The spectral theory of non-holomorphic automorphic forms formally began with Maass (1949). His book (Maass, 1964) has been a source of inspiration to many. Some other references for this material are (Hejhal, 1976), (Venkov, 1981), (Sarnak, 1990), (Terras, 1985), (Iwaniec-Kowalski, 2004).

Maass gave examples of non-holomorphic forms for congruence subgroups of SL(2, ℤ) and took the very modern viewpoint, originally due to Hecke (1936), that automorphicity should be equivalent to the existence of functional equations for the associated L-functions. This is the famous converse theorem given in Section 3.15, and is a central theme of this entire book. The first converse theorem was proved by Hamburger (1921) and states that any Dirichlet series satisfying the functional equation of the Riemann zeta function ζ (s) (and suitable regularity criteria) must actually be a multiple of ζ (s).

Hyperbolic Fourier expansions of automorphic forms were first introduced in (Neunhöffer, 1973). In (Siegel, 1980), the hyperbolic Fourier expansion of GL(2) Eisenstein series is used to obtain the functional equation of certain Hecke L-functions of real quadratic fields with Grössencharakter (Hecke, 1920). When this is combined with the converse theorem, it gives explicit examples of Maass forms. These ideas are worked out in Sections 3.2 and 3.15.

Another important theme of this chapter is the theory of Hecke operators (Hecke, 1937a,b).