Duke Mathematical Journal

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

Takashi Hara

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Abstract

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) noncommutative p-extension F 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 Λ(G) and its canonical Ore localization Λ(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


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