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A Research on Transcendence and Algebraic Independence of Special Values of Automorphic Functions

Research Project

Project/Area Number 09640007
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

AMOU Masaaki  Gunma University, Faculty of Engineering, Lecturer, 工学部, 講師 (60201901)

Co-Investigator(Kenkyū-buntansha) SATO Hisashi  Gunma University, Faculty of Engineering, Research Associate, 工学部, 助手 (60008513)
IKEHATA Masaru  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90202910)
AMANO Kazuo  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90137795)
ONDA Isao  Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (00012906)
SAITO Saburho  Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (10110397)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsIrrational Number / Irrationality Measure / Hypergeometric Function / 超越性 / 代数的独立性 / モジュラー関数 / チャカロフ関数
Research Abstract

We have studied special values of certain q-functions and obtained the following result. Let K be an algebraic number field of finite degree, upsilon a place of K, and q an element of K with |q|upsilon> 1. Let l be a positive integer, and Q(z) and R(z) polynomials with coefficients from K satisfying deg Q <less than or equal> l, Q(0) <double plus> 0. Under these notations, we consider a functional equation of the form Q(z)f(qz) = z^lf(z) + R(z), which has the unique solution f(z) meromorphic on the whole K_<upsilon>, the completion of K with respect to upsilon. Then we proved the following theorem with the aid of Prof. Masanori Katsurada (Kagoshima Univ.) and Prof. Keijo V__n_rien (Univ. of Oulu). (Let alpha be a nonzero element of K which is not a pole of f(z). Then f(alpha) does not belong to K.Moreover, there exists an effectively computable positive constant mu such that the irrationality measure of f(alpha) relative to upsilon is hounded from above by mu. In particular, f(alpha) is not a Liouville number.) This theorem has an important application to certain q-hypergeometric series, by which we can generalize a result of Stihl (Math. Ann., 1984).
By the support of the grant concerned, we invited Prof. Alain Lasjaunias (Bordeaux Univ.) to Gunma Univ. on October, 1998, and asked his opinion about the above mentioned result and also discussed with him about a possibility of further researches extending it. Concerning the present research, we gave a talk at the conference between France and Japan on the theory of transcendental numbers held at Tokyo on November, 1998, and are now preparing a joint paper with Prof. Katsurada and Prof. V__n_nen.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (6 results)

All Other

All Publications (6 results)

  • [Publications] M.Amou: "On algebraic independence of certain functions related to the elliptic modular function" Number Theory and Related Topics. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Amou: "Irrationality results for values of generalized Tschakaloff series" J.Number Theory. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Amou: "On algebraic independence of certain functions related to the elliptic modular function" Number Theory and Related Topics. (to Appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Amou: "Irrationality results for values of generalized Tschakaloff series" J.Number Theory. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Amou: "On algebraic independence of certain functions related to the elliptic modular function" Number Theory and Related Topics. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Amou: "Irrationality results for values of generalized Tschakaloff series" J.Number Theory. (to appear).

    • Related Report
      1998 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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