{
  "id": "1902.01068",
  "version": "v1",
  "published": "2019-02-04T07:55:36.000Z",
  "updated": "2019-02-04T07:55:36.000Z",
  "title": "On the growth of Mordell-Weil ranks in $p$-adic Lie extensions",
  "authors": [
    "Pin-Chi Hung",
    "Meng Fai Lim"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime and $F_{\\infty}$ a $p$-adic Lie extension of a number field $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell-Weil rank of the abelian variety of $A$ in the said $p$-adic Lie extension. Our upper bound can be expressed in terms of invariants coming from the cyclotomic level. Motivated by this formula, we make a conjecture on an asymptotic upper bound of the growth of Mordell-Weil ranks over a $p$-adic Lie extension which is in terms of the Mordell-Weil rank of the abelian variety over the cyclotomic $\\mathbb{Z}_p$-extension. Finally, it is then natural to ask whether there is such a conjectural upper bound when the abelian variety has non-ordinary reduction. For this, we can at least modestly formulate an analogue conjectural upper bound for the growth of Mordell-Weil ranks of an elliptic curve with good supersingular reduction at the prime $p$ over a $\\mathbb{Z}_p^2$-extension of an imaginary quadratic field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-04T07:55:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11R23"
    ],
    "keywords": [
      "adic lie extension",
      "mordell-weil rank",
      "abelian variety",
      "analogue conjectural upper bound",
      "establish asymptotic upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}