Langlands program
Encyclopedia
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

 to automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s and representation theory
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

 of algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

s over local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s and adele
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....

s. It was proposed by .

Overview of the Langlands conjectures

There are a bewildering number of related Langlands conjectures. There are many different groups over many different fields for which they can be stated, and for each field there are several different versions of the conjectures. Some versions of the Langlands conjectures are somewhat vague, or depend on objects such as the Langlands groups whose existence in general is conjectural, or on the L-group that has several inequivalent definitions. Moreover the Langlands conjectures have evolved since Langlands first stated them.

There are different types of groups for which the Langlands conjectures can be stated:
  • Representations of reductive groups 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).
  • Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it.
  • More general fields, such as function fields over the complex numbers.


There are several different ways of stating Langlands conjectures, which are closely related but not obviously equivalent
  • The Langlands reciprocity conjecture. Roughly speaking this gives a correspondence between automorphic representations of a reductive group and certain homomorphisms from a Langlands group
    Langlands group
    In representation theory, a branch of mathematics, the Langlands dual LG of a reductive algebraic group G is a group that controls the representation theory of G. If G is a group over a field k, LG is an extension of the absolute Galois group of k by a complex Lie group...

     to an L-group
    L-group
    In mathematics, L-group may refer to the following groups:* The Langlands dual, LG, of a reductive algebraic group G* A group in L-theory, L...

    . There are numerous variations of this, partly because the definitions of Langlands group and L-group are not fixed. Over local fields this should give a parameterization of L-packet
    L-packet
    In mathematical representation theory, an L-packet is a collections of irreducible representations of a reductive group over a local field, that are L-indistinguishable, meaning they have the same Langlands parameter, and so have the same L-function and ε-factors...

    s of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers this is the Langlands classification
    Langlands classification
    In mathematics, the Langlands classification is a classification of irreducible representations of a reductive Lie group G, suggested by Robert Langlands...

     of representations of real reductive groups. Over global fields it should give a parametrization of automorphic forms, (at least if anyone can figure out what the Langlands group is).
  • The Langlands functoriality conjecture. This states that a suitable homomorphism of L-groups should 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.
  • The geometric Langlands conjectures. These are a more ambitious attempt to give a relation between categories of representations, rather than just the irreducible representations as in the original Langlands conjectures.

Connection with number theory

The starting point of the program may be seen as Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

's reciprocity law
Artin reciprocity
The Artin reciprocity law, established by Emil Artin in a series of papers , is a general theorem in number theory that forms a central part of the global class field theory...

, which generalizes quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...

. The Artin reciprocity law applies to a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

 of algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...

s whose Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

 is abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

, assigns L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...

s to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke character
Hecke character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to...

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, one can still define L-functions in a natural way: Artin L-function
Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...

s.

The setting of automorphic representations

The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

Hecke
Erich Hecke
Erich Hecke was a German mathematician. He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students....

 had earlier related Dirichlet L-functions with automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s (holomorphic functions on the upper half plane of C that satisfy certain functional equations). Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

 GLn over the adele ring
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....

 of Q. (This ring simultaneously keeps track of all the completions of Q, see p-adic numbers
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

.)

Langlands attached automorphic L-function
Automorphic L-function
In mathematics, an automorphic L-function is a function L of a complex variable s, associated to an automorphic form π of a reductive group G over a global field and a finite-dimensional comlplex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a...

s 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 conjecture".

A general principle of functoriality

Langlands then generalized things further: instead of using the general linear group GLn, other connected reductive group
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

s can be used. Furthermore, given such a group G, Langlands constructs an L-group
L-group
In mathematics, L-group may refer to the following groups:* The Langlands dual, LG, of a reductive algebraic group G* A group in L-theory, L...

 LG, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of LG, 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
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...

 construction—what in the more traditional theory of automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s had been called a 'lifting', known in special cases, and so is covariant (whereas a restricted representation
Restricted representation
In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...

 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 Q: algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...

s (the original and most important case), local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s, and function fields (finite extensions
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

 of Fp(t) where p is a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 and Fp(t) is the field of rational functions over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 with p elements).

