Exterior Galois representations in fundamental groups and associated arithmetic phenomena

去年，我们用Grothendieck-Teichmuller群“GT”中的标准参数研究了GT上的Galois作用的主要刻画，特别是引入了GT的一个精化形式。我与L.Schneps就这一主题共同撰写了一篇论文，并将其提交给国际数学期刊《数学发明》。在这一年里，根据裁判对我们试卷的报告的建议，我们对上述结果的描述进行了一些改进，增加了试卷的额外工具。这篇论文已于2000年1月在上述期刊上发表。在这个修正过程中，新的概念-黎曼曲面的被子分解及其形成的2-复形-被证明是非常有用和必要的，我们发现了将其应用于分析映射类群的其他几个方面的新的可能性。另一方面，对于利用Riemann曲面的离化覆盖来产生GT中伽罗华象的新方程的新方法，我们发现了一些新的方面。例如，我们可以得到泛椭圆曲线族中某些切向基点的非平凡几何解释，以及它与参数代数方程族之间的关系。为了获得更全面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。

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.