On the Values of Equivariant Zeta Functions of Curves over Finite Fields

Let K/k be a finite Galois extension of global function fields of characteristic p. Let CK denote the smooth projective curve that has function field K and set G := Gal(K/k). We conjecture a formula for the leading term in the Taylor expansion at zero of the G-equivariant truncated Artin L-functions of K/k in terms of the Weil-´ etale cohomology of Gm on the corresponding open subschemes of CK. We then prove the l-primary component of this conjecture for all primes l for which either l 6 p or the relative algebraic K-group K0(Zl(G), Ql) is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for K/k: if p ∤ |G|, then refined versions of all of Chinburg's '­-Conjectures' in Galois module theory are valid; if the torsion subgroup of K × is a cohomologically-trivial G-module, then Gross's conjectural 'refined class number formula' is valid; if K/k satisfies a certain natural class- field theoretical condition, then Tate's recent refinement of Gross's conjecture is valid.


Introduction
Let K/k be a finite Galois extension of global function fields of characteristic p.Let C K be the unique geometrically irreducible smooth projective curve which has function field equal to K and set G := Gal(K/k).For each finite non-empty set S of places of k that contains all places which ramify in K/k, we write O K,S for the subring of K consisting of those elements that are integral at all places of K which do not lie above an element of S and we set U K,S := Spec(O K,S ).With R denoting either Z or Z ℓ for some prime ℓ and E an extension field of the field of fractions of R, we write K 0 (R[G], E) for the relative algebraic K-group defined by Swan in [46].
In §2 we formulate a conjectural equality C(K/k) between an element of K 0 (Z[G], R) constructed from the leading term in the Taylor expansion at s = 0 of the G-equivariant Artin L-function of U K,S and the refined Euler characteristic of a pair comprising the Weil-étale cohomology of G m on U K,S (considered as an object of an appropriate derived category) and a natural logarithmic regulator mapping.This conjecture is motivated both by the general approach described by Lichtenbaum in [40, §8] and also by analogy to a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (which was formulated by Flach and the present author in [13]).The equality C(K/k) can be naturally reinterpreted as a conjectural equality in K 0 (Z[G], Q) involving the leading term at t = 1 of the G-equivariant Zetafunction of U K,S and in §3 we shall prove the validity, resp.the validity modulo torsion, of the projection of the latter conjectural equality to K 0 (Z ℓ [G], Q ℓ ) for all primes ℓ = p, resp.for ℓ = p (this is Theorem 3.1).If ℓ = p, then our proof combines Grothendieck's formula for the Zeta-function in terms of the action of frobenius on ℓ-adic cohomology together with a non-commutative generalisation of a purely algebraic observation of Kato in [35] (this result may itself be of some independent interest) and an explicit computation of certain Bockstein homomorphisms in ℓ-adic cohomology.In the case that ℓ = p we are able to deduce our result from Bae's verification of the 'Strong-Stark Conjecture' [3] which in turn relies upon results of Milne [43] concerning relations between Zeta-functions and p-adic cohomology.In the remainder of the manuscript we show that C(K/k) provides a universal approach to the study of several well known conjectures.A key ingredient in all of our results in this direction is a previous observation of Flach and the present author which allows an interpretation in terms of Weil-étale cohomology of the canonical extension classes defined using class field theory by Tate in [49].In §4 we consider connections between C(K/k) and the central conjectures of classical Galois module theory.To be specific, we prove that C(K/k) implies the validity for K/k of a strong refinement of the 'Ω(3)-Conjecture' formulated by Chinburg in [18, §4.2].Taken in conjunction with Theorem 3.1 this result allows us to deduce that if K 0 (Z p [G], Q p ) is torsion-free, resp.p ∤ |G|, then the Ω(3)-Conjecture, resp. the Ω(1)-, Ω(2)-and Ω(3)-Conjectures, formulated by Chinburg in loc.cit., are valid for K/k.This is a strong refinement of previous results in this area.We assume henceforth that G is abelian.In this case Gross has conjectured a 'refined class number formula' which expresses an explicit congruence relation between the values at s = 0 of the Dirichlet L-functions associated to K/k [31].This conjecture has attracted much attention and indeed Tate has recently formulated a strong refinement in the case that G is cyclic [51].However, whilst the conjecture of Gross has already been verified in several interesting cases [31,1,47,37,39], much of this evidence is obtained either by careful analysis of special cases or by induction on |G| and, as yet, no coherent overview of or systematic approach to these conjectures of Gross and Tate has emerged.In contrast, in §5 we shall use the general approach of algebraic height pairings developed by Nekovář in [44] to interpret the integral regulator mapping of Gross as a Bockstein homomorphism in Weil-étale cohomology, and we shall also apply this interpretation to prove that if the torsion subgroup µ K of K × is a cohomologically-trivial G-module (a condition that is automatically satisfied if, for example, |µ K | is coprime to |G|), then C(K/k) implies the validity of Gross's conjecture for K/k.Under a certain natural class-field theoretical assumption on K/k we shall also show (in §6) that C(K/k) implies the validity of Tate's refinement of Gross's conjecture.When combined with Theorem 3.1 (and earlier results of Tan concerning p-extensions) these observations allow us to deduce the validity of Gross's conjecture for all extensions K/k for which µ K is a cohomologically-trivial G-module and also to prove the validity of Tate's refinement of Gross's conjecture for a large family of extensions.A further development of the approach used here should allow the removal of any hypothesis on µ K (indeed, in special cases this is already achieved in the present manuscript).However, even at this stage, our results constitute a strong improvement of previous results in this area and also provide a philosophical underpinning to the conjectures of Gross and Tate that was not hitherto apparent.Indeed, the approach presented here leads to the formulation of natural analogues of these conjectures concerning the values of (higher order) derivatives of L-functions that vanish at s = 0.These developments have in turn led to a proof of Tate's conjecture under the hypothesis only that |µ K | is coprime to |G| and have also provided new insight into Gross's 'refined p-adic abelian Stark conjecture' as well as several other conjectures due, for example, to Rubin, to Darmon, to Popescu and to Tan.For more details of this aspect of the theory the reader is referred to [10,34].
Acknowledgements.The author is very grateful to J. Tate and B. H. Gross for their encouragement concerning this project and for their hospitality during his visits to the Universities of Texas at Austin and Harvard respectively.In addition, he is most grateful to M. Kurihara for his hospitality during the author's visit to the Tokyo Metropolitan University, where a portion of this project was completed.The author is also grateful to J. Nekovář for a number of very helpful discussions.
2. The leading term conjecture 2.1.Relative Algebraic K-Theory.In this subsection we quickly recall certain useful constructions in algebraic K-theory.If Λ is any ring, then all modules are to be understood as left modules.We write ζ(Λ) for the centre of Λ, K 1 (Λ) for the Whitehead group of Λ and K 0 (Λ) for the Grothendieck group of the category of finitely generated projective Λ-modules.We also write D(Λ) for the derived category of complexes of Λ-modules with only finitely many non-zero cohomology groups, and we let D fpd (Λ), resp.D perf (Λ),denote the full triangulated subcategory of D(Λ) consisting of those complexes that are quasi-isomorphic to a bounded complex of projective Λ-modules, resp. to a bounded complex of finitely generated projective Λ-modules.We let R denote either Z or Z ℓ for some prime ℓ, E and F denote extension fields of the field of fractions of R and we fix a finite group G.For finitely generated E[G]-modules V and W we write Is E[G] (V, W ) for the set of E[G]-module isomorphisms from V to W .The relative algebraic K-group K 0 (R[G], E) is an abelian group with generators (X, φ, Y ), where X, Y are finitely generated projective R[G]-modules and φ is an element of Is E[G] (X ⊗ R E, Y ⊗ R E).For the defining relations we refer to [46, p. 215].We systematically use the following facts: there is a long exact sequence of relative K-theory (cf.[46,Th. 15.5]) and the product of these homomorphisms over all primes ℓ induces an isomorphism (cf. the discussion following [26, (49.12) Let A be a finite dimensional central simple F -algebra, F ′ an extension of F which splits A and e an indecomposable idempotent of A ⊗ F F ′ .If V is any finitely generated A-module and φ ∈ End A (V ), then the 'reduced determinant' of φ is defined by setting detred A (φ) := det F ′ (φ ⊗ F id F ′ |e(V ⊗ F F ′ )).This is an element of F which is independent of the choices of F ′ and e.This construction extends to finite-dimensional semi-simple F -algebras in the obvious way.In particular, the group is bijective and so there exists a unique homomorphism . (When we need to be more precise we write δ G,ℓ rather than δ ℓ .)The map nr R[G] is not in general surjective, but nevertheless there exists a canonical 'extended boundary' homomorphism δ : [13,Lem. 9]).

