The Brumer--Stark Conjecture over Z

TL;DR

The paper settles the Brumer–Stark conjecture integrally over $\mathbf{Z}$ for CM abelian extensions by constructing generalized Ritter–Weiss modules $\nabla_{\Sigma}^{\Sigma'}(H)$ via a cohomological framework built from class formations and Tate sequences. It couples group-ring valued Hilbert modular forms, Hecke algebras, and associated Galois representations to realize the required congruences and to prove a precise Fitting-ideal formula for $\nabla_{\Sigma}^{\Sigma'}(H)_{p,-}$, namely $\mathrm{Fitt}_{\mathbf{Z}_p[G]_-}(\nabla_{\Sigma}^{\Sigma'}(H)_{p,-})=(\Theta_{\Sigma,\Sigma'}/2^t)$, yielding the Brumer–Stark conjecture and downstream consequences such as ETNC minus, Rubin’s higher rank Brumer–Stark, and related Stark-type conjectures. The work also develops a robust duality theory for class formations, providing a conceptual bridge between Galois cohomology and the arithmetic of class modules, and ensuring the constructions behave well under change of auxiliary sets and under profinite limits. Overall, the paper integrates analytic input from $L$-values with a purely algebraic framework to achieve integral results with wide-ranging consequences in Iwasawa theory and beyond.

Abstract

In this paper we give a complete proof of the Brumer-Stark conjecture over $\mathbf{Z}$.

Theorems & Definitions (95)

• Theorem 1.1: Brumer--Stark Conjecture
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• proof : Proof of Theorem \ref{['t:bs']}
• Lemma 2.5
• Lemma 2.6