Bounded-Degree Property for Reductive Groups

Definitions and mathematical statement

Let $F$ be a number field, meaning a finite extension of ${\mathbb{Q}}$, and let \[ S_{\mathrm{fin}}(F)=\{v: v\text{ is a non-Archimedean place of }F\}. \] Let $G$ be a reductive algebraic group over $F$, with the convention for reductivity intended by the cited source. The supplied record does not specify whether connectedness is required.

The proposed assertion is that every such $G$, over every number field $F$, satisfies property $(\mathrm{BD})$ for the entire set $S=S_{\mathrm{fin}}(F)$. Symbolically, writing $\operatorname{BD}(G,S)$ only as a placeholder for the source's property, \[ \forall F\text{ a number field},\qquad \forall G\text{ reductive over }F,\qquad \operatorname{BD}\left(G,S_{\mathrm{fin}}(F)\right). \] Unresolved definition. The supplied record does not define property $(\mathrm{BD})$. In particular, it does not identify the objects whose degrees are bounded, the bound, its allowed dependencies, or the required uniformity over places. Consequently, this display preserves the conjecture's quantifiers and intended full set of finite places, but is not a self-contained mathematical formulation. The source definition of $(\mathrm{BD})$ must be supplied before a precise standalone statement can be completed.

Short English statement

The conjecture asks whether every reductive group over every number field has the source's bounded-degree property at the full set of finite places. The supplied record omits the definition of that property, so a precise standalone formulation remains incomplete. Its partially solved status is supplied dated curation requiring independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003510$, title "Bounded-Degree Property for Reductive Groups," source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." No individual proposer is identified; the recorded proposed year is 2017.
• Supplied report citation: Harmonic Analysis and the Trace Formula (2018), Oberwolfach Reports, pp. 1551--1630, DOI 10.4171/owr/2017/25: https://doi.org/10.4171/owr/2017/25. This is also the literature-source link labelled "original OWR report." The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The citation's 2018 designation and the record's 2017 proposed year are retained without resolving their relationship.
• Supplied literature source: "Finis and Lapid, local degree bounds," https://doi.org/10.1007/s11856-019-1843-0.
• Supplied literature source: "current Eisenstein-series bounds," https://doi.org/10.1017/fms.2024.59.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem partially solved and states that the universal assertion for all reductive groups and the full finite-place set remains open; that it holds for $\mathrm{GL}_n$, $\mathrm{SL}_n$, and several classical cases; and that every reductive group satisfies the property outside a finite exceptional set of places. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly, with definitions restored and source-dependent wording removed. This is a supplied verification claim, not verification performed for this draft; the supplied record itself contains no definition of $(\mathrm{BD})$.