{
  "id": "math/9512217",
  "version": "v1",
  "published": "1995-12-11T00:00:00.000Z",
  "updated": "1995-12-11T00:00:00.000Z",
  "title": "The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture",
  "authors": [
    "Bjorn Poonen"
  ],
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we show under this assumption that the number of preperiodic points is at most~$9$. Elliptic curves of small conductor and the genus~$2$ modular curves $X_1(13)$, $X_1(16)$, and $X_1(18)$ all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute the rational points on a non-modular genus~$2$ curve by performing a $2$-descent on its Jacobian and afterwards applying a variant of the method of Chabauty and Coleman.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1995-12-11T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational preperiodic points",
      "complete classification",
      "refined conjecture",
      "rational points",
      "curves classifying quadratic polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1995math.....12217P"
    }
  }
}