{
  "abstract": "Let $Ω$ be the curvilinear square bounded by the four circular arcs $|ζ\\mp1|=\\sqrt{2}$, $|ζ\\mp i|=\\sqrt{2}$, and let $W=\\sqrt{1+14ζ^{4}+ζ^{8}}$. Put \\[ θ(ζ)=\\arcsin\\frac{2(1+i)ζ} {\\sqrt{1+4iζ^{2}-ζ^{4}}}, \\] \\[ f=\\tfrac14\\bigl(-iF[θ,\\tfrac14]+F[θ,\\tfrac34]\\bigr),\\quad \\quad g=\\tfrac14\\bigl(\\ \\ iF[θ,\\tfrac14]+F[θ,\\tfrac34]\\bigr), \\] \\[ \\mathfrak h=(2-\\sqrt3)\\, F\\bigl[\\arcsin\\bigl(i(2+\\sqrt3)ζ^{2}\\bigr),(2-\\sqrt3)^{4}\\bigr]. \\] These are, verbatim, the three functions of our companion paper $\\mathrm{[D]}$ on the diamond surface $\\mathrm{D}$: the two papers start from the same point, and differ only in which real parts are taken. With $X=(κ\\,\\operatorname{Re}f,\\ κ\\,\\operatorname{Re}g,\\ \\tfrac12+κ\\,\\operatorname{Im}\\mathfrak{h})$ and $κ=3/(2K[1/9])$, the map $X$ parametrizes a fundamental patch of Schwarz's primitive surface $\\mathrm{P}$, the conjugate of $\\mathrm{D}$. Part I builds the algebraic apparatus $\\ldots$ the normalizing constant is identified exactly: $\\varpi:=κ^{-1}=K[-3]=\\tfrac12 K[3/4]=\\tfrac23 K[1/9]$. Part II carries out the elimination and completes the proof of \\[ (\\star)\\qquad \\operatorname{sn}\\bigl(\\varpi(x{+}y),-3\\bigr)\\,\\operatorname{sn}\\bigl(\\varpi(x{-}y),-3\\bigr) =\\tfrac13\\operatorname{sn}\\bigl(3\\varpi(z-\\tfrac12),\\tfrac19\\bigr). \\] Part III shows that $(\\star)$ is, after all, additively separable $\\ldots$ for an overview of this paper and $\\mathrm{[D]}$, please see arXiv:2609.14206.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22313",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Finch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52mt0hbhtk7ha.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22313",
  "fulltext_url": null,
  "id": "52mt0hbhtk7ha",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22313",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7214015764",
  "pdf_url": "https://arxiv.org/pdf/2609.22313",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52mt0hbhtk7ha.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22313",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22313"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface",
  "title_text": "Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/52mt0hbhtk7ha",
  "venue": "arXiv",
  "versions": []
}