Cohomology of Units and L-values at Zero

Let K/k be a finite Galois extension of number fields with Galois group G, and let S be a finite G-stable set of primes of K containing all archimedean primes. We denote the G-module of S-units of K by E = ES and let ∆S be the kernel of the augmentation map ZS → Z which sends each basis element p ∈ S to 1. We are concerned with invariants of K/k which are associated to a G-homomorphism φ : ∆S → E inducing Q⊗∆S ' Q⊗ E. These invariants were defined by Tate [Ta2] and Chinburg [Ch1] when S is large, i.e. when S contains all ramified primes of K/k and the S-class group cl = clS of K is trivial. There are two of them and each is a function of the complex characters χ ofG. The first, Aφ, hasAφ(χ) equal to the Tate regulator at χ divided by the leading coefficient c(χ) of the Taylor expansion at s = 0 of the Artin L-function L(s, χ) with the Euler factors at primes of S omitted. According to Stark’s conjecture [St] in Tate’s form, Aφ takes algebraic values which satisfy Aφ(χ ) = Aφ(χ) σ