{
  "id": "1306.5284",
  "version": "v2",
  "published": "2013-06-22T02:41:33.000Z",
  "updated": "2014-06-09T14:33:12.000Z",
  "title": "Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians",
  "authors": [
    "T. Shaska"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli $\\mathcal H_3$. We determine this subvariety explicitly. For any given moduli point $\\mathfrak p \\in \\mathcal H_3$ we determine explicitly if the corresponding genus 3 curve $\\mathcal X$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_2$ in terms of the absolute invariants $t_1, \\dots, t_6$ as in \\cite{hyp_mod_3}. This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split. The sublocus of such family when $E_1$ is isomorphic to $E_2$ is a 1-dimensional variety which we determine explicitly. We can also determine $\\mathcal X$ and $E$ starting form the $j$-invariant of $E_1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-09T14:33:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H10",
      "14H37",
      "14Q05"
    ],
    "keywords": [
      "hyperelliptic curves",
      "determine elliptic subcovers",
      "subvariety",
      "hyperelliptic moduli",
      "study degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.5284S"
    }
  }
}