в
Progress in Mathematics Vol. 55 Edited by J. Coates and S. Helgason
Birkhäuser Boston • Basel • Stuttgart
Martin Eichler Don Zagier
The Theory of Jacobi Forms
1985
Birkhäuser Boston • Basel • Stuttgart
Authors M a r t i n Eichler im L e e 27 CH-4144 Arlesheim (Switzerland)
Library
of Congress
D o n Zagier D e p a r t m e n t of Mathematics U n i v e r s i t y of Maryland College P a r k , M D 20742 (USA)
Cataloging
in Publication
Data
Eichler, M . (Martin) T h e t h e o r y of J a c o b i f o r m s . ( P r o g r e s s in m a t h e m a t i c s ; v. 55) Bibliography: p. 1. J a c o b i f o r m s . 2. Forms, Modular. I. Z a g i e r , Don, 1951. II. Title. I I I . S e r i e s : P r o g r e s s in m a t h e m a t i c s ( B o s t o n , M a s s . ) ; v, 5 5 . QA243.E36 1985 512.9'44 84-28250 ISBN 0-8176-3180-1
CIP-Kurztitelaufnahme Eichler,
der Deutschen
Bibliothek
Martin:
T h e t h e o r y of J a c o b i f o r m s / M a r t i n E i c h l e r ; D o n Z a g i e r . Boston ; Basel ; Stuttgart : Birkhäuser, 1985. ( P r o g r e s s in m a t h e m a t i c s ; V o l . 55) I S B N 3 - 7 6 4 3 - 3 1 8 0 - 1 (Basel . . . ) I S B N 0-8176-3180-1 (Boston) N E : Zagier, D o n B.:; G T
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Printed in Germany I S B N 0-8176-3180-1 I S B N 3-7643-3180-1 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
Fourier-Jacobi expansions of Siegel modular forms and the Saito-Kurokawa conjecture
72
Jacobi theta series and a theorem of Waldspurger .
81
The Ring of Jacobi Forms
89
8.
Basic structure theorems
89
9.
Explicit description of the space of Jacobi forms. 100 Examples of Jacobi forms of index greater than 1 . 113
10.
Discussion of the formula for
11.
Zeros of Jacobi forms
dim J, k,m
121 133
Tables
141
Bibliography
146
INTRODUCTION
The functions studied in this monograph are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jaoobi form on SL2(Z) to be a holomorphic function K x & -> <E
(3C = upper half-plane)
satisfying the two transformation equations 2TTimcz^ (1) * ( £ 3 ) - <« +d>k e CT +d" •<*.«> (t 9 e sv«)> /o\ . -\ . \ -2Trim(X2T +2Xz) ,f x , * — _2X (2) <))(T, z+XT + y) = e (j)(T,z) ((X y) G Z ) and having a Fourier expansion of the form oo (3) •(T.Z) - £ £ c(n.r) e2Wi(nT +RZM . n=0 r G I r2 < 4nm Here k and m are natural numbers, called the Weight and index of
(x,z) is a function of the type normally used to embed the elliptic curve (E/2£x + into a projective space. If m= 0, then § is independent of z and the definition reduces to the usual notion of modular forms in one variable. We give three other examples of situations where functions satisfying (l)-(3) arise classically: 1. Theta series. Let Q: 2? TL be a positive definite integer valued quadratic form and B the associated bilinear form. Then for N any vector xQG Z the theta (4) G (T,z) = Eseries e^ Q 'o 2
-1-
Ì(
(x)T+B(x
X
)z)
2
is a Jacobi form (in general on a congruence subgroup of S L ( Z ) ) of 2
weight N/2 and index Q ( x ) ;
2
the condition
Q
r < 4nm
the fact that the restriction of Q to I x + Z x binary quadratic form.
in (3) arises from
is a positive definite
Q
Such theta series (for N=1) were first studied
by Jacobi [ 1 0 ] , whence our general name for functions satisfying (1) and ( 2 ) .
2.
Fourier coefficients of Siegel modular forms.
Siegel modular form of weight (z
T
F
) with
function
z
G
(
C
>
T,T
f
G K
k and degree 2.
(and
Im(z)
2
Fourier expansion with respect to x
1
Then w e can write
Z as
1
I m ( x ) I m ( x ) ) , a n d the
<
x, z
F is periodic in each variable
Let F(Z) be a
and x\
Write its
as
oo
F(Z)
(5)
then for each
=
E Ф (т m=0 т
6
m the function
T
index
m , the condition
4nm £ r
> 2
)
e
2 l T l m T
'
;
is a Jacobi form of weight
m 2
k and
° in (3) now coming from the fact that F 27Ti Tr(TZ)
has a Fourier development of the form
X c(T) e
where
ranges over positive semidefinite symmetric 2x2 matrices.
T
The expan
sion (5) (and generalizations to other groups) was first studied by PiatetskiShapiro expansion
[26] , who referred to it as the Four-ier-Jacobi
of F and to the coefficients
c|>
as Jacobi functions, a word
which we will reserve for (meromorphic) quotients of Jacobi forms of the same weight and index, in accordance with the usual terminology for modular forms and functions. 3. The Weierstrass ff—function.
(6)
|»(T,z)
=
z"
2
+
Yl
The function
2
2
( ( z + 0 J) " Ü J~ )
(0€Z +ZX is a meromorphic Jacobi form of weight 2 and index 0 ;
we will see
-3later how to express it as a quotient of holomorphic Jacobi forms (of index 1 and weights 12 and 1 0 ) . Despite the importance of these examples, however, no systematic theory of Jacobi forms along the lines of Hecke*s theory of modular forms seems to have been attempted previously.*
The authors* interest
in constructing such a theory arose from their attempts to understand and extend Maass
1
beautiful work on the "Saito-Kurokawa conjecture".
This conjecture, formulated independently by Saito and by Kurokawa
[15]
on the basis of numerical calculations of eigenvalues of Hecke operators for the (full) Siegel modular group, asserted the existence of a "lifting" from ordinary modular forms of weight Siegel modular forms of weight
k
2k-2
(and level one) to
(and also level o n e ) ; in a more
precise version, it said that this lifting should land in a specific subspace of the space of Siegel modular forms (the so-called Maass "Spezialschar", defined by certain identities among Fourier coefficients) and should in fact be an isomorphism from
^2k-2^ 2^^ L
onto this space,
mapping Eisenstein series to Eisenstein series, cusp forms to cusp forms, and Hecke eigenforms to Hecke eigenforms.
Most of this conjecture was
proved by Maass [21,22,23], another part by Andrianov remaining part by one of the authors
[40].
[2], and the
It turns out that the
* Shimura [31,32] has studied the same functions and also their higherdimensional generalizations. By multiplication by appropriate elementary factors they become modular functions in x and elliptic (resp. Abelian) functions in z, although non-analytic ones. Shimura used them for a new foundation of complex multiplication of Abelian functions. Because of the different aims Shimura s work does not overlap with ours. We also mention the work of R.Berndt [3,4], who studied the quotient field (field of Jacobi functions) from both an algebraic-geometrical and arithmetical point of view. Here, too, the overlap is slight since the field of Jacobi functions for S L ( Z ) is easily determined (it is generated over C by the modular invariant j (x) and the Weierstrass p-function ^ ( x , z ) ) ; Berndt s papers concern Jacobi functions of higher level. Finally, the very recent paper of Feingold and Frenkel [Math. Ann. 263, 1983] on Kac-Moody algebras uses functions equivalent to our Jacobi forms, though with a very different motivation; here there is some overlap of their results and our §9 (in particular, our Theorem 9.3 seems to be equivalent to their Corollary 7.11). f
2
1
-4-
conjectured correspondence is the composition of three isomorphisms
Maass "Spezialschar" C
M^(Sp^(Z))
\l v Jacobi forms of weight
k and index
1
\l
(7) Kohnen s
(r
~F~" space ([11]) C
?
(4))
l
l
M _ (SL (20) 2 k
;
2
2
the first map associates to each
F
the function
(|> defined by (5) , the i
second is given by
Z n>0
,
\
c(n) e
2nrinT
^
—>
V
L
V
2-,
//
2N
c(4n-r ) e
2TTi(nx + rz)
,
n^O r < 4 n 2
and the third is the Shimura correspondence [29,30] between modular forms of integral and haIf-integral weight, as sharpened by Kohnen
[11]
for the case of forms of level 1. One of the main purposes of this work will be to explain diagram (7) in more detail and to discuss the extent to which it generalizes to Jacobi forms of higher index.
This will be carried out in Chapters I
and II, in which other basic elements of the theory (Eisenstein series, Heeke operators, ...) are also developed.
In Chapter III we will study
the bigraded ring of all Jacobi forms on SL (7Z) Z
.
This is much more
complicated than the usual situation because, in contrast with the classical isomorphism
M (SL„(S)) = C [ E , , E ] , C
(J^ ^ = Jacobi forms of weight
k
and index m)
the ring
J
=
©
J,
is not finitely generated.
Nevertheless, we will be able to obtain considerable information about the structure of bounds for
dim J^ ^
will prove that r
ring
M (SL (S)), i
J^ ^.
?
J
*,m
In particular, we will find upper and lower
which agree for k = © J, k k,m
m
sufficiently large
is a free module of rank
(k^m),
2m over the
and will describe explicit algorithms for finding
-5bases of
J. k,m
M (SL ( Z ) ) . >V
as a vector space over
J *,m
as a module over
The dimension formula obtained has the form
2
(8)
dim J
k
m
=
' for
(C and of
r
k
E
dim \
+ 2
r " N(m
r=0
even (and sufficiently large), where
m j- 2 -) £ 7~ r=0 » »
N(m)
4
We will show that
)
N(m)
is given by
( fx ] = smallest integer > x)
m
N ( m ) can be expressed in terms of class numbers of
imaginary quadratic fields and that (8) is equivalent to the formula
(9)
dim J
where
^2^2
= dim M _ ( r ( m ) ) 2 k
k j m
2
0
~*"S t*ie sPace °^ new f°rms °f weight
which are invariant under the Atkin-Lehner r/ \
^
f(T) — > m
"new"
-k+1
T
-2k+2
_,
N
f(-l/mT)
,
and
_new m
2k-2
on
F0(m)
(or Fricke) involution .
,-
, _.
,
_
a s u i t a b l y d e f i n e d s p a c e of
Jacobi forms. Chapter IV, which will be published as a separate work, goes more
deeply into the Hecke theory of Jacobi forms.
In particular, it is
shown with the aid of a trace formula that the equality of dimensions (9) actually comes from an isomorphism of the corresponding spaces as modules over the ring of Hecke operators. Another topic which will be treated in a later paper (by B.Gross, W.Kohnen and the second author) is the relationship of Jacobi forms to Heegner points.
These are specific points on the modular curve
X (m) = 3C/T (m) U {cusps} Q
o
(namely, those satisfying a quadratic equa-
tion with leading coefficient divisible by each
n
and
r with
r
2
P(n,r) £ Jac(X (m)) (C|) 0
< 4nm
m) .
It turns out that for
one can define in a natural way a class
as a combination of Heegner points and cusps and
-6that the sum
V
p
(n,r) q
£
n
is an element of
r
nr
Jac(X (m))(Q) ®
One final remark.
J
9
^
0
, m
Since this is the first work on the theory of
Jacobi forms, we have tried to give as elementary and understandable an exposition as possible.
This means in particular that we have always
preferred a more classical to a more modern approach (for instance, Jacobi forms are defined by transformation equations in 3Cx €
rather
than as sections of line bundles over a surface or in terms of the representation theory of W e i l s metaplectic group), that we have often 1
given two proofs of the same result if the shorter one seemed to be too uninformative or to depend too heavily on special properties of the full modular group, and that we have included a good many numerical examples. Presumably the theory will be developed at a later time from a more sophisticated point of view.
*
*
* &
This work originated from a much shorter paper by the first author, submitted for publication early in 1980.
In this the Saito-Kurokawa
conjecture was proved for modular (Siegel and elliptic) forms on P ( N ) Q
with arbitrary level
N.
However, the exact level of the forms in the
bottom of diagram (7) was left open. as here in §§4-6.
The procedure was about the same
The second author persuaded the first to withdraw his
paper and undertake a joint study in a much broader frame.
Sections 2
and 8-10 are principally due to the second author, while sections 1, 3-7 and 11 are joint work. The authors would like to thank G. van der Geer for his critical reading of the manuscript.
-7Notations We use
IN to denote the set of natural numbers,
We use Knuth's notation
[xj
greatest-integer function
(rather than the usual
max{nG z|n ^ x}
fx] = min{nGZ|n^>x} = -L"" J* By
d||n
we mean
d|n
and
(d,-j) = 1.
0
for
TN U {0}.
[x] ) for the
and similarly
The symbol •
X
]N
denotes any square number.
In sums of the form
^
£ d
l
or n
/ it is understood that the summation is over positive divisors only. ad=£ The function £ d (d G IN) is denoted O (n) . d|n The symbol e(x) denotes e , while e (x) and e (x) (mG]N) m t
V
m
denote
e(mx) and
e(x/m),
complex variable, but in e (ab
)
1
m
means
We use
e m
M
respectively.
e m
( n ) with and
t
I
(x)
2
bn = a (mod m) , and not
m
is a
Z/mZ ; thus
e(a/bm) .
The symbol
[a,b,c] denotes the quadratic
2
denotes the upper half-plane {xG (C|lm(x) > 0 } .
x = u + iv,
z = x + iy,
often be denoted by weight
k
E^ G
E
e (x) , x
for the transpose of a matrix and for the n x n
z will always be reserved for variables in K
with
has
e(x) and
a x + bxy + c y . K
and
In
it is to be taken in
identity matrix, respectively. form
x
on
T
i
F
x
by
q = e(x),
£ = e(z) .
The letters
and
X
The group
S L ( Z ) will 2
and the space of modular (resp. cusp) forms of (resp.
S^) .
The normalized Eisenstein series
(k > 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
functions
(!)
X
S L ( Z ) and
Z
2
2
on
This action, analogous to the action
(f| M)( ) k
In order to treat
(c
T
(g±i)
(M=C
d)-* f
T +
^er,)
in the usual theory of modular forms, will be important for several later constructions (Eisenstein series, Hecke operators). integers
k
and
m
We fix
and define
(2)
and
W l [ X y])(x,z)
(3)
m
:=
e ( X x + 2Xz) ( ( ) ( T , Z + AT + p) m
2
((X y) e where
e ( x ) = e^ "^"^ m
77
(see "Notations") .
m ),
Thus the two basic transfor-
mation laws of Jacobi forms can be written
<j)| M=<j> 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
(
k m
M
1
=
«k,m
M
I^XM =
>
(MM)'
<j>L
'k,m
(<j>| x)
,
'm
| M ,
((j)| x)
<j)| (x + x )
=
(MM , G
,
1
>m T
K)IN
m
I x*
y
'm
x,x'
r , 1
z ) 2
G
They show that (2) and (3) jointly define an action of the semi-direct :=
product
r
and group law
Kl
x
(= set of products (M,X) with
2
(M,X) (M ,X ) = (MM , XM + X ) ; f
f
1
r
our vectors as row vectors, so Jaoobi group.
1
f
M G ^ ,
X G Z
2
notice that we are writing
acts on the right), the (full)
We will discuss this action in more detail at the end
of this section. We can now give the general definition of Jacobi forms.
Definition. a subgroup
A Jaoobi form of weight
r C r
k
and index
m
(k,m G M)
of finite index is a holomorphic function
on
<J>: Jf x C -> I
satisfying $1.
i) ii)
M = 4>
X 'm
4>|
iii)
(M G T ) ;
(j)
=
for each
(X G
M G F,
2
(t) | ^ ^M
1
form S c ( n , r ) q V unless
% ) ;
(q = e ( x ) , £ = e(z)) with
n > r / 4 m . (If 2
c(n,r) 4 0
n>r /4m, 2
r
J
Remarks.
J, k,m
for
it is called a cusp form.) T
d) is denoted
J, (F) ; k,m
if
J, (F ) . k,m i
1. The numbers n,r
in iii) are in general in
in TL (but with bounded denominator, depending on T
and M ) .
2. We could define Jacobi forms with character, J inserting a factor X(M)
c(n,r) = 0
(J) satisfies the stronger condition
The vector space of all such functions F = r, we write simply . i ^
has a Fourier development of the
fe
m
(F,X),
by
in i) in the usual way.
3. Also, we could replace
TL
1
by any lattice invariant under
e.g. by imposing congruence conditions modulo
N
if
r = T(N).
V,
It would
therefore be more proper to refer to functions satisfying i)-iii)
«10J as Jacob! forms on the Jacobi group
2
T
=
T * %
(rather than on F ) .
However, we will not worry about this since most of the time we will be concerned only with the full Jacobi group. Our first main result is The space
THEOREM 1.1.
J^ ^(T) is
finite-dimensional.
This will follow from two other results, both of independent interest:
THEOREM 1.2. Let xGje,
fixed exactly
1
1
2TTI
J
)
T
Z ••,
Then for zero, has
in any fundamental
domain for
Z x + Z on (C.
It follows easily from the transformation law ii) that
3D)
$ ^'^
(
-r—r 9
8
(counting multiplicity)
of the lattice
Proof.
m.
z I—=>cj)(x,z), if not identically
the function
2m zeros
the action
$ be a Jacobi form of index
Z
— — - dz = 2m
(<j> = T T ,
cp(x,z)
i
F
when one replaces
1
dz
F = fundamental domain for C / Z x + Z ) ,
z
2-nl "cfT
(the expression
of the theorem.
*
z
z by
^
s
invariant under
z -> z+1
and changes by 2m
z + x ) , and this is equivalent to the statement
Notice that the same proof works for cf) meromorphic
(with "number of zeros" replaced by "number of zeros minus number of poles") and any m e Z.
A consequence is that there are no holomorphic
Jacobi forms of negative index, and that a holomorphic Jacobi form of index
0 is independent of
of weight
(and hence simply an ordinary modular form
k in x ) .
THEOREM 1.3. index
z
Let § be a Jacobi form on
m and X,u rational
numbers.
m
of
r
f
k and
Then the function
f(T) = e ( X x ) ( x , Xx + u) is a modular subgroup
T of weight
of finite index depending
form
(of weight
k and on some
only on V and on
X,yJ.
-11For weight
k
A=y=0 on
F.
T -> 4>(x,0)
it is clear that
is a modular form of
We will prove the general case later on in this section
when we have developed the formalism of the action of the Jacobi group further.
f(x)
Note that the Fourier development of
at infinity is
£ e(ry)e(n,r) e((mA + r A + n ) x ) n,r
,
2
so that the conditions
n £ 0,
r
< 4nm
2
in the definition of Jacobi
forms are exactly what is required to ensure the holomorphicity of at
0 0
f
in the usual sense. To deduce 1.1, we pick any 2m pairs of rational numbers
(A ,y ) G q i
f (x)
functions
(A ,y ) f
with
Z
jL
i
(A.. ,y^.) (mod 7L )
2
m
of
F,
Therefore
and the map
dim J^ ( H ' also shows that J, k,m m
for
1
= e ( A x ) c | ) ( x , A^x + y^)
i
subgroups
i
i^j.
lie in M ^ T ^ )
§ -> {f^}^
Then the
for some
Is injective by Theorem 1.2.
< £ dim M ^ X F . ) ; this proves Theorem 1.1 and i is 0 for k < 0 unless k = m = 0, in which case it
reduces to the constants. To prove Theorem 1.3, we would like to apply (3) to (A,y) G (Q) . 2
However, we find that formula (3) no longer defines a group action if we allow non-integral
A
and
(cj)| [A y ] ) | [A
1
'm
and
y,
since
y ])(x,z)
=
?
'm
=
e ( A x + 2 A z + A x + 2A(z + A x + y ) ) ( { ) ( x , z + A ' x + y + Az + y)
=
e(2mAy )(<})| [A+A
m
, 2
2
T
,
m
e(2mA y) f
t
f
T
T
(x,z)
y+y ]) 1
will not in general be equal to
1.
Similarly, the third
2
equation of (4) breaks down if
X
is not in
extend our actions to SL (<$) (or SL (]R)) 2
2
%
and
. Q
2
Hence if we want to (or ]R ) , we must 2
modify the definition of the group action. The verification of the third equation in (4) depends on the two
-12elementary identities
z
^ ат + b
+
ex + d
1
_
+
ex + d
^
H
i
CT + d
»
о
2 X Л
where
, , ат+b , z — + 2Х — cT + d cT + d
(A
1
arbitrary
2 CZ , — + Ау cT + d л
^ .
M = ^
2
1
^
=
и
u ) = (A u) ^
л
c(z + А_т + у ) i ^1 , , — + Xu , ет + d ii '
2 , , X т + 2X,z i i 0
H
Thus to make this equation hold for
G SL (M)
and
X = ( Au ) G ]R
2
we should replace
(3) by
(5)
((j)|[A y])(x,z)
m
2
e ( A x + 2Az + Ay)(j)(x, z + Ax + y)
:=
m
2
((A y) G JR ) ; m
this is compatible with (3) because
e (Ay ) = 1
for
A,y G Z .
Unfortunately, (5) still does not define a group action; we now find
(6)
(cJ)| X)| X m
m
T
=
m
e (Ay
f
A'yHl^X + X ) 1
(X = (A y ) ,
X
f
2
= (A' y') G ]R )
.
To absorb the extra factor, we must introduce a scalar action of the group
IR by
(cj)|
(7)
[K])(X,Z)
:=
and then make a central extension of 1R replace H
2
( K G ]R)
e(mK)(f>(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^) <
0 0
JR • S . 2
1
, and if we
choose the maximal compact subgroup K
then G /K
:=
J
S0(2) x
s
1
C
G
J
=
SL ( JR) ix (]R • S ) 2
can be identified naturally with K x (C
[7 a
b\
.
via
^ /ai + b
1..J
1
2
Xi + y \
The above discussion now gives
THEOREM 1.4. X G I
2
,
|c|
^ G(C,
G
Let
J
=1).
be the set of triples Then
IXX^KM'^ ,^]
-
1
and the
G
d)' =
[MM',XM'+X , C .e(det(™ ))] f
( T
, -k (cx + d)
iii/
^v^F+d
defines an action
of
the transformation
G^
?
Z )
2
.
o
,
. ^ \ /
o-v
z + XT + y \ '
cx+d
on {<j>:
;
x <E -> (E}.
The functions
(J) satisfying
laws i) and ii) of Jacobi forms are precisely
with respect
2 = T k 7L of
'
m / c(z 4- X T + y) e ( -f- d
1N
, /aj + b
j
f
?
( A
C
x
F
2
is a ^powp i;ia
J
formula
*' [(c
invariant
(Me SL (H),
[M,X,£]
to this action under the discrete
those
subgroup
j G
y
and the space of such F(g)
with the set of functions
:=
$ can be identified
via
(<J>|g)(i,0)
F: G
J
trans forming on the right by the
-> (C left invariant representation
under
V
J
and
-15-
'Hues ss).™.']) • of the maximal
compact
K
subgroup
Thus the two integers
J
k "and
= S0(2) x S
m
f ' ™ * ™ G.
of
1
J
in the definition of Jacob! forms
appear, as they should, as the parameters for the irreducible (and here G.
one-dimensional) representations of a maximal compact subgroup of
J
As an application of all this formalism, we now give the proof of 1.3.
The function
(namely
e (Au))
and
f(x)
equal to
m
in that theorem is up to a constant (<J>|x) ( T , 0 ) ,
<j>CO := x
where
X=(Ay)GQ
<J>|X is defined by (5) (from now on we often omit the indices
on the sign
I ).
For
X
f
= (A
h+X
l(T)
by (6) , so <J>
1
y ) G TL
we have
e (Ay
-A y)(f) (x)
f
=
1
m
T
X
(mod N Z ) if
x
X G N
2
k,m
f
depends up to a scalar factor only on
itself depends only on
Z .
_ 1
2
X For
(mod 7L ) and <j> l
M = ^
d)
G
we have
(cx +
so (j)
D ) -
K
4>(f£}) x
=
W|X|M)(T,0)
=
(4>|M|(XM))(T,O)
=
(<J>|(XM))(T,0)
behaves like a modular form with respect to the congruence
subgroup
{MGr|XM=X
of
F
(mods2), m.det|
X x
M
|G
(this group can be written explicitly
|(c
d )
G
r
l ( -D a
X +
2
b A + (d-l)y, m ( c y + ( d - a ) A y - b A ) 2
2
Gz|
F
-16-
T
element of
=
x
ing
(X
U ) = XM,
x
q C n
N
if NX £ Z
(
2
).
Finally, if M
is
any
e (X T + X ^ K ^ M M x , X T + m
2
X
and since (f) | M
1
only for 4nm > r ,
r
\ J
2
then
1
(l M)(T) k where
/
F n T
and hence contains
has a Fourier development contain-
this contains only nonnegative powers of
2
e(x) by the same calculation as given for M = Id
after the statement
of 1.3. We end with one other simple, but basic, property of Jacobi forms.
THEOREM 1.5.
Proof. weight k
ring.
That the product of two Jacobi forms (j> and of i
and
2
The Jacobi forms form a bigraded
k
?
and index m
and m ,
x
a Jacobi form of weight
2
k = kj+
k
respectively, transforms like
and index
2
2
we have to check the condition at infinity.
m = m 4-m 1
is clear;
2
One way to see this is to
use the converse of Theorem 1.3, i.e. to observe that the condition at infinity for a Jacobi form (J)(T,Z) of index condition that
f(x) = e (X x)(f>(Xx + y) m
usual sense) for all
m
is equivalent
be holomorphic at
2
0 0
to the
(in the
X,]i £ q ; this condition is clearly satisfied for
(j)(x,z) = <j> (T,Z)<J> (T,Z).
with
f(x) = f (x)f (x). 1
A more direct proof
2
is to write the (n,r)-Fourier coefficient of as
c(n,r)
=
^ n +n =n 1
r
where the c^ since
n. £ n,
l
C l
(n,>r )c (n ,r ) 1
2
2
?
2
+ R
2=
R
are the Fourier coefficients of
(f^
(the sum is finite
r < 4 n . m . ) and deduce the inequality
r
2
< 4nm
2
from
the identity ( ^ ! n
i
+
N
2 "
4(m
+
R
1 +
2 >
m ) 2
/
2
~
\
/
A\ n
i
Anij /
+
V""
_
4_\
4m / 2
(
+
M
1
R
2 -
M
2
r
i
im^im^m^
)
-17-
This identity also shows that (as for modular forms) the product 4 f ) (
>
1
is a cusp form whenever $ for modular forms) (^^ l
The ring
J
or (f>
2
is one but that (unlike the situation
can be a cusp form even if neither $
2
*>*
l
=
© J, k,m > k
m
2
l
nor <j) is. 2
of Jacobi forms will be the object
of study of Chapter III.
§2.
Eisenstein Series and Cusp Forms As in the usual theory of modular forms, we will obtain our
first examples of Jacobi forms by constructing Eisenstein series.
In
the modular case one sets (for k > 2)
where
1^ = {± Q
1L =1, 'k
where
^
|nG 7L}
is the subgroup of 1^ of elements y
1 denotes the constant function.
Similarly, here we
define
(1)
where
n , l i G Z}
Explicitly, this is
(2)
with
.
-18-
where
a ,b
a re chosen so that ^
€ T .
^
As in the case of modular
l
forms, the series converges a bsolutely for k > 4; it is zero if k odd (repla ce
c,d
by - c , - d ) .
The invariance of
J
T
under
m
clear from the definition a nd the absolute convergence.
is
is
To check the
cusp condition, and in order to have an explicit exa mple of a form in J
k
m' w e must ca lcula te the Fourier development of
^ , which we now
proceed to do. As with as
c
is
E ^ , w e split the sum over
0 or not.
E XGZ
(3)
(q = e
If
2 7 T i T
4nm = r
2
,
£ = e
2 7 r i z
c=0, then
d = ±1;
2
m
c,d
e (X T+2Xz)
=
into two parts, a ccording
these terms give a contribution
E q XGZ
m
X
%
2
m
X
n
).
This is a linea r combina tion of q £
r
with
a nd corresponds to the constant term of the usual Eisenstein
series.
If c ^ 0,
we can assume
c>0
(since k
is e v e n ) ; using the
identity
allk
+
2 X
cT+d
_z
J^L
CT+d
=
- c(z-X/c)l
CT+d
cT+d
^
+
c
we ca n write these terms as
c=l
Note tha t
d
d€l XGZ (d,c)=l d+c
a nd
x
X ->• X+c
correspond to
so this pa rt equa ls
(4) d (mod с) (d,c)=l
with
e
a s in "Notations" and с
A(mod с)
z -> z+1
a nd
T -* T + l ,
-19-
F ,.(T..)
