• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

1991 Fiscal Year Final Research Report Summary

Congruence relations between class numbers of cyclotomic fields

Research Project

Project/Area Number 02640063
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionSaga University

Principal Investigator

UEHARA Tsuyoshi  Saga Univ. Fac. of Science and Engrg, Professor, 理工学部, 教授 (80093970)

Co-Investigator(Kenkyū-buntansha) KAWAI Shigeo  Saga Univ., Fac. of Education, Ass. Prof., 教育学部, 助教授 (30186043)
KOZAKI Masanori  Saga Univ., Fac. of Liberal Arts, Professor, 教養部, 教授 (70039262)
SUGITA Hiroshi  Saga Univ., Fac. of Science and Engrg, Ass. Prof., 理工学部, 助教授 (50192125)
ICHIKAWA Takashi  Saga Univ., Fac. of Science and Engrg, Instructor, 理工学部, 講師 (20201923)
NAKAHARA Toru  Saga Univ., Fac. of Science and Engrg, Professor, 理工学部, 教授 (50039278)
Project Period (FY) 1990 – 1991
KeywordsCyclotomic Field / Class Number / Congruence Relation / Quadratic Field / Ideal Class Group / Unit Group / Factorizing Algorithm / Elliptic Curve
Research Abstract

In this project, we have studied mainly about congruence relations for class numbers and units of cyclotomic fields, in particular of quadratic fields.
We firstly established (cf. [1]) a general linear congruence relation modulo an odd prime number between the class numbers and units of the quadratic fields whose discriminants are divisible by 8m, where m>O is an odd square-free rational integer. From this relation we deduced various congruences for class numbers and units of two quadratic fields and indicated how they included known results by many authors. Using Stickelberger's relation we secondly gave (cf. [2)) a congruence modulo 8 between Jacobi sums and the class number of a real quadratic field, and as an application of this congruence we proved a new type of congruence for the class numbers of a real quadratic field and the corresponding imaginary one. In the proof we developed a new method of calculation of 2-adic logarithm of Jacobi sums. We believe that this method is fundamental to obtain a general congruence modulo some power of a prime number between class numbers of cyclotomic fields. Thirdly we were concerned with algorithm of factorization of large natural numbers into prime numbers, and studied a sieve method using polynomials of degree 3. Finally, during our studies of a factorizing algorithm by elliptic curves, we found two simple proof of the points of an elliptic curve over any field forming an abelian group ; one use only elementary calculus, and the other is done by computer.
Each investigator obtained good result for a respective research plan.

  • Research Products

    (17 results)

All Other

All Publications (17 results)

  • [Publications] T.Uehara: "On linear congruence relations between class numbers of quadratic fields" J.Number Theory. 34. 362-392 (1990)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Uehara: "On congruences for Jacobi sums and class numbers of quadratic fields"

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nakahara: "A construction of quadratic fields whose class numbers are divisible by a power of 3" Colloquia Mathematica Societatis Jahos Bolyai 51.Number Theory VolII(North-Holland Pub.(o.). 889-897 (1990)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Ichikawa: "Algebraic groups associated with abelian varieties" Math.Ann.289. 133-142 (1992)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Sugita: "Various topologies in the Wieher space and Levy's stochastic area" Probability Theory and Related Fields. (1992)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Kozaki: "On mean value theorems for small geodesic snheres in Riemannian manifolds" Cze choslorak Math.Jour.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Uehara: "On linear congruence relations between class numbers of quadratic fields" J. Number Theory. 34. 362-392 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Uehara: "On congruences for Jacobi sums and class numbers of quadratic fields"

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Nakahara: "A construction of quadratic fields whose class numbers are divisible by a power of 3" Colloquia Mathematica Societatis Ja' nos Bolyai. 51. 889-897 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Nakahara: "A simple proof for non-monogenesis of the rings of integral in some cyclic fields" Proc. of the 3rd conference of the Canadian Number Theory Association.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Ichikawa: "Algebraic groups associated with abelian varieties" Math. Ann.288. 133-142 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Ichikawa: "The universal periods of curves and the Schottky problem" Comp. Math.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H. Sugita: "Various topologies in the Wiener space and Levy's stochastic area" Probability Theory and Related Fields. (1992)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Furusho: "Existence of positive entire solutions for weakly coupled semilinear elliptic systems" Proc. Royal Soc. Edinburgh. 120A. 1992

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Kozaki: "On mean value theremes for small geodesic spheres in Riemannian manifolds" Czechoslovak Math. Journ.(1992)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Kitahara & A. Nishi: "Best approximations by vector-valued monotone increasing or convex functions" Journal of Mathematical Analysis and Applications.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Kawai: "Concerning a type of Soboler inequality on the sphere" Math. Ann.42. (1992)

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

URL: 

Published: 1993-03-16  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi