{
  "id": "1703.05420",
  "version": "v1",
  "published": "2017-03-15T23:12:25.000Z",
  "updated": "2017-03-15T23:12:25.000Z",
  "title": "Genus growth in $\\mathbb{Z}_p$-towers of function fields",
  "authors": [
    "Michiel Kosters",
    "Daqing Wan"
  ],
  "comment": "13 pages, this is a short version of arXiv:1607.00523",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $K$ be a function field over a finite field $k$ of characteristic $p$ and let $K_{\\infty}/K$ be a geometric extension with Galois group $\\mathbb{Z}_p$. Let $K_n$ be the corresponding subextension with Galois group $\\mathbb{Z}/p^n\\mathbb{Z}$ and genus $g_n$. In this paper, we give a simple explicit formula $g_n$ in terms of an explicit Witt vector construction of the $\\mathbb{Z}_p$-tower. This formula leads to a tight lower bound on $g_n$ which is quadratic in $p^n$. Furthermore, we determine all $\\mathbb{Z}_p$-towers for which the genus sequence is stable, in the sense that there are $a,b,c \\in \\mathbb{Q}$ such that $g_n=a p^{2n}+b p^n +c$ for $n$ large enough. Such genus stable towers are expected to have strong stable arithmetic properties for their zeta functions. A key technical contribution of this work is a new simplified formula for the Schmid-Witt symbol coming from local class field theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-15T23:12:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "11R37",
      "12F05"
    ],
    "keywords": [
      "function field",
      "genus growth",
      "galois group",
      "explicit witt vector construction",
      "local class field theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}