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Number Theory and Applications to the Related Discrete Mathematics

Research Project

Project/Area Number 14540033
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

NAKAHARA Toru  Saga University, Faculty Science Engineering, Professor, 理工学部, 教授 (50039278)

Co-Investigator(Kenkyū-buntansha) MIYAKE Katsuya  Tokyo Metropolitan University, Department Mathematics, Professor, 大学院・理学研究科, 教授 (20023632)
ICHIKAWA Takashi  Saga University, Faculty Science Engineering, Professor, 理工学部, 教授 (20201923)
UEHARA Tsuyoshi  Saga University, Faculty Science Engineering, Professor, 理工学部, 教授 (80093970)
TAGUCHI Youichiro  Kyushu University, Department Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (90231399)
KATAYAMA Shin-ichi  Tokushima University, Department Math Science, Professor, 総合科学部, 教授 (70194777)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2003: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2002: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsHasse's Problem / Power Intetral basis / Fundamental Unit / Mordell Curve / Teichmueller grounoid / Neron-Tate Height / Elliptic Curve / Algebraic Geometry Code / 整数環の巾底 / 類群及び単数群の構造 / Yokoi-Chowla予想 / 3次体 / Mordell Curves / 代数曲線 / エルミート符号 / タイヒミューラー基本亜群 / Mordell曲線 / 基本単数系と類数 / p進表現
Research Abstract

The three aims of our project have been accomplished by the investigators as follows,
A)Investigation of the power integral bases and the structures of the class groups of abelian fields of finite degree over the rationals
On Hasse's problem, the head investigator, S. I. A. Shah (Univ. Peshawar) and Y. Motoda (Yatsushiro National College of Technology) gave a new characterization of abelian fields whose rings of integers have power integral bases or do not applying a result of M.-N. Gras and F..Tano'e [JNT, 1986 ]. Namely if afield K is an octic field Q(\sqrt{mn}, \sqrt{lm}, sgrt{ι}), where lmn is a square-free integer, then K has no power integral basis except for one field [Arch. Math. To appear]. Thus we proposed an open problem[ibid] ;
Problem. For any octic 2-elementary abelian extensionK of rank 3, if the ring Z K has a power integral basis, then does K coincide with the 24-th cyclotomic field Q(\zeta_{24})? Katayama-Levesque-the head investigator constructed a new family of bicycli … More c biquadratic fields with explicit fundamental system of units[CRM Proceedings, to appear]. S.-I. Katayama and C. Levesque applied these results to the simultaneous diophantine equations and to the construction of certain family of cyclic extensions[Acta Arith., 2003].
Miyake explicitly determined the Mordell curves and described the sets of the Q-rational points of them ascertain subsets of corresponding cubic fields and showed that both of them have Mordell-Weil rank 1 generically. He also explicitly described cubic twists of the Fermat curve of degree three as a family of Mordell curves.
B)Applications of number theory to arithmetic geometry and algebraic geometry
Ichikawa constructed the Teichmueller groupoids in the category of arithmetic geometry, and he described the Galois action and the monodromy representation on the Teichmueller groupoids [J. Reine Angew. Math., 2003]. Next he proved the Bogomolov conjecture which states that if an irreducible curve in an abelian variety is not isomorphic to an elliptic curve, then its algebraic points are distributed uniformly discretely for the Neron-Tate height [J. number Theory, 2004].
Taguchi proved the potentially abelian case of the finiteness conjecture of Fortaine and Mazur and proved some existence/non-existence results on certain 2-dimensional modp (ialois representations of the rational number field(joint work with H. Moon) [Ramanujan J. 2003, Documenta Math. 2003].
C)Applications of number theory to coding theory and discrete mathematics.
Taguchi improved the method of.T. Satoh of the fastp-adic calculation of the number of rational points on elliptic curves over finite fields (joint work with T Satoh and B. Skjernaa) [Finite Field Appl., 2003]. Uehara researched into the determination of the minimum distance of algebraic geometry codes, which are error-correcting codes constructed by algebraic function fields [Kyushu J. Math., 2002]. Less