==
k
(
E
T
+
p , ^ e - ( - i ^ i )
p,qG IL the function
F^
m
r
is periodic in T
and z,
so (4) makes sense.
Now
the usual Poisson summation formula gives
\, - E
Y(n,D
m
q V
with
y(n,r)
j
=
T k e(-nT)
Im(T)=C
(C >0, 1
C
2
arbitrary).
(x/2im)^ e ( r T / 4 m ) . 2
Y(n,r)
|
e(-mz /i - rz) dz dx 2
Im(z)=C
1
2
The inner integral is standard and equals
Hence
=
}
T
-k
( T
/21m)
, 5
e(^fflT)dT
ImCT)^! 0
if
r > 4nm 2
! 2.k- /2
1-k,,
a^m
3
(4nm - r )
2 /
.
r
if
,
r < 4nm
with
.
c-D
'
2
s
k
k / 2
k _ 2
^
r(k-^)
2
r > 4nm, we can deform the path of integration to +i°°, so y = 0 ;
(if if
r < 4nm, 2
we deform it to a path from -i°° to
-i
0 0
circling
0
once
in a clockwise direction and obtain a standard integral representation of
1/r(s) ) .
Substituting the Fourier development of F^ ^
gives the expression
E w*'> <^ r
n,r<E Z 4nm > r with
2
r
into (4)
20
3
k /
OL
(5)
e
k
m
(n,r)
2
dX
1
^ c
(4nm r ) m
(for d
~
2
=
/
1
X
2
- rX + nd)
X,d(mod c) (d,e)=l
c=l
, see "Notations").
^(md
To calculate this, we first replace
in the inner double sum
X
by
(since (d,c) = 1, this simply permutes the
summands); then the summand becomes
e^(dQ(X) ) with
2
Q(X) := mX + rX + n.
We now use the wellknown identity
e
£ c d(mod c) (d,c)=l where y
(
d
N
)
=
y
(
£ t > a | (c,N)
a
•
is the Möbius function (socalled Ramanujan sum; see Hardy
Wright or most other number theory texts); then the inner double sum in (5) becomes
'
X(mod с) Q ( X ) = 0(mod a)
Q(X) = 0(mod a)
Now the condition inner sum is
—
depends only on
X (mod a ) , so the
times N ( Q ) , where a
N (Q) a
:=
#{X(mod a) | Q(X)= 0(mod a)}
.
Hence the triple sum in (5) simplifies to 0 0
c=l
i
i
'
a|c
(the last equality follows by writing
N_(Q)
0 0
a=l
c = ab
a
and using S y ( b ) b
To calculate the Dirichlet series, we first calculate
N (Q) a
S
1
= £(s) ) .
for (a,m)= 1;
this will suffice completely if m = 1 and (using the obvious multiplica tivity of N ) a
will give the Dirichlet series up to a finite Euler
product involving the prime divisors of m
in general.
If (a,m) = 1 ,
-21then N (Q) a
=
#{X(mod a) | m X + r X + n = O(mod a)}
=
#{X(mod a) | ( 2 m X + r )
=
N (r
2
- 4nm)
2
a
= r - 4 n m (mod 4a)}
2
2
,
where
N (D)
:=
a
#{x(mod 2 a ) | x = D(mod 4a)} . 2
It is a classical fact that
(6)
£ N (D)aa=l
if D = 1 or if D
is the discriminant of a real quadratic field,
S
a
where
L^(s) = L ( s , ~ ) )
=
L ( ) D
,
S
is the Dirichlet L-series associated to D.
It was shown in [39, p.130] that the same formula holds for all D ^ Z if L^(s) is defined by
S
L
O
if D ^ 0 , 1 (mod 4) ,
C(2s-1)
if D = 0 ,
D
(s) • £
y(d)(-^)d~
S
2
0 _ (f/d) ±
if
2s
where in the lasto lined| Df has been written as D .-f S
D
Q
D = 0 , 1 (mod 4 ) , D ^ O
with
f G I and
= discriminant of Q ( / D ) (the finite sum in this case can also be
written as a finite Euler product over the prime divisors of f) . Inserting (6) into the preceding equations, we find that we have proved e
k j l
(n,r)
=
a |D| k
k _ 3 / 2
C(2k-2)
_1
L (k-l) D
if m=l and D = r - 4n < 0, while for m arbitrary there is a similar 2
formula (now with
D = r -4nm) 2
but multiplied by an Euler factor
-22involving the prime divisors of L^(s)
and
m.
Using the functional equations of
C(s) we can rewrite this formula in the simpler form e
(n,r)
fc
=
L (2-k)/?(3-2k) D
where now all numerical factors have disappeared. (D < 0 ,
k
,
The values
L^(2-k)
even) are well-known to be rational and non-zero; they have
been studied extensively by Cohen [ 6 ] , who denoted them
H(k-1, | D | ) .
Summarizing, we have proved
THEOREM 2.1. a non-zero
The series
element of
E
J, . k,m
J
fc
^
(k > 4 even) converges
The Fourier
development
of ^
r
and
defines
E, k,m
is
given by
E
k,m > ( T
-
z )
£
e
4nm > r where
e^ ( n , r )
for
m
wise , while for
4nm = r
4nm > r
(n
r
2
1 if
r = 0(mod 2m) and
0
other-
we have
2
x
( e
equals
2
k,m ' >
k,l
U
,
r
H(k-1, 4 n - r ) C(3-2k) 2
=
;
(H(k-1,N) = L_ (2-k) = Cohen s function) and f
N
e
km^
n
' ^ r
=
~
^rk'^r
T
^ * TT
(elementary p-factor) .
p|m
In particular,
e, (n,r) 6 k,m
r
One can in fact complete the calculation of little extra work; the result for m
0"
<> 7
% ^> m
(m)
e^
m
in general with
square-free is
^
=!TF2krL
d -H(k-l.^zr_)
d |(n,r,m)
K 1
•
-23However, we do not bother to give the calculation since this result will follow from the properties of Hecke-type operators introduced in §4 (Theorem 4 . 3 ) . For of
m = 1
H(k-1,N)
E
E
g
i
8
i
k
we find, using the tables
given in [6] , the expansions
=
1 + (£ + 56C + 126+ 56C" + C~ )q
+
(126C + 576C + 756 + 576?"" + 1 2 6 C ~ ) q
+
(56? + 7 5 6 ? + 1512^+ 2072 4-1512C" + 756C~ + 56?~ )q
2
1
2
2
1
3
=
2
2
2
1
1 + ( Ç - 88Ç - 330 - 8 8 Ç 2
+
E
and the first few values of
_ 1
2
3
2
2
1 + (Ç
2
+ ... ,
+ Ç )q
(-330Ç - 4 2 2 4 Ç - 7 5 2 4 - 4224c" - 330Ç~* )q + ...
=
3
1
+ 56Ç 4- 366 4- 56Ç
1
2
+ Ç )q 2
,
2
4- . . .
2
Further coefficients of these and other Jacobi forms of index
1
are
given in the tables on pp.141-143.
In the formula for the Fourier coefficients of striking that
e^ ^(n,r)
depends only on 4 n - r . 2
E^ ^,
it is
We now show that this
is true for any Jacobi form of index 1; more generally, we have
THEOREM 2.2.
Let
§
development
2 c(n,r)q £ .
r(mod 2 m ) .
If
n
k
only on 4nm - r . 2
Proof.
r
be a Jacobi form of index Then
c(n,r)
is even and
m=l
If
k
m = l and
or
depends m
only on
is prime,
is odd, then
m
then
with
Fourier
4nm - r
2
c(n,r)
(j) is identically
and
on'
depends zero.
This is essentially a restatement of the second transfor-
mation law of Jacobi forms: we have
24
n
S c(n,r)q Ç
r
=
<j>(T,z)
т
е ( Л Т + 2Хг)ф(т, + Хт + и)
=
2
2
2
=
m X 2mA п . Х г q Ç Sc(n,r)q (Çq )
=
« z V n + rX + m X S c(n,r) q
v
f
ч
ч
2
^r+2mA Ç
and hence c(n,r)
T
i.e.
r
c(n,r)= c ( n , r ) whenever
as stated in the theorem. c(n,r) = c(n,r)
=
r
f
2
c(n+rX + mX , r + 2 m X )
== r(mod 2m)
T
and
4n mr
,
t 2
2
= 4nm r ,
If k is even, then we also have
(because applying the first transformation law of k
Jacobi forms to I £ r
gives J() ( x , - z ) = (1) 0 ( x z ) ) , ?
2
so if m
is 1
or a prime, then f
4n mr'
2
=
4nmr
Finally, if m = l and k 2
but
4 n m (r) = 4nm r
Remark:
2
r
T
f
is odd then <J) = 0 because 2
and
f
= ±r(mod 2m) => c(n,r) = c ( n , r ) .
r = r(mod 2m)
c(n,r) = c(n,r)
in this case.
Theorem 2.2 is the basis of the relationship between
Jacobi forms and modular forms of halfintegral weight (cf. § 5 ) .
In the definition of Jacobi cusp forms, there were apparently infinitely many conditions to check, namely 2
4nm = r .
c(n,r) = 0
residue classes
r(mod 2m) with
largest square dividing 1
2
2ab I r) . 1
m
r
2
T
è G J k,m
2
with
a
2
is the
squarefree, then
we have
2
c (as , 2abs) =
c u s
in particular, the codimension of j P k,m 5
The number of
= 0 (mod 4m) is b , where b
(namely if m = a b
Thus for
<J> a cusp form
1
with
Theorem 2.2 tells us in particular that we in fact need only
check this for a set of representatives of r (mod 2 m ) .
4m ! r
for all n,r
±
n
0 for s = 0,1,. . . ,bl
j k,m
is t most b. a
;
Using
-25k c(n,-r) = (-1) condition
c(ti,r)
if
THEOREM 2.3. ———
largest integer
k
s = 0,1,
is odd.
(vesp.
of
jf k,m
i
is even and
k
f
in
U S p
J
|_"^2^J if
such that
>["jJ
Hence we have
The oodimension
k is even
+
for
2
s = 1,2,.. ., |^~—J
["2"] l
we see that in fact it suffices to check the
c (as , 2abs) = 0
k
J, k,m
is at most
is odd), where
b
is the
b | m. 2
On the other hand, if k > 2
then for each integer
s
we can
construct an Eisenstein series
E, (T,Z) :k,m,s
(8)
V /
e r 1
(m=ab of
2
q
j
a s 2
S
2 a b s
|Y
J\ J oo r
as above), where the summation is the same as in the definition
E = E, ~. k,m k,m,0 T
Then repeating the beginning of the proof of
Theorem 2.1 we find that
(9) \ , , m
-
s
*
£
r 2 / 4 m q
r
(-i)V ) r
+
+
r G 7L r = 2abs (mod 2m)
where " . . . " (the contribution from all terms in the sum with c ^ 0) has a Fourier development consisting only of terms It is then clear that E. = (-1) k,m,-s (k even) or
k
m
s
depends only on
E. , and that the series k,m,s
0<s
q £ n
r
with
4nm - r
2
> 0.
s(mod b ) , that EL Tc,m,s
b 0 < s < -r~ ~ 2
with
(k odd) are linearly independent.
Comparing this
with 2.3, we see that the bound given there is sharp and that we have proved:
THEOREM 2.4. —
J / k > 2 , then J
is the space of cusp forms in
m
J, k,m
= jf k,m
and
J^ ^
u s p
1
© jf k,m
1 8
,
where
J? K,m
U S P
is the space spanned by
-26-
the functions
E. . k,m,s
J
or
The functions
E, k,m,s
J
0 < s < 77 (k odd) form a basis
J?
2
i s
with
0 < s < ~ (k even) = 2
.
k,m
We will not give the entire calculation of the Fourier developments of the functions the result.
E. k,m,s
here, since it is tedious and we do not need
However, we make some remarks.
certain operators higher index.
and
In §4 we will introduce
which map Jacobi forms to Jacobi forms of
These will act in a simple way on Fourier developments
and will send Eisenstein series to Eisenstein series. combinations of the E, k,m,s
("old
Hence certain
forms") have Fourier coefficients which
can be given in a simple way in terms of the Fourier coefficients of Eisenstein series of lower index (compare equation ( 7 ) , where the coefficients of
E
K. , m
are simple linear combinations of those of
and we need only consider the remaining, "new", forms.
E
),
K. , x
A convenient
basis for these is the set of forms
(10)
E
( X )
k ,m 1
:=
Z
X ( S )
Vm.s
(
m
=
f
2
)
s(mod f) of index
f , where
X
is a primitive Dirichlet character (mod f) with
k X(-l) = (-1)
.
Then a calculation analogous to the proof of Theorem 2.1
for the case m = l shows that the coefficient (11)
e
k*m
if (r,f) = 1 , where and
e(X)
( n
'
r )
L^(s,X)
=
e
(
X
)
X
(
r
)
V_
4 n m
q Ç n
r
in
is given by
(2-k,X)
is the convolution of L^(s)
and
L(s,X)
a simple constant (essentially a quotient of Gauss sums 2
attached to X
and
X
—2
divided by L(3-2k, X
)) ; in particular, the
coefficients are algebraic (in Q(X)) and non-zero.
If
(r,f) > 1 ,
then
(X) e
k m^n'r^ "*"S given by a formula like (11) with the right-hand side
multiplied by a finite Euler product extending over the common prime
-27-
factors of
r
and
f.
If k = 2, then the Eisenstein series fail to converge; however, by the same type of methods as are used for ordinary modular forms ("Hecke's convergence trick") one can show that for X Eisenstein series
non-principal there is an
E» G j having a Fourier development given by ¿-, m, x ^>, m k
the same formula as for k > 2.
Since X
must be even (X(-l) = (-1) ) and
since there exists an even non-principal character or b 2 7, such series exist only for
m
(mod b ) only if b = 5
divisible by 25, 49, 6 4 , . .. .
There is one more topic from the theory of cusp forms in the classical case which we want to generalize, namely the characterization of cusp forms in terms of the Petersson scalar product.
T
=
u + iv
and define a volume element
(12)
dV
:=
(v > 0)
dV
on K
,
t
x
We write
z = x+iy
by
v ~ d x dy du dv 3
It is easily checked that this is invariant under the action of JCx C defined in §1
on
and is the unique G^-invariant measure up to a —2
constant.
(The form v
on 5C; the form normalized
du dv
v ~ dx dy 1
is the usual S L (IR)-invariant volume form 2
is the translation-invariant volume form on
so that the fibre
(C/ZT + Z
transform like Jacobi forms of weight
has volume k
and index
1.) m,
If
(J) and \¡)
then the
expression
k -4™,y /v ^ 2
v
e
( T ) Z )
is easily checked to be invariant under Petersson
(13)
scalar product
(4),*)
of
$
:=
and
J
F \JCx(D J
, so we can define the
ij> by
v
k
e
"
4 T r m y 2 / v
dV
.
C,
-28-
Th en we have
THEOREM 2.5. for
cj),i|j G
^
and at least one of
positive-definite respect
,
The scalar product
Jy^^*,
on
, v . to ( , ) is
$
(13) is well defined
and
and
It is
a cusp form.
and the orthogonal
complement
of
finite
J^m^
with
Eis J. k,m
f
T
This will follow from the results in §5 concerning the connection between Jacobi forms and modular forms of half-integral weight.
§3.
Taylor Expansions of Jacobi Forms The restriction of a Jacobi form
form of the same weight.
^
In §1 we proved an analogous statement for the
z = Ax + u (A,u rational)
restriction to Jk,m^^
<J)(x,z) to z=0 gives a modular
s
^inite-dimensional.
and used it to show that
Another and even more useful way to get
modular forms is to consider the Taylor development of
(f> around z=0 ;
by forming certain linear combinations of the coefficients one obtains (D^
a series of modular forms with
2) $ Q
=
for " v ^
2
development coefficient")
and D^j\> a modular form of weight k + V.
The precise
For
polynomial
result is
THEOREM 3.1. p
2v ^
(1)
°^
variables
Ç2v)Hk-lj\
Then for
p
<> j G j^
y c(n,r)q ç n,r n
r
2v"
v£3N , 0
k6 I
define
a homogeneous
by
1 ) ( r
'
n )
(r)
a Jacobi
, the
function
=
coefficient of
form with Fourier
2 t
V
i n (1 - rt + n t
development
2
) " ^
-29-
(2)
i>
2
is a modular
*
:=
I ( I p^ (r,nm)c(n,r))q n=0 r 1 )
form of weight
k + 2 v on T,
n
If v > 0 , it is a cusp
form.
Explicitly, one has
3)^
=
I(Ic(n,r))q n r
=
1(1 ( k r - 2 n m ) c ( n , r ) ) q n r
=
l h ((k+1) (k+2)r n * r
2
(3)
(k-1) p ^
p - ( r n) P U,n; ( k
2
r is finite since
2
2
n
c (n,r) ^ 0 =» r < 4nm.
is given explicitly by
•= .
1 }
2 v
,
11
- 12(k4-l)r nm+ 1 2 n m ) c ( n , r ) ) q
k
Notice that the summation over The polynomial
,
n
T ^
(-if l i)
(
2
y ! ( 2 v
V
_
)
S
2 l i )
,
(k+2V-y-2) (k + v ~ 2 ) !
r
2v-2
y n
y
r
and is, up to a change of notation and normalization, the so-called Gegenbauer or "ultraspherical" polynomial, studied in any text on ortho(k-1) gonal polynomials;
w e have chosen the normalization so as to make p
2
a polynomial with integral coefficients in k,r,n in a minimal way (actually, ~ - times
p ^ ^
a function of r and n of the polynomial where
Q
p ^
would still have integral coefficients as
for fixed
k G I ) . The characteristic property
is that the function
p ^ "^(B(x,y) ,Q(x)Q(y)) ,
is a quadratic form in 2k variables and B the associated
bilinear form, is a spherical function of x and y with respect to Q (Theorem 7 . 2 ) . There is a similar result involving odd polynomials and giving modular forms 2)^,1)^ and replace
(k+v-2)!
, . . . of weight by
k+l,k+3, . . . (simply take
V G % +1^
(k+v- / ) ! in (1) and (3)), but, as we shall see, 3
2
-30-
this can be reduced to the even case in a trivial way, so we content ourselves with stating the latter case. As an example of Theorem 3.1 we apply it to the function
^
studied in the last section; using the formula given there for the Fourier coefficients of
E. - we obtain k,l
COROLLARY (Cohen [ 6 , Th. 6.2]) . be Cohen s function
Let
k
be even and
H(k-1,N) ( N G I ) 0
-
1
L_ (2-k)
if
N > 0, N = 0
C(3-2k)
if
N=0,
if
N = 1 or
N
or 3(mod 4) ,
! Then for each
v £ ]N
0
0 the
2(mod 4) .
function
r 1 4n is a modular form of weight k + 2y on the full modular group T . If x
it is a cusp
v > 0,
form.
Cohen's proof of this result used modular forms of half-integral weight; the relation of this to Theorem 3.1 will be discussed in §5. Yet another proof was given in [39], where it was shown that
C^ ^ V
has
the property that its scalar product with a Hecke eigenform f = Sa(n)q
n
G S^ 2 +
V
^
s
equal, up to a simple numerical factor, 2
to the value of the Rankin series
—S
2 a(n) n
This property characterizes the form
c£ ^ V
at
s = 2k + 2v - 2.
and also shows (since the
value of the Rankin series is non-zero) that it generates S^ 2-^ (resp. +
M^. if V = 0 ) as a module over the Hecke algebra; an application of this will be mentioned in §7. To prove Theorem 3.1, we first develop expansion around z = 0:
(j) (x ,z)
in a Taylor
-31-
(4)
<(>(T,Z)
=
E
X
(T)Z
V
v=0
and then apply the transformation equation
to get
2 f£\ ( 6 )
X
/ax+b\ v ( ^ J
i.e. X
=
v
, ^.jxk+v/v , . , 2irimc \ V v-2
/^\ , 1 / 27rimc\ 27 v-4
v
(
c
T
+
d
)
T
)
+
X
( T )
+
X
V like a modular form of weight
transforms under
corrections coming from previous coefficients.
( T )
. •• ) '
+
\
k+V modulo
The first three cases
of ( 6 ) are
* (tSr) = 0
^(iff)
X(f^±|) 2
C« d) x (T) k
+
B
d)
=
< «
=
(cx + d )
+
X
k + 1
k
+
l (
T)
X (T) +
2
2
2irimc(cT+d)
k+1
X„(T) .
Differentiating the first of these equations gives X
o (fSJ)
=
kc(cx+d)
k + 1
X (T) + ( c x + d ) o
k + 2
x;(T)
,
and subtracting a multiple of this from the third equation gives
?
2
:
=
x . 2
*g5L ; x
e
M
Proceeding in this way, we find that for each
(7) U
;
E (T) V T
;
•= •
V* /
k
+
V
2
(D
the function
(-2^im) (k+v-y-2)! (k+V-2)! Y!
transforms like a modular form of weight
y
k-fv
.
on T.
(Y) \-2\i (
}
KT>
The algebraic
manipulations required to obtain the appropriate coefficients in (7) directly (i.e. like what we just did for v = 2 ) are not very difficult
-32-
and can be made quite simple by a judicious use of generating series, but w e will in fact prove the result in a slightly different way in a moment.
If <j) is periodic in z and has a Fourier development
£ c(n,r)qV, n,r
then
X
= ~ r l ( l (27Tir) c(n,r))q n r (8) V
V
n
and hence
so
(9)
^ . ( T )
-
( ,IR
2V (
k
2
^ - ; f
)
! ? 2 v
(x)
Thus Theorem 3.1 follows from the following more general result:
THEOREM 3.2. with coefficients holomorphic ?
v
defined
Let (J>(x,z) be a formal power series in z as in (4) X
which satisfy
v
everywhere
(including
by (7) is a modular
Let M^.
Proof.
(O
(6)
aZ-Z ^
d )
the cusps of T).
form of weight
G
^
(Note that M
"
a
p
e
.7%en tTze function
k + v on T.
denote the set of all functions
the conditions of the theorem.
an(2
1
(J> satisfying
(T) is isomorphic to
k,,m M
k 1 ^ via z t—> y/m z .)
we can split up M ^ ^(r) M
k m ^
Since
y
involves only
X^» with
v =v(mod 2 ) , !
into odd and even power series, say,
**k m ^ ® k m ^
=
£
and look at the two parts separately
M
(this corresponds to adjoining
-I
2
to F and looking at the action of (_l) M. (F) = M (F) ) . K.,m ic,m k
-I If
on <j); if T already contains - I . , then ^ ^ d) £ ML (T) , then Tc.m T
^ V ' ^ V
k
d> = zd) i
r
^
O
R
^
A
N
C
*
with d>, G M' , ( T ) i k+l,m
T
Y
and the functions
are the same except for the shift
k + l . Hence it suffices to look at
(F) .
differential operators . 3 3 87Tim -r— - — 7 3Z 2
L
:=
Q
8
T
2
V
->
V-l,
We now introduce the
-33(the heat operator)
and L
_ zkzi f
L
k The operator
L
z
.
8z
is natural in the context of Jacob! forms because it
acts on monomials
q £ n
by multiplication by (27Ti) ( 4 n m - r ) and hence,
r
2
in view of Theorem 2.2, preserves the second transformation law of Jacob! forms; this can also be seen directly by checking that
L(<j>| X)
(10)
=
'm If
L
(X e
(L(J))| X
IR ) 2
,
'm
satisfied a similar equation with respect to the operation of
SL (M),
then it would map Jacobi forms to Jacobi forms.
2
Unfortunately,
this is not quite true; when we compute the difference between L(<j>|.
K,m
M)
and
(L(f))L
M
0
KiZjii
we find that most of the terms cancel but ($1 ^ M ) (T ,z) ,
there is one term, 4Trim(2k-l) ^ ^ c
k = weight
in which case %
L
+
m
left over (unless
really does map Jacobi forms to Jacobi forms of
and the same index m ;
which are annihilated by
L ) .
examples are the Jacobi theta-series, To correct this we replace
L
by L^,,
which no longer satisfies (10) but does satisfy
(11)
L
k *lk,m (
M )
=
( L
k*>lk+2,m
as one checks by direct computation. nator,
CMeSL <*»
M
2
Because of the
z
in the denomi-
acts only on power series with no linear term; in particular
it acts on wf
(D
and (because of (11)) maps
£ i X>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
v.»ff)
V to
A
V>
T ( T (-4) "" (8uim)^ i XH)\y=0 ^ ' V
y
(
U
V
^
)
1
X !
'ii^2v-y-2)l (X+k+v-2)!
(y) \ 2X h v ~ u / (
}
{
T
)
z
+
and composing this with the map
gives
^>2\) ^ ^ k + 2 v *
^^s Proves Theorem 3.2 and hence also Theorem
3.1 except for the assertion about cusp forms.
B ut the latter is (k1)
clearly true, because the constant term of (2) is p £
V
(0,0)c(0,0),
which is 0 for V > 0, and the expansion of jD^jfy at the other cusps is given by a similar formula applied to (j> | ^ ^M,
M £ Tj.
By mapping an even (resp. odd) function
({> €
m
( F ) to
(^ ,? ,^,...) (resp. to (? , C , . . . ) ) , we obtain maps 0
2
x
3
It is clear that these maps are isomorphisms: terms of £
one can express X
( 1 2 )
X
Y (T) v " (
i
)
Ц
V (2тг1ш) (^У-2и1)! Л и ) Zrf (k+v-u1)! y! Ч 2 у Oiy<|
and then the transformation equations (6) of the X v
in
by inverting (7) to get
( 1 2 )
£
y
e M^^CH .
In particular, taking
y
(
) Ш
'
follow from
£ = f, £ = 0 ( v > 1) and m = l we Q
v
obtain the following result, due (independently of one another) to Kuznetsov and Cohen:
-35THEOREM 3.3 (Kuznetsov [16], Cohen [7])., k on T.
form of weight
ffr
(13) satisfies
Then the function CO
-
f(T,Z)
Let f(x) be a modular
\)
V ^
the transformation
(2TTi) (k-l)! (v)., 2v ! (k+v-1)! v
f
f
( T ) z
V
equation
(14) We mention a corollary which will be used later. COROLLARY (Cohen [6, Th. 7.1]). of weight
k
Let f ^ f ^
an
L
2
V
V',-,) •• is a modular
,
t
v
r(k +v)
function
r(k +v)
k +k +2v x
/x
s
ra^T^e "
on V and is a cusp form if v > 0.
2
(We have modified Cohen's definition by a factor (27i*i)
to make
V
the Fourier coefficients of ¥ (f^»f
/ ,
1
^ W O t ^ j
form of weight
forms on T
be modular
) rational in those of f
The corollary follows by computing the coefficient of z ^
V
1
and f . ) 2
in
f (x,z)f (x,iz), which by Theorem 3.3 transforms like a modular form 2
of weight
k +k 1
2
under T.
We observe that the known result 3.3 could also have been used to prove 3.1 and 3.2.
(We preferred to give a direct proof in the
context of the theory of Jacobi forms, especially as the use of the differential operators let
L^. makes the proof rather natural.)
be the subspace of M^.
i.e. have a development
of functions
m
X (T)z V
v
4- X
clear that the leading coefficient
V + 1
X
(T)z
V + 1
Indeed,
(J) which are 0 ( z ) , V
+ ... .
From (6) it is
is then a modular form of weight
-36-
k+V
and we get an exact sequence
o
-»
^
v
+
1
)
->
M^
K,m
- >
V )
M
K,m
K+V
in which the first arrow is the inclusion and the second is On the other hand,
M^ ^ — M^ ^ ^
by division by
V
+
used for V = 1 when we reduced the study of 3.3 gives a map
—
>
M
k+v m
^
v
f
1
—
^
<j) I—> X^.
(this was already
to that of
^( >
>
z
v
>
T
^ ) , and
this shows that
the last map above is surjective and gives an explicit splitting. To get the sequence of modular forms £ , £ j , . .. associated to
(j)GM^
0
w e now proceed by induction:
having found ^
£
V'
, (T,
Z ) Z
,...
