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Properties of mapping class groups rolated to Gdois representations

与 Gdois 表示相关的映射类组的属性

基本信息

批准号：
11640026
负责人：
ASADA Mamoru
金额：
$ 2.3万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640026/
关键词：
Galois nepnesentation fundamental group mapping class group

项目摘要

Let X be a non-singular algebraic curve over a field k of characteristic 0 which is obtained from a complete curve of genus g (【greater than or equal】 0) by removing n (【greater than or equal】 0) k-rational points (2-2g-n<0), and l be a prime number. The absolute Galois group of k acts naturally on the algebraic fundamental group π^<alg>_1 of X【cross product】k^^- (or pro-l fundamental group (the maximal pro-l quotient of π^<alg>_1)) so that we obtain a Galois representation.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>. Then the algebraic fundamental group of M_<g, n> acts naturally on that of the general fiber so that we have a monodromy representation. (A foundation has been given by T.Oda.) This is the Galois representation in the case that the curve X is the universal curve, k being the function field of M_<g, n>. Let π_1(g, n) and Γ^n_g denote the fundamental group and the ma … More pping class group of a Riemann surface of genus g (【greater than or equal】 0) with n (【greater than or equal】 0) punctures respectively. Then the algebraic fundamental group of M_<g, n> 【cross product】 Q^^- and that of the general fiber are isomorphic to Γ^^<^>^n_g and π^^<^>_1 (g, n) respectively (^ : profinite completion). The natural action ρ_<g, n> of Γ^^<^>^n_g on π^^<^>_1 (g, n) is nothing but the (geometric part of) the monodromy representation. The group Γ^^<^>^n_g acts also on the pro-l fundamental group π^<(l)>_1(g, n), which is the pro-l completion of π_1(g, n), and we obtain a Galois representation ρ^<(l)>_<g, n>. In this research, we have investigated the kernels of the representations ρ_<g, n> and ρ^<(l)>_<g, n>. So far, the kernel of ρ^<(l)>_<g, n> has been known only in the case of g=0. In the case that l=2, by applying the method to prove the faithfulness of ρ_<1, 1>, the kernel of ρ^<(2)>_<1, 1> has been determined.On the other hand, whether the center of any open subgroup of Γ^^<^>^n_g is trivial or not is an open problem. (This is related to whether M_<g, n > is "anabelian" or not). We have shown that, if the representation ρ_<g, n> is faithful, then the center of any open subgroup of Γ^^<^>^<n+1>_g is trivial. Less

设X是特征为0的域k上的非奇异代数曲线，它是从亏格为g（0）的完备曲线中去掉n（0）个k-有理点（2-2g-n<0）得到的，l是素数。k的绝对Galois群自然地作用在<alg>X[叉积]k^^-的代数基本群π^1（或pro-l基本群（π^1的最大pro-l商<alg>））上，从而得到一个Galois表示.考虑亏格为g的n点完全曲线的模空间M_<g，n>/Q（Q：有理数）和M_<g，n>上的泛曲线族.然后M_<g，n>的代数基本群自然地作用在一般纤维的代数基本群上，从而得到单值表示。（T. Oda提供了一个基础。）这是曲线X是万有曲线，k是M_<g，n>的函数域时的伽罗瓦表示。设π_1（g，n）和Γ^n_g表示基本群， ...更多信息分别给出了亏格为g（[大于或等于] 0）的Riemann曲面具有n（[大于或等于] 0）个穿孔的一个pping类群。则M_<g，n> [叉积] Q^^-的代数基本群与一般纤维的代数基本群分别同构于Γ^n_g和π^_1（g，n）（^：profinite completion）。r ^^<^>^n_g对π^^<^>_1（g，n）的自然作用ρ_<g，n>只不过是单值表示的（几何部分）。群Γ^n_g也作用在pro-l基本群π^（l）>_1（g，n）上，它是π_1（g，n）的pro-l完备化，我们得到了一个Galois表示ρ^（l）>_<g，n>.在本研究中，我们研究了表示ρ_<g，n>和ρ^<（l）>_<g，n>的核。到目前为止，ρ^<（l）>_<g，n>的核仅在g=0的情况下才被知道。在l=2的情形下，应用证明ρ <1，1>的忠实性的方法，确定了ρ^<（2）><1，1>的核，而关于Γ^^<^>^n_g的任何开子群的中心是否平凡则是一个公开问题. (This与M_<g，n >是否为“Anabelian”有关）。我们证明了，如果表示ρ_g，n>是忠实的，则Γ^^<^>^<n+1>_g的任何开子群的中心是平凡的.少