{
  "id": "2403.14904",
  "version": "v1",
  "published": "2024-03-22T01:50:26.000Z",
  "updated": "2024-03-22T01:50:26.000Z",
  "title": "Improved bounds for integral points on modular curves using Runge's method",
  "authors": [
    "David Zywina"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Consider a modular curve $X_G$ defined over a number field $K$, where $G$ is a subgroup of $GL_2(\\mathbb{Z}/N\\mathbb{Z})$ with $N>2$. The curve $X_G$ comes with a morphism $j: X_G\\to \\mathbb{P}^1_K=\\mathbb{A}^1_K \\cup\\{\\infty\\}$ to the $j$-line. For a finite set of places $S$ of $K$ that satisfies a certain condition, Runge's method shows that there are only finitely many points $P \\in X_G(K)$ for which $j(P)$ lies in the ring $\\mathfrak{O}_{K,S}$ of $S$-units of $K$. We prove an explicit version which shows that if $j(P)\\in \\mathfrak{O}_{K,S}$ for some $P\\in X_G(K)$, then the absolute logarithmic height of $j(P)$ is bounded above by $N^{12} \\log N$. Explicits upper bounds have already been obtained by Bilu and Parent though they are not polynomial in $N$. The modular functions needed to apply Runge's method are constructing using Eisenstein series of weight $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-22T01:50:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G18",
      "11F80"
    ],
    "keywords": [
      "modular curve",
      "integral points",
      "absolute logarithmic height",
      "explicits upper bounds",
      "explicit version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}