2007 Fiscal Year Final Research Report Summary
Research on periods, L-functions and automorphic forms
| Project/Area Number |
16340006
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyoto University |
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Principal Investigator |
YOSHIDA Hiroyuki Kyoto University, Graduate School of Science, Professor (40108973)
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| Co-Investigator(Kenkyū-buntansha) |
IKEDA Tamotsu Kyoto Univ, Graduate School of Science, Professor (20211716)
UMEDA Toru Kyoto Univ, Graduate School of Science, Associate Professor (00176728)
HIRAGA Kaoru Kyoto Univ, Graduate School of Science, Lecturer (10260605)
FURUSAWA Masaaki Osaka City Univ, Graduate School of Science, Professor (50294525)
FUJII Akio Rikkyo Univ, Fuculty of Science, Professor (50097226)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Period / Automorphic form / L-function |
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| Research Abstract |
Yoshida studied, in collaboration with T. Kashio, an p-adic analogue of absolute CM-period, which was introduced by
Yoshida 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.
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