On the Brumer-Stark conjecture

Samit Dasgupta

doi:10.4007/annals.2023.197.1.5

On the Brumer-Stark conjecture

2023-01-01

13

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DOI：10.4007/annals.2023.197.1.5

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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.

Keyword:

class groups

Fitting ideals

Stark conjectures

Hilbert modular forms

On the Brumer-Stark conjecture

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