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1999 Fiscal Year Final Research Report Summary

Studies on the Jacobi sum and its application to the Leopoldt conjecture

Research Project

Project/Area Number 10640021
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

MIKI Hiroo  Kyoto Inst. Tech., Dep. Eng. &Design, Prof., 工芸学部, 教授 (90107368)

Co-Investigator(Kenkyū-buntansha) IWATSUKA Akira  Dep. Tex. Sci., Kyoto Inst. Tech., Prof., 繊維学部, 教授 (40184890)
NAKAOKA Akira  Dep. Eng. &Design, Kyoto Inst. Tech., Prof., 工芸学部, 教授 (90027920)
UCHIYAMA Jun  Dep. Tex. Sci., Kyoto Inst. Tech., Prof., 繊維学部, 教授 (70025401)
ASADA Mamoru  Dep. Eng. &Design, Kyoto Inst. Tech., Assoc. Prof., 工芸学部, 助教授 (30192462)
TSUKAMOTO Chiaki  Dep. Tex. Sci., Kyoto Inst. Tech., Assoc. Prof., 繊維学部, 助教授 (80155340)
Project Period (FY) 1998 – 1999
KeywordsGauss sum / Jacobi sum / Hecke character / Leopoldt conjecture / p-adic L function / Hilbert symbol
Research Abstract

Number theory has been developed relating closely to many areas in mathematics, and recently it is applied to physics and engineering. In the present research, we researched from the integrated standpoint. Investigators in this project attended related conferences, discussed the problem with related researchers, collected many related references, and analyzed the problem using computers. In the process of our research we realized the importance of studies on Gauss sums and Jacobi sums, namely we firmly believed the relation between the Leopoldt conjecture and Gauss sums. The Leopoldt conjecture says that units, which are multiplicatively independent over the ring of rational integers, of a finite algebraic number field are also multiplicatively independent over the ring of p-adic integers. It is very important conjecture and is still a very difficult open problem.
This conjecture is equivalent to the nonvanishing of L function at 1. On the other hand, head investigator in the present research gave an algebraic proof of nonvanishing of L function at 1 using Gauss sums under certain conditions. This implies close and deep relation between the Leopoldt conjecture and Jacobi sums. He also obtained partial affirmative answer to the conjecture from the different standpoint. It seems to be important to pursue the research considering the relation to cohomology theory, integral representation, structure of Galois groups of algrbraic number fields, ramification theory, theory of p-adic L functions, explicit formula for Hilbert norm residue symbols, and its non-abelization.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Masaharu Arai: "On the von Neumann and Wigner Potentials"Journal of differential equations. 157. 348-372 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mamoru Asada: "The faithfulness of the monodromy representations associated with certain families of algebraic curves"Journal of Pure and Applied Algebra. (予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mamoru Asada: "The compatibility of the filtration of mapping class groups of two surfaces pasted along the boundaries"Topology and its Applications. (予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mamoru Asada: "On centerfree quotients of surface groups"Communications in Algebra. (予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mamoru Asada: "On the structure of the braid groups of free nilpotent pro-l groups"Memories of the Faculty of Eng. & Design, Kyoto Inst. Tech.. 48. 23-29 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akira Iwatsuka: "Asymptotic distribution of eigenvalues for Pauli operators with nonconotant magnetic fields"Duke Mathematical Journal. 93巻3号. 535-574 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masahara Arai: "On the von Neumann and Wigner Potentials, Journal of Differential Equations (with Jun Uchiyama)"Journal of Differential Equations. 157. 348-372 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Mamoru Asada: "The faithfulness of the monodromy representations associated with certain families of algebraic curves"Journal of Pure and Applied Algebra. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Mamoru Asada: "The compatibility of the filtration of mapping class groups of two surfaces pasted along the boundaries"Topology and its Applications. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Mamoru Asada: "On centerfree quotients of surface groups"Communications in Algebra. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Mamoru Asada: "On the structure of the braid group of free nilpotent pro-l groups"Memoirs Fac. Engi. Desi. Kyoto Inst. Tech. Ser. Sci. and Tech.. 48. 23-29 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akira Iwatsuka: "Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields (with Hideo Tamura)"Duke Mathematical Journal. 93-3. 535-574 (1998)

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

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Published: 2001-10-23  

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