zbMATH — the first resource for mathematics

Period functions for Maass wave forms. I. (English) Zbl 1061.11021
This paper investigates a connection (first explored by the first author [Invent. Math. 127, No. 2, 271–306 (1997; Zbl 0922.11043)]) between Maass wave forms for \(\text{SL}_2(\mathbb Z)\) and solutions to a certain three term functional equation. We list a special case of their results here.
Theorem: Let \(s\in \mathbb C\) with \(\text{Re}(s)= \frac{1}{2}\). Then there is an isomorphism between the space of Maass cusp forms with eigenvalue \(s(1-s)\) on \(\Gamma\) and the space of real-analytic solutions of the three-term functional equation \[ \psi(x) = \psi(x+1)+ (x+1)^{-2s} \psi \biggl(\frac{x}{x+1}\biggr) \] on \(\mathbb R_+\), which satisfy the growth condition \[ \psi(x)=\text{o}(1/x) \;(x\rightarrow 0), \qquad \psi(x)=\text{o}(1) \;(x\rightarrow \infty). \]

11F37 Forms of half-integer weight; nonholomorphic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
Full Text: DOI EuDML arXiv