THE INVERSE PROBLEM OF GALOIS AND ITS APPLICATION TO NUMBER THEORY

伽罗瓦反问题及其在数论中的应用

• 批准号: 11440013
• 负责人: MIYAKE Katsuya
• 金额: $ 8.45万
• 依托单位: TOKYO METROPOLITAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440013/
• 关键词: INVERSE PROBLEM OF GALOIS; DIHEDRAL EXTENSIONS; GENERIC FAMILY OF POLYNOMIALS; NOETHER PROBLEM; SPACE OF MODULI; ARITHMETIC TRIANGULAR GROUPS; HYPERGEOMETRIC POLYNOMIALS; GREENBERG CONJECTURE; 三面体拡大; generic polynomial; ハセの問題; 二面体群; l進多重対数函数 /

1. The head investigator organized two workshops in 1999, and jointly organized two research meetings in 2000. With these preparations he co-organized an International Conference "Galois Theory and Modular Forms" in the final year, 2001. The proceedings is now under preparation for publication in the DEVM Series of Kluwer Acad. Publ. The purpose of the conference is to collect informations and techniques in various related fields and to announce the results of this research project. During the three years of the project, 14 foreign mathematicians were invited, including the invited speakers of the meetings, whose supports on the research project were fruitful.2. The head investigator obtained the following results on the inverse problem of Galois : (1) for a cyclic group of odd order n he constructed simple and clear generic family of polynomials with one parameter over the maximal real sub field of the nth cylotomic field by utilizing linear fractional transformation representations. Then he generalized the results also for a dihedral group of order 2n with K. Hashimoto ; (2) in case of n=3, he could construct such a family of cubic polynomials with two integral parameters which parametrizes all of the quadratic fields with class numbers divisible by three and all of unramified cyclic cubic extensions of them ; (3) he also prepared a historical exposition on interactions between algebraic number theory and analytic number theory in the final year of the research project.3. The 15 Investigators prepared 73 research papers for the three years. Among them 10 preprints are selected and attached as Appendix II to the main body of the research report.

1.首席调查员于1999年组织了两次讲习班，并于2000年联合组织了两次研究会议。有了这些准备工作，他共同组织了国际会议“伽罗瓦理论和模块化形式”在最后一年，2001年。目前，该论文集正准备在Kluwer Acad的DEVM系列中出版。会议的目的是收集各相关领域的信息和技术，并公布本研究项目的成果。在项目的三年时间里，邀请了14位外国数学家，包括会议的特邀演讲者，他们对研究项目的支持是卓有成效的。首席研究员获得了以下结果的反问题的伽罗瓦：（1）对于一个循环群的奇数阶n，他构造了简单而明确的一般家庭的多项式与一个参数的最大真实的子领域的第n cylotomic域利用线性分式变换表示。然后，他推广的结果也为二面体群的阶为2n与K。桥本;（2）在n=3的情况下，他可以构造这样一个具有两个整参数的三次多项式族，它参数化了所有类数可被3整除的二次域及其所有未分歧的循环三次扩张;（3）在研究项目的最后一年，他还准备了一份关于代数数论和解析数论之间相互作用的历史论述。15名研究人员在三年内编写了73份研究论文。其中10份预印本作为研究报告正文的附件二。

项目成果

Katsuya Miyake: "Class Field Theory-Its Centenary and Prospect"Math. Soc. Japan. 632 (2001)

三宅克也：《类场论——百年纪念与展望》数学。

Katsuya Miyake: "Parametrization of the Quadratic Fields whose Class Numbers are divisible by Three (with Y. Kishi)"J. Number Theory. 80. 209-217 (2000)

Katsuya Miyake：“类数可被三整除的二次场的参数化（与 Y. Kishi）”J.

Ken Nakamula: "Some properties of nonstar steps addition chains and new cases where the Scholz conjecture is true (with Hatem〜M. Bahig)"J. Algorithms. (to appear). (2002)

Ken Nakamula：“非星步骤加法链的一些性质和 Scholz 猜想成立的新案例（与 Hatem〜M. Bahig）”J. 算法。

Ki-ichiro Hashimoto: "On Borumers's family of RM-curves of genus two"Tohoku Math.Jour.. 52. 475-488 (2000)

Ki-ichiro Hashimoto：“关于二属 RM 曲线的 Borumers 族”Tohoku Math.Jour.. 52. 475-488 (2000)

Hiroyoshi Yamaki: "Either 71 : 35 or L_2(71) is a Maximal subgroup of the Monster"Adv. Studies in Pure Math, Math. Sco. Japan, Tokyo. 32. 449-451 (2001)

Hiroyoshi Yamaki：“71 : 35 或 L_2(71) 是怪物的最大子群”Adv.