Langlands–Rapoport Conjecture for Parahoric Level

Definitions and mathematical statement

The supplied title is Langlands--Rapoport Conjecture for Parahoric Level. Write $\mathrm{LR}$ and $\mathrm{LR}\text{-}\tau$ only as names for the original Langlands--Rapoport conjecture and the variant mentioned in the extraction; their mathematical definitions are not supplied.

The extracted passage says that a proof of $\mathrm{LR}\text{-}\tau$ "should suffice for most applications", although that conjecture "is not as strong as the original LR conjecture". This is a comparison and an expectation about applications, rather than a statement of either conjecture. The passage is attributed to Sug Woo Shin's contribution Cohomology of Shimura varieties, reporting joint work with Mark Kisin and Yihang Zhu.

Unresolved mathematical statement. The supplied record does not specify a Shimura datum, a prime, hypotheses on the reductive group, the parahoric level, an integral model, or the objects and compatibility conditions involved in the proposed description. It also does not define the $\tau$-variant. Consequently, no precise quantified formulation of either conjecture, or of a parahoric-level specialization, can be recovered from this record. The title alone does not supply those hypotheses, and the extraction does not determine which conjecture is the intended target.

Supplied status qualifications. The registry labels this entry open, but also labels its statement unrecoverable. Its statement-verification note says that the conjecture discussed in the extraction has been proved and therefore is not an open problem. The supplied dated literature assessment distinguishes the two conjectures: it reports that $\mathrm{LR}\text{-}\tau$ has been proved, with trace-formula consequences, while the original $\mathrm{LR}$ conjecture remains wide open. These are supplied source and curation claims, not independently verified mathematical conclusions. Their distinction must be resolved before a precise open-problem statement can be assigned to this entry.

Short English statement

The passage compares the original Langlands--Rapoport conjecture with an unspecified $\tau$-variant; it does not state either conjecture. The supplied curation reports the variant proved and the original conjecture open. Missing definitions and hypotheses, together with conflicting registry qualifications, prevent recovery of a precise parahoric-level problem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005397$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Registry title: Langlands--Rapoport Conjecture for Parahoric Level. The supplied proposed year is 2023; no proposer is specified.
• Arithmetic of Shimura Varieties (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/5, pp. 261--326. Supplied report reference: https://doi.org/10.4171/owr/2023/5. The extraction attributes Cohomology of Shimura varieties to Sug Woo Shin, joint work with Mark Kisin and Yihang Zhu. No precise page or theorem locator for either conjecture is supplied.
• Supplied literature source: Kisin, Shin, and Zhu, Stable trace formula for Shimura varieties, https://arxiv.org/abs/2110.05381.
• Supplied literature source: Oberwolfach Report 5/2023, https://doi.org/10.4171/owr/2023/5.
• Literature assessment checked at 2026-08-22: the supplied curation reports that the LR-$\tau$ conjecture has been proved and yields important trace-formula consequences, whereas the report describes the original Langlands--Rapoport conjecture as remaining wide open. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The record requests independent verification of time-sensitive status and cited claims.