{
  "id": "1511.00194",
  "version": "v1",
  "published": "2015-11-01T00:51:31.000Z",
  "updated": "2015-11-01T00:51:31.000Z",
  "title": "Finite ramification for preimage fields of postcritically finite morphisms",
  "authors": [
    "Andrew Bridy",
    "Patrick Ingram",
    "Rafe Jones",
    "Jamie Juul",
    "Alon Levy",
    "Michelle Manes",
    "Simon Rubinstein-Salzedo",
    "Joseph H. Silverman"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a finite endomorphism $\\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\\varphi^{-\\infty}(\\alpha)) : = \\bigcup_{n \\geq 1} K(\\varphi^{-n}(\\alpha))$ generated by the preimages of $\\alpha$ under all iterates of $\\varphi$. In particular when $\\varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W \\subseteq X$ such that $\\varphi^{-1}(W) \\subseteq W$ and $\\varphi : W \\to X$ is \\'etale, we prove that $K(\\varphi^{-\\infty}(\\alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X = \\mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = \\mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = \\mathbb{P}^1$. The proof relies on Faltings' theorem and a local argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-01T00:51:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "postcritically finite morphisms",
      "preimage fields",
      "ramification condition characterizes post-critically finite",
      "condition characterizes post-critically finite morphisms",
      "finite ramification condition characterizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151100194B"
    }
  }
}