Perturbation of scattering poles for hyperbolic surfaces and central values of \(L\)-series
type: Item
instance of: scholarly article
title: Perturbation of scattering poles for hyperbolic surfaces and central values of \(L\)-series
zbMATH Open document ID: 0981.11020
DOI: 10.1215/S0012-7094-00-10316-X
author: Yiannis N. Petridis
published in: Duke Mathematical Journal
publication date: 2000-07-24T00:00:00Z
review text: A scattering pole is a pole of the analytic continuation of the determinant of the scattering matrix to \(\Re(s)<1/2\). From \textit{R. S. Phillips} and \textit{P. Sarnak} [J. Am. Math. Soc. 5, 1-32 (1992; Zbl 0743.30039)] it is known that embedded eigenvalues become scattering poles under perturbation. Here under certain hypotheses on the Eisenstein series, integral formulas for the first variation of the weighted mean of the scattering pole are given. Assume the discrete group is \(\Gamma _0 (q)\), for prime \(q\), and the perturbations are generated by a holomorphic cusp form \(g\) of weight 2 for this group, assuming that \(g\) is a Hecke eigenform with real coefficients and eigenvalue \(\varepsilon _q \) for the Hecke involution. Theorem 1.5 relates the nonvanishing of the \(L\)-series of \(g\) at \(s = 1\) to the movement of the scattering poles away from \(\Re(s)=1/4\). In particular, it says that if \(L(g,1) \neq 0\), the scattering poles move. Examples of \(q\) satisfying the condition are \(q=11, 17, 19, 67, 73\). Theorem 1.9 assumes the Riemann hypothesis and a further condition on at least one of the 2 holomorphic cusp forms involved to show that a positive proportion of the scattering poles move off the line \(\Re(s)=1/4\).
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 1463081
zbMATH Keywords: scattering poles
zbMATH Keywords: \(L\)-series
zbMATH Keywords: cusp forms
reviewed by: Audrey A. Terras
describes a project that uses: ecdata
MaRDI profile type: Publication
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full work available at URL: s0012-7094-00-10316-x
OpenAlex ID: W2011376945
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description: scientific article; zbMATH DE number 1463081