The Sign of Fourier Coefficients of Half-Integral Weight Cusp Forms

TL;DR

This paper investigates the sign changes of Fourier coefficients of half-integral weight cusp forms, showing they change sign infinitely often by analyzing related Dirichlet series and addressing a question posed by Kohnen.

Contribution

It provides a partial analytical approach to determine the sign variation of Fourier coefficients, answering a question about their infinite sign changes.

Findings

Fourier coefficients change sign infinitely often.

Partial analytical continuation of a related Dirichlet series.

Addresses a question posed by Kohnen.

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 $\pm \sqrt{L(1/2, f, \chi_m)}$ 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.