Ideas leading up to the Langlands program

In a very broad context, the program built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra
Harish-Chandra
Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...

 and the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula
Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...

 of Selberg
Atle Selberg
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...

 and others.

What initially was very new in Langlands' work, besides technical depth, was the proposed direct connection to number theory, together with the rich organisational structure hypothesised (so-called functoriality).

For example, in the work of Harish-Chandra one finds the principle that what can be done for one semisimple (or reductive) Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

, should be done for all. Therefore once the role of some low-dimensional Lie groups such as GL2 in the theory of modular forms had been recognised, and with hindsight GL1 in class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

, the way was open at least to speculation about GLn 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
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...

 as 'discrete spectrum
Discrete spectrum
In physics, an elementary explanation of a discrete spectrum is that it is an emission spectrum or absorption spectrum for which there is only an integer number of intensities. Atomic electronic absorption and emission spectrum are discrete, as contrasted with, for example, the emission spectrum...

', contrasted with the 'continuous spectrum
Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers \lambda such that A - \lambda I : H \to H is not...

' from Eisenstein series
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...

. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous.

In all these approaches there was no shortage of technical methods, often inductive in nature and based on Levi decomposition
Levi decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra....

s amongst other matters, but the field was and is very demanding.

And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular form
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These stand in relation to the conventional elliptic modular forms as abelian varieties do in relation to elliptic curves; the complex manifolds constructed as in the theory are basic models for what a moduli space for...

s, and theta-series.

The geometric program

The so-called geometric Langlands program, suggested by Gérard Laumon
Gérard Laumon
Gérard Laumon is a French mathematician. He studied at the École Normale Supérieure and Paris-Sud 11 University, Orsay.In 2004 Laumon and Ngô Bảo Châu received the Clay Research Award for the proof of the Langlands and Shelstad's Fundamental Lemma for unitary groups, a key component in the...

 following ideas of Vladimir Drinfel'd
Vladimir Drinfel'd
Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the...

, arises from a geometric reformulation of the usual Langlands program. In simple cases, it relates l-adic representations of the étale fundamental group
Étale fundamental group
The étale fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.-Topological analogue:In algebraic topology, the fundamental group\pi_1...

 of an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

 to objects of the derived category
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...

  of l-adic sheaves on the moduli stack of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s over the curve.

Progress

The Langlands conjectures for GL1(K) follow from (and are essentially equivalent to) class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

.

Langlands proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification
Langlands classification
In mathematics, the Langlands classification is a classification of irreducible representations of a reductive Lie group G, suggested by Robert Langlands...

 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.

Laurent Lafforgue
Laurent Lafforgue
Laurent Lafforgue is a French mathematician.He won 2 silver medals at International Mathematical Olympiad in 1984 and 1985....

 proved Lafforgue's theorem
Lafforgue's theorem
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups....

 verifying the Langlands conjectures for the general linear group GLn(K) for function fields K. This work continued earlier investigations by Vladimir Drinfel'd, who proved the case GL2(K)

proved the local Langlands conjectures
Local Langlands conjectures
In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between representations of the Weil group of a local field and representations of algebraic groups over the local field, generalizing local class field theory from...

 for the general linear group GL2(K) over local fields.

proved the local Langlands conjectures
Local Langlands conjectures
In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between representations of the Weil group of a local field and representations of algebraic groups over the local field, generalizing local class field theory from...

  for the general linear group GLn(K) for positive characteristic local fields K.

proved the local Langlands conjectures
Local Langlands conjectures
In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between representations of the Weil group of a local field and representations of algebraic groups over the local field, generalizing local class field theory from...

 for the general linear group GLn(K) for characteristic 0 local fields K. gave another proof.

Ngô Bảo Châu
Ngo Bao Chau
Ngô Bảo Châu is a Vietnamese and French mathematician at the University of Chicago, best known for proving the fundamental lemma for automorphic forms proposed by Robert Langlands and Diana Shelstad. He is the first Vietnamese to receive the Fields Medal.- Biography :Chau was born in 1972, the...

 proved the so-called "Fundamental Lemma", originally conjectured by Langlands.

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