of this theorem makes no reference to

k

being positive, and hence shows that the weight of a weak Jacobi form can be negative (but not less than - 2 m ) .

As a corollary of Theorem

9.2

we get

COROLLARY.

For all

dim J. k,m

> -

k

and

m

we have the lower

bounds

m /2 dim M, , - N. (m) _£? k+2V +

(k even)

0

,

0

m-1

with by

N (m) +

as in (6).

For k > m the inequality

signs eon be. replaced

equalities.

To prove the corollary, we note that there is an exact sequence

(9)

0

—>

J, k,m

—>

j

k,m

—>

€

N±(m)

in which the last map sends a weak Jacobi form collection of Fourier coefficients

c(n,r) with

\D^'

, ... ,2> j'

Z>^\£>^\

(resp.

2

is odd) are modular forms, since the proof that

used only the Taylor development of (f> up to order (8) is still defined if we replace

J by J . f

surjective is now a question of linear algebra. (the case of odd k forms

is similar).

£ ( T ) = Sa (n)q V

4>' G j!

K,m

n

v

with i) = ^ £\) \) f

•••^ m-3

4 ) ?

i

f

2

!D^> G M ^ ^ +

V.

in §3

Hence the map

To show that it is Suppose

k

is even

Then given a collection of modular

of weight

k+2v

(0 < V < m ) we want to find a

for all V, i.e. to find coefficients

c(n,r)

-107-

satisfying a)

c(n,r) = 0

for

n < 0,

b)

c(ri,r) = c(n ,r')

c)

£ P r

for

T

Because of

r

T

= ± r (mod 2 m ) ,

(r,nm)c(n,r) = a^(n)

2 v

b) , we need only find

for

c (n, r)

r' - 4n m = r 2

f

0 < v < m,

2

- 4nm,

n > 0.

for 0 < r < m, since the rest

are then determined by the periodicity conditions; in terms of this we can rewrite c) as m (12)

E

e

r

p^"

(r,nm)c(n,r) +

1 }

...

=

a (n)

,

v

r=0 where

is 1 or 2 depending whether

a linear combination of coefficients

r

is

0

or not and

11

... " denotes

c(n ,r) with 0 < r < m

and

1

n < n. f

We can assume inductively that the equations have been solved for

n' < n

and hence that the " . . . " denotes a known quantity; then (12) becomes an (m+1) x (m+1) system of linear equations in the

m+1

unknowns

c(n,r)

(k-1) p^

(0 < r < m) with coefficients

(r,nm)

(0 < r, V < m) .

But this

matrix is invertible, since it is the product of the nonsingular matrix

(e 6 ) and the matrix r rv r,V .

K

( p ^ ~^(r,nm)) , 2v 'r,v

reduced to a Vandermonde determinant

(r^ )

diagonal

which can be

by elementary row

V

r, v (k-1) operations because in

r.

(r,nm)

is a polynomial of degree exactly 2v

Hence the equations can be solved inductively for all coeffi-

cients c(n,r). It remains to see that G J* T

and

M G T , l

J =J . f

But this is easy, because if

then the difference

a Jacobi form with respect to translations compatibility of the actions to order

> 2v

at

Z

2

and

(j) - (}) | ^ f

T

transforms like

z -> z + AT + y (because of the

SL (2£) proved in §1) and vanishes 2

z = 0 by (11), so Theorem 1.2 shows that it vanishes

identically and hence that

(J)' transforms correctly under

V^.

We now prove a theorem which determines the structure of the

-108bigraded ring 5

J

© J, k,m k,m k even

ev,*

6

THEOREM 9.3. on two

The ring

completely.

