Jacobi forms, Fourier-Jacobi developments and the ring structure of automorphic forms

Hiroki Aoki (Ritsumeikan University) Sep. 2001, CIRM (Luminy, France)

1 Introduction

Today, I will talk about a new approach to the dimension formula of modular forms, using Jacobi forms. Consider the modular forms, having Fourier-Jacobi development. Every coefficient of the development is in the space of Jacobi forms. By investigating this space explicitly, we have an upper bound of the dimension of the space of modular forms. In fact, this upper bound coincides with the dimension of the space of modular forms, in some cases.

2 Basic idea

Let 𝕄_{k} be a space of modular forms of weight k and let 𝕁_{k,m} be a space of Jacobi forms of weight k and index m. We assume that 𝕄_{k} and 𝕁_{k,m} satisfy the following two conditions: (C1) F ∈ 𝕄_{k} has a Fourier development

F (τ, \underline{z}, ω) = ∑_{m=0}^{∞} φ_{m}(τ, \underline{z})e(mω) (e(∗) := exp (2π\sqrt{−1}∗))

and φ_{m}(τ, \underline{z}) ∈ 𝕁_{k,m}, where τ and ω 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}. (C2) F ∈ 𝕄_{k} satisfies the equation F (τ, \underline{z}, ω) = ±F (ω, \underline{z}, τ). Put

\begin{matrix}𝕄_{k}(r) := \{F (τ, \underline{z}, ω) = ∑_{m=0}^{∞} φ_{m}(τ, \underline{z})e(mω) ∈ 𝕄_{k} | φ_{m}(τ, \underline{z}) ≡ 0 (m < r)\}, \\ 𝕁_{k,m}(r) := \{φ(τ, \underline{z}) = ∑_{n=0}^{∞} g_{n}(\underline{z})e(nτ) ∈ 𝕁_{k,m} | g_{n}(\underline{z}) ≡ 0 (n < r)\}. \end{matrix}

Case(a) In the case of F (τ, \underline{z}, ω) = F (ω, \underline{z}, τ), the condition (C2) induces an 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) In the case of F (τ, \underline{z}, ω) = −F (ω, \underline{z}, τ), the condition (C2) induces an 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).

We can calculate (an upper bound of) the dimension of 𝕁_{k,m}(r) according to the book of Eichler-Zagier [E-Z].

3 Examples

Ex.1: Siegel modular forms of degree 2 [Ao1] We consider Siegel modular forms of degree two with respect to Sp_{2}(ℤ). The even weighted case corresponds to the Case (a) and the odd weighted case corresponds to the Case (b). In any case, 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} of the formal power series expansion of the right hand side. In fact, the both sides are equal. The generators were given by Igusa [Ig1, Ig2], using theta constant. Also even generators can be constructed by Eisenstein series and by Saito-Kurokawa lifting. We remark that P_{1} is Saito-Kurokawa lifting. The odd generator of weight 35 was also given by Ibukiyama [Ib2] (generalized Rankin-Cohen differential operators) and by Gritsenko [Gr] (Borcherds’ infinite product).

By analogous trick, we have an upper bound of the dimension of the space of modular forms, with respect to a usual congruent subgroup Γ_{0}(N) ⊂ Sp_{2}(ℤ). In the case of N = 2, 3, comparing the result by Hashimoto [Ha] and Ibukiyama [Ib1], we know that this estimation coincides the true dimension.

Ex.2: Hermitian modular forms of degree two [Ao2] We consider Hermitian modular forms of degree two with respect to the modular group associated with Gaussian integers. 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 4,8,10,12,12,16, which are symmetric, were given by Freitag [Fr], using theta constant. The ring structure of symmetric forms was determined by Freitag, Nagaoka [Na] and Ibukiyama [Ib3]. Also symmetric generators can be constructed by Eisenstein series and by Maass lifting [Ko, Su]. We constructed the non-symmetric generator of weight 34 by an argument similar to that in Ibukiyama [Ib4].

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

[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.

[Ib1] Ibukiyama, T., On Siegel modular varieties of level 3 Internat. J. Math. 2-1(1991), 17–35.

[Ib2] Ibukiyama, T., On differential operators on automorphic forms and invariant pluri-harmonic polynomials Comm. Math. Univ. St. Pauli, 48(1999), 103–118.

[Ib3] Ibukiyama, T., A remark on the hermitian modular forms, Osaka University Research Reports in Mathematics, 99-19(1999).

[Ib4] 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.