Fundamental groups and applications to arithmetic geometry

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute


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.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509711/1 01/10/2016 30/09/2021
1789793 Studentship EP/N509711/1 01/10/2016 31/03/2020 Alexander Saad
Description A key finding I have made is that the (motivic) periods of the fundamental group of the moduli space of elliptic curves have a faithful action by the Galois group of mixed Tate motives over the integers. At a more concrete level, this means that every multiple zeta value - real numbers defined by certain infinite series that were known to Euler - can be written as an iterated integral of Eisenstein series on the upper half plane, and moreover that this extends to fit in with the "Galois theory of periods" in that it is motivic.

This is a new result in a somewhat orthogonal vein to Oda's Conjecture on the action of the motivic Galois group on certain derivations, and can be seen as explaining Ramanujan's formulae for the odd zeta values in terms of Lambert series.

The proof method relies heavily on the arithmetic fundamental groups that are the focus of this topic; chiefly, I have been studying the relationship between (the motivic Galois actions on) the relative fundamental group of the moduli space of elliptic curves, based at a tangential basepoint corresponding to the Tate curve, and the de Rham fundamental group of the projective line minus three points. The two are related via the intermediary fundamental group of the punctured Tate elliptic curve, on which the former acts via monodromy and the latter injects into. The periods of the fundamental group of P^1 minus three points are multiple zeta values, and those of the relative fundamental group of M_{1,1} are iterated integrals of modular forms. The interplay between the monodromy action and the map of fundamental groups means I can apply a combinatorial argument to detect periods in the fundamental group of M_{1,1} by their presence in the fundamental group of the punctured Tate curve.
Exploitation Route It would be interesting to see if the proof can be made effective (algorithmic). I have made some progress towards a method to do this using motivic Galois theory.
In a different direction, it would be a good project to study varying the basepoint on M_{1,1} and, instead of the Tate curve, take e.g. a CM elliptic curve as the basepoint. We could then study how the periods of this object are related to the CM endomorphism.
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