Stark units and main conjectures for totally real fields

TL;DR

This paper proves the lift of Stark unit Kolyvagin systems to the cyclotomic Iwasawa algebra for totally real fields, advancing the systematic study of Kolyvagin systems in higher core Selmer ranks and linking main Iwasawa conjectures to local Iwasawa theory.

Contribution

It constructs a Kolyvagin system over the Iwasawa algebra from Stark units, enabling a new approach to main conjectures in Iwasawa theory for totally real fields.

Findings

Lifted Kolyvagin systems over Iwasawa algebra for higher core Selmer ranks.

Reduced main Iwasawa conjectures to local Iwasawa theory statements.

Suggested a connection between p-adic and complex Stark conjectures.

Abstract

Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This statement, however, turns out to be interesting in its own right as it suggests a relation between solutions to $p$-adic and complex Stark conjectures.