Q
=
0
m
such that
(mod z )
v
£^(T) as the leading coefficient
expression on the left-hand side; then
(coefficient of
cj> = £ ? V
(T, / m Z ) Z
z )
in the
V
as a formal
V
power series and this is equivalent to the series of identities (12) or (7). We have gone into the meaning of the development coefficients D
$ fairly deeply because they play an important role in the study of
Jacobi forms and because the relation with the identity (14) of Kuznetsov and Cohen concerning striking.
f
(which is not a Jacobi form) seemed
In particular, we should mention that (13) can be written 0 0
f(T,z)
if
f = Sa(n)q
=
a ( 0 ) + (k-1)! £ a(n) n=l
J, _., (47r/n~z) :-=(2ir/n~z)
q
1
X
(this is the form in which Kuznetsov gave the identity).
n
To see where the Bessel functions come from, note that the function J _ (4iTz) :—rk
h(z) = (k-1)!
1
(2 rz) k
satisfies the ordinary differential equation
1
T
2k-l h" +
t 2 h + (4TT) h
=
0
and is the only solution holomorphic at the
-37origin and with
h(0) = 1.
By separation of variables we see that
°° 2TTinT f ( x , z ) = ][ a(n)h(/iiz) e 0
is the unique solution of the partial
differential equation
satisfying the boundary conditions
L^f = 0
f(x4-l,z) = f ( x , z ) and the fact that
f(x,0) = f ( x ) ,
and this uniqueness together with
commutes with the operation of
immediately implies that
SL (]R) (eq. (11)) 2
f has the property (14).
As a first application of the maps 2)^
to Jacobi forms, we have
a second proof and sharpening of Theorem 1.1: THEOREM 3.4. Indeed,
Jfc m ( D
dim
? = ...= £
=0
implies
2m dim M ^ F ) + £ dim Sk+v(I^ *
< X
= . .. = X
0 zm 0 so Theorem 1.2 implies that the map *>
-
v
?
is injective.
0
^
,
J k
(
r
~* V
)
m
r
)
.
\
0
Note that half of the spaces
in particular, for
V =T
or
<J> = 0 ( z
2 m + 1
),
+
i
(
r
)
m
•••
(r)
W
e
r
)
are 0 if - I ^ F ;
we have
l
1
= 0
0
zm
5dimM +dim Sk+2 4- . . . 4-dim S k
dimS,
-4-dxmS.
0
4...4dim
k
+
2
(k
m
(k
S. „ L
even) ,
odd) .
Here the second estimate cank+1 even be strengthened to k+2m-l k+3 (16)
dim
J
K ( F F I
<
dim
S
k + 1
+ ... + dim
S
k + 2 m
_
3
,
because an odd Jacobi form must vanish at the three 2-division points ii X ilX 2 * 2 * 2
a n c
Application:
j hence cannot have more than a (2m-3)-fold zero at z = 0.
Jacobi Forms of Index One
Theorem 3.4 is the basis for the analysis of the structure of J
*>*
=
© J. k,m > k
as given in Chapter III, to which the reader may now m
skip if he so desires (the results of §§4-7 are not used there).
As an
-38-
example, we now treat the ease m = 1, which is particularly easy and will be used in Chapter II.
0
s
Equations (15) and (16) (or Theorem 2.2) give
(k
odd) ,
dim
^ ^
dim M ^ + dim
On the other hand, the Fourier developments of
E^
and
l
^
E
g
even^ *
as given
1
after Theorem 2.1, show that the quotient
6 i '
E
( T
-i I " (144C + 456 + 144? ')q + ...
z ) =
E
( T
z )
>i depends on
z and hence is not a quotient of two modular forms, so the map
\ - 4
e
M
k - 6
~*
(F,G)
is injective.
J
k,l
F(T)E
Since
+ g(T)E
( T , Z )
dim M^_^ + dim M^_^ = dim M ^ + d i m ' S ^ ^
(this follows from the well-known formula for
THEOREM 3.5. free module
(T,Z) b, 1
(2) ,2) Q
E
#2
:
J k
)
l
—
\
3.1) is an
E
= E, • E
H
8,1
&
and 4
+
»
i
E 6
,
.
The map
i
S k
+
2
isomorphism.
In particular, we find that the space with generator
k
2
as in Theorem
2
^
d i m M ^ ) , we deduce
* +
r a
The space of Jacobi forms of index 1 on S L ( Z ) is a
of rank 2 over M , with generators
*>*
f°
,
(E
h,l
J
8, 1
is one-dimensional.
= 1 + 240q + . . .
U
the Eisenstein
n
series in M ^ ) , while the first cusp forms of index 1 are the forms
(17)
*IO,I
*
m
6
=
T77144
Y
i2,i
(
S \ I
E
"
(E E 2
v
k
k,i
V E , I >
- E
6
E
>
6,i
y
)
of weight 10 and 12, respectively (the factor 144 has been inserted to make the coefficients of
<j)
and
(J>
integral and coprime).
We
-39have tabulated the first coefficients of
<J>,
e
v
of
1
E,
(k= 4,6,8) and
1
c
1
(k = 10,12) in Table 1; notice that it suffices to give a single
K, I 1
sequence of coefficients
c(N)
(N > 0, N — 0,3(mod 4)) since by 2.2 any
Jacobi form of index 1 has Fourier coefficients of the form c(n,r) = c(4n - r )
for some
2
{c(N)}.
To compute the
use either assertion of Theorem 3.5, e.g. for
<j> , iQ i
c(N) , we can <J>
^ we can
12
either use (17) and the known Fourier expansions of
and
E^ ^
or
else (what is quicker) use the expansions
(18)
00
(A
=
00
q TT ( l - q ) n=l n
S c 1 rl < 2/n
2 i f
= £ T(n)q ) n=l n
(4n-r ) = 0 ' 2
i n
1
Yl c ]rl<2/n
to obtain the identities
,
I] r c 0
J2 r c 0
( 4 n - r ) = 12T(n) , 2
and then solve these recursively for
< 4 n - r ) = T(n) , 2
1 0
¿ 4 1 1 - r ) = nx(n)
2
c,
2
1 2
'
(N) .
1
K, i
The functions
^10,1
12jl
(C-2 + C )q + (-2C -16C+36-16c"
=
_ 1
=
2
1
(C + 10 + if *)q + (IOC - 88C - 132 - 8 8 f 2
-2C~ )q + 2
1
2
+ l O f )q 2
2
...
+ ...
have several beautiful properties and will play a role in the structure theory developed in Chapter III.
THEOREM 3.6.
The
*io,i is
-3/TT
2
( T
Here we mention only the following:
quotient
» > 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)<()
1
( T , Z ) + g(T)(()
1
(T,Z)
.
Thus the Jacobi forms
E.(T) E,(T) <(). -(T,z) J »1 a
b
h
6
(a,b£0,
JG{10,12},
4a + 6b + j = k)
form an additive basis of the space of Jacobi cusp forms of weight and index 1.
Each of them has a Fourier expansion of the form
2 c(4n - r ) q C ; 2
k
n
r
the coefficients
are given in Table 2.
c(N) for
N £ 2 0 and all weights k < 50
-41§4.
Hecke Operators We define operators
A) (<M V ' (T
= (T,«.Z)
Z)
K>M
(SL > 0 ) on functions
U „ , V„ , T»
(*l
K
>
M
V
(T, ) Z
-
C
by
,
£ " ^M^'^)> (
(2)
: JC* C
I™
+d)_ke
(*5)e \M (D ad-bc = J, r i
- * ~
( 3 )
k
2
<"i ,
zC
4
Mer \M (z) det M= it 1
k
xei /U 2
2
2
M m
L
>
X
2
g.c.d.(M) = Q
where the symbols
|^ ^M,
| X
(except that for M e G L ( M )
one first replaces
2
and
g.c.d.(M) = •
of M
Then we have
THEOREM 4.1.
The operators
J
Proof.
U ,V , T £
£
£
of the ahoioe of representatives)
, m £ ' k,mt 2
k
a
"
d
M by (det M ) M e SL (1R)) 2
2
means that the greatest common divisor of the entries
is a square.
independent J
have the same meanings as in §1
J
k,m>
are well defined (i.e. on
^ and map
m
to
^ope^vely.
The well-def inedness and the fact that
$1 £> '$1V^, '((> JT^ V
transform correctly follow by straightforward calculations from the properties of the Jacobi group given in §1; the conditions at infinity will follow from the explicit Fourier expansions given below. Before proceeding to give the properties of the operators we explain the motivation for the definitions given.
U, V, T,
The operator
is an obvious one to introduce, corresponding to the endomorphism "multiplication by £" on the elliptic curve two,
C/ZT+22.
A S to the other
we would like to define Hecke-operators as in the theory of modular
forms by replacing
<j) by 4> | M,
where
M
runs over a system of represen-
-42-
tatives of matrices of determinant T .
elements of
l
£
modulo left multiplication by
Doing this would produce a new function transforming
like a Jacobi form with respect to
F .
However, since
l
<j)| [M]
is a
multiple of ,/aT + b
/1 \
/1
z
H^FTd ' ^FTd)
( 4 )
(M = ^
d)' ^
=
det ^ »
( | ) |[M] transforms in with
z
anc* ^
is in general irrational, the function
for translations by a lattice incommensurable
Zx +2£, and there is no way to make a Jacobi form of the same index
out of it.
However, by replacing
/£ by
ity; since this is formally the operator index by by
\
£.
£ , 2
£
in (4) restores the rational-
Uy^,
and
multiplies the
we obtain in this way an operator which multiplies indices
This explains the definition (2).
Finally, if
£
is a square
then the function (4) transforms like a Jacobi form with respect to translations in the sublattice
/£~(ZT + Z ) of
ZT + TL,
so we get a
function with the right translation properties by averaging over the quotient lattice.
This explains (3) except for the condition on g.c.d.(M), o
which was introduced for later purposes:
If we define
formula as (3) but with the condition " M
primitive" (i.e.
then
T^
and
T^
T^
by the same g.c.d.(M) = 1),
are related by
Eventually we want to show that the Jacobi-Hecke operators to the usual Hecke operators
T^
T^
correspond
on modular forms of weight 2k-2, and
equation (5) is precisely the relation between these Hecke operators and the corresponding operators defined with primitive matrices. Our main goal is to describe the action of our operators on Fourier coefficients and to give their commutation relations. with
and
since they are much easier to treat.
We start
-43-
THEOREM 4.2.
i)
(6)
<|>|U
§ = 2 c(n,r)q ç . n
km
=
£
(with the convention
<{) e j~ ,
Let
X! n,r
r
fTften
c(n,r/£)qV
c(n,r/£) = 0 -£f Jtfr) anc?
(7) n
ii)
The operators
'
r
X
a|(n,r,£)
U ,V £
8
W
= U-
<>
W
=
(
1
0
)
V
In particular
*
D
O
* -
U
V
r° £ >
D
•
U
K
^
all of these operators
Remark.
the re lotions
satisfy
£
<> 9
7
.
/
'
^
•
w
commute.
Formula (7) nearly makes sense for £ = 0 and suggests
the definition |V
with some constant
0
=
c^.
c(0,0) [ c + E n>l k
Since
(j) | V
Q
V l
(
n
)
q
I
1
]
should belong to
Q=M^,
we
2k c, = - - — so that d> V is a multiple of the Eisenstein series 2k of weight k. This definition will be used later. take
k
Proof.
B
0
Equation (6) is obvious.
For (7) take the standard set
of representatives
(11)
^
for the matrices in (2).
,
Then
a,d>0
,
b(mod d) ,
ad = I
-44W|V )(T,z)
=
A
Y, ad=£
~
A
K
b(mod d) an
=
A*"
1
J
d"
]T
k
ad=£
b(mod d)
k-l V £ 2j
,1-k
0
=
ad=£
E
,
c(n,r) q
Ç
d
a r
e (nb) d
r
\ ^ /
.
v
c(n,r) q
an d ar Ç r
n,r n = 0(mod d)
k-1
^
a|£
/ n£
n,r
which is equivalent to (7).
£ n
V
\
an^ar
'
a
Equation (8) is obvious.
By (6) and ( 7 ) ,
we have
coefficient of
q £ n
r
in (4> I U ) | V^,
=
£
^
^
a
k
c^—j ,
1
a| (n,r,£ ) f
£lH a
0
!
if
£f r
a|(n,|,r) =
coefficient of
q C n
in
r
(0|V^ )|. t
Finally, using (7) we find coefficient of
q £ n
in (<j) | V ) | V^,
r
?
-E
E
a|(n,r,n
-
where
N(e)
I N(e) e ^
b
. f )
,
is the number of ways of writing
e
preceding sum.
1
c
|(^l,f,*)
(
^
as
If such a decomposition exists then
a*b
in the
e ] la
and hence
-45-
a =
v 6 for some integer (e, 36; b = e / a we find the formula
6; writing down the conditions on
e
(
0
6
N(e) = number of divisors
(=0 unless
e |(n£, n £ , £ £ ) , T
coefficient of
q ^ n
r
^ n , £ , £ , e,
2
£
oV
,
!
e |n££ ).
f
calculation just given for U
of
,
a
and
,
On the other hand, using the
f
we find
£ T
in
d
k
((|) | U ) | V
^
1
d
£ £ T
2
d| (£,£')
V*
,,k-l ä\a,V)
where now
?
elnd
we find
1
d
as
,
e
v6
\Ti5 c)
"l
< nM
rN
1
a | ( n , f , ^ )
N (e) counts the decompositions of
the conditions in the sum; from writing
k
we obtain for
N (e) T
e
as a d #
,
e
,
satisfying
divides
d,
and
(n,e) the same formula as for N (e).
This proves (10) (another proof could be obtained by combining the Corollary below with the multiplicative properties of ordinary Hecke operators) and completes the proof of Theorem 4.2. As a consequence of the formula for the action of V
£
on Fourier
coefficients we have the following COROLLARY.
where
T
£
For
on the right denotes
and
v <E IN , £ G IN, o '
the usual Eeeke operator
Notice that this property characterizes V we have
one has
£
on modular
completely, since
forms.
-46-
k,m
k+1
Te
k+N
I
V
T
1
v
^
T
T
v
k,m£
v
v
k+1
k
k+N
and the horizontal maps are infective for N > 2mA
(cf. Theorem 3.4).
To prove it, we calculate
d j n,J6
,k+v-l r
where of
c n
q Ç n
r
denotes "coefficient of q
11
n
in (J>.
geneity of f
Replacing
(k-1)
p^
r by
r/d
(k-1) /
n£ \
in" and c(n,r) '
/n£
\
is the coefficient
in this sum and using the homo-
. we find c
A
d|(n,r,£)
as was to be shown. L. : k
k,m
—
lc+2,m n
Another proof comes from observing that the map used to construct J0 2v O
(cf. proof of Theorem
is equivariant with respect to the action of from (11) of §3 since
maps
F
on
^
3.2)
(this follows
<j) to
£ ((f) | ^ M ) (T ,/£ z)) and that M (|)(T,0) of a form in ,
acts
+
as T^
on the constant term
(
As another application of Theorem 4.2 we have
THEOREM 4.3. for
(12) K
/
m
Let
E, k,m
denote
the Eisenstein
square free we have
E, i | V k,l' m
=
O,
-(m) E. k,m
k-1
series of §2.
The
47
Thus the
Fourie r
ooe ffiaie nts
of
e ar
as in e q. (7) of §2.
Indeed, we have
where M Writing
runs over ^
a
F
1
2
( Z )
| det M = m} f
^
for the product M M
ITT
"V
k1 = m
v
E
\ { M G M
НУ!)
а)
( с
>
k,l' m
and
M
1
over
F ^ M ^ .
gives
/
T ^ 2 ax+b , „,
V >
(cx+d)
e
— 7
X
C T + d
L
zL /a b \
+ 2X
mz
mcz 7
C T + d
2
7 c T + d
»
J
Xez
d
\c ; where now
^
^
squarefree.
r
(c,d) = 6
Then
It follows that of
runs over
^
a
^
divides
m
Now suppose
m
6' = |r is prime to
and
is 6.
can be multiplied on the left by an element a = b = 0(mod 6 ) ,
so as to make
r
I^XJMldet M = m } .
so
-\((" "„)• ««=•-.
- (S
W••«•••=•>M».
and hence
E. J V k, 1 m 1
k _ 1
= m
V ô" /1,
k
6|m
v
V** e
Replacing
^
by
l
x
/
^ ^
(cx+d)~
k
(:SKi?)w« ^'
ax+b , ,
T~
cT+d
l)(c
0
d)
+
2 A
f
2
6 z cx+d
c z m ~| '
cx+d J
*
multiplies the final exponential
48
2
e(X 6'/6),
squarefree).
„ К
f
and this is 1 only if 61X
( since (6,6 ) = 1
Therefore the terms with
6fX
, k1 V olk V , ^ ,.k "ST mfX v = m > о > (cT+d) > e — k,l' m Z, L¿ L ô 6 Im Г00 \ Т 1. * XeôZ
and
5 is
can be omitted, so
2
2
ат+b , X z cz —- + 2 — 7 7 cT+d 6 CT+d ст+d
TT
0
1
= т У к_1
Ô
Z-j
1 k
E, ( T , Z ) = k,m
a, (m)E, (T,Z) k1 k,m
.
1
ôlm
This proves (12) ; the formula for the Fourier coefficients of E k,m
now
1
follows directly from Theorem 4.2. For m not squarefree we find after a similar calculation the more general result
(13)
E
= m
k m
TT ( l + P
k + 1
_
k
+
1
)
_
1
VW
£
)
\
U
5
ll d° J!L V
Eis yet more generally, we see that the space is mapped by U
£
and V
^
defined in Theorem 2.4
to the corresponding spaces with index m £
£
m£ , respectively, the precise formulas for the images of E^. m
U
£
and V
(terms
g
2
and
under
being easily ascertained by looking at the "constant term"
£
n
q ?
r
2
with
4nm=r ).
One then sees by induction that all
Jacobi Eisenstein series can be written uniquely as a linear combination of E ; |u O (£,£'e]N, X a primitive Dirichlet character modulo f K, m Xj Xj A ;
0
with
V ( M
k
X(l) = (l) ,
m=f
THEOREM 4.4.
Le t
2
E ^ as in (10) of § 2 ) . Combining these k,m remarks with the commutation relations of Theorem 4.2, we obtain
l d
J ° k,m 1
and similarly J
J
=
for jcusP>°^ k,m
E £ , . 2; d m d>l
I
0
J, m I U o V ! £ d d —3
£ I d k,
0
T
ancf jEis,old ^ k,m
36
De fine
jcusP>new k,m
as ^e
-49-
orthogonal
complément
of
scalar product)
and
(X
Dirichlet
a primitive
J^ ^ 1
J^^'
1 1
^
j£ ^ u
in
p
(with respect
as the span of the
character
to the
Petersson
functions
(mod f ) ) if
m = f
2
and
0
otherwise.
Then Eis
J
_
k,m
Eis,new
m
"
£
®
,
\,m/£'£ J £ £' 2
U
'
V
£ £'|m 2
ana
J
cusp k,m
_
cusp,new , Z-r k,m/£'£ £ £ £,£' £ £'|m J
2 1
U
V
?
2
Eventually, we will show that the latter decomposition is also a direct sum, but this will be harder and will depend on using a trace formula. It remains to describe the action of the Hecke operators the Fourier coefficients of Jacobi forms.
T
on
£
We do this only for (£,m) = 1.
The formula obtained involves the Legendre symbol
( j - ) if
£
is an
odd prime; in the general case it involves a slightly generalized symbol £^(n) , (D,n £ 7L) , which we now define. If
D
is congruent to 2 or 3 (mod 4) we set
£ (n) = 0 . D
For D = 0
we set
{ Otherwise
D
q
if
n= r ,
0
if
n^d
corresponding to
Then let Q(/D),
X
(p odd) ,
f^ 1
and
D
Q
is the
i.e. the multiplicative function with
X(2) =
1
<-l (
and set
where
be the primitive Dirichlet character
( X(p) = ( — j
,
.
2
Q
Q(/D).
r>0
2
can be written uniquely as D f
discriminant of (mod D )
r
0
B D D
0
0
S
1(8)
= 5(8) ,
o -
°< ) 4
X ( - D = sign D
-50-
e (n) D
The function
-
g|f,
if
n=n g ,
if
(n,f ) i •
2
o
n)
(|,
=1
0
,
o
.
2
e^, which reduces to X
if
D =D ,
occurs naturally
Q
whenever one studies non-fundamental discriminants; it has the properties e ( n + D)
=
D
e (n) D
and (as shown in [13], p.188), »
£ (n)
£ ^ n=l where
L (s)
= vs) •
n
is the L-series already introduced in connection with
Jacobi-Eisenstein series (eq. (6) of
§2 and the following formula).
We can now state the formula for the Fourier coefficients of
THEOREM 4.5. and index 4>|T
£
=
m
and
Let
£
é = X c(n,r)q ? n
a positive
2c*(n,r)qY
be a Jacobi form of Weight
r
integer prime
to
m.
k
Then
with
c*(n,r) = J
(14)
£ 2_ R
( a ) a *" c(n ,r') k
4 n m
2
,
T
a where
the sum is over
a with
a|£ , 2
a |Z (r 2
2
2
- 4nm),
a £ ( r - 4 n m ) = 0,1 2
2
2
(mod 4) and
(15)
r'
In particular,
if
corresponding
(16)
2
= I
- 4n'm
r - 4nm 2
quadratic
(r - 4nm)/a 2
2
,
2
is a fundamental
=
f
discriminant
£r (mod 2m)
and
X
.
the
character > then
c*(n,r)
=
E
X(d) d " k
d|£
To see that (15) makes sense, note that (a,m) = 1,
ar
2
c
n , | r) ^
a|£
2
.
'
d
and (£,m) = 1 imply
so the second equation in (15) defines
r
1
uniquely modulo 2m
-51if m
is even and modulo m
determined
is odd; in the latter case, r
f
is
(mod 2) by the mod 4 reduction of the first of equations (15).
Thus in both cases
r
f
f
2
r
so the number that
if m
c(n ,r*) f
n
f
is uniquely defined modulo 2m and satisfies
=
£ (r - 4nm)/a 2
2
(mod 4m)
2
,
defined by the first of equations (15) is integral;
is independent of the choices made then follows from
Theorem 2.2. Before proving Theorem 4.5 we state two consequences.
COROLLARY 1.
For
T
£ and
£ ' V
V
both prime
Yj
=
d 2 k
"
3
T
to m
UVd
one has
*
2
d|(£,£') In particular, the operators Proof.
the operators T
£
commute with
((£,m) = l) all commute.
£
V .
The spaces
U . £
for
(£,£ m) = 1.
and
m
of the Heche operators
Proof.
and
£
Furthermore>
f
Exercise.
COROLLARY 2. eigenforms
T
T
£
spanned by
common
((£,m) = l ) .
This follows by a standard argument from Corollary 1 and
the easily proved fact that the Hecke operators in question are hermitian with respect to the Petersson scalar product defined in §2.
Proof of Theorem 4.5.
We write the definition of
T
£
as
M where the sum is over all a square and
I A m
MG
r^M^Z)
with
det(M) = £
2
and g.c.d.(M)
is the "averaging operator" which replaces a function
I | ; ( T , Z ) that transforms like a Jacob! form with respect to some sublat-
-52tice
L Ç i
by
2
:=
^| A
Y]
- [Z
:
L]
X
G
Z
2
X /
.
L
(Note that this is independent of the choice of
L
and projects
2
L-invariant functions to Z -invariant functions ; in (3) we took L = £Z .)
As in the usual computation of the action of Hecke operators
2
on Fourier coefficients we choose upper triangular representatives for the left I^-cosets; then
<J) | T
= (f> | A
£
£
0
where
1
* k,m(o
YL
ad=£
-2
!
S)
, / ar+b
£z\
b (mod d) b(mod (a,b,d) = D
k ad=£
M
V* b(mod d) (a,b,d)=D
2
To get rid of the condition " (a,b,d) = • " we use the identity 1
E
(n=D)
*<6) '0
with
n = (a,b,d),
function with
V
where
X
is Liouville s function (the multiplicative 1
A ( p ) = (-1) ). V
T,z)
-
V
^
This gives
a ad=£
V
2
k-1
ad=£
2
(n^CI)
I
k
X(«)
6|(a,d)
V
E
A(6)
6|(a,d)
• ( ^ • T )
b(modd) b=0(mod6) ,
,
/an
r£
\
n,r n = 0 ( m o d d/6)
• I2 ^ E ( ' ' A U d)
ad=£
n,r n = 0(mod d/(a,d))
a7fc-))
c ( n
'
r )
e
(f
T
+
T
z
) '
53
where A(a,B)
:=
X(6)/6
]T]
-
6|a,a6~ |3
Replacing n
(17)
and r by
. T . , )- £
~
-
V
ad=£
k
_
2
gives
£
A(
(
a
x
'
d
)
C
'ror)) (Tf) <« e
+
™>
nGa(a,d) Z
(which, notice, involves
We still must apply the averaging operator A.
nonintegral powers of £) . as A
(a,3€W).
X
This gives the Fourier development of (j)
We can factor A
( f ) ^
Y|(a>3)
and
1
X
~
1
°
where
is the averaging operator with
respect to 0 ^ 1 , i.e.
X
*lm*l
: =
[z7lT]
H 0
XG
L* C Z
for any sublattice 0 x L , and A T
2
containing
such that IF; is invariant with respect to
(for a function invariant under a lattice
by
0.
and r in Z unchanged and to replace all terms Hence we have
$|T
£
=
where ^ ~ ^J^i
^ 2 ^ 2
(^ by omitting all terms in (17) for which condition
rG — Z d
r e — — 7L. (£,d)
by y
4> (T,Z) 2
=
^
a
ad=£
n
q £ ^
s
2
r
1
cf>
ф
2>а
(х, ) В
E
dn = 0 (mod a) d r = 0(mod £)
with
r
r $ ffl,
r ^ Z , i.e. by replacing the
(T,z)
2
with
(18)
n
q £
obtained from
Thus
k
2
Z /0xZ.
is to leave any term
l
n
LCI
0 x ^ ) is the averaging operator with respect to
It is easily seen that the effect of A with
XZ/L'
A ( ( . , d , . ^ ) c ( ^ , f )
4
V
-54-
To compute the coefficients
c*(n,r) of
determine the operation of the operator A
( | )|T ,
on each
\
2 A
we must still
£
.
The operator
a
j
2
acts by
2
(i)\ A )(T,z) m
=
z
|
" X] X
where
N
e ( X T + 2Az)*(T, Z + X T ) M
(mod N )
is any integer such that
that the coefficient of (mod 2 m N ) .
q £ n
|X=
in
r
q ^
2
A E -IN and on
of
in
q C
r
\j)\A
is
X G N1 x i ,
depends only on
C(r, r - 4 n m ) denote the coefficient of
n
for
(For $ = (j> one checks that 2, a
only on
n
4nm - r
i.e. such 2
and on
N = (a,£) works.) in ip
r
r
Letting
(so that C(r,A)
depends
r € Z/2mNZ ) , we easily deduce that the coefficient C*(r, r - 4 m n ) with 2
^2 ' C(R
(19)
,
2
C*(r,A)
=
|
A)
R (mod 2Nm) r (mod 2m)
R=
(which depends, as it should, only on A and on this to
IL» =
^
r (mod 2 m ) ) .
we find that the coefficient of
q Ç
in
Applying <j> | T
£
is
given by
(20)
c*(n,r)
=
a ad=£
with n
C*
k
C*(r, r - 4 n m ) a
1
2
2
related as in (19) to the coefficient
(R -A)/4mçR
i
.
n
^ 2 ,a
C (R,A)
of
a
From eq. (18) we see that this coefficient
is given by „
/
A N
C (R,A) a
«
A / / n A((a,d),
A (a, 3 ) and
with the convention that
R -A 2
W
(
a
?
d
c (n,r)
)
j
are
\
/, R - A c(d ^ r r - , 2
0
T
dR\ ) ,
unless a, 3 , n
and
2
are integral.
Set a = (a,d);
then
ad = £
implies that N
a = xA~ a ,
for some
x,y€ W
with
d -= y„ 2 a ,
A
(x,y) = 1.
£ = xya ,
Then
(j¿,a)
(Ì y
'
(Ä,d)
X
r
-55-
x|R ,
Conversely, if
Hence (taking
x |A , - \ = ( | ) 2
A = x A ,
R = xr ,
A = r
Q
Q
N = (a,£) = xa
Q
Q
2
(mod 4m) .
