Tate sequences and Fitting ideals of Iwasawa modules

C. Greither, M. Kurihara

Research output: Contribution to journalArticle

Abstract

We consider Abelian CM extensions L/k of a totally real field k, and we essentially determine the Fitting ideal of the dualized Iwasawa module studied by the second author in the case where only places above p ramify. In doing so we recover and generalize the results mentioned above. Remarkably, our explicit description of the Fitting ideal, apart from the contribution of the usual Stickelberger element Θ˙ at infinity, only depends on the group structure of the Galois group Gal(L/k) and not on the specific extension L. From our computation it is then easy to deduce that T˙Θ˙ is not in the Fitting ideal as soon as the p-part of Gal(L/k) is not cyclic. We need a lot of technical preparations: resolutions of the trivial module ℤ over a group ring, discussion of the minors of certain big matrices that arise in this context, and auxiliary results about the behavior of Fitting ideals in short exact sequences.

Original languageEnglish
Pages (from-to)941-965
Number of pages25
JournalSt. Petersburg Mathematical Journal
Volume27
Issue number6
DOIs
Publication statusPublished - 2016

Keywords

  • CM-fields
  • Class groups
  • Cohomology
  • Tate sequences
  • Totally real fields

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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