On descent theory and main conjectures in non-commutative Iwasawa theory

TL;DR

This paper establishes a Weierstrass Preparation Theorem and an explicit descent formalism for non-commutative Iwasawa algebras, linking the main conjecture to the equivariant Tamagawa number conjecture.

Contribution

It introduces new algebraic tools and formalism to connect key conjectures in non-commutative Iwasawa theory and Tamagawa numbers.

Findings

Proved a Weierstrass Preparation Theorem for non-commutative Iwasawa algebras.

Developed an explicit descent formalism in Whitehead groups.

Established a precise connection between the main conjecture and Tamagawa number conjecture.

Abstract

We prove a `Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of non-commutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture.