On the ring structure of Siegel modular forms

Hiroki Aoki

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan

aoki hiroki@ma.noda.tus.ac.jp

Abstract. The purpose of this exposition is to overview some recent works on the ring structure of Siegel modular forms. By using the elementary estimation of the dimension of the space of Siegel modular forms, we can determine the ring structure of them. For the sake of simplicity, in this exposition, we treat only the most simple case: even weight ℂ-valued Siegel modular forms with respect to the full modular group Sp(2, ℤ). However, we remark that the same method is available for many other cases.

1 Introduction

On the structure theorem of Siegel modular forms of degree 2, Igusa [11, 12] determined the structure of Siegel modular forms with respect to the full modular group Sp(2, ℤ). There are five generators of weight 4, 6, 10, 12 and 35. First four generators are algebraically independent and the square of the last generator is in the subring generated by first four.

Recently, Aoki and Ibukiyama [4] indicated that the ring of Siegel modular forms with small level has similar structure. That is, on the ring of Siegel modular forms of degree 2 with respect to the congruent subgroup of level N = 1, 2, 3, 4 (for N = 3, 4, taking Neven-type case with character), there are five generators, among which four generators are algebraically independent and the square of the last generator is in the subring generated by first four.

On the structure of vector valued Siegel modular forms of degree 2 with respect to the symmetric tensor of degree 2, Satoh [13] and Ibukiyama [10] determined the structure with respect to the full modular group. There are ten generators with some relations. Recently, Aoki [3] determined the structure of vector valued Siegel modular forms with small level. This structure is similar to the structure with respect to the full modular group.

The original proofs of above structure theorems are various. However, now we can prove all of them by using the elementary estimation of the dimension of the space of Siegel modular forms. In this exposition, we study this method.

2 Notations

We denote the Siegel upper half plane of degree 2 by

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

The symplectic group

Sp(2, ℝ) := \{ M = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ M_{4}(ℝ) | ^{t}MJ_{2}M = J_{2} := (\begin{matrix}0 & −E_{2} \\ E_{2} & 0 \end{matrix}) \}

acts on ℍ_{2} transitively by

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

For M ∈ Sp(2, ℝ), k ∈ ℤ and a holomorphic function F : ℍ_{2} → ℂ, we write

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

Put

Sp(2, ℤ) := Sp(2, ℝ) ∩ M_{4}(ℤ)

and Γ be a finite index subgroup of Sp(2, ℤ). For a holomorphic function F : ℍ_{2} → ℂ and k ∈ ℤ, we say F is a Siegel modular forms of weight k with respect to Γ if F satisfies the condition F (Z) = (F |_{k}M)(Z) for any M ∈ Γ. We remark that this F is bounded at each cusps by Köcher principle. We denote by A_{k} = A_{k}(Γ) the space of all Siegel modular forms of weight k with respect to Γ. Put A_{∗} := ⨁_{k∈ℤ} A_{k}. The space A_{∗} is a graded ring. Our aim is to determine the ring structure of A_{∗}. For the sake of simplicity, in this exposition, we assume that Γ := Sp(2, ℤ) and that k is even.

Let ℍ be a complex upper half plane. For a holomorphic function f : ℍ×ℍ → ℂ and k ∈ ℤ, we say f is a symmetric Witt modular form of weight k if f satisfies the following three conditions:

(1) For any fixed ω_{0} ∈ ℍ, the function f(τ, ω_{0}) on τ ∈ ℍ is an elliptic modular form of weight k.

(2) For any fixed τ_{0} ∈ ℍ, the function f(τ_{0}, ω) on ω ∈ ℍ is an elliptic modular form of weight k.

(3) f is symmetric, namely, f(τ, ω) = f(ω, τ). We denote by M_{k} the space of all symmetric Witt modular forms of weight k. It is well known that the graded ring of all elliptic modular forms is generated by algebraically independent two modular forms of weight 4 and 6. By using the Theorem of Witt [14, Satz A], we know that the graded ring of all symmetric Witt modular forms is generated by algebraically independent three modular forms of weight 4, 6 and 12.

