PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 134, Number 9, September 2006, Pages 2711–2718
S 0002-9939(06)08263-3
Article electronically published on February 8, 2006
THE JOINT WEIGHT ENUMERATORS
AND SIEGEL MODULAR FORMS
Y. CHOIE AND M. OURA
(Communicated by Wen-Ching Winnie Li)
Abstract. The weight enumerator of a binary doubly even self-dual code is an
isobaric polynomial in the two generators of the ring of invariants of a certain
group of order 192. The aim of this note is to study the ring of coefficients of
that polynomial, both for standard and joint weight enumerators.
1. Introduction
It is well known (see [7]) that any modular form whose Fourier coefficients lie in Z
can be written as a polynomial over Z in E4 (τ ), E6 (τ ) and ∆(τ ) = 261·33 (E43 − E62 ),
where Ek is the normalized Eisenstein series of weight k. Furthermore, Igusa [6]
and Nagaoka [8] determined the minimal set of generators over Z of the graded
ring of Siegel modular forms of degree 2 whose Fourier coefficients lie in Z, and of
the√graded ring√of symmetric Hilbert modular forms for the real quadratic fields
Q( 2) and Q( 5), respectively.
On the other hand, recently the connections between the coding theory and
theory of various modular forms have been well studied (see [1], [3]). For instance,
it is well known that the weight enumerator of every doubly even self-dual binary
(1)
code is a polynomial in two generators, the complete weight enumerator We8 of
(1)
the Hamming code and the complete weight enumerator Wg24 of the Golay code.
(1)
The graded ring C[WC ] of the weight enumerators of all doubly even self dual
binary codes is isomorphic to the graded ring C[E4 , ∆] of elliptic modular forms
[3]; explicitly,
(1.1)
(1)
C[WC ] = C[We(1)
, Wg(1)
]∼
= C[E4 , ∆].
8
24
Since the coefficients of any weight enumerators of codes are in Z, a natural
question is if any weight enumerator of all doubly even self-dual binary codes can
(1)
(1)
be written as a polynomial in We8 and Wg24 over a smaller ring than C. It turns
out that we can replace C in the equality in equation (1.1) by the smaller ring
Z[ 12 , 13 , 71 ]. The main result of this paper is that we extend this problem to the
genus 2 case as well.
We start with the definitions and the known facts which are needed in this note.
Let Sn be the Siegel upper-half space of degree n and denote by A(Γn )k the ring
Received by the editors November 1, 2004 and, in revised form, March 14, 2005.
2000 Mathematics Subject Classification. Primary 94B05; Secondary 11F46.
Key words and phrases. Code, weight enumerator, modular form.
c
2006
American Mathematical Society
2711
2712
Y. CHOIE AND M. OURA
of modular forms of weight k on Γn = Sp2n (Z) over C. If f is in A(Γn )k , then f (τ )
can be expanded into a Fourier series of the following form:
⎞
⎛
n
2s
√
sii
⎝
af (s) exp(2π −1trace(sτ )) =
f (τ ) =
qij ij ⎠
qii
,
af (s) ·
sii ≧0
s≧0
i<j
i=1
√
where qij = exp(2π −1τij ) and s runs over the set of half-integral positive (semidefinite) matrices of degree n. For any subring R of C we denote by AR (Γn )k the
R-module consisting
of those f ∈ A(Γn )k such that
af (s) is in R for every s and by
AR (Γn ) :=
k≥0 A(Γn )k ; then AR (Γn ) forms
k≥0 AR (Γn )k taken in A(Γn ) :=
a graded integral ring over R. The explicit structure of the ring AZ (Γn ) is known
only for n = 1, 2, and we shall use them later.
Let m = (m′ m′′ ), with m′ , m′′ ∈ Fn2 ; then the theta constants with characteristic
m are defined as
√
1
1
1
1
1
(p + m′ )τ t (p + m′ ) + (p + m′ )m′′ .
