On the equivariant Tamagawa number conjecture in tame CM-extensions

Abstract

Let L/K be a finite Galois CM-extension with Galois group G. The Equivariant Tamagawa Number Conjecture (ETNC) for the pair $${(h^0({\rm Spec} (L))(0), {\mathbb Z}G)}$$ naturally decomposes into p-parts, where p runs over all rational primes. If p is odd, these p-parts in turn decompose into a plus and a minus part. Let L/K be tame above p. We show that a certain ray class group of L defines an element in $${K_0({\mathbb Z}{_p}G_-, \mathbb Q_p)}$$ which is determined by a corresponding Stickelberger element if and only if the minus part of the ETNC at p holds. For this we use the Lifted Root Number Conjecture for small sets of places which is equivalent to the ETNC in the number field case. For abelian G, we show that the minus part of the ETNC at p implies the Strong Brumer–Stark Conjecture at p. We prove the minus part of the ETNC at p for almost all primes p.