{
  "id": "2305.19969",
  "version": "v1",
  "published": "2023-05-31T15:55:09.000Z",
  "updated": "2023-05-31T15:55:09.000Z",
  "title": "On the $p$-isogenies of elliptic curves with multiplicative reduction over quadratic fields",
  "authors": [
    "George C. Ţurcaş"
  ],
  "comment": "13 pages, comments are welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $q > 5$ be a prime and $K$ a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if $E/K$ is an elliptic curve with potentially multiplicative reduction at all primes $\\mathfrak q \\mid q$, then $E$ does not have prime isogenies of degree greater than $71$ and different from $q$. As an application to our main result, we present a variant of the asymptotic version of Fermat's Last Theorem over quadratic imaginary fields of class number one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-31T15:55:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11F80"
    ],
    "keywords": [
      "elliptic curve",
      "multiplicative reduction",
      "quadratic fields",
      "quadratic imaginary fields",
      "quadratic number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}