On the Equivariant Tamagawa Number Conjecture

for Tate Motives, Part II.

Dedicated to John Coates

David Burns and Matthias Flach

Received: October 26, 2005 Revised: May 1, 2006

Abstract. Let K be any finite abelian extension of ℚ, k any subfield of K and r any integer. We complete the proof of the equivariant Tamagawa Number Conjecture for the pair (h^{0}(Spec(K))(r), ℤ[Gal(K/k)]).

2000 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

1. Introduction

Let K/k be a Galois extension of number fields with group G. For each complex character χ of G denote by L(χ, s) the Artin L-function of χ and let Ĝ be the set of irreducible characters. We call

ζ_{K/k}(s) = (L(χ, s))_{χ∈Ĝ}

the equivariant Dedekind Zeta function of K/k. It is a meromorphic function with values in the center ∏_{χ∈Ĝ} ℂ of ℂ[G]. The ‘equivariant Tamagawa number conjecture’ that is formulated in [9, Conj. 4], when specialized to the motive M := ℚ(r)_{K} := h^{0}(Spec(K))(r) and the order 𝔄 := ℤ[G], gives a cohomological interpretation of the leading Taylor coefficient of ζ_{K/k}(s) at any integer argument s = r. We recall that this conjecture is a natural refinement of the seminal ‘Tamagawa number conjecture’ that was first formulated by Bloch and Kato in [5] and then both extended and refined by Fontaine and Perrin-Riou [18] and Kato [27]. If K = k and r ∈ {0, 1} then the conjecture specializes to the analytic class number formula and is therefore already a theorem. The most succinct formulation of the equivariant Tamagawa number conjecture asserts the vanishing of a certain element T Ω(M, 𝔄) = T Ω(ℚ(r)_{K}, ℤ[G]) in the relative algebraic K-group K_{0}(ℤ[G], ℝ). Further, the functional equation of Artin L-functions is reflected by an equality

(1) T Ω(ℚ(r)_{K}, ℤ[G]) + ψ^{∗}(T Ω(ℚ(1 − r)_{K}, ℤ[G]^{op})) = T Ω^{loc}(ℚ(r)_{K}, ℤ[G])

where ψ^{∗} is a natural isomorphism K_{0}(ℤ[G]^{op}, ℝ) ≅ K_{0}(ℤ[G], ℝ) and T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) is an element of K_{0}(ℤ[G], ℝ) of the form

(2) T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) = L^{loc}(ℚ(r)_{K}, ℤ[G]) + δ_{K/k}(r) + RΩ^{loc}(ℚ(r)_{K}, ℤ[G]).

Here L^{loc}(ℚ(r)_{K}, ℤ[G]) is an ‘analytic’ element constructed from the archimedean Euler factors and epsilon constants associated to both ℚ(r)_{K} and ℚ(1 − r)_{K}, the element δ_{K/k}(r) reflects sign differences between the regulator maps used in defining T Ω(ℚ(r)_{K}, ℤ[G]) and T Ω(ℚ(1 − r)_{K}, ℤ[G]^{op}) and RΩ^{loc}(ℚ(r)_{K}, ℤ[G]) is an ‘algebraic’ element constructed from the various realisations of ℚ(r)_{K}. (We caution the reader that the notation in (1) and (2) is slightly different from that which is used in [9] - see §3.1 for details of these changes.) In this article we shall further specialize to the case where K is an abelian extension of ℚ and prove that T Ω(ℚ(r)_{K}, ℤ[G]) = 0 for all integers r and all subgroups G of Gal(K/ℚ). In fact, taking advantage of previous work in this area, the main new result which we prove here is the following refinement of the results of Benois and Nguyen Quang Do in [1].

Theorem 1.1. If K is any finite abelian extension of ℚ, G any subgroup of Gal(K/ℚ) and r any strictly positive integer, then T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) = 0.

We now discuss some interesting consequences of Theorem 1.1. The first consequence we record is the promised verification of the equivariant Tamagawa number conjecture for Tate motives over absolutely abelian fields. This result therefore completes the proof of [17, Th. 5.1] and also refines the main result of Huber and Kings in [25] (for more details of the relationship between our approach and that of [25] see [11, Intro.]).

Corollary 1.2. If K is any finite abelian extension of ℚ, G any subgroup of Gal(K/ℚ) and r any integer, then T Ω(ℚ(r)_{K}, ℤ[G]) = 0.

Proof. If r ≤ 0, then the vanishing of T Ω(ℚ(r)_{K}, ℤ[G]) is proved modulo powers of 2 by Greither and the first named author in [11, Cor. 8.1] and the argument necessary to cover the 2-primary part is provided by the second named author in [17]. For any r > 0, the vanishing of T Ω(ℚ(r)_{K}, ℤ[G]) then follows by combining Theorem 1.1 with the equality (1). □

Corollary 1.3. The conjecture of Bloch and Kato [5, Conj. (5.15)] is valid for the Riemann-Zeta function at each integer strictly bigger than 1.

Proof. If r is any integer strictly bigger than 1, then [5, Th. (6.1)] proves the validity of [5, Conj. (5.15)] for the leading term of the Riemann Zeta function at s = r, modulo powers of 2 and a certain compatibility assumption [5, Conj. (6.2)] concerning the ‘cyclotomic elements’ of Deligne and Soulé in algebraic K-theory. But the latter assumption was verified by Huber and Wildeshaus in [26] and Corollary 1.2 for K = k = ℚ now resolves the ambiguity at 2. □

For any finite group G the image of the homomorphism δ_{G} : K_{0}(ℤ[G], ℝ) → K_{0}(ℤ[G]) that occurs in the long exact sequence of relative K-theory is equal to the locally-free class group Cl(ℤ[G]). In the following result we use the elements Ω(K/k, 1), Ω(K/k, 2), Ω(K/k, 3) and w(K/k) of Cl(ℤ[Gal(K/k)]) that are defined by Chinburg in [13].

Corollary 1.4. If K is any finite abelian extension of ℚ and k is any subfield of K, then one has Ω(K/k, 1) = Ω(K/k, 2) = Ω(K/k, 3) = w(K/k) = 0. In particular, the Chinburg conjectures are all valid for K/k.

Proof. In this first paragraph we do not assume that K is Galois over ℚ or that G := Gal(K/k) is abelian. We recall that from [10, (31)] one has

δ_{G}(ψ^{∗}(T Ω(ℚ(0)_{K}, ℤ[G]^{op}))) = Ω(K/k, 3) − w(K/k).

Further, [4, Prop. 3.1] implies δ_{G} sends L^{loc}(ℚ(1)_{K}, ℤ[G]) + δ_{K/k}(1) to −w(K/k) whilst the argument used in [4, §4.3] shows that RΩ^{loc}(ℚ(1)_{K}, ℤ[G]) is equal to the element RΩ^{loc}(K/k, 1) defined in [7, §5.1.1]. Hence, from [7, Rem. 5.5], we may deduce that

(3)  δ_{G}(T Ω^{loc}(ℚ(1)_{K}, ℤ[G])) = −w(K/k) + Ω(K/k, 2).

We now assume that G is abelian. Then G has no irreducible complex symplectic characters and so the very definition of w(K/k) ensures that w(K/k) = 0. Hence by combining the above displayed equalities with Theorem 1.1 (with r = 1) and Corollary 1.2 (with r = 0) we may deduce that Ω(K/k, 2) = Ω(K/k, 3) = 0. But from [13, (3.2)] one has Ω(K/k, 1) = Ω(K/k, 2)−Ω(K/k, 3), and so this also implies that Ω(K/k, 1) = 0. □

For finite abelian extensions K/ℚ in which 2 is unramified, an alternative proof of the equality Ω(K/k, 2) = 0 in Corollary 1.4 was first obtained by Greither in [21]. Before stating our next result we recall that, ever since the seminal results of Fröhlich in [19], the study of Quaternion extensions has been very important to the development of leading term conjectures in non-commutative settings. In the following result we provide a natural refinement of the main result of Hooper, Snaith and Tran in [24] (and hence extend the main result of Snaith in [35]).

Corollary 1.5. Let K be any Galois extension of ℚ for which Gal(K/ℚ) is isomorphic to the Quaternion group of order 8 and k any subfield of K. Then one has T Ω^{loc}(ℚ(1)_{K}, ℤ[Gal(K/k)]) = 0.

Proof. We set G := Gal(K/ℚ) and let Γ denote the maximal abelian quotient of G with E the subfield of K such that Γ = Gal(E/ℚ) (so E/ℚ is biquadratic). We set T Ω^{loc} := T Ω^{loc}(ℚ(1)_{K}, ℤ[G]) and T Ω_{E}^{loc} := T Ω^{loc}(ℚ(1)_{E}, ℤ[Γ]). Then from [9, Th. 5.1 and Prop. 4.1] we know that T Ω^{loc}(ℚ(1)_{K}, ℤ[Gal(K/k)]) and T Ω_{E}^{loc} are equal to the images of T Ω^{loc} under the natural homomorphisms K_{0}(ℤ[G], ℝ) → K_{0}(ℤ[Gal(K/k)], ℝ) and K_{0}(ℤ[G], ℝ) → K_{0}(ℤ[Γ], ℝ) respectively. In particular, it suffices to prove that T Ω^{loc} = 0. Now [4, Cor. 6.3(i)] implies that T Ω^{loc} is an element of finite order in the subgroup K_{0}(ℤ[G], ℚ) of K_{0}(ℤ[G], ℝ) and so [10, Lem. 4] implies that T Ω^{loc} = 0 if and only if both T Ω_{E}^{loc} = 0 and δ_{G}(T Ω^{loc}) = 0. But Theorem 1.1 implies T Ω_{E}^{loc} = 0 and, since δ_{G}(T Ω^{loc}) = −w(K/ℚ) + Ω(K/ℚ, 2) (by (3)), the main result of Hooper, Snaith and Tran in [24] implies that δ_{G}(T Ω^{loc}) = 0. □

The following result provides the first generalization to wildly ramified extensions of the algebraic characterization of tame symplectic Artin root numbers that was obtained by Cassou-Noguès and Taylor in [12].

Corollary 1.6. Let K be any Galois extension of ℚ for which G := Gal(K/ℚ) is isomorphic to the Quaternion group of order 8. Then the Artin root number of the (unique) irreducible 2-dimensional complex character of G is uniquely determined by the algebraic invariant RΩ^{loc}(ℚ(1)_{K}, ℤ[G]) in K_{0}(ℤ[G], ℝ).

Proof. This is a direct consequence of combining Corollary 1.5 with a result of Breuning and the first named author [7, Th. 5.8] and the following facts: L^{loc}(ℚ(1)_{K}, ℤ[G])+δ_{K/ℚ}(1) is equal to −1 times the element ˆ∂_{G}^{1}(ϵ_{K/ℚ}(0)) used in [7, §5.1.1] and RΩ^{loc}(ℚ(1)_{K}, ℤ[G]) is equal to the element RΩ^{loc}(K/ℚ, 1) defined in loc. cit. □

To prove Theorem 1.1 we shall combine some classical and rather explicit computations of Hasse (concerning Gauss sums) and Leopoldt (concerning integer rings in cyclotomic fields) with a refinement of some general results proved in [9, §5] and a systematic use of the Iwasawa theory of complexes in the spirit of Kato [27, 3.1.2] and Nekovář [32] and of the generalization of the fundamental exact sequence of Coleman theory obtained by Perrin-Riou in [34]. We would like to point out that, in addition to the connections discussed above, there are also links between our approach and aspects of the work of Kato [28], Fukaya and Kato [20] and Benois and Berger [2]. In particular, the main technical result that we prove (the validity of equality (16)) is closely related to [28, Th. 4.1] and hence also to the material of [20, §3]. Indeed, Theorem 1.1 (in the case r = 1) provides a natural generalization of the results discussed in [20, §3.6]. However, the arguments of both loc. cit. and [28] do not cover the prime 2 and also leave open certain sign ambiguities, and much effort is required in the present article to deal with such subtleties. Both authors were introduced to the subject of Tamagawa number conjectures by John Coates. It is therefore a particular pleasure for us to dedicate this paper to him on the occasion of his sixtieth birthday. Acknowledgements The authors are grateful to Denis Benois for several very helpful conversations regarding this project and also to Laurent Berger and Manuel Breuning for some very helpful advice. Much of the research for this article was completed when the authors held visiting positions in the Mathematics Department of Harvard University and they are very grateful to Dick Gross for making that visit possible.

2. Equivariant local Tamagawa numbers

In this article we must compute explicitly certain equivariant local Tamagawa numbers, as defined in [9]. For the reader’s convenience, we therefore first quickly review the general definition of such invariants. All further details of this construction can be found in loc. cit.

