Study of zeta functions related to Jacobi forms and Siegel modular forms

雅可比形式和西格尔模形式相关的 zeta 函数研究

基本信息

批准号：
12640045
负责人：
ARAKAWA Tsuneo
金额：
$ 1.66万
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：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640045/
关键词：
modular forms Jacobi forms Selberg zeta functions Maass wave forms prime geodesic theorem multiple L-values prehomogeneous vector space b-function Selberg跡公式 Fermat曲線 Hodge予想 Maass wave forms Selberg zeta 関数 Shimura 対応多重L-値

项目摘要

1. On Siegel modular forms : We formulated the converse theorem of Imai in such a manner that it can be applica* to not necessarily cuspidal Siegel modular forms of degree two. As an application we reconstructed the Saito-Kurokawa lifting for Siegel modular forms of degree two by using Duke-Imamogle's method (joint work with the coworker F.Sato and I.Makino).2. Solution of the Hashimoto conjecture : We solved the conjecture presented by K.Hashimoto concerning the that series associated to definite quaternion algebras over Q by using Jacobi forms, the theorem of Deligne-Serre on modular forms of weight one (joint work with S.Boecherer).3. The study of Selberg zeta functions : The head investigator studied the Shimura correspondence for Maass wave forms via Selberg trace formulas and Selberg zeta functions and reduced the problem of bijectivity of the correspondence to a simple relation of the associated two Selberg zeta functions. On the other hand he with the co-operation of S.Koyama and M.Nakasuji obtained explicit forms of the arithmetic Selberg zeta functions attached to indefinite quaternion aogebras over Q and the prime geodesic theorem of Brun-Titchmarsh type.4. On multiple L-values : The head investigator and M.Kaneko formulated the (regularized) double shuffle relations for multiple L-values. We also determined the principal part at the pole s=0 of the associated multiple L-function by using the zeta regularization.5. The study of prehomogeneous vector spaces : The coworker F.Sato and K.Sugiyama showed that the b-function also has the decomposition corresponding to that of the associated representations of the vector space.

1。在西格尔模块化形式上：我们以这种方式制定了iMai的匡威定理，使得不一定是cuspidal siegel siegel模块化的第二学位。作为应用程序，我们通过使用Duke-Mimamogle的方法（与同事F.Sato和I.Makino的联合工作）重建了Siegel模块化二级模块化形式的Saito-Kurokawa举重。2桥本猜想的解决方案：我们解决了K.hashimoto提出的猜想，涉及与Q上确定的四元基因代数相关的该系列的序列，该序列是使用jacobi形式，即DeLigne-serre的定理，在重量的模块化形式上（与S.Beecherer的联合工作））.3。 Selberg Zeta功能的研究：首席研究员通过Selberg Trace Formulas和Selberg Zeta函数研究了Maass Wave形式的Shimura对应关系，并降低了对应关系与相关两个Selberg Zeta功能的简单关系的对应关系的问题。另一方面，他通过s.koyama和M.nakasuji的合作获得了Q的算术Selberg zeta函数的明确形式，该函数附在Q上不确定的Quaternion aogebras和Brun-titchmarsh类型的原始Quodesic定理上。4。在多个L值：首席调查员和M.Kaneko为多个L值制定了（正则）双层式关系。我们还使用Zeta正则化5。固定前矢量空间的研究：同事F.Sato和K.Sugiyama表明，B功能还具有与矢量空间相关表示相对应的分解。