1998 Fiscal Year Final Research Report Summary
Study of Iwasawa Theory for Cyclotomic Fields.
| Project/Area Number |
09640054
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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 | Yokohama City University |
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Principal Investigator |
ICHIMURA Humio Yokohama City Univ., Faculty of Science, Assoc.Prof., 理学部, 助教授 (00203109)
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| Co-Investigator(Kenkyū-buntansha) |
KOYA Yoshihiro Yokohama City Univ., Faculty of Science, Res.Assoc., 理学部, 助手 (50254230)
NAITO Hirotada Kagawa Univ., Faculty of Education, Assoc.Prof., 教育学部, 助教授 (00180224)
NAKAJIMA Shoichi Gakushuin Univ., Faculty of Science, Professor, 理学部, 教授 (90172311)
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| Project Period (FY) |
1997 – 1998
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| Keywords | abelian field / cyclotomic field / ideal class group / class number / Gauss sum / 円分関数体 |
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| Research Abstract |
During 1997 - 1998, I obtained several results on the ideal class groups of real cyclotomic (abelian) fields. Here, I summarize the main ones.
1. Let 1 be a prime number, k an imaginary abelian fields (satisfying some conditions), and k_*/k the cyclotomic Z_l-extension. The l-part A_* of the ideal class group of k_* is decomposed as A_*=A_*^+ <symmetry> A_*^- by the action of complex conjugation. The structure of the odd part A_*^- had been determined by Mazur and Wiles. As for the even part A_*^+, it is conjectured that A_*^+ is a finite group. At present, this conjecture is far to be solved. Let U be the group of semi-local units at 1 of k_*. I defined a subgroup G of U generated by certain Gauss sums, and proved that A_*^+ and the quotient U/G have the "same" Galois module structure. I hope that this result sheds some light on the difficult conjecture on A_*^+.
2. For a prime number 1, let h_l^+ be class number of the real l-cyclotomic field. It is conjectured that for any N, there exist infinitely many primes 1 with h_*^+ > N.But, this is not yet proved to be true. I proved that a function field analogue of this conjecture holds.
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