## Algebra & Number Theory

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

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 7 (2017), 1527-1571.

Dates
Revised: 1 March 2017
Accepted: 10 March 2017
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513096738

Digital Object Identifier
doi:10.2140/ant.2017.11.1527

Mathematical Reviews number (MathSciNet)
MR3697147

Zentralblatt MATH identifier
06775552

#### Citation

Burns, David; Kurihara, Masato; Sano, Takamichi. On Iwasawa theory, zeta elements for $\mathbb{G}_{m}$, and the equivariant Tamagawa number conjecture. Algebra Number Theory 11 (2017), no. 7, 1527--1571. doi:10.2140/ant.2017.11.1527. https://projecteuclid.org/euclid.ant/1513096738

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