Mild pro-p-Galois groups of algebraic number fields

本课题旨在进一步发展John Labute提出的温和亲-对群理论，并研究其在有限分枝代数数域扩展对偶性质中的应用。在一般情况下，一个代数数域在有限位置集S之外的最大p扩展的伽罗瓦群G_S(p)给出了中心目标。提出研究温和的亲-对群的群理论性质，特别是它们与高阶Massey积的关系。此外，目标是给出这些群的算术实现。在加拿大蒙特利尔麦吉尔大学逗留期间，将有机会与约翰·拉布特教授合作。本课题的研究目标如下：1)用伽罗万表示描述G_S(p)形式的亲-p群的高阶Massey积和Milnor不变量，并构造具有任意Zassenhaus不变量的温和亲-p群的显式例子；2)研究FAB亲-p群，利用其相关的梯度李代数的性质给出非解析FAB-群的纯代数或群理论描述；3)给出了Serre关于上同调维数大于2的单相关亲-对群作为自由群的商的p次幂分类问题的答案

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