On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

Abstract

The equivariant Tamagawa number conjecture (hereinafter called the eTNC) predicts close relationships between algebraic and analytic aspects of motives. In this paper, we prove a lot of new cases of the minus component of the eTNC for G m and for CM abelian extensions. One of the main results states that the p-component of the eTNC is true when there exists at least one p-adic prime that is tamely ramified. The fundamental strategy is inspired by the work of Dasgupta and Kakde on the Brumer-Stark conjecture.