The map nr E[G] induces an equivalence relation '∼' on each set Is
In the sequel we shall often not distinguish between an element of Is E[G] (V, W ) and its associated equivalence class in Is E[G] (V, W )/ ∼.For each Z-graded module C • we write C all , C − and C + for the direct sum of C i as i ranges over all, all odd and all even integers respectively.

Documenta Mathematica 9 (2004) 357-399
An 'E-trivialisation' of an object In [9] it is shown that a variant of the classical construction of Whitehead torsion allows one to associate to each such pair (C ).Further details of this construction are recalled in the Appendix.
In the sequel we shall use the following notation and conventions.We abbreviate 'cohomologically-trivial' to 'c-t', 'χ ℓ ' when we need to be more precise); if X is any scheme of finite type over the finite field F p of cardinality p and F is any étale (pro-) sheaf, resp.Weil-étale sheaf, on X, then we abbreviate RΓ(X ét , F), resp.RΓ(X Weil−ét , F) to RΓ(X, F), resp.RΓ W (X, F), and we also use similar abbreviations on cohomology; for any commutative ring Λ we write x → x # for the Λ-linear involution of ζ(Λ[G]) that is induced by setting g # := g −1 for each g ∈ G; for any group H and any H-module M we write M H , resp.M H , for the maximal submodule, resp.quotient, of M upon which H acts trivially; for any abelian group A we let A tors denote its torsion subgroup; unless explicitly indicated otherwise, all tensor products and exterior powers are to be considered as taken in the category of abelian groups.
2.2.Formulation of the conjecture.We assume henceforth that S is a finite non-empty set of places of k containing all places which ramify in K/k.We let Irr C (G) denote the set of irreducible finite dimensional complex characters of G.The leading term θ * K/k,S (0) in the Taylor expansion of θ K/k,S (s) at s = 0 is equal to (L * S (χ, 0)) χ∈Irr(G) and hence belongs to ζ(R[G]) × .In this subsection we follow the philosophy introduced by Lichtenbaum in [40] to formulate a conjectural description of δ(θ * K/k,S (0) # ) in terms of Weil-étale cohomology.For any intermediate field F of K/k we write Y F,S for the free abelian group on the set of places S(F ) of F which lie above those in S and X F,S for the kernel of the homomorphism Y F,S → Z that sends each element of S(F ) to 1.We write O F,S for the ring of S(F )-integers in F and O × F,S for its unit group.We also set U F,S := Spec(O F,S ) and A F,S := Pic(O F,S ).
ii) There exists a natural distinguished triangle in D(Z[G]) of the form where the map induced on cohomology (in degree 2) by the first morphism is the composite of the projection ) that is acyclic outside degrees 0 and 1.One has a canonical identification With respect to the descriptions of cohomology given in iii) (for both K and K J ) the displayed isomorphism induces the natural identification ) J and also identifies X K J ,S with a submodule of X K,S by means of the map that sends each place v of S(K J ) to j∈J j(w) where w is any place of K lying above v.
Since each cohomology group of RΓ W (U K,S , G m ) is finitely generated, a standard argument of homological algebra shows that this complex belongs to [11,proof of Prop. 1.20,Steps 3 and 4]).On the other hand, any G-module that is c-t has finite projective dimension as a Z[G]-module and so it suffices to show that RΓ W (U K,S , G m ) is isomorphic to a bounded complex of G-modules which are each c-t.Now the G-module X K,S ⊗ Q is c-t and so the distinguished triangle of claim ii) implies that we need only prove that RΓ(U K,S , G m ) is isomorphic to a bounded complex of G-modules which are each c-t.But this is true because the natural morphism π :

20, Steps 1 and 2]).
Claim iv) follows from the triangle of claim ii) and the description of cohomology given in iii) (for both K and K J ) together with an explicit computation of the maps induced on cohomology by the natural isomorphism ) (for more details as to the latter see, for example, the proof of [12,Lem. 12]).

Documenta Mathematica 9 (2004) 357-399
For each place w of K we let | • | w denote the absolute value of w normalised as in [50,Chap. 0 We also denote by R K,S the R-trivialisation of RΓ W (U K,S , G m ) that is induced by R K,S and the descriptions of Lemma 1iii).
We can now state the central conjecture of this paper.
Conjecture C(K/k): ). Remark 1. Lemma 1i) shows that C(K/k) can be naturally rephrased in terms of RΓ W (C K , j !Z).We have chosen to work in terms of G m rather than j !Z for the purposes of explicit computations that we make in subsequent sections (see also Remark 3 in this regard).
where the top row is as in Lemma 1ii), Ô× K,S denotes the profinite completion of O × K,S and the second column is a distinguished triangle.This diagram implies that RΓ W (U K,S , G m ) is a precise analogue of the complex Ψ S that occurs in [12,Rem.following Prop.3.1] and [7,Prop. 2.1.1].For this reason, C(K/k) is an analogue of the conjectural vanishing of the element T Ω(K/k, 0) defined in [7,Th. 2.1.2],where K/k is a Galois extension of number fields of group G, and also coincides in the abelian case with the function field case of [8,Conj. 2.1].We recall that the vanishing of the element T Ω(K/k, 0) is equivalent to the validity of the 'Lifted Root Number Conjecture' of Gruenberg, Ritter and Weiss [33] (see [7,Th. 2.3.3] for a proof of this fact) and also to the validity of the 'Equivariant Tamagawa Number Conjecture' of [13,Conj. 4(iv)] as applied to the pair (h 0 (Spec K), Z[G]) where h 0 (Spec K) is considered as a motive that is defined over k and has coefficients Q[G] (see [7,Th. 2.4.1] or [14, §3] for different proofs of this fact).We further recall that [13,Conj. 4(iv)] is itself a natural equivariant version of the seminal conjecture of Bloch and Kato from [6], and that if G is abelian, then it refines the 'Generalized Iwasawa Main Conjecture' formulated by Kato in [35, §3.2] (cf.[14, §2] in this regard).Finally we recall that strong evidence in favour of [13,Conj. 4(iv)] has recently been obtained in [15,16].
By a change of variable we now remove all of the transcendental terms which occur in C(K/k) and then decompose the conjecture according to (1).To do this we set t := p −s and then define a ζ(C[G])-valued function of the complex variable t by means of the equality Z K/k,S (t) := θ K/k,S (s).For each place w ∈ S(K) we write val w and k(w) for its valuation and residue field and let deg(w) denote the degree of the field extension k(w)/F p .We write D K,S : O × K,S → X K,S for the homomorphism which at each u )) → Z denote the algebraic order of Z K/k,S (t) at t = 1 (which we regard as an element of Z π0(Spec(ζ(R[G]))) in the natural way).Then the element Proof.The algebraic order of θ K/k,S (s) # at s = 0 is equal to e.In addition, by an explicit computation one verifies that The validity of this equality follows directly from [7, Prop.1.2.1(ii)] in conjunction with the equality R K,S (u) = log(p) ).In a similar way, if J is any normal subgroup of G, then Lemma 1iv) implies that the validity of the image of the equality of C ℓ (K/k) under the natural coinflation map

Evidence
In this section we shall provide the following evidence in support of C(K/k).
Theorem 3.1.Let K/k be a finite Galois extension of global function fields of characteristic p and set G := Gal(K/k).i) Proof.Clear.
3.1.The descent formalism.In this subsection we prepare for the proof of Theorem 3.1i) by proving a purely algebraic result.This provides a natural generalisation of several results that have already been used elsewhere (cf.Remark 6) and so the material of this subsection may well itself be of some independent interest.We fix an arbitrary rational prime ℓ and for each Z ℓ -module M we set We say that an endomorphism ψ of a finitely generated Z ℓ [G]-module M is 'semi-simple at 0' if the natural composite homomorphism has both finite kernel and finite cokernel.We note that this condition is satisfied if and only if there exists a Let t be an indeterminate.Then for any element )) → Z for the algebraic order of f (t) at t = 1.We identify e f with an element of ) in the natural way and then set In particular, if θ is any endomorphism of a finitely generated Z ℓ [G]-module M for which 1 − θ is semi-simple at 0 and We now suppose given a bounded complex of finitely generated projective We let C(θ) • denote the −1-shift of the mapping cone of the endomorphism of P • induced by 1 − θ.Then from the long exact sequence of cohomology that is associated to the distinguished triangle one obtains in each degree i a short exact sequence Upon combining these sequences with the isomorphisms Proof.We shall argue by induction on the quantity We first assume that |P • | = 0 so that P • has only one non-zero term.To be specific, we assume that where the first term is placed in degree n.In addition, upon choosing a Q ℓ , and using (4) to identify the trivialisation τ θ is induced by the identity map on cohomology.Hence, from Lemma A1, one has ), as required.We now assume that |P • | = n and, to fix notation, that min{j : P j = 0} = 0. We also assume that the claimed formula is true for any pair of the form (Q for the boundaries, cycles and differential in degree i.If necessary, we use the argument of [25,Lem. 7.10] to change θ by a homotopy in order to ensure that, in each degree i, the restriction of 1 − θ i to B i (P • ) induces an automorphism of B i (P • ) Q ℓ .We shall make frequent use of this assumption (without explicit comment) in the remainder of this argument.We henceforth let Q • denote the naive truncation in degree n − 1 of P • (so From the associated long exact cohomology sequence we deduce that H i (Q • ) = H i (P • ) if i < n − 1 and that there are commutative diagrams of short exact sequences of the form ) respectively which are each semi-simple at 0 in all degrees i.

