{
  "id": "0806.0640",
  "version": "v1",
  "published": "2008-06-03T21:33:12.000Z",
  "updated": "2008-06-03T21:33:12.000Z",
  "title": "On the Sum-Product Problem on Elliptic Curves",
  "authors": [
    "Omran Ahmadi",
    "Igor Shparlinski"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\E$ be an ordinary elliptic curve over a finite field $\\F_{q}$ of $q$ elements and $x(Q)$ denote the $x$-coordinate of a point $Q = (x(Q),y(Q))$ on $\\E$. Given an $\\F_q$-rational point $P$ of order $T$, we show that for any subsets $\\cA, \\cB$ of the unit group of the residue ring modulo $T$, at least one of the sets $$ \\{x(aP) + x(bP) : a \\in \\cA, b \\in \\cB\\} \\quad\\text{and}\\quad \\{x(abP) : a \\in \\cA, b \\in \\cB\\} $$ is large. This question is motivated by a series of recent results on the sum-product problem over finite fields and other algebraic structures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-03T21:33:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11L07",
      "11T23"
    ],
    "keywords": [
      "sum-product problem",
      "finite field",
      "ordinary elliptic curve",
      "rational point",
      "residue ring modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.0640A"
    }
  }
}