{
  "id": "1007.4617",
  "version": "v3",
  "published": "2010-07-27T04:27:05.000Z",
  "updated": "2012-10-16T02:13:47.000Z",
  "title": "On elliptic curves with an isogeny of degree 7",
  "authors": [
    "R. Greenberg",
    "K. Rubin",
    "A. Silverberg",
    "M. Stoll"
  ],
  "comment": "The revision gives a complete answer to the question considered in Version 1. Version 3 will appear in the American Journal of Mathematics",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that if $E$ is an elliptic curve over $\\mathbf{Q}$ with a $\\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\\mathbf{Q}(\\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the exceptions are exactly the curves with complex multiplication. As a by-product of one of the key steps in our proof, we determine exactly when there exist elliptic curves over an arbitrary field $k$ of characteristic not 7 with a $k$-rational isogeny of degree 7 and a specified Galois action on the kernel of the isogeny, and we give a parametric description of such curves.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-10-16T02:13:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11D45",
      "11F80",
      "11G18",
      "11G30",
      "14G05"
    ],
    "keywords": [
      "elliptic curve",
      "rational isogeny",
      "complex multiplication",
      "parametric description",
      "first author"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4617G"
    }
  }
}