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 determination 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 Γ_{2,0}(p) ⊂ Sp(2, ℤ). Similar to the full modular 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 integer p ≧ 2. Let

Γ_{1,0}(p) := \{γ̃ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ) | c ≡ 0 (p)\} .

For k ∈ ℕ_{0} := {0, 1, 2, … }, put M_{k} be a ℂ-vector space of all modular forms of weight k with respect to Γ_{1,0}(p). Γ_{1,0}(p) has two conjugacy classes of cusps. Hence f ∈ M_{k} has two Fourier expansions:

f(τ) = ∑_{n=0}^{∞} a_{n}e(nτ),

and

(f|_{k}α̃)(τ) = τ^{−k}f (−\frac{1}{τ}) = ∑_{n=0}^{∞} b_{n}e (\frac{nτ}{p} ) ,

where

e(z) := exp (2π\sqrt{−1}z) , and α̃ := (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

For r, s ∈ ℕ_{0}, put

M_{k}(r, s) := {f ∈ M_{k} | a_{n} = 0 (n < r), b_{n} = 0 (n < s)}

and

d_{k}(r, s) := dim_{ℂ} M_{k}(r, s).

Now we define weak Jacobi forms. Let k, m ∈ ℕ_{0}. For a holomorphic function φ : ℍ × ℂ → ℂ and γ̃ ∈ SL(2, ℝ), define

(φ|_{k,m}γ̃)(τ, z) := (cτ + d)^{−k}e (\frac{−mcz^{2}}{cτ + d} ) φ (\frac{aτ + b}{cτ + d}, \frac{z}{cτ + d}) .

We say that a holomorphic function φ : ℍ × ℂ → ℂ is a weak Jacobi form of weight k and index m, if φ satisfies the following three conditions:

(1) (φ|_{k,m}γ̃)(τ, z) = φ(τ, z) for any γ̃ ∈ Γ_{1,0}(p),

(2) φ(τ, z) = e(m(x^{2}τ + 2xz))φ(τ, z + xτ + y) for any x, y ∈ ℤ,

(3) φ has Fourier expansions

φ(τ, z) = ∑_{n=0}^{∞} f_{n}(z)e(nτ)

and

(φ|_{k,m}α̃)(τ, z) = ∑_{n=0}^{∞} g_{n}(z)e (\frac{nτ}{p} ) .

Put J_{k,m} be a ℂ-vector space of all weak Jacobi forms of weight k and index m. For r, s ∈ ℕ_{0}, put

J_{k,m}(r, s) := {φ ∈ J_{k,m} | f_{n} = 0 (n < r), g_{n} = 0 (n < s)} .

Using the method in the book of Eichler-Zagier [EZ], we have

dim J_{k,m}(r, s) = \{ \begin{matrix}∑_{t=0}^{m} d_{k+2t}(r, s) & (k ≡ 0 (2)) \\ ∑_{t=0}^{m−2} d_{k+2t+1}(r, s) & (k ≡ 1 (2), m ≧ 2) \\ 0 & (k ≡ 1 (2), m = 0, 1) \end{matrix} .

3 Siegel modular forms of degree 2

The Siegel upper half space of degree 2 is defined by

ℍ_{2} := \{Z = ^{t}Z = (\begin{matrix}τ & z \\ z & ω \end{matrix}) ∈ M(2, ℂ) | Im(Z) > 0\} .

Define

Sp(2, ℝ) := \{γ = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ GL(4, ℂ) | ^{t}γJγ = J\} ,

where

J = (\begin{matrix}0 & E_{2} \\ −E_{2} & 0 \end{matrix}) .

The group Sp(2, ℝ) acts on ℍ_{2} by

ℍ_{2} ∋ Z ↦ γ⟨Z⟩ := (AZ + B)(CZ + D)^{−1} ∈ ℍ_{2},

and det(CZ + D) is an automorphic factor of this action. For a holomorphic function F : ℍ_{2} → ℂ and k ∈ ℤ, we define the action of Sp(2, ℝ) by

(F |_{k}γ)(Z) := det(CZ + D)^{−k}F (γ⟨Z⟩).

Let

Γ_{2,0}(p) := \{(\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp(2, ℤ) | C ≡ 0_{2} (p)\}

and

ρ := \frac{1}{\sqrt{p}} (\begin{matrix}0 & −E_{2} \\ pE_{2} & 0 \end{matrix}) ∈ Sp(2, ℝ).

