Mild pro-p-Galois groups of algebraic number fields

代数数域的温和亲 p 伽罗瓦群

• 批准号: 231696112
• 负责人: Dr. Jochen Gärtner
• 金额: --
• 依托单位: Department of Mathematics and Statistics
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2012-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/231696112?language=en
• 关键词: Mild; pro; Galois; groups; algebraic

This research project aims at further developing the theory of mild pro-p-groups as introduced by John Labute as well as studying their applications to duality properties of extensions of algebraic number fields with restricted ramification. The central object is given by the Galois group G_S(p) of the maximal p-extensions of an algebraic number field unramified outside a finite set of places S in the tame case. It is proposed to investigate group theoretical properties of mild pro-p-groups, in particular their relation to higher order Massey products. Furthermore, the objective is to give arithmetic realizations of these groups.The stay at McGill University, Montreal, Canada would provide the opportunity of a collaboration with Professor John Labute.The goals of the research project are the following:1) Give a description of higher order Massey products and Milnor invariants of pro-p-groups of the form G_S(p) in terms of Galois representations and construct explicit examples of mild pro-p-groups with arbitrary Zassenhaus invariant2) Study FAB pro-p-groups, give a purely algebraic or group theoretical description of non-analytic FAB-groups using properties of their associated graded Lie algebras, study applications to FAB-extensions of algebraic number fields3) Give an answer of Serre's question about the classification of one-relator pro-p-groups of cohomological dimension greater than 2 as quotients of free groups by p-th powers

本研究项目旨在进一步发展由John Labute引入的温和亲p-群理论，并研究其在具有限制分支的代数数域扩张的对偶性质中的应用。中心对象由有限个位置集S外非分歧的代数数域的极大p-扩张的伽罗瓦群G_S（p）给出。研究了弱亲p-群的群论性质，特别是它们与高阶Massey积的关系。此外，目标是给这些群体的算术实现。在加拿大蒙特利尔麦吉尔大学的逗留将提供与John Labute教授合作的机会。研究项目的目标如下：1）用Galois表示刻画了G_S（p）型pro-p-群的高阶Massey积和Milnor不变量，构造了具有任意Zassenhaus不变量的轻度pro-p-群的显式例子。2）研究FAB pro-p-群，给出了一个纯粹的代数或群论描述的非解析FAB-群使用的性质，其相关的分次李代数，研究了代数数域的FAB-扩张的应用3）回答了Serre关于单关系pro-p-的分类问题，上同调维数大于2的群作为自由群的p次幂的同调群

项目成果

On $p$-class groups and the Fontaine-Mazur conjecture

关于 $p$ 级群和 Fontaine-Mazur 猜想

• DOI: 10.4310/mrl.2014.v21.n3.a5
• 发表时间: 2014
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: J. Gärtner

Higher Massey products in the cohomology of mild pro-p-groups

轻度 Pro-p 基团的上同调中的更高 Massey 积

• DOI: 10.1016/j.jalgebra.2014.07.023
• 发表时间: 2015
• 期刊: arXiv: Number Theory
• 作者: J. Gärtner