(mod 4 m ) ,
then
in (19))
(21) r (mod 2am) xr = r (mod 2m) r = A (mod 4m) Q
Q
2
Q
if
A = x A 2
Q
and
C*(r,A) = 0
if
x fA 2
Note that, since m is prime to £ = a d 2
r
2
= A (mod 4m) and
The fact that
m
r' := r y = r y x Q
/ number
?
number
2
by hypothesis, the conditions r
2
Q
2
= A
Q
(mod 4m) .
is prime to £ also implies that the two numbers - 1
= r£/d
r -A
r and
determine one another (modulo 2m) and that the \
— 4 m ~ " ' o ^ / in the last formula is the same as the
c(n ,r') f
x ~ A = 2 or 3 (mod 4 ) ) .
r = A (mod 4) already assure that
2
cly
(or if
r
4'^)
in (14) (with
A = r -4nm). 2
]T
(22)
Moreover, we have =
£
A /
a
)
>
r (mod 2am) x r = r (mod 2m) r = A (mod 4) Q
Q
2
0
as we will show in a moment, and substituting this into the last equation and into (20) gives the desired formula (14), if we observe that
^ x
a
e
(a) = - — - e (ax ) o x a o 2
A
A x 2
= ~ e (a) . A
It remains to prove equation (22).
From the definition of A(a,$)
and the fact that a and m are relatively prime, we find that the left-hand side of (22) equals
-56-
Y | ot
r xr r = o
0 E
Q
2
(mod 2am) r (mod 2m) A (mod 4 Y ) o
ot
y N^(A )
The inner sum equals
ing equation (6) of §2.
^ Ç
K
N^(A )
with
q
as in the formula preced-
Q
That equation gives
oo
oo
E N
S
y=l
)
( A
O
) Y "
A
y
in
s
e (a)a
s
à
a=l
0
or (since the coefficient of n ~
E
=
L (s)
=
S
o
Ç(2s)/Ç(s)
is X (n) )
This completes the proof.
We remark that the hypothesis
(£,m) = 1 was not used in the
derivation of equations (20) and (21), so that we have in fact obtained a formula valid for arbitrary
*,
NT
X
a
k
£ and
X
^
m:
^
*( a
(23) *(n,r) = 2 Tz^y
V'"~~^"/ l
i_
a x = —,—jrr- , (a,£)
in (21).
J
y =
£ . (a,£) /
r+2mX\
2
~T~' Ti'
y
y
2
, _ . as before and we have set
r
o
ft
=
r + 2mX x
In particular, we find :
£ prime, £ | m
(24)
n+rX+mX
2
X(mod(a,£)) r + 2mX = 0 (mod x) n 4- rX + mX = 0 (mod x )
2
2
where
/
2
C
c
a|£
n+rX+mX \
S
=>
coefficient of
q Ç n
in <j) | T
r
c(£ n,£r) 2
c(£ n,£r) - £ 2
k _ 2
c(n,r)
c(£ n,£r) + £ - (£-l)c(n,r) + £ 2
k
2
2 k
-
3
^
£
if
£fr
,
if
£.|r, £ f n , (Bll2^1
c
,
X (mod £) | + | X + ^X =0(mod£) if £| r,n . 2
^f^)
CHAPTER II RELATIONS WITH OTHER TYPES OF MODULAR FORMS
§5.
Jacobi Forms and Modular Forms of Half-Integral Weight In §2 we showed that the coefficients
index
m
depend only on the "discriminant"
c(n,r) of a Jacobi form of r - 4nm and on the value of 2
r(mod 2 m ) , i.e.
(1)
c(n,r) = c ( 4 n m - r )
,
2
r
c ,(N) = c ( N ) r
r
for
r = r (mod 2m) . 1
From this it follows very easily, as we will now see, that the space of Jacobi forms of weight
k and index m
is isomorphic to a certain space
of (vector-valued) modular forms of weight
k-^
in one variable; the
rest of this section will then be devoted to identifying this space with more familiar spaces of modular forms of half-integral weight and studying the correspondence more closely. Equation (1) gives us coefficients all integers
0 satisfying
well-defined modulo 4m if
(2)
c (N)
(since y
:=
N = -y (mod 4m) 2
(notice that
y
2
is
c (^J^-
> )
(any
R
r€E Z ,
r = y(mod
2m))
re y
r = y(mod 2m) ; we permit ourselves the slight abuse of We extend the definition to all
N
by setting
c^(N) = 0
N ^ -y (mod 4 m ) , and set 2
oo
(3)
and
is given modulo 2 m ) , namely
is a residue class, one should more properly write
rather than notation).
y
c,^(N) for all y e Z/2mZ
h () T
M
:=
Z c (N) q N=0 u
-57-
N / 4 m
(y € Z/2mZ)
if
58
and z/ \ (4)
r% /_ \ V/ > > T
Z
2
< r /4m?
:=
r
r = y (mod 2m) (The
8^
are independent of the function
)(T
'
Z)
=
E
E
E
2
c (4nmr )q.V y
rGZ n^r /4m r = y(2m) N+r
¿2 E
=
Then
2
y (mod 2m)
(5)
(j).)
li (mod 2m) r = y(2m)
L
h..(T) e
У
N
V> *
I
ro
m,y
2
4M
?
N2 0
(T,Z)
.
y(mod 2m)
Thus knowing the (2m)tuple
(h ) , . _ of functions of one variable y y(mod zm; x
is equivalent to knowing that given any functions
Reversing the above calculation, we see h ^ as in (.3) with
equation (5) defines a function
c^(N) = 0
e
2
N ^ y (mod 4m),
z + X x + y
which transforms like a Jacobi form with respect to (X,y
for
Z ) and satisfies the right conditions at infinity.
In order for
(j) to be a Jacobi form, we still need a transformation law with respect to S L ( Z ) . 2
while
Since the thetaseries (4) have weight
<j> has weight
k
and index
be modular forms of weight
h
k%.
6
^
For the first we have
e
m,y
( T + 1
'
Z )
=
e
*n
( y 2 )
9
m,y
( T
>
z )
and (7)
H (T l) Y
+
=
must
To specify their precise transforma
1
<>
m,
m, we see from (5) that the hy
tion law, it suffices to consider the generators ^ of T .
and index
% (V) M
1> (T) u
,
and
^
-59as one sees either from the invariance of the sum (5) under or from the congruence
N = -y (mod 4m)
in (3).
2
x «—>x+l
For the second we have
as an easy consequence of the Poisson summation formula the identity
(8)
e J-I,f)
= V W M e ^ ^
m
J
e (- v)e 2m
y
m(V
(x, ) , Z
V (mod 2m) so (5) and the transformation law of
(9)
h (-1)
(f> under
-
(x»z) J—> ^- ~ ; , y ^
J e (yv) h (x) v(mod 2m) 2m
V 2 m T / l
give
.
v
We have proved
THEOREM 5.1.
Equation
(5) gives an isomorphism
the space of vector valued modular satisfying
the transformation
forms
( ) y( n
u
moc
between
[ 2m)
o
n
S L
laws (7) and (9) and bounded
J
fc
and
m
2^)
as Im(x)
°°.
When we spak of "vector-valued" forms in Theorem 5.1, we mean that the vector
h(x) = (h ) , , _ satisfies ' V r y (mod 2m) x
(10)
where
h(Mx)
U(M) = (U
=
( c x + d ) ^ U(M)h(x) k
(
M
(M) ) is a certain 2m x 2m matrix (the map
=
(c
d)
G
r
i
}
U: Y ^ -> G L ^ C )
is not quite a homomorphism because of the ambiguities arising from the choice of square-root in (10); to get a homomorphism one must replace r by a double cover). identify weight
k- h
in the cases
The result 5.1 would be more pleasing if we could
^ with a space of ordinary (i.e. scalar) modular forms of on some congruence subgroup of m = 1 and
case a little. of Theorem 5.1.
m prime, k
r .
We will do this below
even, and also discuss the general
First, however, we look at some immediate consequences
-60-
First of all, by combining (5) with the equations
0^
(T,Z) =
k
0
^ (T , - Z )
m
$ (T ,-z) = (-1) (j) (T , z)
and
(11)
h_^
=
(-l) h
we deduce the symmetry property
(y G 2/2mZ)
k
y
(this can also be proved by applying (9) twice), so that in fact (h ) reduces to an (m+1)-tuple of forms to an (m-l)-tuple
(h - h ) _ y -y 0
a finer splitting if (m',m/m ) = 1
m is composite.
(there are 2
T
(h,, + h . if k is even and y -y u S y < m if k is odd. However, we can introduce For each divisor
such divisors, where
t
m
of
?
m
with
t is the number of
distinct prime factors of m) choose an integer £ = £ i satisfying (12) £ = 1 (mod 2m/m ) , £ = -1 (mod 2m') ; f
such a all
£
clearly exists and is unique (mod 2 m ) , and the set of
m'Jm
is precisely
(mod 2m) | £ = 1 (mod 4m) } .
(
1
3
)
( h
Because
£
preserved.
2
r
4) |W^
in
4n m - r' f
2
)
TT
i=l
\y y(mod2m) }
"
it is clear that equations (7) and (9) are
f
For each divisor
W^, is
from
^
c (n ,r' ) 1
= 4nm - r . 2
m
T
of
m with
(m ,m/m') = l T
to itself such that the coefficient
where
r = -r (mod 2 m ) ,
These operators
form a group isomorphic (m =
(
Hence we deduce
is an operator n
( h ) ^ into itself by the permutation
y y(mod2m)
= 1(mod 4 m ) ,
THEOREM 5.2.
q £
Now map the
2
collection of (2m)-tuples
£ , for
1
1
1
by the
and W
V
together
i
P^).
Next, we relate the expansion (5) to the Petersson product introduced in §2.
of
r' = r (mod 2m/m' ) ,
are all involutions
to (Z/2Z) " and generated
there
-61THEOREM 5.3.
be two Jacobi
Let
forms in J, . Then k,m
In other words, the Petersson scalar product of
(j) and
as
defined in §2 is equal (up to a constant) to the Petersson product in the usual sense of the vector-valued modular forms (h ) , (g ) y'y' k-Jg.
V 6
of weight
y y /
&
The assertions of Theorem 2.5 (that (cj),ij;) is well defined and is
finite if
is cuspidal) now follow from the corresponding
statements for modular forms in one variable. Proof.
We first compute the scalar product of
a fixed fiber (T G "K fixed):
Using e(rz-sz)dx 3R/Z
we find that this equals
= 6
e
8
and
9
in
-62(here
6
is the Kronecker delta of
r, s and
6
of
u
and
V modulo 2m).
It immediately follows that
as claimed. Since that
W^,
simply permutes the h^,
W , is Hermitian. m f
commute with all
T^
COROLLARY. all
T
x, 0
((£,m)=l)
From Theorem 4.5 it is clear that the
((£,m) = 1 ) .
^ and
it follows from Theorem 5.3
Hence we deduce
has a basis of simultaneous W. m
W . m
ei gen forms for
(m ||m). " f
T
Theorem 5.2 gives a splitting of
J, k,m
as
±. . .±
© J. ± . . ,± k,m
, where *
the sum is over all t-tuples of signs with product (-1)^; Theorem 5.3 shows that this splitting is orthogonal and that each summand has a basis consisting of Hecke eigenforms. We now discuss the connection between Jacob! forms and scalarvalued modular forms of weight
k-%.
We recall that modular forms of
half-integral weight are defined like forms of integral weight, except that the automorphy factor describing the action of a matrix ^
A
JJ^
involves the Legendre symbol (-7); the easiest way to specify the d automorphy factor exactly is to say that for a modular form F (4m) Q
F (4m). Q
the quotient
h/S^ \
where
6(T) = S q
We denote the space of such forms by
1 1
,
^(T
Q
all primes
p
4m,
that
^(F (4m)) 0
Tp on
on
is invariant under (4m)) .
developed an extensive theory of such forms in [29], [30]. he showed that one can define Hecke operators
h(x)
Shimura
In particular,
M^_^(F (4m)) o
for
is spanned by simultaneous eigen-
forms of these operators, and that the set of eigenvalues of an eigenform is the same as the set of eigenvalues of a certain Hecke eigenform of
-63-
weight 2k-2.
(Shimura used the notation
T(p )
for the Hecke operators
2
in half-integral weight because they are defined using matrices of determinant
p ,
but we prefer to write
2
Tp
since these are the only
naturally definable operators and correspond to the operators weight 2k-2.)
in
His conjecture that the eigenforms of integral weight
obtained in this way have level 2m was proved by Niwa [24]. case m = 1
Tp
(and later for the case of odd, square-free
m
For the
[12]), Kohnen
[11] showed how one could get all the way down to level m by passing to the subspace
of forms in
^ ( T (4m)) whose o
Fourier coefficient vanishes for
k—1 all
N with (-1)
N
congruent to
2 or 3 (mod 4 ) . Following Kohnen s 1
notation in [12], we shall write simply M^_^
instead of M ^ _ ^ ( l ) .
M^_^(m)
for M^_^(4m)
Then Kohnen s main result for m = 1 says that 1
one can define commuting and hermitian Hecke operators all
p
and
Tp
on M^_^
(agreeing with Shimura s operators if p ^ 2) and that
^
?
becomes isomorphic to M^^ 2 as a module
M
all p. (14)
such that the eigenvalues of Explicitly,
Observe also that
h(x) l—> f (4x)h(x)
Tp:
M
^ -> M^_^
, := © M. ,
(h G M
We can now state:
l 9
*—2
h and
"
h G M^._^
h under
Tp
(k even) is given by
is a module over
f G M, ) .
then
over the ring of Hecke operators,
i.e. there is a 1-1 correspondence between eigenforms h G 2k-2
for
M,
by
and
agree for
-64-
THEOREM 5.4.
The
correspondence
n r
.2n
(15)-
gives an isomorphism is compatible
M^_^ and
between
with the Petersson
Hecke operators
9
r
^ (k even).
wtt/z t/ze
scalar products,
and with the structures
of
M
x
№£s
J
and
actions
of
. a s modules
L"k~^2
oyer
isomorphism
#,1
M .
Proof. (15) by
h(x)
D enote the functions defined by the two Fourier series in h (x) = 2 c^(N)q
0r,z) and set
and
N/4
with
y
2
N = -y (mod 4)
c(N) c (N) '
c(N) = c (N) + c (N)
so that
Q
By Theorem 5.1,
(V N)
l
0
otherwise
and
h(x) = h (4T) + h (4T) . Q
<j) is a Jacob! form if and only if
h
Q
and
h
satisfy
the transformation laws (7) and (9), which now become
(16)
These easily imply
(for the three-line calculation, see [40], p.385), and since the matrices ^
^
and
and hence
^ h G M^_^.
h e M^^,
Conversely, if
same calculation shows that Theorem 5.1 that
^ ( 4 ) , it follows that
generate
h
0
<j>(x,z) is in
and ^.
h
T
h G
^ ( ^ ( 4 ) )
then reversing the
satisfy (16) and hence by This establishes the isomorphism
65 claimed.
To see that it preserves the Petersson scalar product
(up to a constant), we combine Theorem 5.3 with the fact that (h,h) = const. [ ( h , h ) + ( h , h ) J Q
0
1
1
for
h
and
h^
related as above.
This fact follows easily from Rankin's method, which expresses (h,h) as ( ^ | c ( N ) | N J and similarly for h and h ~ ^N>0 (for more explicit formulas, look at the proof of Corollary 5 in [13],
a multiple of
2
Res
S
Q
1
S
2
l
/
pp. 189191, and take residues at
s=l
in the identities proved there).
The compatibility with Hecke operators is clear from (14) and Theorem 4.5. The compatibility with the structures of v
over
J
M , , 2*-h
and
0
J
. as modules *,1
is also clear. Theorem 5.4 tells us that the spaces
J
_. and
M„ , , are in
*,1 some sense -identical:
l*~H
they are related by a canonical isomorphism
preserving their Hilbert space structures, their structures as modules over the Hecke algebra and their structures as modules over the ring of modular forms of integral weight, and such that the Fourier coefficients of corresponding forms are the same (up to renumbering).
Combining this
with results on Jacobi forms proved earlier, we can obtain various (previously known) results about forms of halfintegral weight, in particular the following two: COROLLARY 1. Theorem
2.1.
lies in
М^_^ •
Le t
The n the
k^4
be
e ve n and
H(k1,N) = L_ (2k) N
as in
function
This was proved by H.Cohen for both odd and even
k
in [6]; it was
Cohen's discovery of the existence of Eisenstein series, half of whose Fourier coefficients vanish, which led to Kohnen's definition of the space
M^_^(4).
Cohen's calculation of the coefficients of the two
-66Eisenstein series in M ^ _ ^ ( r ( 4 ) )
is of the same order of complexity as
Q
the calculation leading to Theorem 2.1, but our proof makes it clearer where the condition
-N = • (mod 4)
comes from.
does not apply at all to the case of odd
COROLLARY 2. generators
3 f and
EL K
5
.
On the other hand, it
k.
is a free module of rank 2 over M , with t
.
This was proved by Kohnen [11, Prop. 1 ] , who also gave the corresponding result for M
,
(it is also free, with generators
9
and 7C ) ; our proof, which is a restatement of Theorem 3.5, works only 2
for even
k.
Conversely, by combining Theorem 5.4 with Kohnen s results as 1
quoted above, we obtain
COROLLARY 3. operators
If
<j) E
1
is an eigenfunction
T^, then there is a Hecke eigenform
same eigenvalues.
{eigenforms in
is bijective* there is a map
The
in 2 k - 2
More precisely,
> {normalized eigenforms in ^2k-2^
for fixed
: J, -, ->- M
n
Q
and r
Q
with
All of these maps are compatible of them is an
N = 4n -r Q
2
> 0,
given jby
(n'.r'e Z ,
combination
the
M
correspondence
^}/scalars
M
of all Hecke
with Hecke operators,
4n -r f
f 2
= il N/d ). 2
2
and some
isomorphism.
The second statement follows from the explicit description of the
-67-
correspondence
^ -> 2 k - 2
and from the corresponding result for
M
half-integral weight, namely that the map
sends M ^ _ ^ to M ^ _ 2
the g
for all N > 0 and that some linear combination of
2
is an isomorphism ([11], Theorem 1, iii; Kohnen states the
result only for
-N a fundamental discriminant).
However, it also
follows from the first statement of the Corollary and from Theorem 4.5, by noting that the map M
is simply
n
o» o
Finally, by combining Theorem 5.4 with the main theorem of [13], which is a refinement for modular forms of level one of a theorem of Waldspurger [36,37] one gets:
COROLLARY 4. f G 2k-2 S
(f =
§ G j£ ^ U
oorresponding
Then for all
where
Let
n,r with
be an eigenform
P
normalized
of all
T^ and
eigenform as in CoroIlary 3.
r < 4n, 2
L(f, e^, s) denotes the twisted L-series
Xe (n)a(n)n~"
s
D
Sa(n)q ). n
We mention one more result about the correspondences
THEOREM 5.5.
Let
<j> G
to one another as in Theorem
1
and
5.4 and
h G M _^ k
v^O
be forms
an integer.
J^_^ ~>
corresponding
Then
-68D
where 2)^
V
W
=
F (e,h)|u v
is the Taylor development
,
1 (
operator
of §3,
F
is Cohen's
v
2
operator as defined in the Corollary and
is the operator
to Theorem
3.3,
6(x) = 2 q
£ c(n)q »—» 2 c ( 4 n ) q . n
n
This can be checked easily by direct computation. case
(j) =
^,
h=5C^ ^
to Theorem 3.1. defining it as
gives Cohen's function
(Indeed, Cohen proved that F (0,7C _ )|U v
,
k
1
The special as in the Corollary
C ^ ^ belongs to M ^ ^ y V
^
.)
I+
We now turn to the case of general m, where, however, we will not be able to give such precise results as for m = l .
By Theorem 5.2 we have
a splitting
(17) where
e runs over all characters of the group
E = {£ (mod 2m) I £ = 1 (mod 4m)} = (m/2xf 2
with
£ (-1) = (-1)
k
and
£ J. k,m
± . . . ± J. ' * * * ' k,m
(which was denoted
in the remark
following the Corollary to Theorem 5.3) is the corresponding eigenspace, i.e. if to
(h,,) , , s are the modular forms of weight M "M (mod 2m) 0
<J) € J^.
m
then
tiQj = £ ( O h ^
for all ^ € H,
k-%
associated
In particular, the map
(18)
(which generalizes the map
<j>
Theorem 5.4) annihilates all
h ( 4 x ) + h (4x) Q
k,m prove the analogue of Theorem 5.4:
i
with £ ^ 1.
used in the proof of
For this map we can
-69THEOREM 5.6. If
The function
m is prime and
between
m
^th p i ouv
k even, then the map (18) defines an
and the space coefficient
eT
h(f) defined by (18) lies in
isomorphism
M^_^(m) of modular forms in
is zero for all
N with
M^^Cm).
M^^On)
whose
("—") = -!.
More generally, the image of (18) is contained in the subspace M^_^
(m)
whose N
consisting of all modular forms of weight
th
Fourier coefficient vanishes unless
and for m and
r (4m)
on
Q
-N is a square modulo 4m,
square-free the map (18) gives an isomorphism between
M, V * (m) K-h
spaces
k- ^
M^_^
(note that
J, = J. k,m k,m
for m prime and
of modular forms
2 c(N)q^
were studied (for m
k even). '
F
(in), and more generally the spaces M^/^.
^
(m)
The
consisting
with prescribed values of ( )
for
p|m,
odd and square-free) by Kohnen [12], who showed
that M _^,(m) decomposes in a Heeke-invariant way into the sum of these k
spaces.
It is tempting to assume that this decomposition corresponds to
the splitting (17) of
but this is not the case.
m >
We discuss this
in more detail below after proving Theorem 5.6.
Proof (sketch):
It follows from equation (9) that
is a multiple of h ^ ( T ) and hence invariant under the invariance of
h
h G M
We have
k—2"
(F (4m)). o
itself under
h = 2 c(N)q
c^(N) is non-zero only for in M | for m
* . +
k~^2
square-free it is infective on
this shows that
c ^ ( N ) = c^(N)
J,
1
.
jf
K,m
for all
since
h
lies
with £ # 1 , but 6 €E J
Indeed, for
k,m
J
implies that of
li = -N (mod 4 m ) , 2
this and
c(N) = 2 c ( N ) ; y M
with
N
Tf—> T+l;
suffice to show that
The map (18) is clearly zero on all
H
we have
T I—> T+l
T
1
k,m
Y
^ G H , and for m square-free this
Cy(N) is independent of the choice of the square-root
-N (mod 4m);
hence
c(N) = 2 c^(N) t
and therefore
y
h = 0 =>(j> = 0.
Finally, the surjectivity claimed in the theorem results from the formula (19)
dim Jk m
=
dim M2k-2^r*^m^
^m Prime>
k
even)
-70which will be proved in §§9-10 and the isomorphism r*(m)
proved by Kohnen [12], where SL (R).
M^_^(m) =
large, but the inequality with
T (m)
denotes the normalizer of
(Actually, (19) will be proved in §§9-10 only for
2
(m))
= replaced by
k
in
Q
sufficiently
£ will be proved for all
k,
and that is sufficient for the application here; it then actually follows from Theorem 5.6 and Kohnen s work that one has equality in (19) for T
all
k.)
This completes the proof.
Theorem 5.6 and the remarks following it describe the relation between
^ and forms of half-integral weight.
As we said, the
£
situation for the other eigenspaces
^
is more complicated; the
authors are indebted to N.Skoruppa for the following remarks which clarify it somewhat. Suppose that
m
character as above.
{C (mod p) | £
is odd and square-free and let Write
= 1 (mod p)} =
£ = TT P|
£
with
£
£: H •> {±l}
a character on
m
and let
TLllTL
We extend
f
be the product of those
for which
£p is non-trivial.
by taking
£ to be the product of arbitrary odd characters
for
be a
£ to a character
£ (mod f) £p (mod p)
p I f and define
(thus
h~ is our old i)
h
h
for £ = 1 ) .
Then
lies in M ^ C r ^ m O . X ) ,
g
where
X = (^) £
;.
k
£ on all J, k,m f
ii)
the map r
Y
h~ £
is infective on J
iii)
is
0
such that the representation of 2
and
£ J. ; k,m
the image of this map is the set of
SL (Z)) on
for £ ^ £ T
^ Y E SL CZ)
* lY
c
h
h
in
M^_^(r (4mf),X) Q
S L ( Z ) (= double cover of 2
is isomorphic to
C
£
(or to 0 ) ,
2
where
C
£
is a certain explicitly known irreducible
representation of
SL (2£)/Y (4m) . 2
p
-71This set can be characterized in terms of Fourier expansions and its dimension (resp. traces of Hecke operators) explicitly computed; it is contained in but in general not equal to the space of modular forms in M, ^ ( F (4mf) ,X)
2-/ c (N)q^. -N = D(mod4m) Notice that there is a choice involved in extending £ to £ but 0
2
with Fourier expansions of the form
that the different functions twists of one another. arbitrary (if m
h~ which are obtained in this way are
Skoruppa also has corresponding results for
m
is not square-free, then one must restrict to the U-new
part, i.e. the complement of the space / -4 \ then we must take
e
2
to be ( — )
© C
/d l^d »
if m
2
m
is even,
^ ^ >
of conductor 4 rather than 2 and thus
to be twice as large as above; the level of
hg
in general is four times
2
the smallest common multiple of m and f ) and has given a formula for dim jf for arbitrary m and £ [34]. k,m J
We mention one other result of Skoruppa, also based on the correspondence between Jacobi forms and modular forms of half-integral weight:
THEOREM 5.7 ([34]):
J, = {0} for all 1 ,m
m.
The proof uses Theorem 5.1 and a result of Serre and Stark on modular forms of weight 1/2. Finally, the reader might want to see some numerical examples illustrating the correspondence between Jacobi forms and modular forms of half-integral weight. at the end of §3.
Examples of Jacobi forms of index 1 were given
According to Theorem 5.4, these should correspond to
modular forms in M
, , so that, for instance, the coefficients given
in Table 1 should be the Fourier coefficients of the Eisenstein series in
(k = 4,6,8)
and cusp forms in M^_^
(k = 10,12);
one can check
that this is indeed the case by comparing these coefficients with the table in Cohen [7]. Numerical examples for higher
m
illustrating
Theorem 5.6 and the following discussion can be found at the end of §9.
f
-72-
§6.
Fourier-Jacobi Expansions of Siegel Modular Forms and the Saito-Kurokawa Conjecture We recall the definition of Siegel modular forms.
upper half-space of degree symmetric n x n matrices
n
is defined as the set JC
n
The Siegel of complex
Z with positive-definite imaginary part.
The
group
sp
2 n
(i)=
{ M ^ M ^ W l M j / ^ g
= j(£
^
,
J
2 n
=
I A,B,C,D G M ( m ) » A B = B A , C D ^ D C , A D - B C = I | t
t
1 1
t
t
n
n
acts on 3 C by n J
D)
(c
#
Z
(AZ + B M C Z + D ) "
=
a Siegel modular form of degree full Siegel modular group F: JC — > n
€
and weight
= Sp ^(S)
n
k with respect to the
is a holomorphic function
2
satisfying
F(M«Z)
(1)
for all
F
n
;
1
Z G 3C
n
and
=
det(CZ + D )
M = ^
e V^.
^
F(Z)
k
If n > 1, such a function will
automatically possess a Fourier development of the form
(2)
where the summation is over positive semidefinite semi-integral (i.e. 2t.., t. . G z) n x n matrices lj' n
T; if n = 1, of course, this must be posed
as an extra requirement. If n = 2 we can write Im(z)
2
< Im(T)Im(T )
we have
r
Q
and we write
f
T = f \r/z
Z as
^ ) with m / 2
^
with
F(t,z,T )
n,r,m 6 2 ,
?
T,T' G 5 C ,
instead of
n,m 2 0,
r
2
z G C,
F(Z);
< 4nm
similarly
and we write
-73-
A(n,r,m)
for
(3)
A ( T ) , so the Fourier development of
F(T,z,T' )
=
^
^
F
becomes
A(n,r,m) e(nT + r z + m T ) f
n,r,i€I n,m,4nm-r >. 0 2
The relation to Jacob! forms is given by the following result, which, as mentioned in the Introduction, is contained in Piatetski-Shapiro s 1
work
[26].
THEOREM 6 . 1 . degree
2 and write
Let
F be a Siegel modular
the Fourier
development
form of weight
of
k
and
F in the form
oo
(4)
Then
F(T,z,x )
=
f
£ m=0
4> ( > > ^ ' ) T
<j> (T,z) is a Jaoobi form of weight m
Proof.
For
(
ca
" ) G
and
z
e
'
m T
m
k
and index
(X y) G l
2
m.
the matrices
and
(5)
belong to
F
£
and act on 3C
2
(T,Z,T')
respectively.
