Estimation of the dimension of modular forms by using Jacobi forms

Hiroki Aoki (Ritsumeikan University)

October 2, 2000 16:30-17:00

In this report, we introduce a new approach to the dimension formula of modular forms, by using Jacobi forms (cf. [Ao1, Ao2]). According to this approach, we can determine the structure of the ring of some modular forms, for example, Siegel modular forms of degree two and Hermitian modular forms of degree two with respect to the modular group associated with Gaussian integers. The key point of this method is the estimation of the dimensions of Jacobi forms, appearing in the Fourier-Jacobi development of modular forms. This estimation gives the upper bound of the dimension of modular forms. In some case, luckily, we can show this upper bound is the certain dimension of modular forms, by constructing generators explicitly.

1 Sketch of the estimation

Let 𝕄_{k} be a vector space of modular forms of weight k and 𝕁_{k,m} be a vector space of Jacobi forms of weight k and index m. We assume that 𝕄_{k} and 𝕁_{k,m} satisfy the following two conditions:

(1) F ∈ 𝕄_{k} has a Fourier expansion

F (ω, \underline{z}, τ) = ∑_{m=0}^{∞} φ_{m}(\underline{z}, τ) exp(2πimω)

and φ_{m}(\underline{z}, τ) ∈ 𝕁_{k,m}, where ω and τ are of one variable, but \underline{z} may be of several variables. Define the projection P_{m} : 𝕄_{k} → 𝕁_{k,m} by F ↦ φ_{m}.

(2) F ∈ 𝕄_{k} satisfies the equation F (ω, \underline{z}, τ) = ±F (τ, \underline{z}, ω).

Put

𝕄_{k}(r) := \{F (ω, \underline{z}, τ) = ∑_{m=r}^{∞} φ_{m}(\underline{z}, τ) exp(2πimω) ∈ 𝕄_{k}\}

and

𝕁_{k,m}(r) := \{φ(\underline{z}, τ) = ∑_{n=r}^{∞} g_{n}(\underline{z}) exp(2πinτ) ∈ 𝕁_{k,m}\} .

Case (a) First, we consider the case F (ω, \underline{z}, τ) = F (τ, \underline{z}, ω). The condition (2) induces the exact sequence

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

hence we have

dim 𝕄_{k} ≤ ∑_{r=0}^{∞} dim 𝕁_{k,r}(r).

Case (b) Second, we consider the case F (ω, \underline{z}, τ) = −F (τ, \underline{z}, ω). The condition (2) induces the exact sequence

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

hence we have

dim 𝕄_{k} ≤ ∑_{r=0}^{∞} dim 𝕁_{k,r}(r + 1).

In any case, we can calculate an upper bound of dim 𝕁_{k,m}(r) according to the book of Eichler-Zagier [E-Z]. We remark that, these series in the right hand sides may not converge. But, luckily, in some cases these series converge and give the certain dimension of modular forms. In fact, we have the following two lucky cases.

2 Siegel modular forms of degree two

The first lucky case is about Siegel modular forms of degree two. The Siegel upper half plane of degree two is defined by

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

Define

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

be the symplectic subgroup and

Sp(2, ℤ) := Sp(2, ℝ) ∩ M(4, ℤ)

be a discrete subgroup. The group Sp(2, ℝ) acts on ℍ_{2} by

ℍ_{2} ∋ Z ↦ M⟨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}M)(Z) := det(CZ + D)^{−k}F (M⟨Z⟩).

We call F is a modular form of weight k if F |_{k}M = F for any M ∈ Sp(2, ℤ). In 1954, Igusa [Ig1, Ig2] determined the structure of the graded ring of these modular forms.

Theorem 1 (Igusa) The graded ring of Siegel modular forms of degree two has five generators, whose weights are 4, 6, 10, 12 and 35. First four generators are algebraically independent. The square of the last generator is in the ring generated by first four generators.

Now we can prove this theorem easily. The even weighted case corresponds to the Case (a) and the odd weighted case corresponds to the Case (b). In any case, the Jacobi forms 𝕁_{k,m} completely coincide with the Jacobi forms in the book of Eichler-Zagier [E-Z]. According to the book of them, we have

∑_{k∈ℤ}(dim 𝕄_{k})t^{k} ≤ \frac{1 + t^{35}}{(1 − t^{4})(1 − t^{6})(1 − t^{10})(1 − t^{12})},

where ‘≤’ means that dim 𝕄_{k} is not greater than the coefficient of t^{k} on the formal power series expansion of the right hand side. In fact, the both sides are equal! We can easily give the generators and show their algebraic independence. There are some methods to construct the generators. In Igusa [Ig1, Ig2], he gave the generators by using theta constant. The even generators can be constructed by Eisenstein series and by Saito-Kurokawa lifting. The odd generator of weight 35 was constructed by Ibukiyama [Ib1, Ib3] by using generalized Rankin-Cohen differential operators and by Gritsenko [Gr] by using Borcherds’ infinite product.

