{
  "id": "2106.14096",
  "version": "v1",
  "published": "2021-06-26T21:20:31.000Z",
  "updated": "2021-06-26T21:20:31.000Z",
  "title": "Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\\mathbb{Q}$",
  "authors": [
    "Ari Shnidman",
    "Ariel Weiss"
  ],
  "comment": "6 pages. Comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\\equiv 1 \\pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\\langle P \\rangle$. Then the number of positive integers $d \\leq X$, such that the Tate-Shafarevich group of $\\widehat B_d$ has non-trivial $p$-torsion, is $\\gg X/\\log X$, where $\\widehat B_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties of $\\mathrm{GL}_2$-type with a $p$-isogeny satisfying a mild technical condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-26T21:20:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G10",
      "11S25"
    ],
    "keywords": [
      "tate-shafarevich group",
      "prime order",
      "th quadratic twist",
      "geometrically simple abelian varieties",
      "mild technical condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}