{
  "id": "2201.11839",
  "version": "v1",
  "published": "2022-01-27T22:52:30.000Z",
  "updated": "2022-01-27T22:52:30.000Z",
  "title": "The local-global principle for divisibility in CM elliptic curves",
  "authors": [
    "Brendan Creutz",
    "Sheng Lu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of $p$. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with $p > 2d+1$. We also deduce that the local-global principle for divisibility by powers of $7$ holds over quadratic fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-27T22:52:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cm elliptic curve",
      "local-global principle",
      "divisibility",
      "number field",
      "sharp lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}