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Fundamental lemma (Langlands program)
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by Ngô for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. ''Time'' magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009".〔(Top 10 Scientific Discoveries of 2009 ), ''Time''〕 In 2010 Ngô was awarded the Fields medal for this proof.
== Motivation and history ==
Robert Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be related in a particular way. This relationship takes the form of identities between orbital integrals on reductive groups ''G'' and ''H'' over a nonarchimedean local field ''F'', where the group ''H'', called an endoscopic group of ''G'', is constructed from ''G'' and some additional data.
The first case considered was ''G'' = ''SL''2 . then developed the general framework for the theory of endoscopic transfer and formulated specific conjectures. However, during the next two decades only partial progress was made towards proving the fundamental lemma.〔Kottwitz and Rogawski for ''U''3, Wadspurger for ''SL''''n'', Hales and Weissauer for ''Sp''4.〕〔(Fundamental Lemma and Hitchin Fibration ), Gérard Laumon, May 13, 2009〕 Harris called it a "bottleneck limiting progress on a host of arithmetic questions".〔(INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLICATIONS” ), p. 1., Michael Harris〕 Langlands himself, writing on the origins of endoscopy, commented:
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