| Project/Area Number |
09640064
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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 | Chuo University |
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Principal Investigator |
MOMOSE Fumiyuki Chuo University, Faculty of Science & Eng., Professor, 理工学部, 教授 (80182187)
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| Co-Investigator(Kenkyū-buntansha) |
HASEGAWA Yuji Waseda University, J.S.P.S., 理工学部, 学振特別研究員 (30287982)
HASHIMOTO Ki-ichirou Waseda University, Professor, 理工学部, 教授 (90143370)
SEKIGUCHI Tsutomu Chuo University, Faculty of Science & Eng., Professor, 理工学部, 教授 (70055234)
佐武 一郎 中央大学, 理工学部, 教授 (00133934)
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| Project Period (FY) |
1997 – 1999
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| Project Status |
Completed (Fiscal Year 1999)
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| Budget Amount *help |
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Modularity Conjecture / Moduli / Abelian variety / GLィイD22ィエD2-type / Galois representation / モジュラー予想 / モジュラリティ / (1)-curve / QM-curue |
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| Research Abstract |
The modularity conjecture (Serre-Ribet) states that an abelian variety of GLィイD22ィエD2-type over Q is isogenous (over QィイD4-ィエD4) to a Q-simple factor of the jacobian variety of the modular courve XィイD21ィエD2 (N) for some integer N. We study this conjecture. Applying the deformation theory of Galois representation of Wiles, Taylor, Diamond and Gonrad, we proved this conjecture for the objects which have the extra twistings under some conditions. For a given prime ideal, if the objects has potentially "ordinary" reduction, we could improve the condition of modularity, using the results of Skinner-Wiles. If the objects has potentially good reduction of height 2, we proved the modularity under (natural) condition, using the results of Conrad-Diamond-Taylor.
We also got the moduli of the abelian varieties of GLィイD22ィエD2-type over Q (up to isogeny). We made use of the quotient of Shimura varieties of GLィイD22ィエD2-type bye some group of automorphsms (over QィイD4-ィエD4). We generalized the local tree of Elkies, and combined it with the descent up to isogeny of Ribet-Pyle. Further, we studies the arithmetic geometry of the moduli space (over Z). Using these results, we can apply the modularity conditions.
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