Equidistribution of Fourier coefficients of integral weight Maass-Poincaré series
type: Item
instance of: scholarly article
title: Equidistribution of Fourier coefficients of integral weight Maass-Poincaré series
author: Riad Masri
author: Wei-Lun Tsai
published in: Advances in Mathematics
publication date: 2022-10-06T00:00:00Z
review text: One of the breakthrough results in mathematics without any doubt is the proof of the Sato-Tate Conjecture recently. One of the evidences to show its importance, is of course, the great consequences and applications of the Sato-Tate Conjecture. There are dozens of papers in the literature related to these topics. The paper under review concerns the case of integral weight Maass-Poincaré series. It is well known that Fourier coefficients of weak Maass forms have many interesting connections with diverse arithmetic objects. Hence it is not a surprise that it stands at the core of the research recently. In the paper under review, the authors show that the normalized Fourier coefficients of a generic family of Maass-Poincaré series of integral weight \(k\) and prime level \(p\) become quantitatively equidistributed with respect to the Sato-Tate measure as \(p \to \infty\). As a consequence, they deduce similar results for harmonic Maass forms of integral weight \(k \leq 0\) and level \(p\), and weakly holomorphic modular forms of integral weight \(k \geq 2\) and level \(p\). By using SageMath, the authors illustrate their results where the reader can see the equidistribution geometrically. The proof of the main result is a mixture of algebraic and analytic tools.
reviewed by: İlker İnam
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 7597097
zbMATH Keywords: equidistribution
zbMATH Keywords: Fourier coefficient
zbMATH Keywords: weak Maass form
zbMATH Keywords: Sato-Tate measure
Wikidata QID: Q114211419
describes a project that uses: SageMath
describes a project that uses: Digital Library of Mathematical Functions
MaRDI profile type: Publication
full work available at URL: j.aim.2022.108637
OpenAlex ID: W4292771371
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DOI: 10.1016/J.AIM.2022.108637
description: scientific article; zbMATH DE number 7597097