Research on periods, L-functions and automorphic forms

Yoshida与T.Kashio合作研究了绝对CM周期的p-进模拟，这是由Yoshida在1997年引入的。首先，我们定义了一般的p进绝对周期符号。在复数情况下，该符号被猜想为等于下村的周期符号。在p-addy的情况下，我们有了新的特征。如果基域中给出的素数理想完全分裂，则我们可以预测p进绝对CM周期符号的精确值。我们可以证明，这给出了Gross猜想的一个精化，该猜想是Stark-Shintani猜想的p-进类比。在一般情况下，我们详细地研究了我们的符号与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。Ikeda与Hiraga和Atsushi Ichino合作，提出了一个关于p-进还原群表示的伴随L函数的形式度和伽马因子的猜想。他们在几个有趣的案例中证明了这个猜想。Ikeda，Aldo和Ichino合作，给出了Gross-Prasad猜想的一个改进，该猜想涉及到正交群的表示到更小的正交群的限制。这个猜想有一种非常有趣的形式，涉及到L函数的特殊值。Hiraga研究了L包，它是p-ady域上代数群表示理论的基础。他与斋藤博史合作，成功地确定了SL(N)的L包。藤井研究了费雷级数和黎曼假设之间的关系。

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.