{
  "id": "2101.04937",
  "version": "v1",
  "published": "2021-01-13T08:51:12.000Z",
  "updated": "2021-01-13T08:51:12.000Z",
  "title": "Supersingular $j$-invariants and the Class Number of $\\mathbb{Q}(\\sqrt{-p})$",
  "authors": [
    "Guanju Xiao",
    "Lixia Luo",
    "Yingpu Deng"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \\frac{4}{\\sqrt{3}}\\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\\mathbb{F}_p$ if $D$ is not a quadratic residue in $\\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\\mathbb{F}_p$. As a result, we get a deterministic algorithm (Algorithm 3) for computing the class number of $\\mathbb{Q}(\\sqrt{-p})$. The time complexity of Algorithm 3 is $\\tilde{O}(p^{1/2})$ if we use probabilistic factorization algorithms and assume the Generalized Riemann Hypothesis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-13T08:51:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class number",
      "supersingular",
      "class polynomial",
      "invariants",
      "imaginary quadratic order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}