Applying
by
*—>
(1),
(T, Z + À T + y, T
Fourier-Jacobi
expansion
+2XZ +À T)
<j>; the condition at infinity m
Following Piatetski-Shapiro, we call (4) the
of the Siegel modular form
F.
Note that the proof of Theorem 6.1 still applies if by a congruence subgroup.
,
2
we deduce the two transformation laws of
Jacobi forms for the Fourier coefficients follows directly from (3).
!
F
2
is replaced
Note, too, that the first collection of
-74-
matrices in (5) form a group (isomorphic to
S L Z ) but that the second 2
do not; the group they generate has the form
2
so that we again see the necessity of replacing group 3C_. The complete embedding
of
%
by the Heisenberg
SL_ (3R) H 3C
iL
into
TO
IK
Sp (1R) k
is given by
where
(A
1
y ) = (A y) ^
^ ^.
f
However, we shall make no further use
of this. The real interest of the relation between Jacobi and Siegel modular forms, and the way to the proof of the Saito-Kurokawa conjecture, is the following result, due essentially to Maass [21].
THEOREM 6.2. Then the functions coefficients
Proof.
Let §|V
§ be a Jacobi form of weight m
(m^O)
of a Siegel modular
form
V§
in §4 are the of weight
and index
1.
Fourier-Jacobi
k and degree 2.
Reversing the proof of Theorem 6.1, we see that the
function defined by ( 4 ) , where k
defined
k
cj> (m^ 0) are any m
Jacobi forms of weight
and index m, transforms like a Siegel modular form under the action
of the matrices (5).
In particular, this holds for the function
On the other hand, Theorem 4.2 gives the formula
(6)
75 for the Fourier coefficients (defined as in (3)) of V§ n
^ ^ c(n,r)q Ç n ^ r /4m
r
is the Fourier expansion of
, where
2
symmetric in
n
and
we deduce that V§
m
is symmetric in
T and
1
T ,
i.e. transforms like a Siegel modular form with respect to the matrix
(7) K
/0
1
0
0 \
(
1
0
0
0
\ 0 \ 0
0 0
0 1
•
}
as well as the matrices (5). T , it follows that 2
V§
\
1 / 0 /
Since these matrices are known to generate
is indeed a Siegel modular form.
The same proof works for the standard congruence subgroups of
r ,
since these are known to be generated by the matrices (5) (now with ^ in the corresponding subgroup of r
) and (7); nevertheless, it would be
nice to have a "real" proof of Theorem 6.2, i.e. a direct verification of the transformation law of V§
with respect to all elements of
Y. 2
Theorems 6.1 and 6.2 give an infective map
Щ:
М (Г ) к
— >
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 = #/(<^(F)),
i.e. of Siegel modular forms whose Fourier
Jacobi expansion (5) has the property J( > =
(V m ) .
m
that the Fourier coefficients
A(n,r,m)
(defined by (3)) are given by
the formula (6), where ( A(n,0,l) (8)
c(N) =
This means
if
N = 4n
if
N = 4nl
I ( A(n,l,l)
^
-76(The last statement could be omitted, since the validity of (6) for any sequence of numbers je(N)} forces the
c(N)
to be given by this formula.)
Equivalently, we can characterize these functions by the Fourier coefficient identity
(9)
M^(T)
The space
2
of functions satisfying (9) was studied by Maass
([20], [21]; see also [17]) after the existence of Siegel forms satisfying these identities had been discovered experimentally by Resnikoff and Saldana [27] (whose Tables IV and V should be compared to our Tabled 1 and 2 ) ; he called it the "Spezialschar".
We thus have established
inverse isomorphisms
combining these with the isomorphism
COROLLARY 1.
If
h(x) =
^-^-~» M^_^
^
c(N)q
given in §5 we obtain
is a modular
form in
N = 0,3(mod 4) Kohnen s "plus-space" 1
equation
(6) are the coefficients
"Spezialschar"
M£(T ), 2
then the numbers
M^(F ). 2
The map
A(n,r,m) defined by
of a modular
form
F in Maass'
h »-> F is an isomorphism
the inverse map being given in terms of Fourier
by equation
from M^_^ to
coefficients
(8).
From Theorem 3.5 (or Corollary 2 to Theorem 5.4) we also deduce:
COROLLARY 2.
The "Spezialschar"
M*(r
over
M (SL.(Z)) on two generators,
© M*(r k even 4 and 6 .
) = 2
of weights
K
2
)
is free
-77This result was proved by Maass [21], [22]. Finally, we must check that the map V action of Hecke operators in I: T
algebra map of weight weight
•>
g
and index
1 such that
W)|T
=
the chapter we write and
TTj) .
T (£)
T g ( p ) with 2
and
g
g
maps Hecke eigenforms to Hecke
It is well known (cf, Andrianov Tg(p) and
V T G TT .
M<J>|l(T))
This will imply in particular that V
by the operators
i.e. that there is an
2 to the Hecke algebra for Jacobi forms of
(10)
eigenforms.
M^CI^),
from the Hecke algebra for Siegel modular forms
k and degree
k
^ and
is compatible with the
Tj(£)
[1]) that
p
TTg
prime (until the end of
for the Hecke operators in TFg
We use the somewhat more convenient generators
Tg(p) = T ( p ) g
2
T (p) g
and
- T (p ). 2
g
THEOREM 6.3.
The map
Pi
-> M ^ ( F )
1
i: TTg -> TTj. defined
on generators
is
2
(in the sense of (10)) with respect
Heoke-equivariant
to the homomorphism
of Heeke
algebras
by
i(T (p))
=
Tj(p) +
, / w l(Tg(p))
=
/ k-1 , k-2s m / n , o 2k-3 , 2k-4 (p + p ) T ( p ) + 2p + p
s
Proof.
is generated
k P
_
1
+ p ~ k
,
2
J
In [2], Andrianov proves that Maass' "Spezialschar" is
invariant under the Hecke algebra by calculating the Fourier coefficients of the two functions F
for
F G M^
x
=
F|T (p) s
,
F
2
=
F|Tg(p)
and checking that they satisfy the identity (9).
Spezialschar is just the image of V,
this says that for
<j) G
Since the ^,
-78-
for some
b >t
<
1
w
^ *\ 1
( >
2
t
^ ^
1
explicitly given coefficients, so to prove
the theorem we need only check that
Х
=
Ф|(Т (Р) + Р " '
, Ф
=
Ф| ( ( р
Ф
Л
1
К
2
+ Р " )
,
(11) 2
к
"1
. к-2 , ч , 2к-3 ^ 2к-4 + Р )Tj(p) + 2р + р ) Л г г
0
ч
.
The coefficients of a Siegel modular form in the Spezialschar are determined by a single function If
c ( N ) , C j ( N ) , and
way to
(j), (j> and i
c (N) 2
c(N)
(N=0,3(mod 4))
are the coefficients corresponding in this
(f> , then equations (13)- (16) of [2] (with D = - N , t=l, 2
d=0) give
if
p fN,
if
p
2
if
p
2
if
p j N. In a unified notation, this can be written
2
I N,
f N,
as in (6),(8).
and
2I
-79-
with the usual convention
c ^~^r) = 0
from Theorem 4.5 with m = 1, i = p ф I Tj_ (p)
if p f N. 2
On the other hand,
we find that the N ^
coefficient for
is given by
(cf. equation (14) of § 5 ) , and comparing this with the formulas for and
c
2
gives the desired identities (11).
In [1], §1.3, Andrianov associates to any Hecke eigenform F e S (F ) k
where
the Euler product
2
F|T (p) = Y F , s
p
F|T^(p) = YpF.
If
7 =Щ
with
|Tj(p) = Л ф , р
then it follows from Theorem 6.3 that
and hence
By Corollary 3 of Theorem 5.4, there is a 1-1 correspondence between eigenforms in M^^ deduce:
2
and
^,
the eigenvalues being the same.
We
c
1
-80-
COROLLARY 1 (Saito-Kurokawa Conjecture). spanned
by Hecke
normalized
eigenforms.
Hecke eigen forms
The space
S
£(T )
These are in 1-1 correspondence f £
2 » ^he correspondence
is
2
with
being
such
that (12)
Z (s)
=
F
Ç(s-k+l)Ç(s-k+2)L(f,s)
As stated in the Introduction, most of the Saito-Kurokawa was proved by Maass
[21,22,23].
In particular, he found the bijection
between functions in the Spezialschar and pairs of functions satisfying
conjecture
(16) of §5, showed that
dim M^(T^)
= dim 2 k - 2 ' M
(h ,h ) 0
anc* ^2\a-2
established the existence of a lifting from the Spezialschar to satisfying
(12), without, however, being able to show that it was an
isomorphism.
The statement that the Spezialschar is spanned by eigen-
forms was proved by Andrianov
[2] in the paper cited in the proof of
Theorem 6.3, and the bijectivity of M a a s s authors
1
[40].
See also Kojima
[14],
lifting by one of the
1
A detailed exposition of the
proof of the conjecture (more or less along the same lines as the one given here) can be found in [40]. A further consequence of Theorem 6.3 is
COROLLARY 2.
The Fourier
(sum over non-assodated in
M^(T ) 2
coefficients
pairs of coprime
of the Eisenstein
symmetric
matrices
series
C,D e M (2)J 2
are given by
(2) Indeed, since with
A(^|J Q ) ) = 1 ,
is the unique eigenform of all Hecke operators it must equal
j) >
and the result follows.
-81-
The formula for the coefficients
A(T) of Siegel Eisenstein series of
degree 2 was proved by Maass ([18], correction in [19]), but his proof involved much more work, especially in the case of non-primitive
§7.
T.
Jacobi Theta Series and a Theorem of Waldspurger Perhaps the most important modular forms for applications to
arithmetic are the theta series
(i)
where
Q
is a positive-definite rational-valued quadratic form on a
lattice A %rk(A)
of finite rank.
This series is a modular form of weight
and some level; in the simplest case of unimodular
Q
(i.e. Q
is integer-Valued and can be written with respect to some basis of A \ x Ax
with A
t
as
a symmetric matrix with integer entries and determinant 1)
it is a modular form on the full modular group.
As is well known, the
theta-series (1) should really be considered as the restriction to z = 0 ("Thetanullwert") of a function 0 ( T , z ) which satisfies a transformation law for
z i — > z + AT + ]i
and is, in fact, a Jacobi form.
(This fact,
which goes back to Jacobi, is the primary motivation for the definition of Jacobi forms.) modular
We give a precise formulation in the case of uni-
Q.
THEOREM 7.1. form on a lattice with
Let A
Q(x) be a unimodular -positive definite
of rank
Q(x) = h B(x,x).
2k and
B(x,y)
Then for fixed
y£A
the associated the series
quadratic
bilinear
form
-82-
(2)
is a Jacobi form of weight
Proof.
k
and index
m=Q(y)
on
SL (Z) . 2
The transformation law with respect to
is obvious and that for ^ ^
^^
l) ^
0
2
x
The transformation law with respect to
is equally clear:
e (A T + M
( Z )
are particuH »
z i—> Z + A T + U
2
an immediate consequence of the Poisson
summation formula, so that the modular properties of larly easy in this case.
S L
2Az)0
( x , z + A T + p)
xGA
Finally, as pointed out in the Introduction, the conditions at infinity 2
are simply
Q(x)>0,
Q(y)>0,
fact that the restriction of 2 x 4- TLy C A If
Q
is positive
Q
B(x,y)
< 4Q(x)Q(y),
which express the
to the (possibly degenerate) sublattice
(semi-)definite.
is not unimodular, then
0^ Q,x
will have a level and
character which can be determined in a well known manner. We recall two generalizations of the theta series (1) and explain their relation to the Jacobi theta series ( 2 ) . insert a spherical polynomial (recall that "spherical" means with respect to a basis of
P(x)
in front of the exponential in (1)
AP = 0, where
A®3R
for which
homogeneous of degree 2y, then the series (3)
First of all, one can
A Q
is the standard Laplacian is
Sx
2
):
if P
is
-83is a modular form of weight
k + 2v
(2k = rk A ) and the same level as
and a cusp form if V > 0 (see for example Ogg [25]).
THEOREM 7.2. the
Let
A,Q,B,y
be as in Theorem
0^,
We then have
7.1,
v G ]N . Q
Then
polynomial
(4)
(
^
as in Theorem
Proof.
3.1) is a spherical polynomial
2v
and
The final formula is clear from the definition of - ^ v *
and this makes it morally certain that P^ ^ to Q,
of degree
since (3) is never modular unless
is spherical with respect
P
is spherical.
To check that
this is really so, we use eq. (3) of §3 to get
?
v
y (x)
=
(we may assume y ^ 0, (choose a basis so
constant x coefficient of
2 V
in Q ( y - t x )
^
so m ^ 0) , from which the assertion follows easily
Q = Ex.
2
and compute
1
recalling that
t
V
^ ) Q(y - tx) 3x /
,
2
k = % rk A ).
The other generalization of (1) is the Siegel theta series
(5)
where
r^(T)
form T
by
is the number of representations of the binary quadratic Q
(just as the coefficient of
representations of we have
n,
q
11
in (1) is the number of
or of the unary form n x , by 2
Q).
Explicitly,
-84-
Then the following is obvious.
th THEOREM 7.3. Jacobi
Let
coefficient
of
Q be as in Theorem (2) 9^ equals
7.1.
Then the m
Fourier-
Theorems 7.2 and 7.3 show how the Jacobi theta series fit into the theory developed in §3 and §6, respectively. is also easily described: to
0
the modular form of weight
y.
Q
associated
to the orthogonal complement
We illustrate this in the simplest example, with
Then (A,Q) is the
The form
0^
Aut(Q).
and since
J
E
is
4,1
k =4
and
m = 1.
lattice (cf. Serre [ 2 8 , 5 , 1 . 4 . 3 ] ) , i.e.
g
(since
240 choices of y under
k-^
is essentially the theta series associated to the quadratic
r t
form of rank 2k-l obtained by restricting of
Their relation to §5
with
dim
Q(y) = 1,
we can choose
=1);
in particular, there are
but since they are all equivalent
y=(%,...,h).
Then
is one-dimensional this must equal
formula for the coefficient of
q Ç
or, more explicitly
x^
(replacing
n
r
by
in
E
fc
±
h x ),
E
4, l
.
Hence the
(Theorem 2.1) implies
85
(6)
G Z
#{x
= 8n, 2 X = 4r}
2
= . .. = x (mod 2) , 2 x
I
g
= 252 H ( 3 , 4 n r ) . 2
This first seems like an infinite family of identities, parametrized by r E 2£, but in fact the identities depend only on if
r
is even then replacing
x^
x^ 4- \ r
by
of (6) by the same expression with so (6) for any even
n
r (mod 2 ) .
Indeed,
replaces the lefthand side
replaced by
n ^ r
2
and
r by
0,
r is equivalent to the theta series identity
(7) X G F , X X, =
+
1
...+X
. . . =
X.
=0
8
(mod 2) (= l + 1 2 6 q + 756q
Similar remarks hold for 8
replaces
"x e E " by
this condition into for odd
r
r
2
+ 2072q
3
odd except that now replacing
"xe
8
(Jg,...,%)+ Z
8
" x G I , all
x^
" ; doubling the
odd, x == . . . = x x
Q
+
...)
x^
by
x_^+%r
x
then turns
(mod 4 ) " ,
so (6)
is equivalent to
(8) 8
-хЕЖ
..,+х
,х + х
8
=
О
x.¡_ odd, x = . . .= х (mod 4)
Thus the Jacobi form identity
0_ = E Q,y
modularformofhalfintegralweight the integervalued quadratic form 8
{x G Z | S x = 0, x = . . , = X ±
x
V
is equivalent to the k,i
identity 8
2 x
2
0^,=H , 3
where
Q
on the 7dimensional
F
is
lattice
(mod 4) }.
We end this section by combining various of the results of Chapters I and II to obtain a proof (related in content but different in presentation from the original one) of a beautiful theorem of
-86-
Waldspurger s
[35], generalizing Siegel's famous theorem on theta-series.
1
Siegel's theorem [33], in the simplest case of forms of level 1, says
(k e 4 Z )
(9)
Here
(A^,Q^)
(1 < i < h)
denote the inequivalent unimodular positive-
definite quadratic forms of rank of
Q^.
Explicitly
2k
and
the number of automorphisms
(9) says
(10)
(n> 0)
-1
w.
(e. :=
ZTTT
)>
i.e. it gives a formula for the average
w, + . . . + w,
1
h
1
number (with appropriate weights) of representations of an integer the forms
Q^.
n
by
The number of representations by a single form, however,
remains mysterious.
Waldspurger s result in this case is 1
(11)
where
C (x) = fc
(H _ (x)6(x))|is k
1
Cohen's function.
Thus one has
explicit evaluations of weighted linear combinations of theta-series CjJ
with variable weights; in fact, since the
T
are known to span
n
(cf. discussion after the Corollary to Theorem 3 . 1 ) , one gets all modular forms in this way. n=0
(with the convention
Equation (9) follows from (11) by taking f|T
n
= 2C(l-k)
-1
a(0)E
k
for
f =
2a(n)q € n
or by computing the constant term of both sides; taking the coefficient of
q
(12)
m
on both sides gives
-87-
Finally, Waldspurger has a generalization of (11) involving theta series with spherical polynomials. 1
To prove all of these results, we start with Siegel s own generalization of (9) to Siegel modular forms of degree 2 :
(13)
(2) Here 0
is the Eisenstein series of degree 2 and weight k and th the theta-series (5). We then compute the m Fourier-Jacobi
i coefficient of both sides of (13); by Theorem 7.3 and (the proof of) Corollary 2 of Theorem 6.3, this gives (14)
an identity of Jacobi forms of weight the development map
k
and index m.
of §3 to both sides of (14).
We now apply By Theorem 7.2,
the left-hand side of the resulting identity is
where
P?
m
is given by
(15)
and is a spherical polynomial of degree 2v with respect to Q . i
By the
Corollary to Theorem 4.2, the right-hand side is
But
j£> (E, , ) 2v k,l
is Cohen's function
C ^ к
as defined in §3 (Corollary to
-88-
Theorem 3 . 1 ) .
Hence we have the following identity, of which ( 1 1 ) is
the special case V = 0 .
THEOREM 7.4 (Waldspurger inequivalent and
PV
quadratic
(v £ 0 ) the polynomials
m
V
where
unimodular
C^ ^
[35]).
is Cohen's function
Let
Q
(i = 1,. . . ,h) be the
±
forms of rank p^j
(15),
2k ( k > 0 , k = 0 (mod 4))
as in (1) of §3.
(as in the Corollary
to Theorem
Then
3,1)
th and
T
m
the m
Hec k e operator
in
№y 2 * i+
Since, as mentioned in § 3 , the
V
C | T
m
are known to span
S^^^*
one obtains
COROLLARY. P \
spherical if
For fixed
of degree
k
and
V,
2v with respect
the functions to
) span
0^ S
j^ 2 +
p
(i = 1,... h, (respectively
V
v= 0).
In particular, the theta-series associated to the E -lattice and g
to spherical polynomials of degree k-4 span
S^. for every
k.
CHAPTER III THE RING OF JACOBI FORMS
§8.
Basic Structure Theorems The object of this and the following section is to obtain as much
information as possible about the algebraic structure of the set of Jacobi^forms, in particular about i)
the dimension of
J^ ^
(k,m
fixed), i.e. the structure
of this space as a vector space over ii)
the additive structure of
J *,m
module over the graded ring
€;
= © J, k,m
(m
fixed) as a
k
= ©
of ordinary
modular forms; ii!)
the multiplicative structure of the bigraded ring J - © J, of all Jacobi forms. *,* k,m k ) t n
We will study only the case of forms on the full Jacobi group (and usually only the case of forms of even weight), but many of the V.
considerations could be extended to arbitrary
The simplest properties of the space of Jacobi forms were already given in Chapter I. for all k
and m
There we showed that
and zero if k
or m
J^
m
is finite-dimensional
is negative (Theorem 1.1 and its
proof) and obtained the explicit dimension estimate m + £ dim V=l
dim
(1)
dim J
k,m
(k
even)
,
ID-1
E V=l
d
i
m
S
k v-i + 2
-89-
(
k
o
d
d
)
'
-90(Theorem 3.4 and the following remarks). reduces to M, E
if m = 0
for m = 1.
We also proved that
m
and is free over M , on two generators
The very precise result for
J
6,1
E
,
was obtained by k, 1
comparing the upper bound (1) with the lower bound coming from the linear independence of the two special modular forms E and E 6,1
Similarly, we will get information for higher
m
by combining (1) with
the following result.
THEOREM 8.1. independent
over
Proof.
The forms
E^
and
l
E
g
are
1
algebraically
M^ .
Clearly the theorem is equivalent to the algebraic indepen-
dence over M
of the two cusp forms
d>_ , and ' <J>
. defined in §3, (17).
A
10,1
k
Suppose that these forms are dependent.
12 1 9
Since both have index
1 and any
relation can be assumed to be homogeneous, the relation between them has the form m (2)
E
. M
T
*io i
)
for some m, where the the smallest
j
gj
for which
( T
'
z ) 3
* 1 2 i< > > " T
z
m
=
3
0
are modular forms, not all zero. g_, is not identically zero.
Let
j
q
be
Substituting
into.(2) the Taylor expansions given in §3, (19), we find that the left-hand side of (2) equals
(constant)A(T)
m
g . (x) ' o
2 J Z
° +
0(z
2 3
°
+ 2
)
3
and hence cannot vanish identically.
Since the two functions our analysis of
J
E^
J
This proves the theorem.
and
E
g
i
will play a basic role in
, we introduce the abbreviations
A,B
to denote them.
k , k
Thus
A G J
4, i
M [X,Y] -> J k
,
B G J
6,1
sending k
k
X
and the theorem just proved says that the map to A
and Y
to B
is injective.
The (k,m)-
91
graded component of this statement is that the map
^k4m (f ,f 0
is injective.
^k4m2
l S
* * * ^
...,f ) m
X
f ^
^k6m
^
^k,m
+ f ^ - h
+
...
¡ y
+
This implies
COROLLARY 1. —
dim J, k,m
> =
/ . dim M, . „. *l Tc4m2i
We now show how this estimate can be combined with (1) to obtain algebraic information about the ring of Jacobi forms.
COROLLARY 2.
Fix an inte ge r m > 0.
Jacobi forms of inde x
Proof. over dim
m
The n the
and e ve n we ight is a module
0 0
dim ^ + 0 ( 1 ) = dim ^ 4 0(1)
and
•
of
of rank m+1 over M .
The linear independence of the monomials
implies that the rank is at least m+1. >
space
3
A B
m
(0 < j < m )
Using the two facts
(we do not need the more
k precise formula
dim
+0(1))»
we can write Corollary 1 in the
weakened form dim X
If there were m+2
к,m
>
(m+l)dim M. + 0 ( 1 ) к
Jacobi forms of index m
00
(k >• , к
even) .
linearly independent over
M^,
then the same argument used to prove Corollary 1 would show that the factor m+1
in this inequality could be replaced by m + 2 ,
the upper bound
dim J
coming from (1).
fc
m
<
(m+1) dim ^
+ 0(1)
Hence the rank is exactly m+1.
contradicting
-92-
COROLLARY 3. polynomial forms
in A
Every Jacobi form can be expressed
and
B with coefficients
(quotients of holomorphic
Proof.
If
(f> G
as a
which are meromorphic
modular
forms).
, then the forms
be linearly dependent over must involve
modular
uniquely
(f) , A , A m
m
"*"B, ... , B
m
must
by Corollary 2, and this linear relation
<J) by the theorem, i.e. we have the formula
(3)
with
f,f. € M. J *
and
f ^ 0 ; dividing by
f
gives the assertion of the
corollary (the uniqueness follows at once from the algebraic independenc of
A
and B ) .
Looking more carefully at the proof just given, we can obtain an estimate of the minimal weight of the form
f
in (3).
Indeed, the
proof of Corollary 2 depended on the fact that the upper and lower bounds we had obtained for
dim J. (m k,m
fixed)
differ by a bounded
amount, i.e.
for some constant for
dim
C
we see that this holds with
for all even k. bounded by J
C
for all k.
M,[A,B1 n J , * k+h,m
using the explicit formulas C =
Hence the codimension of
of the form (3) with
is < C.
depending only on m;
of
f
Now if of weight
d) G J, k,m h,
m
^
^
and with equality
M [A,B] n j
in J.
is
* k,m k,m and there is no relation
then the subspaces
(f> •M^
and
J, . are disjoint and hence the dimension of M, k+h,m n J
Therefore there is a relation of type (3) at latest in weight
h = 12C = 6m(m-l).
(Later we shall obtain a much better bound.)
Corollary 3 says that
,® K ,
where
K
= C(E. ,E_)
is the
-93-
quotient field of M^,
is a free polynomial algebra
In particular, the quotient field of
J, 7F ,
, is
K^[A,B]
over
€ ( E ,E , A , B ) .
7t
K^.
In view
6
*T
of Theorem 3.6, this is equivalent to the statement that the field of Jacobi functions (= meromorphic Jacobi forms of weight for
SL (Z)
is
2
C(j(T) ,p(x,z)),
0
and index
0)
a fact which is more or less obvious
from the definition of Jacobi functions and the fact that every even elliptic function on
C/ZT+Z
is a rational function of
^-(T,Z).
Before proceeding with the theory we would like to discuss the case of forms of index
2
illustrate our results.
in some detail; this will both motivate and Here, of course, we do not need Theorem 8.1, 2
since we can check the linear independence of monomials
A ,...,B m
m
for any fixed
m)
2
A , AB
and
B
(or of the
directly by looking at the first
few terms of their Fourier expansions, as was done in the case m=l in §3. Thus we obtain the lower bound of Corollary 1 "by hand". the upper bound are given for small
k
k dim 1^. + dim S
+ dim S
k + 2
dim M ^ g + dim \ _
1
Q
f e + 4
+ dim \ _
±
2
This bound and
by the table 2
4
0
1
0
0
6 1
8
10
12
2
3
1
2
2 0
1
Thus the upper and lower bounds no longer agree, as they did for m = l , but now they always differ by
1.
in fact always the correct one. J
even
k>4
and all m > l .
We will see that the upper bound is In §2 we showed that
J
k,m
t
4- 0
Hence there exist non-zero forms
for all
X £ J
Y e J
g
^.
, if , 2
=
By Corollary 2, there must be two linear relations over 2
among the five Jacobi forms
X,
Y,
A , AB
2
and
B
of index
them, we could calculate the leading Fourier coefficients of
2. X
To find and
Y
(we didn't give complete formulas for the coefficients of Eisenstein series of index
>1
in §2, but from §4 we know that
E
and
E
are
-94-
proportional to
^| V
and
£
E
g
1
1V ,
from which the Fourier coeffi-
2
cients can be obtained painlessly).
However, we prefer a different
method which illustrates the use of the Taylor development coefficients .0» d) of §3.
Since
S = S = S, = { 0 } ,
V
V = 1,2; (thus
e
6
we normalize X = E^
2
,
X
Y = -E
6
and Y 2
we must have
) .
Q
0(z ) 6
X,Y
= -E
g
are
since a Jacobi form of index 2 is z
convenience we introduce the abbreviations
Q
(just as we already are using
A,B
for the near-Eisenstein series
for the derivatives of P,
Q
Then equation (12) of §3 shows that the
determined by its Taylor development up to
are those of Ramanujan).
for
V
j2? X = E^, !D Y
by assuming
beginnings of the Taylor expansions of
we do not need to go beyond
JZ>,.X = 2) Y = 0 V
10
8
for
E^
1 - 24 2 0
J
1
(by Theorem 1.2).
4
and and
(n)q
11
R E
g
for ^),
E^
and
For E
as well as
(the notations
g
P
P,Q,R
Then one has the well known identities
Q
and
R,
so the above expansions become
Similarly we find the expansions
for the two basic Eisenstein series of index 1.
Hence the five forms
-95-
2
with
X
2
= X,Y,A ,AB,B
have Taylor expansions
+ X z
Q
+ Xz
2
+ 0(z )
h
2
6
k
given by the table
_Jl_v
*
o
2^
X
Q
R - PQ
Y
-R
PR - Q
Q
QR - P Q
A
X
2
2
AB
QR
B
R
2
2^ P Q 2
%(Q + R ) + p Q 3
2
Q
2
2
2
2
- 2PQR
2
-PQ +Q R
2
3
2
^Q(Q + R ) + P R - 2PQ R 3
R-PR
2
If any linear combination of the rows gives
0
the right, then the corresponding form <j> is zero.
