Fundamental groups and applications to arithmetic geometry

基本群是一个著名的拓扑不变量。它的定义方式是记录拓扑空间内的环的信息，但在空间的覆盖分类中有另一个重要的作用。这种基本群的替代观点使类似的对象能够在代数几何中定义，并与伽罗瓦理论等其他领域联系起来，具有重大的现实意义和理论意义。该理论的一个弧涉及“étale基本群”，这在丢番图几何中具有重要性;特别是，格罗滕迪克的部分猜想将某些代数曲线上的有理点与一个与étale基本群相关的规范精确序列的分裂联系起来。此外，贝利定理提供了一个三个穿孔的黎曼球面的拓扑和有理数的绝对伽罗瓦群之间的联系，通过它的基本群。通过它在这个序列中的位置来理解这个绝对伽罗瓦群在几个领域有着重要的应用：也许最著名的是代数数论，但也是逆伽罗瓦问题，它询问哪些群可以实现为有理数的绝对伽罗瓦群的逆。此外，部分猜想，虽然目前遥不可及，暗示的方式获得深刻的非阿贝尔“丢番图信息”的合理点曲线使用群论。其他形式的“塔纳基基本群”也存在，并以拓扑基本群的表示论为模型。这些群拥有比étale基本群更丰富的结构，因为它们本身是几何对象，但它们也更易于处理，因为它们所构建的几何信息是“线性化的”。他们密切相关的几个领域的算术重要性，如特殊价值的L-功能，动机和伽罗瓦理论的时期。特别是，主要研究的一个领域是投影线减去三个点的motivic基本群和motivic伽罗瓦群对它的作用，这应该会引起对这个群的周期的作用-多重zeta值。这个研究项目的一个新目标是了解更复杂方案的基本群，例如，穿孔椭圆曲线。动机伽罗瓦群对这样一个基本群的作用应该揭示出可以应用于伽罗瓦周期理论的信息，所以一个关键的目标是获得对这种作用的描述。这种作用与椭圆曲线的精细算术结构之间的相互作用也是研究这一数论数据的一个透镜。这个项目属于EPSRC数论研究领域的福尔斯。

The fundamental group is a well-known topological invariant. It is defined in a way that records information about the loops within a topological space, but has another important role in classifying covers of the space. This alternative view of the fundamental group has enabled similar objects to be defined within algebraic geometry and to be linked to other areas such as Galois theory, to great current and conjectural significance. One arc of the theory involves the "étale fundamental group", which has importance in Diophantine geometry; in particular, Grothendieck's section conjecture relates rational points on certain algebraic curves to splittings of a canonical exact sequence associated to étale fundamental groups. In addition, Belyi's Theorem provides a link between the topology of the Riemann sphere with three punctures and the absolute Galois group of the rational numbers via its étale fundamental group. Understanding this absolute Galois group through its place in this sequence has major applications in several fields: perhaps most famously to algebraic number theory, but also to the inverse Galois problem, which asks which groups can be realised as quotients of the absolute Galois group of the rationals. In addition the section conjecture, although currently out of reach, hints at ways of obtaining profoundly nonabelian "Diophantine information" about rational points on curves using group theory. Other forms of "Tannakian fundamental groups" also exist and are modelled on the representation theory of the topological fundamental group. These groups possess a richer structure than the étale fundamental group as they are geometric objects themselves, but they are also more tractable since the geometric information they are built from is "linearised". They are closely related to several areas of arithmetic importance such as special values of L-functions, motives and a Galois theory for periods. In particular, one area of major research has been the motivic fundamental group of the projective line minus three points and the action of the motivic Galois group on it, which ought to induce an action on the periods of this group - multiple zeta values. A novel aim of this research project is to understand the fundamental groups of more complicated schemes; for example, a punctured elliptic curve. The action of the motivic Galois group on such a fundamental group should reveal information that can then be applied towards the Galois theory of periods, so a key objective is to obtain a description of this action. The interplay between this action and the fine arithmetic structure of the elliptic curve - which can vary in a family, unlike the unique projective line - is also a lens to investigate this number-theoretical data. This project falls within the EPSRC Number Theory research area.