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

Abstract

Abstract. We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension F∞ of a totally real number field F such that the finite part of its Galois group G is a p-group with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra Λ(G) and its canonical Ore localisation Λ(G)S 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.

The main conjecture of Iwasawa theory for totally real fields

“…Hence we obtain an algebraic number L(ρ, −n), "the value of complex L-function associated to ρ at −n". For details see section 1.2 in Coates-Lichtenbaum [7] (when ρ is one dimensional see equation (15) below in section 6.3. For ρ of higher dimension one has to use Brauer's induction theorem).…”

“…It uses the integral logarithm of Oliver and Taylor. This strategy was used by the author to prove such algebraic theorem for pro-p meta-abelian groups in [19] and by Hara for the groups of the form H × Z p , where H is a finite group of exponent p, in [15]. The author believes that results in sections 4 and 5 will be useful in handling noncommutative main conjectures for other motives.…”