Estimating the dimension of the space of Siegel modular forms of
genus 2 with level 2 and 3
by Hiroki Aoki (Ritsumeikan University) September 11, 2001
1 Introduction
Using Jacobi forms, we had a new proof of Igusa’s theorem: the determina- tion of the structure of the graded ring of Siegel modular forms of degree 2 with respect to the full modular group Sp(2,Z) (cf. [Ig1, Ig2, Ao]). The key point of the proof is the invariance of forms with respect to the exchange of the diagonal variables. This gives a condition to Jacobi forms appeared in the Fourier-Jacobi expansion. Calculating an upper bound of the dimension of the space of Fourier-Jacobi coefficients, we have an upper bound of the dimension of the space of Siegel modular forms. In fact, this upper bound coincides with the dimension of Siegel modular forms.
In today’s talk, we apply this method to modular forms with level: we choose the congruent subgroup Γ2,0(p)⊂ Sp(2,Z). Similar to the full mod- ular case, we have an upper bound of the dimension of modular forms. In the case p= 2,3, this upper bound coincides with the dimension of modular forms.
2 Weak Jacobi forms
Before investigating Siegel modular forms, we study weak Jacobi forms, which appear in the Fourier-Jacobi expansion of Siegel modular forms. This is an easy application of the theory in the book of Eichler-Zagier [EZ].
We fix an integerp≧2. Let Γ1,0(p) :=
{
˜ γ =
(a b c d
)
∈SL(2,Z)c≡0 (p) }
.
Fork ∈N0 :={0,1,2, . . .}, put Mk be aC-vector space of all modular forms of weightkwith respect to Γ1,0(p). Γ1,0(p) has two conjugacy classes of cusps.
Hence f ∈Mk has two Fourier expansions:
f(τ) =
∑∞ n=0
ane(nτ), and
(f|kα)(τ) =˜ τ−kf (
−1 τ
)
=
∑∞ n=0
bne (nτ
p )
,
where
e(z) := exp( 2π√
−1z)
, and ˜α:=
(0 −1
1 0
) .
For r, s∈N0, put Mk(r, s) :={
f ∈Mk an = 0 (n < r), bn = 0 (n < s)} and
dk(r, s) := dimCMk(r, s).
Now we define weak Jacobi forms. Letk, m∈N0. For a holomorphic function φ:H×C→C and ˜γ ∈SL(2,R), define
(φ|k,mγ)(τ, z) := (cτ˜ +d)−ke
(−mcz2 cτ +d
) φ
(aτ +b cτ +d, z
cτ +d )
.
We say that a holomorphic function φ: H×C → C is a weak Jacobi form of weight k and indexm, if φ satisfies the following three conditions:
(1) (φ|k,mγ)(τ, z) =˜ φ(τ, z) for any ˜γ ∈Γ1,0(p),
(2) φ(τ, z) = e(m(x2τ + 2xz))φ(τ, z+xτ +y) for anyx, y ∈Z, (3) φhas Fourier expansions
φ(τ, z) =
∑∞ n=0
fn(z)e(nτ) and
(φ|k,mα)(τ, z) =˜
∑∞ n=0
gn(z)e (nτ
p )
.
Put Jk,m be a C-vector space of all weak Jacobi forms of weight k and index m. For r, s∈N0, put
Jk,m(r, s) := {
φ∈Jk,mfn= 0 (n < r), gn= 0 (n < s)} . Using the method in the book of Eichler-Zagier [EZ], we have
dim Jk,m(r, s) =
∑m t=0
dk+2t(r, s) (k ≡0 (2))
m−2∑
t=0
dk+2t+1(r, s) (k ≡1 (2), m≧2)
0 (k ≡1 (2), m= 0,1)
.
3 Siegel modular forms of degree 2
The Siegel upper half space of degree 2 is defined by H2 :=
{
Z =tZ =
(τ z z ω
)
∈M(2,C)Im(Z)>0 }
.
Define
Sp(2,R) :=
{ γ =
(A B
C D
)
∈GL(4,C)tγJ γ=J }
,
where
J =
( 0 E2
−E2 0 )
.
The group Sp(2,R) acts on H2 by
H2 ∋Z 7→γ⟨Z⟩:= (AZ +B)(CZ +D)−1 ∈H2,
and det(CZ+D) is an automorphic factor of this action. For a holomorphic function F :H2 →Cand k∈Z, we define the action of Sp(2,R) by
(F|kγ)(Z) := det(CZ+D)−kF(γ⟨Z⟩).
