Editing Langlands program (section)

==Background==
The Langlands program is built on existing ideas: the [[philosophy of cusp forms]] formulated a few years earlier by [[Harish-Chandra]] and {{harvs|txt|authorlink=Israel Gelfand |last=Gelfand|year=1963}}, the work and Harish-Chandra's approach on [[semisimple Lie group]]s, and in technical terms the [[Selberg trace formula|trace formula]] of [[Atle Selberg|Selberg]] and others.

What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called [[functor]]iality).

Harish-Chandra's work exploited the principle that what can be done for one [[Semisimple Lie algebra|semisimple]] (or reductive) [[Lie group]], can 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 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 "[[spectrum (functional analysis)|discrete spectrum]]", contrasted with the "[[spectrum (functional analysis)|continuous spectrum]]" from [[Eisenstein series]]. It becomes much more technical for bigger Lie groups, because the [[Borel subgroup|parabolic subgroups]] are more numerous.

In all these approaches technical methods were available, often inductive in nature and based on [[Levi decomposition]]s amongst other matters, but the field remained demanding.<ref>{{cite book|isbn=978-0-465-05074-1|title=Love & Math|first=Edward|last=Frenkel | author-link=Edward Frenkel| date=2013|quote=All this stuff, as my dad put it, is quite heavy: we've got Hitchin moduli spaces, mirror symmetry, ''A''-branes, ''B''-branes, automorphic sheaves... One can get a headache just trying to keep track of them all. Believe me, even among specialists, very few people know the nuts and bolts of all elements of this construction.|url-access=registration|url=https://archive.org/details/lovemathheartofh0000fren}}</ref>

From the perspective of modular forms, examples such as [[Hilbert modular form]]s, [[Siegel modular form]]s, and [[theta-function|theta-series]] had been developed.