Tate's Brumer–Stark conjecture

Let $F$ be a number field, let $H/F$ be a finite abelian extension with Galois group $G$, and let $S$ and $T$ be disjoint finite sets of places of $F$ such that $S$ contains the archimedean places and the finite places ramified in $H$. Assume that the only root of unity $\zeta\in\mu(H)$ satisfying $\zeta\equiv 1\pmod{\mathfrak p}$ for every prime $\mathfrak p$ of $H$ above a place in $T$ is $1$. Let $\operatorname{Cl}^T(H)$ be the ray class group of $H$ with conductor the product of the primes above $T$, and let $\Theta_{S,T}$ be the corresponding $T$-smoothed, $S$-depleted Stickelberger element. Tate's Brumer–Stark conjecture. One has

$$\Theta_{S,T}\in\operatorname{Ann}_{\mathbf{Z}[G]}\bigl(\operatorname{Cl}^T(H)\bigr).$$

This is a central integral annihilation statement in the Brumer–Stark conjecture, relating special values of equivariant $L$-functions to ray class groups. The source notes that Tate's original formulation is slightly stronger, although the discrepancy disappears after inverting $2$; the status is not resolved in the supplied text.