{
  "id": "2206.14959",
  "version": "v1",
  "published": "2022-06-30T00:34:50.000Z",
  "updated": "2022-06-30T00:34:50.000Z",
  "title": "Explicit open images for elliptic curves over $\\mathbb{Q}$",
  "authors": [
    "David Zywina"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a non-CM elliptic curve $E$ defined over $\\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\\rho_E: Gal(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to GL_2(\\widehat{\\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $\\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\\widehat{\\mathbb{Z}})$. We describe an algorithm that computes the image of $\\rho_E$ up to conjugacy in $GL_2(\\widehat{\\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $\\rho_E(Gal(\\overline{\\mathbb{Q}}/\\mathbb{Q}))\\cap SL_2(\\widehat{\\mathbb{Z}})$ and the indices $[GL_2(\\widehat{\\mathbb{Z}}):\\rho_E(Gal(\\overline{\\mathbb{Q}}/\\mathbb{Q}))]$ for non-CM elliptic curves $E/\\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-30T00:34:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11F80"
    ],
    "keywords": [
      "explicit open images",
      "non-cm elliptic curve",
      "explicit modular curves",
      "serre says",
      "finite index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}