In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."
The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. So to oversimplify, the foundational result and fundamental lemma of the project, posits a direct connection between the generalized fundamental representation of a finite field with its group extension, to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integration, by an equivalence to a certain analytical group as an absolute extension of its algebra. Consequently, this allows an analytical functional construction of powerful invariance transformations for a number field to its own algebraic structure.
Intuitively speaking, the meaning of such a construction is rather nuanced; yet very powerful in its specific solutions and generalizations. The consequence for proof of existence to such theoretical objects, implies an analytical method in constructing the categoric mapping of fundamental structures for virtually any number field. As an analogue to the possible exact distribution of primes, the Langlands program allows a potential general tool for resolution of invariance at generalized algebraic structures. This in turn permits a somewhat unified analysis of arithmetic objects through their automorphic functions. Simply put, the Langlands philosophy allows a general analysis of structuring the abstractions of numbers. Naturally, this description is at once a reduction and over-generalization of the program's proper theorems. But, these mathematical analogues provide the basis of its conceptualization.
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.
There are a 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[which?] of the Langlands conjectures are vague, or depend on objects such as the Langlands groups, whose existence is unproven, or on the L-group that has several inequivalent definitions. Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.
There are different types of objects for which the Langlands conjectures can be stated:
There are several different ways of stating the Langlands conjectures, which are closely related but not obviously equivalent.
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 an algebraic number field whose Galois group is abelian; it 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.
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".
Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L-group. There are numerous variations of this, in part because the definitions of Langlands group and L-group are not fixed.
Over local fields this is expected to 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 correspondence is the Langlands classification of representations of real reductive groups. Over global fields, it should give a parameterization of automorphic forms.
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.
Langlands generalized the idea of functoriality: instead of using the general linear group GL(n), other connected reductive groups can be used. Furthermore, given such a group G, Langlands constructs the Langlands dual 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.
The so-called 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 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.
The Langlands conjectures for GL(1, K) follow from (and are essentially equivalent to) class field theory.
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.
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.
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 use a global argument. Peter Scholze (2013) gave another proof.
In 2008, Ngô Bảo Châu proved the "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.
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 'functoriality' between abstract algebraic representations of number fields and their analytical prime constructions, results in powerful functional tools allowing an exact quantification of prime distributions. This in turn, yields the capacity for classification of diophantine equations and further abstractions of algebraic functions.
Furthermore, if the reciprocity of such generalized algebras for the posited objects exists, and if their analytical functions can be shown to be well defined, some very deep results in mathematics can be within reach of proof; such as rational solutions of elliptic curves, topological construction of algebraic varieties, and the famous Riemann hypothesis, each of which relates to the invariance within structures of number fields.
Additionally some connections between the Langlands program and M theory have been posited, as their dualities connect in 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 analytical methods.