2
P QR-PR
2
2
2
- 2PR
2
- P R - Q R + 2PQ
2
% ( Q + R ) - PQR 3
+Q
2
h
2
2
2
in the three columns on 0(z ) 6
and hence identically
Therefore by linear algebra we find the formulas
expressing
X
and
Y
as polynomials in A
and
B
as in Corollary 3.
2
At this point we can discard AB, B
2
and
2
B
from our collection
X, Y, A ,
since 2AB
=
we can also replace
B
RX - QY
A
Z (Z
AB
2
=
Q X - RY - Q A 2
;
2
by the more convenient basis element
=
2 QX - A
is a cusp form and is equal to
R A 2
2
- 2QRAB + Q B 2
Q -R 3
124>
cusp form of index 1 defined in § 3 ) .
10
Then
, X,Y,Z
2
2
where
4>
iQ
l
is the
are Jacobi forms of
index 2 and weights 4, 6 and 8, respectively, and since they are clearly linearly independent over
we can improve our lower bound to
-96-
dim
2
^
m
=
^k_4
+
c
^
m
^k-6
+
^
M
\
8
'
Since the right-hand side of this inequality is equal to the upper bound (1), we deduce:
THEOREM 8.2. even weight
The space
is free over
and
8, respectively.
and
B by
M
2
°f Jacobi forms of index
on three generators
The functions
X, Y
and
0
0
and
X , Y , Z of weight Z are related
There are two striking aspects to this result: J
2
to
4, 6, A
that the module
is free over M. , i.e. that we need no more generators than are
required by Corollary 2, and that the only modular form we need to invert in order to express these generators in terms of A the discriminant function
A =
1 _
3
0 0
1 / 2o
2
(Q - R ).
and
B
is
We now show that these
two results hold in general.
THEOREM 8.3. In other words, of Theorem
The ring
the meromorphic
8.1 are holomorphic
Proof.
E
6
is contained
modular except at
in
forms occurring
in Corollary
Z
infinity.
Using the fact that
A(x) vanishes nowhere
we find with the same proof that the functions
^(TQJZ)
M ^ [ -j- ][A,B].
In the proof of Theorem 8.1 we used the fact that A ( T )
is not identically zero. in Ky
^
E^ ( T , Z ) i
Q
are algebraically independent for each point T G5C. Q
and Indeed,
-97-
replacing the functions
g^(x)
in (2) by complex numbers
c^, we find
that the left-hand side equals
(const.) I
as
z-*0, where j
because A ( T ) ^ 0.
o
is the first
j with
Now suppose that
c. ^ 0,
d> e j
and this cannot vanish is a non-zero Jacobi
k,m
form and let f (T) be a modular form of minimal weight such that f(j) £
[A,B] ,
i.e. such that there is a relation of the form (3)
(we have already proven that such an of A , then
f
exists)
f vanishes at some point T GdfC, Q
all j by the algebraic independence of
E
If
f
and then
is not a power
f (x ) = 0 for Q
(x .z) and
E
4 , 1 °
(x ,z) . 6 , 1 0
f and f. have a common factor (namely E or E if T is iri/3 3 equivalent to e or i and E ^ - j ( T ) A otherwise) , contradicting But then
Q
the minimality of f.
THEOREM 8.4.
Proof.
J. . is free as a module
over
M .
The proof is similar to those of 8.1 and 8.3.
Let us
assume inductively that for some k > 0 we have found Jacobi forms <j> , . . . ,<J> l
M
*
r
of weight
in weights G
< k,
k ,. . . ,k i r
< k
which are a free basis of J & ,m
i.e. such that every form in J, . . k ,m r '
be written uniquely as
T
\ f _^ ( x ) ^ (x, z)
certainly true if k = 1 or k
with
over
for k* < k can
f^ G M ^ ? ^ .
(This is
is the smallest integer with J^ ^ ^ 0 .)
If we can show that the <j)^ are linearly independent over M^ in weight k
also, i.e. that there is no non-trivial linear combination
S f .
in J, which vanishes, then we are done, for then the subspace k,m ' r
^i^k-k
• • • + ^r^k-k
+
^r+l'*** ^s ,(
,,. . ,<j>
for
°^
m
~*isa
direct
sum, and choosing a C-basis
its complement gives us a new collection of Jacobi forms
satisfying the induction hypothesis with k
So assume that we have a relation
Sf.d). = 0 in J, i
' ~]
k.m
replaced by k + 1 . .
Since k. < k, X
'
-98-
weight(f^) = kk^ > 0 ,
we have
so
lies in the ideal (Q,R) of
i.e. f
for some modular forms
i
=
Q
§
+
i
R
h
±
+ R • Zh.^
i
Jacobi forms
^
0.
=
and
ip
implies that
2
._
-s
= Rip, ip = —Qip 2
of smaller weight
T G H not equivalent to
because
11
e ^^
Q
and
R
i
are invertible near ±
±
for some Jacobi form
=
+R*
»).
±
m
with
e. £ M. i
. ;
k-10k.. '
ZCg.Re.H.
in weights
k-4
and
k-6
not
=
±
Qtf;
and weight k - 1 0 .
k - 1 0 < k,
Since
iji as a linear
combination
then the identities
=
О
,
2(п.+де.)ф.
=
О
(both < k ) imply by the uniqueness part of the
induction assumption that
Remark.
T
Hence we have
X h 4>
,
\Jj of index
and at all
and satisfies the cusp condition
by our induction assumption we can write
T
for some
(to see this, note that
R(T)^0
because
Q(T)^0,
because
2 g <$>
i x
between
l
Q
equivalent to
Se.d).
Qip + Rij^ = 0
transforms like a Jacobi form, is holomorphic at all
R points
(where» of course,
Then our relation becomes
B ut a relation
holomorphic Jacobi form ^
^k-6k.
G
g^ £ M ^ ^ ^ ,
forms of negative weight are zero). Q • 2g <|)
i
g^ = Re_^,
h = Q e ^ and hence
f^ = 0
for all
i.
The method of proof used for Theorem 8 . 4 would equally
show that other spaces of modular forms (e.g. modular forms of half integral weight, or modular forms of level into
M (r >V
M^(T
i
(N)) via either
f(x) H~> f (T)
N, or
with M, (V ) f(t)
i»f(NT))
embedded are free
0
)modules.
This fact, although not at all deep, may be of practical
interest in tabulating modular forms, since it means that all modular
-99-
forms of a given type (e.g. of fixed level but arbitrary even, odd or half-integral weight) can be described by tabulating the Fourier coefficients of a finite system of free generators. The results we have proved up to now have all been additive, i.e. concerned with points i) and ii) in the introduction of this section. We end with a simple result on the multiplicative nature of
THEOREM 8.5.
The ringof
ring of transcendence
Proof. hence 4 over M
Jacobi forms is an infinitely
is clear since
[A,B] = C[Q,R,A,B]
generated
degree 4.
That the transcendence degree of C,
^.
^
^
is 2 over M ^ ,
and
contains the polynomial algebra
and is algebraic over it (the square of any Jacobi
form of odd weight has even weight, and we have proved that a Jacobi form
<j) of even weight actually satisfies a linear equation
over M
[A,B], where in fact we can take
a €= C [Q - R ] ) . 3
We show that
2
J
a € C[Q,R]
acj) + b = 0
or even
, is not finitely generated. k
,
any finite collection of non-constant Jacobi forms f. G J, i k ,m J
T
i
(i=l,...,r).
By the results of §1 we know that tiu > 0 n
It follows that any monomial m/k
Consider
X
i
<|>
n
and
i
k^>0.
r
.. . (j)^
(i^ ,... , n
r
^ 0)
has a ratio
bounded by
Since m/k
is unbounded in J
(in §2 we constructed Eisenstein series k ,k
with
k = 4
and arbitrary
m) , we deduce that ,... , $ x
r
cannot generate
as a ring. Theorem 8.5 is a negative result. show how to embed
J
in a slightly larger ring which is finitely k>*
generated.
In the next section we will
-100§9.
Explicit Description of the Space of Jacobi Forms Our main tool for studying the structure of J ^ ^
fact that we had estimates on
dim J, k,m
in §8 was the
from above and below which
differed by a bounded amount as k->°° with m
fixed (at least in the
case of even weight; for odd weight we have so far given only an upper bound).
In this section we will improve both estimates, obtaining upper
and lower bounds which actually coincide for k
sufficiently large.
This will not only permit a more accurate description of the ring but will also lead to an algorithm for computing algorithms) which is effective and, for modest
J^
m,
J ^ ^,
(actually, two
m
practical.
The
results will apply to both even and odd weight. We begin with the upper bound.
dim (1)
dim
J,
k,m
+
The bound used in § 8 , namely
E dim v=l
£ •
(k
even)
<
~
m-1 d
i
m
S
L
k + 2
v-l
(
k
o
d
d
)
was proved in §3 as a corollary of the injectivity of the map
D
from
J, into the direct sum of the spaces whose dimensions appear on the k,m r
right of *(1).
r
r
We will now give a second proof of (1) which is even
more elementary than this proof (in that it does not make use of the Taylor development operators 3) )
and leads to a sharper result;
however, it gives less precise information than the first proof in that it gives only a filtration
of J^.
m
injectively into spaces M ^ ^ , +
whole of
J, k,m
with successive quotients mapping
rather than an injective map of the
into a direct sum of spaces M,
K+V
as we obtained before.
The sharpening of (1) we will obtain is the following:
THEOREM 9.1.
The dimension
of
J,
is bounded
above by
-101-
(2)
(3)
PROOF.
FOR V > 0
(v)
Jr , K,M
=
VV/
I.E.
J/*^ K,M
L
LET
icj) G J, K,M
I cj>(T,Z) = 0 ( Z )
I S T H E INTERSECTION O F J. K,M
AFTER T H E COROLLARY T O T H E O R E M 3.3.
k,m moreover, '
J^ ^ = k,m V
V +
and
j/ ^ = 0 k,m
for V > 2m
V
R
k,m
z= 0
for all
J
A N D THE SPACE
M , ^ ^ INTRODUCED K,M
k,m
for v = k (mod 2)
vanishing of a Jacobi form at
Z -> 0}
T H E N W E H A V E A FILTRATION
k,m
J5 ~^ k,m
for
V
1
because the order of
has the same parity as the weight,
k
and for V > 2m-3
for
k
odd for
the same reason used to prove (1) in §3 (a Jacobi form has 2m zeros altogether in
C/Z+Zx,
and for
zero 2-division points).
k
odd three of these are at the non-
On the other hand, from the defintion of
Jacobi forms we see that if a Jacobi form of weight
= f(T)z + 0 ( z
of weight
(4)
v
k + v,
V
X
)
near
z= 0
the function
k f
has an expansion is a modular form
so we have an exact sequence
0
-»
J<
V + 1 )
k,m
-»
J<
V )
k,m
4>
similar to (14) of §3.
—
H, Tc+V f
Together this gives filtrations
-102-
for
k
even and T
X
= J < « 3 j<3) 3 ... D J < - > k,m k,m k,m k,m 2
for k
D j<2m-l) k,m
3
=
(2V-1)
k^n (2v+l) k,m
Q
odd, and this immediately gives ( 1 ) (with M
^
R+2V-1
instead of S ) .
To get the sharper estimates ( 2 ) and ( 3 ) , we must say something about the image of the last map in ( 4 ) .
Consider first the case of even k.
The Fourier expansion of a Jacob! form of index m
can be written in
the form
where I* (Ç) is n
a
polynomial in Ç and Ç
of the condition at infinity. Ç
1
for k
< V4nm
by virtue
This polynomial is symmetric in Ç and
even and hence can be written as a polynomial in Ç + Ç — 1
more conveniently, in
Ç+ Ç
-2
Q (T) e c [ T ] ,
_1
é e j / ^ ^ , k,m V
identically if
n
P (e^^ ^ n
then each coefficient
so each polynomial
2
Q^(T) is divisible by T .
deg Q
< V,
deg Q <
.
n
must be 0 ( z ^ ) V
Hence
V
n
or,
(= -4 sin T T Z ) :
n
n
1
2
P (Ç) = Q (^+Ç -2) ,
If
of degree
Q
n
as z-*0,
must vanish
i.e.
(5)
The last map in (4) sends <J> to 2 a q
n
n
with
v and from (5) we see that this is 0 the last map in (4) is contained in
for
2
n < -j— .
Hence the image of
-103-
max ^dim M ^ 2 " " ^
and since the dimension of this space is inequality (2) follows. ence being that now (C - £ ) Q ( C + C 1
1
n
then
G
The case of odd
P (C) n
- 2)
COROLLARY.
k
for some polynomial
For
"j» o ) >
V
is similar, the only differ-
is odd and hence has the form
implies ^
k,m
+
T
k. > m
V
Q
"HQ„(T) and n 1
v
of degree < \ / 4 n m - l ;
n
hence
we ^ai?e tfte upper
V
2
< 4nm
as before.
bounds
(k even)
,
(k odd) where (6)
Proof.
We must show that the "max" in formulas (2) and (3) is
always attained by the first argument.
For
k
odd and 1 < V < m-1 we have
V and the assertion follows. .
for k
even and
0 < V < m-1;
i ^ „ k>m 2 +
k
=
The same argument shows that
m,m+l
_ =>
=>
if V = m
„ d i m M ^
2
dim M ^ ^ ^ >
we use instead >
k+2m-2 - 3 5 -
k+2m ^ 2 (mod 12)
proving the assertion in this case also.
=>
v
>
m ^
dim \
,
+
2
m
>
> f
,
-104-
We now turn to the lower bound; for this we have to construct Jacobi forms.
To do this we begin by enlarging the space of forms under
consideration.
Definition.
A weak Jacobi
form
of weight
k
and index
m
is a
function satisfying the transformation laws of Jacobi forms of this weight and index and having a Fourier expansion of the form
(7)
The space of such forms is denoted
J, k,m
r
The space
J,
is, of course, in general larger than
it is still finite-dimensional.
restrictions to
z = Ax+\x
as in Theorem 1.3 the condition
, but
Indeed, a weak Jacobi form also has 2m C/Z+Zx
zeros in a fundamental domain for transformation law under
J,
z -> z + Ax + ]i (A,11 G Q )
(since it satisfies the same
as a true Jacobi form) , and its
give modular forms in the same way
(the conditions at infinity are satisfied because of
n > 0
in the definition of weak Jacobi forms), so the
proof of Theorem 1.1 carries over unchanged.
Moreover, all of the
content of §3 also still applies, so we get maps
(8) ^
:
~k,m
J
-*
and they are still infective.
\
+
l
9
\
+
3
9
•••
9
\
+
2
m
-3
( k
°
d
d
)
The only point that needs to be checked
in order for, for example, the map
to make sense is that for a given
n
there are only finitely many
r
-105-
with
c(n,r) 4 0.
the coefficients
But this follows from the periodicity condition on c(n,r)
Jacobi forms, for given and
r = ± r (mod 2m) ,
(Theorem 2 . 2 ) , which applies unchanged to weak n
and
r we can choose an
by the condition defining a weak Jacobi form, so r
2
T
with
|r | < m f
and then
f
soon as
r
c(n,r)
vanishes as
- 4nm > m . 2
The basic result on weak Jacobi forms is the following.
THEOREM 9.2.
The map
(8) is an isomorphism
for all
k
and
m.
Notice that the statement of this theorem makes no reference to
k
being positive, and hence shows that the weight of a weak Jacobi form can be negative (but not less than - 2 m ) .
As a corollary of Theorem
9.2
we get
COROLLARY.
For all
dim J. k,m
> -
k
and
m
we have the lower
bounds
m /2 dim M, , - N. (m) _£? k+2V +
(k even)
0
,
0
m-1
with by
N (m) +
as in (6).
For k > m the inequality
signs eon be. replaced
equalities.
To prove the corollary, we note that there is an exact sequence
(9)
0
—>
J, k,m
—>
j
k,m
—>
€
N±(m)
in which the last map sends a weak Jacobi form collection of Fourier coefficients
c(n,r) with
to the
(resp. 0 < r < m
-106if
k
is odd) and
0 < n < r /4m;
if all of these coefficients vanish,
2
then it follows from the periodicity of the coefficients that all c(n,r)
with
4nm < r
vanish and hence that
2
Jacobi form, as asserted in ( 9 ) .
(Theorem 2.2)
<j) is a true
The second statement, of course,
follows from the corollary to Theorem 9.1.
Proof of Theorem 9.2. To do this, we replace
We need only prove the surjectivity of ( 8 ) .
J.
by an a priori
larger space
J*
K., m
the surjectivity of (8) with fact
J=J
f
.
The space r
, prove
K, m
j ' instead of J,
J, k,m T
and then show that in
is defined as the set of all functions
(f> ( T , Z ) satisfying the transformation law of Jacobi forms of index
m
!
with respect to
z + Xx + y
z
(X,y
Z ) and having a Fourier expansion
e
of the form ( 7 ) , but with the transformation law under the action of SL (2Z) replaced by the weaker conditions 2
(10)
CJ>'(T,-z)
-
(-l)
k
CJ) (T,z)
( (mod (11)
($'|
k ) m
M)(x,z)
,
?
2
m
z
+
2
)
even)
'
( (mod z
)
(k odd)
Equation (10) and the transformation law under the coefficients
c(n,r)
(VMerp.
cOCx.z)
-
Z+2£x
show that
of (j) satisfy the periodicity property of 1
Theorem 2.2, and the condition (11) implies that the development coefficients k
!D^>\D^'
, ... ,2> j'
Z>^\£>^\
(resp.
2
is odd) are modular forms, since the proof that
used only the Taylor development of (f> up to order (8) is still defined if we replace
J by J . f
surjective is now a question of linear algebra. (the case of odd k forms
is similar).
£ ( T ) = Sa (n)q V
4>' G j!
K,m
n
v
with i) = ^ £\) \) f
•••^ m-3
4 ) ?
i
f
2
!D^> G M ^ ^ +
V.
in §3
Hence the map
To show that it is Suppose
k
is even
Then given a collection of modular
of weight
k+2v
(0 < V < m ) we want to find a
for all V, i.e. to find coefficients
c(n,r)
-107-
satisfying a)
c(n,r) = 0
for
n < 0,
b)
c(ri,r) = c(n ,r')
c)
£ P r
for
T
Because of
r
T
= ± r (mod 2 m ) ,
(r,nm)c(n,r) = a^(n)
2 v
b) , we need only find
for
c (n, r)
r' - 4n m = r 2
f
0 < v < m,
2
- 4nm,
n > 0.
for 0 < r < m, since the rest
are then determined by the periodicity conditions; in terms of this we can rewrite c) as m (12)
E
e
r
p^"
(r,nm)c(n,r) +
1 }
...
=
a (n)
,
v
r=0 where
is 1 or 2 depending whether
a linear combination of coefficients
r
is
0
or not and
11
... " denotes
c(n ,r) with 0 < r < m
and
1
n < n. f
We can assume inductively that the equations have been solved for
n' < n
and hence that the " . . . " denotes a known quantity; then (12) becomes an (m+1) x (m+1) system of linear equations in the
m+1
unknowns
c(n,r)
(k-1) p^
(0 < r < m) with coefficients
(r,nm)
(0 < r, V < m) .
But this
matrix is invertible, since it is the product of the nonsingular matrix
(e 6 ) and the matrix r rv r,V .
K
( p ^ ~^(r,nm)) , 2v 'r,v
reduced to a Vandermonde determinant
(r^ )
diagonal
which can be
by elementary row
V
r, v (k-1) operations because in
r.
(r,nm)
is a polynomial of degree exactly 2v
Hence the equations can be solved inductively for all coeffi-
cients c(n,r). It remains to see that G J* T
and
M G T , l
J =J . f
But this is easy, because if
then the difference
a Jacobi form with respect to translations compatibility of the actions to order
> 2v
at
Z
2
and
(j) - (}) | ^ f
T
transforms like
z -> z + AT + y (because of the
SL (2£) proved in §1) and vanishes 2
z = 0 by (11), so Theorem 1.2 shows that it vanishes
identically and hence that
(J)' transforms correctly under
V^.
We now prove a theorem which determines the structure of the
-108bigraded ring 5
J
© J, k,m k,m k even
ev,*
6
THEOREM 9.3. on two
The ring
completely.
^ is a polynomial
over
M^
generators I
1
where
d) , 1 - £ kjf1 f ( 'k G
ui
J
Proof.
i Q
i
0
«=
1
p
That and
-2
2
i2
2
i
0,1
1
i
and
<j>
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
0 < n < r /4m)
of our basis
2
as the kernel of the map
X
k,m
N+(m)
—>
€
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
(13)
^12,1
4>
12 , 1
Substituting into this the expressions for of
Ч
l—J * '
(2iri)
=
,1 \
(t z)
d
/
r
=
E
E
l l ,2
E
6,1
E
4,1
6,1
This is a special case of the following easy fact, which gives a general construction for Jacobi forms of odd weight from forms of even weight.
THEOREM 9.5. k
2
and inde x
m
l
Le t
and
m ,
diffe re ntiation
m
and we ight
+ m
£
Proof. and weight
k
and
(J)
be
re spe ctive ly.
2
denotes 1
with re spe ct
to
Jacobi Th e n z,
forms of we ight m2$\§2
k
~ m1^1^2,
is a Jacobi form of
and
l
where inde x
k + k + 1. 2
2
The derivative of a meromorphic Jacobi form of index is a meromorphic Jacobi form of index
Applying this to
m m cf), / (J) z
0
and weight
proves the theorem (the conditions at
0 kf1.
'
-113-
infinity are trivial).
Examples of Jacobi Forms of Index
>1
We end with some numerical examples to illustrate the techniques for computing Jacobi forms implicit in the theorems of this section. We give several types of examples with m > l (the case
m=l
having been
treated in §3) to illustrate the results described at the end of §5. We look only at cusp forms.
1.
k even, m
prime
This situation, the one treated in Theorem 5.6, is the simplest case after
m=1
(because in the decomposition (17) of §5 there is only
the term £ = 1 ) .
From the dimension formulas we see that the first
three cases are
(m,k) = (2,8), (2,10) and (3,10) and that
dim j
P = i k,m c u s
in all three cases. For
(f> = 2 c ( n , r ) q £ n
r
E j
P
c u s
the first two Taylor coefficients
?
8 , 2
3)fl
and j£) <|> are cusp forms of weight 8 and 10 and hence vanish, while
3) § must be a multiple of A ( x ) . Theorem 2.2, so c(N) = c
Also, c(n,r) depends only on 8n - r
c(n,r) = c ( 8 n - r )
by
for some sequence of coefficients
2
(N)
2
satisfying
8,2
I c(8n-r ) = 0 , r 2
for all n
I r c(8n-r ) = 0 , r 2
2
I r c ( 8 n - r ) = 24x(n) r 4
2
(the constant 24 was chosen for convenience).
These
recursions can be solved uniquely (at each stage the numbers c(8n-l)
c(8n),
and c(8n-4) are expressed in terms of c(N) with N < 8(n-l)).
We find the values
c(4) = 1, c ( 7 ) = - 4 ,
c(8) = 6, ...
(cf. Table 3 a ) .
(These coefficients could also be calculated from the fact that 1 (f) = —
2
Z =
$
/A , where
Z
is the form of Theorem 8.2 and
-114(j> ^ G
^ the cusp form given in §3.)
lQ
The most striking thing in
the table is the occurrence of zero entries.
To explain them, we note
N that the form
M+
5 / 2
2 c(N)q
(2)
=
lies in the space
jh M
1 5 / 2
e
(r (8)) |ho
by Theorem 5.6 and is a new form (since
Ç c(N)q N> 0 ) N=0,4,7 (8) ' N
^13/2^
follows from the results of Kohnen [12] that form
4 (8r+3).
" W ) ; It then
=
c(N) = 0
for all N
of the
In Table 3a (and the rest of Table 3) we have also
given the eigenvalues of T^
for £ = 2 , 3 , 5 and 7, computed by means of
Theorem 4.5 and the formulas at the end of §4. For
d> G j
c u s
P
the same method works: now
3) §
const. • A , 3)^
= 0, 3) è =
n
10, 2
0
= 0
and we find
first coefficients given by (see Table 3 b ) .
c
1 0 > 2
(J) = S
T
(8n - r ) q ç 2
?
(4)=l»
c
1 Q
£
n
( 7 ) = 8,
2
,
T
with the
r
c
l Q
2
(8) = -18,
This time there are no zero coefficients because
cj) is not a new form:
it is
<j>
iQ
i
| V .
By Theorem 4.2, this is
2
equivalent to the formula c
(with the convention
1 0 > 2
c
(N)
1 Q
1
=
-%(c
(n) = 0
i 0 > 1
(N) +
2 c 9
l o > 1
(N/4))
for n ^ Z ) , and one can check this
by comparing the values in Table 3b with those in Table 1. Finally, for m = 3 cj) we must have
3)$
the first cusp form has weight 6; for this form
= 3)^
= 3)^
<j) =
I
= 0 and 3)^
c(12n-r )qV
I r
2 V
(° c(12n-r ) 2
=
so
,
2
9u
is a multiple of A,
(
V
=
2
)
I
r ( 720 T(n) (V=3) (again with 720 chosen for convenience), from which we get the values tabulated in Table 3c.
(We could also compute
/A .) 2
Again
-115-
the zero coefficient for N = 36,63,144... can be explained by the fact that 2 c(N)q^ G M ^ y ( 3 )
is a new form and the theory given in [12]; more
2
generally, we would have
2.
k
odd, m
c (N) = 0
for all N
of the form
9 (3r + l ) . S
prime
This is the next simplest example, since again the decomposition of with
m
e
given in (17) of §5 reduces to a single term, but no longer trivial.
The first case occurring, as we saw in the discussion
preceding Theorem 9.4, is m = 2, k = 11.
The corresponding form
(j) = (j) •1.1 , 2
is given explicitly by (13), but although this formula can be used to compute the Fourier coefficients of <)> it is easier to proceed as in the examples above. -e(n,-r)
n
(since the weight is odd) and c(n,r)
and on r (mod 4 ) ; even and that coefficients jZ^cj) G S
<j> = 2 c ( n , r ) q £
Indeed, if we write
1 2
can be written as
c(n,r) = 0
(-~- )c(8n - r ) 2
c(N) which are non-zero at most for
= €*A
2 rc(n,r)
implies that
c(n,r) =
depends only on 8n - r
together these facts imply that
c(n,r)
then
r
2
for r
for some
N = 7 (mod 8 ) .
Then
is a multiple of x ( n ) , so
(with suitable normalization) ^
(^)rc(8n-r ) 2
=
T(n)
.
0 < r
But here we can do more.
Indeed, the equation above is
equivalent to
( \ X
£
c(
N>0 N = 7 (mod 8)
\/j
8 N
)
q
/ \
( f r
>
)
V \ /
0
7
-
J n
>
X(n
n ) q
0
7
The second factor on the left is N ( T ) by Jacob's identity and the 3
expression on the right is
A(x) = T|(x)
.
Hence we get, instead of
.
-116-
just a table, the closed formula
h (T)
:=
?
£
2
Si
(
N
^
)
^ ^
=
8
T
>
2
1
N>0 where
£
as in §5 is an extension of £
to a Dirichlet character modulo 4
/ -4 \ (here necessarily §5.
£ = (—))
The fact that
hg(T/8)
C
of
SL (Z0
the form of weight 1 0 % defined in
is a modular form on all S L &
(with
2
multiplier) rather than on sentation
and hg
F (16)
is due to the fact that the repre-
Q
mentioned at the end of §5 is one-dimensional
2
in this special case (in general for m
prime and k
odd it would have
dimension m - 1 ) . 3.
k
even, m
a product of two primes
This is the first case where the decomposition of than one piece; we are interested in the summand non-trivial
£, since the space
I [ J,
first weight function
k
(j) € n
r
changes sign if y for
y
where
5y
not prime to 12 (since then
(mod 2 4 ) .
In particular,
2A(T),
*• =
E n,r
is 6 and the
To see this, note that a
y
7y.
(mod 12) and
It follows that
c^(N)=0
is congruent to 5y or 7y modulo 12 (—)e(N) for some y
c ( N ) can be written as m
c^(0) = 0
and vanishing unless
for all y,
is determined by
possible weight is k = 12. =
or
c(N) depending only on N
a cusp form, and since
cf)(T,0)
is 12.
m
c ^ ( N ) depends only on y
is replaced by
12) and that in general coefficients
J
has a Fourier development of the form
2 c^(24n - r ) q £ 2
can be understood by Theorem 5.6
The first composite
J^. ^ 4 {0}
with
has more
corresponding to
k,m
r
and the following remarks.
