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.首席研究员佐藤研究了从预齐次向量空间（简称PV）获得的弱球齐次空间（简称WSHS）（SpinD210D2×GLD23D2，半自旋[叉积]ΛD21D2），（SLD25D2×GLD23D2， ΛiiD22ЛD2 [叉积]ΛЛD21ЛD2) 并证明了 Eisenstein 级数所满足的空间方程，这是与筑波大学的 T. Kimura 及其学生合作的成果。上述PV的相对不变量的复数幂的变换这表明WSHS理论是非常富有成果的，因为这两个结构。 PV非常复杂，甚至不能应用微观局部微积分的方法。我们也多次尝试将该理论推广到一般线性群以外的还原群，但结果仍然不令人满意。 2.Hiron … More 又名成功地在相当一般的环境中计算了 p-adic 场上球面齐次空间的球面函数的显式公式，她利用该公式构建了的理论。她计算了基 p-adic 场的未分支二次扩展上的厄米式空间的球函数，并给出了厄米式局部密度的显式公式。此外，她与 Sato 一起计算了非二进 p-adic 场上的二次型局部密度。明确地在最一般的设置中；结果可以扩展到分支二次扩展上的埃尔米特形式的空间。3.荒川主要研究Koecher-Maass zeta 函数附加到 Siegel 模形式并获得了一些结果，包括以下结果：(a) Koecher-Maass zeta 函数到 Jacobi 形式的推广，(b) Koecher-Maass zeta 函数附加到 Nagaoka 形式的显式表达式2 次权重 1 的艾森斯坦级数，由 p 进 Siegel Eisenstein 级数获得。