{
  "id": "2505.06213",
  "version": "v1",
  "published": "2025-05-09T17:43:17.000Z",
  "updated": "2025-05-09T17:43:17.000Z",
  "title": "Determining monogenity of pure cubic number fields using elliptic curves",
  "authors": [
    "Jordi Guàrdia",
    "Francesc Pedret"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study monogenity of pure cubic number fields by means of Selmer groups of certain elliptic curves. A cubic number field with discriminant $D$ determines a unique nontrivial $\\mathbb{F}_3$-orbit in the first cohomology group of the elliptic curve $E^D: y^2 = 4x^3 + D$ with respect to a certain 3-isogeny $\\phi$. Orbits corresponding to monogenic fields must lie in the soluble part of the Selmer group $S^{\\phi}(E^D/\\mathbb{Q})$, and this gives a criterion to discard monogenity. From this, we can derive bounds on the number of monogenic cubic fields in terms of the rank of the elliptic curve. We can also determine the monogenity of many concrete pure cubic fields assuming GRH.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-09T17:43:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pure cubic number fields",
      "elliptic curve",
      "determining monogenity",
      "concrete pure cubic fields",
      "selmer group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}