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

• 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

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.

吉田研究了以绝对cm周期为中心的各种问题。他还通过分解Eichler-Shimura型积分，找到了一种从广义类的自同构形式得到上同调类的直接方法。他把自己的研究成果整理成书，由美国数学学会出版。他与博士生木尾知和（Tomokazu Kashio）合作，研究了绝对cm周期的p进式模拟。有强有力的证据表明，一个p进的类比以一种完美的方式成立。几年前，Ikeda构造了一个从椭圆模形式到多变量西格尔模形式的提升。他以这种举升对角线的限制为核函数，构造了一个新的举升，其中以宫崎举升为特例。他提出了一个猜想，将这种提升的不消失与某些l函数的特殊值联系起来。平贺提出了一个将泽列文斯基对合和阿瑟的a包猜想联系起来的猜想。作为证据，他证明了内窥镜升降机的交换性和Zelevinskii对合。他进一步研究了亚瑟关于自同构表征的猜想。Umeda以经典型李代数的一般包络代数为中心研究了三个问题，即生成系统的具体描述、与其他生成系统的关系和生成系统的显式表示。这些问题起源于Capelli恒等式，即不变微分算子的恒等式。藤原用泰勒-怀尔斯-藤原理论研究了利奥波德猜想，得出了这个数字的新观点。场和双曲流形是类似的。Fujii研究了Riemann zeta函数的零对相关、Montgomery和和高阶矩的Montgomery猜想。

项目成果

Akio Fujii: "On the pair correlation of the zeros of the Riemann zetu function"Analytic number theory (published by Klywer). 127-142 (2002)

Akio Fujii：“论黎曼zetu函数零点的成对相关性”解析数论（Klywer出版）。

Hiroyuki Yashida: "Absolute CM-periods"American Mathmutical Society. (2003)

Hiroyuki Yashida：“绝对 CM 周期”美国数学学会。

Akio Fujii: "Values if the Epstein zeta functions"J. of Math. Kyoto University. 41. 627-667 (2002)

Akio Fujii：“Epstein zeta 功能的价值”J.

藤井昭雄: "On the pair correlation of the zeros of the Riemann zeta function"Analytic Number Theory(published by Kluwer). 127-142 (2002)

Akio Fujii：“论黎曼 zeta 函数零点的配对相关性”解析数论（Kluwer 出版）127-142（2002 年）。

Kaoru Hiraga: "Onfunctoriality of Zelevinski involution"Compositio Moth. (in press).

平贺薰（Kaoru Hiraga）：“论泽列温斯基对合的函数性”复合蛾。