Project/Area Number  10640034 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Tokyo Metropolitan University 
Principal Investigator 
NAKAMURA Hiroaki Tokyo Metropolitan University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (60217883)

CoInvestigator(Kenkyūbuntansha) 
KAWASAKI Takeshi Tokyo Metropolitan University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助手 (40301410)
TAKEDA Yuichirou Tokyo Metropolitan University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助手 (30264584)
MIYAZAKI Takuya Tokyo Metropolitan University, Department of Mathematics, Assistant Professor, 大学院・理学研究科, 助手 (10301409)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 1998 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  Galois Group / fundamental group / Galois representation / anabelian geometry / exterior Galois representation / mapping class group / GrothendieckTeichmuller group / arithemtic fundamental group / ガロア群 / 基本群 / ガロア表現 / 遠アーベル幾何学 / 外ガロア表現 / 写像類群 / グロタンディーク・タイヒミュラー群 / 数論的基本群 / タイヒミュラーモジュラー群 / 楕円曲線 
Research Abstract 
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 GrothendieckTeichmuller 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 2complex 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.