David Burns
We set In addition, since |P n [−n]| = 0, our earlier argument proves that From the commutative diagrams displayed above, one also has Upon combining the last four displayed formulas we obtain an equality and so the claimed result will follow if we can show that Before discussing the proof of this equality we introduce some convenient notation: for any Z ℓ -module A we set A := A ⊗ Z ℓ Q ℓ , and we use similar abbreviations for both complexes and morphisms of Z ℓ -modules.For any complex A we also set The key to proving (6) is the observation (which is itself straightforward to verify directly) that one can choose elements κ 1 , κ 2 and κ 3 of τ θ n [−n] , τ θ and τ φ respectively which together lie in a commutative diagram of short exact sequences of the form (7) Indeed, the equality (6) follows directly upon combining such a diagram with the exact sequence ( 5) and the result of [9, Th. 2.8].However, for the convenience of the reader, we also now indicate a more direct argument.
After taking account of the construction of χ ℓ (•, •) given in the Appendix (the notation of which we now assume) and the definitions of where B C denotes B all (C), and similarly for B D and B E , α ′ : B C → B D and β ′ : B D → B E are the natural homomorphisms that are induced by α and β respectively, κ 1 , κ 2 and κ 3 are as in ( 7) and all unlabelled vertical maps are the isomorphisms induced by a choice of sections to each of the natural homomorphisms and τ φ (E) respectively, and so the commutativity of the diagram formed by the first and fourth rows combines with the exactness of the sequences [46, p. 415] to imply the required equality (6).Thus, upon noting that the rows of this diagram are all exact (the second and third as a consequence of the exactness of the rows of ( 7)), it is enough to prove that sections of the above form can be chosen in such a way that the top and bottom two squares of the diagram commute, and this in turn can be proved by a straightforward and explicit exercise using the following facts.After choosing where S n−1, * denotes the set of elements (π, π ′ ) where π runs over S n−1 and π ′ denotes the unique element of B n (P Remark 6.There are two special cases in which the formula of Proposition 3.1 has already been proved: if is automatically semi-simple at 0 in each degree i and the given formula has been proved by Greither and the present author in [16, 3),( 5)] and [13, Rem.9].) 3.2.Zeta functions of varieties.In this subsection we fix a prime ℓ that is distinct from p.We also fix an algebraic closure F c p of F p , we set Γ := Gal(F c p /F p ) and we write σ for the (arithmetic) Frobenius element in Γ.For any scheme X over F p we write X c for the associated scheme F c p × Fp X over F c p .We let J be a finite group, X and Y separated schemes of finite type over F p and π : X → Y an étale morphism that is Galois of group J.For each ℓ-adic sheaf G on Y ét we follow the approach of Deligne [27, Rem.2.12] to define a J-equivariant Zeta function by setting where y runs over the set of closed points of Y , f y denotes the arithmetic Frobenius of y, deg(y) the degree of y and subscript y denotes taking stalk at a geometric point over y.
We now combine the algebraic approach of the previous subsection with a well known result of Grothendieck from [32] to describe, for each integer r, the image of the leading term To this end we observe that π * is exact and hence that, for each sheaf G as above, there is a natural isomorphism This implies that if G is any étale (pro-)sheaf of finitely generated Z ℓ -modules on Y and we set F := π * G, then the complexes RΓ(X, F) and RΓ(X c , F) both belong to D perf (Z ℓ [J]) (cf.[29, Th. 5.1]).We may therefore fix a bounded complex of finitely generated projective Z ℓ [J]-modules C • for which there exists an isomorphism α : ) and a Z ℓ [J]-endomorphism θ of C • that induces the action of σ on RΓ(X c , F) (the existence of such a θ follows from [41, Chap.VI, Lem.8.17] -but note that the map ψ in loc.cit.need not, in general, be a quasi-isomorphism).In this way we obtain a where the lower row denotes the natural distinguished triangle.Taken in conjunction with the Octahedral axiom, this diagram implies the existence of an isomorphism ).Further, the hypothesis that the composite (4) with ψ = H i (1−θ) and M = H i (C • ) has both finite kernel and finite cokernel is equivalent to the hypothesis that σ acts 'semi-simply' on the space H i (X c , F) ⊗ Z ℓ Q ℓ and is therefore expected to be true under some very general conditions [35,Rem. 3.5.4].In this context, and in terms of the notation of Lemma A2, we write τ X,F ,σ for the Q ℓ -trivialisation of RΓ(X, F) which is equal to (τ θ ) α ′ where τ θ is the Q ℓ -trivialisation of C(θ) • that is defined just prior to Proposition 3.1 (with Remark 7. The trivialisation τ X,F ,σ defined above has an alternative description.To explain this we let C(F) • denote the complex where H 0 (X, F) occurs in degree 0 and κ denotes cup-product with the element of H 1 (X, Z ℓ ) obtained by pulling back the element φ p of H 1 (Spec(F p ), Z ℓ ) = Hom cont (Γ, Z ℓ ) which sends σ to 1. Then the complex C(F) where the first and third maps are induced by the long exact sequence of cohomology associated to the lower row of (8) and the second map is as in (4).Indeed, this equality is a consequence of the description of κ on the level of complexes that is given by Rapaport and Zink in [45, 1.2] (cf.[43, Prop.6.5] and [35, §3.5.2] in this regard).These equalities imply in turn that τ X,F ,σ coincides with the Q ℓ -trivialisation of RΓ(X, F) that is induced by the acyclicity of We now state the main result of this subsection.Theorem 3.2.Let π : X → Y be a finite étale morphism of separated schemes of dimension d over F p .If π is Galois of group J and r is any integer for which σ acts semi-simply on ). Proof.We set r ′ := d−r and make a choice of morphisms θ and α as in diagram (8) with F = Z ℓ (r ′ ).Upon applying Lemma A2 to the induced isomorphism )) and then Proposition 3.1 with For each integer i we set , where subscript 'c' denotes cohomology with compact support.Then, by Poincaré Duality (cf.[41, Chap.VI, Cor.11.2]), in each degree i one has an isomorphism of This isomorphism respects the action of Frobenius in the sense that the action of σ on H i (X c , Q ℓ (r ′ )) corresponds to the inverse of the action of σ that is induced on Hom Q ℓ (V 2d−i , Q ℓ ) by its natural action on V 2d−i (since the linear duality functor is contravariant).Hence one has and the second equality is valid because J acts contragrediently on [7, (2.0.5)]).From the above formula one therefore has To complete the proof it is thus sufficient to observe that, by Grothendieck [32], one has an equality of functions of the complex variable Indeed, the exposition of [41, Chap.VI, proof of Th. 13.3] proves just such an equality with Q ℓ [J] replaced by an arbitrary finite degree field extension Ω of Q ℓ and π * π * Z ℓ (r) ⊗ Z ℓ Q ℓ by any constructible sheaf of vector spaces over Ω, and the last displayed equality can be verified by reduction to such cases since both sides are defined via Galois descent (cf.[27, Rem.2.12]).

