A Main Conjecture in non-commutative Iwasawa theory

TL;DR

This paper introduces a new, more general equivariant Main Conjecture in non-commutative Iwasawa theory for number fields, removing previous restrictions and exploring its properties and implications.

Contribution

It formulates a novel Main Conjecture applicable to arbitrary one-dimensional p-adic Lie extensions, extending previous conjectures without requiring abelian or totally real extensions.

Findings

Proves independence of the conjecture from its parameters

Shows functorial properties of the conjecture

Deduces validity in several cases based on existing conjectures

Abstract

We formulate a new equivariant Main Conjecture in Iwasawa theory of number fields and study its properties. This is done for arbitrary one-dimensional $p$-adic Lie extensions $L_\infty/K$ containing the cyclotomic $\mathbb{Z}_p$-extension $K_\infty$ of the base field. As opposed to existing conjectures in the area, no requirement that $L_\infty/K$ be abelian or that $L_\infty$ be totally real is imposed. We prove the independence of the Main Conjecture of essentially all of its parameters and explore its functorial behaviour. It is furthermore shown that, to a large extent, this new conjecture generalises existing ones of Burns, Kurihara and Sano and Ritter and Weiss, which enables us to deduce its validity in several cases.