In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological groupG to the complex numbers which is invariant under the action of a discrete subgroup of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are automorphic forms defined over the groups SL or PSL 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 G, for an algebraic groupG and an algebraic number fieldF, is a complex-valued function on G that is left invariant under G and satisfies certain smoothness and growth conditions. 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.
Formulation
An automorphic form is a function F on G, subject to three kinds of conditions:
to transform under translation by elements according to the given factor of automorphyj;
to be an eigenfunction of certain Casimir operators on G; and
to satisfy a "moderate growth" asymptotic condition a height function.
It is the first of these that makes Fautomorphic, that is, satisfy an interesting functional equation relating F with F for. 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 automorphyj 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.
History
Before this very general setting was proposed, 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 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 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, 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. Robert Langlands showed how 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. 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 Srinivasa Ramanujan, as the heart of the matter.
Automorphic representations
The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with G an algebraic group, treated as an adelic algebraic group. 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. Inside an L2 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 p-adic groups, with specific enveloping algebra representations for the infinite prime. 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.
One of Poincaré's first discoveries in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician 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 of linear fractional transformations. Automorphic functions then generalize both trigonometric and elliptic functions. Poincaré explains how he discovered Fuchsian functions:
