Automorphic form

Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R), with the discrete subgroup being the modular group or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ${\displaystyle G({\mathbb{A}}_F)}$, for an algebraic group ${\displaystyle G}$ and an algebraic number field ${\displaystyle F}$, is a complex-valued function on ${\displaystyle G({\mathbb{A}}_F)}$ that is left invariant under ${\displaystyle G(F)}$ and satisfies certain smoothness and growth conditions.

Henri Poincaré first discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through the Langlands conjectures, automorphic forms play an important role in modern number theory.

Definition

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group ${\displaystyle G}$ acts on a complex-analytic manifold ${\displaystyle X}$. Then, ${\displaystyle G}$ also acts on the space of holomorphic functions from ${\displaystyle X}$ to the complex numbers. A function ${\displaystyle f}$ is termed an automorphic form if the following holds:

${\displaystyle f(g\cdot x)=j_g(x)f(x)}$

where ${\displaystyle j_g(x)}$ is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of ${\displaystyle G}$.

The factor of automorphy for the automorphic form ${\displaystyle f}$ is the function ${\displaystyle j}$. An automorphic function is an automorphic form for which ${\displaystyle j}$ 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:

1. to transform under translation by elements ${\displaystyle \gamma \in \mathrm{\Gamma }}$ according to the given factor of automorphy j;
2. to be an eigenfunction of certain Casimir operators on G; and
3. to satisfy a "moderate growth" asymptotic condition a height function.

It is the first of these that makes F automorphic, that is, satisfy a functional equation relating F(g) with F(γg) for ${\displaystyle \gamma \in \mathrm{\Gamma }}$. 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 Laplacians 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 but has cusps.

The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-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 groups 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 algebro-geometric properties; and the resultant Langlands program. To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of number fields in an abstract sense, therefore indicating the 'primitivity' of their fundamental structure. It is a mathematical tool for analyzing the invariant constructs of many 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 extensions as Abelian groups.

- Specific generalizations of Dirichlet L-functions as class field-theoretic objects.

- Generally any harmonic analytic object as a functor over Galois groups which is invariant on its ideal class group (or idele).

As a general principle, automorphic forms can be thought of as analytic functions on abstract structures, which are invariant with respect to a generalized analogue of their prime ideal (or an abstracted irreducible fundamental representation). As mentioned, automorphic functions can be seen as generalizations of modular forms (as therefore elliptic curves), constructed by some zeta function analogue on an automorphic structure. In the simplest sense, automorphic forms are modular forms defined on general Lie groups; 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'.