Algebraicity of harmonic Maass forms
type: Item
instance of: scholarly article
title: Algebraicity of harmonic Maass forms
zbMATH Open document ID: 1247.11053
author: Ken Ono
published in: The Ramanujan Journal
publication date: 2010-05-07T00:00:00Z
review text: This excellent expository article describes recent work by the author, J. H. Bruinier and R. Rhoades [\textit{J. H. Bruinier, K. Ono} and \textit{R. C. Rhoades}, Math. Ann. 342, No. 3, 673--693 (2008); corrigendum 345, No. 1, 31 (2009; Zbl 1246.11098); \textit{J. Bruinier} and \textit{K. Ono}, Ann. Math. (2) 172, No. 3, 2135--2181 (2010; Zbl 1244.11046)] on algebraicity of coefficients of certain harmonic weak Maaß forms and the relation of their results to the following two classical topics: {1.} vanishing of Hecke eigenvalues of integer weight modular forms, {2.} vanishing of derivatives of modular \(L\)-functions. As an example, the first result sheds new light on Lehmer's conjecture on the non-vanishing of Ramanujan's \(\tau\)-function. Let \(\Delta(z)\) be the modular discriminant, that is, the unique cusp form of weight 12 for \(\mathrm{SL}_2(\mathbb{Z})\). Then, \[ \Delta(z) = \sum_{n=1}^\infty \tau(n) q^n, \] where \(z \in \mathbb{H} = \{z\in \mathbb C: \text{Im}(z) > 0 \}\), the complex upper half-plane, and \(q=e^{2\pi iz}\). Theorem 1.2 of the article under review implies that if \(\tau(p)=0\) for a prime \(p\), then, for all \(n\) such that \(\mathrm{ord}_p(n)\) is odd, the \(n\)-th Fourier coefficient of a certain weakly holomorphic Poincaré series is rational. In fact, this is equivalent to the rationality of the first Fourier coefficient. The author provides numerical evidence supporting the conjecture that this coefficient is not rational. The second topic is covered by the results of the author and J. H. Bruinier [Zbl 1244.11046] which relates the vanishing of the quadratic twist of discriminant \(D\) of a Hecke \(L\)-function for a weight 2 newform to the algebraicity of the coefficient of index \(D\) of the holomorphic part of a certain harmonic weak Maaß\ form. In fact, the authors show that harmonic weak Maaß\ forms serve as generating functions for central values and derivatives of quadratic twists of \(L\)-functions of weight \(2\) newforms. T…
reviewed by: Stephan Ehlen
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F25
zbMATH DE Number: 5705225
zbMATH Keywords: harmonic weak Maass forms
zbMATH Keywords: Lehmer's conjecture
zbMATH Keywords: L-functions
MaRDI profile type: Publication
full work available at URL: s11139-009-9212-y
OpenAlex ID: W2086134974
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DOI: 10.1007/S11139-009-9212-Y
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description: scientific article; zbMATH DE number 5705225