в
Progress in Mathematics Vol. 55 Edited by J. Coates and S. Helgason
Birkhäuser Boston • Basel • Stuttgart
Martin ...
101 downloads
328 Views
6MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
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 CH4144 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 8428250 ISBN 0817631801
CIPKurztitelaufnahme 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 0817631801 (Boston) N E : Zagier, D o n B.:; G T
All rights reserved. N o part of this p u b l i c a t i o n may be reproduced, s t o r e d i n a
retrieval system, or transmitted, in any form or by any m e a n s , electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright owner. © 1985 B i r k h ä u s e r B o s t o n , Inc.
Printed in Germany I S B N 0817631801 I S B N 3764331801 9 8 7 6 5 4 3 2 1

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
FourierJacobi expansions of Siegel modular forms and the SaitoKurokawa 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 halfplane)
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 "SaitoKurokawa 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
2k2
(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 socalled Maass "Spezialschar", defined by certain identities among Fourier coefficients) and should in fact be an isomorphism from
^2k2^ 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 nonanalytic 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 algebraicgeometrical 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 pfunction ^ ( 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 KacMoody 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(4nr ) 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 haIfintegral 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 AtkinLehner r/ \
^
f(T) — > m
"new"
k+1
T
2k+2
_,
N
f(l/mT)
,
and
_new m
2k2
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 SaitoKurokawa
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 §§46.
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 810 are principally due to the second author, while sections 1, 37 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
greatestinteger function
(rather than the usual
max{nG zn ^ x}
fx] = min{nGZn^>x} = L"" J* By
dn
we mean
dn
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) . dn 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 halfplane {xG (Clm(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 semidirect :=
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 nonintegral
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
=
WXM)(T,0)
=
(4>M(XM))(T,O)
=
(<J>(XM))(T,0)
behaves like a modular form with respect to the congruence
subgroup
{MGrXM=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 + (dl)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 4m 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(zX/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
1k,,
a^m
3
(4nm  r )
2 /
.
r
if
,
r < 4nm
with
.
cD
'
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
'
ac
(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 Lseries 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(2s1)
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(2k2)
_1
L (kl) 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 (2k)/?(32k) D
where now all numerical factors have disappeared. (D < 0 ,
k
,
The values
L^(2k)
even) are wellknown to be rational and nonzero; they have
been studied extensively by Cohen [ 6 ] , who denoted them
H(k1,  D  ) .
Summarizing, we have proved
THEOREM 2.1. a nonzero
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(k1, 4 n  r ) C(32k) 2
=
;
(H(k1,N) = L_ (2k) = Cohen s function) and f
N
e
km^
n
' ^ r
=
~
^rk'^r
T
^ * TT
(elementary pfactor) .
pm
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
squarefree is
^
=!TF2krL
d H(kl.^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 Hecketype operators introduced in §4 (Theorem 4 . 3 ) . For of
m = 1
H(k1,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 41512C" + 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.141143.
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
(2k,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(32k, X
)) ; in particular, the
coefficients are algebraic (in Q(X)) and nonzero.
If
(r,f) > 1 ,
then
(X) e
k m^n'r^ "*"S given by a formula like (11) with the righthand 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
nonprincipal 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 nonprincipal 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 translationinvariant 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)Hklj\
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)
(k1) 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(k4l)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+2Vy2) (k + v ~ 2 ) !
r
2v2
y n
y
r
and is, up to a change of notation and normalization, the socalled Gegenbauer or "ultraspherical" polynomial, studied in any text on ortho(k1) 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+v2)!
, . . . 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(k1,N) ( N G I ) 0

1
L_ (2k)
if
N > 0, N = 0
C(32k)
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 halfintegral 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 nonzero) 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 v2
/^\ , 1 / 27rimc\ 27 v4
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+vy2)! (k+V2)! Y!
transforms like a modular form of weight
y
kfv
.
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)
aZZ ^
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
>
Vl,
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(2kl) ^ ^ 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 thetaseries, 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) (^У2и1)! Л и ) Zrf (k+vu1)! 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 . . . 4dim S k
dimS,
4dxmS.
0
4...4dim
k
+
2
(k
m
(k
S. „ L
even) ,
odd) .
Here the second estimate cank+1 even be strengthened to k+2ml 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 §§47 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 wellknown 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 onedimensional.
= 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
(4nr ) = 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 adbc = J, r i
 * ~
( 3 )
k
2
are well defined (i.e. on
^ and map
m
to
^ope^vely.
The welldef 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 Heckeoperators 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 JacobiHecke operators to the usual Hecke operators
T^
T^
correspond
on modular forms of weight 2k2, 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
44WV )(T,z)
=
A
Y, ad=£
~
A
K
b(mod d) an
=
A*"
1
J
d"
]T
k
ad=£
b(mod d)
kl V £ 2j
,1k
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)
k1
^
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
(0V^ ). 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*
,,kl 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
6m
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
Zj
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 , Zr 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

