The strong Stark conjecture for totally odd characters

Abstract

We prove the p p -part of the strong Stark conjecture for every totally odd character and every odd prime p p . Let L / K L/K be a finite Galois CM-extension with Galois group G G , which has an abelian Sylow p p -subgroup for an odd prime p p . We give an unconditional proof of the minus p p -part of the equivariant Tamagawa number conjecture for the pair ( h 0 ( Spec ⁡ ( L ) ) , Z [ G ] ) (h^0(\operatorname {Spec}(L)), \mathbb {Z}[G]) under certain restrictions on the ramification behavior in L / K L/K .