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The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by Robert Langlands (1967, 1970).

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 to an L-group. 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-packets of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers this is the Langlands classification 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's reciprocity law, which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions 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 characters. 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-functions.
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 had earlier related Dirichlet L-functions with automorphic forms (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 GL(n) over the adele ring of Q. (This ring simultaneously keeps track of all the completions of Q, see p-adic numbers.)

Langlands attached 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 conjecture".
A general principle of functoriality

Langlands then generalized things further: instead of using the general linear group GL(n), other connected reductive groups can be used. Furthermore, given such a group G, Langlands constructs an L-group 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 construction—what in the more traditional theory of automorphic forms had been called a '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 Q: algebraic number fields (the original and most important case), local fields, and function fields (finite extensions of Fp(t) where p is a prime and Fp(t) is the field of rational functions over the finite field 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 and Gelfand (1963) the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula of Selberg 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, should 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 at least 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 'discrete spectrum', contrasted with the 'continuous spectrum' from Eisenstein series. 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 decompositions amongst other matters, but the field was and is very demanding.[citation needed]

And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular forms, and theta-series.
The geometric program

The so-called geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfel'd, arises from a geometric reformulation of the usual Langlands program. In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.
Progress and its lack

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 R and C 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 semi-stable elliptic curves over rationals can be viewed as an exercise in the Langlands conjectures. Unfortunately, his method cannot be extended to arbitrary number fields.

The Langlands conjecture for GL(2, Q) still remains unproved.

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 Vladimir Drinfel'd, who proved the case GL(2, K)

Philip Kutzko (1980) proved the local Langlands conjectures for the general linear group GL(2, K) over local fields.

Gérard Laumon, Michael Rapoport, and Ulrich Stuhler (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.

Richard Taylor and Michael Harris (2001) proved the local Langlands conjectures for the general linear group GL(n, K) for characteristic 0 local fields K. Guy Henniart (2000) gave another proof. Both proofs uses a global argument.

Ngô Bảo Châu proved an auxiliary but difficult statement, the so-called "Fundamental Lemma", originally conjectured by Langlands.

Local Langlands conjectures

References

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Langlands, Robert (1967), Letter to Prof. Weil
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