Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GLm(ℤ)

Abstract

Let f be a full-level cusp form for GLm(ℤ) with Fourier coefficients Af (n1, …, nm−1). In this paper, an asymptotic expansion of Voronoi’s summation formula for Af (n1, …, nm−1) is established. As applications of this formula, a smoothly weighted average of Af (n, 1, …, 1) against (Formula presented.) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/m for a positive integer q, this smooth average has a main term of the size of |Af (1, …, 1, q) + Af (1, …, 1,−q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af (n, 1, …, 1). Similar estimate is also proved for a sharp-cut sum.