3.3.
The case ℓ = p.In this subsection we deduce Theorem 3.1i) from a special case of Theorem 3.2.
To this end we first reinterpret C ℓ (K/k) in the style of Theorem 3.2.We note that the isomorphism ι ℓ constructed in the following result is as predicted by [30,Conj. 7.2] (with X = U K,S and n = 1).
Lemma 3.There exists a natural isomorphism in D perf (Z ℓ [G]) of the form Proof.Following Lemma 1iii) we fix a bounded complex of finitely generated projective Z[G]-modules ) we may also fix a bounded complex of finitely generated projective For each natural number n we consider the following diagram The first two rows of this diagram are the distinguished triangles that are induced by Lemma 1ii) and the isomorphism In addition, all columns of the diagram are distinguished triangles: the first is obviously so, the second is the triangle which is induced by the exact sequence of étale sheaves 1 → 0 in each degree i, and the third column is the distinguished triangle which is induced by the exact sequence of modules 0 → P i ℓ n − → P i → P i /ℓ n → 0 in each degree i.Since the ) and all rows and columns are distinguished triangles, one can deduce the existence of an isomorphism α n : ).Further, as n varies, the isomorphisms α n may be chosen to be compatible with the natural transition morphisms (cf.[12, the proof of Prop.

3.3]
).The inverse limit of such a compatible system of isomorphisms {α n } n then gives an isomorphism in Taken in conjunction with Lemma A2 the quasi-isomorphism ι ℓ implies that where the last equality follows from [9, Th. 2.1(3)].To prove the final assertion of the lemma we need therefore only observe that ρ ℓ (δ # ).Indeed, this equality follows from the fact that on To prove C ℓ (K/k) we need only show that (9) coincides with the formula of Theorem 3.2 in the case is the natural morphism of spectra, J = G and r = 0. We first compare the left hand sides of the respective formulas.If y is any closed point of U k,S , then, after fixing a y point x of U K,S and writing G x for the decomposition subgroup of x in G, the stalk of π * π * Z ℓ (0) ⊗ Z ℓ Q ℓ at y identifies as a (left) G × G x -module with Q ℓ [G] where elements of the form (g, id) ∈ G × G x act via left multiplication by g and elements of the form (id, g x ) ∈ G×G x act via right multiplication by g −1  x (in this regard compare the discussion of [7, beginning of §2]).By using this identification one computes that Z G (U k,S , π * π * Z ℓ (0) ⊗ Z ℓ Q ℓ , t) has the same Euler factor at y as does Z K/k,S (t).Since this is true for all closed points y it follows that there is an equality of functions of the complex variable t This implies that the left hand side of ( 9) is equal to the left hand side of the relevant special case of the formula in Theorem 3.2.Hence our proof of (9) will be complete if we can verify the relevant semi-simplicity hypothesis (in order to apply Theorem 3.2) and then prove that the trivialisation τ UK,S ,Z ℓ (1),σ is induced by the isomorphism (−D K,S,ℓ ⊗ Z ℓ Q ℓ ) −1 .Our proof is therefore completed by combining the description of τ UK,S ,Z ℓ (1),σ in Remark 7 together with the following result.
Proof.Lemma 3 combines with Lemma 1iii) to imply that RΓ(U K,S , Z ℓ (1)) is acyclic outside degrees 1 and 2. Remark 7 therefore implies claim i) is equivalent to asserting that the map β 1 UK,S ,Z ℓ (1),σ ⊗ Z ℓ Q ℓ is bijective and this is an immediate consequence of the explicit description given in claim ii).We now fix an arbitrary place v in S and write c v : Y K,S ⊗Z ℓ → w|v Z ℓ for the homomorphism induced by projecting each element of Y K,S to its respective coefficient at each place w of K above v.Then claim ii) will follow if we show that the composite homomorphism ) ) To prove this we set S ′ := S \ {v}, let Z denote the complement of U K,S in U K,S ′ and write j : U K,S → U K,S ′ , resp.i : Z → U K,S ′ , for the natural open, resp.closed, immersion.Then there exists a natural morphism of étale sheaves j * G m → i * i * Z on U K,S ′ that is induced by taking valuations.In turn this gives rise to a morphism ) and hence, for each non-negative integer n, to a morphism ).These morphisms are compatible with the natural transition maps as n varies and therefore induce, upon passage to the inverse limit, a morphism in ) is induced by the respective valuation map val w .In addition, if we identify w|v Hom cont (Gal(F c p /k(w)), Z ℓ ) with w|v Z ℓ by evaluating each homomorphism at the topological generator σ deg(w) of Gal(F c p /k(w)), then H 2 (λ) • H 1 (ι ℓ ) is induced by projection of an element of X K,S to its respective coefficients at each place w above v.Upon replacing U K,S and Z by U c K,S and Z c one obtains in a similar manner a morphism λ c : RΓ ) that induces a morphism of distinguished triangles of the form After passing to cohomology this diagram induces a commutative diagram ) where the minus sign in the lower row occurs because of the −1-shift in the lower row of the previous diagram.Now the pull-back to H 1 (Z, Z ℓ ) of φ p is the element (φ w ) w|v where φ w (σ deg(w) ) = deg(w) for each w dividing v.
After identifying both H 0 (Z, Z ℓ ) and H 1 (Z, Z ℓ ) with w|v Z ℓ in the manner prescribed above, the description of Remark 7 (with X = Z and F = Z ℓ ) therefore implies that β 0 Z,Z ℓ ,σ is given by component-wise multiplication with the element (deg(w)) w|v .Upon combining the commutativity of this diagram with the explicit descriptions of H 1 (λ) and H 2 (λ) given above, it follows that the composite homomorphism (10) is indeed equal to (− deg(w) • val w (−)) w|v , as required.
3.4.The case ℓ = p.In this subsection we prove Theorem 3.1ii).For each subgroup H of G we let ρ G, * H denote the natural restriction of scalars homomorphism For each abelian group H and each subgroup J of H we also let q H H/J, * denote the natural coinflation homomorphism If now C • is any complex of G-modules which is acyclic outside degrees 0 and 1, then In addition, the tautological exact sequence Proof.An easy consequence of the definition of equivalence of Yoneda extensions.

This result implies that RΓ
In this subsection we relate c W,S (K/k) to the canonical extension class which is defined in terms of class field theory by Tate in [49].
To make such a connection we assume that the G-module A K,S is c-t.In this case the displayed short exact sequence in Lemma 1iii) splits (since Ext 1 G (X K,S , A K,S ) = 0) and also Ext 2 G (A K,S , O × K,S ) = 0 and so there exists a natural isomorphism . We choose a finite set of places W of k which do not belong to S, are each totally split in K/k and are such that A K,S is generated by the classes of places in W (K). We set S ′ := S ∪ W (so that A K,S ′ is trivial) and we observe that there are natural exact sequences of G-modules of the form -module these sequences combine to induce an isomorphism of extension groups ).In the sequel we shall identify Yoneda-Ext-groups with derived functor Extgroups by means of a projective resolution of the first variable (this convention differs from that used in [12] -see in particular [loc.cit., Lem.3]).We also write c S ′ (K/k) for the canonical element of Ext 2 G (X K,S ′ , O × K,S ′ ) which is defined in [49].
Proof.For each w ∈ S ′ (K) we set V w := Spec(K w ).We also let j ′ denote the natural open immersion U K,S ′ → C K and we consider the following diagram in D(Z[G]) . The top row of this diagram is the distinguished triangle from Lemma 1ii) (with S replaced by S ′ ), the first column is the distinguished triangle induced by the tautological exact sequence 0 → X K,S ′ ⊂ − → Y K,S ′ → Z → 0 and the second column is the distinguished triangle from [42, Chap.II, Prop.2.3].Further, under the isomorphism that is induced by [12,Lem. 7(b)], the morphism α corresponds to the composite of the projection Y K,S ′ ⊗Q → Y K,S ′ ⊗Q/Z and the natural identification It is straightforward to show that the square in the above diagram commutes (for example, by using [12,Lem. 7(b)] to reduce to cohomological considerations).By comparing this diagram to the diagrams (85) and (88) from loc. cit., and then using the Octahedral axiom, one may therefore conclude that RΓ W (U K,S ′ , G m ) is equivalent to the complex Ψ S ′ which is defined in [12,Prop. 3.1].From the proof of [12,Prop. 3.5] we may thus deduce that c W,S ′ (K/k) = −c S ′ (K/k).(We remark that whilst the results of [12] are phrased solely in terms of number fields, all of the constructions and arguments of loc.cit.extend directly to the case of global function fields.In addition, we obtain −c S ′ (K/k) rather than c S ′ (K/k) in the present context because we have changed conventions regarding Yoneda-Ext-groups.)To conclude that c W,S (K/k) = ι S • ι S ′ ,S (−c S ′ (K/k)) it suffices to prove that there exists a morphism RΓ [40, the proof of Th. 7.1], the existence of such a morphism can be seen to be a consequence of the morphism of étale sheaves G m → j * G m on U K,S where j : U K,S ′ → U K,S denotes the natural open immersion.
We now state the main results of this section.Proof.Following Remark 4, we may consider C(K/k) with respect to a set S which is large enough to ensure that A K,S is trivial, and in this case Proposition 4.1 (with S = S ′ ) implies that c W,S = ι S (−c S (K/k)).Let now C • and D • be any objects of D perf (Z[G]) which are acyclic outside degrees 0 and 1 and are such that ).This observation combines with the equality c W,S = ι S (−c S (K/k)) and the very definition of Ω(K/k, 3) to imply that the latter element can be computed as the Euler characteristic of )).On the other hand, the same argument as used to prove [7,Lem. 2.3.7]shows that Proof.Claim i) follows directly from Theorem 4.1 and Corollary 1.We now assume that p ∤ |G|.In this case K 0 (Z p [G], Q p ) is torsion-free [13, proof of Lem.11c)] and hence claim i) implies Ω(K/k, 3) = W K/k .In addition, K/k is tamely ramified and so Ch(K/k)ii) has been proved by Chinburg.Indeed, the equality Ω(K/k, 2) = W K/k follows directly upon combining [18, §4.2, Th. 4] with [23,Cor. 4.10].Finally, we observe that the validity of Ch(K/k)i) now follows immediately from the fact that Ω(K/k, 1) = Ω(K/k, 2) − Ω(K/k, 3) [18, §4.1, Th. 2 and the remarks which follow it].
R is equal to the group D p (Z[G]) that arises in [24,Th. 6.13].We recall that the arguments of Chinburg in loc.cit., and of Bae in [3] (the results of which provided the key ingredient in our proof of Theorem 3.1ii) in §3.4), rely crucially upon results of Milne and Illusie concerning p-adic cohomology.In particular, in both cases the occurrence of the term D p (Z[G]) reflects difficulties involved in formulating and proving suitable equivariant refinements of the results of [43].