2.1. We fix a motive M that is defined over ℚ (if M is defined over a general number field as in [9], then we use induction to reduce to the base field ℚ) and we assume that M is endowed with an action of a finite dimensional semisimple ℚ-algebra A. We write H_{dR}(M) and H_{B}(M) for the de Rham and Betti realisations of M and for each prime number p we denote by V_{p} = H_{p}(M) the p-adic realisation of M. We fix a ℤ-order 𝔄 in A such that, for each prime p, if we set 𝔄_{p} := 𝔄⊗_{ℤ}ℤ_{p}, then there exists a full projective Galois stable 𝔄_{p}-sublattice T_{p} of V_{p}. We also fix a finite set S of places of ℚ containing ∞ and all primes of bad reduction for M and then set S_{p} := S ∪ {p} and S_{p,f} := S_{p} \ {∞}. For any associative unital ring R we write D^{perf}(R) for the derived catgeory of perfect complexes of R-modules. We also let Det_{R} : D^{perf}(R) → V (R) denote the universal determinant functor to the Picard category of virtual objects of R (which is denoted by P ↦ [P] in [9]) and ⊗_{R} the product functor in V (R) (denoted by ⊠ in [9]). In particular, if R is commutative, then Det_{R} is naturally isomorphic to the Knudsen-Mumford functor to graded invertible R-modules. We denote by 1_{R} a unit object of V (R) and recall that the group K_{1}(R) can be identified with Aut_{V (R)}(L) for any object L of V (R) (and in particular therefore with π_{1}(V (R)) := Aut_{V (R)}(1_{R})). For each automorphism α : W → W of a finitely generated projective R-module W we denote by Det_{R}(α|W) the element of K_{1}(R) that is represented by α. We let ζ(R) denote the centre of R. If X is any R-module upon which complex conjugation acts as an endomorphism of R-modules, then we write X^{+} and X^{−} for the R-submodules of X upon which complex conjugation acts as multiplication by 1 and −1 respectively. For any ℚ-vector space W we set W_{ℂ} = W ⊗_{ℚ} ℂ, W_{ℝ} = W ⊗_{ℚ} ℝ and W_{p} = W ⊗_{ℚ} ℚ_{p} for each prime p.

2.2. The virtual object

Ξ^{loc}(M) := Det_{A}(H_{dR}(M)) ⊗_{A} Det_{A}^{−1}(H_{B}(M))

is endowed with a canonical morphism

ϑ_{∞}^{loc} : A_{ℝ} ⊗_{A} Ξ^{loc}(M) ≅ 1_{A_{ℝ}}.

To describe this morphism we note that the canonical period isomorphism H_{dR}(M)_{ℂ} ≅ H_{B}(M)_{ℂ} induces an isomorphism of A_{ℝ}-modules

(4)  H_{dR}(M)_{ℝ} = (H_{dR}(M)_{ℂ})^{+} ≅ (H_{B}(M)_{ℂ})^{+}

and that there is also a canonical isomorphism of A_{ℝ}-modules

(5) (H_{B}(M)_{ℂ})^{+} = (H_{B}(M)^{+} ⊗_{ℚ} ℝ) ⊕ (H_{B}(M)^{−} ⊗_{ℚ} ℝ(2πi)^{−1})

≅ (H_{B}(M)^{+} ⊗_{ℚ} ℝ) ⊕ (H_{B}(M)^{−} ⊗_{ℚ} ℝ) = H_{B}(M)_{ℝ}

where the central map results from identifying ℝ(2πi)^{−1} with ℝ by sending (2πi)^{−1} to 1. By applying Det_{A_{ℝ}} to the composite of (4) and (5) one obtains a morphism (ϑ_{∞}^{loc})′ : A_{ℝ}⊗_{A}Ξ^{loc}(M) ≅ 1_{A_{ℝ}} and ϑ_{∞}^{loc} is defined in [9, (57)] to be the composite of (ϑ_{∞}^{loc})′ and the ‘sign’ elements ϵ_{B} := Det_{A}(−1 | H_{B}(M)^{+}) and ϵ_{dR} := Det_{A}(−1 | F^{0}H_{dR}(M)) of π_{1}(V (A_{ℝ})) ≅ K_{1}(A_{ℝ}).

2.3. Following [9, (66), (67)], we set

Λ_{p}(S, V_{p}) := ( ⨂_{ℓ∈S_{p,f}} Det_{A_{p}}^{−1} RΓ(ℚ_{ℓ}, V_{p})) ⊗_{A_{p}} Det_{A_{p}}^{−1}(V_{p}),

and let

θ_{p} : A_{p} ⊗_{A} Ξ^{loc}(M) ≅ Λ_{p}(S, V_{p})

denote the morphism in V (A_{p}) obtained by taking the product of the morphisms θ_{p}^{ℓ-part} for ℓ ∈ S_{p,f} that are discussed in the next subsection.

2.4. There exists a canonical morphism in V (A_{p}) of the form

θ_{p}^{p-part} : A_{p} ⊗_{A} Ξ^{loc}(M) → Det_{A_{p}}^{−1} RΓ(ℚ_{p}, V_{p}) ⊗_{A_{p}} Det_{A_{p}}^{−1}(V_{p}).

This morphism results by applying Det_{A_{p}} to each of the following: the canonical comparison isomorphism H_{B}(M)_{p} ≅ V_{p}; the (Poincaré duality) exact sequence 0 → (H_{dR}(M^{∗}(1))/F^{0})^{∗} → H_{dR}(M) → H_{dR}(M)/F^{0} → 0; the canonical comparison isomorphisms (H_{dR}(M)/F^{0})_{p} ≅ t_{p}(V_{p}) and (H_{dR}(M^{∗}(1))/F^{0})_{p}^{∗} ≅ t_{p}(V_{p}^{∗}(1))^{∗}; the exact triangle

(6)  RΓ_{f}(ℚ_{p}, V_{p}) → RΓ(ℚ_{p}, V_{p}) → RΓ_{f}(ℚ_{p}, V_{p}^{∗}(1))^{∗}[−2] →

which results from [9, (18) and Lem. 12a)]; the exact triangle

(7)  t_{p}(W)[−1] → RΓ_{f}(ℚ_{p}, W) → (D_{cris}(W) ⟶^{1−φ_{v}} D_{cris}(W)) →

of [9, (22)] for both W = V_{p} and W = V_{p}^{∗}(1), where the first term of the last

complex is placed in degree 0 and Det_{A_{p}}(D_{cris}(W) ⟶^{1−φ_{v}} D_{cris}(W)) is identified with 1_{A_{p}} via the canonical morphism Det_{A_{p}}(D_{cris}(W))⊗_{A_{p}} Det_{A_{p}}^{−1}(D_{cris}(W)) → 1_{A_{p}}. For each ℓ ∈ S_{p,f} \ {p} there exists a canonical morphism in V (A_{p}) of the form

θ_{p}^{ℓ-part} : 1_{A_{p}} ≅ Det_{A_{p}}^{−1} RΓ(ℚ_{ℓ}, V_{p}).

For more details about this morphism see Proposition 7.1.

2.5. From [9, (71), (78)] we recall that there exists a canonical object Λ_{p}(S, T_{p}) of V (𝔄_{p}) and a canonical morphism in V (A_{p}) of the form

θ′_{p} : Λ_{p}(S, V_{p}) ≅ A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p})

(the definitions of Λ_{p}(S, T_{p}) and θ′_{p} are to be recalled further in §7.2). We set

ϑ_{p}^{loc} := ϵ(S, p) ∘ θ′_{p} ∘ θ_{p} : A_{p} ⊗_{A} Ξ^{loc}(M) ≅ A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p})

where ϵ(S, p) is the element of π_{1}(V (A_{p})) that corresponds to multiplication by −1 on the complex ⨁_{ℓ∈S_{p,f}} RΓ_{/f}(ℚ_{ℓ}, V_{p}) which is defined in [9, (18)]. If M is a direct factor of h^{n}(X)(t) for any non-negative integer n, smooth projective variety X and integer t, then [9, Lem. 15b)] implies that the data

(∏_{p} Λ_{p}(S, T_{p}), Ξ^{loc}(M), ∏_{p} ϑ_{p}^{loc}; ϑ_{∞}^{loc}),

where p runs over all prime numbers, gives rise (conjecturally in general, but unconditionally in the case of Tate motives) to a canonical element RΩ^{loc}(M, 𝔄) of K_{0}(𝔄, ℝ). For example, if A is commutative, then 1_{A_{ℝ}} = (A_{ℝ}, 0) and K_{0}(𝔄, ℝ) identifies with the multiplicative group of invertible 𝔄-sublattices of A_{ℝ} and, with respect to this identification, RΩ^{loc}(M, 𝔄) corresponds to the (conjecturally invertible) 𝔄-sublattice Ξ of A_{ℝ} that is defined by the equality

ϑ_{∞}^{loc} (⋂_{p}(Ξ^{loc}(M) ∩ (ϑ_{p}^{loc})^{−1}(Λ_{p}(S, T_{p})))) = (Ξ, 0),

where the intersection is taken over all primes p.

2.6. We write L_{∞}(_{A}M, s) and ϵ(_{A}M, 0) for the archimedean Euler factor and epsilon constant that are defined in [9, §4.1]. Also, with ρ ∈ ℤ^{π_{0}(Spec(ζ(A_{ℝ})))} denoting the algebraic order at s = 0 of the completed ζ(A_{ℂ})-valued L-function Λ(_{A^{op}}M^{∗}(1), s) that is defined in loc. cit., we set

E(_{A}M) := (−1)^{ρ}ϵ(_{A}M, 0)\frac{L_{∞}^{∗}(_{A^{op}}M^{∗}(1), 0)}{L_{∞}^{∗}(_{A}M, 0)} ∈ ζ(A_{ℝ})^{×}.

Following [9, §5.1], we define

L^{loc}(M, 𝔄) := δ̂_{𝔄,ℝ}^{1}(E(_{A}M)) ∈ K_{0}(𝔄, ℝ)

where δ̂_{𝔄,ℝ}^{1} : ζ(A_{ℝ})^{×} → K_{0}(𝔄, ℝ) is the ‘extended boundary homomorphism’ of [9, Lem. 9] (so, if A is commutative, then L^{loc}(M, 𝔄) = 𝔄 · E(_{A}M) ⊂ A_{ℝ}). Finally, we let

(8)  T Ω^{loc}(M, 𝔄)′ := L^{loc}(M, 𝔄) + RΩ^{loc}(M, 𝔄) ∈ K_{0}(𝔄, ℝ)

denote the ‘equivariant local Tamagawa number’ that is defined in [9, just prior to Th. 5.1].

3. Normalizations and notation

3.1. Normalizations. In this section we fix an arbitrary Galois extension of number fields K/k, set G := Gal(K/k) and for each integer t write T Ω(ℚ(t)_{K}, ℤ[G])′ for the element of K_{0}(ℤ[G], ℝ) that is defined (unconditionally) by [9, Conj. 4(iii)] in the case M = ℚ(t)_{K} and 𝔄 = ℤ[G]. Let r be a strictly positive integer. Then the computations of [10, 17] show that [9, Conj. 4(iv)] requires that the morphism ϑ_{∞} : ℝ⊗_{ℚ} Ξ(ℚ(1−r)_{K}) → 1_{V (ℝ[G])} constructed in [9, §3.4] should be normalized by using −1 times the Dirichlet (resp. Beilinson if r > 1) regulator map, rather than the Dirichlet (resp. Beilinson) regulator map itself as used in [9]. To incorporate this observation we set

(9)  T Ω(ℚ(1 − r)_{K}, ℤ[G]) := T Ω(ℚ(1 − r)_{K}, ℤ[G])′ + δ_{K/k}(r)

where δ_{K/k}(r) is the image under the canonical map K_{1}(ℝ[G]) → K_{0}(ℤ[G], ℝ) of the element Det_{ℚ[G]}(−1 | K_{2r−1}(O_{K})^{∗} ⊗_{ℤ} ℚ). To deduce the validity of (1) from the result of [9, Th. 5.3] it is thus also necessary to renormalise the definition of either T Ω(ℚ(r)_{K}, ℤ[G])′ or of the element T Ω^{loc}(ℚ(r)_{K}, ℤ[G])′ defined by (8). Our proof of Theorem 1.1 now shows that the correct normalization is to set

T Ω(ℚ(r)_{K}, ℤ[G]) := T Ω(ℚ(r)_{K}, ℤ[G])′

and

(10)  T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) := T Ω^{loc}(ℚ(r)_{K}, ℤ[G])′ + δ_{K/k}(r).

Note that the elements defined in (9) and (10) satisfy all of the functorial properties of T Ω(ℚ(1 − r)_{K}, ℤ[G])′ and T Ω^{loc}(ℚ(r)_{K}, ℤ[G])′ that are proved in [9, Th. 5.1, Prop. 4.1]. Further, with these definitions, the equalities (1) and (2) are valid and it can be shown that the conjectural vanishing of T Ω^{loc}(ℚ(1)_{K}, ℤ[G]) is compatible with the conjectures discussed in both [4] and [7]. Thus, in the remainder of this article we always use the notation T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) as defined in (10).

3.2. The abelian case. Until explicitly stated otherwise, in the sequel we consider only abelian groups. Thus, following [9, §2.5], we use the graded determinant functor of [29] in place of virtual objects (for a convenient review of all relevant properties of the determinant functor see [11, §2]). However, we caution the reader that for reasons of typographical clarity we sometimes do not distinguish between a graded invertible module and the underlying invertible module. We note that, when proving Theorem 1.1, the functorial properties of the elements T Ω^{loc}(ℚ(r)_{K}, ℤ[Gal(K/k)]) allow us to assume that k = ℚ and also that K is generated by a primitive N-th root of unity for some natural number N ≢ 2 mod 4. Therefore, until explicitly stated otherwise, we henceforth fix the following notation:

K := ℚ(e^{2πi/N}); G := Gal(K/ℚ); M := ℚ(r)_{K}, r ≥ 1; A := ℚ[G].