^ is a polynomial

over

M^

generators I

1

where

d) , 1 - £ kjf1 f ( 'k G

ui

J

Proof.

i Q

i



Q1

-

A

0,1

forms constructed

in §3.

are weak Jacob! forms is clear,

^ , being cusp forms, have Fourier developments

containing only positive powers of cj)_ ^ and

'12,1

10,12) a^e the Jaeobi

k=

(j>__

4>

I

t

A

-2,1

since

algebra

q

and

A(T) = q+ 0(q ). 2

Since

are algebraically independent over M^ by Theorem 8.1,

the map

is infective, and combining this with the injectivity of the map (8) in the other direction shows that both are in fact isomorphisms, thus proving Theorems 9.2 (for even weight) and 9.3 at one blow. This proof of Theorem 9,2 i s , of course, much shorter than the one we gave before (and a similar proof works for odd weight, as we shall see in a moment).

Nevertheless, we preferred to give the direct proof

of the surjectivity of D

because

a) In other situations (e.g. for congruence subgroups) one might not happen to have enough explicit generators to deduce the surjectivity of !D purely by dimension considerations, and b) The first proof given shows explicitly how to obtain a weak Jacob! form mapping to a given (m+1)-tuple of modular forms by

-109-

solving a system of recurrence relations. This and the proof just given for Theorem 9.3 then give two algorithms for computing the space of Jacobi forms of given weight and index:

we

construct a basis of

^\+2m

and applying D

1

P

^

either by picking a basis of

or by picking a basis of

as in the proof of Theorem 9.2;

Fourier coefficients elements and obtain

c(n,r) J, k,m

®

© ... © ^\+2m

... ©

and applying

then in both cases we compute the

(0

€

Both methods will be illustrated later. Observe that Theorem 9.3 gives a considerable sharpening of Theorem 8.3.

There we showed that any Jacobi form, multiplied by a

suitable power of

A,

could be expressed as a polynomial in

and that in fact one can take is the index of the form. polynomial in A

and

B,

m

^

^

as the exponent of

Here we show that

A

A, where

find the corresponding result for odd weight.

m

m

Q A - R B , for any Jacobi form or weak Jacobi form J

B,

A can be written as a

and in fact as a polynomial in RA - QB

Theorem 9.3 gives the structure of

and

(j) of index

and

m.

for even weights.

We now

The upper and lower

bounds given in Theorems 9.1 and 9.2 and their corollaries agree for k >m

and also for m < 5 ;

dim J, k,m m

k

3

for m < 6 we obtain the table

5

7

9

11

13

15

17

1

(0 for all k )

±

2

0

0

0

0

1

0

1

3

0

0

0

1

1

1

2

2

4

0

0

1

1

2

2

3

' 3

5

0

1

1

2

3

3

4

5

6

0

0 or 1

1

2

3

3

5

5

In particular, there is a certain Jacobi form

$

n

2

of weight 11 and

­110­

index 2 (we shall see how to construct it explicitly a little later). Since

^ is a cusp form (it follows from §2 that the smallest index

of a non­cusp form of odd weight is 9 ) , the quotient is a weak Jacobi form.

Hence

4

J

(0};

=

2

(j)

/A

u2

of course this also follows

1,2

directly from Theorem 9.2.

Then

J

and

m,

=

0

dim M.

for all

0

+ M. „ 4- . . . + dim M, _

R+l

k+l,m­2

J

(f> • JL C J. ­i,2 k+l,m­2 k,m

k

and since Theorem 9.2 gives

dim J, ­.

for k

r

k+3

odd, this inclusion is in fact an equality.

, , = 4> od,* ­ , 1

2

•J , ev,*'

free module of rank with generator

dim J. k,m

Hence we have 6

over the ring of weak Jacobi forms of even weight,

.

i

=

i.e. the weak Jacobi forms of odd weight form a

1

<J)_

„

Tc+2m­3

This gives the ring structure, too, for cj>

1,2

lies in J and

i

1,2

, and can therefore be expressed as a polynomial in d> r

ev,* with coefficients in

f j -2,i Comparing indices and weights,

.

T

we see that this polynomial has the form

][ f. <J)

(J)^

i=l ^2i­2

'

i

with

f.

in

*

and hence

f, ­ a

,

for some constants D : J

—>

1 - 1 , 2

M

f

,

= 0

a , 3 , Y.

