## Duke Mathematical Journal

- Duke Math. J.
- Volume 158, Number 2 (2011), 247-305.

### Inductive construction of the $p$-adic zeta functions for noncommutative $p$-extensions of exponent $p$ of totally real fields

#### Abstract

We construct the $p$-adic zeta function for a one-dimensional (as a $p$-adic Lie extension) noncommutative $p$-extension ${F}_{\infty}$ of a totally real number field $F$ such that the finite part of its Galois group $G$ is a $p$-group of exponent $p$. We first calculate the Whitehead groups of the Iwasawa algebra $\Lambda (G)$ and its canonical Ore localization $\Lambda (G{)}_{S}$ by using Oliver and Taylor's theory of integral logarithms. This calculation reduces the existence of the noncommutative $p$-adic zeta function to certain congruences between abelian $p$-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne and Ribet's theory and a certain inductive technique. As an application we prove a special case of (the $p$-part of) the noncommutative equivariant Tamagawa number conjecture for critical Tate motives.

#### Article information

**Source**

Duke Math. J., Volume 158, Number 2 (2011), 247-305.

**Dates**

First available in Project Euclid: 31 May 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1306847523

**Digital Object Identifier**

doi:10.1215/00127094-1334013

**Mathematical Reviews number (MathSciNet)**

MR2805070

**Zentralblatt MATH identifier**

1238.11100

**Subjects**

Primary: 11R23: Iwasawa theory

Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R80: Totally real fields [See also 12J15] 19B28: $K_1$of group rings and orders [See also 57Q10]

#### Citation

Hara, Takashi. Inductive construction of the $p$ -adic zeta functions for noncommutative $p$ -extensions of exponent $p$ of totally real fields. Duke Math. J. 158 (2011), no. 2, 247--305. doi:10.1215/00127094-1334013. https://projecteuclid.org/euclid.dmj/1306847523