Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer–Stark conjecture states that the Stickelberger element $\Theta_{S,T}^{H/F}$ annihilates the $T$-smoothed class group $\mathrm{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with ${\bf Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\mathrm{Cl}^T(H) \otimes {\bf Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer--Stark conjecture, again away from 2.

Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross--Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.