Stickelberger elements and Kolyvagin systems

TL;DR

This paper constructs Kolyvagin systems from Stickelberger elements to prove Iwasawa main conjectures for CM fields, offering a new approach that links Stickelberger and Rubin-Stark elements and extends Kolyvagin system techniques.

Contribution

It introduces a novel method to build Kolyvagin systems from Stickelberger elements and applies this to prove key Iwasawa theory conjectures for CM fields.

Findings

Proved the main conjecture of Iwasawa theory for totally real fields.

Established a link between Stickelberger and Rubin-Stark elements.

Provided a new approach to study Kolyvagin systems of core rank r > 1.

Abstract

In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Stark elements) on how to study Kolyvagin systems of core rank r > 1 (in the sense of Mazur and Rubin). Also, by making use of the 'rigidity' of the collection of Kolyvagin systems, we establish a link between the Stickelberger elements and the Rubin-Stark elements.