{
  "id": "1605.03988",
  "version": "v1",
  "published": "2016-05-12T21:07:07.000Z",
  "updated": "2016-05-12T21:07:07.000Z",
  "title": "Modular curves of prime-power level with infinitely many rational points",
  "authors": [
    "Andrew V. Sutherland",
    "David Zywina"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For each open subgroup $G$ of ${\\rm GL}_2(\\hat{\\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arising from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\\ell$, these results provide an explicit classification of the possible images of the $\\ell$-adic Galois representations arising from elliptic curves over $\\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-12T21:07:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35",
      "11G05",
      "11F80"
    ],
    "keywords": [
      "modular curve",
      "rational points",
      "prime-power level",
      "adic galois representations",
      "loosely parametrizes elliptic curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}