The conjecture of Gross
In this section we assume unless explicitly stated otherwise that G is abelian.We set G * := Hom(G, C × ) and for each χ ∈ G * we let e χ denote the associated idempotent In terms of this notation one has θ K/k,S (s) = χ∈G * e χ L S (χ, s).We let I G denote the kernel of the homomorphism ǫ : Z[G] → Z which sends each element of G to 1.

5.1.
Statement of the conjecture.We set n := |S| − 1 and let |n|, resp.|n| * , denote the set of integers j which satisfy 1 ≤ j ≤ n, resp.0 ≤ j ≤ n.We henceforth label (and thereby order) the places in S as {v i : i ∈ |n| * }.For each j ∈ |n| * we fix a place w i of K which restricts to v i on k.For any place v of k which is unramified in K/k we write σ v for its frobenius automorphism in G and Nv for the cardinality of the associated residue field.We also fix a finite non-empty set T of places of k which is disjoint from S and then set This C[G]-valued function is holomorphic at s = 0 and, by using results of Weil, Gross has shown that θ K/k,S,T (0) belongs to Z[G] [31,Prop. 3.7].For any intermediate field F of K/k and any place w of K we let w ′ denote the restriction of w to F and then write f K/F,w for the homomorphism F × → G which is obtained as the composite of the natural inclusion ) and the natural injection Gal(K w /F w ′ ) → G.We also write O × F,S,T for the subgroup of O × F,S consisting of those S(F )-units which are congruent to 1 modulo all places in T (F ).It is known that each such group O × F,S,T is torsion-free.In particular, after choosing an ordered Z-basis {u j : j ∈ |n|} of O × k,S,T , we may define an element of Z[G] by setting Reg G,S,T := det((f K/k,wi (u j ) − 1) 1≤i,j≤n ).
At the same time we also define a rational integer m k,S,T by means of the following equality in where λ k,S denotes the isomorphism Remark 9.The term m k,S,T • Reg G,S,T belongs to I n G and is, when considered modulo I n+1 G , independent of the chosen ordering of S and of the precise choice of the places {w i : i ∈ |n| * } and of the ordered basis {u j : j ∈ |n|}.

Statement of the main results.
At the present time, the best results concerning Gr(K/k) are due to Tan and to Lee.Specifically, it is known that Gr(K/k) is valid if either |G| is a power of p [47] or if |G| is coprime to both |µ K | and the order of the group of divisors of degree 0 of the curve C k [39].However, these results are proved either by reduction to special cases or by induction on |G| and so do not provide an insight into why Gr(K/k) should be true in general.In contrast, in this section we shall show that Gross's integral regulator mapping O × k,S → X k,S ⊗ G [31, (2.1)] arises as a natural Bockstein homomorphism in Weil-étale cohomology and we shall use this observation to prove the following result.
Proof of Corollary 3. It is easily seen to be enough to prove Gr(K/k) in the case that |G| is a prime power.The aforementioned result of Tan therefore allows us to assume that p ∤ |G| (so that K 0 (Z p [G], Q p ) is torsion-free).But since the G-module µ K is assumed to be c-t, in this case the validity of Gr(K/k) follows directly from Theorem 5.1 and Corollary 1.
K is the maximal subgroup of µ K of ℓ-power order.It seems likely that a further development of the method we use to prove Theorem 5.1 will allow the removal of any such hypothesis on µ K .Indeed, in certain special cases this is already achieved in the present manuscript (cf.Corollary 5).
The proof of Theorem 5.1 will be the subject of the next three subsections.

The computation of χ(RΓ
).In this subsection we assume that the G-module µ K is c-t, but we do not assume that G is abelian.We set Tr G := g∈G g ∈ Z[G].For any abelian group A we write A in place of A/A tors and for any extension field E of Q we set A E := A ⊗ E. For any homomorphism of abelian groups φ : A → A ′ we also let φ E denote the induced homomorphism φ ⊗ id E : A E → A ′ E .In the following result we let Cone(α) denote the 'mapping cone' of a particular morphism α in D perf (Z[G]) -our application of this construction can be made rigorous by the same observation as used in [15,Rem. 5

.2].
Lemma 6.There exists an endomorphism φ of a finitely generated free Z[G]module F which satisfies both of the following conditions.Let F • denote the complex F φ − → F , where the first term is placed in degree 0. i) There exists a distinguished triangle in D perf (Z[G]) of the form where α is the morphism where the third, fourth and fifth maps are induced by H 1 (β), the isomorphism of Lemma 1iv) (with J = G) and the short exact sequence of Lemma 1iii) (with Proof.We set ) which is acyclic outside degrees 0 and 1 and that where F , resp.P , is a finitely generated free Z[G]-module, resp.a finitely generated Z[G]-module which is both c-t and Z-free, and P is placed in degree 0. Now any such [26,Th. (32.1)] we may therefore deduce that, for each prime q, the Z q [G]-modules P ⊗ Z q and F ⊗ Z q are isomorphic.We may thus apply Roiter's Lemma [26, (31.6)] to deduce the existence of a Z[G]-submodule P ′ of P for which the quotient P/P ′ is finite and of order coprime to |G| and one has an isomorphism of Z[G]-modules ι : The Z-module im(λ G ) is free and so the exact sequence 0 Hence we may choose a submodule D of F G which λ G maps isomorphically to im(λ G ).We next let T denote the pre-image under the tautological surjection F G → cok(λ G ) of the subgroup cok(λ G ) tors .Then the exact sequence 0 → T → F G → cok(λ G ) → 0 is also split and so we may choose a submodule D ′ of F G which is mapped isomorphically to cok(λ G ) under the natural surjection.Now D ′ and ker(λ G ) have the same Z-rank since ) which corresponds to the morphism from the complex F • (as described in the statement of the Lemma) to F • that is induced by ι in degree 0 and is equal to ψ−1 in degree 1.It is then easily checked that this gives rise to a distinguished triangle of the form stated in i) in which Q := P/P ′ .Now φ G = ψ ′ • λ G and so the above remarks imply both that φ G (D) ⊆ D and that the natural map ker(φ G ) → cok(φ G ) is bijective.We next observe that the decomposition F G = ker(φ G ) ⊕ D can be lifted to a direct sum decomposition F = F 1 ⊕ F 2 in which both F 1 and F 2 are free Z[G]-modules (of ranks n and Documenta Mathematica 9 (2004) 357-399 We let κ denote the composite homomorphism described in claim ii)a).Our earlier observations imply that κ is bijective, and so where id 0 denotes the identity map on the zero space.We also set R Then upon applying Lemma A2 firstly to the distinguished triangle and then to the distinguished triangle in Lemma 6i) we obtain equalities where ι 1 and ι 2 are any choices of R[G]-equivariant sections to the tautological surjections F R → im(φ) R and F R → cok(φ) R and the last equality follows from Lemma A1. 5.4.The connection to Gr(K/k).In this subsection we assume that G is abelian and identify K 0 (Z[G], R) with the multiplicative group of invertible Z[G]-lattices in R[G] (see Remark A1).In particular, we note that if M is any finite G-module which is c-t, then its (initial) Fitting ideal Fitt Z[G] (M ) is an invertible ideal of Z[G] and under the stated identification one has Hence, in this case, (12) implies that the validity of C(K/k) is equivalent to the existence of an element x T of Q[G] × which satisfies both (13) θ * K/k,S,T (0 , and also that for any χ ∈ G * \ G * (0) one has e χ • ker(φ) C = 0 and so e 0 det Z[G] (φ) = det Z[G] (φ).Since θ K/k,S,T (0) # = e 0 θ * K/k,S,T (0) # we therefore deduce from (13) that

