{
  "id": "1705.08777",
  "version": "v1",
  "published": "2017-05-24T14:06:23.000Z",
  "updated": "2017-05-24T14:06:23.000Z",
  "title": "Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians",
  "authors": [
    "Aaron Landesman",
    "Ashvin Swaminathan",
    "James Tao",
    "Yujie Xu"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT",
    "math.AG",
    "math.GR",
    "math.RT"
  ],
  "abstract": "In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-24T14:06:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11G10",
      "11G30",
      "11R32"
    ],
    "keywords": [
      "hyperelliptic curves",
      "maximal galois action",
      "torsion points",
      "maximal adelic galois representations",
      "arbitrary rational families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}