Fundamental groups and applications to arithmetic geometry

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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.