{
  "id": "2101.07658",
  "version": "v1",
  "published": "2021-01-19T14:44:20.000Z",
  "updated": "2021-01-19T14:44:20.000Z",
  "title": "Arithmetic statistics of Prym surfaces",
  "authors": [
    "Jef Laga"
  ],
  "comment": "71 pages. Comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We consider a family of abelian surfaces over $\\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of the Selmer group of this polarization equals $3$. Moreover we show that the average size of the $2$-Selmer group of the abelian surfaces in the same family is bounded above by $5$. This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not principally polarized. The proof is a combination of an analysis of the Lie algebra embedding $F_4\\subset E_6$, invariant theory, a classical geometric construction due to Pantazis and Bhargava's orbit-counting techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-19T14:44:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "14H40",
      "14G25"
    ],
    "keywords": [
      "arithmetic statistics",
      "prym surfaces",
      "prym varieties",
      "selmer group",
      "abelian surfaces carry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}