{
  "id": "1311.5846",
  "version": "v2",
  "published": "2013-11-22T19:22:23.000Z",
  "updated": "2013-12-04T16:22:59.000Z",
  "title": "Generic Newton polygons for curves of given p-rank",
  "authors": [
    "Jeff Achter",
    "Rachel Pries"
  ],
  "comment": "16 pages, to appear in Algebraic Curves and Finite Fields: Codes, Cryptography, and other emergent applications, edited by H. Niederreiter, A. Ostafe, D. Panario, and A. Winterhof",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic) curves. For example, if $0 \\leq f \\leq g-1$, we prove that every component of the $p$-rank $f+1$ stratum of ${\\mathcal M}_g$ contains a component of the $p$-rank $f$ stratum in its closure. We prove that the $p$-rank $f$ stratum of $\\overline{\\mathcal M}_g$ is connected. For all primes $p$ and all $g \\geq 4$, we demonstrate the existence of a Jacobian of a smooth curve, defined over $\\overline{\\mathbb F}_p$, whose Newton polygon has slopes $\\{0, \\frac{1}{4}, \\frac{3}{4}, 1\\}$. We include partial results about the generic Newton polygons of curves of given genus $g$ and $p$-rank $f$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-04T16:22:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "11M38",
      "14H10",
      "14H40",
      "14L05",
      "11G10"
    ],
    "keywords": [
      "generic newton polygons",
      "open questions",
      "geometric results",
      "smooth curve",
      "survey results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.5846A"
    }
  }
}