Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture

Abstract

Let $L/K$ be a finite Galois CM-extension of number fields with Galois group $G$. In an earlier paper, the author has defined a module $SKu(L/K)$ over the center of the group ring $\mathbb Z[G]$ which coincides with the Sinnott-Kurihara ideal if $G$ is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of $SKu(L/K)$ implies the minus part of the equivariant Tamagawa number conjecture at an odd prime $p$ for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above $p$, provided that Iwasawa's $\mu$-invariant vanishes. Here, we prove a relevant part of this integrality conjecture which enables us to deduce the equivariant Tamagawa number conjecture from the vanishing of $\mu$ for the same class of extensions. As an application we prove the non-abelian Brumer and Brumer-Stark conjecture outside the $2$-primary part for any monomial Galois extension of the rationals provided that certain $\mu$-invariants vanish.