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{{short description|Far-reaching conjectures connecting number theory and geometry}}
{{More footnotes|date=June 2022}}

In [[mathematics]],  the '''Langlands program''' is a set of [[conjecture]]s about connections between [[number theory]] and [[geometry]]. It was proposed by {{harvs|txt|authorlink=Robert Langlands|first=Robert|last=Langlands|year1=1967|year2=1970}}. It seeks to relate [[Galois group]]s in [[algebraic number theory]] to [[automorphic form]]s and [[group representation|representation theory]] of [[algebraic group]]s over [[local field]]s and [[Adele ring|adele]]s. It was described by [[Edward Frenkel]] as "[[Grand Unified Theory|grand unified theory]] of mathematics."<ref>{{cite web |title=Math Quartet Joins Forces on Unified Theory |work=[[Quanta Magazine|Quanta]] |date=December 8, 2015 |url=https://www.quantamagazine.org/math-quartet-joins-forces-on-unified-theory-20151208/ }}</ref>

As an explanation to a non-specialist: the program provides constructs for a generalised and somewhat unified framework, to [[fundamental representation|characterise]] the [[algebraic structure|structures]] that underpin numbers and their [[abstract algebra|abstractions]]; thus the [[topological invariant|invariants]] which base them through [[analytic number theory|analytical]] methods.

The Langlands program consists of theoretical abstractions, which challenge even specialist mathematicians. Basically, the [[Fundamental_lemma _(Langlands_program)|fundamental lemma]] of the project links the generalized [[fundamental representation]] of a [[finite field]] with its [[galois group|group extension]] to the automorphic forms under which it is [[invariant theory|invariant]]. This is accomplished through abstraction to [[orbital integral|higher dimensional integration]], by an equivalence to a certain [[langlands dual group|analytical group]] as an [[absolute galois group|absolute extension]] of its [[abstract algebra|algebra]]. This allows an [[l function|analytical functional]] construction of powerful [[artin reciprocity law|invariance transformations]] for a [[number field]] to its own [[automorphism|algebraic structure]].

The meaning of such a construction is nuanced, but its specific solutions and generalizations are far-reaching. The consequence for proof of existence to such theoretical objects, implies an [[functional analysis|analytical method]] for constructing the [[homological algebra|categoric]] mapping of [[ideal class group|fundamental structure]]s for virtually any [[number field]]. As an analogue to the possible [[riemann hypothesis|exact distribution of primes]]; the Langlands program allows a potential [[universal property|general tool]] for the resolution of invariance at the level of generalized [[abstract algebra|algebraic structures]]. This in turn permits a somewhat unified analysis of [[analytic number theory|arithmetic]] objects through their [[automorphic functions]]... The Langlands view allows a general analysis of structuring number-abstractions. This description is at once a reduction and over-generalization of the program's proper theorems – although these mathematical concepts illustrate its basic ideas.

==Background==
The Langlands program is built on existing ideas: the ''[[philosophy of cusp forms]]'' formulated a few years earlier by [[Harish-Chandra]] and {{harvs|txt|authorlink=Israel Gelfand |last=Gelfand|year=1963}}, the work and Harish-Chandra's approach on [[semisimple Lie group]]s, and in technical terms the [[Selberg trace formula|trace formula]] of [[Atle Selberg|Selberg]] and others.

What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called ''[[functor]]iality'').

Harish-Chandra's work exploited the principle that what can be done for one [[Semisimple Lie algebra|semisimple]] (or reductive) [[Lie group]], can be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) in [[class field theory]], the way was open to speculation about GL(''n'') for general ''n'' > 2.

The ''cusp form'' idea came out of the cusps on [[modular curves]] but also had a meaning visible in [[spectral theory]] as "[[spectrum (functional analysis)|discrete spectrum]]", contrasted with the "[[spectrum (functional analysis)|continuous spectrum]]" from [[Eisenstein series]]. It becomes much more technical for bigger Lie groups, because the [[Borel subgroup|parabolic subgroups]] are more numerous.

