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

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Abstract

Let f be a full-level cusp form for GL m (ℤ) with Fourier coefficients A f (n 1, …, n m−1). In this paper, an asymptotic expansion of Voronoi’s summation formula for A f (n 1, …, n m−1) 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 < β < 1/m. When β = 1/m and α equals or approaches ±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.

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, 250100, China

    XiuMin Ren & YangBo Ye

  2. Department of Mathematics, The University of Iowa, Iowa City, 52242-1419, USA

    YangBo Ye

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  1. XiuMin Ren
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  2. YangBo Ye
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Correspondence to XiuMin Ren.

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Ren, X., Ye, Y. Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL m (ℤ). Sci. China Math. 58, 1–20 (2015). https://doi.org/10.1007/s11425-014-4955-3

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  • DOI: https://doi.org/10.1007/s11425-014-4955-3

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