{
  "id": "2401.09396",
  "version": "v1",
  "published": "2024-01-17T18:15:43.000Z",
  "updated": "2024-01-17T18:15:43.000Z",
  "title": "Curves with prescribed rational points",
  "authors": [
    "Katerina Santicola"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a smooth curve $C/\\mathbb{Q}$ with genus $\\geq 2$, we know by Faltings' Theorem that $C(\\mathbb{Q})$ is finite. Here we ask the reverse question: given a finite set of rational points $S\\subseteq \\mathbb{P}^n(\\mathbb{Q})$, does there exist a smooth curve $C/\\mathbb{Q}$ contained in $\\mathbb{P}^n$ such that $C(\\mathbb{Q})=S$? We answer this question in the affirmative by providing an effective algorithm for constructing such a curve.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-17T18:15:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "prescribed rational points",
      "smooth curve",
      "reverse question",
      "finite set",
      "effective algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}