{
  "abstract": "We prove two height inequalities for rational points on abelian varieties over function fields of characteristic 0. The first is the GCD inequality, which shows that the height with respect to a closed subscheme of an abelian variety of codimension at least 2 is much smaller than the usual height function. This can be viewed as a compact and geometric analogue of a theorem of Levin. The second is a special case of the (geometric) abc conjecture proposed by Vojta, which compares the truncated counting function and the usual counting function. This is a compact analogue of a theorem of Guo, Nguyen, Sun and Wang, and also resolves a conjecture of Buium, which was only known for traceless abelian varieties or constant abelian varieties previously. The proofs of these inequalities follow a common three-step pattern: from microscopic local estimates to local estimates, and then to global estimates. Building on the canonical Xie-Yuan heights, we control discrete counting functions by continuous integrals of Betti forms. The main new tools are uniform Nevanlinna-type estimates for families of linear entire curves on abelian varieties; combined with covering arguments based on Besicovitch theory and averaging, these handle overlaps and gaps precisely.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20379",
  "author_total": 1,
  "authors": [
    {
      "name": "Guoquan Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3a3ycvtndam48.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20379",
  "fulltext_url": null,
  "id": "3a3ycvtndam48",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20379",
  "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.CV"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7222075259",
  "pdf_url": "https://arxiv.org/pdf/2610.20379",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3a3ycvtndam48.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20379",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20379"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "GCD and ABC inequalities for abelian varieties over function fields",
  "title_text": "GCD and ABC inequalities for abelian varieties over function fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/3a3ycvtndam48",
  "venue": "arXiv",
  "versions": []
}