3 Estimation

For r ∈ ℕ ∪ {0}, define the map D_{r} : A_{k} → M_{k+2r} by

(D_{r}F) (τ, ω) := (\frac{∂^{2r}F}{∂z^{2r}} ) (\begin{matrix}τ & 0 \\ 0 & ω \end{matrix}) .

Put

A_{k}[r] := {F ∈ A_{k}|D_{s}F = 0 if s < r} .

Easily we have Ker(D_{r}) = A_{k}[r+1]. We put the Fourier coefficients a(n, l, m) of F by

F (\begin{matrix}τ & z \\ z & ω \end{matrix}) = ∑_{n,l,m} a(n, l, m)e(nτ + lz + mω),

where we put e(w) = exp(2π\sqrt{−1}w) and the index (n, l, m) runs all integers satisfying 4nm − l^{2} ≥ 0. Then we have

(D_{r}F) (τ, ω) = (2π\sqrt{−1})^{2r} ∑_{n,l,m} a(n, l, m)l^{2r}e(nτ + lz + mω).

Hence, if we assume F ∈ A_{k}[r], we have

∑_{l} a(n, l, m)l^{2s} = 0 (s = 0, 1, …, r − 1).

It is easy to show that

a(n, l, m) = a(n, −l, m) = a(m, n, l)

and that

a(n, l, m) = a(n + 2lx + mx^{2}, l + 2mx, m)

for any x ∈ ℤ. Then we have

a(n, l, m) = 0 ( if min{m, n} < r).

Put

M_{k}[r] := {f(τ, ω) = ∑_{n,m} c(n, m)e(nτ + mω) ∈ A_{k} | c(n, m) = 0 if min{m, n} < r} .

Then we have an exact sequence

0 ⟶ A_{k}[r + 1] ⟶ A_{k}[r] ⟶^{D_{r}} M_{k+2r}[r]

and we have an estimation

dim A_{k} ≦ ∑_{r=0}^{∞} dim M_{k+2r}[r] = ∑_{r=0}^{∞} dim A_{k−10r}.

By easy calculation, we have

∑_{k≧0}(dim A_{k})x^{k} ≦ \frac{1}{(1 − x^{4})(1 − x^{6})(1 − x^{10})(1 − x^{12})},

where ’≦’ means that each dimension dim A_{k} in the left hand side is not greater than the coefficient of x^{k} on the formal power series development of the right hand side.

Hence, if we construct the algebraically independent modular forms of weight 4, 6, 10 and 12, the upper bound of the dimension of A_{k} in above inequality equals to the true dimension of A_{k}. In fact, we can construct these modular forms easily, for example, by Saito-Kurokawa lifting.

Theorem 1. The graded ring of even weight Siegel modular forms of degree 2 with respect to the full modular group is generated by algebraically independent four modular forms of weight 4, 6, 10 and 12.

We remark that now we have two another elementary proofs of this theorem. One is by Freitag [6], and the other is by Aoki [1].

4 General cases

For more general cases, we know the structure theorem as follows:

Theorem 2. For each Γ = Sp(2, ℤ), Γ_{0}^{(2)}(2), Γ_{0,ψ_{3}}^{(2)}(3) or Γ_{0,ψ_{4}}^{(2)}(4), the graded ring A_{∗}(Γ) is generated by five modular forms. First four generators are algebraically independent and the square of the last generator is in the subring generated by first four.

Γ  The weights of  The weights of  References
first four generators the last generators
Sp(2, ℤ)  4, 6, 10, 12  35  Igusa [11, 12]

Γ_{0}^{(2)}(2)

2, 4, 4, 6

19

Ibukiyama [8]

Γ_{0,ψ_{3}}^{(2)}(3)  1, 3, 3, 4  14  Ibukiyama [8]

Aoki-Ibukiyama [4]

Γ_{0,ψ_{4}}^{(2)}(4)

1, 2, 2, 3

11 Hayashida-Ibukiyama [7]

For these cases, we use the idea of Rankin-Cohen type differential operators given by Eholzer and Ibukiyama [5] and Ibukiyama [9] for constructing modular forms.

