A Main Conjecture in non-commutative Iwasawa theory

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.

arXiv commentAuthor's doctoral dissertation, 190 pages; typo "Novemberber" corrected on the first page