In all these approaches technical methods were available, often inductive in nature and based on [[Levi decomposition]]s amongst other matters, but the field remained demanding.<ref>{{cite book|isbn=978-0-465-05074-1|title=Love & Math|first=Edward|last=Frenkel | author-link=Edward Frenkel| date=2013|quote=All this stuff, as my dad put it, is quite heavy: we've got Hitchin moduli spaces, mirror symmetry, ''A''-branes, ''B''-branes, automorphic sheaves... One can get a headache just trying to keep track of them all. Believe me, even among specialists, very few people know the nuts and bolts of all elements of this construction.|url-access=registration|url=https://archive.org/details/lovemathheartofh0000fren}}</ref>

From the perspective of modular forms, examples such as [[Hilbert modular form]]s, [[Siegel modular form]]s, and [[theta-function|theta-series]] had been developed.

== Objects==
The conjectures have evolved since Langlands first stated them. Langlands conjectures apply across many different groups over many different fields for which they can be stated, and each field offers several versions of the conjectures.<ref>{{citation|title=Love and Math: The Heart of Hidden Reality|first=Edward|last=Frenkel|author-link=Edward Frenkel|publisher=Basic Books|year=2013|isbn=9780465069958|page=77|url=https://books.google.com/books?id=sb0PAAAAQBAJ&pg=PT77|quote=The Langlands Program is now a vast subject. There is a large community of people working on it in different fields: number theory, harmonic analysis, geometry, representation theory, mathematical physics. Although they work with very different objects, they are all observing similar phenomena.}}</ref> Some versions{{which|date=September 2012}} are vague, or depend on objects such as [[Langlands group]]s, whose existence is unproven, or on the ''L''-group that has several non-equivalent definitions.

Objects for which Langlands conjectures can be stated:
*Representations of [[reductive group]]s over local fields (with different subcases corresponding to archimedean local fields, ''p''-adic local fields, and completions of function fields)
*Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields).
*Analogues for finite fields.
*More general fields, such as function fields over the complex numbers.

==Conjectures==
The conjectures can be stated variously in ways that are closely related but not obviously equivalent.

===Reciprocity===
The starting point of the program was [[Emil Artin]]'s [[Artin reciprocity|reciprocity law]], which generalizes [[quadratic reciprocity]]. The [[Artin reciprocity law]] applies to a [[Galois extension]] of an [[algebraic number field]] whose [[Galois group]] is [[abelian group|abelian]]; it assigns [[L-function|''L''-functions]] to the one-dimensional representations of this Galois group, and states that these ''L''-functions are identical to certain [[Dirichlet L-series|Dirichlet ''L''-series]] or more general series (that is, certain analogues of the [[Riemann zeta function]]) constructed from [[Hecke character]]s. The precise correspondence between these different kinds of ''L''-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, ''L''-functions can be defined in a natural way: [[Artin L-function|Artin ''L''-functions]].

Langlands' insight was to find the proper generalization of [[Dirichlet L-function|Dirichlet ''L''-functions]], which would allow the formulation of Artin's statement in Langland's more general setting. [[Erich Hecke|Hecke]] had earlier related Dirichlet ''L''-functions with [[automorphic form]]s ([[holomorphic function]]s on the upper half plane of the [[complex number|complex number plane]] <math>\mathbb{C}</math>  that satisfy certain [[Functional equation|functional equations]]). Langlands then generalized these to [[automorphic cuspidal representation]]s, which are certain infinite dimensional irreducible representations of the [[general linear group]] GL(''n'') over the [[adele ring]] of <math>\mathbb{Q}</math> (the [[rational number]]s). (This ring tracks all the completions of <math>\mathbb{Q},</math> see [[p-adic number|''p''-adic numbers]].)

