Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture

Abstract

Abstract Let L / K ${L/K}$ 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 ) ${\mathrm{SKu}(L/K)}$ over the center of the group ring ℤ ⁢ G ${\mathbb{Z}G}$ 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 ) ${\mathrm{SKu}(L/K)}$ 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 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 ℚ ${\mathbb{Q}}$ provided that certain μ-invariants vanish.