= € 0

2

f ­ BE„

,

9

f„

­ YE

6

To find these constants, we observe that

by Theorem 9.2, so

J

this constant determines

2

.

.2) d>

is a constant and

r

'

l -l,2

We normalize by choosing ^) c^_

i

2

=

2;

then Ф_

2 ' ' ( T

х

z )

=

n

I I c(n,r)q ç n>0 r

í

í rc(n,r) r>0

=

1 < (O

(n=0)

r

,

,

( n ¿ 0)

By the argument preceding the statement of Theorem 9.2, r

2

> 8n+l;

in particular

c(0,r) = 0

for |r| > 1

c(n,r)=0

and therefore

for

­111­

(j>

1

(Ç ­ C ) + X] (polynomial in Ç,Ç n>l

=

This already suffices to determine a, coefficients of

1

(ç­çf )

2

1

1

(ç­2+ç~ )

T]

)q

for comparing the

in the two expressions for (j> — 1,2 1

Y

and Y;

2

q°

= a ( ç ­ 2 + ç~ )(ç + l 0 + ç " ) +

3

l

gives

+ 3 ( ç ­ 2 + ç" ) (ç + 1 0 + ç ~ )

3

1

3

1

i f

or T

X

(T = Ç ­ 2 + Ç

2

+ 4T

),

aT(T+12)

from which

9.4. The ring

THEOREM

J

where

=

,

+ 3T (T+12)+YT 3

, 3 = ­3a,

a=

has the

J

2

b G J

­2, 1

,

t f

Y = 2a.

We have proved:

structure

M [a,b,c]/c = t ~ a ( b

=

a £ J

3

3

Z

3

­ 3Qa b + 2 R a )

,

c G J

0 ,1

­1,2

We make one remark about the relation just obtained.

We can write

it in the form

1

2 c

=

i,

432

a

On the other hand, by Theorem 3.6 we have b a

where p

=

*0,1 6

=

__3_

=

<J>

is the Weierstrass

2

*12,1

а

ц

­IT

function.

/, з

2

^

'

Hence

4

4ïï _

6

л

8тг _ \

where the expression in parentheses is equal to ^ ' equation.

2

by Weierstrass*

Thus the relation in Theorem 9.4 is just Weierstrass

1

­112­

12

equation for p-

as a cubic in fi

as saying that

^

and the theorem can be interpreted

is the coordinate ring of the universal elliptic J

curve over the modular curve (we leave this purposely vague);

itself, of course, is not finitely generated and hence not the coordi­ nate ring of anything. is

The square root of the relation just obtained

c = — ­ — ­ ¿2.' (T,Z) (27fi)

or

3

- l i ^10

7

TZ^T -1

2

2

x

12(2Tfi)A

— 12A

^ » ,

2

1

8 0

)

1

3

(*' 2lTi

V Y

A­E^ 2

12~ ——— 2iTi

and

1

B = E

1 T (A B ­AB ). A l

T

6

,

x

z

*io,i

-

(J>

1Q

t

Z )

z

> >

*

10,l

and

1

'

^ Y

M0,1

(

( T

Y

<j>

)

12,l'

i2 1

in terms

we find that the right­hand side is equal tc

This proves the relation F

288îri

12 , 1

Substituting into this the expressions for of

Ч

l—J * '

(2iri)

=

,1 \

(t z)

d

/

r

=

E

E

l l ,2

­

E

6,1

E

4,1

6,1

This is a special case of the following easy fact, which gives a general construction for Jacobi forms of odd weight from forms of even weight.

THEOREM 9.5. k

2

and inde x

m

l

Le t

and

m ,

diffe re ntiation

m

and we ight

+ m

£

Proof. and weight

k

and

(J)

be

re spe ctive ly.

2

denotes 1



with re spe ct

to

Jacobi Th e n z,

forms of we ight m2$\§2

k

~ m1^1^2,

is a Jacobi form of

and

l

where inde x

k + k + 1. 2

2

The derivative of a meromorphic Jacobi form of index is a meromorphic Jacobi form of index

Applying this to

m m cf), / (J) z

0

and weight

proves the theorem (the conditions at

0 k­f1.

