On the Brumer-Stark conjecture

• 成果类型: Article
• 署名作者: Dasgupta, Samit; Kakde, Mahesh
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2023.197.1.5
• 发表日期: 2023
• 页码: 289-388
• 关键词: abelian l-functions; fitting ideals; iwasawa theory; cm extensions; zeta elements; VALUES; FIELDS

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 Theta H/F S,T annihilates the T-smoothed class group ClT (H). We prove this conjecture away from p = 2, that is, after tensoring with 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 ClT (H) circle times 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 Rib et'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.