Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series
type: Item
instance of: scholarly article
title: Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series
zbMATH Open document ID: 1322.11050
author: Yiannis N. Petridis
author: Nicole Raulf
author: Morten S. Risager
published in: Algebra \& Number Theory
publication date: 2014-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.1894
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F68
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 58J51
zbMATH DE Number: 6361763
zbMATH Keywords: spectral theory
zbMATH Keywords: quantum unique ergodicity
zbMATH Keywords: half-integral weight Eisenstein series
zbMATH Keywords: multiple Dirichlet series
zbMATH Keywords: convexity bound
MaRDI profile type: Publication
OpenAlex ID: W3101634942
arXiv ID: 1209.1894
DOI: 10.2140/ANT.2014.8.1539
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review text: Let \(d\mu(z)=dx\,dy/y^2\) denote the volume element corresponding to the hyperbolic metric on the upper half plane \(\mathcal{H}\) and let \(E(z,s)\) be the weight \(1/2\) Eisenstein series for \(\Gamma_0(4)\). Then the authors study the limiting behavior of \(d\mu_t(z)=|E(z,1/2+it)|^2\,d\mu(z)\) on compact sets as \(t\to\infty\). The similar question for weight \(0\) Eisenstein series was studied by \textit{W. Luo} and \textit{P. Sarnak} [Publ. Math., Inst. Hautes Étud. Sci. 81, 207--237 (1995; Zbl 0852.11024)]. However, this case requires new ingredients since the Fourier coefficients of the weight \(1/2\) Eisenstein series \(E(z,s)\) are themselves expressed in terms of quadratic Dirichlet \(L\)-functions. Dirichlet series given as sums of such Fourier coefficients are typically not Eulerian.NEWLINENEWLINETo adapt the method of Luo and Sarnak, the authors study the growth of a double Dirichlet series \(Z(s,w)\) which may be realized as a sum of quadratic twists of a \(\mathrm{GL}_2\) Maass form; this series arose in work of the reviewer and \textit{J. Hoffstein} [Ann. Math. (2) 142, No. 2, 385--423 (1995; Zbl 0847.11026)], following earlier related work of \textit{D. Bump} et al. [Ann. Math. (2) 131, No. 1, 53--127 (1990; Zbl 0699.10039)]. The authors derive the continuation of this series, following the method outlined in [\textit{D. Bump} et al., Bull. Am. Math. Soc., New Ser. 33, No. 2, 157--175 (1996; Zbl 0865.11043)], give an exact functional equation, and then establish an estimate on the growth of \(Z(s,w)\) for \(s,w\) on the critical line as the imaginary parts of \(s,w\) approach \(\infty\) (independently). They call this bound the convexity bound. They then show that a stronger bound, that is a subconvexity bound, would imply that for any two compact Jordan measurable sets of \(\Gamma_0(4)\backslash \mathcal{H}\), the ratio of their measures with respect to \(d\mu_t(z)\) approaches the ratio of their volumes as \(t\to \infty\). An appendix gives bound…
description: scientific article; zbMATH DE number 6361763