{
  "abstract": "Let $E$ be an elliptic curve over $\\mathbb{Q}$. We prove that the degree of the Shimura parametrization of $E$ arising from the optimal quotient $J^D(M)\\to E$ is equal to the $D$-new modular degree of $E$, under mild assumptions on the local Galois representations of $E$. As an application, we prove a conjecture of Deines asserting the equality of the $D$-new modular degree, the $D$-new congruence number, and the Shimura degree of $E$. We introduce the Shimura congruence number into this framework and prove, under the same assumptions, that all four quantities coincide; moreover, the two congruence numbers are always equal. We establish new cases of multiplicity one for Shimura Jacobians and use them to prove the main theorems, following a strategy introduced by Agashe, Ribet, and Stein. Building on these results, we obtain new examples of the failure of multiplicity one. Finally, we give a negative answer to a question of Papikian and Rabinoff asking whether the functorial map between the component groups of the Néron models of $J^D(M)$ and $E$ is surjective when $D>1$, and we relate this phenomenon to the failure of multiplicity one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.23244",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad Masih Hamidi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5v4bd49rxyqne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.23244",
  "fulltext_url": null,
  "id": "5v4bd49rxyqne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.23244",
  "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": "W7171541701",
  "pdf_url": "https://arxiv.org/pdf/2607.23244",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5v4bd49rxyqne.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2607.23244",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.23244"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties",
  "title_text": "On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/5v4bd49rxyqne",
  "venue": "arXiv",
  "versions": []
}