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.

我们已经研究了代数数字段的代数结构，即课程组和单位组以及相关的二磷剂方程。在A1中，我们构建了具有明确基本单元的某些实际二次字段和四分之一的字段。这些构造在数量理论中非常有用，实际上，在B2）中，我们使用这些结构来表明具有带有Y.Kishi的规定类群的代数循环数字的无限家族的存在。在A2）和B1）中，研究了同时佩尔方程的正整数溶液，并改善了ABC猜想下正整数溶液的数量。我们还验证了某些实际二次领域的基本单元与ABC猜想下同时佩尔方程的正整数解决方案之间的关系。我们已经发表了以下3篇论文A1），A2），A3），A3），A3），并给了2个受邀的演讲B2），B2），B2），在International Choreces.a1）上。 （2004）a2）在同时进行双志他的方程式上，Acta Arithmetica 108（2003）A3）在与有限群体相关的Zeta函数上，当代数学4（2002）B1的高级研究对同时佩尔方程的家族对9^<th>日本korea infinite in Inflinite in Inflinitiite b2 k k 2 k. 2004 ocke in Infrinite b2。 P-1的假想循环场具有理想的P级级别的p-ranks群体，Yokoi-Chowla的猜想和相关问题（2003年10月在Nagoya）我们注意到，我们已经在Nagoya教授的帮助下，在Nagahara的帮助下，Yokoi-Chowla猜想及相关问题在Nagoya的帮助下举行 大学。研讨会受益于科学补助金（140030，140033）。我们已经发表了Furukawa Total Printing的研讨会（ISBN 4-921090-99-8）的会议记录。

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。