For any natural number n we also set ζ_{n} := e^{2πi/n} and denote by σ_{n} the resulting complex embedding of the field ℚ(ζ_{n}). For each complex character η of G we denote by e_{η} = \frac{1}{|G|} ∑_{g∈G} η(g^{−1})g the associated idempotent in A_{ℂ}. For each ℚ-rational character (or equivalently, Aut(ℂ)-conjugacy class of ℂ-rational characters) χ of G we set e_{χ} = ∑_{η∈χ} e_{η} ∈ A and denote by ℚ(χ) = e_{χ}A the field of values of χ. There is a ring decomposition A = ∏_{χ} ℚ(χ) and a corresponding decomposition Y = ∏_{χ} e_{χ}Y for any A-module Y. We make similar conventions for ℚ_{p}-rational characters of G.

4. An explicit analysis of T Ω^{loc}(ℚ(r)_{K}, ℤ[G])

In this section we reduce the proof of Theorem 1.1 to the verification of an explicit local equality (cf. Proposition 4.4).

4.1. The archimedean component of T Ω^{loc}(ℚ(r)_{K}, ℤ[G]). In this subsection we explicate the morphism ϑ_{∞}^{loc} defined in §2.2 and the element E(_{A}M) ∈ A_{ℝ}^{×} defined in §2.6. The de Rham realization H_{dR}(M) of M identifies with K, considered as a free A-module of rank one (by means of the normal basis theorem). The Betti realisation H_{B}(M) of M identifies with the ℚ-vector space Y_{Σ} with basis equal to the set Σ := Hom(K, ℂ) of field embeddings and is therefore also a free A-module of rank one (with basis σ_{N}). We set Y_{Σ}^{−1} := Hom_{A}(Y_{Σ}, A). Then, by [9, Th. 5.2], we know that (ϑ_{∞}^{loc})^{−1}((E(_{A}M)^{−1}, 0)) belongs to Ξ^{loc}(M) = (K ⊗_{A} Y_{Σ}^{−1}, 0) and we now describe this element explicitly.

Proposition 4.1. We define an element ϵ_{∞} := ∑_{χ} ϵ_{∞,χ}e_{χ} of A^{×} by setting

ϵ_{∞,χ} := \begin{cases}−2 & if χ(−1) = (−1)^{r} \\ −\frac{1}{2} & if χ(−1) = −(−1)^{r} and (χ ≠ 1 or r > 1) \\ \frac{1}{2} & if χ = 1 and r = 1. \end{cases}

Then

(ϑ_{∞}^{loc})^{−1}((E(_{A}M)^{−1}, 0)) = (ϵ_{∞}β_{N} ⊗ σ_{N}^{−1}, 0) ∈ (K ⊗_{A} Y_{Σ}^{−1}, 0)

where σ_{N}^{−1} is the (unique) element of Y_{Σ}^{−1} which satisfies σ_{N}^{−1}(σ_{N}) = 1 and β_{N} is the (unique) element of K = ∏_{χ} e_{χ}K which satisfies

e_{χ}β_{N} := [K : ℚ(ζ_{f_{χ}})]^{−1}(r − 1)!f_{χ}^{r−1} · e_{χ}ζ_{f_{χ}}

for all ℚ-rational characters χ of G.

Proof. For each Dirichlet character η of G the functional equation of L(η, s) is

L(η, s) = \frac{τ(η)}{2i^{δ}} (\frac{2π}{f_{η}} )^{s} \frac{1}{Γ(s) cos(\frac{π(s−δ)}{2})}L(η̄, 1 − s)

where f_{η} is the conductor of η and

(11)  τ(η) = ∑_{a=1}^{f_{η}} η(a)e^{2πia/f_{η}}; η(−1) = (−1)^{δ}, δ ∈ {0, 1}

Documenta Mathematica · Extra Volume Coates (2006) 133–163

(cf. [36, Ch. 4]). Thus, by its very definition in §2.6, the η-component of the element E(_{A}M)^{−1} of A_{ℂ} = ∏_{η} ℂ is the leading Taylor coefficient at s = r of the meromorphic function

(−1)^{ρ_{η}} \frac{2i^{δ}}{τ(η)} (\frac{f_{η}}{2π} )^{s} Γ(s) cos(\frac{π(s − δ)}{2}); ρ_{η} = \begin{cases}1 & r = 1, η = 1 \\ 0 & else. \end{cases}

Hence we have

E(_{A}M)_{η}^{−1} = \begin{cases}\frac{2i^{δ}}{τ(η)} (\frac{f_{η}}{2π} )^{r} (r − 1)!(−1)^{\frac{r−δ}{2}} , & r − δ even \\ (−1)^{ρ_{η}} \frac{2i^{δ}}{τ(η)} (\frac{f_{η}}{2π} )^{r} (r − 1)!(−1)^{\frac{r−δ+1}{2}} \frac{π}{2} , & r − δ odd \end{cases}

which, after collecting powers of i and using the relation τ(η)τ(η̄) = η(−1)f_{η}, can be written as

E(_{A}M)_{η}^{−1} = \begin{cases}2 τ(η̄)(2πi)^{−r}f_{η}^{r−1}(r − 1)!, & r − δ even \\ (−1)^{ρ_{η}+1} \frac{1}{2} τ(η̄)(2πi)^{−(r−1)}f_{η}^{r−1}(r − 1)!, & r − δ odd. \end{cases}

Lemma 4.2. The isomorphism Y_{Σ,ℂ}^{+} = (H_{B}(M)_{ℂ})^{+} ≅ H_{B}(M)_{ℝ} = Y_{Σ,ℝ} in (5) is given by

∑_{g∈G} α_{g}g^{−1}σ_{N} ↦ ∑_{g∈G/<c>}(ℜ(α_{g})(1 + (−1)^{r}c) − 2πℑ(α_{g})(1 − (−1)^{r}c))g^{−1}σ_{N}

where c ∈ G is complex conjugation, G acts on Σ via (gσ)(x) = σ(g(x)) and ℜ(α), resp. ℑ(α), denotes the real, resp. imaginary, part of α ∈ ℂ.

Proof. An element x := ∑_{g∈G} α_{g}g^{−1}σ_{N} of Y_{Σ,ℂ} belongs to the subspace Y_{Σ,ℂ}^{+} if and only if one has α_{gc} = (−1)^{r}ᾱ_{g} for all g ∈ G. Writing

α_{g} = ℜ(α_{g}) − (2πi)^{−1}(2π)ℑ(α_{g}), ᾱ_{g} = ℜ(α_{g}) + (2πi)^{−1}(2π)ℑ(α_{g})

we find that

x = ∑_{g∈G/<c>}(ℜ(α_{g})(1 + (−1)^{r}c) − (2πi)^{−1}2πℑ(α_{g})(1 − (−1)^{r}c))g^{−1}σ_{N}.

But ∑_{g∈G/<c>}(2πi)^{−1}2πℑ(α_{g})(1 − (−1)^{r}c)g^{−1}σ_{N} ∈ H_{B}(M)^{−} ⊗_{ℚ} ℝ · i and the central map in (5) sends (2πi)^{−1} to 1. This implies the claimed result. □

The canonical comparison isomorphism K_{ℂ} = H_{dR}(M)_{ℂ} ≅ H_{B}(M)_{ℂ} = Y_{Σ,ℂ} which occurs in (4) sends any element β of K to

∑_{g∈G} σ_{N}(gβ)(2πi)^{−r}g^{−1}σ_{N} = ∑_{a∈(ℤ/Nℤ)^{×}} σ_{N}τ_{a}(β)(2πi)^{−r}τ_{a}^{−1}σ_{N}

where τ_{a}(ζ) = ζ^{a} for each N-th root of unity ζ. In particular, after composing this comparison isomorphism with the isomorphism of Lemma 4.2 we find that ζ_{f} is sent to the following element of Y_{Σ,ℝ}

∑_{a}(ℜ(e^{2πia/f}(2πi)^{−r})(1 + (−1)^{r}c) − 2πℑ(e^{2πia/f}(2πi)^{−r})(1 − (−1)^{r}c))τ_{a}^{−1}σ_{N}

where the summation runs over all elements a of (ℤ/Nℤ)^{×}/ ± 1. For each Dirichlet character η the η-component of this element is equal to e_{η}σ_{N} multiplied by

∑_{a∈(ℤ/Nℤ)^{×}/±1}(2π)^{−r}ℜ(e^{2πia/f}i^{−r})\overline{η(a)} · 2

= ∑_{a∈(ℤ/Nℤ)^{×}/±1}(2πi)^{−r}(e^{2πia/f} + (−1)^{r}e^{−2πia/f})\overline{η(a)}
=(2πi)^{−r} ∑_{a∈(ℤ/Nℤ)^{×}} e^{2πia/f} \overline{η(a)}

if η(−1) = (−1)^{r} (so δ − r is even), resp. by

− 2π ∑_{a∈(ℤ/Nℤ)^{×}/±1}(2π)^{−r}ℑ(e^{2πia/f}i^{−r})\overline{η(a)} · 2

= − 2π ∑_{a∈(ℤ/Nℤ)^{×}/±1}(2πi)^{−r} \frac{e^{2πia/f} − (−1)^{r}e^{−2πia/f}}{i} \overline{η(a)}
=(2πi)^{−(r−1)} ∑_{a∈(ℤ/Nℤ)^{×}} e^{2πia/f} \overline{η(a)}

if η(−1) = −(−1)^{r} (so δ − r is odd). Taking f = f_{η} we find that the (η-part of the) morphism (ϑ_{∞}^{loc})′ : A_{ℝ} ⊗_{A} Ξ^{loc}(M) ≅ (A_{ℝ}, 0) defined in §2.2 sends

(12) e_{η}ζ_{f_{η}} ⊗_{ℂ} e_{η}σ_{N}^{−1} ↦ \begin{cases}(2πi)^{−r}[K : ℚ(f_{η})]τ(η̄) & if η(−1) = (−1)^{r} \\ (2πi)^{−(r−1)}[K : ℚ(f_{η})]τ(η̄) & if η(−1) = −(−1)^{r}. \end{cases}

Now ϑ_{∞}^{loc} is defined to be the composite of (ϑ_{∞}^{loc})′ and the sign factors ϵ_{dR} and ϵ_{B} that are defined at the end of §2.2. But it is easily seen that e_{dR} = 1, that (ϵ_{B})_{χ} = −1 for χ(−1) = (−1)^{r} and that (ϵ_{B})_{χ} = 1 otherwise. Thus, upon comparing (12) with the description of E(_{A}M)_{η}^{−1} before Lemma 4.2 one verifies the statement of Proposition 4.1. □

4.2. Reduction to the p-primary component. By [9, Th. 5.2] we know that T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) belongs to the subgroup K_{0}(ℤ[G], ℚ) of K_{0}(ℤ[G], ℝ). Recalling the direct sum decomposition K_{0}(ℤ[G], ℚ) ≅ ⨁_{ℓ} K_{0}(ℤ_{ℓ}[G], ℚ_{ℓ}) over all primes ℓ from [9, (13)], we may therefore prove Theorem 1.1 by showing that, for each prime ℓ, the projection T Ω^{loc}(ℚ(r)_{K}, ℤ[G])_{ℓ} of T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) to K_{0}(ℤ_{ℓ}[G], ℚ_{ℓ}) vanishes. Henceforth we therefore fix a prime number p and shall analyze T Ω^{loc}(ℚ(r)_{K}, ℤ[G])_{p}. We denote by

T_{p} := Ind_{K}^{ℚ} ℤ_{p}(r) ⊂ V_{p} := Ind_{K}^{ℚ} ℚ_{p}(r) = H_{p}(M)

the natural lattice in the p-adic realisation V_{p} of M. Then by combining the definition of T Ω^{loc}(ℚ(r)_{K}, ℤ[G]) from (10) (and (8)) together with the explicit description of Proposition 4.1 one finds that T Ω^{loc}(ℚ(r)_{K}, ℤ[G])_{p} = 0 if and only if

ℤ_{p}[G] · ϵ(r)ϵ(S, p) · θ′_{p} ∘ θ_{p}((ϵ_{∞}β_{N} ⊗ σ_{N}^{−1}, 0)) = Λ_{p}(S, T_{p})

where θ_{p} is as defined in §2.3, Λ_{p}(S, T_{p}), θ′_{p} and ϵ(S, p) ∈ A_{p}^{×} are as discussed in §2.5 and we have set ϵ(r) := Det_{A}(−1 | K_{2r−1}(O_{K})^{∗} ⊗_{ℤ} ℚ) ∈ A^{×}.

Lemma 4.3. We set

ϵ_{p} := Det_{A_{p}}(2|V_{p}^{+}) Det_{A_{p}}(2|V_{p}^{−})^{−1} ∈ A_{p}^{×}.

Then, with ϵ_{∞} as defined in Proposition 4.1, there exists an element u(r) of ℤ_{p}[G]^{×} such that ϵ(r)ϵ(S, p)ϵ_{∞} = u(r)ϵ_{p}.