exp 2π −1
θm (τ ) =
2
2
2
2
2
n
p∈Z
Let C be a (linear) code of length k over F2 . The weight enumerator
(n)
WC (xa : a ∈ Fn2 ) of degree n is defined as
(n)
(n)
xana (v1 ,...,vk ) ,
WC = WC (xa : a ∈ Fn2 ) =
v1 ,...,vk ∈C a∈Fn
2
where na (v1 , . . . , vk ) denotes the number of i such that a = (v1i , . . . , vki ). We
(n)
note that WC is a homogeneous polynomial of degree k with non-negative in(n)
tegers as its coefficients. For any subring R of C we denote by R[WC ] the
graded ring generated by the weight enumerators of degree n of all doubly even
self-dual codes of any length over R. It is known that the Broué-Enguehard map
(n)
T h : xa → θa0 (2τ ), a ∈ Fn2 , gives the C–algebra homomorphism from C[WC ]
(4)
to A(Γn )
=
k≧0,k≡0 (mod 4) A(Γn )k . In particular, it gives the isomorphisms
(n) ∼
C[WC ] = A(Γn )(4) when n = 1, 2 (see [10]). In the next section we explain our
problem dealing with the case when n = 1. The main theme of this note is to
investigate this in the case when n = 2.
2. The case when n = 1
In this section, we discuss the case when n = 1 (and may omit n = 1 in the
notation of the weight enumerator for the sake of simplicity). Before proving the
assertion, we modify our setting. We started from the fact (see [3]), called the
Gleason Theorem, that C[WC ] is generated by We8 and Wg24 over C, where
We8 = x80 + 14x40 x41 + x81 ,
16 8
12 12
8 16
24
Wg24 = x24
0 + 759x0 x1 + 2576x0 x1 + 759x0 x1 + x1 .
The doubly even self-dual code of length 8 is unique (up to isomorphism), however,
we may take another doubly even self-dual code of length 24 instead of g24 . There
exist 7 indecomposable doubly even self dual codes of length 24 (see [11]):
d212 , d10 e27 , d38 , d46 , d24 , d64 , g24 .
JOINT WEIGHT ENUMERATORS AND SIEGEL MODULAR FORMS
2713
We call them C24,1 , . . . , C24,7 . The following table gives the values ai , bi , if we write
WC24,i = ai We38 + bi Wg24 , i = 1, 2, . . . , 7:
ai
bi
C24,1
5/7
2/7
C24,2
4/7
3/7
C24,3
3/7
4/7
C24,4
2/7
5/7
C24,5
11/7
−4/7
C24,6
1/7
6/7
C24,7
0
1
We state the following proposition.
Proposition 2.1. Let R be a ring such that Z ⊆ R ⊆ C. Then we have
1 1 1
R[WC ] = R[We8 , WC24,4 ] if and only if Z[ , , ] ⊆ R,
2 3 5
1 1 1
R[WC ] = R[We8 , WC24,7 ] if and only if Z[ , , ] ⊆ R,
2 3 7
and for i = 1, 2, 3, 5, 6,
1 1
R[WC ] = R[We8 , WC24,i ] if and only if Z[ , ] ⊆ R.
2 3
Before proceeding to the proof, we recall the modular forms for Γ1 over Z. If
we denote by Ek the Eisenstein series of even weight k normalized as Ek (τ ) =
∞
2k
n
2πiτ
1− B
, and if we put ∆ = 2−6 3−3 (E43 − E62 ), then it is
n=1 σk−1 (n)q , q = e
k
well known (see [7]) that AZ (Γ1 ) = Z[E4 , E6 , ∆]. Moreover, we have AZ (Γ1 )(4) =
Z[E4 , ∆].
Proof of Proposition 2.1. First we consider the case when C24,7 ∼
= g24 . Suppose
that we have the equality R[WC ] = R[We8 , Wg24 ]. We pick the doubly even selfdual code C32,50 of length 32, which is No. 50 in the list taken from Sloane’s
homepage (http://www.research.att.com/ njas/). Direct computation gives
1 4
41
WC32,50 =
W + We Wg .
42 e8 42 8 24
Therefore R must contain 21 , 13 , 71 .
Conversely, suppose that Z[ 21 , 13 , 17 ] ⊆ R. The inclusion R[WC ] ⊇ R[We8 , Wg24 ]
is trivial, and we show the converse. Let C be a doubly even self-dual code of length
k. Then T h(WC ) is in A(Γ1 ) k and T h(WC ) can be expressed in the form
2
T h(WC ) = WC (θ00 (2τ ), θ10 (2τ )) =
cab E4a ∆b , for some cab ∈ Z.
Since
E4 (τ ) = T h(We8 ),
we get
T h(WC ) =
=
=
∆(τ ) =
1
T h(We8 )3 − T h(Wg24 ) ,
25 3 · 7
cab E4a ∆b
cab T h(We8 )a
′
1
T h(We8 )3 − T h(Wg24 )
5
2 3·7
b
′
ca′ b′ T h(We8 )a T h(Wg24 )b ,
in which ca′ b′ ’s are elements of Z[ 21 , 31 , 17 ]. Therefore WC is contained in R[We8 ,Wg24 ].
