An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications

Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for every admissible one-dimensional p-adic Lie extension whose Galois group has an abelian Sylow p-subgroup. Crucially, this result does not depend on the vanishing of any μ-invariant. As applications, we deduce the Coates--Sinnott conjecture away from its $2$-primary part and new cases of the equivariant Tamagawa number conjecture for Tate motives.