Proof. We recall that ϵ(S, p) is a product of factors Det_{A_{p}}(−1|RΓ_{/f}(ℚ_{ℓ}, V_{p})). Further, the quasi-isomorphism RΓ_{/f}(ℚ_{ℓ}, V_{p}) ≅ RΓ_{f}(ℚ_{ℓ}, V_{p}^{∗}(1))^{∗}[−2] from [9, Lem. 12a)] implies that each such complex is quasi-isomorphic to a complex of the form W → W (indeed, this is clear if ℓ ≠ p and is true in the case ℓ = p because the tangent space of the motive ℚ(1 − r)_{K} vanishes for r ≥ 1) and so one has ϵ(S, p) = 1. We next note that if ϵ(r) := ∑_{χ} ϵ(r)_{χ}e_{χ} with ϵ(r)_{χ} ∈ {±1}, then the explicit structure of the ℚ[G]-module K_{2r−1}(O_{K})^{∗} ⊗_{ℤ} ℚ (cf. [17, p. 86, p. 105]) implies that ϵ(r)_{χ} = 1 if either r = 1 and χ is trivial or if χ(−1) = (−1)^{r}, and that ϵ(r)_{χ} = −1 otherwise. Thus, after recalling the explicit definitions of ϵ_{∞} and ϵ_{p}, it is straightforward to check that the claimed equality ϵ(r)ϵ(S, p)ϵ_{∞} = u(r)ϵ_{p} is valid with u(r) = −(−1)^{r}c where c ∈ G is complex conjugation. □

The element ϵ_{p} in Lemma 4.3 is equal to the element ϵ_{V_{p}} that occurs in Proposition 7.2 below (with V_{p} = Ind_{K}^{ℚ}ℚ_{p}(r)). Hence, upon combining Lemma 4.3 with the discussion which immediately precedes it and the result of Proposition 7.2 we may deduce that T Ω^{loc}(ℚ(r)_{K}, ℤ[G])_{p} = 0 if and only if

(13) ℤ_{p}[G]·θ_{p}((β_{N} ⊗σ_{N}^{−1}, 0)) = (⨂_{ℓ|Np} Det_{ℤ_{p}[G]}^{−1} RΓ(ℚ_{ℓ}, T_{p}))⊗_{ℤ_{p}[G]} (T_{p}^{−1}, −1).

Here we have set T_{p}^{−1} := Hom_{ℤ_{p}[G]}(T_{p}, ℤ_{p}[G]) and also used the fact that, since T_{p} is a free rank one ℤ_{p}[G]-module, one has Det_{ℤ_{p}[G]}^{−1}(T_{p}) = (T_{p}^{−1}, −1). Now Shapiro’s Lemma allows us to identify the complexes RΓ(ℚ_{ℓ}, T_{p}) and RΓ(ℚ_{ℓ}, V_{p}) with RΓ(K_{ℓ}, ℤ_{p}(r)) and RΓ(K_{ℓ}, ℚ_{p}(r)) respectively. Further, the complex RΓ(K_{p}, ℚ_{p}(r)) is acyclic outside degree 1 for r > 1, and for r = 1 one has a natural exact sequence of ℚ_{p}[G]-modules

(14) 0 → Ô_{K_{p}}^{×} → K̂_{p}^{×} ≅ H^{1}(K_{p}, ℚ_{p}(1)) ⟶^{val} ∏_{v|p} ℚ_{p} ≅ H^{2}(K_{p}, ℚ_{p}(1)) → 0

where the first isomorphism is induced by Kummer theory and the second by the invariant map on the Brauer group. Our notation here is that M̂ := (\underleftarrow{lim}_{n} M/p^{n}M) ⊗_{ℤ_{p}} ℚ_{p} for any abelian group M. We let

K_{p} = D_{dR}(V_{p}) ⟶^{exp} H_{f}^{1}(K_{p}, ℚ_{p}(r))

denote the exponential map of Bloch and Kato for the representation V_{p} of Gal(ℚ̄_{p}/ℚ_{p}). This map is bijective (since r > 0) and H_{f}^{1}(K_{p}, ℚ_{p}(r)) coincides with Ô_{K_{p}}^{×} for r = 1 and with H^{1}(K_{p}, ℚ_{p}(r)) for r > 1 (cf. [5]). Also, both source and target for the map exp are free A_{p}-modules of rank one. By using the sequence (14) for r = 1 we therefore find that for each r ≥ 1 there exists an isomorphism of graded invertible A_{p}-modules of the form

(15) ẽxp : (K_{p}, 1) ⟶^{exp} (H_{f}^{1}(K_{p}, ℚ_{p}(r)), 1) ≅ Det_{A_{p}}^{−1} RΓ(K_{p}, ℚ_{p}(r)).

For any subgroup H ⊆ G we set

e_{H} := ∑_{χ(H)=1} e_{χ} = \frac{1}{|H|} ∑_{g∈H} g.

Also, for each prime ℓ we denote by J_{ℓ} and D_{ℓ} the inertia and decomposition groups of ℓ in G. For x ∈ A_{p} we then set

e_{ℓ}(x) := 1 + (x − 1)e_{J_{ℓ}} ∈ A_{p}

(so x ↦ e_{ℓ}(x) is a multiplicative map that preserves the maximal ℤ_{p}-order in A_{p}) and we denote by Fr_{ℓ} ∈ G ⊂ A any choice of a Frobenius element.

Proposition 4.4. We define an element e_{p}^{∗}(1 − p^{r−1} Fr_{p}^{−1}) of A_{p}^{×} by setting

e_{χ}e_{p}^{∗}(1 − p^{r−1} Fr_{p}^{−1}) = \begin{cases}e_{χ}e_{p}(1 − p^{r−1} Fr_{p}^{−1}), & if r > 1 or χ(D_{p}) ≠ 1 \\ |D_{p}/J_{p}|^{−1}e_{χ}, & otherwise. \end{cases}

Then one has T Ω^{loc}(ℚ(r)_{K}, ℤ[G])_{p} = 0 if and only if

(16) ℤ_{p}[G] · ∏_{\begin{matrix}ℓ|N \\ ℓ≠p \end{matrix}} e_{ℓ}(− Fr_{ℓ}^{−1}) e_{p}(1 − \frac{Fr_{p}}{p^{r}})^{−1}e_{p}^{∗}(1 − p^{r−1} Fr_{p}^{−1}) ẽxp((β_{N}, 1))

= Det_{ℤ_{p}[G]}^{−1} RΓ(K_{p}, ℤ_{p}(r)).

Proof. It suffices to prove that (16) is equivalent to (13). Now, by its definition in §2.3, the morphism θ_{p} which occurs in (13) is induced by taking the tensor product of the morphisms

θ_{p}^{p-part} : A_{p} ⊗_{A} Ξ^{loc}(M) ≅ Det_{A_{p}}^{−1} RΓ(K_{p}, ℚ_{p}(r)) ⊗_{A_{p}} (V_{p}^{−1}, −1),

where we set V_{p}^{−1} := Hom_{A_{p}}(V_{p}, A_{p}), and for each prime ℓ | N with ℓ ≠ p

θ_{p}^{ℓ-part} : (A_{p}, 0) ≅ Det_{A_{p}}^{−1} RΓ(K_{ℓ}, ℚ_{p}(r)).

In addition, for W = V_{p} the exact triangle (7) identifies with

K_{p}[−1] → RΓ_{f}(ℚ_{p}, V_{p}) → (D_{cris}(V_{p}) ⟶^{1−p^{−r} Fr_{p}} D_{cris}(V_{p}))

(with this last complex concentrated in degrees 0 and 1), and there is a canonical quasi-isomorphism

RΓ_{f}(ℚ_{p}, V_{p}^{∗}(1))^{∗}[−2] ≅ (D_{cris}(V_{p}) ⟶^{1−p^{r−1} Fr_{p}^{−1}} D_{cris}(V_{p})) ,

where the latter complex is concentrated in degrees 1 and 2. The identity map on D_{cris}(V_{p}) therefore induces isomorphisms of graded invertible A_{p}-modules

(17) (K_{p}, 1) ≅ Det_{A_{p}}^{−1} RΓ_{f}(ℚ_{p}, V_{p}); (A_{p}, 0) ≅ Det_{A_{p}} RΓ_{f}(ℚ_{p}, V_{p}^{∗}(1))^{∗}[−2].

The morphism θ_{p}^{p-part} is thus induced by (17) and (6) together with the (elementary) comparison isomorphism

γ : Y_{Σ,p} = H_{B}(M)_{p} ≅ H_{p}(M) = V_{p}

between the Betti and p-adic realizations of M. On the other hand, the isomorphism ẽxp arises by passing to the cohomology sequence of (6) and then also using the identifications in (14) if r = 1. Hence, from [8, Lem. 1, Lem. 2], one has

(18)  θ_{p}^{p-part} = e_{p}(1 − p^{−r} Fr_{p})^{−1}e_{p}^{∗}(1 − p^{r−1} Fr_{p}^{−1})ẽxp ⊗_{A_{p}} γ^{−1}.

Now if ℓ ≠ p, then Proposition 7.1 below implies that

Det_{A_{p}}(−σ_{ℓ}ℓ^{−1}|(V_{p})_{I_{ℓ}})^{−1} · θ_{p}^{ℓ-part}((ℤ_{p}[G], 0)) = Det_{ℤ_{p}[G]}^{−1} RΓ(K_{ℓ}, ℤ_{p}(r)).

Thus, since γ(σ_{N}) is a ℤ_{p}[G]-basis of T_{p}, we find that (13) holds if and only if the element

∏_{\begin{matrix}ℓ|N \\ ℓ≠p \end{matrix}} Det_{A_{p}}(−σ_{ℓ}ℓ^{−1}|(V_{p})_{I_{ℓ}}) e_{p}(1 − p^{−r} Fr_{p})^{−1}e_{p}^{∗}(1 − p^{r−1} Fr_{p}^{−1}) ẽxp((β_{N}, 1))

is a ℤ_{p}[G]-basis of Det_{ℤ_{p}[G]}^{−1} RΓ(K_{p}, ℤ_{p}(r)). But

Det_{A_{p}}(−σ_{ℓ}ℓ^{−1}|(V_{p})_{I_{ℓ}}) = Det_{A_{p}}(− Fr_{ℓ}^{−1} ℓ^{r−1}|A_{p} · e_{J_{ℓ}}) = e_{ℓ}(− Fr_{ℓ}^{−1})e_{ℓ}(ℓ^{r−1})

and so Proposition 4.4 is implied by Lemma 4.5 below with u equal to the function which sends 0 to ℓ^{r−1} and all non-zero integers to 1. □

Lemma 4.5. Fix a prime number ℓ ≠ p. If u : ℤ → ℤ_{p}[G]^{×} is any function such that ℓ−1 divides u(0)−u(1) in ℤ_{p}[G], then the element ∑_{χ} u(ord_{ℓ}(f_{χ}))e_{χ} is a unit of ℤ_{p}[G].

Proof. If ℓ − 1 divides u(0) − u(1), then ℓ − 1 divides (u(1) − u(0))/u(1)u(0) = u(0)^{−1} −u(1)^{−1}. It follows that the function u^{−1} also satisfies the hypothesis of the lemma and so it suffices to prove that the element x_{u} := ∑_{χ} u(ord_{ℓ}(f_{χ}))e_{χ} belongs to ℤ_{p}[G]. To this end, we let J_{ℓ} = J_{ℓ,0} ⊆ G denote the inertia subgroup at ℓ and J_{ℓ,k} ⊆ J_{ℓ,k−1} ⊆ ⋯ ⊆ J_{ℓ,1} ⊆ J_{ℓ,0} its canonical filtration, so that a character χ satisfies ord_{ℓ}(f_{χ}) = k if and only if χ(J_{ℓ,k}) = 1 (and χ(J_{ℓ,k−1}) ≠ 1 if k > 0). Then

x_{u} = ∑_{k=0}^{k=K} u(k)(e_{J_{ℓ,k}} − e_{J_{ℓ,k−1}}) = ∑_{k=0}^{k=K−1}(u(k) − u(k + 1))e_{J_{ℓ,k}} + u(K)e_{J_{ℓ,K}}

where K = ord_{ℓ}(N) and we have set e_{J_{ℓ,−1}} := 0. For k ≥ 1 one has e_{J_{ℓ,k}} ∈ ℤ_{p}[G] since J_{ℓ,k} is an ℓ-group and ℓ ≠ p. If K = 0, then e_{J_{ℓ,0}} = e_{J_{ℓ,K}} = 1 also lies in ℤ_{p}[G]. Otherwise the assumptions that ℓ − 1 divides u(0) − u(1) and that ℓ ≠ p combine to imply that

(u(0) − u(1))e_{J_{ℓ,0}} = \frac{u(0) − u(1)}{(ℓ − 1)ℓ^{K−1}} ∑_{g∈J_{ℓ,0}} g ∈ ℤ_{p}[G],

as required. □

5. Local Iwasawa theory

As preparation for our proof of (16) we now prove a result in Iwasawa theory. We write

N = N_{0}p^{ν}; ν ≥ 0, p ∤ N_{0}.

For any natural number n we set G_{n} := Gal(ℚ(ζ_{n})/ℚ) ≅ (ℤ/nℤ)^{×}. We also let ℚ(ζ_{Np^{∞}}) denote the union of the fields ℚ(ζ_{Np^{m}}) over m ≥ 0 and set G_{Np^{∞}} := Gal(ℚ(ζ_{Np^{∞}})/ℚ). We then define

Λ := ℤ_{p}[[G_{Np^{∞}}]] = \underleftarrow{lim}_{n} ℤ_{p}[G_{Np^{n}}] ≅ ℤ_{p}[G_{N_{0}p̃}][[T]].

