Editing Langlands program (section)

===Functoriality===
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.

====Generalized functoriality====
Langlands generalized the idea of functoriality: instead of using the general linear group GL(''n''), other connected [[reductive group]]s can be used. Furthermore, given such a group ''G'', Langlands constructs the [[Langlands dual]] group ''<sup>L</sup>G'', and then, for every automorphic cuspidal representation of ''G'' and every finite-dimensional representation of ''<sup>L</sup>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 form]]s had been called a '[[Lift (mathematics)|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 <math>\mathbb{Q}</math>: [[algebraic number field]]s (the original and most important case), [[local field]]s, and function fields (finite [[field extension|extensions]] of '''F'''<sub>''p''</sub>(''t'') where ''p'' is a [[prime number|prime]] and '''F'''<sub>''p''</sub>(''t'') is the field of rational functions over the [[finite field]] with ''p'' elements).