Study on automorphic forms on algebraic groups and associated zeta functions

代数群自守形式及相关zeta函数的研究

基本信息

批准号：
13440016
负责人：
MURASE Atsushi
金额：
$ 7.81万
依托单位：
Kyoto Sangyo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

1. Metaplectic representations of unitary groups :We studied metaplectic representations of unitary groups over local fields and gave their "universal" splitting, which are in particular useful in the study of theta lifting. As an application of this result, we gave an explicit character formula for metaplectic representations.2. Fourier-Jacobi expansion of automorphic form on unitary groups of degree three :We reformulated Shintani' s theory on Fourier-Jacobi expansion of automorphic forms on unitary groups of degree three in adelic language, and calculated explicit form for Fourier-Jacobi expansion of Eisenstein series and Kudla lifts, theta lifts from elliptic modular forms. As an application, we gave a criterion for the non-vanishing of Kudla lifts.3. Siegel-Weil formula :We studied a non-regularized Siegel-Weil formula in the case of the dual reductive pair (U(2,2), U(2, 1)).4. Inner product formula for Kudla lifts :Using the formula stated in 3, we gave an explicit formula for the Petersson norms of Kudla lifts in term of special values of automorphic L-functions. As an application, we gave a criterion for the non-vanishing of Kudla lifts different from the one given in 2. (The studies 2- 4 are joint works with Takashi Sugano).5. Support for the Summer School of Number Theory :We supported financially the Summer School of Number Theory held annually.The themes were as follows : "Zeta functions" in 2001, "Prehomogeneous vector spaces" in 2002, "Iwasawa theory" in 2003 and "Fundamental groups and Galois representations" in 2004.

1。统一群体的互惠表示：我们研究了本地领域的单一群体的元容器表示，并进行了“普遍”分裂，这在theta提升的研究中特别有用。作为此结果的应用，我们给出了一个明确的字符公式用于元容器表示2。自动形式的傅里叶 - 雅各比在三分之二的群体上扩展：我们在单一的三分之一群体上以阿德语语言的单一群体对傅里叶 - 雅各比的扩展进行了重新制定的理论，并计算出Eellier-jacobi扩展Eisenstein Series and Kudla Lifts的傅立叶 - 雅各比的明确形式。作为应用程序，我们给出了非kudla升降机的标准。3。 Siegel-Weil公式：在双还原对（U（2,2），U（2，1））的情况下，我们研究了一个非规范化的Siegel-Weil公式。4。 Kudla Lifts的内部产品公式：使用3中陈述的公式，我们为kudla升降机的彼得森型规范提供了明确的公式，以特殊的自动形态L功能的特殊值。作为应用程序，我们给出了与2所给出的库德拉举报的不变的标准。对暑期数理论的支持：我们在财务上支持每年举行的暑期数学理论。主题如下：2001年的“ Zeta函数”，“ 2002年的“均匀矢量空间”，“ 2003年的iWasawa Theory”，2003年的iWasawa Theopart和“基本团体和Galois表示”。