Conditional Harder-Congruence Theorem

Definitions and mathematical statement

The supplied local registry identifies permanent UnsolvedMath ID $30004343$, source code catalogue entry, as a conditional Harder-congruence theorem. Its supplied verification note identifies the extraction as a proved theorem under four explicit hypotheses, rather than a separate open problem; the unrestricted conjecture is recorded separately as OWR-17300-006.

The recorded mathematical content is that, under four hypotheses concerning Fourier coefficients, an $\ell$-unit condition, denominators, and exclusion of endoscopic congruences, a holomorphic Jacobi-form version of Harder's conjecture holds. The record additionally reports the divisibility assertion \[ \ell\mid L^{\mathrm{alg}}(2k+j-3,f). \] This display reproduces a claimed consequence in the supplied record; it is not a self-contained mathematical statement.

The four hypotheses are not supplied. The record also does not define $f$, the domains or restrictions on $k,j,\ell$, the algebraic normalization $L^{\mathrm{alg}}$, the coefficient field, or the ring or valuation in which divisibility is intended. The actual Jacobi-form congruence, including its forms, level, weights, modulus, and quantified coefficients or eigenvalues, is absent. Thus neither the conditional theorem nor its reported divisibility consequence can be formulated precisely from this record alone. These omissions remain unresolved; no hypotheses, definitions, or quantifiers are supplied by inference.

Short English statement

For permanent ID $30004343$, the supplied record describes a conditional theorem concerning a Jacobi-form version of Harder's conjecture and reports $\ell\mid L^{\mathrm{alg}}(2k+j-3,f)$. Its four hypotheses and essential definitions are missing. The record marks the statement as unrecoverable and identifies it as an extracted proved theorem, not a separate open problem. A complete mathematical statement requires human source review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004343$, source code catalogue entry, title "Conditional Harder-Congruence Theorem," collection "Oberwolfach Reports Open Problems." The registry assigns proposed year $2019$ and supplies no named proposer.
• Supplied report citation: Modular Forms (2020), Oberwolfach Reports, pp. 3529--3606, DOI 10.4171/owr/2019/57. https://doi.org/10.4171/owr/2019/57. The supplied background describes these reports as documenting workshops and their open-problem discussions; its verification note specifically classifies this extraction as a proved conditional theorem.
• Supplied literature-source label: "Katsurada, congruence for Klingen--Eisenstein series." https://doi.org/10.4171/owr/2019/57. This attribution is retained as supplied; no more specific theorem locator or full bibliographic entry is provided.
• Supplied literature-source label: "Harder's conjecture I." https://doi.org/10.2969/jmsj/87988798.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the record "partially solved," describes a Jacobi-form congruence proved under conditions (1)--(4), and states that the underlying Harder conjecture is not known in full generality. The supplied background calls this assessment dated literature triage and requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note, in the record updated 2026-09-16T12:06:53Z: the official report is said to identify a proved theorem under four explicit hypotheses; the unrestricted conjecture is assigned source code OWR-17300-006. No separate clean statement is promoted, and the original record is retained for provenance. These are supplied verification claims, not independent source verification performed for this draft.