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).

The Rankin–Selberg convolution for L-functions associated to automorphic forms on GL(2) was independently discovered by Rankin (1939) and Selberg (1940). The method was discussed in Section 7.2 in connection with the Gelbart–Jacquet lift and a generalization to GL(3) was given in Section 7.4. A much more general interpretation of the original Rankin–Selberg convolution for GL(2) × GL(2) in the framework of adeles and automorphic representations was first obtained in (Jacquet, 1972). The theory was subsequently further generalized in (Jacquet and Shalika, 1981).

The Rankin–Selberg convolution for the case GL(n) × GL(n′), (1 ≤ n < n′), requires new ideas. Note that this includes GL(1) × GL(n′) which is essentially the Godement–Jacquet L-function whose holomorphic continuation and functional equation was first obtained by Godement and Jacquet (1972). A sketch of the general Rankin–Selberg convolution for GL(n) × GL(n′) in classical language was given in (Jacquet, 1981). The theory was further extended in the context of automorphic representations in (Jacquet, Piatetski-Shapiro and Shalika, 1983). In this chapter, we shall present an elementary and self contained account of both the meromorphic continuation and functional equation of Rankin–Selberg L-functions associated to GL(n) × GL(n′). In particular, this will give the meromorphic continuation and functional equation of the Godement–Jacquet L-function.

The fact that Rankin–Selberg L-functions have Euler products is a consequence of the uniqueness of Whittaker functions for local fields.

The theory of automorphic forms and L-functions for the group of n × n invertible real matrices (denoted GL(n, ℝ)) with n ≥ 3 is a relatively new subject. The current literature is rife with 150+ page papers requiring knowledge of a large breadth of modern mathematics making it difficult for a novice to begin working in the subject. The main aim of this book is to provide an essentially self-contained introduction to the subject that can be read by someone with a mathematical background consisting only of classical analysis, complex variable theory, and basic algebra – groups, rings, fields. Preparation in selected topics from advanced linear algebra (such as wedge products) and from the theory of differential forms would be helpful, but is not strictly necessary for a successful reading of the text. Any Lie or representation theory required is developed from first principles.

This is a low definition text which means that it is not necessary for the reader to memorize a large number of definitions. While there are many definitions, they are repeated over and over again; in fact, the book is designed so that a reader can open to almost any page and understand the material at hand without having to backtrack and awkwardly hunt for definitions of symbols and terms.

The philosophy of the exposition is to demonstrate the theory by simple, fully worked out examples.

Maass forms for SL(2, ℤ) were introduced in Section 3.3. An important objective of this book is to generalize these functions to the higher-rank group SL(n, ℤ) with n ≥ 3. It is a highly non-trivial problem to show that infinitely many even Maass forms for SL(2, ℤ) exist. The first proof was given by Selberg (1956) where he introduced the trace formula as a tool to obtain Weyl's law, which in this context gives an asymptotic count (as x → ∞) for the number of Maass forms of type ν with |ν| ≤ x. Selberg's methods were extended by Miller (2001), who obtain Weyl's law for Maass forms on SL(3, ℤ) and Müller (2004), who obtained Weyl's law for Maass forms on SL(n, ℤ).

A rather startling revelation was made by Phillips and Sarnak (1985) where it was conjectured that Maass forms should not exist for generic non-congruence subgroups of SL(2, ℤ), except for certain situations where their existence is ensured by symmetry considerations, see Section 4.1. Up to now no one has found a single example of a Maass form for SL(2, ℤ), although Maass (1949) discovered some examples for congruence subgroups (see Section 3.15). So it seemed as if Maass forms for SL(2, ℤ) were elusive mysterious objects and the non-constructive proof of their existence (Selberg, 1956) suggested that they may be unconstructible.

Poincaré series and Kloosterman sums associated to the group SL(3, ℤ) were introduced and studied in (Bump, Friedberg and Goldfeld, 1988) following the point of view of Selberg (1965). A very nice exposition of the GL(2) theory is given in (Cogdell and Piatetski-Shapiro, 1990). The method was first generalized to GL(n) in (Friedberg, 1987), (Stevens, 1987). In (Bump, Friedberg and Goldfeld, 1988) it is shown that the SL(3, ℤ) Kloosterman sums are hyper Kloosterman sums associated to suitable algebraic varieties. Non-trivial bounds were obtained by using Hensel's lemma and Deligne's estimates for hyper-Kloosterman sums (Deligne, 1974) in (Larsen, 1988), and later (Dabrowski and Fisher, 1997) improved these bounds by also using methods from algebraic geometry following (Deligne, 1974). Sharp bounds for special types of Kloosterman sums were also obtained in (Friedberg, 1987a,c). In (Dabrowski, 1993), the theory of Kloosterman sums over Chevalley groups is developed. Important applications of the theory of GL(n) Kloosterman sums were obtained in (Jacquet, 2004b) (see also (Ye, 1998)).

Another fundamental direction for research in the theory of Poincaré series and Kloosterman sums was motivated by the GL(2) Kuznetsov trace formula, (see (Kuznecov, 1980) and also (Bruggeman, 1978)). Generalizations of the Kuznetsov trace formula to GL(n), with n ≥ 3 were obtained in (Friedberg, 1987), (Goldfeld, 1987), (Ye, 2000), but they have not yet proved useful for analytic number theory. The chapter concludes with a new version of the GL(n) Kuznetsov trace formula derived by Xiaoqing Li.