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Properties of mapping class groups related to Galois representations

Research Project

Project/Area Number 15540025
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKyoto Institute of Technology

Principal Investigator

ASADA Mamoru  Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (30192462)

Co-Investigator(Kenkyū-buntansha) MIKI Hiroo  Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90107368)
MAITANI Fumio  Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (10029340)
YAGASAKI Tatsuhiko  Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (40191077)
NAKAMURA Hiroaki  Okayama University, Faculty of Science, professor, 理学部, 教授 (60217883)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsGalois group / cyclotomic field / Iwasawa Theory
Research Abstract

Let κ_0 be a finite algebraic number field, κ_∞ be the field obtained by adjoining to κ_0 all roots of unity, and L be the maximal unramified abelian extension of κ_∞. Let κ_1 be the field obtained by adjoining ζ_4 and ζ_p for all odd prime p to κ_0 and consider the subgroup g=Gal(κ_∞/κ_1) of Gal(κ_∞/κ_0). In this research, we have investigated the structure of Gal(L/κ_∞) and the ideal class group C_∞ of κ_∞ with this g-action.
As for Gal(L/κ_∞), we have shown that it is, as modules over the completed group algebra Z^^^[[g]], isomorphic to the direct pruduct of countable number of copies of Z^^^[[g]]. (Z^^^: the profinite completion of the ring of rational integers Z.)
On the other hand, the ideal class group C_∞ is a discrete g-module. Assume that κ_0 is totally real and p is an odd prime. Let C_∞ (p)^- denote the minus part of C_∞)(p) under the action of the complex conjugation. (A result of Kurihara indicates that the plus part is {0}.) In general, for a pro-p g-module X and the group W(p) of all p-powerth roots of unity, let Hom(X,W(p)) denote the set of continuous homomorphisms from X to W(p). Then this is naturally a discrete g-module. As for C_∞(p)^-, we have shown that it is isomorphic to Hom(Π^∞_<N=1> Z_p[[g]], W(p)). (Z_p[[g]] : the completed group algebra g over the ring of p-adic integers Z_p.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (10 results)

All 2005 2004 Other

All Journal Article (10 results)

  • [Journal Article] 最大円分体のイデアル類群への円分体のガロア群の作用について2005

    • Author(s)
      朝田 衞
    • Journal Title

      仙台数論及び組み合わせ論小研究集会2004報告集

      Pages: 1-10

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the cyclotomic Galois action on the ideal class group of the maximal cyclotomic field (in Japanese).2005

    • Author(s)
      M.Asada
    • Journal Title

      Proceeding of the conference Sendai mini symposium on Number Theory and Combinatorial Theory 2004

      Pages: 1-10

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Variation of Bergman metrics on Riemann surfaces2004

    • Author(s)
      F.Maitani, H.Yamaguchi
    • Journal Title

      Mathematische Annalen 330

      Pages: 477-489

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Variation of Bergman metrics on Riemann surfaces.2004

    • Author(s)
      F.Maitani, H.Yamaguchi
    • Journal Title

      Math.Ann. 330

      Pages: 477-489

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Variation of Bergman metrics on Riemann surfaces2004

    • Author(s)
      F.Maitani, H.Yamaguchi
    • Journal Title

      Mathmatische Annalen 330

      Pages: 477-489

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Homotopy types of the components of spaces of embeddings of compact polyhedra into 2-manifolds

    • Author(s)
      T.Yagasaki
    • Journal Title

      Topology and its applications (to appear)

    • NAID

      110001127618

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Eigenloci of 5 point configurations on the Riemann sphere and the Grothendieck-Teichmuller group

    • Author(s)
      P.Lochak, H.Nakamura, L.Schneps
    • Journal Title

      Mathematical Journal of Okayama University (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Homotopy types of the components of spaces of embeddings of compact polyhedra into 2-manifolds.

    • Author(s)
      T.Yagasaki
    • Journal Title

      Topology and its applications (To appear)

    • NAID

      110001127618

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Eigenloci of 5 point configurations on the Riemann sphere and the Grothendieck-Teichmuller group.

    • Author(s)
      P.Lochak, H.Nakamura, L.Schneps
    • Journal Title

      Math.J.Okayama Univ. (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Homotopy types of the components of the spaces of embeddings of compact polyhedra into 2-manifolds

    • Author(s)
      T.Yagasaki
    • Journal Title

      Topology and its Applications in press

    • NAID

      110001127618

    • Related Report
      2004 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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