Langlands program

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.

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 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 a Langlands group ^L"G", and then, for every automorphic cuspidal representation of "G" and every finite-dimensional representation of ^L"G", 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 (where 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 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).

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 Israel Gelfand [I.M. Gelfand, `Automorphic functions and the theory of representations', in Proceedings, International Congress of Mathematicians, Stockholm, 1962, pp. 74-85.] , 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 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.

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.

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 Drinfeld, 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.

Prizes

Langlands received the Wolf Prize in 1996 and the Nemmers Prize in Mathematics in 2006 for his work on these conjectures.

Laurent Lafforgue received the Fields Medal in 2002 for his work on the function field case. This work continued earlier investigations by Vladimir Drinfeld; he had been honored with the Fields Medal in 1990, in part for this work.

References

* James Arthur: "The Principle of Functoriality", Bulletin of the AMS v.40 no. 1 October 2002
* 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", [http://www.arxiv.org/abs/hep-th/0512172 hep-th/0512172]
*J. Bernstein, S. Gelbart, "An Introduction to the Langlands Program", ISBN 3-7643-3211-5

Notes