Equivariant Iwasawa theory and non-abelian Stark-type conjectures

TL;DR

This paper explores three equivalent formulations of the equivariant Iwasawa main conjecture for totally real fields, establishing their equivalence and proving non-abelian versions of key conjectures under certain conditions.

Contribution

It introduces three equivalent formulations of the equivariant Iwasawa main conjecture and proves non-abelian generalizations of several important conjectures assuming w-invariant vanishes.

Findings

Proved equivalence of three formulations of the main conjecture.

Established non-abelian versions of Brumer's and Stark conjectures.

Demonstrated results under the condition that w-invariant is zero.

Abstract

We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawa's \mu-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that \mu = 0.