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1998 Fiscal Year Final Research Report Summary

Study of Iwasawa Theory for Cyclotomic Fields.

Research Project

Project/Area Number 09640054
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionYokohama City University

Principal Investigator

ICHIMURA Humio  Yokohama City Univ., Faculty of Science, Assoc.Prof., 理学部, 助教授 (00203109)

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)
Project Period (FY) 1997 – 1998
Keywordsabelian field / cyclotomic field / ideal class group / class number / Gauss sum / 円分関数体
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.

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] 市村文男: "Class numbers of real quadratic function fields of genus one" Fimite Fields and Their Applications. 3. 181-185 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 市村文男: "On the Iwasawa invariants of certain real abelian fields" Tohoku Mathematical Journal. 49. 203-215 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 市村文男: "A note on Greenberg's conjecture and the abc conjecture" Proceedings of the American Mathematical Society. 126. 1315-1320 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 市村文男: "Local units modulo Gauss sums" Journal of Number Theory. 68. 36-56 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 八森祥隆: "Semi-local units modulo Gauss sums" Manusripta Mathematica. 95. 377-395 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 市村文男: "On the class numbers of the maximal real subfields of cyclotomic function fields" Finite Fields and Their Applications. 4. 167-174 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 市村文男: "On the class numbers of the maximal real subfields of cyclotomic function fields II" Journal of Number Theory. 72. 140-149 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小屋良祐: "The Block-Kato conjecture for good reduction curues over local fields" K-theory. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ichimura, H.: "Class numbers of real quadratic function fields of genus one." Finite Fields and Their Appl.3. 181-185 (1977)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ichimura, H.: "On the Iwasawa invariants of certain real abelian fields." Tohoku Math.J.49. 203-215 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ichimura, H.: "A note on Greenberg's conjecture and the abc conjecture." Proc.Amer.Math.Soc.126. 1315-1320 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ichimura, H.: "Local units modulo Gauss sums." J.Number Theory. 68. 36-56 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hachimori, Y.: "Semi-local units modulo Gauss sums." Manuscripta Math.65. 377-395 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ichimura, H.: "On the class numbers of the maximal real subfields of cyclotomic function fields." Finite Fields and Their Appl.4. 167-174 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ichimura, H.: "On the class numbers of the maximal real subfields of cyclotomic function fields II." J.Number Theory. 72. 140-149 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Koya, Y.: "The Block-Kato conjecture for good reduction curves over local fields." K-Theroy. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1999-12-08  

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