{
  "abstract": "We construct explicit elements in the group K_4^(3) of the Fermat curves x^N+y^N=1 for all N >= 3. The construction, which is uniform in N, uses polylogarithmic complexes and a map of de Jeu to K-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to 3/2 * ζ(3) * N^2 as N tends to infinity. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for K_2 groups of Fermat curves. Finally, we numerically verify some cases of Beilinson's conjectures on special values of L-functions at s=3 for N = 3, 4, 6.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.24532",
  "author_total": 3,
  "authors": [
    {
      "name": "François Brunault"
    },
    {
      "name": "David Lilienfeldt"
    },
    {
      "name": "Yusuke Nemoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5fkh3009jk5w.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "c5fkh3009jk5w",
  "issns": [],
  "landing_url": "https://hal.science/hal-05803673v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05803673/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5fkh3009jk5w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:58Z",
      "native_id": "hal-05803673",
      "source": "hal",
      "url": "https://hal.science/hal-05803673v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Elements in K_4 and regulator maps of Fermat curves",
  "title_text": "Elements in K_4 and regulator maps of Fermat curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:58Z",
  "url": "https://research.draconitic.com/paper/c5fkh3009jk5w",
  "venue": "HAL",
  "versions": []
}