{
  "id": "1403.5635",
  "version": "v2",
  "published": "2014-03-22T09:36:34.000Z",
  "updated": "2015-04-01T08:00:37.000Z",
  "title": "Elliptic curves and their Frobenius fields",
  "authors": [
    "Manisha Kulkarni",
    "Vijay M. Patankar",
    "C. S. Rajan"
  ],
  "comment": "15 pages. This is a revised, corrected and expanded version. A new author has been added",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let \\( E \\) be an elliptic curve defined over a number field \\( K \\). For a place \\( v \\) of \\( K \\) of good reduction for \\( E \\), let \\( F(E, v) \\) denote the Frobenius field of \\( E \\) at \\( v \\), given by the splitting field of the characteristic polynomial of the Frobenius automorphism at \\( v \\) acting on the Tate module of \\( E \\). We show that the set of finite places \\( v \\) of \\( K \\) such that \\( F(E,v) \\) equals a fixed imaginary quadratic field \\( F \\) has positive upper density if and only if \\( E \\) has complex multiplication by \\( F \\). Given two elliptic curves \\( E_1 \\) and \\( E_2 \\) defined over a number field \\( K \\), with at least one of them without complex multiplication, we prove that the set of places \\( v \\) of \\( K \\) of good reduction such that \\( F(E_1, v) = F (E_2, v) \\) has positive upper density if and only if \\( E_1 \\) and \\( E_2 \\) are isogenous over some extension of \\( K \\).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-22T09:36:34.000Z",
      "abstract": "Let \\(E_1 \\) and \\(E_2 \\) be two elliptic curves over a number field \\(K \\). For a place \\(v \\) of \\(K \\) of good reduction for \\(E_1 \\) and for \\(E_2 \\), let \\(F(E_1, v) \\) and \\(F(E_2, v) \\) denote the splitting fields of the characteristic polynomials of the Frobenius automorphism at \\(v \\) acting on the Tate modules of \\(E_1 \\) and \\(E_2 \\) respectively. \\(F(E_1, v) \\) and \\(F(E_2, v) \\) are called the Frobenius fields of \\(E_1 \\) and \\(E_2 \\) at \\(v \\). Assume that at least one of the two elliptic curves is without complex multiplication. Then, we show that the set of places \\(v \\) of \\(K \\) of good reduction such that \\(F(E_1, v) = F (E_2, v) \\) has positive upper density if and only if \\(E_1 \\) and \\(E_2 \\) are isogenous over some extension of \\(K \\). We use this result to prove that, for an elliptic curve \\(E \\) over a number field \\(K \\), the set of finite places \\(v \\) of \\(K \\) such that \\(F(E,v) \\) equals a fixed imaginary quadratic field \\(F \\) has positive upper density iff \\(E \\) has complex multiplication by \\(F \\).",
      "comment": "7 pages",
      "journal": null,
      "doi": null,
      "authors": [
        "Manisha Kulkarni",
        "Vijay M. Patankar"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-04-01T08:00:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "14K22"
    ],
    "keywords": [
      "elliptic curve",
      "frobenius fields",
      "positive upper density",
      "number field",
      "complex multiplication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.5635K"
    }
  }
}