Stickelberger elements and Kolyvagin systems

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.