Now we define vector valued Siegel modular forms of degree 2. Let s be a non-negative integer, V be a (s + 1)-dimensional ℂ-vector space and ρ : GL(2, ℂ) → GL(V) be a rational irreducible representation ρ = ρ_{k,s} := Sym^{s} ⊗ det^{k}. For M ∈ Sp(2, ℝ) and a holomorphic function F : ℍ_{s} → ℂ, we write

(F |_{ρ}M)(Z) := ρ(CZ + D)^{−1}F (M⟨Z⟩).

We say F is a Siegel modular forms of weight ρ with respect to Γ if F satisfies the condition F (Z) = (F |_{ρ}M)(Z) for any M ∈ Γ. We remark that this F is bounded at each cusps by Köcher principle. We denote by A_{k,s}(Γ) the space of all Siegel modular forms of weight ρ_{k,s} with respect to Γ. We remark A_{k,0}(Γ) = A_{k}(Γ). It is easy to show that if s is odd and if −E_{4} ∈ Γ, then A_{k,s}(Γ) = {0}. Put A_{∗,s}(Γ) := ⨁_{k∈ℤ} A_{k,s}(Γ). The space A_{∗,s}(Γ) is a graded module of A_{∗}(Γ). In vector valued case, we know the structure theorem as follows:

Theorem 3. For each Γ = Sp(2, ℤ), Γ_{0}^{(2)}(2), Γ_{0,ψ_{3}}^{(2)}(3) or Γ_{0,ψ_{4}}^{(2)}(4), the graded module A_{∗,2}(Γ) is generated by ten modular forms.

Γ  The weights of generators The weights of generators  References
(Type 1)  (Type 2)
Sp(2, ℤ) 10, 14, 16, 16, 18, 22  21, 23, 27, 29  Satoh [13]

Ibukiyama [10]

Γ_{0}^{(2)}(2)  6, 6, 8, 8, 10, 10  11, 13, 13, 15  Aoki [3]
Γ_{0,ψ_{3}}^{(2)}(3)  4, 4, 5, 6, 7, 7  8, 9, 9, 11
Γ_{0,ψ_{4}}^{(2)}(4)  3, 3, 4, 4, 5, 5  6, 7, 7, 8

Finally, we remark that this method is available for some Hilbert modular forms with real quadratic field. (c.f. Aoki [2])

References

[1] Aoki, H., Estimating Siegel modular forms of genus 2 using Jacobi forms, J. Math. Kyoto Univ. 40-3(2000), 581–588.

[2] Aoki, H., Estimate of the dimension of Hilbert modular forms by means of differential operator, Automorphic forms and Zeta functions (Rikkyo, 2004), Proceedings of the Conference in Memory of Tsuneo Arakawa, World Scientific, 20-28.

[3] Aoki, H., Vector valued Siegel modular forms of degree 2 with small levels, in preparation.

[4] Aoki, H., Ibukiyama, T., Simple graded rings of Siegel modular forms of small levels, differential operators and Borcherds products Int. J. Math. 16-3(2005), 249–279.

[5] Eholzer, W., Ibukiyama, T., Rankin-Cohen type differential operators for Siegel modular forms, Int. J. Math. 9-4(1998), 443–463.

[6] Freitag, E., Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper (German), Sitzungsber. Heidelb. Akad. Wiss., (1967), 3–49.

[7] Hayashida, S., Ibukiyama, T., Siegel modular forms of half integral weights and a lifting conjecture, preprint.

[8] Ibukiyama, T., On Siegel modular varieties of level 3, Int. J. Math. 2-1(1991), 17–35.

[9] Ibukiyama, T., On differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comm. Math. Univ. St. Pauli, 48(1999), 103–118.

[10] Ibukiyama, T., Differential operators and structures of vector valued Siegel modular forms (Japanese), Algebraic number theory and related topics (Japanese) (Kyoto, 2000), RIMS Kokyuroku 1200(2001), 71–81

[11] Igusa, J., On Siegel modular forms of genus two, Amer. J. Math. 84(1962), 175-200.

[12] Igusa, J., On Siegel modular forms of genus two (II), Amer. J. Math. 86(1964), 392-412.

[13] Satoh, T., On certain vector valued Siegel modular forms of degree two. Math. Ann., 274-2(1986), 335–352.

[14] Witt, E., Eine Identität zwischen Modulformen zweiten Grades Abh. Math. Sem. Hansischen Univ., 14(1941), 323–337.