Local Determination of Serre Weights for Unitary Groups

Definitions and mathematical statement

The supplied question concerns the weight part of Serre's conjecture for unitary groups of semisimple rank greater than one. Write $W(r)$ for the set of modular Serre weights attached to the representation $r$, as intended by the source. For a place $w$ of the underlying field $F$, write $F_w$ for its completion and $G_{F_w}=\operatorname{Gal}(\overline{F_w}/F_w)$ for its absolute Galois group. The notation $r|_{G_{F_w}}$ denotes the corresponding local restriction, and $W^{?}(r|_{G_{F_w}})$ denotes the source's predicted local set of Serre weights.

Source question, with unresolved specifications. Is $W(r)$ determined by the local restrictions $r|_{G_{F_w}}$ through the predicted sets $W^{?}(r|_{G_{F_w}})$ for unitary groups of semisimple rank greater than one?

The supplied record does not specify the prime, the field $F$, the unitary group, the domain and coefficient field of $r$, the admissible representations, the relevant places $w$, the definitions of the weight sets, or the rule assembling the local predictions into a global set. It also does not supply the hypotheses or quantifiers under which the proposed determination should hold. Consequently, this is a qualified transcription of the question, not a fully specified conjectural equality. The supplied background expressly describes the extracted passage as fragmentary and insufficient to specify a complete conjecture.

Short English statement

For unitary groups of semisimple rank greater than one, the question asks whether modular Serre weights are determined by predicted weight sets attached to local Galois representations. The supplied fragment leaves the precise objects, hypotheses, and local-to-global rule unspecified, so a complete mathematical conjecture cannot yet be stated.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002948$, source problem code catalogue entry, title: Local Determination of Serre Weights for Unitary Groups. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2015 and names no proposer.
• Supplied report citation: Reductions of Shimura Varieties (2016), Oberwolfach Reports, pp. 2265--2328, DOI 10.4171/owr/2015/39. https://doi.org/10.4171/owr/2015/39. This is also the supplied literature source labelled "original OWR report." No passage-specific page or problem locator is supplied.
• Supplied literature source labelled "later local model results": https://arxiv.org/abs/2007.05398. The record supplies no authors, title, or theorem locator for this reference.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, describes the extracted passage as fragmentary, reports that generic unitary group cases are proved, and says that the broader Serre weight landscape remains only partially resolved. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited mathematical claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming that the cleanup is self-contained and adds no unsupported content. These are supplied editorial assertions; they do not resolve the missing mathematical specifications identified in the supplied literature assessment.