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Properties of mapping class groups related to Galois representations

与伽罗瓦表示相关的映射类组的属性

基本信息

批准号：
13640020
负责人：
ASADA Mamoru
金额：
$ 2.11万
依托单位：
Kyoto Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640020/
关键词：
Galois representation fundamental group mapping class group

项目摘要

Let us consider the moduli space M_<g, n>/Q (Q : the rationals) of n-pointed complete curves of genus g and the universal family of curves over M_<g, n>. The algebraic fundamental group π^<alg>_1 (M_<g, n>) of M_<g, n> acts naturally on the pro-l fundamental group (l : prime) of the general fiber so that we have a monodromy representation. Let Γ^n_g denote the mapping class group of a n-pointed Riemann surface of genus g. The algebraic fundamental group of M_<g, n> 【cross product】 Q^^- is isomorphic to Γ^^^^n_g (^ : profinite completion). In this research, for the purpose of investigating the monodromy representation in the case that (g, n) = (1, 1), we have tried to determine the weighted completion of π^<alg>_1 (M_<1, 1>). We have applied the general theory of the weighted completion (Hain-Matsumoto) to a former result of Ihara, the structure theorem of the projective limit of l-adic Tate modules of Jacobian varieties of modular curves. This leads us to the determination of weighted … More completion of the subgroup π^<alg>_1 (M<1, 1>【cross product】 Q^^-) of π^<alg>_1 (M<1, 1>).Let X be a non-singular algebraic curve over a field k which is obtained from a complete curve of genus g by removing n k-rational points. In the case 2 - 2g- n < 0, the algebraic fundamental group π^<alg>_1 (【cross product】 k^^-) of 【cross product】 k^^- has the following property ; every subgroup with finite index is centerfree. Whether the group Γ^^^^n_g also has this property or not is an open problem. In order for it to have this poperty, it is necessary that its dense subgroup Γ^n_g also has the same property, and this is known. We have given, under the assumption that n 【greater than or similar】 1, an alternative proof of this fact.On the other hand, let k be a finite algebraic number field and k_∞ denote the field obtained by adjoining all roots of unity to k. Let M be the maximum unramified Galois extension of k_∞. The Galois group Gal (M/k_∞) is regarded as an analogue, in algebraic number fields, to the group π^<alg>_1 (X 【cross product】 k^^-). In this research, we have shown that Gal (M/k_∞) and Gal (M/k) both have the above property. Less

我们考虑g属的n点完全曲线的模空间M_<g, n>/Q （Q：有理数）和M_<g, n>上的曲线全称族。M_<g, n>的代数基群π^<alg>_1 （M_<g, n>）自然作用于一般纤维的前1基群（l: prime），从而得到一个单态表示。设Γ^n_g表示g属的n点黎曼曲面的映射类群。M_<g, n>【外积】Q^^-的代数基本群同构于Γ^^^^n_g（^：无限补全）。在本研究中，为了研究(g, n) =(1,1)情况下的单态表示，我们试图确定π^<alg>_1 （M_< 1,1 >）的加权补全。我们将加权补全的一般理论（Hain-Matsumoto）应用于Ihara先前的一个结果，即模曲线Jacobian变体的l-adic Tate模的投影极限的结构定理。进一步完成π^<alg bbb_1 （M< 1,1 >）的子群π^<alg bbb_1 （M< 1,1 >）的【外积】Q^^-。设X为域k上的非奇异代数曲线，该曲线由g属的完全曲线去掉n个k个有理点得到。在2 - 2g- n < 0的情况下，（外积）k^^-的代数基本群π^<alg>_1（外积）k^^-有如下性质；具有有限索引的子群是无中心的。组Γ^^^^n_g是否也具有此属性是一个开放的问题。为了使它具有这种性质，它的密集子群Γ^n_g也必须具有同样的性质，这是已知的。在n大于或类似于1的假设下，我们给出了这个事实的另一种证明。另一方面，设k为有限代数数域，k_∞表示将k的所有单位根相邻得到的域。设M为k_∞的最大无分支伽罗瓦扩展。在代数数域中，伽罗瓦群Gal （M/k_∞）与群π^<alg>_1 （X【外积】k^^-）是类似的。在本研究中，我们证明了Gal （M/k_∞）和Gal （M/k）都具有上述性质。少