For k ∈ ℕ_{0}, put 𝕄_{k} be a ℂ-vector space of all modular forms of weight k with respect to Γ_{2,0}(p):

𝕄_{k} := {F : ℍ_{2} → ℂ : holomorphic | F |_{k}γ = F for any γ ∈ Γ_{2,0}(p)} .

ρ gives an isomorphism 𝕄_{k} ∋ F ↦ F |_{k}ρ ∈ 𝕄_{k}. Hence we have two Fourier-Jacobi expansions of F ∈ 𝕄_{k}:

F (Z) = ∑_{m=0}^{∞} φ_{m}(τ, z)e(mω),

and

(F |_{k}ρ)(Z) = ∑_{m=0}^{∞} ψ_{m}(τ, z)e(mω).

For r, s ∈ ℕ_{0}, put

𝕄_{k}(r, s) := {f ∈ M_{k} | φ_{m} = 0 (m < r), ψ_{m} = 0 (m < s)} .

We remark that dim 𝕄_{k}(r, s) = dim 𝕄_{k}(s, r). The projection P_{r} : 𝕄_{k}(r, s) ∋ F ↦ φ_{r} ∈ J_{k,r} gives an exact sequence

0 ⟶ 𝕄_{k}(r + 1, s) ⟶ 𝕄_{k}(r, s) ⟶^{P_{r}} J_{k,r}.

Now, we investigate Image(P_{r}). Put

σ := (\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) ∈ Γ_{2,0}(p), α := (\begin{matrix}0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) ∈ Sp(2, ℤ),

β := (\begin{matrix}\sqrt{p} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{−1}{\sqrt{p}} \\ 0 & 0 & \frac{1}{\sqrt{p}} & 0 \\ 0 & \sqrt{p} & 0 & 0 \end{matrix}) ∈ Sp(2, ℝ).

(1) σ gives a condition

F (\begin{matrix}τ & z \\ z & ω \end{matrix}) = (−1)^{k}F (\begin{matrix}ω & z \\ z & τ \end{matrix}) .

Hence we have

Image(P_{r}) ⊂ \{ \begin{matrix}J_{k,r}(r, 0) & (k ≡ 0 (2)) \\ J_{k,r}(r + 1, 0) & (k ≡ 1 (2)) \end{matrix} .

(2) On the one hand,

(F |_{k}α)(Z) = ∑_{m=r}^{∞}(φ_{m}|_{k,m}α̃)(τ, z)e(mω).

On the other hand, because α = ρβ^{−1} and F |_{k}ρ = F |_{k}ρσ, we have

(F |_{k}α) (Z) = (F |_{k}ρσβ^{−1}) (Z)

= (pω)^{−k} ∑_{m=s}^{∞} ψ_{m} (−\frac{1}{pω} , −\frac{z}{pω} ) e (\frac{mτ}{p} − \frac{mz^{2}}{pω} ) .

Hence we have

Image(P_{r}) ⊂ J_{k,r}(0, s).

Finally, from (1) and (2), we have an estimation

Image(P_{r}) ⊂ \{ \begin{matrix}J_{k,r}(r, s) & (k ≡ 0 (2)) \\ J_{k,r}(r + 1, s) & (k ≡ 1 (2)) \end{matrix} .

Now we have our main theorem. Theorem 1. The following estimation gives an upper bound of the dimension of the space of Siegel modular forms: if k is even,

dim 𝕄_{k} ≦ ∑_{r=0}^{∞} ∑_{s=0}^{r} {d_{k+2s}(r, r) + d_{k+2s}(r, r + 1)}

and if k is odd,

dim 𝕄_{k} ≦ ∑_{r=0}^{∞} ∑_{s=0}^{r} {d_{k+2s+1}(r + 3, r + 2) + d_{k+2s+1}(r + 3, r + 3)} .

We remark that, in the case p = 2 or p = 3, this upper bound coincides with the dimension of 𝕄_{k}, which is calculated by the dimension formula (cf. [Ha, Yo]). When p = 2, it is

∑_{k∈ℤ}(dim 𝕄_{k}) x^{k} = \frac{1 + x^{19}}{(1 − x^{2}) (1 − x^{4})^{2} (1 − x^{6})}.

When p = 3, it is

∑_{k∈ℤ}(dim 𝕄_{k}) x^{k} = \frac{(1 + 2x^{4} + x^{6}) + x^{15} (1 + 2x^{2} + x^{6})}{(1 − x^{2}) (1 − x^{4}) (1 − x^{6})^{2}}.

The generators of even weights were determined by Ibukiyama [Ib] and the generators of odd weights were determined by Ibukiyama and Aoki (in preparation).

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, Birkhäuser, 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.