Riemann–Hilbert Correspondence for Étale and v-Sheaves

Definitions and mathematical statement

Let $p$ denote the prime occurring in the intended correspondence, and let $\mathbb F_p=\mathbb Z/p\mathbb Z$. An étale $\mathbb F_p$-sheaf means a sheaf of $\mathbb F_p$-vector spaces on an étale site; the underlying geometric object and site must be specified to determine its category. Such a sheaf is annihilated by $p$. The supplied record does not define the overconvergence condition or specify the intended meaning of $v$-sheaves in the proposed extension.

The extracted question is the following: in the geometric setting of the Riemann--Hilbert correspondence referred to in the source passage, extend that correspondence from overconvergent étale $\mathbb F_p$-sheaves to all étale $\mathbb F_p$-sheaves in that setting. The source further conjectures an extension to $v$-sheaves. Each extension is required to recover the preceding correspondence on its original domain, up to the appropriate natural identification.

This is an unresolved formulation qualification: the supplied passage refers to "the above Riemann--Hilbert correspondence," but does not supply that correspondence. In particular, its ambient spaces, target category, functor, definition of overconvergence, and equivalence or other correspondence properties are absent. The passage also does not determine whether the final extension concerns sheaves of $\mathbb F_p$-vector spaces for the $v$-topology, geometric objects called $v$-sheaves, or another construction specified in the omitted context. Consequently, the requested extension cannot be stated as a fully specified functorial conjecture from this record alone. No constructibility, finiteness, boundedness, or geometric hypotheses are added here.

The registry labels the problem partially solved. Its supplied dated assessment says that a $p$-torsion Riemann--Hilbert correspondence is established for overconvergent sheaves and small $v$-stacks, while the conjectural extension to all étale sheaves and $v$-sheaves is incomplete. This assessment is supplied curation, not a mathematical result independently verified in this draft; its relation to the two requested extensions requires review.

Short English statement

Extend the source's $p$-torsion Riemann--Hilbert correspondence from overconvergent étale $\mathbb F_p$-sheaves to all étale $\mathbb F_p$-sheaves, and conjecturally to $v$-sheaves, compatibly with the original correspondence. The supplied excerpt omits the ambient setting, target category, and meaning of the final extension, so a fully precise conjecture requires the preceding source context. The partially solved status is a supplied dated assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005001$, title: Riemann--Hilbert Correspondence for Étale and v-Sheaves; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is identified. The proposed year is recorded as 2022.
• Supplied report citation: Non-Archimedean Geometry and Applications (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/5, pp. 231--302. https://doi.org/10.4171/owr/2022/5. The supplied excerpt refers to an earlier correspondence without reproducing its definition. The record describes the statement as reconstructed and checked against the official report and labels it corrected_verified; these are supplied verification claims, not independent verification by this draft.
• Mann, A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry. Supplied literature link: https://arxiv.org/abs/2206.02022.
• Bhatt--Lurie, A Riemann-Hilbert correspondence in positive characteristic. Supplied literature link: https://arxiv.org/abs/1711.04148.
• Literature assessment checked at 2026-08-22, exactly as recorded. This is supplied dated curation derived from the OWR record, not independently verified. It labels the problem partially solved and reports an established correspondence for overconvergent sheaves and small $v$-stacks, with the extension to all étale sheaves and $v$-sheaves incomplete. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.