Noether's Problem for Cremona Groups over algebraic number fields and its application to Number theory

代数数域上克雷莫纳群的诺特问题及其在数论中的应用

• 批准号: 19340011
• 负责人: HASHIMOTO Kiichiro
• 金额: $ 8.74万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19340011/
• 关键词: ガロア理論; ガロアの逆問題; ネーター問題; クレモナ変換群; 有理関数体; 生成多項式; 代数曲線; ヤコビ多様体; ガロア群; 不変式

We studied the Noether's Problem, which asks the rationality of the fixed field of the rational function field of several variables over a given field, with respect to a given finite subgroup G of the Cremona group. We solved this problem affirmatively in the case where G is one of the transitive permutation groups of degree six, and obtained the explicit description of the fixed field as expected. The results are now being collected and prepared in some papers, although it will take some time before the completion. During the period of the research, we had in each year a workshop entitled as "Galois theory and related topics", and discussed the various related problems.. They were held in the university of Yamagata (2007), Tokushima (2008), and Kanazawa (2009). We also had a conference on number theory each year at Waseda university and communicated with many experts of this subject, including those from foreign countries.

本文研究了关于Cremona群的有限子群G的Noether问题，该问题要求给定域上的多元有理函数域的固定域的合理性。在G是一个六次可迁置换群的情况下，我们肯定地解决了这个问题，并得到了预期的固定域的明确描述。目前正在收集调查结果，并将其编入一些文件，不过完成调查还需要一些时间。在研究期间，我们每年都有一个名为“伽罗瓦理论和相关主题”的研讨会，并讨论了各种相关问题。分别在山形大学（2007年）、德岛大学（2008年）、金泽大学（2009年）举办。我们每年还在早稻田大学举行数论会议，与许多这方面的专家，包括来自国外的专家进行交流。

项目成果

Weber's Class Number Problem in the Cyclotomic Z_2-Extension of QII

QII 循环 Z_2 扩展中的韦伯类数问题

• 发表时间: 2010
• 期刊: Journal de Theorie des Nombres de Bordeaux 22
• 作者: Takashi Fukuda;Keiichi Komatsu
• 通讯作者: Keiichi Komatsu

General form of Humbert's modular equation for curves with real multiplication of D=5

实数乘法 D=5 的曲线亨伯特模方程的一般形式

• 发表时间: 2009
• 期刊: Proceedings of the Japan Academy Series A 85, no.10
• 作者: Kiichiro Hashimoto;Yukiko Sakai
• 通讯作者: Yukiko Sakai

Noether's problem and Q-generic polymials for the normalizer the 8-cycle in s_8 and its subgroups

诺特问题和标准化器 s_8 及其子群中的 8 周期的 Q 泛多项式

• 发表时间: 2008
• 期刊: Mathematics of Computation 77
• 作者: Kiichiro Hashimoto;Akinari Hoshi;Yuichi Rikuna
• 通讯作者: Yuichi Rikuna

Iwasawa lambda-invariants and Mordell-Weil rank of Jacobian varieties with complex multiplication

复杂乘法雅可比簇的 Iwasawa lambda 不变量和 Mordell-Weil 等级

• 发表时间: 2007
• 期刊: Acta Arithmetica 127
• 作者: Takashi Fukuda;Keiichi Komatsu;Shuji Yamagata
• 通讯作者: Shuji Yamagata

Ponceletの閉形定理とDelta=8の実乗法を持つ種数2の超楕円曲線について

关于Poncelet的闭式定理和具有Delta=8实数定律的属2超椭圆曲线

• 发表时间: 2007
• 作者: Kiichiro Hashimoto;Yukiko Sakai
• 通讯作者: Yukiko Sakai