# Automorphic form

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

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

[Modular form](/source/Modular_form)s are [holomorphic](/source/holomorphic_function) automorphic forms defined over the groups [SL(2, '''R''')](/source/SL2(R)) or [PSL(2, '''R''')](/source/PSL2(R)), with the discrete subgroup being the [modular group](/source/modular_group) or one of its [congruence subgroup](/source/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 [adelic](/source/Adele_ring) approach as a way of dealing with the whole family of [congruence subgroup](/source/congruence_subgroup)s at once. From this point of view, an automorphic form over the group <math>G(\mathbb{A}_F)</math>, for an algebraic group <math>G</math> and an [algebraic number](/source/algebraic_number) field <math>F</math>,  is a complex-valued function on <math>G(\mathbb{A}_F)</math> that is left invariant under <math>G(F)</math> and satisfies certain smoothness and growth conditions.

[Henri Poincaré](/source/Henri_Poincar%C3%A9) first discovered automorphic forms as generalizations of [trigonometric](/source/Trigonometric_functions) and [elliptic function](/source/elliptic_function)s. Through the [Langlands conjectures](/source/Langlands_conjectures), automorphic forms play an important role in modern number theory.<ref name=Freidberg2013>{{cite web|author1-link=Solomon Friedberg|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>

==Definition==
{{Confusing section|date=July 2023|reason=the connections between multiple attempts at definition are unclear; see talk "Definition section is confusing"}}
In [mathematics](/source/mathematics), the notion of '''factor of automorphy''' arises for a [group](/source/group_(mathematics)) [acting](/source/Group_action_(mathematics)) on a [complex-analytic manifold](/source/complex-analytic_manifold). Suppose a group <math>G</math> acts on a complex-analytic manifold <math>X</math>. Then, <math>G</math> also acts on the space of [holomorphic function](/source/holomorphic_function)s from <math>X</math> to the complex numbers. A function <math>f</math> is termed an ''automorphic form'' if the following holds:

: <math>f(g\cdot x) = j_g(x)f(x)</math>

where <math>j_g(x)</math> is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of <math>G</math>.

The ''factor of automorphy'' for the automorphic form <math>f</math> is the function <math>j</math>. An ''automorphic function'' is an automorphic form for which <math>j</math> is the identity.

An automorphic form is a function ''F'' on ''G'' (with values in some fixed finite-dimensional [vector space](/source/vector_space) ''V'', in the vector-valued case), subject to three kinds of conditions:

# to transform under translation by elements <math>\gamma \in \Gamma </math> according to the given [factor of automorphy](/source/factor_of_automorphy) ''j'';
# to be an [eigenfunction](/source/eigenfunction) of certain [Casimir operator](/source/Casimir_operator)s on ''G''; and
# to satisfy a "moderate growth" asymptotic condition a [height function](/source/height_function).

It is the first of these that makes ''F'' ''automorphic'', that is, satisfy an interesting [functional equation](/source/functional_equation) relating ''F''(''g'') with ''F''(''γg'') for <math>\gamma \in \Gamma </math>. In the vector-valued case the specification can involve a finite-dimensional [group representation](/source/group_representation) ρ acting on the components to 'twist' them. The Casimir operator condition says that some [Laplacian](/source/Laplacian)s{{Citation needed|reason=unveriviable and unsufficient citation about the source|date=May 2017}} have ''F'' as eigenfunction; this ensures that ''F'' has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where ''G''/Γ is not [compact](/source/compact_space) but has [cusp](/source/Cusp_form)s.

The formulation requires the general notion of ''factor of automorphy'' ''j'' for Γ, which is a type of 1-[cocycle](/source/Group_cohomology) in the language of [group cohomology](/source/group_cohomology). The values of ''j'' may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when ''j'' is derived from a [Jacobian matrix](/source/Jacobian_matrix), by means of the [chain rule](/source/chain_rule).

A more straightforward but technically advanced definition using [class field theory](/source/class_field_theory), constructs automorphic forms and their correspondent functions as embeddings of [Galois group](/source/Galois_group)s to their underlying [global field](/source/global_field) extensions. In this formulation, automorphic forms are certain finite invariants, mapping from the [idele class group](/source/idele_class_group) under the [Artin reciprocity law](/source/Artin_reciprocity_law). Herein, the analytical structure of its [L-function](/source/L-function) allows for generalizations with various [algebro-geometric](/source/Algebraic_geometry) properties; and the resultant [Langlands program](/source/Langlands_program). To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of [number fields](/source/Algebraic_number_field) in a most abstract sense, therefore indicating the ['primitivity'](/source/Primitive_root_of_unity) of their [fundamental structure](/source/Ideal_class_group). Allowing a powerful mathematical tool for analyzing the invariant constructs of virtually any numerical structure.

Examples of automorphic forms in an explicit unabstracted state are difficult to obtain, though some have directly analytical properties:

- The [Eisenstein series](/source/Eisenstein_series) (which is a prototypical [modular form](/source/modular_form)) over certain [field extension](/source/field_extension)s as [Abelian group](/source/Abelian_group)s.

- Specific generalizations of [Dirichlet L-function](/source/Dirichlet_L-function)s as [class field-theoretic](/source/Class_field_theory) objects.

- Generally any [harmonic analytic](/source/Harmonic_analysis) object as a [functor](/source/functor) over [Galois group](/source/Galois_group)s which is invariant on its [ideal class group](/source/ideal_class_group) (or [idele](/source/idele)).