Report

(3 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • Research Products

    (52 results)

All Other

All Publications (52 results)

  • [Publications] S.-I.Katayama, C.Levesque, T.Nakahara: "On a family of bicyclic biquadratic fields"CRM Proceedings and Lecture Notes. 36. 165-175 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.I.A.Shah, T.Nakahara: "Monogenesis of the Rings of Integers in Certain Imaginary Abelian Fields"Nagoya Math.J.. 168. 85-92 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Motoda, T.Nakahara, S.I.A.Shah: "On a Problem of Hasse for Certain Imaginary Abelian Fields"J.Number Theory. 96. 326-334 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Motoda, T.Nakahara: "Monogenesis of Algebraic Number Fields whose Galois Groups are $2$-elementary Abelian"Proceedings of the 2003 Nagoya Conference ; Yokoi-Chowla Conjecture and Related Problems. 91-99 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Motoda, T.Nakahara: "Power integral bases in algebraic number fields whose Galois groups are $2$-elementary abelian"Arch.Math.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama: "On zeta functions associated to finite groups"Adv.Stud.Contemp.Math.. 4・2. 115-125 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama, C.Levesque: "On simultaneous diophantine equations"Acta Arithmetica. 108・4. 369-377 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyake: "Some Families of Mordell Curves associated to Cubic Fields"Jour.of Computational and Applied Math.. 160. 217-231 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyake: "Cubic Fields and Mordell Curves"The Proceedings of Confence au MAROC dans le cadre des Grandes F'etes de L'Universit'e Laval. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Ichikawa: "Teichmueller groupoids and Galois action"J.Reine Angew.Math.. 559. 95-114 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Ichikawa: "Heights on a subvariety of an abelian variety"J.Number Theory. 104. 170-176 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Ichikawa: "Teichmueller groupoids, and monodromy in conformal field theory"Commun.Math.Phys.. 246. 1-18 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Satoh, Y.Taguchi: "Computing zeta functions for ordinary formal groups over finite fields"Discrete Appl.Math.. 130・1. 51-60 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Satoh, B.Skjernaa, Y.Taguchi: "Fast computation of canonical lifts of elliptic curves and its application to point counting"Finite Fields Appl.. 9・1. 89-101 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Taguchi: "On potentially abelian geometric representations"Ramanujan J.. 7. 477-483 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Moon, Y.Taguchi: "Refinement of Tate's discriminant bound and non-existence theorems for mod $p$ Galois representations"Documenta Math.Extra Volume. 641-654 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.K.Song, T.Uehara: "On the Feng-Rao bound for the minimum destance of certain algebraic geometry codes"Kyushu J.Math.. 56. 405-418 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Hashimito, H.Nakamura, K.Miyake: "Galoas Theory and Modular Forms"DEVM 11,Kluwer Acad.Publ.,Dordrecht/Boston/London(To appear). 390

