2004 Fiscal Year Final Research Report Summary
Arithmetic study of Shimura varieties
| Project/Area Number |
13440004
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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 | Nagoya University |
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Principal Investigator |
FUJIWARA Kazuhiro Nagoya University, Graduate school of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (00229064)
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| Co-Investigator(Kenkyū-buntansha) |
UZAWA Tohru Nagoya University, Graduate school of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (40232813)
KONDO Shigeyuki Nagoya University, Graduate school of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (50186847)
SAITO Shuji University of Tokyo, Graduate school of Mathematical Sciences, professor, 大学院・数理科学研究科, 教授 (50153804)
SAITO Takeshi University of Tokyo, Graduate school of Mathematical Sciences, professor, 大学院・数理科学研究科, 教授 (70201506)
MUKAI Shigeru Kyoto University, Research Institute of Mathematical Sciences, professor, 数理解析研究所, 教授 (80115641)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Shumura variety / non-abelian class field theory / rigid geometry / hyperbolic geometry / Taniyama-Shimura conjecture / automorphic forms |
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| Research Abstract |
The head investigator has been studying a non-abelian class field theory (Langlands program) via arithmetic geometry of Shimura varieties. Here is a brief summary of the research during the period.
1.By applying the non-abelian class field theory to algebraic number theory, we have got a new insight on Leopoldt conjecture in algebraic number theory. Especially, a connection between A.Wiles' work on Taniyama-Shimura conjecture for elliptic curves and W.Thurston's three dimensional hyperbolic geometry was found. This connection is closely related to an analogy between knots and primes dur to B.Mazer and M.Morishita, showing the variety of the related research areas. This is reported at international conferences hold at Paris 13(2002), Nagoya(2003). The abstract is published, and more detailed article is under preparation.
2.During the research period, p-adic description of Shimura varieties has been developed worldwide. Motivated by this, the author is trying to establish a new foundation of rigid geometry with F.Kato(Kyoto). This work generalizes the framework of rigid geometry, giving more solid foundation for the application. The detail is under preparation, and will be published as a book in near future.
3.The head investigator stayed one month and half at University Paris 7 from May 2002,and also University Paris 13 in September 2002. There were many research activities including discussions with M.F.Vigneras(Paris 7), J.P.Labesse(Paris 7), M.Harris, (Paris 7), A.M.Aubert(Ecole Normal)J.Tilouine(Paris 13), especially on automorphic forms and representation theory of p-adic groups. Our group is also making collaboration with P.Colmez(Paris 6) and J.Nekovar(Paris 7) from more number theoretical viewpoint using p-adic methods. For these collaborations, the grant is used effectively.
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