The sign of Fourier coefficients of half-integral weight cusp forms

DOI10.1142/S179304211250042XzbMATH Open1290.11081arXiv1106.3606WikidataQ60108236 ScholiaQ60108236MaRDI QIDQ2880334

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Publication date: 13 April 2012

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Abstract: From a result of Waldspurger, it is known that the normalized Fourier coefficients $a(m)$ of a half-integral weight holomorphic cusp eigenform $f$ are, up to a finite set of factors, one of $pmsqrtL(1/2,f,ch{i}_m)$ when $m$ is square-free and $f$ is the integral weight cusp form related to $f$ by the Shimura correspondence. In this paper we address a question posed by Kohnen: which square root is $a(m)$? In particular, if we look at the set of $a(m)$ with $m$ square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so.

Full work available at URL: https://arxiv.org/abs/1106.3606

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