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Inmathematics, theLanglands program is a set ofconjectures about connections betweennumber theory, the theory ofautomorphic forms, andgeometry. It was proposed by the Canadian mathematicianRobert Langlands (1967,1970). It seeks to relate the structure ofGalois groups inalgebraic number theory toautomorphic forms and, more generally, therepresentation theory ofalgebraic groups overlocal fields andadeles.
The Langlands program is built on existing ideas: thephilosophy of cusp forms formulated a few years earlier byHarish-Chandra andGelfand (1963), the work and Harish-Chandra's approach onsemisimple Lie groups, and in technical terms thetrace formula ofSelberg 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-calledfunctoriality).
Harish-Chandra's work exploited the principle that what can be done for onesemisimple (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) inclass field theory, the way was open to speculation about GL(n) for generaln > 2.
The 'cusp form' idea came out of the cusps onmodular curves but also had a meaning visible inspectral theory as "discrete spectrum", contrasted with the "continuous spectrum" fromEisenstein series. It becomes much more technical for bigger Lie groups, because theparabolic subgroups are more numerous.
In all these approaches technical methods were available, often inductive in nature and based onLevi decompositions amongst other matters, but the field remained demanding.[1]
From the perspective of modular forms, examples such asHilbert modular forms,Siegel modular forms, andtheta-series had been developed.
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.[2] Some versions[which?] are vague, or depend on objects such asLanglands groups, whose existence is unproven, or on theL-group that has several non-equivalent definitions.
Objects for which Langlands conjectures can be stated:
The conjectures can be stated variously in ways that are closely related but not obviously equivalent.
The starting point of the program wasEmil Artin'sreciprocity law, which generalizesquadratic reciprocity. TheArtin reciprocity law applies to aGalois extension of analgebraic number field whoseGalois group isabelian; it assignsL-functions to the one-dimensional representations of this Galois group, and states that theseL-functions are identical to certainDirichletL-series or more general series (that is, certain analogues of theRiemann zeta function) constructed fromHecke characters. The precise correspondence between these different kinds ofL-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:ArtinL-functions.
Langlands' insight was to find the proper generalization ofDirichletL-functions, which would allow the formulation of Artin's statement in Langland's more general setting.Hecke had earlier related DirichletL-functions withautomorphic forms (holomorphic functions on the upper half plane of thecomplex number plane that satisfy certainfunctional equations). Langlands then generalized these toautomorphic cuspidal representations, which are certain infinite dimensional irreducible representations of thegeneral linear group GL(n) over theadele ring of (therational numbers). (This ring tracks all the completions of seep-adic numbers.)
Langlands attachedautomorphicL-functions to these automorphic representations, and conjectured that every ArtinL-function arising from a finite-dimensional representation of the Galois group of anumber field is equal to one arising from an automorphic cuspidal representation. This is known as hisreciprocity conjecture.
Roughly speaking, this conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from aLanglands group to anL-group. This offers numerous variations, in part because the definitions of Langlands group andL-group are not fixed.
Overlocal fields this is expected to give a parameterization ofL-packets of admissible irreducible representations of areductive group over the local field. For example, over the real numbers, this correspondence is theLanglands classification of representations of real reductive groups. Overglobal fields, it should give a parameterization of automorphic forms.
The functoriality conjecture states that a suitable homomorphism ofL-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.
Langlands generalized the idea of functoriality: instead of using the general linear group GL(n), other connectedreductive groups can be used. Furthermore, given such a groupG, Langlands constructs theLanglands dual groupLG, and then, for every automorphic cuspidal representation ofG and every finite-dimensional representation ofLG, he defines anL-function. One of his conjectures states that theseL-functions satisfy a certain functional equation generalizing those of other knownL-functions.
He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved)morphism between their correspondingL-groups, this conjecture relates their automorphic representations in a way that is compatible with theirL-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of aninduced representation construction—what in the more traditional theory ofautomorphic forms had been called a 'lifting', known in special cases, and so is covariant (whereas arestricted 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:algebraic number fields (the original and most important case),local fields, and function fields (finiteextensions ofFp(t) wherep is aprime andFp(t) is the field of rational functions over thefinite field withp elements).
The geometric Langlands program, suggested byGérard Laumon following ideas ofVladimir Drinfeld, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relatesl-adic representations of theétale fundamental group of analgebraic curve to objects of thederived category ofl-adic sheaves on themoduli stack ofvector bundles over the curve.
In 2024, a 9-person collaborative project led byDennis Gaitsgory announced a proof of the (categorical, unramified) geometric Langlands conjecture leveragingHecke eigensheaves as part of the proof.[3][4][5][6]
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 (thereal numbers) and (thecomplex numbers) by giving theLanglands 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 remains unproved.
In 1998,Laurent Lafforgue provedLafforgue's theorem verifying the global Langlands correspondence for the general linear group GL(n,K) for function fieldsK. 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.[7][8][9]
Philip Kutzko (1980) proved thelocal Langlands correspondence for the general linear group GL(2,K) over local fields.
Gérard Laumon, Michael Rapoport, and Ulrich Stuhler (1993) proved the local Langlands correspondence for the general linear group GL(n,K) for positive characteristic local fieldsK. 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.
Michael Harris and Richard Taylor (2001) proved the local Langlands conjectures for the general linear group GL(n,K) for characteristic 0 local fieldsK.Guy Henniart (2000) gave another proof. Both proofs use a global argument of a similar flavor to the one mentioned in the previous paragraph.Peter Scholze (2013) gave another proof.
In 2008,Ngô Bảo Châu proved the "fundamental lemma", which was conjectured initially by Langlands and Shelstad in 1983 and is required in the proof of some essential conjectures in the Langlands program.[10][11]
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.
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.
