The fundamental lemma and the Hitchin fibration [after Ngo Bao Chau]

TL;DR

This paper reviews Ngo Bao Chau's proof of the fundamental lemma, a key identity in number theory linked to the cohomology of Hitchin fibration fibers, with major implications for automorphic representations.

Contribution

It explains Ngo Bao Chau's novel geometric approach to proving the fundamental lemma, connecting it to the Hitchin fibration and cohomological methods.

Findings

Proof of the fundamental lemma established

Connections to automorphic representations clarified

Implications for number theory demonstrated

Abstract

This article is a Bourbaki seminar report on Ngo Bao Chau's proof of the fundamental lemma. About thirty years ago, R. P. Langlands conjectured a collection of identities to hold among integrals over conjugacy classes in reductive groups. Ngo Bao Chau has proved these identities (collectively called the fundamental lemma) by interpreting the integrals in terms of the cohomology of the fibers of the Hitchin fibration. The fundamental lemma has profound consequences for the theory of automorphic representations. Significant recent theorems in number theory use the fundamental lemma as an ingredient in their proofs.