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.

本课题研究的主要问题是：(1)用自同构形式的zeta函数来确定与预齐次向量空间相关的zeta函数；(2)构造一个未被预齐次向量空间理论涵盖的（局部）泛函方程理论。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划分为5个经典型系列和24个零星型空间。我们用标准l函数或某些实解析爱森斯坦级数的koecher - mass zeta函数确定了与经典型4个预齐次向量空间级数相关的zeta函数。证明这些结果所必需的结果之一是西格尔级数的一个新的积分表示（=爱森斯坦级数的傅里叶系数的p部分）。作为新的积分表示的另一个应用，我们证明了在正交群的p进半单对称空间上Siegel级数与球函数的联系公式。(2)证明了给定Ra上的一对满足局部泛函方程的齐次多项式和一对R^m到R^n的非退化对偶二次映射，则多项式的二次映射的回拉也满足局部泛函方程。这推广了Faraut-Koranyi-Clerc的结果，我们可以构造许多未被预齐次向量空间理论涵盖的泛函方程的例子。我们还对二次空间上的非退化对偶二次映射进行了分类，并证明了这种二次映射与两个Clifford代数的张量积的表示是一一对应的。

项目成果

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

局所密度の一次独立性とその保型形式の数論への応用

局域密度线性无关及其自守形式在数论中的应用

• 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;広中 由美子
• 通讯作者: 広中 由美子

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

Simple graded rings of Siegel modular forms, differential operators and Borcherds products.

Siegel 模形式、微分算子和 Borcherds 产品的简单分级环。

• 发表时间: 2005
• 期刊: Internat.J.Math. 16-no.3
• 作者: H.Aoki;T.Ibukiyama
• 通讯作者: T.Ibukiyama

Siegel modular forms of half-integral weight and conjectures on Shimura type correspondence and liftings

半积分重量的西格尔模形式以及志村式对应和提升的猜想

• 发表时间: 2006
• 作者: 雨宮;昭彦;H. Katsurada;簡 施儀;雨宮昭彦;柳 到亨;T. Ibukiyama;Eiichi Bannai;柳 到亨;坂内英一;柳 到亨;Eiichi Bannai;柳 到亨;柳 到亨;Eiichi Bannai;Eiichi Bannai and Etsuko Bannai,;石井 淳蔵;Eiichi Bannai,;加藤 司;T. Ibukiyama;T. Ibukiyama
• 通讯作者: T. Ibukiyama