Local global principles for torsors under fundamental groups

数论和算术几何中一个非常重要的原理是局部整体原理，特别是代数变量有理点的局部整体原理，更一般地说是各种群下环量的局部整体原理。给定在数域上定义的各种数学对象和结构，目的是将这些结构的属性与在数域的不同定位上观察的相同结构的属性联系起来；所谓的局部领域。Christopher Haines的研究项目旨在建立若干非阿贝尔伽罗瓦上同群的局部全局原理，这些非阿贝尔伽罗瓦上同群是数域上曲线的非阿贝尔基群。在数域上双曲曲线的可解基本群下，这种局部全局原理是已知的。本课题的主要目标是建立一个类似的原理，用于数域上仿射曲线的亲可解基本群。

A much important principle in number theory and arithmetic geometry is the local global principle, especially the local global principle for rational points of algebraic varieties and more generally the local global principle for torsors under various groups. Given various mathematical objects and structures defined over number fields one aims to relate properties of these structures to those of the same structure but viewed over the various localisations of the number field; the so-called local fields. The research project of Christopher Haines aims to establish new local global principles for certain non abelian Galois cohomology groups with coefficients which are non abelian fundamental groups of curves over number fields. Such a local global principle is known for torsors under pro-solvable fundamental groups of hyperbolic curves over number fields. The main objective of this project is to establish a similar principle for pro-solvable fundamental groups of affine curves over number fields.