Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(\mathbb Z)$

TL;DR

This paper derives an asymptotic expansion of Voronoi's summation formula for $GL_m$ Maass forms and investigates the resonance phenomena in Fourier coefficients through weighted averages, revealing rapid decay and main term behaviors.

Contribution

It establishes a new asymptotic expansion of Voronoi's formula for $GL_m$ Maass forms and analyzes resonance effects in Fourier coefficients through weighted sums.

Findings

Weighted averages decay rapidly when $0<eta<1/m$

Resonance occurs at $eta=1/m$ with main terms of size $|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$

Similar estimates are proved for sharp-cut sums

Abstract

Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a smoothly weighted average of $A_f(n,1,...,1)$ against $e(\alpha|n|^\beta)$ is proved to be rapidly decayed when $0<\beta<1/m$. When $\beta=1/m$ and $\alpha$ equals or approaches $\pm mq^{1/m}$ for a positive integer $q$, this smooth average has a main term of the size of $|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients $A_f(n,1,...,1)$. Similar estimate is also proved for a sharp-cut sum.