On the structure of the Selberg class. III: Sarnak's rigidity conjecture
type: Item
instance of: scholarly article
title: On the structure of the Selberg class. III: Sarnak's rigidity conjecture
zbMATH Open document ID: 0997.11072
DOI: 10.1215/S0012-7094-00-10137-8
author: J. Kaczorowski
author: A. Perelli
published in: Duke Mathematical Journal
publication date: 2002-11-12T00:00:00Z
review text: In the third paper of this series the authors deal with a problem, raised by Peter Sarnak, about rigidity in the Selberg class \(\mathcal S\) [Part I, cf. Acta Math. 182, 207-241 (1999; Zbl 1126.11335), Part II, J. Reine Angew. Math. 524, 73-96 (2000; Zbl 0982.11051), Part IV, Acta Arith. 104, 97-116 (2002; Zbl 0996.11053)]. Roughly speaking, the rigidity conjecture states that every continuous one-parameter family \({\mathcal F} =\{F(s;\xi)\}_{\xi\in\mathbb R}\) of functions in \(\mathcal S\) is a shifted family, that is, \[ F(s;\xi)=\prod_{j=1}^k F(s+ih_j(\xi)) \tag \(*\) \] with \(F_j\in\mathcal S\) and \(h_j(\xi)\) continuous. In the first theorem the authors prove the rigidity conjecture, assuming the Selberg orthonormality conjecture and the countability conjecture. This theorem has been independently obtained by \textit{U. Vorhauer} and \textit{E. Wirsing} in [On Sarnak's rigidity conjecture, J. Reine Angew. Math. 531, 35-47 (2001; Zbl 0960.11044)]. Further, they define an analytic family \({\mathcal F} =\{F(s;\xi)\}_{\xi\in\mathcal U}\), where \({\mathcal U}\supset{\mathbb R}\) is a domain in \(\mathbb C\), the function \(F(s;\xi)\) belongs to \(\mathcal S\) for \(\xi\in \mathbb R\) and \({\mathcal F}\) satisfies some natural holomorphicity conditions. For the analytic family \({\mathcal F}\) it is expected that the functions \(h_j(\xi)\) in \((*)\) can be chosen to be holomorphic on a suitable domain \({\mathcal V}\supset\mathbb R\). In this case, \({\mathcal F}\) is called an analytic shifted family. The second theorem states that, assuming the same conditions as in the first theorem, every analytic family is an analytic shifted family. Also the authors show that both theorems hold unconditionally in the class of automorphic \(L\)-functions.
reviewed by: Ramūnas Garunkštis
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 1441964
zbMATH Keywords: Selberg class
zbMATH Keywords: rigidity conjecture
zbMATH Keywords: Selberg orthonormality conjecture
zbMATH Keywords: countability conjecture
zbMATH Keywords: analytic family
zbMATH Keywords: anlaytic shifted family
zbMATH Keywords: automorphic \(L\)-functions
MaRDI profile type: Publication
Wikidata QID: Q122949921
cites work: Q4234990
cites work: Q4161937
cites work: Q3322387
cites work: Q4252240
cites work: Holomorphic functions of several variables. An introduction to the fundamental theory. With the assist. of Gottfried Barthel transl. by Michael Bridgland
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cites work: On Sarnak's rigidity conjecture
full work available at URL: s0012-7094-00-10137-8
OpenAlex ID: W1990960763
description: scientific article; zbMATH DE number 1441964