2003 Fiscal Year Final Research Report Summary
Research of Iwasawa Theory fof Formal Groups
| Project/Area Number |
12640038
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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 | Gakushuin University |
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Principal Investigator |
NAKAJIMA Shoichi Gakushuin Univ., Fac.of Sci., Professor, 理学部, 教授 (90172311)
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| Co-Investigator(Kenkyū-buntansha) |
ICHIMURA Humio Yokohama City Univ., Fac.of Sci., Professor, 理学部, 教授 (00203109)
NAITO Hirotada Kagawa Univ., Fac.of Education, Professor, 教育学部, 教授 (00180224)
NAKANO Shin Gakushuin Univ., Fac.of Sci., Associate Professor, 理学部, 助教授 (40180327)
KAWAMOTO Fuminori Gakushuin Univ., Fac.of Sci., Assistant, 理学部, 助手 (50195161)
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| Project Period (FY) |
2000 – 2003
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| Keywords | Iwasawa Theory / Formal Group / Elliptic Curve |
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| Research Abstract |
Iwasawa Theory started as a method for investigating (a tower of) cyclotomic fields and soon was generalized to arbitrary number fields. Recently, Iwasawa Theory for elliptic curves has been constructed and has developed extensively. Iwasawa Theory for number fields is related to multiplicative group and Iwasawa Theory for elliptic curves is naturally related to the group structure of elliptic curves. Our aim in this project was to generalize Iwasawa Theory for formal groups, which include multiplicative and elliptic curve groups as special cases. For that purpose, we first investigated
(1) general theory of formal groups
and, as important examples of the theory,
(2) Iwasawa Theory of number fields (multiplicative group),
(3) Iwasawa Theory of elliptic curves (elliptic curve group).
For (1) we mainly considered Honda's theory which classified formal groups over the integer ring. We also tried numerical computation for formal groups using Maple. For (2) our main concern was Greenberg's conjecture and its relation to normal integral bases. For (3) we obtained results for mu-invariants of elliptic curves, extending the former results of R.Greenberg.
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