Editing Langlands program (section)

==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 eigensheaves]] 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>