The arithmetic of p-adic automorphic forms and Galois representations

One of the fundamental problems in number theory is to understand linear representations of the Galois groups of number fields. These encode structure and symmetries of the number fields (for example the rational numbers).The Langlands programme seeks to study these Galois representations using automorphic forms - these are special kinds of analytic functions which are conjecturally (and sometimes provably) related to Galois representations. Initiated in the 1970s, the work of Langlands and others has been extremely influential, playing a crucial part in Wiles and Taylor's celebrated proof of Fermat's last theorem, as well as more recent results on conjectures of Serre and Sato-Tate.In recent years number theorists have been to develop an extension of the Langlands programme which seeks to understand the finer p-adic structure of automorphic forms and Galois representations (particularly the way they move in p-adic families). This proposal focuses on developing this `p-adic Langlands programme', which is currently only understood in special cases.Specific aims of the proposal include proving some cases of p-adic Langlands functoriality, allowing one to move between p-adic automorphic forms on different groups, by studying the arithmetic of Shimura varieties for unitary groups. Secondly, studying p-adic Banach space representations of GL_2(K), where K is a finite extension of Q_p, using tools from arithmetic geometry and the completed cohomology of Shimura curves.

As with many projects in pure mathematics, my proposed research will primarily have an impact on the academic research community. Having said that, work in pure mathematics can have unforeseen applications in the long term (for example the role of elliptic curves over finite fields in modern cryptography). In order to maximise the impact of my research I will ensure that it is appropriately communicated to the widest possible range of other mathematicians. I will publish my research both in freely accessible online preprint repositories (e.g. the arXiv) and widely-circulated peer-reviewed journals. I will also take opportunities to present my work at seminars and conferences, and endeavour to discuss my work in a way which is informative for non-specialists (if this is appropriate for the audience). An important role of research mathematicians is also to pass skills and knowledge on to students and other researchers. As a member of the London number theory group, I will have the opportunity to organise regular study groups, which bring together graduate students and established researchers to learn about current developments in number theory, as well as more foundational material. For example in the last year I organised a study group for graduate students to learn about class field theory. There will also be opportunities to gain more formal teaching experience at UCL.