As a general principle, automorphic forms can be thought of as [analytic function](/source/analytic_function)s on [abstract structures](/source/Abstract_algebra), which are invariant with respect to a generalized analogue of their [prime ideal](/source/prime_ideal)  (or an abstracted [irreducible fundamental representation](/source/Fundamental_representation)). As mentioned, automorphic functions can be seen as generalizations of modular forms (as therefore [elliptic curve](/source/elliptic_curve)s), constructed by some [zeta function](/source/Riemann_zeta_function) analogue on an [automorphic](/source/Automorphism_group) structure. In the simplest sense, automorphic forms are [modular form](/source/modular_form)s defined on general [Lie group](/source/Lie_group)s; because of their symmetry properties. Therefore, in simpler terms, a general function which analyzes the invariance of a structure with respect to its prime ['morphology'](/source/Morphism_of_algebraic_varieties).

==History==
Before this very general setting was proposed (around 1960), there had already been substantial developments of automorphic forms other than modular forms. The case of Γ a [Fuchsian group](/source/Fuchsian_group) had already received attention before 1900 (see below). The [Hilbert modular form](/source/Hilbert_modular_form)s (also called Hilbert-Blumenthal forms) were proposed not long after that, though a full theory was long in coming. The [Siegel modular form](/source/Siegel_modular_form)s, for which ''G'' is a [symplectic group](/source/symplectic_group), arose naturally from considering [moduli space](/source/moduli_space)s and [theta function](/source/theta_function)s. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by [Ilya Piatetski-Shapiro](/source/Ilya_Piatetski-Shapiro), in the years around 1960, in creating such a theory. The theory of the [Selberg trace formula](/source/Selberg_trace_formula), as applied by others, showed the considerable depth of the theory. [Robert Langlands](/source/Robert_Langlands) showed how (in generality, many particular cases being known) the [Riemann–Roch theorem](/source/Riemann%E2%80%93Roch_theorem) could be applied to the calculation of dimensions of automorphic forms; this is a kind of ''post hoc'' check on the validity of the notion. He also produced the general theory of [Eisenstein series](/source/real_analytic_Eisenstein_series), which corresponds to what in [spectral theory](/source/spectral_theory) terms would be the 'continuous spectrum' for this problem, leaving the [cusp form](/source/cusp_form) or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since [Srinivasa Ramanujan](/source/Srinivasa_Ramanujan), as the heart of the matter.

==Automorphic representations==
{{See also|Cuspidal representation}}
The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with ''G'' an [algebraic group](/source/algebraic_group), treated as an [adelic algebraic group](/source/adelic_algebraic_group). It does not completely include the automorphic form idea introduced above, in that the [adelic](/source/Adele_ring) approach is a way of dealing with the whole family of [congruence subgroup](/source/congruence_subgroup)s at once. Inside an ''L''<sup>2</sup> space for a quotient of the adelic form of ''G'', an automorphic representation is a representation that is an infinite [tensor product](/source/tensor_product) of representations of [p-adic group](/source/p-adic_group)s, with specific [enveloping algebra](/source/Universal_enveloping_algebra) representations for the [infinite prime](/source/infinite_prime)(s). One way to express the shift in emphasis is that the [Hecke operator](/source/Hecke_operator)s are here in effect put on the same level as the Casimir operators; which is natural from the point of view of [functional analysis](/source/functional_analysis){{Citation needed|reason=unveriviable and unsufficient citation about the source|date=May 2017}}, though not so obviously for the number theory. It is this concept that is basic to the formulation of the [Langlands philosophy](/source/Langlands_philosophy).

==Poincaré on discovery and his work on automorphic functions==
One of [Poincaré](/source/Henri_Poincar%C3%A9)'s first discoveries in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician [Lazarus Fuchs](/source/Lazarus_Fuchs), because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré actually developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a discrete [infinite group](/source/infinite_group) of linear fractional transformations. Automorphic functions then generalize both [trigonometric](/source/Trigonometric_functions) and [elliptic function](/source/elliptic_function)s.

Poincaré explains how he discovered Fuchsian functions:
{{blockquote|For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the [hypergeometric series](/source/hypergeometric_series); I had only to write out the results, which took but a few hours.}}

==See also==
* [Automorphic factor](/source/Automorphic_factor)
* [Automorphic function](/source/Automorphic_function)
* [Maass cusp form](/source/Maass_cusp_form)
* ''[Automorphic Forms on GL(2)](/source/Automorphic_Forms_on_GL(2))'', a book by H. Jacquet and Robert Langlands
* [Jacobi form](/source/Jacobi_form)

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* {{springer|id=a/a014160|authorlink=Alexey Parshin|author=A. N. Parshin|title=Automorphic Form}}
* [Henryk Iwaniec](/source/Henryk_Iwaniec), ''Spectral Methods of Automorphic Forms, Second Edition'', (2002) (Volume 53 in ''[Graduate Studies in Mathematics](/source/Graduate_Studies_in_Mathematics)''), American Mathematical Society, Providence, RI {{ISBN|0-8218-3160-7}}
* [Daniel Bump](/source/Daniel_Bump), "Automorphic Forms and Representations", 1998, Cambridge University Press
{{refend}}

* Stephen Gelbart (1975), "Automorphic forms on Adele groups", {{ISBN|9780608066042}}
* {{PlanetMath attribution|id=3793|title=Jules Henri Poincaré}}

==External links==
*{{Wikiquote-inline|Automorphic form}}
*{{Commons category-inline|Automorphic forms}}

{{Authority control}}

Category:Lie groups
Category:Automorphic forms

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