Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

TL;DR

This paper investigates the sign behavior, non-vanishing, and distribution of Fourier coefficients of half-integral weight cusp forms, providing new insights into their oscillatory nature and sign changes.

Contribution

It introduces new results on the sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms, extending understanding of their oscillatory properties.

Findings

Existence of the first negative coefficient in the sequence

Quantitative results on the number of coefficients with same signs

Non-vanishing of coefficients in short intervals and arithmetic progressions

Abstract

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form $\mathfrak{f}$ with half-integral weight:\begin{itemize}\item[--]The first negative coefficient of the sequence $\{\mathfrak{a}\_{\mathfrak{f}}(tn^2)\}\_{n\in \N}$,\item[--]The number of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ of same signs,\item[--]Non-vanishing of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ in short intervals and in arithmetic progressions,\end{itemize}where $\mathfrak{a}\_{\mathfrak{f}}(n)$ is the $n$-th Fourier coefficient of $\mathfrak{f}$ and $t$ is a square-free integersuch that $\mathfrak{a}\_{\mathfrak{f}}(t)\not=0$.