Some New Maps and Ideals in Classical Iwasawa Theory with Applications

TL;DR

This paper develops new algebraic ideals in the Galois group-ring of abelian fields using Stark units and pairings, unifying aspects of Iwasawa theory and establishing connections with classical theorems and conjectures.

Contribution

It introduces novel ideals {} and {} in Iwasawa theory, linking Stark units, reciprocity, and Fitting ideals, and provides a new framework including a real analogue of Stickelberger's theorem.

Findings

Defined new ideals {} and {} in Galois group-rings.

Established explicit reciprocity relations involving Stark units.

Connected ideals to Fitting ideals and the Main Conjecture.

Abstract

We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a Z_p-tower of abelian fields and {\mathfrak J} is linked by explicit reciprocity to a third ideal {\mathfrak S} studied more generally in a previous work. This leads to a new and unifying framework for the Iwasawa Theory of such fields including a real analogue of Stickelberger's Theorem, links with certain Fitting ideals and \Lambda-torsion submodules, and a new exact sequence related to the Main Conjecture.