Invariant theoretical approaches to the ring of Siegel modular
forms and the related topics(a survey)
Manabu Oura1 (Kochi University, Japan)
This is a fairly precise reproduction of my talk at the conference. My
intention is to present some of our results without going into details, gathering the related ones. We are concerned with the theory of modular forms,
coding theory, invariant theory and theta series.
1. Preliminaries. The notations are standard. Let Sg denote theSiegel upper-half space and Γg = Sp2g (Z) the Siegel modular group. In thefollowing, we shall express a typical element M of Sp2g (R) in the form
Ã
!
a b
M=
c d
by four g × g matrices a, b, c, d. If n is a positive integer, a subgroupΓg (n) of Γg defined by the condition M ≡ 1 (mod n) is called the principal
congruence group of level n. For an even positive integer n, we shall define
a subgroup Γg (n, 2n) of Γg (n) by the condition (b)0 ≡ (d)0 ≡ 0 (mod 2l),
in which ( · )0 denotes a vector of its diagonal coefficients.
We consider a discrete subgroup Γ of Sp2g (R) which is “commensurable”
with Γg . A holomorphic function ψ on Sg satisfying
ψ((aτ + b)(cτ + d)−1 ) = det(cτ + d)k ψ(τ )for every element M in Γ is called a modular form of weight k for Γ. In thisdefinition, we have to assume that ψ is holomorphic at cusps in the caseg = 1. The set of such functions forms a vector space A(Γ)k over C andthe graded ring of modular forms of integral weights for Γ is denoted by
M
A(Γ) =
A(Γ)k , A(Γ)0 = C.
0≤k<∞

For a graded ring S = S0 ⊕ S1 ⊕ · · · , we put S (d) = S0 ⊕ Sd ⊕ S2d ⊕ · · ·.

We shall write vectors in Q2g by m and use m0 , m00 to denote the firstand second g coefficients in m. If τ is a point of Sg , the function
¾
½
X
√
m0
m0
m0 m001t(p +)τ (p +
) + t(p +)θm (τ ) =exp 2π −1
2
2
2
2 2
g
p∈Z

is called the theta-constants of characteristic m.
1 This research was partially supported by the Ministry of Education, Science, Sports
and Culture, Grant-in-Aid for Young Scientists (B).

1

2. History on A(Γg ), g = 2, 3. First we collect some papers.
[’39 Siegel] Siegel, C.L., Einführung in die Theorie der Modulfunktionen
n-ten Grades, Math. Ann. 116, 617-657 (1939).
[’56 Satake] Satake, I., On the compactification of the Siegel space, J.
Indian Math. Soc., n. Ser. 20, 259-281 (1956).
[’58 Baily] Baily, W.L., Satake’s compactification of Vn , Am. J. Math.
80, 348-364 (1958).
[’57-’58 Cartan] Séminaire de Cartan, Fonctions automorphes, E.N.S. (19571958).
[’60 Igusa] Igusa, J., Arithmetic variety of moduli for genus two, Ann.
Math. (2) 72, 612-649 (1960).
[’62 Igusa] Igusa, J., On Siegel modular forms of genus two, Am. J. Math.
84, 175-200 (1962). The ring A(Γ2 )(2) is determined.
[’64-1 Igusa] Igusa, J., On the graded ring of theta-constants, Am. J.
Math. 86, 219-246 (1964). A “fundamental lemma” is given.
[’64-2 Igusa] Igusa, J., On Siged modular forms of genus two II, Am. J.
Math. 86, 392-412 (1964). The fundamental lemma in [’64-1 Igusa] isused to determine A(Γ2 ).
[’66 Igusa] Igusa, J., On the graded ring of theta-constants II, Am. J.
Math. 88, 221-236 (1966). The fundamental lemma in [’64-1 Igusa] isgeneralized.
[’67 Igusa] Igusa J., Modular forms and projective invariants, Am. J.
Math. 89, 817-855 (1967). The ρ-homomorphism is introduced. The
ring structures A(Γg ) for g = 1, 2 are reproved.
[’86 Tsuyumine] Tsuyumine, S., On Siegel modular forms of degree three,
Am. J. Math. 108, 755-862; Addendum 1001-1003 (1986). Usingthe ρ-homomorphism, he determined the dimension formula and thegenerators of A(Γ3 ). The relations among the generators are not given.

