On the Brumer-Stark Conjecture

DOI10.4007/ANNALS.2023.197.1.5arXiv2010.00657WikidataQ122972712 ScholiaQ122972712MaRDI QIDQ6350345

Samit Dasgupta, Mahesh Kakde

Publication date: 1 October 2020

Abstract: 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 $Thet{a}_{S,T}^{H/F}$ annihilates the $T$-smoothed class group $ext{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $mathbfZ[1/2]$. 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 $ext{Cl}^T(H)otimesmathbfZ[1/2]$ 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 $p$.

Mathematics Subject Classification ID

Class numbers, class groups, discriminants (11R29) Zeta functions and (L)-functions of number fields (11R42) Iwasawa theory (11R23)

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