Arithmetic Topology over Rings of S-Integers

Definitions and mathematical statement

A global field $F$ is a finite extension of ${\mathbb{Q}}$ or a finite extension of ${\mathbb{F}}_q(t)$, where $q$ is a prime power. Let $S$ be a finite set of places of $F$; if $F$ is a number field, require $S$ to contain every archimedean place. For each nonarchimedean place $v$, write $v:F^\times\to{\mathbb{Z}}$ for its normalized additive valuation and set $v(0)=+\infty$. The ring of $S$-integers is \[ \mathcal O_{F,S}=\{x\in F: v(x)\geq 0\text{ for every nonarchimedean place }v\notin S\}. \] The supplied record imposes no nonemptiness condition on $S$ in the function-field case.

Source question. For a global field $F$ and a finite set $S$ satisfying these conditions, how does the intended arithmetic--topology correspondence change when its base ring is replaced by $\mathcal O_{F,S}$?

Unresolved scope. This is an extension question, not a specified theorem or equivalence. The supplied record does not identify the original base ring, define the arithmetic--topology correspondence, or specify the objects, constructions, or properties to be compared after replacement. Consequently, it does not determine a unique precise conjecture beyond the question above. The original excerpt uses $F$ for the field but $K$ in the number-field qualification; the supplied corrected statement interprets both as $F$, and this draft follows that interpretation.

Short English statement

For a global field $F$, the ring $\mathcal O_{F,S}$ permits denominators at the nonarchimedean places in a finite set $S$, which must also include all archimedean places in the number-field case. The question asks how the intended arithmetic--topology correspondence extends to this ring. The supplied excerpt leaves the correspondence and the desired conclusions unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30006312$, title: Arithmetic Topology over Rings of S-Integers; source problem code catalogue entry, in the supplied Oberwolfach Reports Open Problems collection.
• Arbeitsgemeinschaft: Relative Langlands Duality (2025), Oberwolfach Reports, pp. 811--882, DOI 10.4171/owr/2025/18: https://doi.org/10.4171/owr/2025/18. The supplied original excerpt includes a report-title marker followed by page number 829. No individual proposer is named; the registry supplies proposed year 2025.
• Supplied literature source, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2025/18. The literature assessment dated 2026-08-22 labels the question open and reports that no general theorem answering the extension to rings of $S$-integers was located. This is supplied dated curation, not an independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The registry assigns statement status corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired for self-contained publication. These are supplied provenance claims, not verification performed for this draft.