2

In the paper [’62 Igusa], he says, after describing the importance of A(Γg ),“..., we knew very little about this ring beyond the facts that it is finitely
generated and that an operator Φ introduced by Siegel is almost an epimorphism of the graded rings of degree n to degree n − 1.” At the time of his
writing [’62 Igusa], it is known
dim A(Γ2 )2 = 0, dim A(Γ2 )4 = dim A(Γ2 )6 = dim A(Γ2 )8 = 1,
which are proved by Maass [22] and partly by Witt [39]. Applying themoduli theory [’60 Igusa] of curves in genus two, he obtains that A(Γ2 )(2)
can be generated by Eisenstein series of weights four, six, ten and twelve.
These four Eisenstein series are algebraically independent over C and thedimension formula of A(Γ2 )(2) also follows. In [’64-1 Igusa], he showed thatthe graded ring A(Γg (4, 8)) is the normalization of the graded ring C[θm θn ]generated over C by the products of theta-constants with m, n ∈ Z2g i.e.,N

A(Γg (4, 8)) = (C[θm θn ]) ,in which N denotes the integral closure of the ring in its field of fractions.
This is called by him a fundamental lemma. In [’64-2 Igusa], it is shown thatthe ring C[θm θn ] is normal in g = 2, which coincides with A(Γ2 (4, 8)) by thefundamental lemma. Finally Going-down process, i.e., taking successively
the invariant subrings of finite groups, provides A(Γ2 ). As a consequence,
we know that the ring A(Γ2 ) can be generated by five elements and that
there is essentially a unique relation among the generators. The explicit
formula of this relation can be found in [’67 Igusa].
The fundamental lemma is generalized in [’66 Igusa] asNA(Γg (r2 , 2r2 )) = (C[θm θn ])rn
2g
with rm
for any even positive integer r. This generalization,
2 , 2 ∈ Z
again with Going-down process, enables us to investigate A(Γg (l)) for anyl.
In [’67 Igusa], he defined the ρ-homomorphism from a subring of A(Γg )to S(2, 2g + 2). We give a quick view on S(2, 2g + 2) later. This ρhomomorphism gives a bijection from A(Γ1 ) to S(2, 4), an injection fromA(Γ2 ) to S(2, 6). In this way, he obtains the structure theorems of A(Γg ), g =1, 2. In g = 3, he determines the kernel of ρ, which is a principal ideal, andthe vector spaces of low weights. The latter leads to the affirmative answer of a problem of Witt [39], which asks whether two theta series of even+
unimodular lattices E82 and D16
in genus 3 coincide, i.e., whether we haveϑE82 = ϑD+ in genus 3. This problem is independently settled by Kneser
163

[21]. See also [5]. The problem of Witt in the case of rank 24 is consideredin [12], [5]. Their theorem says that theta series of twenty-four even unimodular lattices of rank 24 are linearly independent if and only if g ≥ 12.

We will come back to this problem in the next section.
We conclude this section by describing the graded ring S(2, r) of projective
invariants of a binary formf = u0 xr + u1 xr−1 y + u2 xr−2 y 2 + · · · + ur y r .

We refer to [36] in more details. Putting f (u, σx) = f (ũ, x) for σ ∈ SL2 (C),
we have an irreducible representation of SL2 (C) of degree r + 1. Under thisrepresentation, we get the invariant ring S(2, r). For example, we consider
f = u0 x2 + u1 xy + u2 y 2 . For
Ã
!
a b
σ=
c d
we have

σf = u0 (ax + by)2 + u1 (ax + by)(cx + dy) + u2 (cx + dy)2= (u0 a2 + u1 ac + u2 c2 )x2 + (2u0 ab + u1 (ad + bc) + 2u2 cd)xy+ (u0 b2 + u1 bd + u2 d2 )y 2
= u˜0 x2 + u˜1 xy + u˜2 y 2 ,i.e.,   2
 
u˜0
a
ac
c2
u0
  
 
=
u˜1  2ab ad + bc 2cd u1  .
u˜2
b2
bd
d2
u2

The ring S(2, 4) is a set of polynomials f (u0 , u1 , u2 ) such that f (u˜0 , u˜1 , u˜2 ) =f (u0 , u1 , u2 ) for any
Ã
!
a b
c

d

in SL2 (C). Investigation (finite generation, ring structures, etc.) of S(n, r)
is one of the main topics in the 19th century. For example,¡¢
S(2, 6) = C[A2 , B4 , C6 , D10 , E15 ]/ E 2 − P (A, B, C, D) − Q(A, B, C, D)E ,{z}|alg. indep.

in which P and Q are certain polynomials. This ring plays an importantrole in [’60 Igusa], [’62 Igusa], [’67 Igusa] The ring S(2, 8) is determined by
Shioda [37] as follows:
S(2, 8) = C[J2 , J3 , J4 , J5 , J6 , J7 , J8 , J9 , J10 ]/{5 relations}.
|{z}
alg. indep.

4

This ring is used to determine A(Γ3 ) in [’86 Tsuyumine]. In the above two
examples, the subscripts denote the degrees of homogeneous polynomials.
3. Coding theory, theta relations, and some graded rings. First
we recall the theory of binary codes. We shall denote by F2 the field of twoelements 0, 1. A subspace of Fn2 is called a linear code of length n. In therest of this note, we omit “linear”. A code C is called self-dual if it coincidesPwith its dual C ⊥ = {x ∈ Fn2 | x · y =xi yi = 0, ∀y ∈ C}. A weight wt(x)of x ∈ Fn2 is the number of the non-zero coordinates of x. A code C is called
doubly-even if the weight of every element in C is a multiple of 4. If a code
is self-dual and doubly-even, it is said to be Type II. It is known that TypeII codes exist if and only if the length n is a multiple of 8. Two codes issaid to be equivalent if one is obtained from another under some coordinatechange. Classifications of non-equivalent Type II codes are known up ton = 32 (Pless, Conway, Sloane, see [8] and its references).

n=8

unique e8n = 16 2 classes e28 , d+16n = 24 9 classes
n = 32 85 classesn ≥ 40 open
We give the first example of Type II code. That is a vector space e8 oflength 8 generated by four elements(1, 1, 1, 1, 0, 0, 0, 0),(0, 0, 1, 1, 1, 1, 0, 0),(0, 0, 0, 0, 1, 1, 1, 1),(1, 0, 1, 0, 1, 0, 1, 0).
The weight enumerator of C of length n is defined byX
WC (x, y) =xn−wt(v) y wt(v) .
v∈CFor example, the weight enumerator of e8 is
We8 (x, y) = x8 + 14x4 y 4 + y 8 .
The weight enumerator of a code C and that of its dual C ⊥ are related as
WC ⊥ (x, y) =1
WC (x + y, x − y),|C|5in which |C| denotes the cardinality of C. This is called the MacWilliams

identity.
Next we shall consider the graded ring over C generated by the weight
enumerators of Type II codes. This ring coincides with the invariant ringof the finite group generated by
Ã
! Ã
!
1 0
1 1 1
√
,
.
2 1 −10 iThis is from [17]. This group is a finite unitary reflection group of order
192, known as No.9 in [38]. The map
x 7−→ θ00 (2τ ), y 7−→ θ10 (2τ )induces the isomorphism between this invariant ring and A(Γ1 )(4) , see [6].
In particular, the weight enumerator WC (x, y) of any Type II code C oflength n gives a modular form of weight n/2 for Γ1 . If we consider an
index 2 subgroup of No.9, known as No.8 in [38], the invariant ring can
not be generated by the weight enumerators, however, the above map gives
the isomorphism between the invariant ring of this subgroup and the ringA(Γ1 ) = A(Γ1 )(2) .
We introduce the concept of genus in the weight numerators. What wehave stated above corresponds to the case genus one. The weight enumerator of a code C in genus g is defined by(g)

WC (xa : a ∈ Fg2 ) =X

x1 ,··· ,xg ∈CY

xna a (x1 ,··· ,xg ) ,

a∈Fg
2

where
na (x1 , · · · , xg ) = ]{i|a = (x1i , x2i , · · · , xgi )}.
Here, as we promised, we consider the problem of Witt in coding theory.
The general method to attack this problem is given by Nebe [23]. See
also [32], [10], [26]. There exist 1, 2, 9, 85 classes of Type II codes oflength 8, 16, 24, 32, respectively. For length 16, two weight enumerators
are linearly independent if and only if g ≥ 3. For length 24, nine weight

enumerators are linearly independent if and only if g ≥ 6. For length 32,
eighty-five weight enumerators are linearly independent if and only if g ≥ 10.
(g)
The Gleason type theorem for WC also holds, i.e., the ring over C gener(g)ated by the weight enumerators WC of all Type II codes coincides with theinvariant ring of the finite group Gg for any positive integer g. If we consider
a index 2 subgroup Hg of Gg , the invariant ring of Hg is not generated by

6

the weight enumerators any more, but Runge [32] showedN

A(Γg )(2) ∼= ( C[xa ] g /{theta relations}) .
H

We omit the precise definitions of these groups. For a Type II code C of(g)length n, WC (θa0 (2τ )) is a modular form of weight n/2 for Γg .
Runge([32]. cf. [15]) obtained the following theorem:
A(Γ3 ) ∼= C[xa ]H3 /(Wd+ − We2 ).
(3)

(3)

16

8

From this, the dimension formula of A(Γ3 ) reduces to the computation of
the dimension formula of the invariant ring of H3 , i.e.,


X ¡

X
¢
0
(dimC A(Γ3 )d ) t2d =dimC (C[xa ]H3 )d0 td × (1 − t16 ).

 0
d ≥0

d≥0

Note that an exponent of t in the left-hand side is 2d. In [33], the dimension
formula of the invariant ring of H3 is computed and the dimension formulaof A(Γ3 ) in [’86 Tsuyumine] is reproved.
If we view the works of Igusa and of Runge, we can find the “normalization
type” theorems in both. However, Igusa starts from θm (τ ), while Runge
from θm0 0 (2τ ).
Next we shall give some results in genus 4. The degree 24 part of theinvariant ring of H4 is spanned by the weight enumerators of Type II codesof length 24. There are 9 Type II codes of this length and the dimension of
the vector space over C spanned by the weight enumerators of those codes
is 7, see [25]. On the other hand, the dimension of A(Γ4 )12 is 6, see [28]. In[16], we analysed the map
(4)

(4)

(4)

(C[xa ]H4 )24 = hWC1 , WC2 , · · ·, WC9 iC −→ A(Γ4 )12
given by xa 7→ θa0 (2τ ). This linear map is surjective and the kernel is
explicitly described. The obtained relation gives a non-trivial cusp form ingenus 5. It is an open problem if this relation comes from Riemann’s theta
relations.
The same problem in the case of length 32 is treated in [26]. The dimension of the degree 32 part of the invariant ring C[xa ]H4 is 19, while
the dimension of the corresponding vector space of modular forms is 14.
See [25], [30], respectively. Finally we get 5 relations among theta series
of weight 16 in genus 4. In the course of these computations, we use the
Restriction technique ([29], [30]) to overcome a difficulty which does notappear in [16].
7The following picture is helpful for understanding what we have said.
code

(g) 
C7→WC y“Construction A”−−−−−−−−−−−→

invariant polynomial

xa 7→θa0 (2τ )−−−−−−−−→

latticeythetamodular form

For this diagram, see Conway-Sloane [9], Runge [34], Ebeling [11], BannaiOzeki [2], Bannai-Dougherty-Harada-Oura [3], Bannai-Harada-IbukiyamaMunemasa-Oura [1], Betsumiya-Choie [4], etc. For Gleason type theorem,
see Runge [34], Rains-Sloane [31] Nebe-Rains-Sloane [24], Chiera [7], etc.
As far as we concerned with even unimodular lattices, A(Γg )(4) might besuitable. This ring is the normalization of the ring generated over C byN
theta series of even unimodular lattices, i.e., A(Γg )(4) = C[ϑ∧ ] , see [14].
No normalization is necessary if g ≤ 4, see [35]. Explicit structures are
known as follows.

A(Γ1 )(4) = C[E4 , ∆] by Hecke [18],A(Γ2 )(4) = C[ϑE8 , ϑA24
, ϑD+ , ϑD+ , ϑD+ ] by Ozeki [27].
1
24

32

40

Next we consider the ring AZ (Γg ) over Z generated by the modular forms
whose Fourier coefficients are all integral. It is known that this ring is
finitely generated over Z, see [13]. Explicit structures are known as follows.
AZ (Γ1 ) = Z[E4 , E6 , ∆],AZ (Γ2 ) = Z[X4 , · · · , X48 ] by Igusa [20].
{z}|
15 generators

We conclude this note by describing a result with Prof.Choie concerning
to the weight enumerators in genus 2. Actually this investigation comesfrom a question raised by Prof.Skoruppa when I visited Bordeaux in December 1998. It is known that the ring over C by the weight enumerators
(2)
(2)
(2)
(2)
(2)
of Type II codes is generated by five elements We8 , Wg24 , Wd+ , Wd+ , Wd+ .
24
32
40
Therefore the weight enumerator of any Type II code C can be expressedby these elements. Now, we are interested in the coefficients of polynomial
expressions of the weight enumerators by these five elements. The result
could be stated as follows. Let R be a subring of C. Then the equality
(2)

(2)

(2)

(2)

24

32

40

R[WC ] = R[We(2), Wg(2), Wd+ , Wd+ , Wd+ ]8
24
holds if and only if
1 1 1 1 1 1
Z[ , , , , , ] ⊆ R.
2 3 5 7 11 43
In the proof of this theorem, the above identity on AZ (Γ2 ) is applied.
8

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