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