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

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.

💡 Research Summary

The paper addresses the long‑standing problem of constructing a non‑commutative p‑adic zeta function for a one‑dimensional p‑adic Lie extension L/K of a totally real number field K, where the finite part G₀ of the Galois group G = Gal(L/K) is a p‑group of exponent p. The author proceeds in four major stages.

First, the algebraic K‑theory of the Iwasawa algebra Λ(G)=ℤₚ