Geometric Structures in the p-adic Langlands programme

The goal of my proposed research is the development of new geometric tools in the p-adic part of the Langlands program, including the beginnings of a p-adic theory of endoscopy.

The Langlands program is a central theme in modern mathematics, with its roots as a bridge between harmonic analysis and number theory. It has deep links to dualities in mathematical physics, and to enumerative geometry.

The "p-adic Langlands program" is an outgrowth of the Langlands program that focuses on the connection to algebraic number theory.

It has produced many of the recent highlights in number theory, including the proof of Fermat´s Last Theorem and progress on the Birch--Swinnerton-Dyer conjecture, but its basic structure remains a mystery.

The proposal consists of two main themes, the construction a (geometric) p-adic local Langlands correspondence for SL(2,Qp) and the study of eigenvarieties for GL(n).

We intend to show how geometry naturally produces a p-adic theory of endoscopy, which is an aspect of harmonic analysis on non-abelian groups that is key in the Langlands program, with far-reaching consequences for algebraic number theory. So far no counterpart exists in the p-adic program.

This would provide a key step in our understanding of the p-adic Langlands program, and help guide us towards a fundamental understanding of the nature of the p-adic Langlands program. Through the study of eigenvarieties, we will also make progress on the global Langlands correspondence.