Exterior Galois representations in fundamental groups and associated arithmetic phenomena

点击翻译按钮获取中文摘要

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.