On the graded ring of Siegel modular forms of degree

The aim of this paper is to give the dimension of the space of Siegel modular forms M k (Γ(3)) of degree 2, level 3 and weight k for each k. Our main result is Theorem dim M k (Γ(3)) = 1 2 (6k 3 − 27k 2 + 79k − 78) k ≥ 4. In other words we have the generating function : ∞ k=0 dim M k (Γ(3))t k = 1 + t + t 2 + 6t 3 + 6t 4 + t 5 + t 6 + t 7 (1 − t) 4. About the space of cusp forms, the dimension formula of weight k ≥ 4 is shown by Morita ([Morita]) Christian ([Christian 1,2]) and Yamazaki ([Yamazaki]), by using the Selberg trace formula or Riemann-Roch theorem. Therefore for weight k ≥ 5, the calculation of the dimension of M k (Γ(3)) is not so hard using Eisenstein series and the Siegel Φ-operator. In the case of low weights, the fact that M k (Γ(3)) are representation spaces of the finite group Sp(2, F 3) is crucial. Fortunately all the irreducible characters of Sp(2, F 3) are given by Srinivasan ([Srinivasan]). By utilizing this table we can determine these representations, hence all the more the dimensions. Now we review the contents of our paper in detail. Firstly we recall some basic facts on elliptic modular forms in §1. In §2 the dimension formula of the spaces of Siegel cusp forms of higher weights is given, and we examine the boundaries of the Satake compactification of Γ(3)\H 2 in §4. We determine the dimensions in §5. The space of modular forms M k (Γ(3)) is decomposed into the space of cusp forms S k (Γ(3)), the space of Siegel-Eisenstein series E k (Γ(3)) and the space of Klingen-Eisenstein series K k (Γ(3)) in § §5.1, 5.2. In § §5.3 we define some elements of M 1 (Γ(3)) by using the theory of theta series of quadratic forms, and we calculate the dimensions of M 1 (Γ(3)) and M 2 (Γ(3)) in § §5.4. Finally in §6 the dimensions of M 3 (Γ(3)) and M 4 (Γ(3)) are determined exactly by using the theory of non-holomorphic Eisenstein series. 1 Acknowledgements : The author would like to thank to Professor Ken-ichi Shinoda of Sophia University for advices about the character table of Sp(2, F q), and to Professor Shin-ichiro …