gf,
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 nonfundamental 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 Lseries already introduced in connection with
JacobiEisenstein 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
xR ,
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 lefthand 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 HalfIntegral 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 (vectorvalued) 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 halfintegral weight and studying the correspondence more closely. Equation (1) gives us coefficients all integers
0 satisfying
welldefined 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 JI,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 "vectorvalued" 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 squareroot 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 (ml)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 11 correspondence between eigenforms h G 2k2
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 threeline 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 ^2k2^
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
halfintegral 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 2k2 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 Lseries
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,
squarefree the map (18) gives an isomorphism between
M, V * (m) Kh
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
pm,
odd and squarefree) by Kohnen [12], who showed
that M _^,(m) decomposes in a Heekeinvariant 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 nonzero only for in M  for m
* . +
k~^2
squarefree 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 squarefree this
Cy(N) is independent of the choice of the squareroot
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 M2k2^r*^m^
^m Prime>
k
even)
70which will be proved in §§910 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 §§910 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 halfintegral 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 squarefree 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 nontrivial.
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 squarefree, then one must restrict to the Unew
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 halfintegral 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 halfintegral 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.
FourierJacobi Expansions of Siegel Modular Forms and the SaitoKurokawa Conjecture We recall the definition of Siegel modular forms.
upper halfspace of degree symmetric n x n matrices
n
is defined as the set JC
n
The Siegel of complex
Z with positivedefinite 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 semiintegral (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,4nmr >. 0 2
The relation to Jacob! forms is given by the following result, which, as mentioned in the Introduction, is contained in PiatetskiShapiro 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
FourierJacobi
expansion
+2XZ +À T)
<j>; the condition at infinity m
Following PiatetskiShapiro, 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 SaitoKurokawa 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.
FourierJacobi
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 "plusspace" 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
Heokeequivariant
to the homomorphism
of Heeke
algebras
by
i(T (p))
=
Tj(p) +
, / w l(Tg(p))
=
/ k1 , k2s m / n , o 2k3 , 2k4 (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
=
FT (p) s
,
F
2
=
FTg(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
. к2 , ч , 2к3 ^ 2к4 + 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
FT (p) = Y F , s
p
FT^(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 11 correspondence between eigenforms in M^^ deduce:
2
and
^,
the eigenvalues being the same.
We
c
1
80
COROLLARY 1 (SaitoKurokawa Conjecture). spanned
by Hecke
normalized
eigenforms.
Hecke eigen forms
The space
S
£(T )
These are in 11 correspondence f £
2 » ^he correspondence
is
2
with
being
such
that (12)
Z (s)
=
F
Ç(sk+l)Ç(sk+2)L(f,s)
As stated in the Introduction, most of the SaitoKurokawa 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\a2
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 nonassodated 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 nonprimitive
§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 positivedefinite rationalvalued 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 integerValued 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
thetaseries (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 2kl 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 onedimensional 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 thetaseries.
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 thetaseries 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 fT
n
= 2C(lk)
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 thetaseries (5). We then compute the m FourierJacobi
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 lefthand 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 righthand 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 thetaseries associated to the E lattice and g
to spherical polynomials of degree k4 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 finitedimensional
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)
,
ID1
E V=l
d
i
m
S
k vi + 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 lefthand 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(ml).
(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 nonzero 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 nearEisenstein 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
RPR
2
If any linear combination of the rows gives
0
the right, then the corresponding form <j> is zero.
2
P QRPR
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
^k6
+
^
M
\
8
'
Since the righthand 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 lefthand 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 nonzero 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 nontrivial linear combination
S f .
for
°^
m
~*isa
direct
sum, and choosing a Cbasis
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 > 2m3
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 2division 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+2m2  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 finitedimensional.
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
m1
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
•••^ m3
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)
(k1) 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 (k1) 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(8nr ) = 0 , r 2
for all n
I r c(8nr ) = 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(8nl)
c(8n),
and c(8n4) are expressed in terms of c(N) with N < 8(nl)).
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(12nr )qV
I r
2 V
(° c(12nr ) 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 nonzero 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(8nr ) 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 onedimensional
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 nontrivial
£, 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(24nr )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)
onedimensional). E a(n)q
or some
n
G M
S i m i l a r l y , if
g £ k12'
(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)) zkz 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 nontrivial 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 ) )  fW 2
2
o
= f}
m
(cf. proof of Theorem 5 . 6 ) .
m = 37 both spaces are onedimensional, 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
Vl
+ 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 2k2 on
FrickeAtkinLehner
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 nontrivial Diriehlet characters modulo 3
k
3
and modulo 4 , and calculate: V* / k + 2 v + 5
vf_
x
(2k3)(m+1) 24
\
v=0
V (3)
V
3
£
X (k+2V1) 3
=
3
{ Vb
^ ^
%
\ (k+2vl)
=
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
4m
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 ' " dN h
(
•
d )
In particular j (k,m) 5
Proof.
=
% ^ h (d) d4m
.
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 ) ,
lefthand 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)