{
  "id": "1111.4679",
  "version": "v2",
  "published": "2011-11-20T21:08:41.000Z",
  "updated": "2014-12-10T18:51:17.000Z",
  "title": "Heuristics for $p$-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst",
  "authors": [
    "Nigel Boston",
    "Michael R. Bush",
    "Farshid Hajir"
  ],
  "comment": "Revised Version. Section 2 has been reorganized and divided into five subsections; the main results appear in Section 2.4. The numerical data has been greatly expanded using a modified technique of computing generating polynomials for unramified cubic extensions of imaginary quadratic fields",
  "categories": [
    "math.NT"
  ],
  "abstract": "Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\\mathrm{Gal}(K_\\infty/K)$ where $K_\\infty$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\\geq 1$, we give a heuristic of Cohen-Lentra type for the maximal $p$-class $c$ quotient of $\\G_K$ and thereby give a conjectural formula for how frequently a given $p$-group of $p$-class $c$ occurs in this manner. In particular, we predict that every finite Schur $\\sigma$-group occurs as $G_K$ for infinitely many fields $K$. We present numerical data in support of these conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-20T21:08:41.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-10T18:51:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11R11"
    ],
    "keywords": [
      "imaginary quadratic field",
      "class tower",
      "jonathan blackhurst",
      "class group",
      "class field theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.4679B"
    }
  }
}