TY - JOUR

T1 - Inductive construction of the p-adic zeta functions for noncommutative p-extensions of exponent p of totally real fields

AU - Hara, Takashi

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/6/1

Y1 - 2011/6/1

N2 - 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 groupGis a p-group of exponent p. We first calculate theWhitehead 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.

AB - 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 groupGis a p-group of exponent p. We first calculate theWhitehead 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.

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U2 - 10.1215/00127094-1334013

DO - 10.1215/00127094-1334013

M3 - Article

AN - SCOPUS:79959902068

VL - 158

SP - 247

EP - 305

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -