  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 $\mathbb{G}_m$ 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.
