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.