Documenta Mathematica 9 (2004) 357-399
Now |Q| is coprime to |G| and I n G /I n+1 G is annihilated by a power of |G|, and so (14) implies that x T acts naturally on I n G /I n+1 G .In addition, Lemma 6ii) implies that the matrix of φ with respect to the ordered Z[G]-basis {b i : 1 ≤ i ≤ d} of F is a block matrix of the form (15) A B C D where ) and all entries of both B and C belong to G ) and so the above matrix representation combines with the previous displayed equality to imply that ( 16) To compute the term (−1) n ǫ(x T )ǫ(det(D)) we first multiply ( 13) by Tr G and obtain an equality For convenience we fix the sections ι 1 and ι 2 so that ι G 1 is equal to the inverse of the automorphism of where σ is the bijection induced by the injection X k,S → X K,S described in Lemma 1iv) (with J = G), and the final arrow denotes the inverse of the isomorphism induced by the displayed map in Lemma 6ii)a).Now, with respect to the ordered Z-basis {Tr G (b i ) : n < i ≤ d} of F G 2 , each component of the matrix of ψ 2 is the image under ǫ of the corresponding component of D and so lim s→0 On the other hand, the commutative diagram [50, Chap.I, §6.5]) combines with the above description of ψ 1 to imply that det R (ψ 1 ) is equal to the determinant of the map Comparing this equality with (11)  In turn, upon substituting this equality into ( 16) we obtain a congruence  17) is equal to Reg G,S,T .The key to our proof of this equality will be the observation that the 'regulator map' O × k,S → X k,S ⊗ G introduced by Gross in [31, (2.1)] arises as a natural Bockstein homomorphism in Weil-étale cohomology (this is Lemma 8).The material in this subsection is strongly influenced by the general philosophy of algebraic height pairings that is developed by Nekovář in [44, §11].At the outset we let Γ be any finite abelian group and C • any object of D fpd (Z[Γ]).Then, upon tensoring C • with the tautological exact sequence 0 → I Γ → Z[Γ] → Z → 0 we obtain a distinguished triangle in D(Z) of the form In addition, if C • is acyclic outside degrees 0 and 1, then there are natural identifications In this case the canonical identification I Γ /I 2 Γ ∼ = Γ therefore combines with the cohomology sequence of the above triangle to induce a 'Bockstein homomorphism' ⊗ Γ and also an associated pairing In the remainder of this subsection we shall use these constructions in the cases that Γ = G and C • is equal to both F • (as described in Lemma 6) and RΓ W (U K,S , G m ), and also in the case that Γ is equal to a given decomposition subgroup of G and C • is a local analogue of RΓ W (U K,S , G m ).In the course of so doing we shall always use the Z-basis {v i − v 0 : i ∈ |n|} to identify X k,S with Hom Z (X k,S , Z).Before stating our first result we observe that the action of Tr G (in each degree) induces an isomorphism in D(Z) between which the first term is placed in degree 0. We shall use this isomorphism to identify H 1 (F • G ) with X k,S by means of the map F G → X k,S described in Lemma 6ii)a).
). Proof.The homomorphism β F • ,G can be computed as the composite of the connecting homomorphism in the following commutative diagram Upon computing the above connecting homomorphism by using the matrix representation of φ given in (15), and observing Lemma 6ii) implies that the tautological surjection ).This implies the stated result.The construction of the pairing  17) is equal to the discriminant of the restriction of ρ RΓW (UK,S ,Gm),G to O × k,S,T × X k,S as computed with respect to the ordered Z-bases {u i : i ∈ |n|} and {v i − v 0 : i ∈ |n|}.To prove Theorem 5.1 we therefore need only show that the homomorphism β RΓW (UK,S ,Gm),G coincides with the regulator mapping O × k,S → X k,S ⊗ G defined by Gross in [31, (2.1)].In turn, this is achieved by the following result (which, we observe, does not assume that the G-module µ K is c-t).
Proof.We fix an index i ∈ |n| and set v := v i , w := w i and D := Gal(K w /k v ).We let We set V w := Spec(K w ).Then the result of [12,Lem. 7(b)] combines with the fact that H 1 (V w , G m ) = 0 to imply that there exists a unique morphism |D| (recall that, following the approach of §4.1, we are here using a different convention regarding Yoneda-Ext-groups than that used in [12], and hence e(C • w ) is equal to −e w rather than e w .)The natural localisation morphism where We remark that the upper row of this diagram is exact since the D-module A is c-t.Our proof now concludes by means of an explicit diagram chase showing that the connecting homomorphism in the second of these diagrams induces the inverse of the connecting homomorphism in the first diagram.

The conjecture of Tate
In this section we provide evidence for Tate's refinement of Gr(K/k).To do so we continue to use the notation of §5.1.In addition, we fix a prime number ℓ and assume henceforth that G has order ℓ m with m ≥ 1.For each index j in |n| * we let G j denote the decomposition subgroup of w j in G and we define an integer m j by the equality |G j | = ℓ m−mj .
6.1.Statement of the conjecture.In this subsection we assume S to be ordered so that Conjecture Ta(K/k, S, T ) (Tate, [51]): If G is cyclic of order ℓ m , m 0 = 0 and m n = m − 1, then one has For a further discussion of this conjecture see, for example, [38, §4].

Statement of the main results
, where S ′ is any set as described in §4.1 (and c S (K/k) is indeed independent of the choice of S ′ ).For each index j in |n| we write I j for the kernel of the natural projection map We consider the following hypothesis on K/k.

Hypothesis (S,T):
There exist finite non-empty sets S and T of places of k which satisfy each of the following conditions: i) S contains all places which ramify in K/k, ii) the G-module A K,S is c-t, iii) G 0 = G, n > 0 and G j is cyclic for each j ∈ |n|, iv) T is disjoint from S and c S (K/k) lies in the image of the map If K/k is cyclic, then there always exists a set of places S which satisfies conditions i), ii) and iii) above.In general however, for a given field K there are restrictions on the abstract structure of the decomposition group G 0 and therefore (under condition iii)) also on G. Nevertheless, the validity of Hypothesis (S,T) does not itself imply, for example, that G is abelian.If ℓ ∤ |µ k |, then (since |G| is a power of ℓ) one has ℓ ∤ |µ K | and so [50, Chap.IV, Lem.1.1] implies that there exists a set T which is disjoint from S and satisfies ℓ ∤ [O × K,S : O × K,S,T ] and hence also condition iv).In fact, condition iv) can be shown to be satisfied under reasonably general conditions even if ℓ | |µ k | (cf.[17,Lem. 2]).
The following result will be proved in §6.4.

