{
  "id": "2310.03709",
  "version": "v1",
  "published": "2023-10-05T17:35:55.000Z",
  "updated": "2023-10-05T17:35:55.000Z",
  "title": "The Effect of Quadratic Base Change on Torsion",
  "authors": [
    "Irmak Balçık",
    "Burton Newman"
  ],
  "comment": "14 pages, comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a quadratic field and $E$ an elliptic curve defined over $K$ such that $E(K)[2]\\simeq C_2.$ In this paper, we study the effect of quadratic base change on $E(K)_{\\text{tor}}.$ In particular, we examine the growth of $E(K)_{\\text{tor}}$ upon quadratic base change when $K$ is any quadratic cyclotomic field. In addition, for a given elliptic curve $E/K$ with prescribed torsion group over $K,$ (no restriction on its $2$-torsion part) we describe an algorithm to find all quadratic extensions $L/K$ in which $E(K)_{\\text{tor}} \\subsetneq E(L)_{\\text{tor}}$ and describe $E(L)_{\\text{tor}}$ in each such case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-05T17:35:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic base change",
      "elliptic curve",
      "quadratic cyclotomic field",
      "quadratic extensions",
      "torsion part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}