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama, C.Levesque, T.Nakahara: "Proceedings of the 2003 Nagoya Conference Yokoi-Chowla Conjecture and Related Problems"Furukawa Total Printing Co.LTD, Saga, Japan. 148 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama, C.Levesque, T.Nakahara: "On a family bicyclic biquadratic fields"CRM Proceedings and Lecture Notes. 36. 165-175 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.I.A.Shah, T.Nakahara: "Monogenesis of the Rings of Integers in Certain Imaginary Abelian Fields"Nagoya Math.J.. 168. 85-92 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Motoda, T.Nakahara, S.I.A.Shah: "On a Problem of Hasse for Certain Imaginary Abelian Fields"J.Number Theory. 96. 326-334 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Motoda, T.Nakahara: "Monogenesis of Algebraic Number Fields whose Galois Groups are 2-elementary Abelian"Proceedings of the 2003 Nagoya Conference ; Yokoi-Chowla Conjecture and Related. Problems. 91-99 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Motoda, T.Nakahara: "Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian"Arch.Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama: "On zeta functions associated to finite, groups"Adv.Stud.Contemp.Math.. 4-2. 115-125 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama, C.Levesque: "On simultaneous diophantine equations"Acta Arithmetica. 180-4. 369-377 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyake: "Some Families of Mordell Curves associated to Cubic Fields"Jour.of Computational and Applied Math. 160. 217-231 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyake: "Cubic Fields aiid Mordell Curves"The Proceedings of Conf'ence au MAROO dans le cadre des Grandes F'etes de L'Universit' e Laval. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Ichikawa: "Teichmueller groupoids and Galois action"J.Reine Angew.Math.. 559. 95-114 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Ichikawa: "Heights on 'a subvariety of an abelian variety"J.Number Theory. 104. 170-176 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Ichikawa: "Teichmueller groupoids, and monodromy In conformal field theory"Commun.Math.Phys.. 246. 1-18 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Satoh, Y.Taguchi: "Computing zeta functions for ordinary formal groups over finite fields"Discrete Appl.Math.. 130-1. 51-60 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Satoh, B.Skjernaa, Y.Taguchi: "Fast computation of canonical lifts of elliptic curves and its application to point counting"Finite Fields Appl.. 9-1. 89-101 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Taguchi: "On potentially abelian geometric representations"Ramanujan J.. 7. 477-483 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Moon, Y.Taguchi: "Refinement of Tate's discriminant bound and non-existence theorems for mod p Galois representations"Documenta Math. Extra Volume. 641-654 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.K.Song, T.Uehara: "On the Feng-Rao bound for. the minimum, destance of certain algebrai geometry codes"Kyushu J. Math.. 56. 405-418 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Hashimoto, H.Nakamura, K.Miyake: "Galos Theory and Modular Forms DEVM11"Kluwer Acad. Publ., Dordrecht/Boston/London (to appear). 390

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.-I.Katayama, C.Levesque, T.Nakahara: "Proceedings of the 2003 Nagoya Conference ; Yokoi-Chowla Conjecture and Related Problems"Furukawa Total Printing Co. LTD. Saga, Japan. 148 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Katayama: "On a family of real bicyclic biquadratic fields"Procce.of the 2002 Canadian Number Theory Associ.Confer.. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Katayama: "On simultaneous Diophantine equations"Acta Arithmetica. 108.4. 369-377 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Miyake: "Some Families of Mordell Curves associated to Cubic Fields"J.of Computational and Applied Math.. 160. 217-231 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ichikawa: "Teichmueller groupoids and Galois action"J.Reine Angew.Math.. 559. 95-114 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ichikawa: "Heights on a subvariety of an abelian variety"J.Number Theory. 104. 170-176 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y, Taguchi: "On potentially abelian geometric representations"Ramanujan J.. 7. 477-483 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Katayama: "Proceedings of the 2003 Nagoya Conference ; Yokoi-Chowla Conjecture and Related Probrems"Saga University, Furukawa Total Printing Co.Ltd.. 140 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Nakahara, Toru: "Monogenesis of the Rings of Integers in Certain Imaginary Abelian Fields, with Shah, S.I.A."Nagoya Math. J.. 168. 85-92 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nakahara, Toru: "On a Problem of Hasse for Certain Imaginary Abelian Fields, with Motoda, Y and Shah, S.I.A."J. Number Theory. 96. 326-334 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Uehara, Tsuyoshi: "On the Feng-Rao bound for the minimum distance of certain algebraic geometry codes, with Song, H.K."Kyushu J. Math.. 56. 405-418 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ichikawa, Takashi: "Teichmueller groupoids and Galois action"J. Reine Angew. Math.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Katayama, Shin-Ichi: "On simultaneous diophantine equations, with Levesque, C."Acta Arith. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Taguchi, Yuichiro: "Induction formula for the Artin conductors of mod1 Galois representations"Proc. Amer. Math. Soc.. 130. 2865-2869 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Miyake, Katsuya(Ed. in Chief): "Galois Theory and Modular Forms, with Hashimoto, Kiichiro and Nakamura, Hiroaki"Kluwer Academic Publishers (to appear).

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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