Automorphic form
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.
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[citation needed] 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 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 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.
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 a most abstract sense, therefore indicating the 'primitivity' of their fundamental structure.
Allowing a powerful mathematical tool for analyzing the invariant constructs of virtually any numerical structure.
Therefore, in simpler terms, a general function which analyzes the invariance of a structure with respect to its prime 'morphology'.
The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions.
Robert Langlands showed how (in generality, many particular cases being known) the Riemann–Roch 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.
From the point of view of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan, as the heart of the matter.
It does not completely include the automorphic form idea introduced above, in that the adelic approach is a way of dealing with the whole family of congruence subgroups at once.
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.
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; I had only to write out the results, which took but a few hours.