This completes the proof of the case when C24,7 ∼
= g24 .
2714
Y. CHOIE AND M. OURA
For other cases in the proposition, a similar method can be applied, and so we
omit the detailed proof.
3. The case when n = 2
In this section, we shall discuss the case when n = 2 (and may omit n = 2 in the
notation of the weight enumerator). Our starting point is the following equality
given in [4]:
C[WC ] = C[We8 , Wg24 , Wd+ , Wd+ , Wd+ ],
24
32
40
where
Wg24 = (24) + 759(16, 8) + 2576(12, 12) + 212520(12, 4, 4, 4)
+ 340032(10, 6, 6, 2) + 22770(8, 8, 8) + 1275120(8, 8, 4, 4)
Wd+
k
+ 4080384(6, 6, 6, 6),
⎞ k2
⎛
1
⎝
= 2
(−1)α·β xα+γ xα ⎠ , k = 8, 24, 32, 40,
2
2
2
α∈F2
β,γ∈F2
with the usual inner product · of F22 . Here we write e8 instead of d+
8 and use the
convention (∗, ∗, . . .) to express the symmetric polynomials, such as (24) = x24
00 +
24
24
+ , W + are algebraically
x24
+
x
+
x
,
etc.
In
[4]
it
was
shown
that
W
,
W
,
W
e
g
8
24
01
10
11
d24
d40
independent over C and there exists a unique relation, which is explicitly given in
[12]:
Wd2+ = −113 · 32621 · 3−4 5−1 7−2 41−1 We88 −28 60289 · 3−4 5−1 7−2 11−1 41−1 We58 Wg24
32
+ 24 821477 · 3−4 5−1 7−1 11−1 41−1 We58 Wd+ + 2 · 751 · 3−2 7−1 41−1 We48 Wd+
24
9
2
−3 −1 −1
− 2 11 · 3
5
7
−1
41
11
−4 −1
7
11
6
−4
−2
+ 2 73 · 79 · 3
− 2 107 · 499 · 3
We38 Wd+
40
−2
11
−1
41
14
32
−4 −2
+ 2 163 · 3
7
−2
11
−1
41
We28 Wg224
We28 Wg24 Wd+
24
41−1 We28 Wd2+
24
− 28 389 · 3−2 7−1 11−1 41−1 We8 Wg24 Wd+
32
+ 24 5 · 197 · 3−2 11−1 41−1 We8 Wd+ Wd+
24
32
+ 212 3−1 5−1 7−1 41−1 Wg24 Wd+ + 29 3−1 5−1 41−1 Wd+ Wd+ .
40
24
40
So, we finally state our main result.
Theorem 3.1. Let R be a ring such that Z ⊆ R ⊆ C. Then we have
R[WC ] = R[We8 , Wg24 , Wd+ , Wd+ , Wd+ ]
24
32
40
1
1
if and only if Z[ 12 , 13 , 15 , 17 , 11
, 41
] ⊆ R.
The proof of this theorem is carried out by a method similar to that of Proposition 2.1. We recall that A(Γ2 ) is generated by homogeneous elements ψ4 , ψ6 , χ10 ,
χ12 , χ35 over C, each with the subscript as its weight. The normalization is made
JOINT WEIGHT ENUMERATORS AND SIEGEL MODULAR FORMS
2715
as follows (we follow the notation in [6]):
ψ4 (τ ) = 1 + · · · ,
ψ6 (τ ) = 1 + · · · ,
χ10 (τ ) = (q11 q22 + · · · )(πτ12 )2 + · · · ,
χ12 (τ ) = (q11 q22 + · · · ) + · · · ,
2 2
q22 (q11 − q22 ) + · · · )(πτ12 ) + · · · .
