| Project/Area Number |
11640040
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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 | Tokyo Metropolitan University |
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Principal Investigator |
KURIHARA Masato TMU, Faculty of Science, Associate Professor, 理学(系)研究科(研究院), 助教授 (40211221)
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| Co-Investigator(Kenkyū-buntansha) |
KURANO Kazuhiko TMU, Faculty of Science, Associate Professor, 理学(系)研究科(研究院), 助教授 (90205188)
NAKAMULA Ken TMU, Faculty of Science, Professor, 理学(系)研究科(研究院), 教授 (80110849)
MIYAKE Tatsuya TMU, Faculty of Science, Professor, 理学(系)研究科(研究院), 教授 (20023632)
MATSUNO Kazuo TMU, Faculty of Science, Assistant, 理学(系)研究科(研究院), 助手 (40332936)
TAKEDA Yuichiro TMU, Faculty of Science, Assistant, 理学(系)研究科(研究院), 助手 (30264584)
中村 博昭 東京都立大学, 理学研究科, 助教授 (60217883)
宮崎 琢也 東京都立大学, 理学研究科, 助手 (10301409)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | Iwasawa theory / elliptic curve / Tate-Shafarevich group / supersingular reduction / テイトーシャファレビッチ群 / 岩澤不変量 / 岩澤主予想 / ティトーシャファレビッチ群 / スーパーシンギュラーリダクション |
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| Research Abstract |
B. Mazur began to construct in 1970's Iwasawa theory for elliptic curves which generalizes the work of Iwasawa on ideal class groups to Selmer groups and Tate Shafarevich groups. If an elliptic curves has ordinary reduction at every prime above p, we have a sufficient Iwasawa theory which describes the relation between the Selmer group and the p-adic L-function. But if it does not have ordinary reduction, nothing had been known for a long time. In our research, at first we considered an elliptic curve over the rational number field which has supersingular reduction at p. We determined the Galois wodule structure of the Selmer groups over the intermediate fields of the cyclotomic Z-extension of the rational number field, in particular we determined the orders of the p-components of the Tate-Shafarevich groups over them. Our new discovery is that they can be described by using fractional invariants though in usual Iwasawa theory one uses integer invariants. We constructed a conjectuve which describes how the orders grow in general case, and proved it in some special cases. We found that the distribution relation and the Galois module structure of the local Mordell Weil group are important to understand this phenomenon. We also constructed a homomovphism which sends the zeta element by K. Kato to the modular element by Mazuv and Tate, which gives the relation between the p-adic analytic side and the p-adic algebraic side.
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