Here we have set p̃:= p for odd p and p̃:= 4 for p = 2, and the isomorphism depends on a choice of topological generator of Gal(ℚ(ζ_{Np^{∞}})/ℚ(ζ_{N_{0}p̃})) ≅ ℤ_{p}. We also set

T_{p}^{∞} := \underleftarrow{lim}_{n} Ind_{ℚ(ζ_{Np^{n}})}^{ℚ} ℤ_{p}(r).

This is a free rank one Λ-module upon which the absolute Galois group G_{ℚ} := Gal(ℚ̄/ℚ) acts by the character (χ_{cyclo})^{r}τ^{−1} where χ_{cyclo} : G_{ℚ} → ℤ_{p}^{×} is the cyclotomic character and τ : G_{ℚ} → G_{Np^{∞}} ⊆ Λ^{×} is the tautological character. In this section we shall describe (in Proposition 5.2) a basis of the invertible Λ-module Det_{Λ}^{−1} RΓ(ℚ_{p}, T_{p}^{∞}). We note first that the cohomology of RΓ(ℚ_{p}, T_{p}^{∞}) is naturally isomorphic to

(19)  H^{i}(ℚ_{p}, T_{p}^{∞}) ≅ \begin{cases}(\underleftarrow{lim}_{n} ℚ(ζ_{Np^{n}})_{p}^{×}/p^{n}) ⊗_{ℤ_{p}} ℤ_{p}(r − 1) & i = 1 \\ ∏_{v|p} ℤ_{p}(r − 1) & i = 2 \\ 0 & otherwise \end{cases}

where the limit is taken with respect to the norm maps (and ℚ(ζ_{Np^{n}})_{p} = ℚ(ζ_{Np^{n}}) ⊗_{ℚ} ℚ_{p} is a finite product of local fields). The valuation map induces a natural short exact sequence

(20)  0 → Z̃ := \underleftarrow{lim}_{n} O_{ℚ(ζ_{Np^{n}})_{p}}^{×}/p^{n} → \underleftarrow{lim}_{n} ℚ(ζ_{Np^{n}})_{p}^{×}/p^{n} ⟶^{val} ∏_{v|p} ℤ_{p} → 0

and in addition Perrin-Riou has constructed an exact sequence [34, Prop. 4.1.3]

(21)  0 → ∏_{v|p} ℤ_{p}(r) → Z̃(r − 1) ⟶^{θ_{r}^{P R}} R → ∏_{v|p} ℤ_{p}(r) → 0

where

R := {f ∈ ℤ[ζ_{N_{0}}]_{p}[[X]] | ψ(f) := ∑_{ζ^{p}=1} f(ζ(1 + X) − 1) = 0}

and ℤ[ζ_{N_{0}}]_{p} denotes the finite étale ℤ_{p}-algebra ℤ[ζ_{N_{0}}] ⊗_{ℤ} ℤ_{p}. We remark that, whilst p is assumed to be odd in [34] the same arguments show that the sequence (21) exists and is exact also in the case p = 2. The ℤ_{p}-module R carries a natural continuous G_{Np^{∞}}-action [34, 1.1.4], and with respect to this action all maps in (19), (20) and (21) are Λ-equivariant. In addition, if r = 1, then the exact sequence (21) is due to Coleman and the map θ_{1}^{P R} is given by

(22)  θ_{1}^{P R}(u) = (1 − \frac{ϕ}{p}) log(f_{u})

where f_{u} is the (unique) Coleman power series of the norm compatible system of units u with respect to (ζ_{p^{n}})_{n≥1} and one has ϕ(f_{u})(X) = f_{u}^{Fr_{p}}((1+X)^{p} −1).

Lemma 5.1. The Λ-module R is free of rank one with basis

β_{N_{0}}^{∞} := ξ_{N_{0}}(1 + X); ξ_{N_{0}} := ∑_{N_{1}|d|N_{0}} ζ_{d}

where N_{1} := ∏_{ℓ|N_{0}} ℓ.

Proof. The element ξ_{N_{0}} is a ℤ_{p}[G_{N_{0}}]-basis of ℤ[ζ_{N_{0}}]_{p}. Indeed, this observation (which is due originally to Leopoldt [30]) can be explicitly deduced from [31, Th. 2] after observing that the idempotents ε_{d} of loc. cit. belong to ℤ_{p}[G_{N_{0}}]. On the other hand, Perrin-Riou shows in [33, Lem. 1.5] that if W is the ring of integers in any finite unramified extension of ℤ_{p}, then W [[X]]^{ψ=0} is a free rank one W [[G_{p^{∞}}]]-module with basis 1 + X (her proof applies for all primes p, including p = 2). Since ℤ[ζ_{N_{0}}]_{p} is a finite product of such rings W and G_{Np^{∞}} ≅ G_{N_{0}} × G_{p^{∞}}, the result follows. □

Proposition 5.2. Let Q be the total ring of fractions of Λ (so Q is a finite product of fields). Using Lemma 5.1, we regard β_{N_{0}}^{∞} as a Q-basis of

R ⊗_{Λ} Q ≅ Z̃(r − 1) ⊗_{Λ} Q ≅ H^{1}(ℚ_{p}, T_{p}^{∞}) ⊗_{Λ} Q ≅ (Det_{Λ}^{−1} RΓ(ℚ_{p}, T_{p}^{∞})) ⊗_{Λ} Q,

where the first isomorphism is induced by (θ_{r}^{P R} ⊗_{Λ} Q)^{−1}, the second by (19) and the (r − 1)-fold twist of (20) and the third by (19). Then one has

Λ · β_{N_{0}}^{∞} = Det_{Λ}^{−1} RΓ(ℚ_{p}, T_{p}^{∞}) ⊂ (Det_{Λ}^{−1} RΓ(ℚ_{p}, T_{p}^{∞})) ⊗_{Λ} Q.

Proof. We note first that, since Λ is noetherian, Cohen-Macauley and semilocal, it is enough to prove that β_{N_{0}}^{∞} is a Λ_{𝔮}-basis of Det_{Λ_{𝔮}}^{−1} RΓ(ℚ_{p}, T_{p}^{∞})_{𝔮} for all height one prime ideals 𝔮 of Λ (see, for example, [17, Lem. 5.7]). In view of (19),

(20) and (21) this claim is immediate for prime ideals 𝔮 which are not in the support of the (torsion) Λ-modules ∏_{v|p} ℤ_{p}(r − 1) and ∏_{v|p} ℤ_{p}(r). On the other hand, since these modules are each p-torsion free, any prime 𝔮 which does lie in their support is regular in the sense that p ∉ 𝔮 (see, for example, [17, p. 90]). In particular, in any such case Λ_{𝔮} is a discrete valuation ring and so it suffices to check cancellation of the Fitting ideals of the occurring torsion modules. But the Fitting ideal of H^{2}(ℚ_{p}, T_{p}^{∞})_{𝔮} cancels against that of the module (∏_{v|p} ℤ_{p}(r − 1))_{𝔮} which occurs in the (r − 1)-fold twist of (20), whilst the Fitting ideals of the kernel and cokernel of θ_{r}^{P R} obviously cancel against each other. □

6. Descent calculations

In this section we deduce equality (16) as a consequence of Proposition 5.2 and thereby finish the proof of Theorem 1.1. At the outset we note that the natural ring homomorphism

(23)  Λ → ℤ_{p}[G] ⊆ ℚ_{p}[G] = ∏_{χ} ℚ_{p}(χ)

induces an isomorphism of perfect complexes of ℤ_{p}[G]-modules

RΓ(ℚ_{p}, T_{p}^{∞}) ⊗_{Λ}^{𝕃} ℤ_{p}[G] ≅ RΓ(ℚ_{p}, T_{p})

and hence also an isomorphism of determinants

Det_{Λ}^{−1} RΓ(ℚ_{p}, T_{p}^{∞}) ⊗_{Λ} ℤ_{p}[G] ≅ Det_{ℤ_{p}[G]}^{−1} RΓ(ℚ_{p}, T_{p}).

Taken in conjunction with Proposition 5.2, this shows that (β_{N_{0}}^{∞} ⊗_{Λ} 1, 1) is a ℤ_{p}[G]-basis of the graded module Det_{ℤ_{p}[G]}^{−1} RΓ(ℚ_{p}, T_{p}). Hence, if we define an element u of ℚ_{p}[G]^{×} by means of the equality

(24) ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1}) e_{p}(1 − \frac{Fr_{p}}{p^{r}})^{−1}e_{p}^{∗}(1 − p^{r−1} Fr_{p}^{−1}) ẽxp((β_{N}, 1))

= (u · β_{N_{0}}^{∞} ⊗_{Λ} 1, 1)

then it is clear that the equality (16) is valid if and only if u ∈ ℤ_{p}[G]^{×}.

6.1. The unit u′. To prove that the element u defined in (24) belongs to ℤ_{p}[G]^{×} we will compare it to the unit described by the following result.

Lemma 6.1. There exists a unit u′ ∈ ℤ_{p}[G]^{×} such that for any integer k with

0 ≤ k ≤ ν and any ℚ_{p}-rational character χ of G the element e_{χ}(ζ_{p^{k}}ξ_{N_{0}}^{Fr_{p}^{−k}}) is equal to

χ(u′) ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ−1} ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})e_{χ}ζ_{f_{χ}},  if k = ord_{p}(f_{χ})
χ(u′)(− Fr_{p}^{−1}) ∏_{ℓ|N,ℓ∤f_{χ}} \frac{1}{ℓ−1} ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})e_{χ}ζ_{f_{χ}}, if k = 1, ord_{p}(f_{χ}) = 0
0,  otherwise.

Proof. For d | N_{0} and k ≥ 0 we set d_{k} := p^{k}d and

a(d) := (d, 1) ∈ (ℤ/p^{ν}ℤ)^{×} × (ℤ/N_{0}ℤ)^{×} ≅ (ℤ/Nℤ)^{×} ≅ G so that ζ_{p^{k}}ζ_{d}^{Fr_{p}^{−k}} = ζ_{d_{k}}^{a(d)}. Since ξ_{N_{0}} = ∑_{N_{1}|d|N_{0}} ζ_{d} Lemma 6.2 below implies

(25) e_{χ}(ζ_{p^{k}}ξ_{N_{0}}^{Fr_{p}^{−k}}) = ∑_{N_{1}|d|N_{0},f_{χ}|d_{k}} \frac{ϕ(f_{χ})}{ϕ(d_{k})}μ(\frac{d_{k}}{f_{χ}})χ^{−1}(\frac{d_{k}}{f_{χ}})χ(a(d))e_{χ}ζ_{f_{χ}}.

The only non-vanishing summands in (25) are those for which the quotient d_{k}/f_{χ} is both square-free and prime to f_{χ}. Given the nature of the summation condition there is a unique such summand corresponding to

d_{k}/f_{χ} = \begin{cases}∏_{ℓ|N_{0},ℓ∤f_{χ}} ℓ, & if k = ord_{p}(f_{χ}) \\ ∏_{ℓ|N,ℓ∤f_{χ}} ℓ, & if k = 1 and ord_{p}(f_{χ}) = 0. \end{cases}

If neither of these conditions on k and ord_{p}(f_{χ}) is satisfied, then e_{χ}(ζ_{p^{k}}ξ_{N_{0}}^{Fr_{p}^{−k}}) = 0. By using the multiplicativity of μ, ϕ and χ and the equalities μ(ℓ) = −1 and ϕ(ℓ) = ℓ − 1 we then compute that (25) is equal to

χ(a(d_{χ})) ∏_{ℓ|N_{0},ℓ∤f_{χ}} (\frac{1}{ℓ−1}(−χ^{−1}(ℓ))) e_{χ}ζ_{f_{χ}}, if k = ord_{p}(f_{χ})
χ(a(d_{χ})) ∏_{ℓ|N,ℓ∤f_{χ}} (\frac{1}{ℓ−1}(−χ^{−1}(ℓ))) e_{χ}ζ_{f_{χ}}, if k = 1, ord_{p}(f_{χ}) = 0
0,  otherwise

where d_{χ} is the index of the unique nonvanishing summand in (25), i.e. d_{χ} = f_{χ,0} ∏_{ℓ|N_{0},ℓ∤f_{χ}} ℓ with f_{χ,0} the prime to p-part of f_{χ}. Now the element

u′ := ∑_{χ} χ(a(d_{χ}))e_{χ} ∈ ℚ_{p}[G]^{×}

belongs to ℤ_{p}[G]^{×} by Lemma 4.5 (indeed the function d ↦ a(d) is multiplicative, d_{χ} = d(ord_{ℓ}(f_{χ})) is a function of ord_{ℓ}(f_{χ}) only and satisfies d(0) = d(1) as such). From here the explicit description of Lemma 6.1 follows because the definition of e_{ℓ} ensures that ∏_{ℓ|N_{0},ℓ∤f_{χ}}(−χ^{−1}(ℓ)) = ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})e_{χ}. □

Lemma 6.2. For any ℚ-rational (resp. ℚ_{p}-rational) character χ of G ≅ (ℤ/Nℤ)^{×}, any d | N and any primitive d-th root of unity ζ_{d}^{a} we have

(26)  e_{χ}ζ_{d}^{a} = \{ \begin{matrix}0, & if f_{χ} ∤ d \\ \frac{ϕ(f_{χ})}{ϕ(d)}μ(\frac{d}{f_{χ}})χ^{−1}(\frac{d}{f_{χ}})χ(a)e_{χ}ζ_{f_{χ}}, & if f_{χ} | d \end{matrix}

in K (resp. K_{p}). Here ϕ(m) is Euler’s ϕ-function, μ(m) is the Möbius function and χ(m) = 0 if (m, f_{χ}) > 1.

