Brumer–Stark Class Group Annihilation Conjecture

Definitions and mathematical statement

Let $K/{\mathbb{Q}}$ be a finite abelian extension and put $G=\operatorname{Gal}(K/{\mathbb{Q}})$. Let $\mu_K=\{\zeta\in K^\times:\zeta^n=1\text{ for some }n\geq1\}$ and $e=|\mu_K|$. Write $\operatorname{Cl}_K$ for the group of nonzero fractional ideals of the ring of integers of $K$, modulo principal fractional ideals. The natural action of $G$ on ideals makes $\operatorname{Cl}_K$ a ${\mathbb{Z}}[G]$-module.

The supplied record does not explicitly specify the underlying field of the places in $S$, the finiteness of $S$, or the Frobenius convention. To give a precise candidate formulation, use the conventional interpretation in which $S$ is a finite set of places of ${\mathbb{Q}}$ containing the infinite place and every finite prime ramified in $K/{\mathbb{Q}}$. For a finite prime $p\notin S$, let $\sigma_p\in G$ be arithmetic Frobenius, characterized on residue fields by $x\mapsto x^p$. Since $G$ is abelian, this element is independent of the prime of $K$ chosen above $p$. Here the norm of the base-field prime is $Np=p$. These conventions require confirmation against the source; they are not additional hypotheses explicitly supplied by the record.

For $\operatorname{Re}(s)>1$, define the equivariant Euler product \[ \theta_S(s)=\prod_{p\notin S}\left(1-\sigma_p^{-1}p^{-s}\right)^{-1}\in{\mathbb{C}}[G], \] where the product runs over finite primes of ${\mathbb{Q}}$ outside $S$. Interpret $\theta_S(0)$ as the value at zero obtained by analytic continuation, rather than by direct substitution in the Euler product.

Annihilation question under these conventions. For every finite abelian extension $K/{\mathbb{Q}}$ and every such finite set $S$, is \[ e\theta_S(0)\in\operatorname{Ann}_{{\mathbb{Z}}[G]}(\operatorname{Cl}_K)? \] Here \[ \operatorname{Ann}_{{\mathbb{Z}}[G]}(\operatorname{Cl}_K) =\{\alpha\in{\mathbb{Z}}[G]:\alpha\cdot c=0\text{ for every }c\in\operatorname{Cl}_K\}, \] using additive notation for the class group. Equivalently, integral group-ring annihilation means that $e\theta_S(0)=\sum_{\sigma\in G}n_\sigma\sigma$ with $n_\sigma\in{\mathbb{Z}}$, and that for every nonzero fractional ideal $\mathfrak a$ of $K$, the fractional ideal \[ \prod_{\sigma\in G}\sigma(\mathfrak a)^{n_\sigma} \] is principal. Integrality is implicit in this interpretation of annihilation; the supplied statement does not discuss it separately.

The record calls this the Brumer--Stark conjecture, but supplies only the class-group annihilation assertion. No further condition on generators of principal ideals is specified, so no stronger conjectural formulation is included here.

Short English statement

For a finite abelian extension $K/{\mathbb{Q}}$, the supplied question asks whether multiplying the value at zero of an equivariant Euler product by the number of roots of unity in $K$ gives an integral group-ring element annihilating every ideal class. The precise candidate uses a finite set of base-field places and arithmetic Frobenius; these conventions need source confirmation. The supplied partially solved label and dated literature assessment are curation claims, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001645$, exact source problem code catalogue entry, title: Brumer--Stark Class Group Annihilation Conjecture. No named proposer is supplied.
• Supplied report citation: 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 describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The same URL is supplied under the literature-source label "original Oberwolfach problem report."
• Supplied literature source, labeled "Brumer-Stark away from 2": https://annals.math.princeton.edu/2023/197-1/p05.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the Brumer--Stark conjecture is proved for the minus part away from $2$, while the full integral conjecture in the stated generality remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status, cited claims, and applicability to the displayed annihilation question require independent verification.