{
  "abstract": "We study the divisibilities of Tamagawa numbers $ c(E) $ of elliptic curves $ E $ over global function fields $ K $ in terms of their torsion subgroups $ E(K)_{\\operatorname{tors}} $. In particular, for a non-isotrivial elliptic curve $ E / k(t) $, where $ k $ is a finite field of characteristic greater than $ 3 $, we prove that $ |E(k(t))_{\\operatorname{tors}}|^2 $ divides $ c(E) $, except possibly in four exceptional torsion families. More specifically, we give a complete characterisation of divisibilities for $ c(E) $ in each torsion family, and provide explicit examples to prove that they are best possible. Over a general global function field, we also prove that a rational point of prime order $ 5 \\le N \\le 101 $ on $ E / K $ forces $ N^2 $ to divide $ c(E) $, which motivates our result. Finally, we formulate a conjecture on the leading coefficient of the $ L $-function of $ E / K $, motivated by the integrality of Birch--Swinnerton-Dyer quotients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31910",
  "author_total": 2,
  "authors": [
    {
      "name": "David Kurniadi Angdinata"
    },
    {
      "name": "Mentzelos Melistas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dj4112c108khq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31910",
  "fulltext_url": null,
  "id": "dj4112c108khq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31910",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214711689",
  "pdf_url": "https://arxiv.org/pdf/2609.31910",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dj4112c108khq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31910",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31910"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Tamagawa numbers and torsion of elliptic curves over function fields",
  "title_text": "Tamagawa numbers and torsion of elliptic curves over function fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/dj4112c108khq",
  "venue": "arXiv",
  "versions": []
}