{
  "id": "1412.3252",
  "version": "v1",
  "published": "2014-12-10T10:46:54.000Z",
  "updated": "2014-12-10T10:46:54.000Z",
  "title": "Linear relations in families of powers of elliptic curves",
  "authors": [
    "Fabrizio Barroero",
    "Laura Capuano"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_\\lambda$ of equation $Y^2=X(X-1)(X-\\lambda)$, we prove that, given $n$ linearly independent points $P_1(\\lambda), ...,P_n(\\lambda)$ on $E_\\lambda$ with coordinates in $\\bar{\\mathbb{Q}(\\lambda)}$, there are at most finitely many complex numbers $\\lambda_0$ such that the points $P_1(\\lambda_0), ...,P_n(\\lambda_0)$ satisfy two independent relations on $E_{\\lambda_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-10T10:46:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G50",
      "11U09",
      "14K05"
    ],
    "keywords": [
      "linear relations",
      "legendre elliptic curve",
      "complex numbers",
      "linearly independent points",
      "abelian varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3252B"
    }
  }
}