Around Talking, the reciprocity conjecture gives a correspondence involving automorphic representations of the reductive group and homomorphisms from a Langlands group to an L-team. You'll find many versions of this, in part since the definitions of Langlands team and L-group are certainly not fixed.
In an exceptionally broad context, the program constructed on current Concepts: the philosophy of cusp types formulated a couple of years earlier by Harish-Chandra and Gelfand (1963), the work and approach of Harish-Chandra on semisimple Lie teams, and in complex terms the trace system of Selberg and others.
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Ordinarily, a contribution that could be attributed to a specific new observing program is normally modest. In that context, the magnitude of your affect of your DWL is exceedingly scarce.
In straightforward cases, it relates l-adic representations from the étale basic team of the algebraic curve to objects with the derived classification of l-adic sheaves to the moduli stack of vector bundles about the curve. Present standing[edit]
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The so-named geometric Langlands method, suggested by Gérard Laumon subsequent Tips of Vladimir Drinfeld, arises from a geometrical reformulation of the standard Langlands software that attempts to relate additional than just irreducible representations.
will get nearer and closer to 1. So when you understand that every one of these functorial lifts are in truth automorphic, as Langlands predicts, you’ve just solved Ramanujan. What a clever trick!
Broadly witnessed as The one most important challenge in modern day mathematical analysis, the Langlands application is described by Edward Frenkel as "a type of grand unified theory of arithmetic."[one]
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Automorphic varieties on reductive teams above world wide fields (with subcases akin to number fields or perform fields).
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