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p-adic analysis of algebraic numer fields and algorithm on algebraic number fields or finite fields

Research Project

Project/Area Number 12640029
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNAGASAKI UNIVERSITY

Principal Investigator

KUDO Aichi  Nagasaki Univ., Fac. of Engineering, Professor, 工学部, 教授 (00112285)

Co-Investigator(Kenkyū-buntansha) MARUYAMA Yukihiro  Nagasaki Univ., Fac. of Economics, Professor, 経済学部, 教授 (30229629)
SUEYOSHI Yutaka  Nagasaki Univ., Fac. of Education, Associate Professor, 工学部, 助教授 (80128040)
WASHIO Tadashi  Nagasaki Univ., Fac. of Education, Professor, 教育学部, 教授 (60039435)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsBernoulli number / p-adic zeta-funciton / Carmichael number / elliptic curve / Hasse invariant / 4-class rank / 2-class field tower / decision process / 代数体 / 一般Bernoulli数 / P進L関数 / 有限体 / 楕円関数体 / 類数 / 類体塔 / アルゴリズム / 代数的数体 / イデアル類群 / 超楕円関数体 / L関数 / P進体
Research Abstract

A. Kudo studied algorithms for computing p-adic properties of Bernoulli numbers, for prime factor ization of integers and for calculation of Carmichael numbers. He proved that the p-adic Euler constant γ_p of p-adic zeta function is equivalent to -(B_<p-1>/(p-1) - 1/p) for modulo p (p【greater than or equal】5), where B_n denotes the n-th Bernoulli number, and calculated the value (modp) of them for p【less than or equal】20,000,000. It is derived that in this range γ_p*0 (modp) for p≠5, 13, 563. Also he calculated all Carmichael numbers less than 10^<17> with their prime factorizations.
T. Washio studied class number and Hasse invariants of elliptic curves over finite fields. By means of determination of the number of rational points, he derived a sufficient condition for an elliptic curve over a finite prime field GF(p) to be supersingular, and gave a new equality of binomial coefficients.
Y. Sueyoshi studied 4-class ranks of quadratic number fields and matrices over finite field GF(2). Using Redei matrices, he proved new inequality relations between the narrow 4-class ranks of quadratic number fields. He further gave a characterization of Redei matrices with minimal rank. As an application of these results, he also proved the fact that if the ideal class group of imaginary quadratic number field K contains a subgroup of type (4,4, 2,2) and 4 is not contained in the prime discriminant of K, then the 2-class field tower of K is infinite.
Y. Maruyama studied discrete optimization problems using the theory of finite automata. He intro duced the notion of a new sequential decision process called bitone sequential decision process and gave a strong representation theorem for a discrete decision process.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] Aichi Kudo: "A computation of Carmichael numgers (Carmichael数の一つの計算法)"本報告書所収. 14 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tadashi Washio, Tetsuo Kodama: "On a certain supersingular elliptic curve"Bull.Fac.Educ., Nagasaki Univ.. vol.66(in press). 5 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yutaka Sueyoshi: "Relations between the narrow 4-class ranks of quadratic number fields"Adv.Stud.Contemp.Math.. vol.2. 47-58 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yutaka Sueyoshi: "On Redei matrices with minimal rank"Far East J.Math.Sci.. vol.3. 121-128 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yutaka Sueyoshi: "Infinite 2-class field towers of some imaginary quadratic number fields"preprint. 3 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yukihiro Maruyama: "Strong representation of a discrete decision process by a bitone sequential decision prosess"submitted to Proceedings of NACA '01. 10 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aichi Kudo: "A congruence of generalized Bernoulli number for the character of the first kind"Adv. Stud. Contemp. Math.. 2. 1-8 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aichi Kudo: "A computation of Carmichael numgers (Japananese)"In the report of this project.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Washio and Tetsuo Kodama: "On a certain supersingular elliptic curve"Bull. Fac. Educ., Nagasaki Univ.. 66(preprint). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tadashi Washio and Tetsuo Kodama: "Note on a certain supersingular elliptic curve"(Preprint). 1-3

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yutaka Sueyoshi: "Relations between the narrow 4-class ranks of quadratic number fields"Adv. Stud. Contemp. Math.. 2. 47-58 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yutaka Sueyoshi: "On Redei matrics with minimal rank"Far East J. Math. Sci. 3. 121-128 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yutaka Sueyoshi: "Infinite 2-class field towers of some imaginary quardratic number fields"(Preprint). 1-3

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yukihiro Maruyama: "Strong representation of a discrete decision process by a bitone sequential decision prosess"Proc. of NACA '01. (Preprint). 1-10

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aichi Kudo: "A computation of Carmichael numbers"平成12-13年度科学研究費補助金研究成果報告書所収予定. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Tadashi Washio, Tetsuo Kodama: "On a certain supersingular elliptic curve"Bulletin of Faculty of Education, Nagasaki University, Natural Science. 66. 1-3 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yutaka Sueyoshi: "Infinite 2-class field towers of some imaginary quadratic number fields"preprint. 1-3 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Aichi Kudo: "A congruence of generalized Bernoulli number for the character of the first kind"Adv.Stud.Contemp.Math.. 2. 1-8 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yutaka Sueyoshi: "Relations between the narrow 4-class ranks of quadratic number fields"Adv.Stud.Contemp.Math.. 2. 47-58 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yutaka Sueyoshi: "On Redei matrices with minimal rank"Far East J.Math.Sci.. 3. 121-128 (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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