Exterior Galois representations in fundamental groups and associated arithmetic phenomena

基本群和相关算术现象中的外伽罗瓦表示

• 批准号: 10640034
• 负责人: NAKAMURA Hiroaki
• 金额: $ 1.92万
• 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 (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640034/
• 关键词: Galois Group; fundamental group; Galois representation; anabelian geometry; exterior Galois representation; mapping class group; Grothendieck-Teichmuller group; arithemtic fundamental group; タイヒミュラーモジュラー群; 楕円曲線

Last year, we investigated main descriptions of the Galois action on the profinite Teichmuller modular groups of higher genera in terms of standard parameters in the Grothendieck-Teichmuller group "GT", especially introduced a refined version of GT. I wrote up a joint paper on this subject with L. Schneps, and submitted it to an international mathematics journal "Inventiones Mathematicae". In this yea, according to the advice of the referee's report on our paper, we had made a number of improvements of the description of the above result by increasing the paper with additional implements. This paper has been accepted for publication in the above journal in January 2000. In this revision process, the new notion - the quilt decompositions of Riemann surfaces and a 2-complex formed by them - turns out to be very useful and essential, and we find new possibilities of applying it to analyze several other aspects of the mapping class groups. This should be one of the important themes of future studies.On the other hand, concerning the new method of using remified covering of Riemann surfaces to produce new equations of the Galois images in GT, we found a few more new aspects. For example, one can get a nontrivial geometric interpretation of certain tangential base points arising in the universal family of elliptic curves and its relations with parametric family of algebraic equations. To get a more synthetic viewpoint for describing these phenomena, it is necessary to continue comparative studies of several related areas and information available from various sources.

去年，我们研究了Grothendieck-Teichmuller群“GT”中基于标准参数的高属无限Teichmuller模群上伽罗瓦作用的主要描述，特别介绍了GT的一个改进版本。我与L. Schneps就这一主题共同撰写了一篇论文，并提交给了国际数学期刊《Inventiones Mathematicae》。在这一年中，根据我们论文的推荐人报告的建议，我们对上述结果的描述做了一些改进，增加了论文的附加工具。该论文已于2000年1月被上述期刊接受发表。在这个修正过程中，新的概念——黎曼曲面的被子分解和由它们形成的2-复形——变得非常有用和必要，并且我们发现了应用它来分析映射类群的其他几个方面的新的可能性。这应该是未来研究的重要主题之一。另一方面，关于利用黎曼曲面的简化覆盖生成伽罗瓦像新方程的新方法，我们又发现了一些新的方面。例如，可以得到椭圆曲线泛族中某些切向基点的非平凡几何解释及其与代数方程参数族的关系。为了获得一个更综合的观点来描述这些现象，有必要继续对几个相关领域和各种来源的信息进行比较研究。

项目成果

Hiroaki, Nakamura, Stavros Garoufalidis: "Some IHX-type relations on trivalent graphs and symplectic representation theory"Marh. Res. Letters. 5. 391-402 (1998)

Hiroaki、Nakamura、Stavros Garoufalidis：“三价图和辛表示论上的一些 IHX 型关系”Marh。

Hiroaki Nakamura: "Limits of Galois representations in fundamental groups I"American Journal of Mathematics. 121. 315-358 (1999)

Hiroaki Nakamura：“基本群中伽罗瓦表示的极限 I”美国数学杂志。

Hiroaki Nakamura: "On a swbgroup of the Grothendieck-Teichmuller group"Inventiones mathematicae. (発売予定).

Hiroaki Nakamura：“关于 Grothendieck-Teichmuller 小组的一个 swbgroup”数学发明（待发布）。

Hiroaki Nakamura: "Tangential base points and Eisenstein power series"London Math. SOC. Lecture Note Series. 256. 202-217 (1999)

Hiroaki Nakamura：“切向基点和爱森斯坦幂级数”伦敦数学。

Hiroaki Nakamura: "Limits of Galois representations in fundamentel groups I"American Journal of Mathematics. 121. 315-358 (1999)

Hiroaki Nakamura：“基础群 I 中伽罗瓦表示的极限”美国数学杂志。