{
  "id": "0911.2833",
  "version": "v2",
  "published": "2009-11-15T06:00:28.000Z",
  "updated": "2011-03-17T14:40:59.000Z",
  "title": "Primes of the form x^2+n*y^2 in function fields",
  "authors": [
    "Piotr Maciak"
  ],
  "comment": "The paper has been withdrawn by the author",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let n be a square-free polynomial over F_q, where q is an odd prime power. In this paper, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary (and almost sufficient) condition is that the ideal generated by p splits completely in the Hilbert class field H of K = F_q(x,sqrt{-n}) (for the appropriate notion of Hilbert class field in this context). In order to get explicit conditions for p to be of the form X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for H/K. This algorithm generalizes to all situations an algorithm of D.S. Dummit and D.Hayes for the case where -n is monic of odd degree.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-17T14:40:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D85",
      "11E12"
    ],
    "keywords": [
      "function fields",
      "hilbert class field",
      "odd prime power",
      "class field theory",
      "sgn-normalized rank-one drinfeld modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.2833M"
    }
  }
}