{
  "id": "2104.13639",
  "version": "v1",
  "published": "2021-04-28T08:50:53.000Z",
  "updated": "2021-04-28T08:50:53.000Z",
  "title": "Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication",
  "authors": [
    "Jared Asuncion"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a particular family $\\{F_K(m) : m \\in \\mathbb{Z} >0 \\}$ of abelian extensions of $K$, and showed that the Hilbert class field $H_K$ of $K$ is contained in $F_K(m)$ for some positive integer m. We make this m explicit. We then give an algorithm that computes a set of defining polynomials for the Hilbert class field using the field $F_K(m)$. Our proof-of-concept implementation of this algorithm computes a set of defining polynomials much faster than current implementations of the generic Kummer algorithm for certain examples of quartic CM fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-28T08:50:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert class field",
      "quartic cm field",
      "complex multiplication",
      "real quadratic number field",
      "totally imaginary quadratic extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}