#### 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