{
  "id": "1611.05623",
  "version": "v1",
  "published": "2016-11-17T10:17:56.000Z",
  "updated": "2016-11-17T10:17:56.000Z",
  "title": "Supersingular zeros of divisor polynomials of elliptic curves of prime conductor",
  "authors": [
    "Matija Kazalicki",
    "Daniel Kohen"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $\\overline{\\mathbb{F}_p}$. We show that these supersingular zeros are in bijection with zeros modulo $p$ of an associated quaternionic modular form $v_E$. This allows us to prove that if the root number of $E$ is $-1$ then all supersingular $j$-invariants of elliptic curves defined over $\\mathbb{F}_{p}$ are zeros of the corresponding divisor polynomial. If the root number is $1$ we study the discrepancy between rank $0$ and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in $\\mathbb{F}_p$ seems to be larger. In order to partially explain this phenomenon, we conjecture that when $E$ has positive rank the values of the coefficients of $v_E$ corresponding to supersingular elliptic curves defined over $\\mathbb{F}_p$ are even. We prove this conjecture in the case when the discriminant of $E$ is positive, and obtain several other results that are of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-17T10:17:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisor polynomial",
      "supersingular zeros",
      "prime conductor",
      "supersingular elliptic curves",
      "higher rank elliptic curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}