{
  "abstract": "Let $C/\\mathbb{Q}$ be a Picard curve and let $\\ell\\equiv1\\pmod3$ be a prime such that the $λ$-torsion representation of the Jacobian of $C$ is reducible for a prime $λ$ of $\\mathbb{Q}(ζ_3)$ above $\\ell$. We describe the constraints imposed by complex conjugation on the $2$-dimensional subquotient $ρ$, in particular, showing that if it is absolutely irreducible, then up to twist it descends to $\\mathbb{Q}$. We then give criteria for ruling out the irreducible case, using Dickson's classification of subgroups of $\\operatorname{GL}_2(\\mathbb{F}_{\\ell})$, under the natural hypothesis that the characteristic polynomials of all elements in the image of $ρ$ split. We also develop algorithms to compute the set of $1$-dimensional Jordan--Hölder constitutents of the $\\ell$-torsion representation over $\\mathbb{Q}(ζ_3)$, and Frobenius characteristic polynomials of the $2$-dimensional subquotient $ρ$. We apply these methods to $381$ Picard curves with possibly reducible $7$-torsion representations, and to five corresponding $13$-torsion examples. Under the reducibility and splitting hypotheses, we rule out an irreducible $2$-dimensional subquotient in all of the examples considered, showing that the semisimplification of the $\\ell$-torsion representation is a direct sum of $6$ characters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24751",
  "author_total": 5,
  "authors": [
    {
      "name": "Xiyao Chen"
    },
    {
      "name": "Shiva Chidambaram"
    },
    {
      "name": "Andrew Coursin"
    },
    {
      "name": "Erik Reed"
    },
    {
      "name": "Leo Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9gjabacspdhzm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24751",
  "fulltext_url": null,
  "id": "9gjabacspdhzm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24751",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223023390",
  "pdf_url": "https://arxiv.org/pdf/2610.24751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9gjabacspdhzm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On reducible $\\ell$-torsion representations of Picard curves",
  "title_text": "On reducible ℓ-torsion representations of Picard curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/9gjabacspdhzm",
  "venue": "arXiv",
  "versions": []
}