{
  "id": "2310.08493",
  "version": "v1",
  "published": "2023-10-12T16:50:58.000Z",
  "updated": "2023-10-12T16:50:58.000Z",
  "title": "The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2",
  "authors": [
    "Niven Achenjang"
  ],
  "comment": "71 pages; comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2\\zeta_B(2)\\zeta_B(10)$, where $\\zeta_B$ is the zeta function of the curve $B/k$. In particular, in the limit as $q=\\#k\\to\\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in ``bad'' characteristics. Along the way, we also produce new bounds on 2-Selmer groups of elliptic curves over characteristic $2$ global function fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-12T16:50:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05"
    ],
    "keywords": [
      "elliptic curves",
      "average size",
      "global function fields",
      "smooth curve",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}