Serre-Weight Modularity Theorem

Definitions and mathematical statement

The supplied record does not determine a complete mathematical statement. It refers to a residual representation $\bar r$, a Serre weight $\sigma$, and the weights predicted by the Buzzard--Diamond--Jarvis conjecture, but does not specify the representation's domain, dimension, coefficient field, ambient arithmetic setting, or hypotheses.

For a fixed setting in which these notions have been specified, write schematically \[ W_{\mathrm{mod}}(\bar r) :=\{\sigma:\bar r\text{ occurs modularly in Serre weight }\sigma\}, \qquad W_{\mathrm{BDJ}}(\bar r) :=\{\sigma:\sigma\text{ is predicted for }\bar r\text{ by the BDJ recipe}\}. \] These expressions name the sets mentioned in the record; they do not supply the missing definitions of modular occurrence, the admissible Serre weights, or the applicable prediction recipe. The extracted question asks whether \[ W_{\mathrm{mod}}(\bar r)=W_{\mathrm{BDJ}}(\bar r), \] or, equivalently, whether for every Serre weight $\sigma$ in the intended setting, \[ \bar r\text{ occurs modularly in weight }\sigma \quad\Longleftrightarrow\quad \sigma\in W_{\mathrm{BDJ}}(\bar r). \] No range of representations or missing hypotheses is supplied, so this equivalence is only a schematic transcription, not a fully quantified conjecture or theorem.

The supplied statement-verification note says that the official contribution states and proves the modularity criterion under its hypotheses, and that the extraction removed those hypotheses and incorrectly presented the theorem as an open problem. Those hypotheses are absent from this record. Accordingly, the record marks the statement as unrecoverable and supplies no clean replacement. Neither a precise theorem nor a distinct remaining open problem can be recovered from the supplied evidence alone.

Short English statement

The extracted question asks whether the modular Serre weights of $\bar r$ are exactly its Buzzard--Diamond--Jarvis predicted weights. The supplied verification says this was a theorem under omitted hypotheses, incorrectly extracted as an open problem. The missing setting and hypotheses prevent a precise standalone formulation; human mathematical review is required.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002156$, exact source code catalogue entry, title "Serre-Weight Modularity Theorem," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied report citation: Arithmetic Geometry (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/38, pp. 2335--2388. https://doi.org/10.4171/owr/2012/38. The record describes these reports as documenting mathematical workshops and their open-problem discussions. The contribution's complete theorem and hypotheses are not supplied here.
• Supplied literature reference: Gee--Liu--Savitt, "BDJ conjecture for unitary groups." https://doi.org/10.1090/S0894-0347-2013-00775-4.
• Supplied literature reference: Gee--Liu--Savitt, "weight part of Serre." https://arxiv.org/abs/1309.0527.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the record "partially solved" and reports that the BDJ weight recipe is proved in major unitary and $\mathrm{GL}_2$ settings under standard residual, local, and automorphy hypotheses, while the extracted unqualified formulation is broader than those results. This is supplied dated curation, not an independently verified assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note separately reports that the official contribution proves its criterion under its hypotheses, identifies the open-problem extraction as erroneous, and retains the original record for public provenance without promoting a clean statement. This source-verification claim is supplied metadata and has not been independently checked.