{
  "id": "2210.16977",
  "version": "v1",
  "published": "2022-10-30T23:05:22.000Z",
  "updated": "2022-10-30T23:05:22.000Z",
  "title": "Growth of torsion groups of elliptic curves upon base change from number fields",
  "authors": [
    "Tyler Genao"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a number field $F_0$ which contains no Hilbert class field of any imaginary quadratic field, we show that under GRH there exists an effectively computable constant $B:=B(F_0)\\in\\mathbb{Z}^+$ for which the following holds: for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup $E(L)[\\textrm{tors}]=E(F_0)[\\textrm{tors}]$. This generalizes a previous result of Gonz\\'{a}lez-Jim\\'{e}nez and Najman over $F_0=\\mathbb{Q}$. Towards showing this, we also prove a result on relative uniform divisibility of the index of a mod-$\\ell$ Galois representation of an elliptic curve over $F_0$. Additionally, we show that the main result's conclusion fails when we allow $F_0$ to have rationally defined CM, due to the existence of $F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain congruency conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-30T23:05:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G15"
    ],
    "keywords": [
      "elliptic curve",
      "number field",
      "base change",
      "torsion groups",
      "large prime degrees satisfying"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}