Brumer–Stark Conjecture for Ray Class Groups

Definitions and mathematical statement

The supplied record concerns a proposed Brumer--Stark variant for a $T$-class group. It does not supply enough hypotheses or definitions for a self-contained conjecture. The following retains its formula and records its intended annihilation assertion conditionally.

Let $S$ and $T$ be disjoint finite sets of places in the unspecified arithmetic setting of the source. The displayed expression in the record is \[ \theta_{S,T}(s) = \left(\prod_{P\in T} \left(1-\sigma_P^{-1}(NP)^{1-s}\right)^{-1}\right) \theta_S(s). \] The record does not specify the underlying field or extension, the group containing $\sigma_P$, the definition and normalization of $\theta_S(s)$, or the meaning of $NP$. Nor does it specify the algebra in which the displayed inverses and the value at $s=0$ are defined. The inverse Euler factors above are reproduced exactly as supplied.

Write $A=\operatorname{Cl}(\mathcal O_T)$ for the object called the $T$-class group in the record. Its precise definition is not supplied; in particular, the title's reference to ray class groups does not determine its modulus or local conditions. For an element $\alpha$ of a ring acting on an abelian group $A$, to say that $\alpha$ annihilates $A$ means \[ \alpha\cdot a=0\qquad\text{for every }a\in A. \] Thus the proposed assertion, whenever its arithmetic data, acting ring, action, and special value are defined, is \[ \theta_{S,T}(0)\cdot a=0 \qquad\text{for every }a\in\operatorname{Cl}(\mathcal O_T). \] This assertion is subject to an unspecified "minor technical condition on $T$." The supplied source-verification note reports that the cited report does not state that condition. Consequently, the assertion cannot be quantified over all disjoint finite $S,T$, and no precise admissible family can be extracted from the supplied record. Additional restrictions on $S$, including any requirements concerning archimedean or ramified places, are also not supplied. This is a conditional transcription of the intended assertion, not a certified self-contained conjecture.

Short English statement

The proposed variant says that a modified special value $\theta_{S,T}(0)$ kills every element of a $T$-class group. The underlying arithmetic setting, several definitions, and a required condition on $T$ are missing, so the supplied record does not determine a precise unconditional formulation. Its supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001646$, exact source problem code catalogue entry, title "Brumer--Stark Conjecture for Ray Class Groups." The proposed year is recorded as 2010; no individual proposer is supplied.
• Supplied source citation and literature-source label "original Oberwolfach problem report": Mini-Workshop: 1-Motives (2011), Oberwolfach Reports, pp. 3237--3267, DOI 10.4171/owr/2010/55. https://doi.org/10.4171/owr/2010/55. The background attributes the collection to workshop open-problem discussions documented in Oberwolfach Reports. The citation is reproduced as supplied, including its year designation.
• Supplied literature source, "Brumer-Stark away from 2": https://annals.math.princeton.edu/2023/197-1/p05.
• Supplied literature source, "equivariant Iwasawa applications": https://doi.org/10.1007/s40687-025-00578-8.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the function-field analogue is a theorem, while the number-field $T$-class-group conjecture is known in substantial partial cases, including the odd-prime part. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The supplied statement-verification field labels the formulation unrecoverable, reports that the official report leaves the technical condition on $T$ unspecified, and says that no separate clean statement was promoted. This draft preserves that source limitation and does not claim to have inspected the report.