Rankin-Selberg method for real analytic cusp forms of arbitrary real weight
type: Item
instance of: scholarly article
title: Rankin-Selberg method for real analytic cusp forms of arbitrary real weight
zbMATH Open document ID: 0738.11042
DOI: 10.1007/BF02571424
author: Roland Matthes
published in: Mathematische Zeitschrift
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174437
review text: Let \(\Xi\) be a unitary \(m\)-dimensional multiplier system of real weight \(r\) for \(\Gamma=SL(2,\mathbb{Z})\) and \(F(z)\) be a real-analytic cusp form with respect to \(\Xi\) having values in \(\mathbb{C}^ m\) and which is normalized, i.e. \(\int_{D_ \Gamma}| F(z)|^ 2\;d\mu(z)=1\), \(d\mu\) is the hyperbolic measure, \(D_ \Gamma\) a fundamental domain. We establish a meromorphic continuation for the Rankin-Selberg zeta functions \[ R_{\pm,F}(s):=\sum^ m_{j=1}\quad\sum_{n\equiv\alpha_ j\bmod 1, \pm n>0}|\rho_{j,F}(n)|^ 2/| n|^{s-1} \] where \(\rho_{j,F}(n)\) are the Fourier coefficients of \(F\) and \(\alpha_ j\) are the cusp parameters. This is done via a modification of the Rankin- Selberg method. We obtain a functional equation of the form \(\mathbb{R}_ F(s)=\mathbb{C}_ F(s)\mathbb{R}_ F(1-s)\), where \(\mathbb{C}_ F(x)\) is a 2-by-2 matrix and \(\mathbb{R}_ F(s)\) is the vector \((R_{+,F}(s),R_{-,F}(s))^ T\). \(\zeta(2s)R_{\pm,F}(s)\) are holomorphic with the exception of a simple pole at \(s=1\). The components of \(\mathbb{C}_ F(s)\) contain Mellin transforms of the square of some Whittaker functions. The analytic properties of these are studied to some extent.
reviewed by: Roland Matthes
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 4216102
zbMATH Keywords: multiplier system
zbMATH Keywords: real-analytic cusp form
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: Rankin-Selberg zeta functions
MaRDI profile type: Publication
cites work: Q4760447
cites work: Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind
cites work: Prime geodesic theorem.
cites work: Q3879185
cites work: The special functions and their approximations. Vol. I, II
cites work: Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise
cites work: Lectures on modular functions of one complex variable
cites work: Q5518575
cites work: Q5774823
cites work: Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II. (The eigenvalue problem of automorphic forms in the hyperbolic plane. I. II)
cites work: Q5777299
cites work: Q5517939
OpenAlex ID: W2089393953
description: scientific article; zbMATH DE number 4216102