{
  "id": "1706.04963",
  "version": "v1",
  "published": "2017-06-15T16:53:10.000Z",
  "updated": "2017-06-15T16:53:10.000Z",
  "title": "Abelian varieties isogenous to a power of an elliptic curve over a Galois extension",
  "authors": [
    "Isabel Vogt"
  ],
  "comment": "6 pages, comments welcome!",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\\rm End}(E_{k'})$ with a semilinear ${\\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-15T16:53:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K02"
    ],
    "keywords": [
      "elliptic curve",
      "galois extension",
      "abelian varieties isogenous",
      "complex multiplication",
      "exact functor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}