{
  "id": "1410.6798",
  "version": "v1",
  "published": "2014-10-24T19:32:32.000Z",
  "updated": "2014-10-24T19:32:32.000Z",
  "title": "Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)",
  "authors": [
    "Patrick Morton"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Explicit solutions of the cubic Fermat equation are constructed in ring class fields $\\Omega_f$, with conductor $f$ prime to $3$, of any imaginary quadratic field $K$ whose discriminant satisfies $d_K \\equiv 1$ (mod $3$), in terms of the Dedekind $\\eta$-function. As $K$ and $f$ vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function $T(z)$ and its inverse defined on natural subsets of the maximal unramified, algebraic extension $\\textsf{K}_3$ of the $3$-adic field $\\mathbb{Q}_3$. This is used to give a dynamical proof of a class number relation of Deuring. This also determines all the periodic points of the algebraic function T(z), considered as a multi-valued function on $\\mathbb{C}$; all such periodic points, except for $z=3$, generate ring class fields over an imaginary quadratic field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-24T19:32:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "imaginary quadratic field",
      "ring class fields",
      "cubic fermat equation",
      "periodic points",
      "algebraic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.6798M"
    }
  }
}