On Iwasawa theory, zeta elements for 𝔾m, and the equivariant Tamagawa number conjecture

Burns, David; Kurihara, Masato; Sano, Takamichi

doi:10.2140/ant.2017.11.1527

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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

DOI: 10.2140/ant.2017.11.1527

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

Vol. 11, No. 7, 2017