'

-113-

infinity are trivial).

Examples of Jacobi Forms of Index

>1

We end with some numerical examples to illustrate the techniques for computing Jacobi forms implicit in the theorems of this section. We give several types of examples with m > l (the case

m=l

having been

treated in §3) to illustrate the results described at the end of §5. We look only at cusp forms.

1.

k even, m

prime

This situation, the one treated in Theorem 5.6, is the simplest case after

m=1

(because in the decomposition (17) of §5 there is only

the term £ = 1 ) .

From the dimension formulas we see that the first

three cases are

(m,k) = (2,8), (2,10) and (3,10) and that

dim j

P = i k,m c u s

in all three cases. For

(f> = 2 c ( n , r ) q £ n

r

E j

P

c u s

the first two Taylor coefficients

?

8 , 2

3)fl

and j£) are cusp forms of weight 8 and 10 and hence vanish, while

3) § must be a multiple of A ( x ) . Theorem 2.2, so c(N) = c

Also, c(n,r) depends only on 8n - r

c(n,r) = c ( 8 n - r )

by

for some sequence of coefficients

2

(N)

2

satisfying

8,2

I c(8n-r ) = 0 , r 2

for all n

I r c(8n-r ) = 0 , r 2

2

I r c ( 8 n - r ) = 24x(n) r 4

2

(the constant 24 was chosen for convenience).

These

recursions can be solved uniquely (at each stage the numbers c(8n-l)

c(8n),

and c(8n-4) are expressed in terms of c(N) with N < 8(n-l)).

We find the values

c(4) = 1, c ( 7 ) = - 4 ,

c(8) = 6, ...

(cf. Table 3 a ) .

(These coefficients could also be calculated from the fact that 1 (f) = —

2

Z =

$

/A , where

Z

is the form of Theorem 8.2 and

-114(j> ^ G

^ the cusp form given in §3.)

lQ

The most striking thing in

the table is the occurrence of zero entries.

To explain them, we note

N that the form

M+

5 / 2

2 c(N)q

(2)

=

lies in the space

jh M

1 5 / 2

e

(r (8)) |ho

by Theorem 5.6 and is a new form (since

Ç c(N)q N> 0 ) N=0,4,7 (8) ' N

^13/2^

follows from the results of Kohnen [12] that form

4 (8r+3).

" W ) ; It then

=

c(N) = 0

for all N

of the

In Table 3a (and the rest of Table 3) we have also

given the eigenvalues of T^

for £ = 2 , 3 , 5 and 7, computed by means of

Theorem 4.5 and the formulas at the end of §4. For

d> G j

c u s

P

the same method works: now

3) §

const. • A , 3)^

= 0, 3) è =

n

10, 2

0

= 0

and we find

first coefficients given by (see Table 3 b ) .

c

1 0 > 2

(J) = S

T

(8n - r ) q ç 2

?

(4)=l»

c

1 Q

£

n

( 7 ) = 8,

2

,

T

with the

r

c

l Q

2

(8) = -18,

This time there are no zero coefficients because

cj) is not a new form:

it is

<j>

iQ

i

| V .

By Theorem 4.2, this is

2

equivalent to the formula c

(with the convention

1 0 > 2

c

(N)

1 Q

1

=

-%(c

(n) = 0

i 0 > 1

(N) +

2 c 9

l o > 1

(N/4))

for n ^ Z ) , and one can check this

by comparing the values in Table 3b with those in Table 1. Finally, for m = 3 cj) we must have

3)$

the first cusp form has weight 6; for this form

= 3)^

= 3)^

<j) =

I

= 0 and 3)^

c(12n-r )qV

I r

2 V

(° c(12n-r ) 2

=

so

,

2

9u

is a multiple of A,

(

V

=

2

)

I

r ( 720 T(n) (V=3) (again with 720 chosen for convenience), from which we get the values tabulated in Table 3c.

