Research on periods, L-functions and automorphic forms

周期、L-函数和自守形式的研究

基本信息

批准号：
16340006
负责人：
YOSHIDA Hiroyuki
金额：
$ 10.25万
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：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

Yoshida studied, in collaboration with T. Kashio, an p-adic analogue of absolute CM-period, which was introduced byYoshida in 1997. First we defined p-adic absolute period symbol in general. In the complex case, this symbol is conjectured to be equal to Shimura's period symbol. In the p-adic case, we have new features. If the prime ideal given in the base field splits completely, then we can predict the exact value of the p-adic absolute CM-period symbol. We can prove that this gives a refinement of a conjecture of Gross, which is a p-adic analogue of the Stark-Shintani conjecture. In the general case, we studied the relation of our symbol to the p-adic periods in detail. Yoshida studied the problem of generalizing the Shimura-Taniyama conjecture to an arbitrary motive and formulated a precise conjecture.Ikeda, in collaboration with Hiraga and Atsushi Ichino, formulated a conjecture relating the formal degree and the gamma factor of the adjoint L-function of a representation of a p-adic reductive group. They proved the conjecture in several interesting cases. Ikeda, aldo in collaboration with Ichino, gave a refinement of the Gross-Prasad conjecture, which concerns the restriction of a representation of an orthogonal group to a smaller orthogonal group. This conjecture has a very interesting form involving special values of L-functions.Hiraga studied L-packet which is basic in representation theory of an algebraic group over a p-adic field. He succeeded to determine the L-packets for SL(n) in collaboration with Hiroshi Saito. Fujii studied relations between Farey series and the Riemann hypothesis.

Yoshida 与 T. Kashio 合作研究了绝对 CM 周期的 p-adic 类似物，由 Yoshida 于 1997 年提出。首先我们一般定义了 p-adic 绝对周期符号。在复杂的情况下，这个符号被推测等于志村的时期符号。在 p-adic 情况下，我们有新的特征。如果基域中给出的素理想完全分裂，那么我们可以预测 p 进绝对 CM 周期符号的精确值。我们可以证明这给出了 Gross 猜想的改进，它是 Stark-Shintani 猜想的 p-adic 类似物。在一般情况下，我们详细研究了符号与 p 进周期的关系。吉田研究了将 Shimura-Taniyama 猜想推广到任意动机的问题，并提出了一个精确的猜想。池田与 Hiraga 和 Atsushi Ichino 合作，提出了一个与形式度和伴随 L 函数的伽马因子相关的猜想。 p-adic 还原基团的表示。他们在几个有趣的案例中证明了这个猜想。池田、奥尔多与一野合作，对格罗斯-普拉萨德猜想进行了改进，该猜想涉及将正交群的表示限制为更小的正交群。这个猜想有一个非常有趣的形式，涉及L-函数的特殊值。平贺研究了L-包，它是p进数域上代数群表示论的基础。他与 Hiroshi Saito 合作成功确定了 SL(n) 的 L 数据包。藤井研究了法雷级数和黎曼假设之间的关系。