Documenta
In particular, in this case Ta(K/k, S, T ) is valid.
Proof.Since, by assumption, m 0 = 0 the sets S and T satisfy all parts of Hypothesis (S,T).The first assertion thus follows directly from Theorem 6.1.
To prove the second assertion we recall that if ℓ = p, then Ta(K/k, S, T ) has been proved by Tan [48].We may therefore assume that ℓ = p so that C(K/k) is valid by Corollary 1.It thus suffices to deduce the validity of Ta(K/k, S, T ) from the stated congruence for θ K/k,S,T (0) and this is true because j∈|n| I j ⊆ . Indeed, since m n = m − 1, the required inclusion follows directly from the criterion of [8,Lem. 5.11].
The next result improves upon Corollary 3 and also the main result of Lee in [37].
Corollary 5.If G has prime exponent, then Gr(K/k) is valid.
Proof.In this case, the functorial properties of θ K/k,S,T (0) and Reg G,S,T under change of K/k combine with results on the structure of I G /I n+1 G to show that it is enough to prove Gr(L/k) for each sub-extension L/k of K/k which is of prime degree.The theorem of Tan [47] also allows us to assume that [L : k] is a prime number different from p, and in this case the required congruence can be proved by combining the result of Corollary 4 (with K = L) together with arguments of Gross from [31, §6].The precise details of this argument are presented in joint work of the author with Lee [17].
In this subsection we prepare for the proof of Theorem 6.1 by using Hypothesis (S,T) to refine the computation of χ(RΓ W (U K,S , G m ), R K,S ) given in §5.3.We do not assume here that G is abelian or that the G-module µ K is c-t.At the outset we fix sets S and T as in Hypothesis (S,T).Since S is fixed we abbreviate O × K,S,T , X K,S and O × K,S to O × K,T , X K and O × K respectively.We also set A K := Pic(O K,S ) and write A K,T for the quotient of the group of fractional ideals of O K,S that are prime to T by the subgroup of principal ideals with a generator congruent to 1 modulo all places in T (K).For each j ∈ |n| we fix a generator g j of G j and a set of representatives S(j) of the orbits of G j on the set of places of K lying above {v i : i ∈ |n|}.We assume that S(j) contains w i for each i ∈ |n|.For each place w in S(j) we define δ jw to be 1 if w = w j and to be 0 otherwise.For each j ∈ |n| we also set Tr j := g∈Gj g ∈ Z[G] and K j := K Gj .If d is any strictly positive integer, then in the sequel we shall use the canonical basis Proposition 6.1.Let S and T be as in Hypothesis (S,T).Assume also that G j is not trivial for any j ∈ |n|.Then, for each j ∈ |n| there exists an element ǫ j of O × Kj ,T which satisfies all of the following conditions.i) For each w ∈ S(j) one has f K/Kj ,w (ǫ j ) = g δjw j .
ii) For each pair of integers i, j in |n| let y ji denote the (unique Then the matrix M To prove this result we let Ψ• denote any complex of G-modules of the form Ψ0 d − → Ψ1 where Ψ0 occurs in degree 0 and e( Ψ• ) = c W,S (K/k) in the notation of §4.1.We write Ψ 1 for the pullback of the natural surjection Ψ1 → H 1 W (U K,S , G m ) and a choice of section γ to the surjection H 1 W (U K,S , G m ) → X K provided by Lemma 1iii) (such a section always exists under Hypothesis (S,T)ii)).In this way we obtain a complex Ψ and lies in a distinguished triangle in D perf (Z[G]) of the form where H 0 (α) is the identity map and H 1 (α) = γ.Upon applying Lemma A2 to this triangle we obtain an equality multiplication by (g j − 1) on each summand R[G] • (g j − 1).It follows that, with respect to a suitable ordered R[G]-basis of F ⊗R, the matrix of λ, φ ι1,ι2 is equal to M T and hence that M T is invertible, as required by claim ii).In addition, in this case Lemma A1 implies that χ(F • , R K,S • φ) = δ(detred R[G] (M T )).Our proof of Proposition 6.1 is thus completed by combining this equality with ( 18), (19) and the following two results.
→ 0 of G-modules.These sequences combine to imply that the G-module v∈T F × (v) is c-t and moreover that χ ).At this stage we know that all of the modules which occur in (20) are c-t, except possibly for A K,T .The exactness of this sequence therefore implies that A K,T is also c-t.Finally, the claimed equality follows upon decomposing (20) into short exact sequences and then using Lemma A2 (repeatedly).To prove this we choose for each i, j in |n|, an integer a ij such that f K/k,wi (u j ) = g aji i , we set a := (a ij ) ∈ M n (Z) and we show that b • a ≡ I n (mod ℓ • M n (Z)).For each intermediate field F of K/k we write J F for the idele group of F and f F : J F → Gal(K/F ) for the global reciprocity map.For each j ∈ |n| we write f F,j : F × → Gal(K/F ) for the composite of f F and the natural inclusion of F × into s F × s ⊂ J F where the product is taken over the set of places s of F which lie above v j .We note that if F = k, then f F,j = f K/k,wj .

Documenta Mathematica 9 (2004) 357-399
For each pair of elements i, j of |n| we set S(ij) := {w ∈ S(i) : w | v j }.Then property i) in the statement of the Proposition implies f Ki,j (ǫ i ) = w∈S(ij) f K/Ki,w (ǫ i ) = w∈S(ij) g δiw i = g δij i .After taking account of the functorial behaviour of Artin maps, this implies g δij i = f K/k,wj (N i (ǫ i )) = s∈|n| f K/k,wj (u s ) bis = s∈|n| g bisasj j = g s∈|n| bisasj j and hence, since by assumption no element g j is trivial, that s∈|n| b is a sj ≡ δ ij (mod ℓ).It follows that b • a ≡ I n (mod ℓ • M n (Z)), as required.
6.4.The proof of Theorem 6.1.In this subsection we use Proposition 6.1 to prove Theorem 6.1.We assume throughout that G is abelian.Our argument is similar to that used in §5.4 and so we continue to use the notation G * (0) and e 0 introduced in that subsection.At the outset we observe that if G j is trivial for any j ∈ |n|, then θ K/k,S,T (0) and Reg G,S,T are both equal to 0 and so the congruence of Theorem 6.1 is valid trivially.In the sequel we shall therefore assume that G j is not trivial for any j ∈ |n|, as is required by Proposition 6.1.Now, since G is abelian, Proposition 6.1iii) shows that C(K/k) implies the existence of an element x T of Q[G] × which satisfies both This combines with (21) to imply that θ K/k,S,T (0) # = x T • e 0 det(M T ) = x T i∈|n| (g i − 1).In addition, (22) combines with Lemma 10 to imply that x T ∈ Z ℓ [G] and hence one has where m k,S,T is as defined in (11).In addition, with the matrix b as defined in the proof of Lemma 10, one has and so it suffices to show that ǫ(x T ) • det(b) = (−1) n m k,S,T .But, just as in the deduction of ( 17) from ( 13), this can be proved by first multiplying (21) by Tr G and then comparing the resulting equality to (11).This completes our proof of Theorem 6.1.

Appendix
We recall some relevant properties of the refined Euler characteristic construction discussed in §2.1 (the notation of which we continue to use).For further details we refer the reader to [9] (or to [7, §1] for a fuller review than that given here).We let R denote either Z or Z ℓ for some prime ℓ and E an extension of the field of fractions of R. For any R[G]-module M , resp.homomorphism of R-modules φ, we set M E := M ⊗ R E, resp.φ E := φ ⊗ R id E .Let P • be a bounded complex of finitely generated projective R[G]-modules.
For each integer i we let B i , resp.Z i , denote the submodules of coboundaries, resp.cocycles, of P • E in degree i.After choosing E[G]-equivariant splittings of the tautological exact sequences 0 → Z i → P i E → B i+1 → 0 and 0 → B i → Z i → H i (P • E ) → 0 one obtains non-canonical isomorphisms By using the identity map on B all one can therefore extend each element φ of Is E[G] (H + (P • ) E , H − (P • ) E ) to give an element φ(P • E ) of Is E[G] (P + E , P − E ).This construction clearly depends upon the above choice of splittings but nevertheless induces a well-defined map from Is E[G] (H + (P • ) E , H − (P • ) E )/ ∼ to Is E[G] (P + E , P − E )/ ∼ which is independent of all such choices.We denote this map by τ → τ (P • E ) and we obtain a well-defined element of K 0 (R[G], E) by setting χ R[G],E (P • , τ ) := (P + , φ, P − ) for any (and therefore every) φ ∈ τ (P • E ).In the following result we record this construction in a special case.
For each χ ∈ Irr C (G) we write L S (χ, s) for the associated S-truncated Artin L-function and L * S (χ, 0) for the leading term in the Taylor expansion of L S (χ, s) at s = 0. Recalling that ζ(C[G]) identifies with Irr C (G) C, we define a ζ(C[G])-valued meromorphic function of a complex variable s by setting θ K/k,S (s) := (L S (χ, s)) χ∈Irr C (G) .