Proof. Recall that we view a ℚ-rational character χ as the tautological homomorphism G → A^{×} → ℚ(χ)^{×} to the field ℚ(χ) := e_{χ}A which is a direct ring factor of A. Thus, any complex embedding j : ℚ(χ) → ℂ induces a complex character jχ = η : G → ℂ^{×}. We set b := N/d. Then under the ℂ-linear map σ_{N} : K_{ℂ} → ℂ the element

(j e_{χ})ζ_{d}^{a} = e_{η}ζ_{N}^{ab} = \frac{1}{|G|} ∑_{g∈G} η(g^{−1})g ζ_{N}^{ab} = \frac{1}{ϕ(N)} ∑_{x modN} η̄(x)ζ_{N}^{xab} ∈ K_{ℂ}

is sent to the general Gaussian sum ϕ(N)^{−1}τ(η̄_{N}|ζ_{N}^{ab}) in the notation of Hasse [22, §20.1]. By [22, §20.2.IV] we have

τ(η̄_{N}|ζ_{N}^{ab}) = \begin{cases}0, & f_{η} ∤ d \\ \frac{ϕ(N)}{ϕ(d)}μ(\frac{d}{f_{χ}})η̄(\frac{d}{f_{χ}})η(a)τ(η̄), & f_{η} | d \end{cases}

where the Gaussian sum τ(η) attached to the character η is as defined in (11). For d = f_{χ} and ζ_{d}^{a} = ζ_{f_{χ}} we find τ(η̄) = ϕ(f_{χ})σ_{N}((j e_{χ})ζ_{f_{χ}}). This yields the image of (26) under σ_{N}. Note that K_{ℂ} ≅ ∏_{g∈G} ℂ via x ↦ (σ_{N}gx)_{g∈G} and both sides of (26) are multiplied by χ(g) after applying g. Since jχ(g) = η(g) is a scalar and σ_{N} is ℂ-linear we find that (26) holds in K_{ℂ}, hence in K, hence also in K_{p} for all p. □

Given Lemma 6.1, our proof of Theorem 1.1 will be complete if we can show that uu′ ∈ ℤ_{p}[G]^{×}. Recalling Lemma 4.5 it thus suffices to prove that for each ℚ_{p}-rational character χ one has

(27)  χ(uu′) = \frac{f_{χ,0}^{r−1} ∏_{ℓ|N_{0},ℓ∤f_{χ}}(ℓ − 1)}{[ℚ(ζ_{N_{0}}) : ℚ(ζ_{f_{χ,0}})]}

where f_{χ,0} denotes for the prime to p-part of f_{χ}. (In this regard note that the expression on the right hand side of (27) belongs to ℤ_{p}^{×}.) We shall use explicit descent computations to prove that (27) is a consequence of the definition of u in (24). To this end, for each ℚ_{p}-character χ of G we let 𝔮_{χ} denote the kernel of the homomorphism Λ → ℚ_{p}(χ) in (23). Then 𝔮_{χ} is a regular prime ideal of Λ and Λ_{𝔮_{χ}} is a discrete valuation ring with residue field ℚ_{p}(χ). To apply [17, Lem. 5.7] we need to describe a Λ_{𝔮_{χ}}-basis of H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}} and for this purpose we find it convenient to split the argument into several different cases.

6.2. The case r > 1 or χ(D_{p}) ≠ 1. In this subsection we shall prove (27) for all characters χ except those which are trivial on D_{p} in the case that r = 1. In particular, the material of this section completes the proof of Theorem 1.1 in the case r > 1. We note first that if either r > 1 or χ(D_{p}) ≠ 1, then 𝔮_{χ} does not lie in the support of either ∏_{v|p} ℤ_{p}(r − 1) or ∏_{v|p} ℤ_{p}(r). Hence, modulo the identifications made in Proposition 5.2, it follows from (19), (20) and (21) that β_{N_{0}}^{∞} is a Λ_{𝔮_{χ}}-basis of H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}} = (Det_{Λ}^{−1} RΓ(ℚ_{p}, T_{p}^{∞}))_{𝔮_{χ}} and that β_{N_{0}}^{∞} ⊗_{Λ_{𝔮χ}} 1 is equal to the image of β_{N_{0}}^{∞} under the composite map

H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}} → H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}} ⊗_{Λ_{𝔮χ}} ℚ_{p}(χ) ≅ H^{1}(K_{p}, ℚ_{p}(r)) ⊗_{A_{p}} ℚ_{p}(χ)

where the isomorphism is induced by the vanishing of H^{2}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}} (cf. [17, Lem. 5.7]).

6.2.1. The descent diagram. By an obvious semi-local generalization of the argument of [1, §2.3.2] there exists a commutative diagram of Λ-modules

(28)  \begin{matrix}Z̃(r − 1) & ⟶^{θ_{r}^{P R}} & R \\ ↓ & ^{Ξ_{r,ν}} ↓ \\ H^{1}(K_{p}, ℚ_{p}(r)) & ⟵^{(r−1)! exp} & K_{p} \end{matrix}

where ν = ord_{p}(N) is as defined at the beginning of §5,

Ξ_{r,ν}(f) = \begin{cases}∑_{k=1}^{ν} p^{rk−ν}f^{Fr{−k \atop p}}(ζ_{p^{k}} − 1) + p^{−ν}(1 − \frac{Fr_{p}}{p^{r}})^{−1}f(0), & ν ≥ 1 \\ Tr_{K(ζ_{p})/K}(Ξ_{r,1}(f)), & ν = 0 \end{cases}

is the map of [1, Lem. 2.2.2] and the choice of Frobenius element Fr_{p} ∈ G ≅ G_{N_{0}} × G_{p^{ν}} is that which acts trivially on p-power roots of unity. (We are grateful to Laurent Berger for pointing out that the methods of [3] show that the diagram (28) commutes even in the case p = 2.) Now for f = β_{N_{0}}^{∞} = ξ_{N_{0}}(1 + X) this formula gives

Ξ_{r,ν}(β_{N_{0}}^{∞}) = \{ \begin{matrix}∑_{k=1}^{ν} p^{rk−ν}ζ_{p^{k}} Fr_{p}^{−k} ξ_{N_{0}} + p^{−ν}(1 − \frac{Fr_{p}}{p^{r}} )^{−1}ξ_{N_{0}}, & ν ≥ 1 \\ Tr_{K(ζ_{p})/K} (p^{r−1}ζ_{p} Fr_{p}^{−1} +\frac{1}{p}(1 − \frac{Fr_{p}}{p^{r}})^{−1}) ξ_{N_{0}} = \\ (−p^{r−1} Fr_{p}^{−1} +\frac{p − 1}{p}(1 − \frac{Fr_{p}}{p^{r}})^{−1}) ξ_{N_{0}} = \\ (1 − \frac{Fr_{p}}{p^{r}})^{−1}(1 − p^{r−1} Fr_{p}^{−1})ξ_{N_{0}}, & ν = 0. \end{matrix}

In addition, since either r > 1 or χ(D_{p}) ≠ 1, one has e_{χ} exp = e_{χ}ẽxp and so the commutativity of (28) implies that the e_{χ}-projection of the defining equality (24) is equivalent to an equality in e_{χ}K_{p} of the form

∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1}) e_{p}(1 − \frac{Fr_{p}}{p^{r}})^{−1}e_{p}(1 − p^{r−1} Fr_{p}^{−1})e_{χ}β_{N} (29)

=χ(u)(r − 1)! (∑_{k=1}^{ν} p^{rk−ν}e_{χ}(ζ_{p^{k}}ξ_{N_{0}}^{Fr_{p}^{−k}}) + p^{−ν} (1 − \frac{Fr_{p}}{p^{r}} )^{−1} e_{χ}ξ_{N_{0}})

if ν ≥ 1, resp.

(30)  ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})e_{χ}β_{N} = χ(u)(r − 1)!e_{χ}ξ_{N_{0}}

if ν = 0.

6.2.2. The case ord_{p}(f_{χ}) > 0. In this case ν > 0 and e_{p}(x)e_{χ} = e_{χ} for all x ∈ A_{p}^{×} and so we may leave out all factors of the form e_{p}(−) on the left hand side of (29). In addition, Lemma 6.1 implies that the only non-vanishing term in the summation on the right hand side of (29) is that corresponding to k = ord_{p}(f_{χ}) and moreover that (29) is equivalent to an equality

e_{χ}β_{N} = χ(uu′)(r − 1)!p^{rk−ν} ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1}e_{χ}ζ_{f_{χ}}.

Now, since k = ord_{p}(f_{χ}) and ν = ord_{p}(N), we have

p^{rk−ν} ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1} = \frac{[ℚ(ζ_{N_{0}}) : ℚ(ζ_{f_{χ,0}})]}{f_{χ,0}^{r−1} ∏_{ℓ|N_{0},ℓ∤f_{χ}}(ℓ − 1)} \frac{f_{χ}^{r−1}}{[K : ℚ(ζ_{f_{χ}})]}.

To deduce the required equality (27) from the last two displayed formulas one need only substitute the expression for e_{χ}β_{N} given in Proposition 4.1.

6.2.3. The case ord_{p}(f_{χ}) = 0 and ν > 0. In this case Lemma 6.1 shows that the only non-zero terms in the summation on the right hand side of (29) are those which correspond to k = 0 and k = 1. Moreover, one has e_{p}(x)e_{χ} = xe_{χ} for x ∈ A_{p}^{×}. By Lemma 6.1, equation (29) is thus equivalent to

(31) (1 − \frac{Fr_{p}}{p^{r}} )^{−1} (1 − p^{r−1} Fr_{p}^{−1})e_{χ}β_{N} =

χ(uu′)(r − 1)! ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1} (\frac{p^{r−ν}}{p − 1}(− Fr_{p}^{−1}) + p^{−ν} (1 − \frac{Fr_{p}}{p^{r}} )^{−1}) e_{χ}ζ_{f_{χ}}.

But

\frac{p^{r−ν}}{p − 1}(− Fr_{p}^{−1}) + p^{−ν} (1 − \frac{Fr_{p}}{p^{r}} )^{−1}

=\frac{p^{−ν+1}}{p − 1} (1 − \frac{Fr_{p}}{p^{r}} )^{−1} (p^{r−1}(− Fr_{p}^{−1}) (1 − \frac{Fr_{p}}{p^{r}} ) + \frac{p − 1}{p})

=\frac{1}{ϕ(p^{ν})} (1 − \frac{Fr_{p}}{p^{r}} )^{−1} (1 − p^{r−1} Fr_{p}^{−1})

and so (31) implies that

e_{χ}β_{N} = χ(uu′)(r − 1)!\frac{1}{ϕ(p^{ν})} ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1}e_{χ}ζ_{f_{χ}}.

The required equality (27) follows from this in conjunction with the equality

\frac{1}{ϕ(p^{ν})} ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1} = \frac{[ℚ(ζ_{N_{0}}) : ℚ(ζ_{f_{χ,0}})]}{f_{χ,0}^{r−1} ∏_{ℓ|N_{0},ℓ∤f_{χ}}(ℓ − 1)} \frac{f_{χ}^{r−1}}{[K : ℚ(ζ_{f_{χ}})]}

and the expression for e_{χ}β_{N} given in Proposition 4.1.

Documenta Mathematica · Extra Volume Coates (2006) 133–163

6.2.4. The case ν = ord_{p}(N) = 0. In this case (27) results directly upon substituting the formulas of Proposition 4.1 and Lemma 6.1 (with k = 0) into (30).

6.3. The case r = 1 and χ(D_{p}) = 1. In this case 𝔮_{χ} lies in the support of ∏_{v|p} ℤ_{p}(r − 1) (but not of ∏_{v|p} ℤ_{p}(r)) and β_{N_{0}}^{∞} is not a Λ_{𝔮_{χ}}-basis of H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}}. We fix a generator γ of ℤ_{p} ≅ Gal(ℚ(ζ_{Np^{∞}})/K(ζ_{p})) ⊆ G_{Np^{∞}} and then define a uniformizer of Λ_{𝔮_{χ}} by setting

ϖ := 1 − γ.

The p-adic places of the fields K = ℚ(ζ_{N_{0}p^{ν}}) and ℚ(ζ_{N_{0}p^{∞}}) are in natural bijection. We fix one such place v_{0} and set

η^{∞} := (1 − ζ_{p^{n}})_{n≥1} ∈ \underleftarrow{lim}_{n}(ℚ(ζ_{N_{0}p^{n}})_{v_{0}}^{×})/p^{n}
⊆ \underleftarrow{lim}_{n} ∏_{v|p}(ℚ(ζ_{N_{0}p^{n}})_{v}^{×})/p^{n} = H^{1}(ℚ_{p}, T_{p}^{∞}).