^
^
so cj) is automatically
2)^§ = < K T , 0 )
For the form with
N = 23
k = 12,
the first
e
and normalized by
we have (^)c(24n-r )qV 2
,
£ r>0
( ^ ) c ( 2 4 n - r
a
)
-
T(n) ,
-117-
which can be solved recursively to give the numerical values given in Table 3e or explicitly in the same way as was done for the last example: we write the recursion for
c
as
and o b s e r v e t h a t t h e first f a c t o r i s n ( T )
hg(T)
:=
^
c(N)q
1
n(24x)
=
N
(Euler S i d e n t i t y ) , so
(Z =
2 3
(^))
N > 0 N = 23(24) (again
is a m o d u l a r f o r m on a l l S L ^ ( Z )
hg(T/24)
one-dimensional). E a(n)q
or some
n
G M
S i m i l a r l y , if
g £ k-12'
(f>(x,z) is
g(x)
^
a n C
f (T) .
1
* ^ t
times
Since ^gCy^)
i e n
cb
f
h<$>(i,0) = f ( T ) =
=
r
(x,z) ,
is a cusp form we have
l^ ^ 8^ ^* T
with generator
2 3
i.e.
T
J
I t :
f = Ag
for
follows that
, is a free module of X,D
12,6
rank 1 over
with
is
then the recursion relation for the c(N) gives
h ~ ( - ~ ) = X] (T) M
<J) £
because
6
•
We could also have seen this
by a direct argument; more generally, it is not hard to modify the proof of Theorem 8.4 to show that
J
is a free module over x,m
every character
4.
£
M.
for
*
and to give a formula for its rank.
k= 2 The case of Jacob! forms of weight 2 is particularly
interesting,
both because this is the smallest weight occurring and because of the connection with Heegner points mentioned briefly in the Introduction. However, it is also the most difficult case, since we do not even have
-118the Eisenstein series bound for J„ 2,m
E
- to get things off the ground.
K., 1
The lower
obtained from the corollary to Theorem 9.2 is J
(14) m
25
37
43
49
53
61
64
67
73
79
81
83
85
89
91
93
97
98
100
bound
1
1
1
2
1
1
1
2
2
1
2
1
1
1
1
1
3
1
1
the bounds for the omitted
m
49, 6 4 , 81
being zero or negative.
The cases m = 2 5 ,
are accounted for by the Eisenstein series of weight 2 (cf.
the remarks following (11) of § 2 ) . those of
S
less when
m
2
(r*(m))
The other dimensions are equal to
(given in [ 5 ] , Table 5) when
m
is prime and one
is a product of two primes, the first in accordance with
the relation between
X k,m
and
M (F*(m)) zk-z 0 1
0
discussed in the Introduc-
tion (eq. (9)) and in §10. This relation will be proved in Chapter IV by trace formula methods for k > 2, but the case k = 2 presents extra difficulties, so the numbers (14) are not necessarily the true values of
J^ ' m
In any case, they are lower bounds, so we must be able to
find, for instance, a Jacobi cusp form of weight 2 and index 37. Let us look for a cusp form coefficient of q £ n
r
(j) £ J
2
3 ?
.
By Theorem 2.2, the
in depends only on 148n - r ,
i.e. we have
2
<J>(T,Z)
=
]T
c(148n - r ) q C
n,r 148n > r with some coefficients
2
n
r
2
c(N) (N > 0, N = 0
or 3 (mod 4) , ( — )
= 0 or 1 ) .
However, to find these coefficients by either of the algorithms mentioned after Theorem 9.3 would involve an impossible amount of computing.
For
instance, using the Taylor development operator in (8) would involve constructing an explicit map into the space
S
2
©
©
... © S
y 6
of
dimension 102 and then solving a 1 0 2 x 102 system of equations with coefficients which are very large and hard to compute. use another method.
The function
We therefore
-119-
h(T)
belongs to
M . (37)
=
£ c(N)q N>0
N
by Theorem 5.6, and an easy calculation shows that
3/2
the form CO
h(T)9(T)|U.
=
£
a(n)q
,
n
n=l (15)
r
is a cusp form of level 37 (and weight 2 ) .
Since
S (1^(37)) 2
is only
two-dimensional, and the Fourier coefficients of a basis have been tabulated (e.g. in [5]), we have a good chance to get the coefficients a ( n ) , but at first sight this seems insufficient to determine the c ( N ) , since the identity (15) gives a recursion which at each new step involves two new coefficients
c(4n) and c(4n-l).
What saves us is that we know
a priori that about one-half of the c(N) are 0
(namely those with
a quadratic non-residue of 3 7 ) , so that on the average for two coefficients works in practice.
c(4n), c(4n-l) will suffice.
N
this one recursion
Let us see how this
The first cases of our recursion, combined with the
vanishing of c(N) for
\~yf) ~
•> give
c(4) + 2c(3)
=
a(l)
2c(7) + 2c(4)
=
a(2)
c(12) + 2c(ll) + 2c(3) =
a(3)
c(16) + 2c(12) + 2c(7) =
a(4)
2c(16) + 2c(ll) + 2c(4) =
a(5)
0
=
a(6)
Looking at the tables in [ 5 ] , we see that the last equation determines the form
Sa(n)q
n
E S (1^(37))
up to a constant:
2
it must be a multiple
of the eigenform
f^z)
= q + q -2q" -q -2q + 3 q 3
7
9
U
-2q
12
-4q" +4q
16
+ ... .
-120-
At this stage we have five non-trivial equations involving seven unknowns
a(l), c(3), c(4), c(7), c(ll),
further, we find that at
n = 15
c(12),
n,
Proceeding
the number of equations overtakes the
number of unknowns, so that we suddenly get all For various higher
c(16).
c(N)
up to N = 6 0 .
we again have enough equations.
However, eventu-
ally we will have too few, since (15) provides an average of one equation for every two
c(N) with
N = 0
or
3 (mod 4 ) , while the proportion of
c(N) known a priori to vanish is only extra information.
Namely, for
m
18
/ 3 7 < V2 •
However, we have some
prime the space
M^^(m)
as defined
in Theorem 5.6 is by Kohnen s work isomorphic as a Hecke module to T
M ( r * ( m ) ) = {f G M ( F ( m ) ) | f|W 2
2
o
= f}
m
(cf. proof of Theorem 5 . 6 ) .
m = 37 both spaces are one-dimensional, so our form
h(x) G M ^ ( 3 7 ) 2
For is a
Hecke eigenform with the same eigenvalues as the second eigenform
f (z) 2
in
S (T (37)). 2
Q
=
q - 2q - 3q + 2q* - 2q + 6q - q 2
3
5
6
7
+
...
This means that we can write all coefficients
multiples of those with
-N
c(N)
as
a fundamental discriminant, thus reducing
the number of unknown coefficients by a further factor of
£(2)
1
~ 0.61.
Using this, we find that we now have a surfeit of equations to determine the coefficients
c (N)
(so many, in fact, that we can simultaneously
solve, for the coefficients of
q , p
p
prime,
dispense with the Antwerp tables entirely!).
in
f
1
and
f
2
and hence
We have tabulated the
resulting values (in Table 4) up to N = 250, since the example is of interest in connection with recent work of B.Gross and the second author on "Heegner points" and will be cited in a forthcoming paper of theirs.
-121-
§10.
Discussion of the Formula for For
IN
k,m E
dim J, k,m
set
L|)
k v-
(
dimM
j(k,m)
+2
= < f 2 ~] \
fll—l y S .
(
d i m M
k
^
) 1
dim J,
> j (k,m)
k,m k
V-l-
+ 2
V=l V
In §9 we showed that
for
'
(keven)
(
d
d
•
)
'
for all
k
and
dim J,
= i (k,m)
k,m k > m;
this will be improved to
k ^ 3 in Chapter IV by means of a trace formula). j(k,m)
°
'
J
sufficiently large (namely for
section is to write
k
The purpose of this
in terms of standard arithmetical functions
and to relate it to the dimensions of known spaces of modular forms. More precisely, w e will prove the following result.
THEOREM 10.1.
For k > 2 and
(1)
n 6 I
j(k,m) = £ j (k,m) Q
equals
new forms of weight
the dimension 2k-2 on
Fricke-Atkin-Lehner
Similarly, replaced j^
u s p
by
(k,m)
ponding
if
j
m (k,m)
-—^ j + 1, equals
s \
^
: f (T)
*—>
is defined
and
j^
s
-k+1 m
(-l)
-2k+2 (
^
under
k
°^ the
like
m
'
1 \
r
f(
T
i
\ mx / but with
j(k,m)
by the obvious
u s p
dim 2 ^ - 2 ^ 0 ( " ~
space of cusp
O^V^m
M ^ ^ ^ o ^ ^
with eigenvalue
Q
analogue
^he dimension
"| of ( 1 ) , then
of the
corres-
forms.
Note that the difference with
/
2
of the space
T (m)
•
k
\£ |d
by
involution
W c u s p
j (k,m) inductively
A V >f >
( E
dim Then
define
(resp. 0 < V < m
if
j(k,m) - j k
C U S p
(k,m)
is odd) and
V
2
is the number of == 0 (mod 4 m ) ,
V
which
122 b+2 y
equals
, (resp.
l.
I), where
i_
-J
b
is the largest integer whose
J
m,
2.3 and 2 . 4 ) .
In view of Theorem 4.4, Theorem 10.1 has a plausible
n
and this is just the dimension of
Eis J, k,m
square divides
J
(Theorems
interpretation as saying that the decomposition
k,m
,,
„21
,
Jo
Id
Jm
a
k,m
I &l
1
given there is direct and that there is an isomorphism between ° m
and
^2k2^o^ ^
preserving cusp forms.
jnew k,m
In Chapter IV we will prove
that this is indeed the case by proving the analogue of Theorem 10.1 with dimensions replaced by traces of Hecke operators.
Here we restrict
ourselves to the equality of dimensions.
Proof.
We carry out the proof only for
entirely analogous case of odd
k
k
even, leaving the
to the reader.
Our first objective is to write the formula defining terms of familiar arithmetic functions.
in
It is convenient to replace
the different rounding functions in the definitions of by their average.
j(k,m)
j
and of
j
c u s
We therefore use the notation
Í
a - h
(
a - h
if if
x G 2Z 4 a , 0 < a < 1 , x G 2Z 4- a , О < a < 1 ,
О
if
x G Ж
;
then
(2)
j
(
k
>
n
>
=
£
{
d
l
m
M
k
+
2
v
- ^ - %
+
( ( ^ ) ) }
as above) and
j
c u s p
L
(k,m)
sign of the last term reversed.
dim ^
^
+
V=0
(b
P
=
J
is given by a similar formula with the We substitute into this the formula
"
1 / 3
(
k
1
> < > ~ ^ 3
X^kl)
-123-
where X ,
and X,
denote the non-trivial Diriehlet characters modulo 3
k
3
and modulo 4 , and calculate: V* / k + 2 v + 5
vf_
x
(2k-3)(m+1) 24
\
v=0
V (3)
-V
3
£
X (k+2V-1) 3
=
3
{ -Vb
^ ^
-%
\ (k+2v-l)
=
V=0
(mod 3 )
,
if m
< - %
if m
I
((x))
k = 2
U
is even, k = 0 (mod 4 )
,
is even, k = 2 (mod 4 )
,
,,
.
n
0 Also, since
,
otherwise
X
m (4)
(mod 3 )
if m ^
0
B
if m = k f 2
if m
is odd
is periodic with period 1, the function
(("^)) I
s
periodic with period 2m, so we can write
V
V(mod 2m)
U
h i Finally, the last term in ( 2 ) whether or not
4
equals
(5)
^ + ~
Hence ( 2 )
divides m.
j(k,m)
-
h i or
— + —
depending
can be written
£ j.(k,m) 1=1
with
j^k,*)
= ^
j (k,m)
=
RHS of ( 3 )
j (k,m)
=
h
3
5
m
,
J
J 2
(k,m)
,
=
(k,m) =
>
V > > k
m
=
^ 3
RHS of ( 4 )
«(f»
=
,
-VsX^m),
V (mod 2m) j (k,m) 7
=
hb
j (k,m)
h
if
4fm
h
if
4|m
8
-124-
and
j
C U S p
and j
(k,m)
reversed.
i s given by the same formula but with the signs of
We have now p r a c t i c a l l y achieved our goal of writing
j (k,m) in terms of familiar a r i t h m e t i c a l functions, for a l l except j
5
j
are extremely simple functions of m and k
the
(periodic
functions of m and k or products of a polynomial in k and a m u l t i p l i cative function of m).
We s h a l l now see t h a t the function j , 5
which
depends only on m, can also be expressed in terms of a well known — though l e s s elementary — arithmetic function.
LEMMA. Define b*(d) for d 4 - 3 , - 4
as the class number of
positive definite binary quadratic forms of discriminant d (so, h (d) = f
if d> 0 or d 4 0,1 (mod 4 ) ) ,
h (-4) = 7 .
h (-3) = % , f
f
2
Then for any
natural number N (6)
J «t» v(mod N)
= - I ' " d|N h
(
•
d )
In particular j (k,m) 5
Proof.
=
-% ^ h (-d) d|4m
.
f
Suppose N = p i s a prime.
If p = 2 or p = 1 (mod 4 ) , then
both sides of (6) are zero, the l e f t because
((
)) i s an odd function
and - 1 i s a quadratic residue of p , the r i g h t side because p has no d i v i s o r s congruent to 0 or 3 (mod 4 ) .
If
p = 3 (mod 4 ) ,
left-hand side of (6) equals
E «f» E n(modp) V(modp) V = n(mod p) 1
E « P n (mod p)
=
2
=
E n (mod p)
<
i +
then the
-125-
P-
1
P
V
= | r
and this equals
-h'(-p)
of composite N of n (mod N)
E
n=l
2 '
P'
v
,
(f )n r
by Dirichlet s class number formula. T
The case
can be handled similarly by expressing the square-roots £ X(n),
as
where the sum is over all quadratic Dirichlet
X X (mod N)
characters
(this formula must be modified if
applying Dirichlet's formula to each X
(n,N) > 1)
and
separately.
Equation (5) and the Lemma give a very explicit formula for j(k,m); to prove Theorem 10.1 we must relate it to the formula for the dimension of
and write M^ . e w
d
Q
^2k-2
, w
Q
*
Set
d(k,m)
=
dim 1^(1^ (m))
w(k,m)
=
tr(W , M ^ r ^ m ) ) ) k
for the corresponding numbers with
replaced by
We have to show that
j (k,m) Q
=
%(d (2k-2,m) + w (2k-2,m)) o
Q
since this is the dimension of the (+1)-eigenspace of W
m
.
Equivalently,
if we define j (k,m) T
=
hH
( E
d|mV|d
l ) d (2k-2, m/d) ;
(7) j"(k,m)
=
hH
( E
d|*V|d
l ) w (2k-2, m/d) ;
(which equal one-half of the dimension and trace of W lying between that
j
M^^I^On))
and
is the sum of j ' and
M 2 k - 2 ^ o ^ m ^ ^'
j".
m
on a space
then we must show
Rather than work out everything
-126-
in terms of new forms, it is convenient to write (7) directly in terms of
d
and
w
instead of
d
and w .
Q
As is well known, d
Q
and
d
are
Q
related by
d(k,m)
=
E
( E
m'|m indeed, this is clear because functions dividing
f ( T ) = h(£x) m
and
£
1
)
d
o
( k
>
m T )
JLlp-
T (m)) has a basis consisting of the Q
with
h
a new (eigen)form of some level
a divisor of -^7- .
Similarly, w
and
w
m
T
are
Q
related by
w(k,m)
=
w (k,m )
,
f
Q
m |m m/m* = • because for
h
and
f
as above we have k/2
h|W
and therefore
h
t
«
eh
-
f|w
=
gives a contribution
nY-^-x)
e^-V)
e
to
tr(W ) ffi
£ = -^j—
if
k/2 and
0
otherwise (since then the sum and difference of h(-i~- x ) m £
("~rV m £ ' v
v
eigenvalues).
7
are non-zero eigenfunctions of W m
h(£x)
£
and
with opposite
Substituting these two formulas into (7) and carrying
out an easy exercise in multiplicative functions, we deduce
(8)
j'(k,m)
=
h
(9)
j"(k,m)
=
h
E A(—-)d(2k-2,m') m* | m
E
w(2k-2,m ) f
m'
where
X
is the Liouville function
Im
(X(IIp^ ) = (-1) 1
formulas we have to substitute the known formulas for and then compare
j'+j"
,
with our formula for
3.
).
Into these
d(k,m)
and
w(k,m)
-127-
is well known (see e.g. [8]);
The formula for d(k,m)
for k > 2
we can write it as (10)
d(k,m)
4 £ c (k) f (m) i=l
=
where
(k)
=
c (k)
=
Cl
3
c (k) = V
,
~
-y
X (k-1) ,
3
3
S
,
2
2
(k)
-y
=
X^k-1)
4
and the f^ are multiplicative functions:
f^m)
m]T
=
(1+J) P
,
pirn
£,W
-
TT
(p
pL(v-i)/2J)
L v / 2 j +
T
p !lm V>0 V
. f (m)
=
f (m)
/ 1
TT ( 1 + X ( )) p|m °
if
9 fu
P
-
I 0
(1 ( 1 + X (p)) x {
IT
if
9 | m
if
4fm
4
p|m
I0
if
4 | m
These formulas can be written more uniformly as
f,
= T T f r W p
f
= TT(x (P ) p
3
3
1
)
.
f
= TTJ^ p
2
4-x (P ~ )),
V
V
3
f (m) ±
-
TT P
( S
±
v
P
Thus each
( P
V
)
+
^
= TT(x^( ) + p
1
where in each case the product is over primes the exponent of p in m.
+p
f
g^" )) 1
p dividing
m
has the form
Y,
-
8i
m*[m m/m squarefree f
( m )
)
,
x^" )) , 1
and V is
-128-
for some much simpler multiplicative function
g^m)
=
m
> (m)
=
X (m)
3
(b
,
:
g (m)
=
T T P
g (m)
=
X (m)
2
,
3
4
as before the largest integer such that
= b
,
I>
b |m). 2
On the other hand,
the inversion of (8) is
(11)
d(2k-2,m)
=
2
^ j'(k,m) m Jm m/m' square free f
so this means that
j'(k,m)
is given by 4
(12)
j'Oc.m)
=
V
2
£ c.(2k-2) (m) i=l gi
m + y b + y 4
6
X (k) X (m) - y 3
3
8
X^m)
The first two terms and the last are the functions respectively.
and
j
of (5),
(Note how much simpler (12) is than the more familiar
formula (10); the fact that each of the four multiplicative functions f^(m) occurring in the dimension formula (10) has a natural decomposition parallel to (11) suggests that, in some respects at least, the theory of Jacob! forms is simpler of higher level.
than the usual theory of modular forms
We shall see the same phenomenon again in Chapter IV,
where almost every term of the trace formula is simpler for J
r
for either M _ ( T ( m ) ) 2 k
2
o
or
M^^O^On))
J. k,m
than
.)
We still have to look at the formula for
tr(W ) . m
Since this is
given in the literature only for cusp forms, we first consider the Eisenstein series contribution separately. the contribution from the cusps ( f ( ) m
2
The term
i=2 in (10) is
is the number of cusps of ^ ( m ) )
-129-
and would change sign if we replaced just as the term when we replace
j (k,m)
S^^(T (m)),
dim
Q
in (5) to which it corresponded changes sign
7
j by j
d(k,m) by
c u s
P.
The number
f ( m ) can be written 2
£ ^[))> where $ is the Euler function; this formula arises djm by attaching to a cusp — € (Q U {°°})/r (m) ( x,y coprime integers) o
the invariants
d = (y,m)
y m x • -j- (mod(d, ) ) •
and
To each cusp corresponds
an Eisenstein series, so the trace of W on M^?"S is just the number ' m 2k-2 J
of fixed points of Wm on the set of cusps. Since "W sends — m y to x ., , y i m ,, / I N m . ^ — = — r with x = - — , y = -j x, d = (y , m) = — , w e see that a mx y d d d m 2 y* y fixed point occurs only if d = -j- (so m = d ) and x — j - = x -j- (mod d) ; r
-y
1
u
J
T
y since on the other hand
xy = -x y f
and
f
can only happen for d = 1 or d = 2.
"(w , M ^ ( r ( l s
m
o
m )
x —
is prime to d, this
Hence
))
m = l or 4 ,
(l
if
\ 0
otherwise
= I
,
and putting this into (9) gives a contribution if < ( 1
to j"(k,m),
4
f
m =
if
2j
(k,m)
4 | m
as required (notice that
j
E
l
s
= 2(j + j ) ?
g
in the notation
of equation (5)). Finally, the formula for the trace of W
on S ( r (m)) f c
m
(given in
[38] and for k = 2 essentially going back to Fricke, who computed the on K/T (m))
number of fixed points of ^
tr(W , M _ ( r ( m ) ) ) m
2 k
2
0
-
can be written
Q
%tr(W , M ^ ( r ( m ) ) ) ffi
2
o
( -%h (-4m) - ^h (-m)
if
(-%h (-4m)
otherwise
f
f
f
m = 3 (mod 4)
-130-
y» 2k-3 12 + / X (k) 1
3
3
(the correction terms for m = 2 and
3
3j k
if
m = 3,
if
m = 3,
3\k
if
m = 2,
k = 0 (mod 4)
if
m = 2,
k = 2 (mod 4)
if
m= 1
come from extra fixed points of
W ; the large correction for m = 1, of course, comes from the fact that m
W, = 1 and hence
Inserting this into ( 9 ) we find
w(k,l) = d(k, 1) ) .
(-l) /4 k/2
j"(k,m)
=
^j"
E i S
(k,m) - h L
h'(-4d> +
( 0
d|4m
The first term on the right is j , g
equals
j
s
if
2 I m
if
2f m
\
by the discussion above, the second
by the Lemma, and the third and fourth terms are
respectively.
and j ^ ,
Finally, the last two terms together with the so-far-
unaccounted-for term as one checks easily.
Notice that for
/e X (k) X ( m )
in (12) together add up to j ,
l
3
3
3
This completes the proof of Theorem 10.1.
m
j (k,m)
prime the Theorem gives
J (k,m) + j (k,l) o
0
% ( d ( 2 k - 2 , m ) + w(2k-2, m)) + d ( 2 k - 2 , 1) 0
Q
%(d(2k-2, m ) + w(2k-2, m))
dimM _ (r ( )) 2 k
2
o
m
dim M _ ( r * ( m ) ) 2k
2
+ 1
131 where
r*(m) =
(m) U r ( m ) W o
is the normalizer of
m
^(m)
SL (]R) ;
in
2
this is the equation which we used in §4 (eq. (19)) . Finally, we say something about the case k = 2.
(13)
j*(2,m)
=
j(2,m) +
^
We set
1
d
Then Theorem 10.1 remains true for k = 2 with
j
replaced by j * .
Indeed, in the proof of the theorem only two things are different for k = 2: i)
There is one less "new" Eisenstein series of level 1 (compare the remarks near the end of §3, where the only new (X) Eisenstein series
E
the one with
which fails to exist for k=2 is
0
k
f
> x
X = trivial character,
f = 1) ; A
ii)
The formulas for
dim S ( F ( m ) )
and
1
ifi
must be increased by
1
tr(W , S (F (m)))
0
ID.
for k = 2
JK.
v
(compare [8],[38]).
The effect of i) is to decrease the righthand side of (1) by k = 2 (where as usual
b
2
is the largest integer with b | m
and
G ( b ) if Q
G (b) Q
the number of positive divisors of b ) , while that of ii) is to increase it by
h E H^r) m' I m
I Ц if m = square + h E i = ¡if.W + j m' |m ' 0 if m ф square
(cf. (8) and (9)), so the two effects together give the correction term in (13),
Since we would like the identity new dim J, k,m T
to be true for k=2 that
dim
given by
m
j(2,m)
=
_.new N(1) dim M. (r.(m)) K o /p
as well as for k > 2,
is in fact given by
f
k
N
it is reasonable to conjecture
j*(2,m)
(it certainly cannot be
in general since this is sometimes negative).
In any
-132-
case, it would be nice to prove at least the lower bound dim J
2
m
=
j*(2,m)
by elementary methods like those used in §9 to establish j(k,m).
dim J-^
m
=
Looking at the proof of that inequality, we see that this is
a question of finding
#{d < i/m d|m, d fm}
linearly independent linear
9
relations among the numbers
{c(n,r)} ^ 0
(Ic(n,r V
m
) q
e JJ
.
2
0 ^ n < r /4m 2
For m
square-free we can do this as follows: n
the function
(|W, ) (T,0)
for
d) G J
belongs to M ( S L ( Z ) )
m
2
and m | m f
2,m
1
and is therefore
2
identically zero; taking the constant term of this relation gives the identity
rem where
r
f
=
-r (mod 2 m ) ,
W . = W , . , this gives 2 m m/m T
t
_
f
1
T
c(n,r) with 2
r
f
t-1
4nm < r
2
= r (mod 2 ^ ) (cf. Theorem 5 . 2 ) .
Since
relations involving c(0,0) and coefficients
(t = number of prime factors of m) , and hence 2
- 1 relations among the c(n,r) with
4nm < r , as desired.
something similar works for non-squarefree
m
using the Hecke operators
T„ (&|m) defined at the end of §4 as well as the involutions % 1
The first few values of
j*(2,m)
Presumably
W , . m
are given in the following table
(in which omitted values are zero), which is to be compared with (14) of §9 and with Table 5 of [5]: m
25
37
43
49
50
53
57
58
61
64
65
67
73
74
75
77
j*(2,m)
1
1
1
2
1
1
1
1
1
1
1
2
2
1
1
1
m
79
81
82
83
85
86
88
89
91
92
93
97
98
99
100
j*(2,m)
1
2
1
1
2
1
1
1
2
1
2
3
2
1
2
-133-
§11.
Zeros of Jacobi Forms To complete the theory developed in this chapter, we discuss the
sets of zeros of Jacobi forms. surface
(3C x C ) / T ^ ,
These zeros are divisors on the algebraic
and w e could give a discussion in algebraic-
geometric terms, but we prefer to take a function-theoretic point of view more in accordance with the rest of the exposition. Let, then,
$ £
be a non-zero Jacobi form.
vanishes identically in where
z
x
for some
0
G JC ,
f (T) is a modular form vanishing at
holomorphic Jacobi form. contains no fibres of values of
z
many-valued function.
(j)(x,z) =
0
Q
and
IT: (5Cx e ) / r ^
-> "K/Y . l
( x ) ,..., ± V ( x ) m
<=>
1
l
is still a
We denote by
By Theorem 1.2, we know that
J
<$>
(J> = f
Hence we can assume that the zero-divisor of
(counting multiplicity) modulo Z x + Z ; ±V
Q
then we can write
(J)(x,z) vanishes; then V ( x )
at which
can number these
X
<{> ( x , z)
If
z =
v(x)
the
is an infinitely v(x)
near a given point
has 2m values X
Q
of "K we
so that
Ax + y ± V . ( x ) for some
E H,
jE{l,...,m}
(this numbering cannot be done globally because the restriction of tt to the divisor of functions
V
<j) is ramified) .
and the
V_. are.
We want to see what kind of
In particular, we will show that they
are well-behaved at infinity and that their second derivations are algebraic modular forms (i.e. satisfy algebraic equations over the ring of modular forms) of weight 3.
134
THEOREM 1 1 . 1 .
(1)
As
v.(x)
3
/02» seme a G Q, 3
E
T ->
by 3 + ^
T]
ax + 3 +
=
c
€, and 5 £ Q>q
e
has
an
expans ion
27Tin5T
n
n=lt
t/zen there are M-l further replaced
, each branch of V ( T )
0 0
If № is the denominator
of a,
of v(x) it?it?z tfoe same a £ut 3
branches
(L = 1,...,M-1).
In other words, the branches of v(x) near infinity tend with exponential rapidity to a subset of € of the form
U ((a +
for some numbers
M
g
Z ) X
e U,
a
+
3
+ M
S
1
Z )
1
g
€ M g ^ and
3
e S
€
with
][ M s
g
= 2m ;
in a fundamental parallelogram for the action of ET+ Z on € such a set could look as in Figure 1 (where m = 5 a
i *> Ot =-V ). = 1/
2
Proof.
s
the dots \ are the ^ { limiting ^ positions of the zeros of (j)
The method of proof will
be to take the expansion (1) as an "Ansatz", substitute it into the expansion of <j)(T,z) = 0 ,
and show
that we obtain the full number Figure 1 (i.e. 2m modulo translations by
ZT+Z)
of zeros in this way. Let 4>(T,z)
=
^
n
c(n,r)q £
r
(r,n)e
C=
{(r,n) G.
| c(n,r) ^ 0 } .
