Period functions for Maass wave forms

Contents Introduction Chapter I. The period correspondence via L-series 1. The correspondences u ↔ Lε ↔ f ↔ ψ 2. Periodicity, L-series, and the three-term functional equation 3. Even and odd 4. Relations between Mellin transforms; proof of Theorem 1 Chapter II. The period correspondence via integral transforms 1. The integral representation of ψ in terms of u 2. The period function as the integral of a closed 1-form 3. The incomplete gamma function expansion of ψ 4. Other integral transforms and intermediate functions 5. Boundary values of Maass wave forms Chapter III. Periodlike functions 1. Examples 2. Fundamental domains for periodlike functions 3. Asymptotic behavior of smooth periodlike functions 4. " Bootstrapping " Chapter IV. Complements 1. The period theory in the noncuspidal case 2. Integral values of s and connections with holomorphic modular forms 3. Relation to the Selberg zeta function and Mayer's theorem References Introduction Recall that a Maass wave form 1 on the full modular group Γ = PSL(2, Z) is a smooth Γ-invariant function u from the upper half-plane H = {x+iy | y > 0} to C which is small as y → ∞ and satisfies ∆u = λ u for some λ ∈ C, where ∆ = −y 2 ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is the hyperbolic Laplacian. These functions give a basis for L 2 on the modular surface Γ\H, in analogy with the usual trigonometric 1 We use the traditional term, but one should really specify " cusp form. " Also, the word " form " should more properly be " function, " since u is simply invariant under Γ, with no automorphy factor. We often abbreviate " Maass wave form " to " Maass form. " 192 J. LEWIS AND D. ZAGIER waveforms on the torus R 2 /Z 2 , which are also (for this surface) both the Fourier building blocks for L 2 and eigenfunctions of the Laplacian. Although therefore very basic objects, Maass forms nevertheless still remain mysteriously elusive fifty years after their discovery; in particular, no explicit construction exists for any of these functions for the full modular group. The basic information about them (e.g. their existence and the density of the eigenvalues) comes mostly from the Selberg trace formula; the rest is conjectural with support from extensive numerical computations. Maass forms arise naturally in such diverse fields as number theory, dy-namical …