Signs of Fourier coefficients of half-integral weight modular forms

DOI10.1007/S00208-020-02123-0zbMATH Open1467.11047arXiv1903.05811OpenAlexW3118680539MaRDI QIDQ2664162

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Publication date: 20 April 2021

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Abstract: Let $g$ be a Hecke cusp form of half-integral weight, level $4$ and belonging to Kohnen's plus subspace. Let $c(n)$ denote the $n$th Fourier coefficient of $g$, normalized so that $c(n)$ is real for all $ngeq1$. A theorem of Waldspurger determines the magnitude of $c(n)$ at fundamental discriminants $n$ by establishing that the square of $c(n)$ is proportional to the central value of a certain $L$-function. The signs of the sequence $c(n)$ however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that $c(n)<0$ and respectively $c(n)>0$ holds for a positive proportion of fundamental discriminants $n$. Moreover we show that the sequence $c(n)$ where $n$ ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of $c(n)$ which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms $g$ of level $4N$ with $N$ odd, square-free.

Full work available at URL: https://arxiv.org/abs/1903.05811

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