(We could also compute

= 2 c ( n , r ) q £

Indeed, if we write

1 2

can be written as

c(n,r) = 0

(-~- )c(8n - r ) 2

c(N) which are non-zero at most for

= €*A

2 rc(n,r)

implies that

c(n,r) =

depends only on 8n - r

together these facts imply that

c(n,r)

then

r

2

for r

for some

N = 7 (mod 8 ) .

Then

is a multiple of x ( n ) , so

(with suitable normalization) ^

(^)rc(8n-r ) 2

=

T(n)

.

0 < r 0 N = 7 (mod 8)

\/j

8 N

)

q

/ \

( f r

>

)

V \ /

0

7

-

J n

>

X(n

n ) q

0

7

The second factor on the left is N ( T ) by Jacob's identity and the 3

expression on the right is

A(x) = T|(x)

.

Hence we get, instead of

.

-116-

just a table, the closed formula

h (T)

:=

?

£

2

Si

(

N

^

)

^ ^

=

8

T

>

2

1

N>0 where

£

as in §5 is an extension of £

to a Dirichlet character modulo 4

/ -4 \ (here necessarily §5.

£ = (—))

The fact that

hg(T/8)

C

of

SL (Z0

the form of weight 1 0 % defined in

is a modular form on all S L &

(with

2

multiplier) rather than on sentation

and hg

F (16)

is due to the fact that the repre-

Q

mentioned at the end of §5 is one-dimensional

2

in this special case (in general for m

prime and k

odd it would have

dimension m - 1 ) . 3.

k

even, m

a product of two primes

This is the first case where the decomposition of than one piece; we are interested in the summand non-trivial

£, since the space

I [ J,

first weight function

k

(j) € n

r

changes sign if y for

y

where

5y

not prime to 12 (since then

(mod 2 4 ) .

In particular,

2A(T),

*• =

E n,r

is 6 and the

To see this, note that a

y

7y.

(mod 12) and

It follows that

c^(N)=0

is congruent to 5y or 7y modulo 12 (—)e(N) for some y

c ( N ) can be written as m

c^(0) = 0

and vanishing unless

for all y,

is determined by

possible weight is k = 12. =

or

c(N) depending only on N

a cusp form, and since

cf)(T,0)

is 12.

m

c ^ ( N ) depends only on y

is replaced by

12) and that in general coefficients

J

has a Fourier development of the form

2 c^(24n - r ) q £ 2

can be understood by Theorem 5.6

The first composite

J^. ^ 4 {0}

with

has more

corresponding to

k,m

r

and the following remarks.

^

^

so cj) is automatically

2)^§ = < K T , 0 )

For the form with

N = 23

k = 12,

the first

e

and normalized by

we have (^)c(24n-r )qV 2

,

£ r>0

( ^ ) c ( 2 4 n - r

a

)

-

T(n) ,

-117-

which can be solved recursively to give the numerical values given in Table 3e or explicitly in the same way as was done for the last example: we write the recursion for

c

as

and o b s e r v e t h a t t h e first f a c t o r i s n ( T )

hg(T)

:=

^

c(N)q

1

n(24x)

=

N

(Euler S i d e n t i t y ) , so

(Z =

2 3

(^))

N > 0 N = 23(24) (again

is a m o d u l a r f o r m on a l l S L ^ ( Z )

hg(T/24)

one-dimensional). E a(n)q

or some

n

G M

S i m i l a r l y , if

g £ k-12'

(f>(x,z) is

g(x)

^

a n C

f (T) .

1

* ^ t

times

Since ^gCy^)

i e n

cb

f

h(i,0) = f ( T ) =

=

r

(x,z) ,

is a cusp form we have

l^ ^ 8^ ^* T

with generator

2 3

i.e.

