Studies related with special values of various arithmetic functions

与各种算术函数的特殊值相关的研究

基本信息

批准号：
09640072
负责人：
ARAKAWA Tsuneo
金额：
$ 1.92万
依托单位：
RIKKYO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
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-09640072/
关键词：
Jacobi forms Eisenstein series Siegel modular forms Koecher Maass series functional equation multiple zeta values prehomogeneous vector space spherical homogeneous space Maass波形式保型形式 Siegel保型形式 Saito-Kurokawa lift Eisenstein級数 s

项目摘要

(1) (a) We defined the Koecher-Maass series attached to Jacobi forms following the case of Sieegel modular forms and proved the analytic continuation, the functional equation, and the explicit residue formula for them. (b) Using the relationship between Siegel modular forms of half integral weights and Jacobi forms we established the structure theorem for the so called plus space. This relationship also enabled us to formulate basic properties of the Koecher-Maass weries attached to modular forms in the plus space. (2) By the joint work with S. Bocherer we established the isomorphism from certain space of modular forms of weight one onto certain subspace of modular forms of weight 4 satisfying the Weierstrass condition, and also onto certain subspace of Jacobi forms of weight two. (3) By the joint work with M. Kaneko we extended multiple zeta values to multiple zeta funtions and expressed poly-Bernoulli numbers as special values of those multiple zeta functions at negative integer arguments. (4) Our cooperator, Sato succeeded in computing the functional equation corresponding to the prehomogeneous vector space obtained from the tensor product of certain representation of SLィイD25ィエD2 and that of GLィイD23ィエD2 from a view point of weakly spherical homogeneous spaces. This space is the case in which micro local calculus, an effective tool to obtain the functional equation of various prehomogeneous vector spaces, cannot be applicable. We could make clear the effectiveness of our viewpoint.

（1）（a）我们定义了Siegel模块化形式的情况下，定义了与雅各比形式相关的koecher-maass系列，并提供了分析延续，功能方程以及为其提供明确的保留公式。（b）使用一半积分权重和雅各比形式的siegel模块化形式之间的关系，我们建立了所谓的Plus空间的结构定理。这种关系还使我们能够通过与S. Bocherer的联合合作来制定与模块化（2）相关的Koecher-maass Weies的基本特性，我们确定了从模块化的某些重量的某些空间中的同构，一种重量的某些空间，即某些模块空间的重量的某些子空间4，重量4，以及满足Weierstrass条件的某些子空间，以及对Jacobi形式的某些jacobi形式的子空间。（3）通过与M. Kaneko的联合合作，我们将多个Zeta值扩展到多个Zeta函数，并将poly-Bernoulli数字作为负整数参数的这些多个Zeta函数的特殊值。（4）我们的教练SATO成功地计算了从SLI D25的某些表示和Glii D23 D2的张量产物中获得的近似矢量空间，从弱球体均质空间的角度来看。在这个空间中，微微积分是获得各种均匀矢量空间的功能方程的有效工具，因此不能适用。我们可以清楚地表明我们观点的有效性。