权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Zeta functions on weakly spherical homogeneous spaces

弱球形均匀空间上的 Zeta 函数

基本信息

批准号：
09440024
负责人：
SATOU Fumihiro
金额：
$ 3.01万
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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440024/
关键词：
zeta function functional equation weakly spherical homogeneous space local density automorphic form Koecher-Maass zeta function Eisenstein series prehomogeneous vector spaces 二次形式 p-進球関数 Koecher-Maass級数 Siegel保型形式 Jacobi Eisenstein

项目摘要

1.Sato, the head investigator, studied the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneous vector spaces (PV for short) (SpinィイD210ィエD2 × GLィイD23ィエD2, half-spin 【cross product】 ΛィイD21ィエD2), (SLィイD25ィエD2 × GLィイD23ィエD2, ΛィイD22ィエD2 【cross product】 ΛィイD21ィエD2) and proved the functional equations satisfied by Eisenstein series attached to the spaces, which is a joint work with T. Kimura of Tsukuba Univ. and his students. As an application we can calculate explicitly the Fourier transforms of the complex powers of relative invariants of the PV's above. This shows that the theory of WSHS is very fruitful, since the structure of these two PV's is very complicated and even the method of micro local calculus can not be applied to them. We also made several attempts to generalize the theory to WSHS of reductive groups other than the general linear group. The result is still unsatisfactory ; however several suggestive partial results have been obtained.2.Hiron … More aka succeeded in calculating the explicit formula for spherical functions of spherical homogeneous spaces over p-adic fields in a rather general setting. Using the formula, she constructed the theory of spherical functions of the space of hermitian forms over unramified quadratic extensions of the base p-adic field and gave an explicit formula for local densities of hermitian forms. Moreover, jointly with Sato, she calculated local densities of quadratic forms over nondyadic p-adic fields explicitly in the most general setting ; the results can be extended to the space of hermitian forms over ramified quadratic extensions.3.Arakawa studied mainly Koecher-Maass zeta functions attached to Siegel modular forms and obtained some results including the following : (a) a generalization of Koecher-Maass zeta functions to Jacobi forms, (b) an explicit expression of the Koecher-Maass zeta function attached to the Nagaoka Eisenstein series of weight 1 of degree 2, which is obtained from p-adic Siegel Eisenstein series. Less

1.佐藤,研究员,研究了弱球面均匀空间(wsh)从prehomogeneous获得向量空间(PV)(自旋ィイD210ィエD2×GLィイc15ィエD2, half-spin【积】ΛィイD21ィエD2), (SLィイD25ィエD2×GLィイc15ィエD2,ΛィイD22摊位ィエD2【积】ΛィイD21ィエD2)和证明了功能方程满足艾森斯坦级数附加到空间,这是一个与t .木村共同工作的筑波大学和他的学生。作为一种应用，我们可以明确地计算上述PV的相对不变量的复幂的傅里叶变换。这表明WSHS理论是非常富有成效的，因为这两个PV的结构非常复杂，甚至连微局部微积分的方法都无法应用于它们。我们还尝试将该理论推广到除一般线性群以外的约化群的WSHS。结果仍然不令人满意；然而，已经获得了一些具有启发性的部分结果。Hiron…More aka在相当一般的情况下成功地计算了p进域上球面齐次空间的球面函数的显式公式。利用该公式，她在基p进域的未分枝二次扩展上构造了厄米形式空间的球函数理论，并给出了厄米形式局部密度的显式公式。此外，她与Sato一起，在最一般的情况下，明确地计算了非二进p进域上二次型的局部密度；所得结果可推广到二次分岔扩展上的厄米形式空间。Arakawa主要研究了附加在Siegel模形式上的Koecher-Maass zeta函数，得到了以下结果：(a)将Koecher-Maass zeta函数推广到Jacobi形式，(b)由p进Siegel Eisenstein级数得到附加在2次权值为1的Nagaoka Eisenstein级数上的Koecher-Maass zeta函数的显式表达式。少