{
  "id": "1411.6022",
  "version": "v1",
  "published": "2014-11-21T21:11:31.000Z",
  "updated": "2014-11-21T21:11:31.000Z",
  "title": "Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(\\mathbb Z)$",
  "authors": [
    "Xiumin Ren",
    "Yangbo Ye"
  ],
  "comment": "Accepted to appear in SCIENCE CHINA Mathematics",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-21T21:11:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L07",
      "11F30"
    ],
    "keywords": [
      "fourier coefficients",
      "rapid decay",
      "exponential sums",
      "maass form",
      "full-level cusp form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.6022R"
    }
  }
}