Vol. 11, No. 7, 2017

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On Iwasawa theory, zeta elements for $\mathbb{G}_{m}$, and the equivariant Tamagawa number conjecture

David Burns, Masato Kurihara and Takamichi Sano

Vol. 11 (2017), No. 7, 1527–1571
Abstract

We develop an explicit “higher-rank” Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of number fields. We show this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture and, as a first application of this approach, we prove new cases of the conjecture over natural families of abelian CM-extensions of totally real fields for which the relevant p-adic L-functions possess trivial zeroes.

Keywords
Rubin–Stark conjecture, higher-rank Iwasawa main conjecture, equivariant Tamagawa number conjecture
Mathematical Subject Classification 2010
Primary: 11S40
Secondary: 11R23, 11R29, 11R42
Milestones
Received: 3 May 2016
Revised: 1 March 2017
Accepted: 10 March 2017
Published: 7 September 2017
Authors
David Burns
Department of Mathematics
King’s College London
London
United Kingdom
Masato Kurihara
Department of Mathematics
Keio University
Yokohama
Japan
Takamichi Sano
Department of Mathematics
Osaka City University
Osaka
Japan