Let
Γ2,0(p) :=
{(A B
C D
)
∈Sp(2,Z)C≡02 (p) }
and
ρ:= 1
√p
( 0 −E2 pE2 0
)
∈Sp(2,R).
For k ∈ N0, put Mk be a C-vector space of all modular forms of weight k with respect to Γ2,0(p):
Mk:={
F :H2 →C: holomorphic F|kγ =F for any γ ∈Γ2,0(p)} . ρ gives an isomorphism Mk ∋F 7→F|kρ∈ Mk. Hence we have two Fourier- Jacobi expansions of F ∈Mk:
F(Z) =
∑∞ m=0
φm(τ, z)e(mω), and
(F|kρ)(Z) =
∑∞ m=0
ψm(τ, z)e(mω).
For r, s∈N0, put Mk(r, s) :={
f ∈Mk φm = 0 (m < r), ψm = 0 (m < s)} .
We remark that dimMk(r, s) = dimMk(s, r). The projectionPr :Mk(r, s)∋ F 7→φr∈Jk,r gives an exact sequence
0−→Mk(r+ 1, s)−→Mk(r, s)−→Pr Jk,r. Now, we investigate Image(Pr). Put
σ:=
0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
∈Γ2,0(p), α:=
0 0 −1 0
0 1 0 0
1 0 0 0
0 0 0 1
∈Sp(2,Z),
β :=
√p 0 0 0 0 0 0 −√1p
0 0 √1p 0
0 √p 0 0
∈Sp(2,R).
(1) σ gives a condition F
(τ z z ω
)
= (−1)kF
(ω z z τ
) .
Hence we have
Image(Pr)⊂
{ Jk,r(r,0) (k ≡0 (2)) Jk,r(r+ 1,0) (k ≡1 (2)) . (2) On the one hand,
(F|kα)(Z) =
∑∞ m=r
(φm|k,mα)(τ, z)e(mω).˜
On the other hand, because α =ρβ−1 and F|kρ=F|kρσ, we have (F|kα) (Z) =(
F|kρσβ−1) (Z)
= (pω)−k
∑∞ m=s
ψm (
− 1 pω,− z
pω )
e (mτ
p − mz2 pω
) .
Hence we have
Image(Pr)⊂Jk,r(0, s).
Finally, from (1) and (2), we have an estimation Image(Pr)⊂
{ Jk,r(r, s) (k≡0 (2)) Jk,r(r+ 1, s) (k≡1 (2)) . Now we have our main theorem.
Theorem 1. The following estimation gives an upper bound of the dimen- sion of the space of Siegel modular forms: if k is even,
dimMk ≦
∑∞ r=0
∑r s=0
{dk+2s(r, r) +dk+2s(r, r+ 1)} and if k is odd,
dimMk≦∑∞
r=0
∑r s=0
{dk+2s+1(r+ 3, r+ 2) +dk+2s+1(r+ 3, r+ 3)}.
We remark that, in the case p = 2 or p = 3, this upper bound coincides with the dimension ofMk, which is calculated by the dimension formula (cf.
[Ha, Yo]). When p= 2, it is
∑
k∈Z
(dimMk)xk = 1 +x19
(1−x2) (1−x4)2(1−x6). When p= 3, it is
∑
k∈Z
(dimMk)xk = (1 + 2x4 +x6) +x15(1 + 2x2+x6) (1−x2) (1−x4) (1−x6)2 .
The generators of even weights were determined by Ibukiyama [Ib] and the generators of odd weights were determined by Ibukiyama and Aoki (in prepa- ration).
References
[Ao] Aoki, H., Estimating Siegel modular forms of genus 2 using Jacobi forms, J. Math. Kyoto Univ.40-3(2000), 581–588.
[EZ] Eichler, M., Zagier D., The Theory of Jacobi Forms, Progress in Math.
55, Birkh¨auser, Boston, 1985.
[Ha] Hashimoto, K., The dimension of the spaces of cusp forms on Siegel upper half-plane of degree two. I. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30-2(1983), 403–488.
[Ib] Ibukiyama, T., On Siegel modular varieties of level 3 Internat. J. Math.
2-1(1991), 17–35.
[Ig1] Igusa, J., On Siegel modular forms of genus two, Amer. J. Math.
84(1962), 175-200.
[Ig2] Igusa, J., On Siegel modular forms of genus two (II), Amer. J. Math.
86(1964), 392-412.
[Yo] Yoshida, H., On representation of finite groups in the space of Siegel modular forms and theta series, J. Math. Kyoto Univ. 28(1998), 343- 372.