Proof.
Claim i) is proved by the argument of [40, proof of Th. 6.5].The existence of the distinguished triangle in claim ii) can be proved by comparing the spectral sequences of [40, Prop.2.3(f)] or by using the approach of Geisser in [30, Th. 6.1].The descriptions of the groups H i W (U K,S , G m ) given in claim iii) are proved by Lichtenbaum in [40, Th. 7.1c)].They follow from the long exact sequence of cohomology associated to the triangle in claim ii), the canonical identifications H 0 in the Appendix).By using this observation in conjunction with Remark 1 it can be shown that C(k/k) is equivalent to a special case of the conjecture formulated by Lichtenbaum in [40, Conj.8.1e)].Remark 3. Let j : U K,S → C K denote the natural open immersion.Then the Poincaré Duality Theorem of [42, Chap.II, Th. 3.1] gives rise to a commutative diagram in D(Z[G]) of the form ։ cok(ψ) Documenta Mathematica 9 (2004) 357-399

1 . 4 . 1 .
where the intersection runs over all cyclic subgroups H of G and over all subgroups J of H which are such that p ∤ |H/J|[9, Th. 4.1].Taken in conjunction with the functorial properties of C p (K/k) under change of group (Remark 4), the above displayed equality implies that Cp (K/k) is valid modulo K 0 (Z p [G], Q p ) tors if and only if C p (F/E) is valid for each cyclic extension F/E with k ⊆ E ⊆ F ⊆ Kand p ∤ [F : E].But, for each such extension F/E, the argument of [7, Lem.2.2.7] shows that C p (F/E) is implied by the Strong-Stark Conjecture for F/E, as formulated by Chinburg (cf.[3, §3.1]).The required result therefore follows directly from Bae's proof of the Strong-Stark Conjecture in this case [3, Th. 3.5.4].This completes our proof of Theorem 3.The conjectures of Chinburg 4.Canonical 2-extensions.In the sequel we shall say that two complexes of G-modules C • and D • are 'equivalent' if H i (C) = H i (D) in each degree i and there exists an isomorphism in D(Z[G]) from C • to D • which induces the identity map in all degrees of cohomology.

Lemma 5 .
Let C • and D • be any complexes of G-modules which are acyclic outside degrees 0 and 1 and are also such that H i (C • ) = H i (D • ) for i = 0, 1.Then C • and D • are equivalent if and only if one has e(C • ) = e(D • ).

Theorem 4 . 1 .
The image under ∂ 0 Z[G],R of the equality of C(K/k) is equivalent to the equality of Ch(K/k)iii).
there exists an integer d with d ≥ n and an ordered Z[G]-basis {b i : 1 ≤ i ≤ d} of F which satisfies both of the following conditions.a) The Z[G]-module F 1 which is generated by {b 0) denote the set of characters χ ∈ G * at which L S (χ, 0) = 0, and we set e 0 := χ∈G * (0) e χ .Then the criterion of [50, Chap.I, Prop.3.4] implies T as a subgroup of O × k,S in the natural way, and define elements a := (a ij ) 1≤i,j≤n and b := (b ij ) 1≤i,j≤n of M n (Z) by the equalities u i = j∈|n| a ij d j and Tr G (H 0 (β)(b i )) = j∈|n| b ij d j for each i ∈ |n|, then the last displayed formula implies that (lim

Lemma 7 .
With respect to the ordered Z-bases {Tr G (b i ) : i ∈ |n|} and {v When taken in conjunction with the computation of Lemma 7 and the fact that multiplication by det(b) is invertible on I n G /I n+1 G (since |Q G | is coprime to Documenta Mathematica 9 (2004) 357-399 |G|), this observation implies that the term det(a)det(b) −1 • det(A) (mod the remark just prior toLemma 6).Then, by an argument similar to that used in the proof of Lemma 1iii), one shows thatC • w is an object of D fpd (Z[D]) which is acyclic outside degrees 0 and 1 and is such that H 0 (C • w ) and H 1 (C • w ) identify canonically with K × w and Z respectively.Further, in the notation of §4.1, the result of [12, Prop.3.5(a)] implies that the associated Yoneda extension class e(C • w ) is equal to the element −e w of Ext 2 D (Z, K × w ) ∼ = H 2 (D, K × w ) where inv kv (e w ) = 1 C • w and by consideration of this morphism one finds that β C • ,G,v is equal to the composite of the embedding O × k,S → k × v , the homomorphism β C • w ,D and the natural injection D ⊆ G.It is therefore enough for us to prove that β C • w ,D is equal to the reciprocity map rec w : k × v → D of the extension K w /k v .To this end we first recall that rec w is defined to be the map induced by the inverse of the isomorphism D ∼ = Ĥ0 (D, K × w ) which results from the canonical identifications D ∼ = I D /I 2 D = Ĥ−1 (D, I D ), the isomorphism Ĥ−1 (D, I D ) ∼ = Ĥ−2 (D, Z) which is induced by the connecting homomorphism associated to the tautological exact sequence 0 → I D → Z[D] → Z → 0 and the isomorphism Ĥ−2 (D, Z) ∼ = Ĥ0 (D, K × w ) which is given by cup-product with e w .To proceed we choose an extension of D-modules Documenta Mathematica 9 (2004) 357-399 of Yoneda extension class −e w .Then C • w is equivalent to the complex A • which is given by A ψ − → Z[D], where the modules are placed in degrees 0 and 1 and the cohomology is identified with K × w and Z by means of the given maps.Taken in conjunction with the description of rec w in the preceding paragraph and the compatibility of cup-products with connecting homomorphisms in Tate cohomology (cf.[2, Th. 3 and Th.4(iii),(iv)]), this observation implies that rec w is induced by the canonical isomorphism D ∼ = I D /I 2 D = (I D ) D together with the inverse of the connecting homomorphism in the following commutative diagram Tr w := d∈D d ∈ Z[D].On the other hand, the fact that C • w is equivalent to A • combines with the definition of β C • w ,D to imply that the latter homomorphism can be computed as the composite of the natural identification D ∼ = (I D ) D and the connecting homomorphism in the following commutative diagram k

Corollary 4 .
Mathematica 9 (2004) 357-399 Theorem 6.1.If S and T are as in Hypothesis (S,T) and G is abelian, then C(K/k) implies that θ K/k,S,T (0) ≡ m k,S,T • Reg G,S,T (mod I G • j∈|n| I j ).Assume the notation and hypotheses of Ta(K/k, S, T ).If the G-module Pic(O K,S ) is c-t and c S (K/k) lies in the image of the map

Lemma 9 .
If the G-module cok(φ T ) = O × K,T /E is c-t, then so also are cok(φ) = O × K /E, A K and A K,T , and in K 0 (Z[G], R) one has χ(cok(φ)) − χ(A K ) = χ(cok(φ T )) − χ(A K,T ) − δ(detred R[G] (∆ # T )).Proof.We use the natural exact sequence of finite G-modules(20) 0 → cok(φ T ) ⊆ − → cok(φ) → v∈T F × (v) → A K,T → A K → 0where F (v) denotes the direct sum of the residue fields F w of each place w of K which lies above v[31, (1.5)].Let G w denote the decomposition group of w in G.Then, if η is any generator of the cyclic group F × w , there exists a G wequivariant surjection Z[G w ] → F × w which sends 1 to η.In this way one obtains an exact sequence 0 → Z[G]

Lemma 10 .
The G-module cok(φ T ) is c-t.Indeed, one has ℓ ∤ | cok(φ T )|.Proof.It suffices to prove thatℓ ∤ | cok(φ T ) G |. Now E ∼ = Σ K so H 1 (G, E) ∼ = H 1 (G, Σ K ) = 0.This implies cok(φ T ) G = (O × K,T /E) G ∼ = O × k,T /E G and also that E G is generated by {N j (ǫ j ) : j ∈ |n|} where, for each j ∈ |n|, we write N j for the field theoretic norm map K × j → k × .We fix an ordered Z-basis{u i : i ∈ |n|} of U k,T and define an element b := (b ij ) of M n (Z) by the equalities N i (ǫ i ) = n j=1 u bij j for each i, j in |n|.Then |O × k,T /E G | = ±det(b)and so we must show that ℓ ∤ det(b).

Lemma A1 .
Let P be a finitely generated projective R[G]-module, φ an R[G]endomorphism of P and λ : ker(φ) E → cok(φ) E an E[G]-isomorphism.Choose Documenta Mathematica 9 (2004) 357-399 proof of Prop.4.1]; if G is abelian, then Proposition 3.1 can be reinterpreted in terms of graded determinants and in this case the given formula has been proved to within a sign ambiguity by Kato in [35, Lem.3.5.8].(This sign ambiguity arises because Kato uses ungraded determinants -for more details in this regard see [loc cit., Rem.3.2.3(3)and 3.2.6(