Stark units and the main conjectures for totally real fields

Abstract

The main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed in an earlier paper also by the author, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this indeed is true. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) 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, assuming the truth of the Rubin-Stark conjecture and Leopoldt’s conjecture. This statement of local Iwasawa theory turns out to be interesting in its own right as it suggests a relation between solutions to p-adic and complex Stark conjectures.