Then the image η̄^{∞} of η^{∞} in

H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}}/ϖ ⊆ H^{1}(K_{p}, ℚ_{p}(1)) ⊗_{A_{p}} ℚ_{p}(χ) =: H^{1}(K_{p}, ℚ_{p}(1))_{χ}

coincides with that of p ∈ ℚ(ζ_{N_{0}})_{v_{0}}^{×} and so is non-zero. In particular therefore, η^{∞} is a Λ_{𝔮_{χ}}-basis of H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}}. Now, by [17, Lem. 5.7] there is an exact sequence

0 → H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}}/ϖ → H^{1}(K_{p}, ℚ_{p}(1))_{χ} ⟶^{β} H^{2}(K_{p}, ℚ_{p}(1))_{χ} → 0

where β is the χ-projection of the composite homomorphism

H^{1}(K_{p}, ℚ_{p}(1)) ≅ K̂_{p}^{×} → ∏_{v|p} ℚ_{p} ≅ H^{2}(K_{p}, ℚ_{p}(1)); u_{v} → \frac{Tr_{K_{v}/ℚ_{p}}(log_{p}(u_{v}))}{log_{p}(χ_{cyclo}(γ))}

(see [17, Lem. 5.8] and its proof). This exact sequence induces an isomorphism

ϕ_{ϖ} : Det_{ℚ_{p}(χ)}^{−1} RΓ(ℚ_{p}, V_{p})_{χ} ≅ H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}}/ϖ

and [17, Lem. 5.7] implies that, modulo the identifications made in Proposition 5.2, one has

(32)  λ̄ · β_{N_{0}}^{∞} ⊗_{Λ_{𝔮χ}} 1 = ϕ_{ϖ}^{−1}(η̄^{∞})

where the elements λ ∈ Λ_{𝔮_{χ}}^{×} and e ∈ ℤ are defined by the equality

(33)  (θ_{1}^{P R})_{𝔮_{χ}}^{−1}(λ · β_{N_{0}}^{∞}) = ϖ^{e}η^{∞} ∈ H^{1}(ℚ_{p}, T_{p}^{∞})_{𝔮_{χ}}

and λ̄ denotes the image of λ in Λ_{𝔮_{χ}}/ϖ. This description of η̄^{∞} implies that

val(η̄^{∞}) = β(exp(e_{χ}b)),

where ‘val’ is the normalized valuation map which occurs in (14) and

b := |D_{p}|^{−1} log_{p}(χ_{cyclo}(γ)) ∈ ℚ_{p} ⊆ K_{v_{0}} ⊆ ∏_{v|p} K_{v} = K_{p}.

Lemma 6.3. The element λ that is defined in (33) belongs to Λ and in ℚ(ζ_{N_{0}})_{p} ≅ ∏_{v|p} ℚ(ζ_{N_{0}})_{v} one has

b = −|D_{p}|^{−1} (1 − \frac{1}{p})^{−1} λ · ξ_{N_{0}}.

This formula uniquely determines the image λ̄ of λ in ℚ_{p}(χ).

Proof. Since β_{N_{0}}^{∞} is a basis of the free rank one Λ-module R (by Lemma 5.1) we have

(34)  λ · β_{N_{0}}^{∞} = θ_{1}^{P R}((η^{∞})^{ϖ})

for some element λ of Λ, which then also satisfies the condition (33). The map θ_{1}^{P R} is described explicitly by (22). Further, with respect to the system (ζ_{p^{n}})_{n≥1}, the Coleman power series that is associated to the norm compatible system of units (η^{∞})^{ϖ} = (η^{∞})^{(1−γ)} is equal to

f(X) := \frac{X}{(1 + X)^{χ_{cyclo}(γ)} − 1} ≡ χ_{cyclo}(γ)^{−1} mod (X).

Thus, by computing constant terms in the power series identity (34) we obtain equalities

λ · ξ_{N_{0}} = (1 − \frac{ϕ}{p}) log(f(X))|_{X=0}

= (1 − \frac{1}{p}) log_{p}(χ_{cyclo}(γ)^{−1})

= −(1 − \frac{1}{p})|D_{p}|b

as required to finish the proof of the first sentence of the lemma. On the other hand, the second sentence of the lemma is clear because ξ_{N_{0}} is a ℚ_{p}[G_{N_{0}}]-basis of ℚ(ζ_{N_{0}})_{p} and ℚ_{p}(χ) = Λ_{𝔮_{χ}}/ϖ is a quotient of ℚ_{p}[G_{N_{0}}]. □

With ẽxp denoting the map in (15), the last lemma implies that

ϕ_{ϖ}^{−1}(η̄^{∞}) = η̄^{∞} ∧ exp(e_{χ}b) ⊗ β(exp(e_{χ}b))^{−1}
= − exp(e_{χ}b) ∧ η̄^{∞} ⊗ val(η̄^{∞})^{−1}
= ẽxp(−e_{χ}b)
= ẽxp (|D_{p}|^{−1} (1 − \frac{1}{p})^{−1} λ̄ · e_{χ}ξ_{N_{0}}) ,

and hence, using (32), that

ẽxp^{−1}(β_{N_{0}}^{∞} ⊗_{Λ_{𝔮χ}} 1)
=|D_{p}|^{−1}(1 − \frac{1}{p})^{−1}e_{χ}(ξ_{N_{0}})
=|D_{p}|^{−1}(1 − \frac{1}{p})^{−1}χ(u′) ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1} ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})e_{χ}ζ_{f_{χ}}

=|D_{p}|^{−1}(1 − \frac{1}{p})^{−1}χ(u′) ∏_{ℓ|N_{0},ℓ∤f_{χ}} \frac{1}{ℓ − 1} ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})[K : ℚ(ζ_{f_{χ}})]e_{χ}β_{N}

=|D_{p}/J_{p}|^{−1}(1 − \frac{1}{p})^{−1}χ(u′)\frac{[ℚ(ζ_{N_{0}}) : ℚ(ζ_{f_{χ,0}})]}{f_{χ,0}^{r−1} ∏_{ℓ|N_{0},ℓ∤f_{χ}}(ℓ − 1)} ∏_{ℓ|N_{0}} e_{ℓ}(− Fr_{ℓ}^{−1})e_{χ}β_{N}

where the second equality follows from Lemma 6.1, the third from Proposition 4.1 and the fourth from the fact that r = 1, f_{χ} = f_{χ,0} and

\frac{[K : ℚ(ζ_{f_{χ}})]}{|J_{p}| ∏_{ℓ|N_{0},ℓ∤f_{χ}}(ℓ − 1)} = \frac{[ℚ(ζ_{N_{0}}) : ℚ(ζ_{f_{χ,0}})]}{f_{χ,0}^{r−1} ∏_{ℓ|N_{0},ℓ∤f_{χ}}(ℓ − 1)}.

The required equality (27) is now obtained by comparing the above formula for ẽxp^{−1}(β_{N_{0}}^{∞} ⊗_{Λ_{𝔮χ}} 1) to the definition of u in (24). This completes our proof of Theorem 1.1.

7. Some remarks concerning T Ω^{loc}(M, 𝔄)

In this section we prove two results that were used in the proof of Theorem 1.1 but which are most naturally formulated in a more general setting. In particular, these results extend the computations made in [9, §5]. We henceforth fix notation as in §2. Thus, we stress, M is no longer assumed to be a Tate motive and the (finite dimensional semisimple) ℚ-algebra A is not assumed to be either commutative or a group ring.

7.1. The contribution from primes ℓ ≠ p. We first recall the following basic fact about the cohomology of the profinite group ℤ̂ (for distinction we shall denote the canonical generator 1 ∈ ℤ̂ by σ). Let R be either a pro-p ring, or a localization of such a ring, and let C be a perfect complex of R-modules with a continuous action of ℤ̂. Then

RΓ(ℤ̂, C) ≅ Tot(C ⟶^{1−σ} C)

is a perfect complex of R-modules where ‘Tot’ denotes the total complex of a double complex. The identity map of C induces a morphism

id_{C,triv} : 1_{R} ≅ Det_{R}^{−1} RΓ(ℤ̂, C)

in V (R) which is functorial for exact triangles in the variable C and also commutes with scalar extension.

Proposition 7.1. For a prime number ℓ ≠ p we let σ_{ℓ} denote the Frobenius automorphism in Gal(ℚ_{ℓ}^{ur}/ℚ_{ℓ}). If

θ_{p}^{ℓ-part} : 1_{A_{p}} ≅ Det_{A_{p}}^{−1} RΓ(ℚ_{ℓ}, V_{p})

denotes the morphism in V (A_{p}) which occurs in [9, (67)], then

Det_{A_{p}}(−σ_{ℓ}ℓ^{−1}|(V_{p})_{I_{ℓ}})^{−1} θ_{p}^{ℓ-part}

is induced by a morphism 1_{𝔄_{p}} ≅ Det_{𝔄_{p}}^{−1} RΓ(ℚ_{ℓ}, T_{p}) in V (𝔄_{p}).

Proof. Recall the exact triangle of complexes of A_{p}-modules

(35)  RΓ_{f}(ℚ_{ℓ}, V_{p}) → RΓ(ℚ_{ℓ}, V_{p}) → RΓ_{/f}(ℚ_{ℓ}, V_{p})

from [9, (18)] as well as the isomorphism

AV : RΓ_{/f}(ℚ_{ℓ}, V_{p}) ≅ RΓ_{f}(ℚ_{ℓ}, V_{p}^{∗}(1))^{∗}[−2]

from [9, Lem. 12a)]. The triangle (35) is obtained by applying RΓ(ℤ̂, −) to the exact triangle

(36)  H^{0}(I_{ℓ}, V_{p}) → RΓ(I_{ℓ}, V_{p}) → H^{1}(I_{ℓ}, V_{p})[−1]

together with the isomorphism

RΓ(ℤ̂, RΓ(I_{ℓ}, V_{p})) ≅ RΓ(ℚ_{ℓ}, V_{p}).

According to the convention [9, (19)] the generator σ we use here is σ_{ℓ}^{−1}. The isomorphism AV is more explicitly given by the diagram

(V_{p})_{I_{ℓ}}(−1) ⟶^{1−σ_{ℓ}^{−1}} (V_{p})_{I_{ℓ}}(−1)
∥  ↓^{−σ_{ℓ}}
(V_{p})_{I_{ℓ}}(−1) ⟶^{1−σ_{ℓ}} (V_{p})_{I_{ℓ}}(−1)
↓≅  ↓≅
((V_{p}^{∗})^{I_{ℓ}}(1))^{∗} ⟶^{1−σ_{ℓ}^{−1}} ((V_{p}^{∗})^{I_{ℓ}}(1))^{∗}.

Note here that H^{1}(I_{ℓ}, V_{p}) is naturally isomorphic to (V_{p})_{I_{ℓ}}(−1) and that in the isomorphism ((V_{p}^{∗})^{I_{ℓ}})^{∗} ≅ (V_{p})_{I_{ℓ}} the first dual is the contragredient σ_{ℓ}- representation whereas the second is simply the dual. From this last diagram we deduce

id_{(V_{p})_{I_{ℓ}} (−1),triv} = Det_{A_{p}}(−σ_{ℓ}ℓ^{−1}|(V_{p})_{I_{ℓ}}) id_{((V_{p}^{∗})^{I_{ℓ}} (1))^{∗},triv}

and by the discussion above with R = A_{p} the exact triangle (36) gives

id_{V_{p}^{I_{ℓ}} ,triv} ⊗ id_{(V_{p})_{I_{ℓ}} (−1),triv}^{−1} = id_{RΓ(I_{ℓ},V_{p}),triv} .

By the definition of [9, (67)] the morphism θ_{p}^{ℓ-part} is induced by the triangle (35), the isomorphism AV and the morphisms id_{V_{p}^{I_{ℓ}} ,triv} and id_{((V_{p}^{∗})^{I_{ℓ}} (1))^{∗},triv}. Hence

Det_{A_{p}}(−σ_{ℓ}ℓ^{−1}|(V_{p})_{I_{ℓ}})^{−1} θ_{p}^{ℓ-part}

is the scalar extension of the morphism id_{RΓ(I_{ℓ},T_{p}),triv} in V (𝔄_{p}) and this finishes the proof of the Proposition. □

7.2. Artin-Verdier Duality. In this subsection we extend [9, Lem. 14] to include the case p = 2 and hence resolve the issue raised in [9, Rem. 16]. Before stating the main result we recall that [9, (78)] defines a morphism in V (A_{p}) of the form

(37) θ′_{p} : ( ⨂_{ℓ∈S_{p,f}} Det_{A_{p}}^{−1} RΓ(ℚ_{ℓ}, V_{p})) ⊗_{A_{p}} Det_{A_{p}}^{−1}(V_{p}) ≅ A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p})

where

Λ_{p}(S, T_{p}) := Det_{𝔄_{p}} C(ℚ, T_{p})

with C(ℚ, T_{p}) a certain canonical perfect complex of 𝔄_{p}-modules (as occurs in the diagram (39) below with E = ℚ). We set

ϵ_{V_{p}} := Det_{A_{p}}(2|V_{p}^{+}) Det_{A_{p}}(2|V_{p}^{−})^{−1} ∈ K_{1}(A_{p}).

Proposition 7.2. The morphism ϵ_{V_{p}} · θ′_{p} is induced by a morphism in V (𝔄_{p}) of the form

(38)  ( ⨂_{ℓ∈S_{p,f}} Det_{𝔄_{p}}^{−1} RΓ(ℚ_{ℓ}, T_{p})) ⊗_{𝔄_{p}} Det_{𝔄_{p}}^{−1}(T_{p}) ≅ Λ_{p}(S, T_{p}).

