On twisted zeta-functions at s=0 and partial zeta-functions at s=1

Let K be an abelian extension of a totally real number field k, K+ its maximal real subfield and G = Gal(K/k). We have previously used twisted zeta-functions to define a meromorphic CG-valued function Phi(K/k)(s) in a way similar to the use of partial zeta-functions to define the better-known function Theta(K/k)(s). For each prime number p, we now show how the value Phi(K/k)(0) combines with a p-adic regulator of semilocal units to define a natural Z(p)G-submodule of Q(p)G which we denote S-K/k. If p is odd and splits in k, our main theorem states that S-K/k is (at least) contained in Z(p)G. Thanks to a precise relation between Phi(K/k)(1 - s) and Theta(K/k)(s), this theorem can be reformulated in terms of (the minus part of) Theta(K/k)(s) at s = 1, making it an analogue of Deligne-Ribet and Cassou-Nogues' well-known integrality result concerning Theta(K/k)(0). We also formulate some conjectures including a congruence involving Hilbert symbols that links S-K/k with the Rubin-Stark Conjecture for K+/k. (C) 2007 Elsevier Inc. All rights reserved.