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 $\Theta ^{H/F}_{S, T}$ 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. \par 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 $p$.

设 $H/F$ 是数域的有限阿贝尔扩展，其中 $F$ 是全实数，$H$ 是 CM 域。设 $S$ 和 $T$ 是满足标准条件的 $F$ 的不相交有限位置集。Brumer–Stark 猜想表明 Stickelberger 元素 $\Theta ^{H/F}_{S, T}$ 湮灭了 $T$ 平滑类群 $\mathrm {Cl}^T(H)$。我们证明这个猜想远离 $p=2$，即在用 ${\bf Z}[1/2]$ 张量之后。我们证明了 Kurihara 推测的这个结果的更强版本，它给出了 $\mathrm {Cl}^T(H) \otimes {\bf Z}[1/ 的 Pontryagin 对偶的负部分的第 0 次拟合理想的公式2]$ 以 Stickelberger 元素表示。我们还表明，这个更强的结果意味着 Rubin 的 Brumer-Stark 猜想的更高等级版本，再次远离 2。\par 我们的技术是 Ribet 方法的推广，建立在我们早期关于 Gross-Stark 猜想的工作之上。在这里，我们使用 Wiles 引入的群环值 Hilbert 模形式。我们方法的一个关键方面是尖点形式和 Eisenstein 级数之间的同余构造比通常预期的更强，作为 $p$-adic $L$-函数的平凡零点的影子出现。这些更强的同余对于证明我们构建的上同调类在 $p$ 处是无分支的至关重要。