T

J

I t :

f = Ag

for

follows that

, is a free module of X,D

12,6

rank 1 over

with

is

then the recursion relation for the c(N) gives

h ~ ( - ~ ) = X] (T) M

<J) £

because

6

•

We could also have seen this

by a direct argument; more generally, it is not hard to modify the proof of Theorem 8.4 to show that

J

is a free module over x,m

every character

4.

£

M.

for

*

and to give a formula for its rank.

k= 2 The case of Jacob! forms of weight 2 is particularly

interesting,

both because this is the smallest weight occurring and because of the connection with Heegner points mentioned briefly in the Introduction. However, it is also the most difficult case, since we do not even have

-118the Eisenstein series bound for J„ 2,m

E

- to get things off the ground.

K., 1

The lower

obtained from the corollary to Theorem 9.2 is J

(14) m

25

37

43

49

53

61

64

67

73

79

81

83

85

89

91

93

97

98

100

bound

1

1

1

2

1

1

1

2

2

1

2

1

1

1

1

1

3

1

1

the bounds for the omitted

m

49, 6 4 , 81

being zero or negative.

The cases m = 2 5 ,

are accounted for by the Eisenstein series of weight 2 (cf.

the remarks following (11) of § 2 ) . those of

S

less when

m

2

(r*(m))

The other dimensions are equal to

(given in [ 5 ] , Table 5) when

m

is prime and one

is a product of two primes, the first in accordance with

the relation between

X k,m

and

M (F*(m)) zk-z 0 1

0

discussed in the Introduc-

tion (eq. (9)) and in §10. This relation will be proved in Chapter IV by trace formula methods for k > 2, but the case k = 2 presents extra difficulties, so the numbers (14) are not necessarily the true values of

J^ ' m

In any case, they are lower bounds, so we must be able to

find, for instance, a Jacobi cusp form of weight 2 and index 37. Let us look for a cusp form coefficient of q £ n

r

(j) £ J

2

3 ?

.

By Theorem 2.2, the

in depends only on 148n - r ,

i.e. we have

2

<J>(T,Z)

=

]T

c(148n - r ) q C

n,r 148n > r with some coefficients

2

n

r

2

c(N) (N > 0, N = 0

or 3 (mod 4) , ( — )

= 0 or 1 ) .

However, to find these coefficients by either of the algorithms mentioned after Theorem 9.3 would involve an impossible amount of computing.

For

instance, using the Taylor development operator in (8) would involve constructing an explicit map into the space

S

2

©

©

... © S

y 6

of

dimension 102 and then solving a 1 0 2 x 102 system of equations with coefficients which are very large and hard to compute. use another method.

The function

We therefore

-119-

h(T)

belongs to

M . (37)

=

£ c(N)q N>0

N

by Theorem 5.6, and an easy calculation shows that

3/2

the form CO

h(T)9(T)|U.

=

£

a(n)q

,

n

n=l (15)

r give

c(4) + 2c(3)

=

a(l)

2c(7) + 2c(4)

=

a(2)

c(12) + 2c(ll) + 2c(3) =

a(3)

c(16) + 2c(12) + 2c(7) =

a(4)

2c(16) + 2c(ll) + 2c(4) =

a(5)

0

=

a(6)

Looking at the tables in [ 5 ] , we see that the last equation determines the form

Sa(n)q

n

E S (1^(37))

up to a constant:

2

it must be a multiple

of the eigenform

f^z)

= q + q -2q" -q -2q + 3 q 3

7

9

U

-2q

12

-4q" +4q

16

+ ... .

-120-

At this stage we have five non-trivial equations involving seven unknowns

a(l), c(3), c(4), c(7), c(ll),

further, we find that at

n = 15

c(12),

n,

Proceeding

the number of equations overtakes the

number of unknowns, so that we suddenly get all For various higher

c(16).

c(N)

up to N = 6 0 .

we again have enough equations.

However, eventu-

ally we will have too few, since (15) provides an average of one equation for every two

c(N) with

N = 0

or

3 (mod 4 ) , while the proportion of

c(N) known a priori to vanish is only extra information.