The proof of this result will occupy the remainder of this subsection. We note first that if p is odd, then ϵ_{V_{p}} ∈ im(K_{1}(𝔄_{p}) → K_{1}(A_{p})) and so the above claim is equivalent to asserting that θ′_{p} itself is induced by a morphism in V (𝔄_{p}) of the form (38). Since this is precisely the statement of [9, Lem. 14] we shall assume henceforth that p = 2. Now if E is any number field, then [9, (81)] gives a true nine term diagram

(39)

⨁_{v∈S∞} RΓ_{Δ}(E_{v}, T_{p}^{∗}(1))^{∗}[−4]  ⨁_{v∈S∞} RΓ_{Δ}(E_{v}, T_{p}^{∗}(1))^{∗}[−4]
↓  ^{α(E)}↓
R̃Γ_{c}(O_{E,Sp} , T_{p}^{∗}(1))^{∗}[−4] ⟶  _{3}L(S_{p}, T_{p})[−1]  ⟶ RΓ_{c}(O_{E,Sp} , T_{p})
↓  ↓  ∥
RΓ_{c}(O_{E,Sp} , T_{p}^{∗}(1))^{∗}[−4] ⟶  C(E, T_{p})  ⟶ RΓ_{c}(O_{E,Sp} , T_{p})

where the complex _{3}L(S_{p}, T_{p}) is endowed with a natural quasi-isomorphism

β(E) : _{3}L(S_{p}, T_{p}) ≅ ⨁_{v∈S_{p}} RΓ(E_{v}, T_{p}).

To prove the Proposition we shall make an explicit study of the composite morphism β(ℚ) ∘ α(ℚ). To do this we observe that if E is any Galois extension of ℚ with group Γ, then (39), resp. β(E), is a true nine-term diagram, resp. quasi-isomorphism, of complexes of 𝔄_{p}[Γ]-modules and the same arguments as used in [8, Lem. 11] show that application of R Hom_{ℤ_{p}[Γ]}(ℤ_{p}, −) to (39), resp. β(E), renders a diagram which is naturally isomorphic to the corresponding diagram for E = ℚ, resp. a quasi-isomorphism which identifies naturally with β(ℚ). We now fix E to be an imaginary quadratic field and set Γ := Gal(E/ℚ) and RΓ_{Tate}(E_{v}, −) := RΓ(E_{v}, −) for each non-archimedean place v. Then for each v_{0} ∈ S one has a natural morphism RΓ_{Tate}(E_{v_{0}}, −) → RΓ(E_{v_{0}}, −) and we let γ_{v_{0}}(E) denote the following composite morphism in D(𝔄_{p}[Γ])

RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1))^{∗}[−3] ⟶^{β(E)∘α(E)[1]} ⨁_{v∈S_{p}} RΓ(E_{v}, T_{p}) → RΓ(E_{v_{0}}, T_{p}).

Now if v_{0} is non-archimedean, then γ_{v_{0}}(E) is equal to the composite

(40) RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1))^{∗}[−3] → ⨁_{v∈S_{p}} RΓ_{Tate}(E_{v}, T_{p}) → RΓ(E_{v_{0}}, T_{p}),

where the first arrow denotes the diagonal morphism in the following commutative diagram in D(𝔄_{p}[Γ])

RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1))^{∗}[−3] ⟶ R̃Γ_{c}(O_{E,Sp} , T_{p}^{∗}(1))^{∗}[−3]  ⟵^{AV}  RΓ(O_{E,Sp} , T_{p})
∥  ↓  ↓
RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1))^{∗}[−3] ⟶ ⨁_{v∈Sp} RΓ_{Tate}(E_{v}, T_{p}^{∗}(1))^{∗}[−2] ⟵^{⊕ AVv} ⨁_{v∈Sp} RΓ_{Tate}(E_{v}, T_{p})

in which the left, resp. right, hand square comes directly from the definition of RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1)) in [9, (80)], resp. from the compatibility of local and global Artin-Verdier duality as in [9, Lem. 12]. Since (v_{0} is assumed for the moment to be non-archimedean and) the image of the lower left hand arrow in this diagram is contained in the summand RΓ_{Tate}(E_{∞}, T_{p}^{∗}(1))^{∗}[−2] it is therefore clear that (40) is the zero morphism. Hence, there exists a natural isomorphism in D(𝔄_{p}[Γ]) of the form

C(E, T_{p}) ≅ C_{∞}(E, T_{p})[−1] ⊕ ⨁_{v∈S_{p,f}} RΓ(E_{v}, T_{p})[−1]

where C_{∞}(E, T_{p}) is a complex which lies in an exact triangle in D(𝔄_{p}[Γ]) of the form

(41)  RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1))^{∗}[−3] ⟶^{γ_{∞}(E)} RΓ(E_{∞}, T_{p}) → C_{∞}(E, T_{p}) → .

Now, via the canonical identifications RΓ_{Δ}(E_{∞}, T_{p}^{∗}(1))^{∗}[−3] ≅ T_{p}(−1)[−3] and RΓ(E_{∞}, T_{p}) ≅ T_{p}[0], we may regard γ_{∞}(E) as an element of

Hom_{D(𝔄_{p}[Γ])}(T_{p}(−1)[−3], T_{p}[0]) ≅ Ext_{𝔄_{p}[Γ]}^{3}(T_{p}(−1), T_{p}).

With respect to this identification, C_{∞}(E, T_{p}) represents γ_{∞}(E) viewed as a Yoneda 3-extension and so can be obtained via a push-out diagram of 𝔄_{p}[Γ]- modules of the form

(42)  \begin{aligned}0 & ⟶ & T_{p} & ⟶ & T_{p}[Γ] & ⟶^{1−c} & T_{p}[Γ] & ⟶^{1+c} & T_{p}[Γ] & ⟶ & T_{p}(−1) & ⟶ & 0 \\ & & ^{μ}↓ & & ↓ & & ∥ & & ∥ & & ∥ \\ 0 & ⟶ & T_{p} & ⟶ & B_{μ} & ⟶ & T_{p}[Γ] & ⟶^{1+c} & T_{p}[Γ] & ⟶ & T_{p}(−1) & ⟶ & 0. \end{aligned}

Here we write c for the natural diagonal action of the generator τ of Gal(ℂ/ℝ), the second arrow in the upper row is the map t ↦ t + τ(t) · γ where γ is the generator of Γ and the fifth arrow in both rows is the map t + t′ · γ ↦ (t − τ(t′)) ⊗ ξ^{−1} with ξ := (ζ_{p^{n}})_{n≥1} (regarded as a generator of ℤ_{p}(1)). For any 𝔄_{p}[Γ]-module X the above diagram induces a commutative diagram of the form

Ext_{𝔄_{p}[Γ]}^{i}(T_{p}, X) ⟶ Ext_{𝔄_{p}[Γ]}^{i+3}(T_{p}(−1), X)
μ^{∗,i}↑  ∥
Ext_{𝔄_{p}[Γ]}^{i}(T_{p}, X) ⟶ Ext_{𝔄_{p}[Γ]}^{i+3}(T_{p}(−1), X).

But C_{∞}(E, T_{p}) belongs to D^{perf}(𝔄_{p}[Γ]) (since the lower row of (39) belongs to D^{perf}(𝔄_{p}[Γ])) and so the projective dimension of the 𝔄_{p}[Γ]-module B_{μ} is finite and therefore at most 1. This implies that the upper (resp. lower) horizontal map in the last diagram is bijective for i ≥ 2 and surjective for i = 1 (resp. bijective for i ≥ 1). The map μ^{∗,i} is therefore bijective for each i ≥ 2 and surjective for i = 1 and so a result of Holland [23, Th. 3.1] implies that there exists an automorphism α ∈ Aut_{𝔄_{p}[Γ]}(T_{p}) and a projective 𝔄_{p}[Γ]-module P such that μ − α is equal to a composite of the form T_{p} → P → T_{p}. Now the Γ-module

Hom_{𝔄_{p}}(T_{p}(−1), P) ≅ Hom_{𝔄_{p}}(T_{p}(−1), 𝔄_{p}) ⊗_{𝔄_{p}} P =: T^{∗} ⊗_{𝔄_{p}} P

is cohomologically trivial (indeed, it suffices to check this for P = 𝔄_{p}[Γ] in which case T^{∗} ⊗_{𝔄_{p}} 𝔄_{p}[Γ] = T^{∗} ⊗_{ℤ_{p}} ℤ_{p}[Γ] ≅ ℤ_{p}[Γ]^{d} with d = rank_{ℤ_{p}}(T^{∗})) and so Ext_{𝔄_{p}[Γ]}^{3}(T_{p}(−1), P) ≅ H^{3}(Γ, Hom_{𝔄_{p}}(T_{p}(−1), P)) = 0. In the diagram (42) we may therefore assume that μ ∈ Aut_{𝔄_{p}[Γ]}(T_{p}) and hence can use this diagram to identify C_{∞}(E, T_{p}) with the complex

T_{p}[Γ] ⟶^{1−c} T_{p}[Γ] ⟶^{1+c} T_{p}[Γ],

where the first term is placed in degree 0 (and the cohomology is computed via the maps in upper row of (42)). Writing C_{∞}(T_{p}) for the complex

T_{p} ⟶^{1−c} T_{p} ⟶^{1+c} T_{p}

(where the first term is placed in degree 0), we may thus deduce the existence of a composite isomorphism in D(𝔄_{p}) of the form

C_{∞}(T_{p})[−1] ⊕ ⨁_{ℓ∈S_{p,f}} RΓ(ℚ_{ℓ}, T_{p})[−1]

≅ R Hom_{ℤ_{p}[Γ]}(ℤ_{p}, C_{∞}(E, T_{p})[−1] ⊕ ⨁_{v∈S_{p,f}} RΓ(E_{v}, T_{p})[−1])
≅ R Hom_{ℤ_{p}[Γ]}(ℤ_{p}, C(E, T_{p}))
≅ C(ℚ, T_{p}).

When taken in conjunction with the natural morphism

j(T_{p}) : Det_{𝔄_{p}} C_{∞}(T_{p})[−1] = Det_{𝔄_{p}}^{−1}(T_{p}) ⊗_{𝔄_{p}} (Det_{𝔄_{p}}(T_{p}) ⊗_{𝔄_{p}} Det_{𝔄_{p}}^{−1}(T_{p}))

≅ Det_{𝔄_{p}}^{−1}(T_{p}) ⊗_{𝔄_{p}} 1_{𝔄_{p}} = Det_{𝔄_{p}}^{−1}(T_{p})

the above composite isomorphism induces a morphism in V (𝔄_{p}) of the form

θ′′_{p} : ( ⨂_{ℓ∈S_{p,f}} Det_{𝔄_{p}}^{−1} RΓ(ℚ_{ℓ}, T_{p}))⊗_{𝔄_{p}} Det_{𝔄_{p}}^{−1}(T_{p}) ≅ Det_{𝔄_{p}} C(ℚ, T_{p}) =: Λ_{p}(S, T_{p}).

Now A_{p} ⊗_{𝔄_{p}} θ′′_{p} differs from the morphism θ′_{p} in (37) only in the following respect: in place of A_{p} ⊗_{𝔄_{p}} j(T_{p}) the morphism θ′_{p} involves the composite morphism

j(V_{p}) : A_{p} ⊗_{𝔄_{p}} Det_{𝔄_{p}} C_{∞}(T_{p})[−1] ≅

Det_{A_{p}}^{−1}(A_{p} ⊗_{𝔄_{p}} H^{0}(C_{∞}(T_{p}))) ⊗_{A_{p}} Det_{A_{p}}^{−1}(A_{p} ⊗_{𝔄_{p}} H^{2}(C_{∞}(T_{p})))
≅ Det_{A_{p}}^{−1} V_{p}

where the first morphism is the canonical ‘passage to cohomology’ map and the second is induced by combining the isomorphisms A_{p} ⊗_{𝔄_{p}} H^{0}(C_{∞}(T_{p})) ≅ V_{p}^{+} and A_{p} ⊗_{𝔄_{p}} H^{2}(C_{∞}(T_{p})) ≅ V_{p}(−1)^{+} that are induced by the upper row of (42) with the isomorphism V_{p}^{+} ⊕ V_{p}(−1)^{+} ≅ V_{p}^{+} ⊕ V_{p}^{−} = V_{p} (where the second component of the first map sends each element v of V_{p}(−1)^{+} to v ⊗ ξ ∈ V_{p}^{−}). But the complex A_{p} ⊗_{𝔄_{p}} C_{∞}(T_{p}) identifies with

V_{p}^{+} ⊕ V_{p}^{−} ⟶^{(0,2)} V_{p}^{+} ⊕ V_{p}^{−} ⟶^{(2,0)} V_{p}^{+} ⊕ V_{p}^{−}

and so, by an explicit computation, one has A_{p} ⊗_{𝔄_{p}} j(T_{p}) = ϵ_{V_{p}} · j(V_{p}) where ϵ_{V_{p}} := Det_{A_{p}}(2|V_{p}^{+}) Det_{A_{p}}(2|V_{p}^{−})^{−1} ∈ K_{1}(A_{p}). The induced equality

A_{p} ⊗_{𝔄_{p}} θ′′_{p} = ϵ_{V_{p}} · θ′_{p}

then completes the proof of the Proposition.

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David Burns

King’s College London Dept. of Mathematics London WC2R 2LS United Kingdom Matthias Flach Dept. of Mathematics Caltech Pasadena CA 91125 USA 164

Documenta Mathematica · Extra Volume Coates (2006)