### 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 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.

Original language | English |
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Pages (from-to) | 247-305 |

Number of pages | 59 |

Journal | Duke Mathematical Journal |

Volume | 158 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2011 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)