{
  "id": "2008.13158",
  "version": "v1",
  "published": "2020-08-30T13:02:40.000Z",
  "updated": "2020-08-30T13:02:40.000Z",
  "title": "The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3",
  "authors": [
    "Jef Laga"
  ],
  "comment": "48 pages. Comments welcome",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We show that the average size of the $2$-Selmer group of the family of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\\mathbb{Z}/2\\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-30T13:02:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G25",
      "14G05",
      "14H45",
      "11E72"
    ],
    "keywords": [
      "average size",
      "non-hyperelliptic curves",
      "marked rational hyperflex point",
      "simple curve singularity",
      "mumford theta group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}