χ35 (τ ) = (q11
We put
X4 = ψ4 , X6 = ψ6 , X10 = −22 χ10 , X12 = 22 3χ12 , X35 = 22 iχ35 ,
and
Y12 = 2−6 3−3 (X43 − X62 ) + 24 32 X12 , X16 = 2−2 3−1 (X4 X12 − X6 X10 ),
2
2
− X4 X10
),
X18 = 2−2 3−1 (X6 X12 − X42 X10 ), X24 = 2−3 3−1 (X12
X28 = 2−1 3−1 (X4 X24 − X10 X18 ), X30 = 2−1 3−1 (X6 X24 − X4 X10 X16 ),
2
X16 ), X40 = 2−2 (X4 X36 − X10 X30 ),
X36 = 2−1 3−2 (X12 X24 − X10
2
X42 = 2−2 3−1 (X12 X30 − X4 X10 X28 ), X48 = 2−2 (X12 X36 − X24
).
Igusa [6] showed that the fifteen elements
X4 , X6 , X10 , X12 , Y12 , X16 , X18 , X24 , X28 , X30 , X35 , X36 , X40 , X42 , X48
form a minimal set of generators of AZ (Γ2 ) over Z. For our purpose, we deduce
the following lemma.
Lemma 3.2. The ring AZ (Γ2 )(4) can be generated over Z by the following thirty
elements:
X4 , X12 , Y12 , X16 , X24 , X28 , X36 , X40 , X48 ,
and
2
X62 , X6 X10 , X6 X18 , X6 X30 , X6 X42 , X6 X35
,
2
2
,
X10 , X10 X18 , X10 X30 , X10 X42 , X10 X35
2
2
,
X18 X30 , X18 X42 , X18 X35
X18
2
2
X30
,
X30 X42 , X30 X35
,
2
2
,
X42 X35
X42 ,
4
X35
.
Proof. This is derived from the usual argument on the graded ring. See Chapter
III in [5].
We note that the thirty elements in Lemma 3.2 do not form a minimal set of
generators of AZ (Γ2 )(4) , however, it is enough for our purpose. We put
1 1 1 1 1 1
Z = Z[ , , , , , ][T h(We8 ), T h(Wg24 ), T h(Wd+ ), T h(Wd+ ), T h(Wd+ )].
24
32
40
2 3 5 7 11 41
By the following two lemmas, we show that the thirty elements in Lemma 3.2 are
in Z.
2
are in Z, then the remaining
Lemma 3.3. If the elements X4 , X12 , X62 , X6 X10 , X10
twenty five elements in Lemma 3.2 are also in Z.
2716
Y. CHOIE AND M. OURA
Proof. This is derived from the definition of each element and the formula
2
2
3
3
X35
= (−22 X42 X16 + Y12
)X36 X10 + (−26 X4 X12
+ 23 Y12 X28 )X10
4
5
6
7
+ (2Y12 X18 + 210 X30 )X10
+ 3 · 61X42 X12 X10
− 2 · 73X4 X6 X10
+ 210 55 X10
,
which was given in [6]. For example, by the assumption that X4 , X62 , X12 are in Z,
we conclude that Y12 = 2−6 3−3 (X43 − X62 ) + 24 32 X12 is in Z. Since the assertion
can be checked directly, we omit the detailed proof.
2
are in Z.
Lemma 3.4. The elements X4 , X12 , X62 , X6 X10 , X10
Proof. It is known that the Broué-Enguehard map gives an isomorphism
C[We8 , h12 , F20 , Wg24 , Wd+ ] ∼
= A(Γ2 )(2) ,
40
where
h12 = (12) − 33(8, 4) + 330(4, 4, 4) + 792(6, 2, 2, 2),
F20 = (20) − 19(16, 4) − 336(14, 2, 2, 2) − 494(12, 8) + 716(12, 4, 4)
+ 1038(8, 8, 4) + 7632(10, 6, 2, 2) + 106848(6, 6, 6, 2) + 129012(8, 4, 4, 4).
The relations among the polynomials and Siegel modular forms can be given explicitly as follows (cf. [9]):
Wd+ = 112 3−2 7−1 We38 + 2 · 3−2 h212 − 23 7−1 Wg24 ,
24
Wd+ = 43 · 53 · 3−4 7−1 We48 + 24 5 · 23 · 3−5 11−1 We8 h212
32
− 26 43 · 3−2 7−1 11−1 We8 Wg24 + 26 3−5 h12 F20 ,
Wd+ = 3 · 19 · 7−1 We58 + 2 · 5 · 7 · 557 · 3−7 11−1 We28 h212
40
2
− 23 5 · 19 · 7−1 11−1 We28 Wg24 + 26 52 3−7 We8 h12 F20 + 22 5 · 41 · 3−7 F20
,
and
T h(We8 ) = ψ4 , T h(h12 ) = ψ6 , T h(F20 ) = ψ4 ψ6 + 212 34 χ10 ,
T h(Wg24 ) = 11 · 2−1 3−2 ψ43 + 7 · 2−1 3−2 ψ62 − 210 32 7 · 11χ12 .
