{
  "abstract": "We prove two supercongruence statements proposed by Z.-W. Sun for Domb numbers. The first concerns the unweighted truncated sum $$S_p=\\sum_{n=0}^{p-1}D_n$$ and is governed by the two proper ideal classes of the order of discriminant $-60$ in $\\mathbb{Q}(\\sqrt{-15})$. We prove Sun's conjectured congruences modulo $p^2$ and obtain stronger formulas modulo $p^3$ in the split cases. The second concerns Domb sums weighted by the Lucas sequences associated with $X^2-11X+1$. Sun's original formulation has the exceptional counterexample $p=3$; after excluding this prime, we prove the corrected statement, including its modulo-$p^3$ refinement when both relevant quadratic characters are $+1$. Both results arise from one modular mechanism. The twisted Domb series is a modular period on $Γ_0(6)+\\langle 3\\rangle$ and is related by a quadratic pullback to Zagier's sporadic $C$-sequence. At primes inert in the imaginary quadratic field underlying the relevant complex multiplication, the truncated $C$-period is the Hasse invariant and vanishes at supersingular reduction; a finite pullback then gives divisibility modulo $p^2$. At split primes, explicitly constructed determinant-$1$ or determinant-$3$ matrices identify the Beukers unit root, yielding congruences modulo $p^3$. The two applications correspond respectively to orders of complex multiplication of discriminants $-60$ and $-156$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04772",
  "author_total": 1,
  "authors": [
    {
      "name": "Kevin Calderon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dez9mdx55sreg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04772",
  "fulltext_url": null,
  "id": "dez9mdx55sreg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04772",
  "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",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7220721972",
  "pdf_url": "https://arxiv.org/pdf/2610.04772",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dez9mdx55sreg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04772",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04772"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Supercongruences for Domb numbers",
  "title_text": "Supercongruences for Domb numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dez9mdx55sreg",
  "venue": "arXiv",
  "versions": []
}