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

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 $\{\mathfrak{a}\_\mathfrak{f}(t{n}^2)\}\_n\in \mathbb{N}$,\item[--]The number of coefficients $\mathfrak{a}\_\mathfrak{f}(t{n}^2)$ of same signs,\item[--]Non-vanishing of coefficients $\mathfrak{a}\_\mathfrak{f}(t{n}^2)$ 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 $\mathfrak{a}\_\mathfrak{f}(t)\ne 0$.