{
  "id": "1909.07468",
  "version": "v1",
  "published": "2019-09-16T20:31:29.000Z",
  "updated": "2019-09-16T20:31:29.000Z",
  "title": "Uniform bounds on the image of arboreal Galois representations attached to non-CM elliptic curves",
  "authors": [
    "Michael Cerchia",
    "Jeremy Rouse"
  ],
  "comment": "While the main result is similar to some results of Lombardo and Tronto (1909.05376), our results were obtained independently and using somewhat different methods",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\\alpha \\in E(F)$ of infinite order. Attached to the pair $(E,\\alpha)$ is the $\\ell$-adic arboreal Galois representation $\\omega_{E,\\alpha,\\ell^{\\infty}} : {\\rm Gal}(\\overline{F}/F) \\to \\mathbb{Z}_{\\ell}^{2} \\rtimes {\\rm GL}_{2}(\\mathbb{Z}_{\\ell})$ describing the action of ${\\rm Gal}(\\overline{F}/F)$ on points $\\beta_{n}$ so that $\\ell^{n} \\beta_{n} = \\alpha$. We give an explicit bound on the index of the image of $\\omega_{E,\\alpha,\\ell^{\\infty}}$ depending on how $\\ell$-divisible the point $\\alpha$ is, and the image of the ordinary $\\ell$-adic Galois representation. The image of $\\omega_{E,\\alpha,\\ell^{\\infty}}$ is connected with the density of primes $\\mathfrak{p}$ for which $\\alpha \\in E(\\mathbb{F}_{\\mathfrak{p}})$ has order coprime to $\\ell$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-16T20:31:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11G05",
      "12G05"
    ],
    "keywords": [
      "non-cm elliptic curve",
      "uniform bounds",
      "adic arboreal galois representation",
      "adic galois representation",
      "order coprime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}