Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p

TL;DR

This paper constructs p-adic zeta functions for specific non-commutative p-extensions of totally real fields, using algebraic K-theory, congruences, and inductive methods, and applies these to a case of the Tamagawa number conjecture.

Contribution

It introduces an inductive method for constructing non-commutative p-adic zeta functions in new cases involving totally real fields.

Findings

Calculated Whitehead groups of Iwasawa algebras using Oliver-Taylor theory.

Reduced existence problem to congruences among abelian p-adic zeta pseudomeasures.

Verified congruences via Deligne-Ribet theory and inductive techniques.

Abstract

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet's theory and certain inductive technique. As an application we shall prove a special case of (the p-part of) the non-commutative equivariant Tamagawa number conjecture for critical Tate motives. The main results of this paper give generalisation of those of the preceding paper of the author.