Integral Structures in the Langlands Programme

Number Theory is the study of the integers and their arithmetic applications. While problems in Number Theory can be easy to state, their solutions often become extremely intricate. For example, Fermat's Last Theorem - was formulated in the 17th century, yet only resolved in the 1990's.

A fundamental approach in mathematics is to transform a seemingly difficult problem from one area to another, where it becomes tractable or even obvious. A famous example, is Wiles' proof of Fermat's Last Theorem; the key change in perspective transforming a problem about certain arithmetic objects (Galois representations of elliptic curves) into one about analytic objects (modular forms).

This correspondence established by Wiles completing the proof of Fermat's Last Theorem is a very special case of a broad web of predicted correspondences and connections between analysis and arithmetic, collectively known as the Langlands Programme. The Local Langlands Programme is the specialization of the Langlands Programme at a prime number, and this is where the bulk of the research of our project takes place.

The language of the Local Langlands Programme is in a branch of Algebra called Representation Theory, which deals with symmetries of spaces. The Local Langlands Programme is a deep statement that certain fundamental symmetries of finite dimensional spaces which arise in Number Theory can be understood in terms of completely different symmetries of infinite dimensional spaces, and conversely.

The finite and infinite dimensional spaces considered are built on top of the complex numbers. A natural question now arises, why the complex numbers? Is there a more fundamental arithmetic connection hiding behind this? In this project, using explicit constructions of representations, we study integral structures in the Local Langlands Programme and their relation.

The research programme will build expertise in UK mathematics in the Langlands Programme, including the training of a PhD student and a postdoctoral research associate. The Langlands Programme is a web of open problems and partial results in number theory connecting arithmetic, analysis, algebra, and geometry. Even partial results in the Langlands Programme have strong consequences, for example Fermat's Last Theorem, and establishing the results in the programme would represent significant progress and increase the profile of UK mathematics forging new ties between Sheffield and researchers worldwide

In recent years, number theory has come to the forefront in applications to modern cryptography. While the proposal does not address any applications to cryptography, the research will increase our understanding of structures involved (for example elliptic curves).