{
  "id": "2107.06803",
  "version": "v1",
  "published": "2021-07-14T16:03:25.000Z",
  "updated": "2021-07-14T16:03:25.000Z",
  "title": "Ranks of abelian varieties in cyclotomic twist families",
  "authors": [
    "Ari Shnidman",
    "Ariel Weiss"
  ],
  "comment": "30 pages. Comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $A$ be an abelian variety over a number field $F$, and suppose that $\\mathbb Z[\\zeta_n]$ embeds in $\\mathrm{End}_{\\bar F} A$, for some root of unity $\\zeta_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group $A[1-\\zeta_n]$ is sufficiently reducible, we bound the average rank of the Mordell--Weil groups $A_d(F)$, as $A_d$ varies through the family of $\\mu_{2n}$-twists of $A$. Combining this with the recently proved uniform Mordell-Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves $y^3 = f(x^2)$, as well as in twist families of theta divisors of cyclic trigonal curves $y^3 = f(x)$. Our main technical result is the determination of the average size of a $3$-isogeny Selmer group in a family of $\\mu_{2n}$-twists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-14T16:03:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11E76",
      "11S25",
      "14G05"
    ],
    "keywords": [
      "cyclotomic twist families",
      "abelian variety",
      "uniform mordell-lang conjecture",
      "bicyclic trigonal curves",
      "isogeny selmer group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}