Fundamental groups and applications to arithmetic geometry

基本群是一个众所周知的拓扑不变量。它以一种记录拓扑空间内环路信息的方式定义，但在对空间的覆盖进行分类方面具有另一个重要作用。这种对基本群的另一种看法，使代数几何中的类似对象得以定义，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推测意义。该理论的一个弧线涉及到“<s:1>基本群”，这在丢番图几何中很重要；特别地，Grothendieck的截面猜想将某些代数曲线上的有理点与与<s:1>基本群有关的标准精确序列的分裂联系起来。此外，Belyi定理通过其<s:1>基本群（samtale fundamental group）提供了黎曼球面三穿孔拓扑与有理数绝对伽罗瓦群之间的联系。通过其在数列中的位置来理解绝对伽罗瓦群在几个领域有重要的应用：也许最著名的是代数数论，但也适用于反伽罗瓦问题，该问题要求哪些群可以作为绝对伽罗瓦群的有理数的商。此外，截面猜想虽然目前还无法实现，但它暗示了利用群论获得曲线上有理点的深刻非abel“丢芬图信息”的方法。其他形式的“Tannakian基本群”也存在，并以拓扑基本群的表示理论为模型。这些群体拥有比<s:1>基本群体更丰富的结构，因为它们本身就是几何对象，但它们也更容易处理，因为它们所依据的几何信息是“线性化的”。它们与几个重要的算术领域密切相关，如l函数的特殊值、动机和周期的伽罗瓦理论。特别是，一个主要的研究领域是投影线减去三点的动机基本群和动机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-多个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，一个穿孔的椭圆曲线。动机伽罗瓦群在这样一个基本群上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这种作用的描述。这种作用和椭圆曲线的精细算术结构之间的相互作用——椭圆曲线可以在一个家族中变化，不像唯一的投影线——也是研究这个数论数据的一个镜头。该项目属于EPSRC数论研究领域。

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.