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Research on ideal class groups of algebraic number fields and number theoretic functions and its applications

Research Project

Project/Area Number 16540007
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTohoku University

Principal Investigator

TAYA Hisao  Tohoku University, Graduate School of InformationSciences, Research Associate, 大学院情報科学研究科, 助手 (40257241)

Co-Investigator(Kenkyū-buntansha) MUNEMASA Akihiro  Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (50219862)
IMAI Hideo  Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院情報科学研究科, 助教授 (10093668)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordsalgebraic number fields / class numbers / ideal class groups / Iwasawa invariants / Zp-extensions / natural density / abelian extensions / central extensions / アーベル体 / 組合せ構造 / アルゴリズム / 格子 / p進L関数 / デザイン / 局所体
Research Abstract

The purpose of this research is to develop the fundamental theory of ideal class groups of algebraic number fields further, especially taking account of Iwasawa theory. Though we did not find out a theoretical behavior of p-ambiguous ideal class groups in the first year of this research, we verified that our estimate on the density of real quadratic fields whose Iwasawa invariants for p=3 are all zero is very near to the conjectural ratio obtained by modifying Cohen-Lenstra Heuristics on class numbers. As compared with the case p>3 in which known estimates are too far from the conjectural one, our analysis is interesting. Next year, we showed that the density of real quadratic fields whose Iwasawa invariants for p=2 are all zero is zero. This is also interesting, because the density for p=3 is more than 0.7 by our previous result and the one for p>3 is conjectured to be positive. From that time to the last year of this research, we had an opportunity to have a joint research with Dr. Gen Yamamoto (Tokyo Denki University) and determined all real abelian 2-extension fields whose Iwasawa invariants for p=2 are all zero, by using genus theory and the theory of central extensions. The results obtained in this research are presented in some of conferences like as Korea-Japan Number Theory Seminar and ICM2006. Though we did not get a satisfying result about mutual application between number theory and combinatorial theory unfortunately, we played an important role in posing and solving common problems in both fields by organizing mini conferences where researchers in both fields gathered

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (20 results)

All 2007 2006 2005 2004

All Journal Article (20 results)

  • [Journal Article] λ_2=μ_2=ν_2=0となる実アーベル2-拡大体について2007

    • Author(s)
      田谷 久雄
    • Journal Title

      第1回・第2回室蘭数論研究集会報告集 1&2

      Pages: 146-161

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On real abelian 2-extension fields with λ2=μ 2= v 2=0 (Japanese)2007

    • Author(s)
      Hisao Taya
    • Journal Title

      Proceedings of the 1st and 2nd Muroran Number Theory Symposium Vol.1&2

      Pages: 146-161

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] λ_2=μ_2=ν_2=0となる実アーベル2-拡大体について2007

    • Author(s)
      田谷 久雄
    • Journal Title

      第1回・第2回室蘭数論研究集会報告集 1

      Pages: 146-161

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Notes on certain real abelian 2-extension fields with λ_2=μ_2=ν_2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Trends in Mathematics 9・1

      Pages: 81-89

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Certain real abelian 2-extensions with λ_2=μ_2=ν_2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Abstracts of International Congress of Mathematicians 2006, Madrid, Poster sessions

      Pages: 29-30

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On certain real abelian fields with λ_2=μ_2=ν_2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Proceedings of Korea-Japan Number Theory Seminar 2006, KAIST

      Pages: 77-81

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Notes on certain real abelian 2-extensions fields with λ2= μ2= v2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Trends in Mathematics 9(2006), no.1

      Pages: 81-89

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Certain real abelian 2-extensions with λ2=μ2 = v 2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Abstracts of International Congress of Mathematicians 2006,Madrid, Poster sessions 2006

      Pages: 29-30

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On certain real abelian fields with λ2=μ2= v2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Proceedings of Korea-Japan Number Theory Seminar 2006, KAIST 2006

      Pages: 77-81

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Notes on certain real abelian 2-extension fields with λ_2=μ_2=ν_2=02006

    • Author(s)
      Hisao Taya
    • Journal Title

      Trends in Mathematics 9

      Pages: 81-89

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Hyperplane partitions and difference systems of sets2006

    • Author(s)
      Ryoh Fuji-Hara
    • Journal Title

      Journal of Combinatorial Theory, Ser. A 113

      Pages: 1689-1698

    • Related Report
      2006 Annual Research Report
  • [Journal Article] 代数体の類数の非可除性について2005

    • Author(s)
      田谷 久雄
    • Journal Title

      第3回北陸数論小研究集会報告集 3

      Pages: 60-83

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] 全単数群を用いたイデアル類群の汎用アルゴリズムについて2005

    • Author(s)
      田谷 久雄
    • Journal Title

      日本応用数理学会2005年度年会講演予稿集

      Pages: 298-299

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On indivisibility of class numbers of algebraic number fields (Japanese)2005

    • Author(s)
      Hisao Taya
    • Journal Title

      Proceedings of the 3rd Workshop on Number Theory, Hokuriku (Kanazawa) Vol.3

      Pages: 60-83

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] An algorithm for computing ideal class groups and unit groups (Japanese)2005

    • Author(s)
      Hisao Taya
    • Journal Title

      Proceedings of the 2005 Annual Meeting of Japan Society for Industrial and Applied Mathematics

      Pages: 298-299

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 全単数群を用いたイデアル類群の汎用アルゴリズムについて2005

    • Author(s)
      田谷 久雄
    • Journal Title

      日本応用数理学会2005年度年会講演予稿集 2005年度版

      Pages: 298-299

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A characterization of designs related to an extremal doubly-even self-dual code of length 482005

    • Author(s)
      Masaaki Harada
    • Journal Title

      Annals of Combinatorics 9

      Pages: 189-198

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 代数体の類数の非可除性について2005

    • Author(s)
      田谷 久雄
    • Journal Title

      第3回北陸数論小研究集会報告集 3(発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Spherical 5-designs obtained from finite unitary groups2004

    • Author(s)
      Akihiro Munemasa
    • Journal Title

      European Journal of Combinatorics 25

      Pages: 261-267

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A new quasi-symmetric 2-(56,16,6) design obtained from codes2004

    • Author(s)
      Akihiro Munemasa
    • Journal Title

      Discrete Mathematics 29

      Pages: 231-234

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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