{
  "id": "1504.07618",
  "version": "v1",
  "published": "2015-04-28T19:55:46.000Z",
  "updated": "2015-04-28T19:55:46.000Z",
  "title": "Computing images of Galois representations attached to elliptic curves",
  "authors": [
    "Andrew V. Sutherland"
  ],
  "comment": "preliminary draft, 36 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let E be an elliptic curve without complex multiplication defined over a number field K. Let G_E(\\ell) denote the image of the Galois representation induced by the action of the absolute Galois group of K on the \\ell-torsion subgroup of E. We present two probabilistic algorithm to simultaneously determine G_E(\\ell) up to local conjugacy for all primes \\ell by sampling images of Frobenius elements. In each case we determine G_E(\\ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(\\ell), each of which occurs for an elliptic curve that is isogenous to E and has the same semisimplification. Under the generalized Riemann hypothesis, both algorithms run in time polynomial in the bit-size n of an integral Weierstrass equation for E. Our first algorithm is a Las Vegas algorithm with expected running time polynomial in n, while the second is a Monte Carlo algorithm with one-sided error whose running time is quasi-linear in n. We have applied our Monte Carlo algorithm to all the non-CM elliptic curves in Cremona's tables and the Stein-Watkins database, some 140 million curves with conductors ranging up to 10^{12}, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(\\ell) that arise for non-CM elliptic curves E/Q. We also give several examples of exceptional Galois images for non-CM elliptic curves defined over various quadratic fields K that do not occur for non-CM elliptic curves over Q.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-28T19:55:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11Y16",
      "11F80",
      "11G20",
      "20G40",
      "14H52"
    ],
    "keywords": [
      "galois representations",
      "computing images",
      "exceptional galois images",
      "monte carlo algorithm",
      "non-cm elliptic curves e/q"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150407618S"
    }
  }
}