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

• 批准号: 09640033
• 负责人: ASADA Mamoru
• 金额: $ 1.73万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640033/
• 关键词: Galois representation; fundamental group; mapping class group

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 1 be a prime number. The absolute Galois group of k acts naturally on the algebraic fundamental group pi_1^<alg> of X <cross product> k (or pro-1 fundamental group (the maximal pro-l quotient of pi_1^<alg>)) 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>. In this research, we have investigated the faithfulness of this monodromy representation and have shown that, in the case that g = 0, 1, it is faithful for all n.In investigating Galois representations, one of basic properties of the group pi_1^<alg> (*pi_1^<alg>(g, n) ) is that it is center free (M.P.Anderson). In this research we have shown, by a comparatively elementary way, a property which is stronger than this.The braid group of a finitely generated free nilpotent pro-I group can be regarded as approximating the genus 0 pro-l mapping class groups. In this research we have shown that this group has a structure an algebraic group which is independent of l.

设X是特征为0的域k上的一条非奇异代数曲线，它是亏格g(&lt；大于或等于&gt；0)的完全曲线去掉n(&lt；大于或等于&gt；0)k-有理点(2-2g-n.&lt；0)而得到的，1是素数。K的绝对伽罗瓦群自然作用于X&lt；叉积&gt；k的代数基础群pi_1^&lt；alg&gt；(或PRO-1基本群(pi_1^&lt；alg&gt；的最大PRO-L商)，从而得到伽罗瓦表示。让我们考虑亏格G的n点完全曲线的模空间M_&lt；g，n&gt；/q(q为有理数)和M_&lt；g，n&gt；上的泛曲线族。然后，M&lt；g，n&gt；的代数基本群自然地作用于一般纤维的代数基本群，因此我们有一个单行表示。(T.Oda已经提供了一个基金会。)这是在曲线X是泛曲线的情况下的伽罗瓦表示，k是M&lt；g，n&gt；的函数域。在研究Galois表示时，群pi_1^&lt；alg&gt；(*pi_1^&lt；alg&gt；(g，n))的一个基本性质是它是无中心的(M.P.Anderson)。在这一研究中，我们用一个比较初等的方法证明了一个比这更强的性质：有限生成的自由幂零PRO-I群的辫子群可以看作近似于亏格0的PRO-L映射类群。在这一研究中，我们证明了这个群有一个独立于L的代数群结构。

项目成果

T.Yagasaki: "The groups of quasiconformal homeomorphisms of Riemann surfaces" Proc.Amer.Math.Soc.(to appear).

T.Yagasaki：“黎曼曲面的拟共形同胚群”Proc.Amer.Math.Soc.（即将出现）。

A.Nakaoka (with A.Masuda): "A method of wavelet construction from n-MRA" Mem.Fac.Eng.and Design, Kyoto Inst.Tech.(to appear).

A.Nakaoka（与 A.Masuda）：“一种来自 n-MRA 的小波构造方法”Mem.Fac.Eng.and Design，Kyoto Inst.Tech（待发表）。

T.Yagasaki: "The groups of quasiconformal homeomorphisms of Riemann surfaces" Proc.Amer.Math.Soc.to appear.

T.Yagasaki：“黎曼曲面的拟共形同胚群”Proc.Amer.Math.Soc. 出现。

F.Maitani(with K.Nishikawa): "Moduli of ring domains obtained by a conformal welding" Kodai Math.J.20. 161-171 (1997)

F.Maitani（与 K.Nishikawa）：“通过保形焊接获得的环域模量”Kodai Math.J.20。

A.Nakaoka(with A.Masuda): "A method of wavelet construction from n-MRA" Mem.Eac.Eng.and Design, Kyoto Inst.Tech.(to appear).

A.Nakaoka（与 A.Masuda）：“一种来自 n-MRA 的小波构造方法”Mem.Eac.Eng.and Design，Kyoto Inst.Tech（即将出现）。