Riemann hypothesis for period polynomials attached to the derivatives of \(L\)-functions of cusp forms for \(\Gamma_0(N)\)
type: Item
instance of: scholarly article
title: Riemann hypothesis for period polynomials attached to the derivatives of \(L\)-functions of cusp forms for \(\Gamma_0(N)\)
zbMATH Open document ID: 1500.11034
author: Bo-Hae Im
author: Hojin Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-02-14T00:00:00Z
review text: Period polynomials play a critical role in the theory of modular forms. They encode the critical \(L\)-values of modular forms, and they form spaces which determine modular forms via the Eichler-Shimura isomorphism. There has been a great deal of work developing their arithmetic structure, for example, Manin's Theorem on the transcendence of their coefficients, and the seminal work of \textit{W. Kohnen} and \textit{D. Zagier} [in: Modular forms, Symp. Durham/Engl. 1983, 197--249 (1984; Zbl 0618.10019)] on the structure of rational period polynomials. Recently, their analytic structure, as polynomials, has received much attention. Specifically, their roots have been shown to be tightly constrained in various situations. A few particular examples include the work of Conrey-Farmer-Imamoglu [\textit{J. B. Conrey} et al., Int. Math. Res. Not. 2013, No. 20, 4758--4771 (2013; Zbl 1305.11030)] showing that all but 5 of the roots of the odd parts of all level \(1\) cusp forms lie on the unit circle and of \textit{A. El-Guindy} and \textit{W. Raji} [Bull. Lond. Math. Soc. 46, No. 3, 528--536 (2014; Zbl 1311.11027)] showing that all roots of the full period polynomials lie on the unit circle when the level is \(1\). This was greatly generalized by Jin-Ma-Ono-Soundararajan [\textit{S. Jin} et al., Proc. Natl. Acad. Sci. USA 113, No. 10, 2603--2608 (2016; Zbl 1412.11075)], who showed that the same is true in all levels for newforms of any level, where unit circle is replaced by a circle of appropriate radius dictated by the Fricke involution. This has been termed the Riemann Hypothesis for Period Polynomials (RHPP). Afterwards, several papers have considered RHPPs in other contexts. For instance, \textit{N. Diamantis} and \textit{L. Rolen} [Res. Math. Sci. 5, No. 1, Paper No. 9, 15 p. (2018; Zbl 1457.11032); J. Reine Angew. Math. 770, 1--25 (2021; Zbl 1472.11152)] considered analogues of period polynomials which are built instead out of higher derivatives of modular form \(L\)-f…
reviewed by: Larry Rolen
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 7472992
zbMATH Keywords: period polynomials
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: modular forms
zbMATH Keywords: \(L\)-derivatives
Wikidata QID: Q115188930
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125971
OpenAlex ID: W4200369901
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DOI: 10.1016/J.JMAA.2021.125971
description: scientific article; zbMATH DE number 7472992