{
  "id": "2304.07715",
  "version": "v1",
  "published": "2023-04-16T07:52:59.000Z",
  "updated": "2023-04-16T07:52:59.000Z",
  "title": "Almost ordinary Abelian surfaces over global function fields with application to integral points",
  "authors": [
    "Ruofan Jiang"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $A$ be a non-isotrivial almost ordinary Abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $\\Delta$ is an infinite set of positive integers, such that $\\left(\\frac{m}{p}\\right)=1$ for $\\forall m\\in \\Delta$. If $A$ doesn't admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\\mathbb{Z}[x]/(x^2-m)$ for some $m\\in \\Delta$. This generalizes the $S$-integrality conjecture for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of Abelian surfaces over global function fields. As a corollary, we show that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-16T07:52:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global function field",
      "ordinary abelian surface",
      "integral points",
      "application",
      "places modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}