ABC Conjecture and the Structure of Algebraic Number Fields

ABC猜想与代数数域的结构

• 批准号: 14540030
• 负责人: KATAYAMA Shin-ichi
• 金额: $ 1.15万
• 依托单位: The University of Tokushima
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540030/
• 关键词: ABC Conjecture; Unit groups; Class Groups; 連立ペル方程式; 類数; 単数; 不足方程式; 不定方程式

We have investigated the algebraic structure of algebraic number fields, namely, class groups and unit groups, and related diophantine equations. In A1), we have constructed certain real quadratic fields and quartic fields with explicit fundamental units. These constructions are very useful in number theory, and actually, in B2), we have used these constructions to show the existence of infinite family of algebraic cyclic number fields with prescribed class groups with Y.Kishi. In A2) and B1), we have investigated the positive integer solutions of simultaneous Pell equations, and improved the number of positive integer solutions under the ABC conjecture. We also verified a relation between the fundamental units of certain real quadratic fields and the positive integer solutions of simultaneous Pell equations under the ABC conjecture.We have published the following 3 papers A1),A2),A3) and gave 2 invited lectures B1), B2) at international conferences.A1)On a family of real bicyclic biquadratic fields, CRM Proceedings 36 (2004)A2)On simultaneous Diophantine equations, Acta Arithmetica 108 (2003)A3)On zeta functions associated to finite groups, Advanced Studies in Contemporary Mathematics 4 (2002)B1)On a family of simultaneous Pell equations, The 9^<th> Japan-Korea Joint Seminar on Number theory (2004 Oct. at Kujyu)B2)An infinite family of imaginary cyclic fields of degree p-1 which have ideal class groups of p-ranks greater than one, Yokoi-Chowla Conjecture and Related Problems (2003 Oct. at Nagoya)We note that we have held the above symposium "Yokoi-Chowla Conjecture and Related Problems" at Nagoya University with the help of Prof. Toru Nakahara of Saga University and Prof. Hideo Yokoi of Aichi Gakuin University. The symposium benefited from the scientific grants (140030, 140033). We have published the proceedings of the symposium (ISBN 4-921090-99-8) from Furukawa Total Printing.

我们研究了代数数域的代数结构，即类群和单位群，以及相关的丢芬图方程。在A1)中，我们构造了若干具有显式基本单位的实数二次场和四次场。这些构造在数论中是非常有用的，实际上，在B2课中，我们已经用这些构造证明了具有规定类群的无穷代数循环数域族的存在性。在A2)和B1)中，我们研究了联立Pell方程的正整数解，并改进了ABC猜想下正整数解的个数。在ABC猜想下，我们还证明了某些实二次域的基本单位与联立Pell方程的正整数解之间的关系。发表论文3篇（A1），A2),A3)，在国际会议上做过2次特邀演讲（B1）， B2)。A1)关于实双环双二次域族，数学学报36 （2004)A2)关于联列Diophantine方程，数学学报108 （2003)A3）关于与有限群相关的zeta函数，当代数学高级研究4 （2002)B1）关于联列Pell方程族，9^< >日本-韩国数论联合研讨会（2004年10月，Kujyu）B2）具有p阶大于1的理想类群的p-1次虚循环域的无限族Yokoi- chowla猜想及相关问题（2003年10月，名古屋）我们注意到，在佐贺大学中原彻教授和爱知学院大学横井秀夫教授的帮助下，我们在名古屋大学举办了上述研讨会“Yokoi- chowla猜想及相关问题”。本次研讨会得益于科学资助（140030,140033）。我们已经从Furukawa Total Printing出版了研讨会的会议记录（ISBN 4-921090-99-8）。

项目成果

On a family of real bicyclic biquadratric fields

关于一族实双环双二次域

• 发表时间: 2004
• 期刊: CRM Proceedings and Lecture Notes 36
• 作者: Shin-ichi Katayama

On zeta functions associated to finite groups

与有限群相关的 zeta 函数

• 发表时间: 2002
• 期刊: Advanced Studies in Contemporary Mathematics 4. no2
• 作者: Shin-ichi Katayama;Shin-ichi Katayama;Shin-ichi Katayama
• 通讯作者: Shin-ichi Katayama

Proceedings of the 2003 Nagoya Conference, Yokoi-Chowla Conjecture and Related Problems

2003年名古屋会议论文集、横井-秋罗猜想及相关问题

• 发表时间: 2004
• 作者: Shinnichi Katayama;Claude Levesque;Toru Nakahara
• 通讯作者: Toru Nakahara

Proceedings of the Nagoya Conference, Yokoi-Chowla Conjecture and Related Problems

名古屋会议论文集、横井秋罗猜想及相关问题

• 发表时间: 2004
• 作者: Shin-ichi Katayama;Shin-ichi Katayama;Shin-ichi Katayama;Shin-ichi Katayama;Shin-ichi Katayma (Ed.);Shin-ichi Katayama (Ed.)
• 通讯作者: Shin-ichi Katayama (Ed.)

片山 真一: "On simultaneous diophantine equations"Acta Arithme tcca. (to appear).

Shinichi Katayama：“论联立丢番图方程”Acta Arithme tcca。