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

自守函数特殊值的超越性和代数独立性研究

基本信息

批准号：
09640007
负责人：
AMOU Masaaki
金额：
$ 1.47万
依托单位：
Gunma University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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.

我们研究了某些q函数的特殊值，得到了以下结果。设K是有限次代数数域，上取K的一个位置，q是K的一个元素，|Q| 1.设l是正整数，且Q（z）和R（z）多项式的系数从K开始满足deg <less than or equal>Ql，Q（0）<double plus>0。在这些符号下，我们考虑了形式为Q（z）f（qz）= z^lf（z）+ R（z）的函数方程，它有唯一解f（z）在整个K_<upsilon>（K关于上凸的完备化）上是亚纯的.然后我们在Masanori Katsurada教授（鹿儿岛大学）的帮助下证明了以下定理Keijo V_n_rien教授（欧卢大学）。(Letα是K的非零元素，它不是f（z）的极点。则f（alpha）不属于K。此外，存在一个有效的可计算的正常数μ，使得f（alpha）相对于upperity的非理性测度由μ从上而下确定。特别地，f（alpha）不是刘维尔数。该定理对某些q-超几何级数有重要的应用，推广了Stihl（Math. Ann.，1984年）。在有关赠款的支持下，我们邀请了Alain Lasjaunias教授（波尔多大学）。1998年10月，我们访问了群马大学，询问了他对上述结果的看法，并与他讨论了进一步研究的可能性。关于目前的研究，我们于1998年11月在东京举行的法日超越数理论会议上作了报告，现在正在准备与Katsurada教授和V__n_nen教授的联合论文。