Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture.

Let be a finite Galois CM-extension of number fields with Galois group G. In an earlier paper, the author has defined a module over the center of the group ring 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 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 μ-invariant vanishes. Here, we prove a relevant part of this integrality conjecture which enables us to deduce the minus- p-part of the equivariant Tamagawa number conjecture from the vanishing of μ 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 every monomial Galois extension of provided that certain μ-invariants vanish. [ABSTRACT FROM AUTHOR]