Langlands attached [[automorphic L-function|automorphic ''L''-functions]] to these automorphic representations, and conjectured that every Artin ''L''-function arising from a finite-dimensional representation of the Galois group of a [[number field]] is equal to one arising from an automorphic cuspidal representation. This is known as his [[reciprocity law|reciprocity conjecture]].

Roughly speaking, this conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a [[Langlands group]] to an [[Langlands dual group|''L''-group]]. This offers numerous variations, in part because the definitions of Langlands group and ''L''-group are not fixed.

Over [[local field]]s this is expected to give a parameterization of [[L-packet|''L''-packets]] of admissible irreducible representations of a [[reductive group]] over the local field. For example, over the real numbers, this correspondence is the [[Langlands classification]] of representations of real reductive groups. Over [[global field]]s, it should give a parameterization of automorphic forms.

===Functoriality===
The functoriality conjecture states that a suitable homomorphism of ''L''-groups is expected to give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.

====Generalized functoriality====
Langlands generalized the idea of functoriality: instead of using the general linear group GL(''n''), other connected [[reductive group]]s can be used. Furthermore, given such a group ''G'', Langlands constructs the [[Langlands dual]] group ''<sup>L</sup>G'', and then, for every automorphic cuspidal representation of ''G'' and every finite-dimensional representation of ''<sup>L</sup>G'', he defines an ''L''-function. One of his conjectures states that these ''L''-functions satisfy a certain functional equation generalizing those of other known ''L''-functions.

He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved) [[morphism]] between their corresponding ''L''-groups, this conjecture relates their automorphic representations in a way that is compatible with their ''L''-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an [[induced representation]] construction—what in the more traditional theory of [[automorphic form]]s had been called a '[[Lift (mathematics)|lifting]]', known in special cases, and so is covariant (whereas a [[restricted representation]] is contravariant). Attempts to specify a direct construction have only produced some conditional results.

All these conjectures can be formulated for more general fields in place of <math>\mathbb{Q}</math>: [[algebraic number field]]s (the original and most important case), [[local field]]s, and function fields (finite [[field extension|extensions]] of '''F'''<sub>''p''</sub>(''t'') where ''p'' is a [[prime number|prime]] and '''F'''<sub>''p''</sub>(''t'') is the field of rational functions over the [[finite field]] with ''p'' elements).

===Geometric conjectures===
{{main|Geometric Langlands correspondence}}

The geometric Langlands program, suggested by [[Gérard Laumon]] following ideas of [[Vladimir Drinfeld]], arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates {{mvar|l}}-adic representations of the [[étale fundamental group]] of an [[algebraic curve]] to objects of the [[derived category]] of {{mvar|l}}-adic sheaves on the [[moduli stack of bundles|moduli stack]] of [[vector bundle]]s over the curve.

A 9-person collaborative project led by [[Dennis Gaitsgory]] announced a proof of the (categorical, unramified) geometric Langlands conjecture leveraging [[Hecke eigensheaf]] as part of the proof.<ref>{{cite web
| url=https://people.mpim-bonn.mpg.de/gaitsgde/GLC/
| last1=Gaitsgory | first1=Dennis | authorlink1=Dennis Gaitsgory
| title=Proof of the geometric Langlands conjecture
| access-date=August 19, 2024}}</ref><ref>{{cite arXiv
| last1=Gaitsgory | first1=Dennis | authorlink1=Dennis Gaitsgory
| last2=Raskin | first2=Sam
| date=May 2024
| title=Proof of the geometric Langlands conjecture I: construction of the functor
| eprint=2405.03599
| class=math.AG}}</ref><ref>{{cite arXiv
| last1=Arinkin | first1=D.
| last2=Beraldo | first2=D.
| last3=Campbell | first3=J.
| last4=Chen | first4=L.
| last5=Faergeman | first5=J.
| last6=Gaitsgory | first6=D.
| last7=Lin | first7=K.
| last8=Raskin | first8=S.
| last9=Rozenblyum | first9=N.
| date=May 2024
| title=Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE
| eprint=2405.03648
| class=math.AG}}</ref><ref>{{cite web |url=https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/ |title= Monumental Proof Settles Geometric Langlands Conjecture|date=July 19, 2024 |publisher=Quanta Magazine}}</ref>