-135-
(2)
0 =
cj>(T, A T + 3 + o(l))
where we have used small as T ->
0 0
.
^
e
c(n,r)q
2 7 r l r 3
a r + n
( l + o(l))
to denote any function which is exponentially
is attained for at least two distinct pairs (n,r) e 6?,
n = y - ar
is a supporting line of G.
Hence
-a must be
the slope of one of the segments bounding the convex hull G By the properties of the coefficients
we know that G
c(n,r),
G.
of
satisfies
the two properties
i) ii)
e
c,{< ,n) | n z £•} r
,
G is invariant under the map
(cf. Figure 2).
0: (r,n)
(P ,^ g
s +
j)
s
G
S E 3 N and the numbering chosen so that
.. . < r < r .-,<... s s+1 segment, i.e.
(r+2m, n+r+m)
It follows that the boundary of G
infinite sequence of line segments for some
,
This equation can hold only if the minimum value Y of
ar + n ((n,r) £ G) i.e. only if
o (1)
=
.
g
consists of an with
P
g
0(P ) = P g
= (r ,n ) g
g
with
Let O L , denote the negative slope of this s
a
n
tr
s+l - s n
(the dots indicate _ points of (?)
Figure 2
g +
g
-136-
Then ( n
a
s^l
s-M " s s > (r + 2 m ) - (r +2m)
+ r
s+S
so the numbers
a
+
m
)
( n
+
r
s + 1
repeat periodically
g
+
m
a
g
s "
'
1
(modulo 1) with period
S.
If we set
then the right-hand side of (2) with
a =a
equals
g
(3) qV J2 ' C(N
\(r,n)e[P ,P s
This can vanish only if
r
s+1
•
R)E2TÌRB +
]
e
/
is a root of the polynomial
s l" s r
+
(4)
^
c ( n - a j , r + j)x g
g
J
g
j=0
of degree
r
- - r_ . S+X
6
and c
n
r
3 by
$, we can choose the
in (1) to make the o(l) terms in (3) vanish (Puiseux expansion
principle). of
Conversely, given any such
«3
Thus we get exactly
r
g +
^ -r
Z ) of the form (1) with
„ - r = 2m, s+S s '
g
solutions (modulo translation
a =a . Since £ (r - r ) = s (mod S) this accounts for all branches of V ( T ) (mod Z T + Z ) .
The translation invariance
g
3 —-> 3+ rr~
g + 1
(M
= denom(a ))
g
follows from
s s the fact that the only non-zero coefficients in (4) are those with M
n
g
-a j E Z
or
g
r
of degree
j = 0 (mod M ) , g
s+l " s — s
M in x
r
G
so that (4) is actually a polynomial
g
and its roots therefore invariant under
M
27ri/M multiplication by
e
s
.
This completes the proof.
It is not hard to see from the proof of Theorem 11.1 that we have the following converse: invariant under
(a,3)
any collection of 2m points (-a,-3) and
(a,3)
(a, 3+a)
(a,3)
e
Q
x
C/Z
occurs as the
2
-137-
llmiting position of the zeros of some Jacobi form of index converse is that any m-valued section of function
V ( T ) G (C/2LT + Z ) / ( ± 1 )
(5C C)/F^ X
m.
Another
(i.e. any m-valued
satisfying the transformation
equation
(5) below) whose behavior at infinity is as described in Theorem 11.1 is the zero-divisor of some holomorphic Jacobi form Indeed, if the branches of ± V ( T ) (mod l T + 1 ) V
(x),...,V
( T ) with
(J) of index m.
are numbered
V . ( T ) identically zero for j = m + 1 , . ..,m, f
i m j then Theorem 3.6 and a little thought imply that we can take m1
f
f(T)<J>
=
for some modular form
1 0 j l
(T,z)
T.^-
m _ m
l l > 1
^,z) +
^^(t,y> MH ^,z i
f.
We now consider the behavior of v ( x ) in the interior of 5C. r
the transformation law of cf) under (ft^Mx,
^^ = 0
of many-valued
1Btl
for any M = ^
it is clear that
d )
G
^i'
From
(j)(T,z) = 0
^
so we have the equality
functions
(5)
V
( t ? T ! )
( « + d)
=
_ 1
V(T)
or, written out explicitly,
(6)
for some
V..(Mx)
j ' €= {l,...,m}
=
fCcT + d ) "
and X,y
1
V. , (T) + XMT + ]i
(j = l,...,m)
2£. Since the infinitely-many-
valuedness of V arises from the addition of linear* functions A T + li (X,\i e Z ) , the second derivative many-valued.
V " ( T ) = -p
is only finitely
An easy calculation from (5) or (6) gives
v"(f|i|)
=
(cx + d ) v ( T )
-
±(cT + d )
3
or
3
cT + d
3
V",(T)
3
.
-138-
r
In other words, V " ( T ) transforms under form of weight
3
and
v" ( T )
2
like a (2m-valued) modular
like an m-valued modular form of weight
6.
We can eliminate the remaining many-valuedness by setting
(7)
=
* (t)
a ( v " ( t ) , ... ,v^(t) ) 2
(j = l,...,m)
2
,
th where
G_. denotes the j
elementary-symmetric polynomial; then I{K is
single-valued and transforms under However, ^
F
is not necessarily holomorphic in general:
tion point of V ( T ) Q the function order
>-1
like a modular form of weight 6 j .
1
and
V (T) ?
at a ramifica-
can have a pole of rational
V " ( T ) a pole of rational order
>-2
(locally,
V = c + c (T ~ l* ) ^ + ... with X a positive rational number, n —2+X v" ~ c X(A-l)(x - T ) ) . Hence the function (7) can have a pole of Q
1
q
Q
order
>-4j at such a point
11.1 that all
(T)
given by
z = ±v
Let
(|)(T,Z)
( T ) (mod Z T + Z )
be a Jacobi form of index
m
(j = 1,... , m ) ,
functions
( T ) (j = l,...,m) by (7) . Then each
form of weight
We of
are exponentially small, so the same holds for
We have proved:
THEOREM 11.2.
^
Near infinity, it follows from Theorem
Q
9
ii
TJJ^ ( T ) .
T .
index
can m
6j which vanishes
also have
state the
this
at
by
^ (x)
and define
is a meromorphic
with
modular
and has no poles of order
00
saying
that
the
zeros
of
any
^ 4j .
Jacob!
form loo
z
=
(X + a ) x + p + 3
±j
(t-T)
V p ( t ) dt
T ,,
where
2
p(t) (= V ( T ) ) p(T)
m
is a root of
- i|i (T)p(T) L
m
1
+
... ± * ( T ) m
=
0
zeros
(X,y e 7L)
form
139
i.e.
has weight 6 and is algebraic of degree
p
field of modular functions. p(T) = ^ ( T )
m
over the graded
In the particular case m = 1,
the function
is itself a meromorphic modular form of weight 6.
L
In
this case w e can make Theorem 11.2 more precise:
THEOREM 1 1 . 3 .
Le t- <J)(T,Z)
which is not divisible of -positive weight.
KT,Z)
=
J
K
be
X
«=*
ze ro-se t
z = + ( ат + 8 +
a e {0 , % } and
k
and
G
a modular Proof.
'
k = 12,
multiples of
(
T
3
( 3k 6
otherwise
form
.
dt)
/ 2
(modZT+2
/
is a modular
if
3
form of we ight
and
E
( 3k 30
i
k' = 6 (since G
G
H(t)
1
form of
we ight
(j) is a cusp form
=
/ k'+3
A,
ord^CG) > / o r d ^ H ) . 3
with
2
The particular case
E
(tT) {
e e wh r
(note that the zeros of There
cj> has the form
of
\
for some
a Jacobi form of inde x
in the ring of Jacobi forms by any modular Th e n the
О
E
<j) = <J>
12 i
and of p
2
was treated in [9]
are the same, by Theorem 3 . 6 ) :
(f> is a cusp form), so
respectively.
H
and
G
must be
The formula actually obtained
in [9] was
(A,u G The general case is similar.
(Theorem 3.5)
=
to get
.
We write
f(T)E (T,z) + S I
t )
g(T)E
6?I
(T,z)
(f G M _ , k
4
g e
M _ ) k
6
140
(Theorem 3 . 6 ) . ±V(T)
The only poles of
v"(T)
(mod TLT + Z ) meet, i.e. at points
at a 2division point
z G %(2£T+2£);
2
T
occur where the two branches Q
where the zero of
(j)(x,z) is
at such a point, the cubic
polynomial
vanishes.
Substituting this into the above formula and clearing
denominators, we see that this is equivalent to the vanishing of
where
a modular form of weight triple (since v
k' = 3 k - 6 .
2
Any pole of v" (T) 3
V
looks like (T T )
annihilates the poles of v " ( T )
2
2
or
/
2
(TT ) q
is simple or
locally), so
or turns them into double zeros; away
from ramification points, v(x) is locally holomorphic, so V " ( T ) zeros of even order.
Hence
V"(T)
2
has the form
3 +
H(i°°) ^ 0
3
T
/2k .
+
(j) is not a cusp form, then
has G
of
3
f (i°°) ^ ~g(i°°)
so
and we are done ( G must be a cusp form because V " ( T )
vanishes at °°). i
If
2
for some H(T)
weight
3
H
g^,
If
(j) (T , 0)
with
f €= M,
i n
is a cusp form, then we can write ,
g ^ M
9
§
as
(cf. comments at the end
of § 3 ) and repeat the argument to get (T,z) = where now
H
0, ramified
has degree 3k30.
«
G
in terms of
f
and
g
3
З Е ^
2
2E g g
3
=
О ,
This completes the proof, and we have
even given an explicit formula for for
H := f
H
in terms of
f
and
g;
and their derivatives of order
given implicitly in the last paragraph of [9].
a formula 1 2 is
-141TABLE 1 .
Coefficients
of Jacobi
forms
of index
1.
Coefficients of the
Eisenstein series of index 1 and weights 4, 6 and 8 and the cusp forms of index 1 and weights 10 and 12. The table gives c(N) for N < 1 0 0 , where
<J> =
][ c ( 4 n - r ) q ç 2
n
r
n.r
n
e (n) k
1 56 126 576 T 756 8 1512 11 2072 12 1+032 15 1+158 16 55M+ 19 7560 20 12096 23 2h 11592 13661+ 27 28 1670I+ 21+192 31 21+91+8 32 27216 35 31878 36 M+352 39 1+0 39816 1+3 1+1832 5591+1+ 1+1+ hi 72576 1+8 6658I+ . 51 67536 52 76101+ 55 100800 56 99792 59 101301+ 60 H6928 63 11+5728 61+ 133182 67 126501+ 68 160272 71 205632 72 177660 75 176U56 76 205128 79 2I+998U 80 2I+9I+80 83 23I+360 81+ 87 26510I+ 88 326592 91 281736 92 277200 95 350781+ 96 1+23360 382536 99 355320 I 100 390726 0 3 k
£ J, - . K , J
e (n) 6
-
e (n) 8
c (n) 1Q
c (n) 12
1 0 0 1 56 1 -88 1 366 10 -330 -2 -88 ll+0l6 -1+221+ -16 -132 -752I+ 33156 36 -306OO 1275 260712 99 -I+6552 1+62392 736 -272 -13091+1+ 1987392 -80I+O -2I+0 -I6929O 2998638 -2880 1056 -35508O 9090981+ 21+035 -253 -1+61+901+ 13080 -1800 -899712 -1I+136 2736 -10520I+0 -5I+120 -11+61+ -1732192 -12881+1+ -1+281+ -2099328 115!+56 1251+1+ -3I+21I+U0 389520 -6816 -3859812 38016 -19008 -559310I+ -256I+IO 27270 -6522I+50 -697950 -h55h -96518I+O -806520 -6861+ -10I+335I+I+ 963160 39880 -1I+00282I+ I892363 -66013 -161871+00 9381+00 -26928 -22I+29I+I+O -1227600 1+1+061+ -23836120 -2309120 12 51+1+ -30320I+OO -813I+50 . 108102 -33965I+I+8 -2813096 -9370I+ -I+51I+1888 231161+0 -22000 -I+7828880 55I+90I+O 80781+ -58659I+80 -3336015 . -2819I+3 -65079168 105I+8I+80 I88l60 -83I+87360 61I+1960 -36I+32 -866768IO -295I+2I+ -201I+2080 -10302362I+ -II65I+893 659651 -I1I+521616 -10887888 193392 -1I+3637120 5100360 -81+816 -1I+7I+92972 2I+801876 -3901+20 -171930088 311+06575 -635225 -187837320 17689760 68816 -230331+720 -1+7059760 -IO9088 -238I+95752 -37670I+O 9501+00 -272322072 -37381+71 -22I+55 -29533I+160 -I+8I+368 -61+883280 -35680550I+ -5321I+I+8 IO50768 -362360328 26020696 11+3176 -I+08875280 66711190 195910 -I+I+715686I+ 1851+61+32 -21I+502I+ -5323Ö8736 96031320 -3708OO -539696520 15586560 772992 -599851800 -IO73655 -239563575 -61+1+325330 2832950 H8753250
142 TABLE 2.
Coe fficie nts
of Jacobi
cusp
forms
Table 1; the coefficients c ( N ) , N ^ 20, of index 1 and weight < 50. к a b
j
c(3) 'c<4)'
of inde x
1.
Similar to
are given for all cusp forms
c(7)
c(8)
c(il)
cU2)
c(15î e e u )
240
"c(19)
cC20)
10 0 0 10
1
2
16
36
99
272
12 Ö 0 12
1
10
88
132
1275
736
8Ô40
2880
24035
13080
14
1 0 ' 10
Í
2
224
444
1581
4048
4320
96
65987
129000
16 0 1 10 16 i 0 12
1 1
2 10
520 152
1044 2268
8469 17635
14848 9344
93000 114600
214656 44160
90973 274595
628200 199RÖ0
18 2 0 10 18 0 1 12
1 1
2 10
464 592
924 5172
54339 28995
106832 99056
106800 690000
864 591840
2862973 6840445
5619000 129000
20 1 1 10 20 2 0 12
12 1 10
280 392
564 131109 4668 20955
261088 556576
3056040 3794760
5590464 2679360
459Í07 9382045
11836920 15165480
22 3 0 ÎO 22 0 2 10 22 1 1 12
1 1 1
2 704 1404 167859 332912 2 1024 2052 236979 478064 10 352 2772 110925 1318736
14157120 12837040 6376800
27649824 24815904 34285920
143276867 157308227 217428035
231589800 265458600 106371000
24 2 1 10 24 3 0 12 24 0 2 12
1 1 1
2 40 84 196149 10 632 7068 117195 10 1096 10212 310731
392128 35120280 1698496 2087380 2341312 4200312
69456384 146407680 135307008
1018050013 872209885 764576221
1897579560 1861629240 1980045624
26 4 0 10 26 1 2 10 26 2 1 12
1 1 1
2 2 10
944 1884 338979 674Î92 55970640 110594784 784 1572 6621 10096 67556880 135132384 И2 372 193245 1962416 20998800 356702880
3908309507 3755208707 1505914115
7596107400 7240139400 11201914200
28 28 28 28
3 1 0 3 4,0 1 2
10 10 12 12
1 1 1 1
2 200 396 203589 407968 82275720 2 1523 3060 736443 1479008 89642184 10 872 9468 271035 3416416 31586520 10 856 7812 49851 87968 68014488
163735104 9870850333 182245440 10153799965 569380800 113712925 675231168 1108819741
19414637640 19941636168 40715895960 39442512024
30 30 30 30
5 2 3 0
0 2 1 3
10 10 12 12
1 1 1 1
2 1184 2364 567699 1130672 139371360 2 544 1092 192621 383056 64281120 10 128 2028 217965 2030096 67612800 10 1600 15252 846483 7321840 142703040
32 32 32 32
4 1 13 5 0 2 2
10 10 12 12
1 1 1 1
2 440 876 2 1288 2580 ÎO 1112 11868 10 616 5412
34 34 34 34 34
6 3 0 4 1
0 2 4 1 3
10 10 10 12 12
1 1 1 1 1
2 1424 2844 854019 1702352 278183280 552964704 52763149187 104422077000 2 304 612 321021 640816 16833760 35043544 34957076867 69843414600 2 2032 4068 1489923 29B 7984 435518736 877017504 22964715395 47686445640 10 368 4428 185085 1521776 119641200 1311194400 20648940925 304243522200 10 1360 12852 464643 3682960 57003600 848266464 36112508797 298455551640
36 36 36 36 36
5 2 6 3 0
1 3 0 2 4
10 10 12 12 12
1 1 1 1 1
2 680 1356 45669 94048 166564200 332938944 61751452573 2 1048 2100 64923 134048 170286936 340303680 9173338205 10 1352 14268 751515 8530256 216727800 2806021440 32754039395 10 376 3012 299109 3218528 39989638 159789888 33733829219 10 2104 20292 1636251 14842528 542842248 4313363872 54358407779
1056
276483744 129327264 828419040 876098016
253
»800
18079318947 35607805800 19949307587 39639575400 3951942205 101051748600 3681447997 103904883960
153429 308608 130698360 260778624 30055413853 371883 748928 83810376 166120320 30087457885 482475 5710336 98525160 1409838720 ' 8058364835 153429 1941248 78136483 637312128 15316600739
59589577320 60506412648 184810546680 201255660024
%
122837118600 19037153928 535536093400 349981281624 217741975896
-143TABLE 2 (continued). к a b
j
с(3) с(4)
c(?î
t(8)
cfli)
cíi2)
CÜ5)
c(16)
c(l9)
c(20>
38 38 38 38 38
7 4 1 5 2
Ô 2 4 1 3
10 10 10 12 12
1 1 1 1 1
2 1664 3324 1197939 2389232 486230400 967685664 121381404227 240829833000 2 64 132391821 783376 60811200 120055776 38313743547 76866825000 2 1792 3588 1004403 2015984 82319616 168674734 78355185085 156371011800 10 608 6828 94605 437456 163260000 1666783960 49760122045 622187283000 10 1120 10452 140403 620080 165587040 1704483744 21437137597 87002596920
40 40 40 40 40 40
6 3 0 7 4 1
1 3 5 Ö 2 4
10 10 10 12 12 12
1 1 1 1 1 1
2 920 1836 2 808 1620 2 2536 5076 1Ü 1592 16668 10 136 612 10 1864 17892
119691 235712 176049240 352568064 101820918493 202936461480 184437 365632 183611496 367952640 31823735075 629Ш95992 2497419 50049921152766680 2315548416 217935113507 440506318680 1078155 12026176 400018440 489616B 960 86401086755 1227660648120 387189 3919808 32601912 619095552 42682769699 381358768824 1133451 9994048 154679928 795420672 72403550941 785661679944
42 42 42 42 42 42
8 5 2 6 3 0
0 2 4 1 3 5
10 10 10 12 12 12
1 1 1 1 1 1
2 1904 3804 2 176 348 2 1552 3108 10 848 9228 10 880 8052 10 2608 25332
1599459 405021 576483 53475 126237 2680035
44 44 44 44 44 44
7 4 1 8 5 2
1 3 5 0 2 4
10 10 10 12 12 12
1 1 1 1 1 1
2 1160 2316 2 568 1140 2 2296 4596 10 1832 19068 10 104 1788 10 1624 15492
342651 680672 145329480 292017984 143808003613 287031285960 376197 750112 137608056 276715200 75486697955 150419213112 1890939 3791072 558357160 1125301056 53351489053 104443594120 1462395 16048096 662221080 7818521280 184745042915 2428829922840 417669 4045088 125814312 1561104192 34021086179 224305113624 • 688251 5721568 113328792 1564571328107090988061 955095556584
46 46 46 46 46 46 46
9 6 3 0 7 4 1
0 2 4 6 1 3 5
10 10 10 10 12 12 12
1 1 1 1 1 1 t
2 2144 4284 2 416 828 2 1312 2628 2 3040 6084 10 1038 11628 10 640 5652 10 2368 22932
2058579 360621 206163 3758931 259155 335277 2056275
48 48 48 48 48 48 48
8 5 2 9 6 3 0
1 3 5 0 2 4 6
10 10 10 12 12 12 12
1 1 1 1 1 1 1
2 1400 2796 623211 2 328 660 510357 2 2056 4116 1342059 10 2072 21463 1904235 10 344 4188 390549 10 1384 13092 300651 10 3112 30372 3977335
1240832 60580920 123640704 1019392 46100616 94239360 2692352 109984440 225357696 20646016 1017159720 11711318400 3594368 225823512 2527996032 2025088 275007912 2904352128 37504334264012636023539440640
177939139933 107696246435 183408190813 346849203875 2924202659 78416351581 928392397475
50 50 50 50 50 50 50
10 7 4 1 8 5 2
0 2 4 6 1 3 5
10 10 10 10 12 12 12
1 1 1 1 1 1 î
2 2384 4764 2 656 1308 2 1072 2148 2 2800 5604 10 1328 14028 10 400 3252 10 2128 20532
5141072 1664022960 519856 337441630 208816 336535344 60741921473326320 6271504 105418800 5112560 82838160 13363216 206662320
714843158147 1423055929800 76404102013 151461036600 12458463875 23569742280 197144242307 400214160840 173562759305 1780966968600 104364S4S003 1078154321640 175456509565 1836900807960
2575299 258621 106557 3031491 522435 486717 1490115
3191312 777336720 1548294624 810736 154979760 308337696 1159184 154873104 307424736 1222864 184645200 1756962720 1866800 196871280 1830793824 24903376133264200011458231200
240675443067 478257505800 2287231462? 46360494600 95954407165 192522510840 89142765565 1023115479000 18599513603 323342812200 300994982915 2147558543400
4108592 1165326240 2322439584 430703904707 856767047400 722896 251797920 502149216 15196472893 29389368600 417534 239833424 578929056 57550070845 116257587480 75300322369410080 4753886304 75770378182? 1524922854120 3459184 169972300 1443475680 133336391485 1442082958200 3777680 164680320 1365436704 65570052803 758647107720 18845296 695060160 5536070880 13067729725 548795708280
3317768544 673842336 673486176 2959306624 588087840 446652384 1062665760
355629752040 215202993432 366362982120 4340067229560 159035091976 567343732024 7511424019320
144 Coe fficie nts
TABLE 3.
of Jacobi
cusp
forms
of inde x
> 1 . This table gives
the first few coefficients c(N) of the forms with (k,m)= (8,2), (10,2), (6,3), (11,2) and (12,6) discussed in § 9 , where c(n,r) = c ( 4 n m r ) in the first three cases and c(n,r) = (~4/r)c(8n r ) , c(n,r) = (12/r)c(24n r ) for the last two forms. The eigenvalues of T , T , T and T are also given. The number N is the sum of numbers above and to the left of the table. 2
2
2
2
3
4
5
7
?
8
12
15
16
20
23
24
28
0 0
64 Ô 5376
84 3724 1944
HB4 -33B8 -2268
252 792 27676
512 -4608 10752
ÏÏS
28 -15390
36 -504 -7496
4428 -4668
-384 4096 -16128
-20748 -107464 -68616 117690
0 0 0 0
20412 12012 -78460 215460
-32768 70848 0 -238336
-31836 -41492 215460 -236796
-10968 140996 -109836 -38892
79704 -20748 175640 -462672
21504 -124416 64512 433664
-7520 -35028 109752 -143920
24576 -1792 0 -50688
28 ^2176 -32640 372096 -926976 -933504
31 3408 18216 185400 -159680 3000552
32 288 -122624 -11712 2207232 319040
i
~
1
2 4
-1107 -168
972 4332
1 B ) 2
-11635 60984 63504 -315783
10108 -66096 -12012 -17556
0 I
a)
Jg
4 î Î 2 2277 I -96696 6 -1416475 * 2912712 0 1
-143356 8990793
T
15 ÏÏÔ -22032 54544 -793272 1431144
-2172456 2603304
11322168 7921560 11434050 -12972240
-5081300 14835960
J*
Q
.
2
T
8 ^6 -300 864 120 1092 -7488 0
î
-120 -689 -600 -6B81 3000 17340
c)
J*
11 15 144 -1059 1680 192 1200 -9981
.
3
= -272,
2
T
T
T
7
15
23
31
-21 46683 -3603805 -13691664 -20752011
189 -98287 -1391733 4004343 44496424
-910 -264915 478170 -8355270 116926740
12284811
283682483
-27484758
67323690
d)
. 0
0 0 0
J
1
1
>
2
T
-
=
il 1 -23 354706 -123763 -682088 -11372603 7731427 -119019296 2
0
62601078 -577640492
T
= 3
"
5
3
0
20 900 46852 242134 824076 -160124
T
:
5
20 24
0 1248 -4704 4320 4134 -9300
T
3037848 13100760
23 24 465 -840 -705 -4584 18816 -720
I
27 8Ï
0 -2430 -1944 0 3838 6561
32 216 -480 -1140 2520 3120 -7752 2040
35 Ï26 -120 1521 1128 -7905 2136 3345 :
36
F 972 1620 0 3888 -4860 0
_55
63
Jl
-13321 -1113588 -7889805 39441969 116695530
33345 -1066527 -12745348 30902949 -60695838
-10395 1042055 -1009470 -22027005 -129044790
-86870 536025 6926850 4899895 45585540
-82198368
417708354
111666555
-52005240
-214302315 8
>
T
5
-371932239
= -5556930,
-396525635
-820862617
77872135
599025570
-18765677
= П
= 3225992.
47
!5
T
?
2205 -378 96390 1163064 -562555 13742379 -30343950 5444712 78003135 -226108554
-164778900
= 0 .
T
5992704 -6656112 -17725440 -2992000 -15189280 646272
y
248467505
T
32
T =-4480.
284313649 -1384290765 .
-5316040 -3135888
24 60 -1248 -2160 -3264 -1176 -10200 32640
U9 3795 817190 -5451115 -22347260
Il
3Î
= -433432.
23 24 ^îiii 732 11000 -40392 -525384 -71588 -523728 -601800 30584 -9881964
I
= -1314,
5
y
=-1025850,
39
2
T
-1265 -48530 13479425 46163645
221269499
.T
,
= 3990,
230 -407560 5678535 17214120
3
0
0
2
:
6272640 -9760500 -749632 -10001268
= -162,
1 i 122703 2287467 • 21064883 -91000161
5
16 ÏÏ6 63360 -14400 -948224 -36432
15 24~ 510 792 8358 840 -8772 -5040 3
T
=-4284,
12 ^20 208 1680 -1272 6240 2752 -19776
= -36,
2
3
:
= -1836,
3
8 12 ïïi ^120 -19940 -11880 195210 30360 359892 514080 1778744 -3272400
3
I 3 \ 3 >
=-128,
2
7 8 3432 42408 1175176 -514080
b)
) I h
T
r
"Ô
T
143 -3519 1464341 16579596 134763417
=-9^9AÇnqn
T
79__
T =-44496424. ?
167 -16445 -4376693 -48805655 9991982
191 64285 -135355 -11515065 -115146395
215 -64515 6303955 61570080 -208431980
1040502519
303756860
-310026775
2191229389
470056290 -2021367210 -2588660810
= - 1 *Ì9 0 Q 7 7 7 A Q
239__ -120175 -1232710 21234520 72814665 615008615
-145TABLE 4. where
Coefficients
of the form
Z c(148n - r ) q £ 2
c
3 4 7 11 12 16 27 28 36 40 44 47 48 63 64 67 71 75 83 84 95 99 100 104 107 108 111 112 115 120 123
2 > 3 7
1 1 -1 1 -1 -2 -3 3 -2 2 -1 -1 0 2 2 6 1 -1 -1 -1 0 -4 -3 0 0 3 1 -4 -6 -2 3
n
r
(n)
G
in J
2
37.
Table of c(N) for N < 250,
3-7.
n
127 132 136 139 144 147 148 151 152 155 159 160 164 175 176 184 188 192 195 196 211 212 215 219 223 231 232 243 247 248
^ ( n )
1 3 4 О 4 -2 -3 -2 -2
2 1 -4 -1 1 0 0 3 2 2 0 3 3 О 1 -3 -1 6 6 -4 0
-146-
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(1980)
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