{
  "abstract": "In this paper, we propose a method to study the {\\it Uniform Boundedness Conjecture} and the {\\it Lang-Silverman Conjecture} for abelian varieties $A$ defined over a global field $K$; the latter is a uniform lower bound on the heights of non-torsion rational points. Our method is inspired by Vojta's proof of the Mordell Conjecture (Faltings's Theorem). Over function fields of characteristic $0$, a recent breakthrough of Looper-Yap (arXiv:2603.23396) proves both conjectures with inexplicit bounds. In our paper, we give a new proof of both conjectures with explicit bounds, which also depend polynomially on the field $K$ unless $A/K$ admits a factor of good reduction everywhere. We also prove explicit bounds for elliptic curves over function fields of characteristic $p>0$. Over number fields, we prove both conjectures under a suitable high-dimensional Szpiro conjecture (weaker than Hindry's version, Conjecture 3.4 of https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf) that we propose.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22285",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziyang Gao"
    },
    {
      "name": "Kaiyuan Gu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cmxy3qftavzqw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22285",
  "fulltext_url": null,
  "id": "cmxy3qftavzqw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22285",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204156536",
  "pdf_url": "https://arxiv.org/pdf/2608.22285",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cmxy3qftavzqw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.22285",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22285"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Uniformity in rational torsion and small points on abelian varieties",
  "title_text": "Uniformity in rational torsion and small points on abelian varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/cmxy3qftavzqw",
  "venue": "arXiv",
  "versions": []
}