Iwasawa theory and zeta elements for $\mathbb{G}_m$

TL;DR

This paper develops a higher rank Iwasawa theory for zeta elements related to the multiplicative group over abelian extensions of number fields, providing a new approach to the equivariant Tamagawa number conjecture and proving new cases for Tate motives.

Contribution

It introduces an explicit higher rank Iwasawa theory for zeta elements and applies it to establish new cases of the equivariant Tamagawa number conjecture for Tate motives.

Findings

New cases of the equivariant Tamagawa number conjecture proved.

Development of a concrete strategy for relating zeta elements to special values of p-adic L-functions.

Application to abelian CM extensions with trivial zeroes in p-adic L-functions.

Abstract

We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture. As a first application of this approach, we use it to prove new cases of the conjecture for Tate motives over natural families of abelian CM extensions of totally real fields for which the relevant $p$-adic $L$-functions possess trivial zeroes.