==Status==
The Langlands conjectures for GL(1, ''K'') follow 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 Langlands conjectures for the general linear group GL(''n'', ''K'') for function fields ''K''. This work continued earlier investigations by Drinfeld, who proved the case GL(2, ''K'') in the 1980s.

In 2018, [[Vincent Lafforgue]] established 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 conjectures]] 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 conjectures for the general linear group GL(''n'', ''K'') for positive characteristic local fields ''K''. Their proof uses a global argument.

{{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. {{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>

===Implications===
To a lay reader or even nonspecialist mathematician, abstractions within the Langlands program can be somewhat impenetrable. However, there are some strong and clear implications for proof or disproof of the fundamental Langlands conjectures. 

As the program posits a powerful connection between [[analytic number theory]] and generalizations of [[algebraic geometry]], the idea of [[homological algebra#Functoriality|'Functoriality']] between abstract algebraic [[Galois group|representations]] of [[number field]]s and their analytical [[ideal class group|prime]] constructions results in powerful [[functional analysis|functional tools]] allowing an exact quantification of [[distribution of primes|prime distributions]]. This, in turn, yields the capacity for classification of [[diophantine equations]] and further abstractions of [[algebraic functions]].
 
Furthermore, if the [[Artin reciprocity law|reciprocity]] of such generalized [[algebraic variety|algebras]] for the posited objects exists, and if their [[L function|analytical functions]] can be shown to be well-defined, some very deep results in mathematics could be within reach of proof. Examples include: [[Birch and Swinnerton-Dyer conjecture|rational solutions of elliptic curves]], [[hodge conjecture|topological construction of algebraic varieties]], and the famous [[Riemann hypothesis]].<ref>{{cite arXiv |last=Milne |first=James |title=The Riemann Hypothesis over Finite Fields: From Weil to the Present Day |date=2015-09-02 |class=math.HO |eprint=1509.00797}}</ref> Such proofs would be expected to utilize abstract solutions in objects of [[Artin L-function|generalized analytical series]], each of which relates to the [[invariant theory|invariance]] within [[abstract algebra|structures]] of number fields. 

Additionally, some connections between the Langlands program and [[M theory]] have been posited, as their [[duality (mathematics)|dualities]] connect in [[fundamental representation|nontrivial]] ways, providing potential exact solutions in [[superstring theory]] (as was similarly done in [[group theory]] through [[monstrous moonshine]]).

Simply put, the Langlands project implies a deep and powerful framework of solutions, which touches the most fundamental areas of mathematics, through high-order generalizations in exact solutions of algebraic equations, with analytical functions, as embedded in geometric forms. It allows a unification of many distant mathematical fields into a formalism of powerful [[automorphic form|analytical methods]].

==See also==
*[[Jacquet–Langlands correspondence]]
*[[Erlangen program]]

