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

TL;DR

This paper provides an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields in certain $p$-adic Lie extensions, leading to new results in number theory without relying on $$-invariant assumptions.

Contribution

It offers the first unconditional proof of the conjecture in this setting, independent of $$-invariant vanishing, and applies it to prove related conjectures.

Findings

Proves the equivariant Iwasawa main conjecture unconditionally for specific extensions.

Deduces the Coates-Sinnott conjecture away from 2-primary parts.

Establishes new cases of the equivariant Tamagawa number conjecture.

Abstract

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 $\mu$-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.