{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0077",
  "title": "Semisimple Torsion Fields of Modular Jacobians at Powers of 11 and 13",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For ell in {11,13} and every m >= 1, we identify the semisimple ell-torsion field of the Jacobian of the connected principal modular curve X(ell^m) over K_m = Q(zeta_(ell^m)). It is the compositum of K_m with the field cut out by the residual Ramanujan Delta representation, and its Galois group over K_m is SL_2(F_ell). The full torsion field has an ell-group kernel over this quotient; the prime-to-ell part of its degree is exactly 120 or 168, respectively. The result is an explicit structural corollary of level-one modularity, the modular decomposition of Jacobians, and Hoshi-Iijima's non-pro-ell theorem. We give a cyclotomic-field moduli construction of the covering needed in the argument and reproducible exact certificates for the two residual Delta images. We do not determine the full torsion field or claim to resolve the unrestricted AIM splitting-field question. This note is an unrefereed synthesis; absolute priority is not asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "keywords": [
    "modular curves",
    "torsion fields",
    "semisimplification",
    "Ramanujan Delta",
    "Galois representations",
    "modular Jacobians"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-number-theory-0077/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-number-theory-0077/paper.pdf?v=b0122a1c01b8",
  "doi": "10.5281/zenodo.23322977",
  "zenodo_record_url": "https://zenodo.org/records/23322977",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete scoped semisimple-field corollary for powers of 11 and 13 under the explicitly chosen principal-curve and cyclotomic-base conventions. The full torsion field and Hall's general AIM question are not determined. Khare, Hoshi-Iijima and classical modular results are credited. Priority confidence is low; no independent review, formal verification or certified novelty is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "b0122a1c01b8c827c1555aa00212fd0eb1703d3c4037bb17fb735be15c596450"
    },
    "source.zip": {
      "sha256": "4759f5a32a3338b8fca5cacc4698b189a2b20a52ff9e90e4399912cc85bd6aeb"
    },
    "verification_report.md": {
      "sha256": "8b182db69edec26cf5b5520a015616e38b9303c88399aca6124c78889c1af32b"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}