In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. It was proposed by Robert Langlands beginning in 1967. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory. ... 1967 (MCMLXVII) was a common year starting on Sunday of the Gregorian calendar. ...

Connection with number theory

The starting point of the program may be seen as the Artin reciprocity law which generalizes quadratic reciprocity. The Artin reciprocity law applies to an algebraic number field whose Galois group over Q 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 (that is, the analogues of the Riemann zeta function constructed from Dirichlet characters). The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law. Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... In mathematics, reciprocity is applied to a number of theorems, and at times certain relationships. ... In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... In mathematics, Artin reciprocity refers to various results connecting Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to Heckes grossencharacters of that number field. ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ... In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions. 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 1920s by Emil Artin, in connection with his research into class field theory. ...

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.) Erich Hecke (September 20, 1887 – February 13, 1947) was a German mathematician. ... 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. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... 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. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... ... The p-adic number systems were first described by Kurt Hensel in 1897. ...

Langlands attached 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". In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

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 a complex Lie 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. In mathematics, a reductive group is an algebraic group G such that the unipotent radical of the identity component of G is trivial. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ...

He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a 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 (where a restricted representation is contravariant). Attempts to specify a direct construction have only produced some conditional results. 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 (whole) group G itself. ... 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. ... In mathematics, if G is a group and H a subgroup, then for any linear representation ρ of G, we can define the restricted representation ρ|H by simply setting ρ|H(h) = ρ(h). ...

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 F_p(t) where p is a prime and F_p(t) is the field of rational functions over the finite field with p elements). In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many 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 Israel Gelfand, the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula of Selberg and others. Israel Moiseevich Gelfand (Russian: ) (born in 1913) is a prolific mathematician in the field of functional analysis, which he interprets in a broad sense as the mathematics of quantum mechanics. ... See Harishchandra for the character in Hindu mythology Harish-Chandra (11 October 1923-16 October 1983) was an Indian mathematician, who did fundamental work in representation theory. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, the Selberg trace formula is a central result, or area of research, in non-commutative harmonic analysis. ... Atle Selberg (born June 17, 1917) is 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. ...

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 GL₂ in the theory of modular forms had been recognised, and with hindsight GL₁ in class field theory, the way was open at least to speculation about GL_n for general n > 2. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, class field theory is a major branch of algebraic number theory. ...

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 mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as H/Γ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ... In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ... In mathematics and physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. ... In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. ... In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. ... In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ...

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. In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. ...

And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular forms, and theta-series. In mathematics, a Hilbert modular form is a generalization of the elliptic modular forms, to functions of two or more variables. ... In mathematics, theta functions are special functions of several complex variables. ...

Prizes

Parts of the program for local fields were completed in 1998 and for function fields in 1999. Laurent Lafforgue received the Fields Medal in 2002 for his work on the function field case. This work continued earlier investigations by Vladimir Drinfeld, which had been honored with the Fields Medal in 1990. Only special cases of the number field case have been proven, some by Langlands himself. Laurent Lafforgue (born November 6, 1966) is a French mathematician. ... The Fields Medal is a prize awarded to up to four mathematicians (not over forty years of age) at each International Congress of International Mathematical Union (therefore once every four years), since 1936 and regularly since 1950 at the initiative of the Canadian mathematician John Charles Fields. ... Vladimir Gershonovich Drinfeld (Владимир Гершонович Дринфельд) is a mathematician born February 14, 1954 in Ukraine. ...

Langlands received the Wolf Prize in 1996 for his work on these conjectures. Past winners of the Wolf Prize in Mathematics: 1978 Israel M. Gelfand, Carl L. Siegel 1979 Jean Leray, André Weil 1980 Henri Cartan, Andrei Kolmogorov 1981 Lars Ahlfors, Oscar Zariski 1982 Hassler Whitney, Mark Grigoryevich Krein 1983/4 Shiing S. Chern, Paul Erdős 1984/5 Kunihiko Kodaira, Hans... 1996 (MCMXCVI) is a leap year starting on Monday of the Gregorian calendar, and was designated the International Year for the Eradication of Poverty. ...

References

• Stephen Gelbart: An Elementary Introduction to the Langlands Program, Bulletin of the AMS v.10 no. 2 April 1984.
• Edward Frenkel: Lectures on the Langlands Program and Conformal Field Theory, hep-th/0512172