# Signs of Fourier coefficients of half-integral weight modular forms

## TL;DR

This paper investigates the signs of Fourier coefficients of half-integral weight modular forms, showing under GRH that these signs change frequently and a positive proportion are positive or negative, with some unconditional results on their sign behavior.

## Contribution

It establishes new results on the sign distribution of Fourier coefficients of half-integral weight modular forms, including positive proportion results under GRH and unconditional sign results.

## Key findings

- Under GRH, positive proportion of coefficients are positive or negative.
- The sequence of coefficients changes sign a positive proportion of the time.
- Unconditionally, results on the sign of coefficients match the strength of known non-vanishing results.

## 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 $n \geq 1$. 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 $4 N$ with $N$ odd, square-free.