{
  "id": "1210.6085",
  "version": "v1",
  "published": "2012-10-22T23:05:07.000Z",
  "updated": "2012-10-22T23:05:07.000Z",
  "title": "The growth of the rank of Abelian varieties upon extensions",
  "authors": [
    "Peter Bruin",
    "Filip Najman"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\\Gal(L/K)$ does not have an index 2 subgroup and $A/K$ is an Abelian variety, then $\\rk A(L)-\\rk A(K)$ can never be 1. We obtain more precise results when $\\Gal(L/K)$ is of odd order, alternating, $\\SL_2(\\F_p)$ or $\\PSL_2(\\F_p)$. This implies a restriction on $\\rk E(K(E[p]))-\\rk E(K(\\zeta_p))$ when $E/K$ is an elliptic curve whose mod $p$ Galois representation is surjective. Similar results are obtained for the growth of the rank in certain non-Galois extensions. Second, we show that for every $n\\ge2$ there exists an elliptic curve $E$ over a number field $K$ such that $\\Q\\otimes_\\Q\\Res_{K/\\Q} E$ contains a number field of degree $2^n$. We ask whether every elliptic curve $E/K$ has infinite rank over $K\\Q(2)$, where $\\Q(2)$ is the compositum of all quadratic extensions of $\\Q$. We show that if the answer is yes, then for any $n\\ge2$, there exists an elliptic curve $E/K$ admitting infinitely many quadratic twists whose rank is a positive multiple of $2^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-22T23:05:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G10",
      "14K15"
    ],
    "keywords": [
      "abelian variety",
      "elliptic curve",
      "number field",
      "finite galois extension",
      "galois representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.6085B"
    }
  }
}