Namely, for

m

18

/ 3 7 < V2 •

However, we have some

prime the space

M^^(m)

as defined

in Theorem 5.6 is by Kohnen s work isomorphic as a Hecke module to T

M ( r * ( m ) ) = {f G M ( F ( m ) ) | f|W 2

2

o

= f}

m

(cf. proof of Theorem 5 . 6 ) .

m = 37 both spaces are one-dimensional, so our form

h(x) G M ^ ( 3 7 ) 2

For is a

Hecke eigenform with the same eigenvalues as the second eigenform

f (z) 2

in

S (T (37)). 2

Q

=

q - 2q - 3q + 2q* - 2q + 6q - q 2

3

5

6

7

+

...

This means that we can write all coefficients

multiples of those with

-N

c(N)

as

a fundamental discriminant, thus reducing

the number of unknown coefficients by a further factor of

£(2)

1

~ 0.61.

Using this, we find that we now have a surfeit of equations to determine the coefficients

c (N)

(so many, in fact, that we can simultaneously

solve, for the coefficients of

q , p

p

prime,

dispense with the Antwerp tables entirely!).

in

f

1

and

f

2

and hence

We have tabulated the

resulting values (in Table 4) up to N = 250, since the example is of interest in connection with recent work of B.Gross and the second author on "Heegner points" and will be cited in a forthcoming paper of theirs.

-121-

§10.

Discussion of the Formula for For

IN

k,m E

dim J, k,m

set

L|)

k v-

(

dimM

j(k,m)

+2

= < f 2 ~] \

fll—l y S .

(

d i m M

k

^

) 1

dim J,

> j (k,m)

k,m k

V-l-

+ 2

V=l V

In §9 we showed that

for

'

(keven)

(

d

d

•

)

'

for all

k

and

dim J,

= i (k,m)

k,m k > m;

this will be improved to

k ^ 3 in Chapter IV by means of a trace formula). j(k,m)

°

'

J

sufficiently large (namely for

section is to write

k

The purpose of this

in terms of standard arithmetical functions

and to relate it to the dimensions of known spaces of modular forms. More precisely, w e will prove the following result.

THEOREM 10.1.

For k > 2 and

(1)

n 6 I

j(k,m) = £ j (k,m) Q

equals

new forms of weight

the dimension 2k-2 on

Fricke-Atkin-Lehner

Similarly, replaced j^

u s p

by

(k,m)

ponding

if

j

m (k,m)

-—^ j + 1, equals

s \

^

: f (T)

*—>

is defined

and

j^

s

-k+1 m

(-l)

-2k+2 (

^

under

k

°^ the

like

m

'

1 \

r

f(

T

i

\ mx / but with

j(k,m)

by the obvious

u s p

dim 2 ^ - 2 ^ 0 ( " ~

space of cusp

O^V^m

M ^ ^ ^ o ^ ^

with eigenvalue

Q

analogue

^he dimension

"| of ( 1 ) , then

of the

corres-

forms.

Note that the difference with

/

2

of the space

T (m)

•

k

\£ |d

by

involution

W c u s p

j (k,m) inductively

A V >f >

( E

dim Then

define

(resp. 0 < V < m

if

j(k,m) - j k

C U S p

(k,m)

is odd) and

V

2

is the number of == 0 (mod 4 m ) ,

V

which

­122­ b+2 ­y­

equals

, (resp.

l.

I), where

i_

-J

b

is the largest integer whose

J

m,

2.3 and 2 . 4 ) .

In view of Theorem 4.4, Theorem 10.1 has a plausible

n

and this is just the dimension of

Eis J, k,m

square divides

J

(Theorems

interpretation as saying that the decomposition

k,m

,,

„21

,

Jo

Id

Jm

a

k,m

I &l

1

given there is direct and that there is an isomorphism between ° m

and

^2k­2^o^ ^

preserving cusp forms.

jnew k,m

In Chapter IV we will prove

that this is indeed the case by proving the analogue of Theorem 10.1 with dimensions replaced by traces of Hecke operators.

