Editing Automorphic form (section)

==Definition==
{{Confusing section|date=July 2023|reason=the connections between multiple attempts at definition are unclear; see talk "Definition section is confusing"}}
In [[mathematics]], the notion of '''factor of automorphy''' arises for a [[group (mathematics)|group]] [[Group action (mathematics)|acting]] on a [[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]]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]] ''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]] ''j'';
# to be an [[eigenfunction]] of certain [[Casimir operator]]s on ''G''; and
# to satisfy a "moderate growth" asymptotic condition a [[height function]].

It is the first of these that makes ''F'' ''automorphic'', that is, satisfy an interesting [[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]] ρ acting on the components to 'twist' them. The Casimir operator condition says that some [[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 space|compact]] but has [[Cusp form|cusp]]s.

The formulation requires the general notion of ''factor of automorphy'' ''j'' for Γ, which is a type of 1-[[Group cohomology#Cochain complexes|cocycle]] in the language of [[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]], by means of the [[chain rule]].

A more straightforward but technically advanced definition using [[class field theory]], constructs automorphic forms and their correspondent functions as embeddings of [[Galois group]]s to their underlying [[global field]] extensions. In this formulation, automorphic forms are certain finite invariants, mapping from the [[idele class group]] under the [[Artin reciprocity law]]. Herein, the analytical structure of its [[L-function]] allows for generalizations with various [[Algebraic geometry|algebro-geometric]] properties; and the resultant [[Langlands program]]. To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of [[Algebraic number field|number fields]] in a most abstract sense, therefore indicating the [[Primitive root of unity|'primitivity']] of their [[Ideal class group|fundamental structure]]. 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]] (which is a prototypical [[modular form]]) over certain [[field extension]]s as [[Abelian group]]s.

- Specific generalizations of [[Dirichlet L-function]]s as [[Class field theory|class field-theoretic]] objects.

- Generally any [[Harmonic analysis|harmonic analytic]] object as a [[functor]] over [[Galois group]]s which is invariant on its [[ideal class group]] (or [[idele]]).

As a general principle, automorphic forms can be thought of as [[analytic function]]s on [[Abstract algebra|abstract structures]], which are invariant with respect to a generalized analogue of their [[prime ideal]]  (or an abstracted [[Fundamental representation|irreducible fundamental representation]]). As mentioned, automorphic functions can be seen as generalizations of modular forms (as therefore [[elliptic curve]]s), constructed by some [[Riemann zeta function|zeta function]] analogue on an [[Automorphism group|automorphic]] structure. In the simplest sense, automorphic forms are [[modular form]]s defined on general [[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 [[Morphism of algebraic varieties|'morphology']].