| Project/Area Number |
08640023
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 Univ.graduate school of Mathematics, Associate prof., 大学院・多元数理科学研究科, 助教授 (00229064)
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| Co-Investigator(Kenkyū-buntansha) |
MUKAI Shigeru Nagoya Univ.graduate school of Math.prof., 大学院・多元数理科学研究科, 教授 (80115641)
NAMIKAWA Yukihiko Nagoya Univ.graduate school of Math.prof., 大学院・多元数理科学研究科, 教授 (20022676)
KITAOKA Yoshiyuki Nagoya Univ.graduate school of Math.prof., 大学院・多元数理科学研究科, 教授 (40022686)
UMEMURA Hiroshi Nagoya Univ.graduate school of Math.prof., 大学院・多元数理科学研究科, 教授 (40022678)
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| Project Period (FY) |
1996 – 1997
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| Project Status |
Completed (Fiscal Year 1997)
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| Budget Amount *help |
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1997: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1996: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | Galois representation / Automorphic representation / Hecke algebra / 楕円曲線 |
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| Research Abstract |
I have studied the Iwasawa theory and Langlands conjecture over number fields, motivated by A.Wiles' work. Here the identification of deformation rings of Galois representations and Hecke algebras (called Mazur conjecture) plays a central role. I have shown that the Hecke algebra of GL (2) over a totally real field is a local complete intersection ring, and is identified with a universal deformation ring of a mod p modular representation.
The essential step in the work is, that the freeness property of a cohomology group of a modular curve over a Hecke ring (used essentially in Taylor-Wiles work) is a consequence of axioms which are easier to verify. I have named the axioms Taylor-Wiles system, in analogy with Euler systems (I should note that a similar idea was found independently by F.Diamond). By constructing a Taylor-Wiles system by Shimura curves, the Mazur conjecture over general totally real fields is proved. By combining the result with a level optimization argument (the even degree case of the Mazur principle is most difficult), many two dimensional 1-adic Galois representations correspond to automorphic representations, thus verifying the Langlands correspondence in these cases. Especially, a generalization of Taniyama-Shimura conjecture is shown fairly generally. There is a report on this work, including recent results. The detail is distributed as a preprint (Deformation rings and Hecke algebras in the totally real case), submitted to a journal, and 2 other articles are under preparation.
Even in case of reducible residual representations, it is shown that there are infinitely many reducible representations which satisfy the Mazur conjecture, when the totally real field is fixed.
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