Editing Automorphic form (section)

==History==
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 (see below). The [[Hilbert modular form]]s (also called Hilbert-Blumenthal forms) were proposed not long after that, though a full theory was long in coming. The [[Siegel modular form]]s, for which ''G'' is a [[symplectic group]], arose naturally from considering [[moduli space]]s and [[theta function]]s. 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 (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. He also produced the general theory of [[real analytic Eisenstein series|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.