==Notes==
{{reflist}}

==References==
*{{Citation | last1=Arthur | first1=James | title=The principle of functoriality | doi=10.1090/S0273-0979-02-00963-1 | mr=1943132 | year=2003 | journal=Bulletin of the American Mathematical Society |series=New Series | issn=0002-9904 | volume=40 | issue=1 | pages=39–53| doi-access=free }}
*{{Cite book |editor-first1=J. |editor-last1=Bernstein |editor-first2=S. |editor-last2=Gelbart | editor-link2= Stephen Gelbart |title=An Introduction to the Langlands Program |location=Boston |publisher=Birkhäuser |year=2003 |isbn=978-3-7643-3211-2 }}
*{{Citation | last1=Gelbart | first1=Stephen | author1-link = Stephen Gelbart | title=An elementary introduction to the Langlands program | doi=10.1090/S0273-0979-1984-15237-6 | mr=733692 | year=1984 | journal=Bulletin of the American Mathematical Society |series=New Series | issn=0002-9904 | volume=10 | issue=2 | pages=177–219| doi-access=free }}
*{{cite arXiv |first=Edward |last=Frenkel |author-link=Edward Frenkel |title=Lectures on the Langlands Program and Conformal Field Theory |year=2005 |eprint=hep-th/0512172}}
*{{Citation | last1=Gelfand | first1=I. M. | title=Proc. Internat. Congr. Mathematicians (Stockholm, 1962) | chapter-url=http://mathunion.org/ICM/ICM1962.1/ | publisher=Inst. Mittag-Leffler | location=Djursholm | mr=0175997 | year=1963 | chapter=Automorphic functions and the theory of representations | pages=74–85}}
*{{Citation | last1=Harris | first1=Michael | last2=Taylor | first2=Richard | title=The geometry and cohomology of some simple Shimura varieties | url=https://books.google.com/books?id=sigBbO69hvMC | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-09090-0 | mr=1876802 | year=2001 | volume=151}}
*{{Citation | last1=Henniart | first1=Guy | title=Une preuve simple des conjectures de Langlands pour GL(''n'') sur un corps ''p''-adique | doi=10.1007/s002220050012 | mr=1738446 | year=2000 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=139 | issue=2 | pages=439–455| bibcode=2000InMat.139..439H | s2cid=120799103 }}
*{{Citation|last=Kutzko |first=Philip |year=1980 |title=The Langlands Conjecture for Gl<sub>2</sub> of a Local Field |journal=[[Annals of Mathematics]] |volume=112 |issue=2 |pages=381–412 |jstor=1971151 | doi = 10.2307/1971151 |url=http://projecteuclid.org/euclid.bams/1183546356 }}
*{{citation|last=Langlands|first=Robert|title=Letter to Prof. Weil|year=1967|url=http://publications.ias.edu/rpl/section/21}}
*{{Citation | last1=Langlands | first1=R. P. | title=Lectures in modern analysis and applications, III | chapter-url=http://publications.ias.edu/rpl/section/21 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series= Lecture Notes in Math | isbn=978-3-540-05284-5 | doi=10.1007/BFb0079065 | mr=0302614 | year=1970 | volume=170 | chapter=Problems in the theory of automorphic forms | pages=18–61}}
*{{Citation | last1=Laumon | first1=G. | last2=Rapoport | first2=M. | last3=Stuhler | first3=U. | title=''D''-elliptic sheaves and the Langlands correspondence | doi=10.1007/BF01244308 | mr=1228127 | year=1993 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=113 | issue=2 | pages=217–338| bibcode=1993InMat.113..217L | s2cid=124557672 }}
* {{cite book |editor-first1=Julia  |editor-last1=Mueller  |editor-first2=Freydoon  |editor-last2=Shahidi  | editor-link2=Freydoon Shahidi | title=The Genesis of the Langlands Program | publisher=Cambridge University Press | publication-place=Cambridge | date=2021 | isbn=978-1-108-71094-7 }}
*{{Citation |last1=Scholze |first1=Peter |year=2013 |title=The Local Langlands Correspondence for GL(''n'') over ''p''-adic fields |journal=[[Inventiones Mathematicae]] |volume=192 |issue=3 |pages=663–715 |doi=10.1007/s00222-012-0420-5 |arxiv=1010.1540 |bibcode=2013InMat.192..663S |s2cid=15124490 }}

==External links==
{{Sister project links| wikt=no | commons=no | b=no | n=no | q=Langlands program | s=no | v=no | voy=no | species=no | d=no}}
*[http://publications.ias.edu/rpl/ The work of Robert Langlands ]

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