Editing Automorphic form (section)

{{Short description|Type of generalization of periodic functions in Euclidean space}}
[[Image:Dedekind Eta.jpg|right|thumb|500px|The [[Dedekind eta-function]] is an automorphic form in the complex plane.]]

In [[harmonic analysis]] and [[number theory]], an '''automorphic form''' is a well-behaved function from a [[topological group]] ''G'' to the [[complex number]]s (or complex [[vector space]]) which is invariant under the [[group action (mathematics)|action]] of a [[discrete subgroup]] <math>\Gamma \subset G</math> of the topological group. Automorphic forms are a generalization of the idea of [[periodic function]]s in [[Euclidean space]] to general topological groups.

[[Modular form]]s are holomorphic automorphic forms defined over the groups [[SL2(R)|SL(2, '''R''')]] or [[PSL2(R)|PSL(2, '''R''')]] with the discrete subgroup being the [[modular group]], or one of its [[congruence subgroup]]s; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the [[Adele ring|adelic]] approach as a way of dealing with the whole family of [[congruence subgroup]]s at once. From this point of view, an automorphic form over the group ''G''('''A'''<sub>''F''</sub>), for an algebraic group ''G'' and an [[algebraic number]] field ''F'',  is a complex-valued function on ''G''('''A'''<sub>''F''</sub>) that is left invariant under ''G''(''F'') and satisfies certain smoothness and growth conditions.

[[Henri Poincaré]] first discovered automorphic forms as generalizations of [[Trigonometric functions|trigonometric]] and [[elliptic function]]s. Through the [[Langlands conjectures]], automorphic forms play an important role in modern number theory.<ref name=Freidberg2013>{{cite web|last=Friedberg|first=Solomon|title=Automorphic Forms: A Brief Introduction|url=https://icerm.brown.edu/materials/Slides/sp-s13-off_weeks/Automorphic_Forms-_A_Brief_Introduction_]_Solomon_Friedberg,_Boston_College.pdf|archive-url=https://web.archive.org/web/20130606222417/http://icerm.brown.edu/materials/Slides/sp-s13-off_weeks/Automorphic_Forms-_A_Brief_Introduction_%5D_Solomon_Friedberg,_Boston_College.pdf|url-status=dead|archive-date=6 June 2013|access-date=10 February 2014}}</ref>