Here we restrict

ourselves to the equality of dimensions.

Proof.

We carry out the proof only for

entirely analogous case of odd

k

k

even, leaving the

to the reader.

Our first objective is to write the formula defining terms of familiar arithmetic functions.

in

It is convenient to replace

the different rounding functions in the definitions of by their average.

j(k,m)

j

and of

j

c u s

We therefore use the notation

Í

a - h

(

a - h

if if

x G 2Z 4­ a , 0 < a < 1 , x G 2Z 4- a , О < a < 1 ,

О

if

x G Ж

;

then

(2)

j

(

k

>

n

>

=

£

{

d

l

m

M

k

+

2

v

- ^ - %

+

( ( ^ ) ) }

as above) and

j

c u s p

L

(k,m)

sign of the last term reversed.

dim ^

^

+

V=0

(b

P

=

J

is given by a similar formula with the We substitute into this the formula

"

1 / 3

(

k

1

> < ­ > ~ ^ 3

X^k­l)

-123-

where X ,

and X,

denote the non-trivial Diriehlet characters modulo 3

k

3

and modulo 4 , and calculate: V* / k + 2 v + 5

vf_

x

(2k-3)(m+1) 24

\

v=0

V (3)

-V

3

£

X (k+2V-1) 3

=

3

{ -Vb

^ ^

-%

\ (k+2v-l)

=

V=0

(mod 3 )

,

if m

< - %

if m

I

((x))

k = 2

U

is even, k = 0 (mod 4 )

,

is even, k = 2 (mod 4 )

,

,,

.

n

0 Also, since

,

otherwise

X

m (4)

(mod 3 )

if m ^

0

B

if m = k f 2

if m

is odd

is periodic with period 1, the function

(("^)) I

s

periodic with period 2m, so we can write

V

V(mod 2m)

U

h i Finally, the last term in ( 2 ) whether or not

4

equals

(5)

^ + ~

Hence ( 2 )

divides m.

j(k,m)

-

h i or

— + —

depending

can be written

£ j.(k,m) 1=1

with

j^k,*)

= ^

j (k,m)

=

RHS of ( 3 )

j (k,m)

=

h

3

5

m

,

J

J 2

(k,m)

,

=

(k,m) =

>

V > > k

m

=

^ 3

RHS of ( 4 )

«(f»

=

,

-VsX^m),

V (mod 2m) j (k,m) 7

=

hb

j (k,m)

h

if

4fm

h

if

4|m

8

-124-

and

j

C U S p

and j

(k,m)

reversed.

i s given by the same formula but with the signs of

We have now p r a c t i c a l l y achieved our goal of writing

j (k,m) in terms of familiar a r i t h m e t i c a l functions, for a l l except j

5

j

are extremely simple functions of m and k

the

(periodic

functions of m and k or products of a polynomial in k and a m u l t i p l i cative function of m).

We s h a l l now see t h a t the function j , 5

which

depends only on m, can also be expressed in terms of a well known — though l e s s elementary — arithmetic function.

LEMMA. Define b*(d) for d 4 - 3 , - 4

as the class number of

positive definite binary quadratic forms of discriminant d (so, h (d) = f

if d> 0 or d 4 0,1 (mod 4 ) ) ,

h (-4) = 7 .

h (-3) = % , f

f

2

Then for any

natural number N (6)

J «t» v(mod N)

= - I ' " d|N h

(

•

d )

In particular j (k,m) 5

Proof.

=

-% ^ h (-d) d|4m

.

f

Suppose N = p i s a prime.

If p = 2 or p = 1 (mod 4 ) , then

both sides of (6) are zero, the l e f t because

((

)) i s an odd function

and - 1 i s a quadratic residue of p , the r i g h t side because p has no d i v i s o r s congruent to 0 or 3 (mod 4 ) .

If

p = 3 (mod 4 ) ,

left-hand side of (6) equals

E «f» E n(modp) V(modp) V = n(mod p) 1

E « P n (mod p)

=

2

=

E n (mod p)