Automorphic form
From Academic Kids

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. It is in terms of a Lie group G, to generalise the groups SL_{2}(R) or PSL_{2}(R) of modular forms, and a discrete group Γ in G, to generalise the modular group, or one of its congruence subgroups. The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1cocycle 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 vectorvalued 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.
In the general setting, then, an automorphic form is a function F on G (with values in some fixed finitedimensional vector space V, in the vectorvalued case), subject to three kinds of conditions:
 to transform under translation by elements γ of Γ according to the given automorphy factor j;
 to be an eigenfunction of certain Casimir operators on G; and
 to satisfy some conditions on growth at infinity.
It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) with F(γg) for γ in Γ. In the vectorvalued case the specification can involve a finitedimensional 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 complexanalytic function depends on the particular case. The third condition is to handle the case where G/Γ is not compact but has cusps.
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 had already received attention before 1900. The Hilbert modular forms (HilbertBlumenthal, as one should say) were proposed not long after that, though a full theory was long in coming. The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions. The postwar interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complexanalytic. Much work was done, in particular by PyatetskiiShapiro, in the years around 1960, in creating such a theory. The theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Langlands showed how (in generality, many cases being known) the RiemannRoch 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, which corresponds to what in spectral theory terms would be the 'continuous spectrum' for this problem, leaving the cusp form or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since Ramanujan, as the heart of the matter.
The subsequent notion of automorphic representation has proved of great technical value for dealing with G an algebraic group, treated as an adelic algebraic group. It does not actually completely include the automorphic form idea introduced above, in that the adele approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L^{2} space for a quotient of the adelic form of G, an automorphic representation is a representation that is an infinite tensor product of representations of padic groups, with specific enveloping algebra representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis, though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.
Poincare on his work on automorphic functions
Poincaré's first area of interest in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician Lazarus Fuch, because Fuch was known for being a good teacher and had researched differential equations and the theory of functions heavily. (Obviously, the functions did not keep the name Fuchsian). Poincaré actually developed the concept of these functions as part of his doctoral thesis.
Under Poincaré's definition, an automorphic function is a function where which is analytic under its domain and which is invariant under a denumerable infinite group of linear fractional transformations; they are the generalizations of trigonometric functions and elliptic functions.
Poincaré explains how he discovered Fuchsian functions:
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; I had only to write out the results, which took but a few hours.
Poincaré communicated a lot with Klein, another mathematician working on Fuchsian functions. They were able to discuss and further the theory of automorphic/Fuchsian functions. Apparently, Klein became jealous of Poincaré's high opinion of Fuch's work and ended their relationship on bad terms.