{
  "id": "2409.02202",
  "version": "v1",
  "published": "2024-09-03T18:12:10.000Z",
  "updated": "2024-09-03T18:12:10.000Z",
  "title": "The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\\mathbb{Z}_p$-extensions at inert primes",
  "authors": [
    "Erman Isik",
    "Antonio Lei"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E$ be an elliptic curve defined over $\\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell-Weil ranks of $E$ are bounded over any subextensions of the anticyclotomic $\\mathbb{Z}_p$-extension of $K$. Additionally, we provide an asymptotic formula for the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-03T18:12:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11G05",
      "11R20"
    ],
    "keywords": [
      "supersingular elliptic curves",
      "tate-shafarevich groups",
      "inert primes",
      "anticyclotomic",
      "imaginary quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}