Editing Langlands program (section)

==Status==
The Langlands correspondence for GL(1, ''K'') follows from (and are essentially equivalent to) [[class field theory]].

Langlands proved the Langlands conjectures for groups over the archimedean local fields <math>\mathbb{R}</math> (the [[real number]]s) and <math>\mathbb{C}</math> (the [[Complex number|complex numbers]]) by giving the [[Langlands classification]] of their irreducible representations.

Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.

[[Andrew Wiles]]' proof of modularity of semistable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for <math>\text{GL}(2,\mathbb{Q})</math> remains unproved.

In 1998, [[Laurent Lafforgue]] proved [[Lafforgue's theorem]] verifying the global Langlands correspondence for the general linear group GL(''n'', ''K'') for function fields ''K''. This work continued earlier investigations by Drinfeld, who previously addressed the case of GL(2, ''K'') in the 1980s.

In 2018, [[Vincent Lafforgue]] established one half of the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields.<ref>{{cite web |website=icm2018.org |url=http://www.icm2018.org/portal/program-at-a-glance.html |first=V. |last=Lafforgue |title=Shtukas for reductive groups and Langlands correspondence for function fields |year=2018 |arxiv=1803.03791}} {{cite web |title=alternate source |website=math.cnrs.fr |url=http://vlafforg.perso.math.cnrs.fr/files/cht-ICM-lafforgue.pdf}}</ref><ref>{{cite journal |first=V. |last=Lafforgue |journal=Journal of the American Mathematical Society |volume=31 |pages=719–891 |year=2018 |title=Chtoucas pour les groupes réductifs et paramétrisation de Langlands |doi=10.1090/jams/897 |url=https://www.ams.org/journals/jams/2018-31-03/ |arxiv=1209.5352|s2cid=118317537 }}</ref><ref>{{cite conference |first=B. |last=Stroh |date=January 2016 |title=La paramétrisation de Langlands globale sur les corps des fonctions (d'après Vincent Lafforgue) |conference=Séminaire Bourbaki 68ème année, 2015–2016, no.&nbsp;1110, Janvier&nbsp;2016 |url=https://webusers.imj-prg.fr/~benoit.stroh/bourbaki.pdf}}</ref>

===Local Langlands conjectures===
{{main|local Langlands conjectures}}

{{harvs|txt|last=Kutzko|first=Philip |authorlink=Philip Kutzko|year=1980}} proved the [[local Langlands correspondence]] for the general linear group GL(2, ''K'') over local fields.

{{harvs|txt|last=Laumon|first=Gérard |last2=Rapoport |author1-link=Gérard Laumon |first2=Michael |author2-link=Michael Rapoport |last3=Stuhler |first3=Ulrich |author3-link=Ulrich Stuhler |year=1993}} proved the local Langlands correspondence for the general linear group GL(''n'', ''K'') for positive characteristic local fields ''K''. Their proof uses a global argument, realizing smooth admissible representations of interest as the local components of automorphic representations of the group of units of a division algebra over a curve, then using the point-counting formula to study the properties of the global Galois representations associated to these representations.

{{harvs |txt |author2-link=Richard Taylor (mathematician) |first2=Richard |last2=Taylor |first1=Michael |last1=Harris |author1-link=Michael Harris (mathematician) |year=2001}} proved the local Langlands conjectures for the general linear group GL(''n'', ''K'') for characteristic 0 local fields ''K''. {{harvs |txt |last=Henniart |first=Guy |author-link=Guy Henniart |year=2000}} gave another proof. Both proofs use a global argument of a similar flavor to the one mentioned in the previous paragraph. {{harvs |txt |author-link=Peter Scholze |last=Scholze |first=Peter |year=2013}} gave another proof.

===Fundamental lemma===
{{main|Fundamental lemma (Langlands program)}}

In 2008, [[Ngô Bảo Châu]] proved the "[[Fundamental lemma (Langlands program)|fundamental lemma]]", which was originally conjectured by Langlands and Shelstad in 1983 and being required in the proof of some important conjectures in the Langlands program.<ref>{{cite journal |first=Ngô Bảo |last=Châu |author-link=Ngô Bảo Châu |year=2010 |title=Le lemme fondamental pour les algèbres de Lie |journal=Publications Mathématiques de l'IHÉS |volume=111 |pages=1–169|doi=10.1007/s10240-010-0026-7 |arxiv=0801.0446 |s2cid=118103635 }}</ref><ref>{{cite journal |last1=Langlands |first1=Robert P. |title=Les débuts d'une formule des traces stable |url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy.html#debuts |publisher=Université de Paris |department=U.E.R. de Mathématiques |location=Paris |journal=Publications Mathématiques de l'Université Paris [Mathematical Publications of the University of Paris] |volume=VII |mr=697567 |year=1983 |issue=13}}</ref>