On the ring of Hilbert modular forms over Z
On the ring of Hilbert modular forms over Z
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在 Z 上的希尔伯特模形式环上
DOI：
10.2969/jmsj/03540589
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发表时间：
1983
影响因子：
0.7
通讯作者：
S. Nagaoka
中科院分区：
文献类型：
--
作者：
S. Nagaoka
$E_{4},$ $E_{6},$ $\Delta$ with coefficients in $Z$, where $E_{k}$ is the normalized Eisenstein series of weight $k$ and $\Delta=2^{-6}\cdot 3^{-3}(E_{4}^{3}-E_{6}^{2})$ . On the other hand, in his paper [7], J. Igusa gave a minimal set of generators over $Z$ of the graded ring of Siegel modular forms of degree two whose Fourier coefficients lie in $Z$. Also, some related topics and problems on the finite generation of an algebra of modular forms were discussed by W. L. Baily, Jr. in his recent paper [2]. In this paper, we give analogous results for symmetric Hilbert modular forms for the real quadratic fields $Q(\sqrt{}\overline{2})$ and $Q(\sqrt{5})$ . Let $K$ be a real quadratic field and $A_{z}(\Gamma_{K})_{k}$ denote the Z-module of symmetric Hilbert modular forms of even weight $k$ with rational integral Fourier coefficients and we put $A_{Z}(\Gamma_{K})=\oplus A_{Z}(\Gamma_{K})_{k}$ . Denote by $G_{k}$ the normalized Eisenstein series for the Hilbert modular group
$E_{4},$ $E_{6},$ $\Delta$ with coefficients in $Z$, where $E_{k}$ is the normalized Eisenstein series of weight $k$ and $\Delta=2^{-6}\cdot 3^{-3}(E_{4}^{3}-E_{6}^{2})$ . On the other hand, in his paper [7], J. Igusa gave a minimal set of generators over $Z$ of the graded ring of Siegel modular forms of degree two whose Fourier coefficients lie in $Z$. Also, some related topics and problems on the finite generation of an algebra of modular forms were discussed by W. L. Baily, Jr. in his recent paper [2]. In this paper, we give analogous results for symmetric Hilbert modular forms for the real quadratic fields $Q(\sqrt{}\overline{2})$ and $Q(\sqrt{5})$ . Let $K$ be a real quadratic field and $A_{z}(\Gamma_{K})_{k}$ denote the Z-module of symmetric Hilbert modular forms of even weight $k$ with rational integral Fourier coefficients and we put $A_{Z}(\Gamma_{K})=\oplus A_{Z}(\Gamma_{K})_{k}$ . Denote by $G_{k}$ the normalized Eisenstein series for the Hilbert modular group