Fitting invariants in equivariant Iwasawa theory

TL;DR

This paper extends the algebraic framework of Fitting invariants in equivariant Iwasawa theory, providing new noncommutative and multi-variable results that deepen understanding of Iwasawa modules and $p$-adic $L$-functions.

Contribution

It develops the theory of shifts of Fitting invariants and applies it to generalize Greither and Kurihara's work to noncommutative and two-variable contexts.

Findings

Established a noncommutative version of Fitting invariants.

Extended the theory to two-variable Iwasawa modules.

Provided explicit descriptions of Fitting ideals in new settings.

Abstract

The main conjectures in Iwasawa theory predict the relationship between the Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved formulation of the main conjecture, Greither and Kurihara described explicitly the (initial) Fitting ideals of the Iwasawa modules for the cyclotomic $\mathbb{Z}_p$-extensions of finite abelian extensions of totally real fields. In this paper, we generalize the algebraic theory behind their work by developing the theory of ``shifts of Fitting invariants.'' As applications to Iwasawa theory, we obtain a noncommutative version and a two-variable version of the work of Greither and Kurihara.