{
  "schema_version": 1,
  "problem_number": "OWR-782-001",
  "title": "Extremal Modular Forms Modulo p and a Conjecture of Bannai, Koike, Shinohara and Tagami",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For even k ≥ 4 let f_k = 1 + O(q^{dim M_k}) be the extremal modular form of weight k for SL_2(ℤ). Bannai, Koike, Shinohara and Tagami conjectured that if k = 12μ and f_k mod p is a nonconstant power series in q^p (their Case (2)), then f_k(τ) ≡ g(p^r τ) (mod p) for an extremal modular form g of smaller weight and some r ≥ 1. We observe that this is a corollary of the classical theory of modular forms modulo p: the key input is the equality w(h^p) = p·w(h) for the filtration of a p-th power, which follows from Swinnerton-Dyer's structure theorem and underlies the theorem of Serre and Katz on the kernel of θ. For every even k ≥ 4 and every prime p in Case (2) we get f_k(τ) ≡ f_{k'}(pτ) (mod p) with k' = w(f_k mod p)/p, where 4 ≤ k' ≤ k/p and k' ≡ k (mod p − 1). So the conjecture holds with r = 1, and by iteration also with the largest possible r. For k = 12μ every prime in Case (2) satisfies 11 ≤ p ≤ 3μ and p ≠ 13, and Case (2) is decided by a finite test. Exact computations for all weights 12μ ≤ 7200 and all even k ≤ 2400, covering 9433 pairs (k, p) in Case (2), agree with these statements. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NT"
  ],
  "keywords": [
    "extremal modular forms",
    "modular forms modulo p",
    "filtration",
    "theta operator",
    "Swinnerton-Dyer congruences",
    "Hecke operator",
    "extremal lattices",
    "Oberwolfach Reports",
    "OWR-782-001",
    "OWR-782-002",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-01",
  "publication_date": "2026-10-01",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-01",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-782-001/",
  "pdf_url": "https://eulersolve.org/papers/owr-782-001/paper.pdf?v=f51a36957ec5",
  "doi": "10.5281/zenodo.23071924",
  "zenodo_record_url": "https://zenodo.org/records/23071924",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Proves Conjecture 4 of Bannai, Koike, Shinohara and Tagami (OWR 1/2005, p. 14; duplicate record OWR-782-002) for level one, with r = 1 and with the largest r, as a corollary of the classical theory of modular forms modulo p. The journal version's full text was not read; the argument is short and may be known to experts.",
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    "verification_report.md": {
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  "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."
}