In mathematics, the Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by Robert Langlands.
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- Arithmetic automorphic representation theory is one of the most active areas in current mathematical research, centering around what is known as the Langlands program.
- The functoriality conjecture is at the heart of the Langlands program and will undoubtedly remain as a challenge to number theorists for many decades to come. Shortly after formulating his program, however, Langlands proposed to test it in two interdependent settings. The first was the framework of Shimura varieties, already understood by Shimura as a natural setting for a non-abelian generalization of the Shimura-Taniyama theory of complex multiplication. The second was the phenomenon of endoscopy, which can be seen alternatively as a classification of the obstacles to the stabilization of the trace formula or as an opportunity to prove the functoriality conjecture in some of the most interesting cases. After three decades of research, much of it by Langlands and his associates, these two closely related experiments are coming to a successful close, at least for classical groups, thanks in large part to the recent proof of the so-called Fundamental Lemma by Waldspurger, Laumon, and especially Ngô.