3 Hermitian modular forms of degree two

The second lucky case is about Hermitian modular forms of degree two with respect to the modular group associated with Gaussian integers. The Hermitian upper half plane of degree two is defined by

H_{2} := \{Z = (\begin{matrix}ω & z_{1} \\ z_{2} & τ \end{matrix}) ∈ M(2, ℂ) | \frac{Z −^{t}\overline{Z}}{2i} > 0\} .

Define

Ω := \{M = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ M(4, ℂ) | ^{t}\overline{M}JM = J, J = (\begin{matrix}0 & E_{2} \\ −E_{2} & 0 \end{matrix})\}

and

Γ := Ω ∩ M(4, ℤ[i])

be a discrete subgroup. The group Ω acts on H_{2} by

H_{2} ∋ Z ↦ M⟨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 Ω by

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

We call F is a modular form of weight k if F |_{k}M = F for any M ∈ Γ. A modular form F is said to be symmetric if F (Z) = F (^{t}Z). Freitag [Fr], Nagaoka [Na] and Ibukiyama [Ib2] determined the structure of the graded ring of symmetric modular forms.

Theorem 2 (Freitag-Nagaoka-Ibukiyama) The graded ring of symmetric modular forms has six generators, whose weights are 4, 8, 10, 12, 12 and 16. First five generators are algebraically independent. The square of the last generator is in the ring generated by first five generators.

In this case, we can also apply the above idea and we can determine the structure of the modular forms, including non-symmetric forms. The calculation is a little complicated, but we have

∑_{k∈ℤ}(dim 𝕄_{k})t^{k} ≤ \frac{(1 + t^{16})(1 + t^{34})}{(1 − t^{4})(1 − t^{8})(1 − t^{10})(1 − t^{12})^{2}} .

In fact, the both sides are equal!! The generators of weights 4, 8, 10, 12, 12 and 16 are symmetric. In Freitag [Fr], he gave the symmetric generators by using theta constant. We remark that the symmetric generators can be constructed by Eisenstein series and by Maass lifting [Ko, Su]. The generator of weight 34 is non-symmetric and it is constructed by an argument similar to that in Ibukiyama [Ib3]. The conclusion is as follows.

Theorem 3 The graded ring of modular forms has seven generators, whose weights are 4, 8, 10, 12, 12, 16 and 34. First six generators are symmetric and the last generator is skew-symmetric.

References

[Ao1] Aoki, H., Estimating Siegel modular forms of genus 2 using Jacobi forms, J. Math. Kyoto Univ. 40-3(2000), 581–588.

[Ao2] Aoki, H., The graded ring of Hermitian modular forms of degree two, in preparation.

[E-Z] Eichler, M., Zagier D., The Theory of Jacobi Forms, Progress in Math. 55, Birkhäuser, Boston, 1985.

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

[Gr] Gritsenko, V.A., The Igusa modular forms and “the simplest” Lorentzian Kac-Moody algebras, Yukawa e-Print, alg-geom, 9603010

[Ib1] Ibukiyama, T., On differential operators on automorphic forms and invariant pluri-harmonic polynomials Comm. Math. Univ. St. Pauli, 48(1999), 103–118.

[Ib2] Ibukiyama, T., A remark on the hermitian modular forms, Osaka University Research Reports in Mathematics, 99-19(1999).

[Ib3] Ibukiyama, T., The Siegel modular forms of weight 35, in preparation.

[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.

[Ko] Kojima, H., An Arithmetic of Hermitian Modular Forms of Degree two, Invent. Math. 69(1982), 217-227.

[Na] Nagaoka, S., A note on the structure of the ring of symmetric Hermitian modular forms of degree 2 over the Gaussian field, J. Math. Soc. Japan 48-3(1996), 525-549.

[Su] Sugano, T., The Maass space for SU(2,2) (Japanese), Res. Inst. Math. Sci. Kokyuroku 546(1985), 1-16.