{
  "abstract": "In this paper, we provide an algebraic proof of the condition for the existence of modular form solutions to the Kaneko-Zagier differential equation of weight $k$. Unlike previous representation-theoretic approaches relying on $SL_2(\\mathbb{Z})$, our method employs the connection matrices of the principal congruence subgroup $Γ(N)$. By deriving a condition for simultaneous triangularizability from the commutators of the representation matrices and applying Galois theory, we prove that the associated two-dimensional representation of $Γ(N)$ is irreducible when the denominator $m$ of the fraction $(k+1)/6 = n/m$ satisfies $m=1$ or $m \\ge 7$. Consequently, we identify the weights that admit no modular form solutions and obtain a complete classification of the dimensions of the spaces of such solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05668",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuichi Sakai"
    },
    {
      "name": "Hiroyuki Tsutsumi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46bzpd66c3x5z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05668",
  "fulltext_url": null,
  "id": "46bzpd66c3x5z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05668",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7220714126",
  "pdf_url": "https://arxiv.org/pdf/2610.05668",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46bzpd66c3x5z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.05668",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05668"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An Algebraic Proof of the Non-Modularity of Solutions to the Canonical Modular Differential Equation",
  "title_text": "An Algebraic Proof of the Non-Modularity of Solutions to the Canonical Modular Differential Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/46bzpd66c3x5z",
  "venue": "arXiv",
  "versions": []
}