{
  "id": "1711.11545",
  "version": "v1",
  "published": "2017-11-30T18:07:43.000Z",
  "updated": "2017-11-30T18:07:43.000Z",
  "title": "On Fourier Coefficients of GL(n)-Automorphic Functions over Number Fields",
  "authors": [
    "Eleftherios Tsiokos"
  ],
  "comment": "In Subsection 1.4 I mention: relations of the present paper with arXiv:1511.09374v1, and two mistakes in that paper",
  "categories": [
    "math.NT",
    "math.GR",
    "math.RT"
  ],
  "abstract": "We study Fourier coefficients of $GL_n(\\A)$-automorphic functions $\\phi$, for $\\A$ being the adele group of a number field $\\kkk$. Let FC be an abbreviation for such a Fourier coefficient (and FCs for plural). Roughly speaking, in the present paper we process FCs by iteratively using the operations: Fourier expansions, certain exchanges of Fourier expansions, and conjugations. In Theorem \\ref{general} we express any FC in terms of---degenerate in many cases---Whittaker FCs. For FCs obtained from the trivial FC by choosing a certain \"generic\" term in each Fourier expansion involved, we establish a shortcut (Main corollary \\ref{maincor}) for studying their expressions of the form in Theorem \\ref{general}. The shortcut gives considerably less information, but it remains useful on finding automorphic representations so that for appropriate choices of $\\phi$ in them, the FC is factorizable and nonzero. Then in Theorems \\ref{thD1}, \\ref{thD2}, and \\ref{thD3}, we study examples of FCs on which this shortcut applies, with many of them turning out to \"correspond\" to more than one unipotent orbit in $GL_n.$ For most of the paper, no knowledge on automorphic forms is necessary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-30T18:07:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "number field",
      "fourier expansion",
      "study fourier coefficients",
      "automorphic forms",
      "adele group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}