On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\mathrm{GL}_3$

TL;DR

This paper investigates the sign changes of Fourier coefficients of Hecke-Maass cusp forms on GL_3, establishing lower bounds and positive proportions of sign changes, and introduces a new effective Sato-Tate theorem for these forms.

Contribution

It provides new quantitative results on sign changes of Fourier coefficients for GL_3 cusp forms, including a novel effective Sato-Tate theorem and non-vanishing results under Ramanujan-Petersson.

Findings

Sign changes occur at least on the order of X^{5/6- ext{epsilon}} for self-dual forms.

A positive proportion of coefficients exhibit sign changes for many self-dual forms.

New effective Sato-Tate theorem for a family of GL_3 cusp forms.

Abstract

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\mathrm{GL}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.