论文标题
在Brumer-Stark的猜想上
On the Brumer-Stark Conjecture
论文作者
论文摘要
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 $Θ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$.我们证明了库里亚拉（Kurihara）猜想的更强大版本，该结果为$ \ text {cl}^t（h）\ otimes \ otbf {z} [z} [1/2] $的pontryagin dual的0拟合理想提供了一个公式。 We also show that this stronger result implies Rubin's higher rank version of the Brumer-Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross-Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles.我们方法的一个关键方面是尖峰形式和爱森斯坦系列之间的一致性，它们比通常预期的要强，这是$ p $ adic $ l $ contuntions的琐事零的阴影。 These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.
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 $Θ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$. 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 $\text{Cl}^T(H) \otimes \mathbf{Z}[1/2]$ 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 $p$.
