{
  "id": "math/0604034",
  "version": "v1",
  "published": "2006-04-03T12:28:11.000Z",
  "updated": "2006-04-03T12:28:11.000Z",
  "title": "Power residues of Fourier coefficients of elliptic curves with complex multiplication",
  "authors": [
    "Tom Weston",
    "Elena Zaurova"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m=4 and complex multiplication by Q(i).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-04-03T12:28:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11G15"
    ],
    "keywords": [
      "complex multiplication",
      "elliptic curve",
      "power residues",
      "mth power modulo",
      "pth fourier coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4034W"
    }
  }
}