{
  "id": "math/0309475",
  "version": "v2",
  "published": "2003-09-30T13:03:17.000Z",
  "updated": "2003-10-09T14:36:20.000Z",
  "title": "Power residues of Fourier coefficients of modular forms",
  "authors": [
    "Tom Weston"
  ],
  "comment": "Expanded to include applications to abelian extensions of imaginary quadratic fields. Several incorrect statements also eliminated",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let r : G_Q -> GL_n Q_l be a motivic l-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under r is an m^th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of r is open. We further conjecture that for such r the set of these primes p is independent of any set defined by Cebatorev-style Galois theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-10-09T14:36:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11G15",
      "11A15"
    ],
    "keywords": [
      "modular forms",
      "fourier coefficients",
      "imaginary quadratic fields unramified away",
      "cebatorev-style galois theoretic conditions",
      "motivic l-adic galois representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......9475W"
    }
  }
}