Relation between automorphic forms and zeta functions associated with prehomogeneous vector spaces

自同构形式和与预齐次向量空间相关的 zeta 函数之间的关系

• 批准号: 16340012
• 负责人: SATO Fumihiro
• 金额: $ 10.48万
• 依托单位: Rikkyo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340012/
• 关键词: prehomogeneous vector space; zeta function; automorphic form; Eisenstein series; Clifford algebra; spherical function; 局所密度; 局所関数等式; Eisenstein級数; 二次形式; Koecher-Maassゼータ関数; b-関数

The main problems we investigated in this research project are(1) To identify the zeta functions associated with prehomogeneous vector spaces with some kind of zeta functions attached to automorphic forms(2) To construct a theory of(local) functional equations which is not covered by the theory of prenomogeneous vector spaces. The results we obtained are as follows :(1) According to the classification theory due to Sato and Kimura, irreducible regular prehomogeneous vector spaces are classified into 5 series of classical type and 24 spaces of sporadic type. We identified the zeta functions associated with 4 series of prehomogeneous vector spaces of classical type with the standard L-functions or Koecher-Maass zeta functions of certain real analytic Eisenstein series. One of the results which are necessary for the proof of these results is a new integral representation of the Siegel series (= p-part of the Fourier coefficients of Eisenstein series). As another application of the new integral representation, we proved a formula which connects the Siegel series to spherical functions on a p-adic semisimple symmetric space of the orthogonal groups.(2) We proved that, given a pair of homogeneous polynomials on Ra satisfying a local functional equation and a pair of nondegenerate dual quadratic mappings of R^m to R^n, then, the pull backs of the polynomials by the quadratic mappings also satisfy a local functional equation. This generalizes a result due to Faraut-Koranyi-Clerc and we can construct many examples of functional equations which are not covered by the theory of prehomogeneous vector spaces. We also classified nondegenerate dual quadratic mappings over quadratic spaces and proved that such quadratic mappings are in one to one correspondence to representations of a tensor product of 2 Clifford algebras.

我们在这个研究项目中研究的主要问题是（1）识别与预齐次向量空间相关的zeta函数以及附加到自守形式的某种zeta函数（2）构建未涵盖的（局部）函数方程理论通过先生向量空间理论。我们得到的结果如下：(1)根据Sato和Kimura的分类理论，将不可约正则预齐次向量空间分为5个系列经典型和24个零星型空间。我们用某些实解析爱森斯坦级数的标准 L 函数或 Koecher-Maass zeta 函数识别了与经典类型的 4 个系列预齐次向量空间相关的 zeta 函数。证明这些结果所必需的结果之一是西格尔级数的新积分表示（=爱森斯坦级数的傅立叶系数的 p 部分）。作为新积分表示的另一个应用，我们证明了一个将 Siegel 级数与正交群的 p 进半单对称空间上的球函数联系起来的公式。 (2) 我们证明，给定 Ra 上的一对齐次多项式满足一个局部函数方程和一对 R^m 到 R^n 的非简并对偶二次映射，则二次映射对多项式的回拉也满足局部函数方程。这概括了 Faraut-Koranyi-Clerc 的结果，并且我们可以构造预齐次向量空间理论未涵盖的许多函数方程的示例。我们还对二次空间上的非简并对偶二次映射进行了分类，并证明了这种二次映射与 2 个 Clifford 代数的张量积的表示一一对应。

项目成果

Quadratic maps and nonprehomogeneous local functional equations

二次映射和非齐次局部函数方程

• 发表时间: 2007
• 期刊: Comment.Math.Univ.St.Pauli 56
• 作者: Kazuhiro;Konno;Nguyen;Khac-Viet;Kazuhiro Konno and Viet Nguyen Khac;Tomoyoshi Ibukiyama;Tatsuo Kimura;Fumihiro Sato
• 通讯作者: Fumihiro Sato

On the functional equations of shpherical functions on certain spherical homogeneous space

关于某球齐次空间上球函数的泛函方程

• 发表时间: 2004
• 作者: Tomoyoshi Ibukiyama;Hidenori Katsurada;Yumiko Hironaka;木村 達雄;Tatsuo Kimura;佐藤 文広;佐藤 文広;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;佐藤 文広;Fumihiro Sato;広中 由美子;伊吹山 知義;Tomoyoshi Ibukiyama;広中 由美子;広中 由美子;Yumiko Hironaka;木村 達雄;Tatsuo Kimura;佐藤 文広;木村 達雄;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;Fumihiro Sato;佐藤 文広;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;Fumihiro Sato;広中 由美子;Yumiko Hironaka;佐藤 文広;Fumihiro Sato;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;Fumihiro Sato;広中 由美子;Yumiko Hironaka;広中 由美子;Yumiko Hironaka;佐藤 文広;Fumihiro Sato;広中 由美子;Yumiko Hironaka;広中 由美子;Yumiko Hironaka;佐藤 文広;Fumihiro Sato;伊吹山 知義;Tomoyoshi Ibukiyama;広中 由美子;Yumiko Hironaka;伊吹山 知義;Tomoyoshi Ibukiyama;広中 由美子;Yumiko Hironaka
• 通讯作者: Yumiko Hironaka

Siegel modular forms of weight three and conjectural correspondence of Shimura type and Langlands type

权重三的西格尔模形式及志村型和朗兰兹型的猜想对应

• 发表时间: 2007
• 期刊: The conference on L-functions(World Scientific) 国際会議報告集
• 作者: Kazuhiro;Konno;Nguyen;Khac-Viet;Kazuhiro Konno and Viet Nguyen Khac;Tomoyoshi Ibukiyama;Tatsuo Kimura;Fumihiro Sato;Tomoyoshi Ibukiyama;Tatsuo Kimura;Fumihiro Sato;Fumihiro Sato;Tomoyoshi Ibukiyama
• 通讯作者: Tomoyoshi Ibukiyama

Quadratic mappings over(GO(p, q) , R^p+q) and functional Equations

(GO(p, q) , R^p q) 和函数方程的二次映射

• 发表时间: 2007
• 作者: Tomoyoshi Ibukiyama;Hidenori Katsurada;Yumiko Hironaka;木村 達雄;Tatsuo Kimura;佐藤 文広;佐藤 文広;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;佐藤 文広;Fumihiro Sato;広中 由美子;伊吹山 知義;Tomoyoshi Ibukiyama;広中 由美子;広中 由美子;Yumiko Hironaka;木村 達雄;Tatsuo Kimura;佐藤 文広;木村 達雄;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;Fumihiro Sato;佐藤 文広;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;Fumihiro Sato
• 通讯作者: Fumihiro Sato

有限軌道をもつ(G×GL_n,ρ×Λ_1)(n≧2)の分類

(G×GL_n,ρ×Λ_1)(n≧2)有限轨道分类

• 发表时间: 2008
• 作者: Tomoyoshi Ibukiyama;Hidenori Katsurada;Yumiko Hironaka;木村 達雄
• 通讯作者: 木村 達雄