Editing Langlands program (section)

===Reciprocity===
The starting point of the program was [[Emil Artin]]'s [[Artin reciprocity|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 group|abelian]]; it assigns [[L-function|''L''-functions]] to the one-dimensional representations of this Galois group, and states that these ''L''-functions are identical to certain [[Dirichlet L-series|Dirichlet ''L''-series]] or more general series (that is, certain analogues of the [[Riemann zeta function]]) constructed from [[Hecke character]]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, ''L''-functions can be defined in a natural way: [[Artin L-function|Artin ''L''-functions]].

Langlands' insight was to find the proper generalization of [[Dirichlet L-function|Dirichlet ''L''-functions]], which would allow the formulation of Artin's statement in Langland's more general setting. [[Erich Hecke|Hecke]] had earlier related Dirichlet ''L''-functions with [[automorphic form]]s ([[holomorphic function]]s on the upper half plane of the [[complex number|complex number plane]] <math>\mathbb{C}</math>  that satisfy certain [[Functional equation|functional equations]]). Langlands then generalized these to [[automorphic cuspidal representation]]s, which are certain infinite dimensional irreducible representations of the [[general linear group]] GL(''n'') over the [[adele ring]] of <math>\mathbb{Q}</math> (the [[rational number]]s). (This ring tracks all the completions of <math>\mathbb{Q},</math> see [[p-adic number|''p''-adic numbers]].)

Langlands attached [[automorphic L-function|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 law|reciprocity conjecture]].

Roughly speaking, this conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a [[Langlands group]] to an [[Langlands dual group|''L''-group]]. This offers numerous variations, in part because the definitions of Langlands group and ''L''-group are not fixed.

Over [[local field]]s this is expected to give a parameterization of [[L-packet|''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 field]]s, it should give a parameterization of automorphic forms.