{
  "id": "2309.09938",
  "version": "v1",
  "published": "2023-09-18T16:55:53.000Z",
  "updated": "2023-09-18T16:55:53.000Z",
  "title": "On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves",
  "authors": [
    "Anish Ray"
  ],
  "comment": "19 pages, 1 Table. Comments and suggestions are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "In 2021, Daqing Wan and Ping Xi studied the equivalence of the Lang-Trotter conjecture for CM elliptic curves and the Hardy-Littlewood conjecture for primes represented by a quadratic polynomial. Wan and Xi provided an alternative description of the Lang-Trotter conjecture under the Hardy-Littlewood conjecture. They obtained an explicit constant $\\overline{\\omega}_{E,r}$ in the asymptotics of the Lang-Trotter conjecture. They further conjectured that this particular constant would be equal to the constant $C_{E,r}$ in the asymptotics of the original Lang-Trotter conjecture. In this paper, we verify the same for $20$ CM elliptic curves, which also establishes the equivalence of the Lang-Trotter Conjecture and the Hardy-Littlewood Conjecture with respect to $r$, for these CM elliptic curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-18T16:55:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G15",
      "11F80"
    ],
    "keywords": [
      "cm elliptic curves",
      "hardy-littlewood conjecture",
      "original lang-trotter conjecture",
      "quadratic polynomial",
      "explicit constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}