A Research on Transcendence and Algebraic Independence of Special Values of Automorphic Functions
Project/Area Number  09640007 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  Gunma University 
Principal Investigator 
AMOU Masaaki Gunma University, Faculty of Engineering, Lecturer, 工学部, 講師 (60201901)

CoInvestigator(Kenkyūbuntansha) 
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 Fiscal Year 
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)

Keywords  Irrational Number / Irrationality Measure / Hypergeometric Function / 無理数 / 無理数度 / 超幾何級数 / 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 qfunctions 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 qupsilon> 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 qhypergeometric 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
(4results)
Research Output
(6results)