Equivariant Local Epsilon Constants and Etale Cohomology

Let L/K be a Galois extension of number fields with Galois group G. For each character ofG the functional equation of the Artin L-function gives rise to an epsilon constant. These epsilon constants are conjecturally related to natural algebraic invariants. For tamely ramified extensions such a relation was conjectured by Fröhlich and proved by Taylor [10]. A generalisation of Fröhlich’s conjecture to wildly ramified extensions is given by Chinburg’s Ω(2)-conjecture [9] of which so far only special cases are known to be valid. Chinburg’s conjecture was recently refined by Bley and Burns [2] who conjectured that in the relative algebraicK-group K0(Z[G],R) the equivariant global epsilon constant of L/K is equal to the sum of an equivariant discriminant and certain terms coming from étale cohomology. It is shown in [4] that this conjecture of Bley and Burns fits into the general framework of the equivariant Tamagawa number conjecture [8]. In this paper we formulate a corresponding local conjecture. Let L/K be a Galois extension of p-adic fields and G = Gal(L/K). We define an invariant RL/K in the relative algebraic K-group K0(Zp[G],Qp) which incorporates the equivariant local epsilon constant of L/K and a natural algebraic invariant coming from the étale cohomology of the sheaf Zp(1) on Spec(L). We conjecture that RL/K = 0. This local conjecture and the global conjecture of Bley and Burns satisfy many similar properties, and indeed these conjectures are closely linked. Before we can give the precise statement we must introduce some notation. Again let L/K be a Galois extension of number fields and G = Gal(L/K). For a prime number p let Sp(K) be the set of places of K above p and for each v ∈ Sp(K) fix a place w of L above v. We identify the Galois group of the completion Lw/Kv with the decomposition group Gw of w and write iGGw :K0(Zp[Gw],Q c p)−→K0(Zp[G],Qp) for the induction map of the relative algebraic K-groups. We will show that the p-part of the global conjecture for the extension L/K is valid if and only if ∑