Research on special values and zeros of L-funcions and on automorphic forms

L-函数的特殊值和零点以及自守形式的研究

基本信息

批准号：
13440007
负责人：
YOSHIDA Hiroyuki
金额：
$ 7.49万
依托单位：
Kyoto 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 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440007/
关键词：
CM-period Automorphic form L-function zeros of L-functions 周期モティーフ

项目摘要

Yoshida studied various problems centered around the absolute CM-period. Also he found a direct method to obtain cohomology classes from automorphic forms of a wide class, by decomposing the integral of Eichler-Shimura type. He organized his research results in a book, which was published by American Mathematical Society. He studied a p-adic analogue of the absolute CM-period, in collaboration with Ph. D student Tomokazu Kashio. There are strong evidences that a p-adic analogue holds in a perfect manner. A few years ago, Ikeda constructed a lifing from elliptic modular forms to Siegel modular forms of several variables. Taking as the kernel function the restriction to the diagonal of this lifting, he constructed a new lifing, which contains Miyawaki's lifting as a apecial case.. He formulated a conjecture which relates the nonvanishing of this lifting to special values of certain L-functions. Hiraga formulated a conjecture which relates the Zelevinskii involution and A-packet conjectured by Arthur. As an evidence, he proved the commutativity of endoscopic lifts and the Zelevinskii involution. He further studied Arthur's conjecture 'on automorphic representations. Umeda studied three problems on the center of the univeral enveloping algebra of a Lie algebra of classical type, namely concrete description of generating system, relations to the other generating system and explicit representations of a generation system. These problems originated from the Capelli identity which is the identity of invariant differential operators. Fujiwara studied Leopoldt's conjecture using the Taylor-Wiles-Fujiwara theory and arrived at the new point of view that number. fields and hyperbolic manifolds are analogous. Fujii studied on the Montgomery conjecture on the pair correlations of zeros, the Montgomery sum and higher 'moments of the argument of the Riemann zeta function.

吉田研究了围绕绝对CM周期的各种问题。他还发现了一种直接方法，通过分解 Eichler-Shimura 型积分，从宽类的自守形式中获得上同调类。他将自己的研究成果整理成书，由美国数学会出版。他与博士生 Tomokazu Kashio 合作，研究了绝对 CM 周期的 p-adic 类似物。有强有力的证据表明 p 进类似物以完美的方式成立。几年前，池田构建了从椭圆模形式到多个变量的西格尔模形式的生命周期。以这种提升对角线的限制为核函数，他构造了一种新的升维，其中包含宫胁提升作为特殊情况。他提出了一个猜想，将这种提升的不为零与某些 L 函数的特殊值联系起来。平贺提出了一个猜想，将 Zelevinskii 对合和 Arthur 猜想的 A 包联系起来。作为证据，他证明了内窥镜升降机和 Zelevinskii 对合的交换律。他进一步研究了亚瑟的“自守表征猜想”。梅田研究了经典型李代数的普遍包络代数中心的三个问题，即生成系统的具体描述、与其他生成系统的关系以及生成系统的显式表示。这些问题源于卡佩利恒等式，即不变微分算子的恒等式。藤原利用泰勒-怀尔斯-藤原理论研究了利奥波德猜想，得出了新的观点：数字。域和双曲流形是类似的。藤井研究了关于黎曼 zeta 函数论证的零点对、蒙哥马利和和高矩的蒙哥马利猜想。