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

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 image-valued function ΦK/k(s) in a way similar to the use of partial zeta-functions to define the better-known function ΘK/k(s). For each prime number p, we now show how the value ΦK/k(0) combines with a p-adic regulator of semilocal units to define a natural image-submodule of image which we denote image. If p is odd and splits in k, our main theorem states that image is (at least) contained in image. Thanks to a precise relation between ΦK/k(1−s) and ΘK/k(s), this theorem can be reformulated in terms of (the minus part of) ΘK/k(s) at s=1, making it an analogue of Deligne–Ribet and Cassou-Noguèsʹ well-known integrality result concerning ΘK/k(0). We also formulate some conjectures including a congruence involving Hilbert symbols that links image with the Rubin–Stark Conjecture for K+/k.