So, we have
X4 = T h(We8 ),
X12 = T h(−2−10 3−1 7−1 We38 + 2−8 3−1 7−1 11−1 Wg24 + 2−10 3−1 11−1 Wd+ ),
24
2
2 −1 −1
3
2 2 −1
2 −1
X6 = T h −11 2 7 We8 + 2 3 7 Wg24 + 3 2 Wd+ ,
24
−16 −1 −1
X6 X10 = T h(−5 · 53 · 2
3
−13 −1
2
X10
+ 53 · 2
7
We48
−1
3
−9 −1 −1
+5·2
11 We8 Wd+ −
24
−25 −1 −1 −1 −1
= T h(−461 · 2
3
5
7
41
3
−16
3·2
We58
7
−1
11
We8 Wg24
Wd+ ),
32
+ 2−18 3−1 7−1 11−1 41−1 We28 Wg24
+ 13 · 2−21 3−1 11−1 41−1 We28 Wd+ − 3 · 2−25 41−1 We8 Wd+
24
32
+ 2−22 3−1 5−1 41−1 Wd+ ).
40
This shows Lemma 3.4.
JOINT WEIGHT ENUMERATORS AND SIEGEL MODULAR FORMS
2717
Proof of Theorem 3.1. Suppose that R[WC ] = R[We8 , Wg24 , Wd+ , Wd+ , Wd+ ].
24
32
40
Since the weight enumerator Wd+ is uniquely expressed with our fixed generators
48
as
Wd+ = 23 · 22229 · 2−2 3−2 5−1 7−2 41−1 We68 − 25 13 · 23 · 3−2 7−2 11−1 41−1 We38 Wg24
48
+ 2 · 23 · 113 · 3−2 7−1 11−1 41−1 We38 Wd+ − 32 5 · 23 · 2−2 41−1 We28 Wd+
24
32
+ 24 7 · 23 · 3−1 5−1 41−1 We8 Wd+ − 29 19 · 3−2 7−2 11−2 Wg224
40
+ 26 23 · 3−2 7−1 11−2 Wg24 Wd+ + 2 · 23 · 37 · 3−2 11−2 Wd2+ ,
24
24
1
1
Z[ 21 , 31 , 15 , 17 , 11
, 41
].
1 1 1 1 1
1
R contains Z[ 2 , 3 , 5 , 7 , 11 , 41
].
we see that R must contain
Conversely, suppose that
It is enough to show
that WC is in R[We8 , Wg24 , Wd+ , Wd+ , Wd+ ] for any doubly even self-dual code C.
24
32
40
Take any doubly even self-dual code C of length k. Then T h(WC ) is in AZ (Γ2 )k ,
with weight k2 and k ≡ 0 (mod 8). By Lemma 3.2, T h(WC ) is expressed as the
polynomial of the thirty elements X4 , . . . , X62 , . . . over Z, say
T h(WC ) =
ca···b··· X4a · · · X62b · · · ,
a,...,b,...
in which ca···b··· ’s are integers. By Lemmas 3.3 and 3.4, all thirty elements are in Z
and we have
c′
a′
b′
d′
e′
′ b′ c′ d′ e′ W
T h(WC ) = T h(
ca
e8 Wg24 Wd+ Wd+ Wd+ ),
24
32
40
a′ ,b′ ,c′ ,d′ ,e′ ∈Z
1
1 1 1 1 1
′ b′ c′ d′ e′ are in Z[ ,
in which the coefficients ca
2 3 , 5 , 7 , 11 , 41 ]. At any rate, WC is in
R[We8 , Wg24 , Wd+ , Wd+ , Wd+ ]. This completes the proof of Theorem 3.1.
24
32
40
Acknowledgement
This work was partially supported by ITRC, KOSEF R01 − 2003 − 00011596 − 0
and the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young
Scientists (B), No. 14740081, 2004.
This work was done during the second author’s stay in Pohang. He thanks
Professor Choie and the Pohang University of Science and Technology for their
great hospitality.
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Department of Mathematics, Pohang University of Science and Technology, Pohang,
790–784, Korea
E-mail address:
[email protected]
Department of Mathematics, Kochi University, Kochi, 780–8520, Japan
E-mail address:
[email protected]