Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $G{L}_m(\mathbb{Z})$

Abstract

Let $f$ be a full-level cusp form for $G{L}_m(\mathbb{Z})$ 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 $\pm m{q}^{1/m}$ 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.

Accepted to appear in SCIENCE CHINA Mathematics

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