On the leading terms of

Zeta isomorphisms and p-adic L-functions

in non-commutative Iwasawa theory

Dedicated to John Coates

David Burns and Otmar Venjakob

Abstract. We discuss the formalism of Iwasawa theory descent in the setting of the localized K_{1}-groups of Fukaya and Kato. We then prove interpolation formulas for the ‘leading terms’ of the global Zeta isomorphisms that are associated to certain Tate motives and of the p-adic L-functions that are associated to certain critical motives.

1991 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

1. Introduction

In the last few years there have been several significant developments in noncommutative Iwasawa theory. Firstly, in [10], Coates, Fukaya, Kato, Sujatha and the second named author formulated a main conjecture for elliptic curves without complex multiplication. More precisely, if F_{∞} is any Galois extension of a number field F which contains the cyclotomic ℤ_{p}-extension F_{cyc} of F and is such that Gal(F_{∞}/F) is a compact p-adic Lie group with no non-trivial p-torsion, then Coates et al formulated a Gal(F_{∞}/F)-equivariant main conjecture for any elliptic curve which is defined over F, has good reduction at all places above p and whose Selmer group (over F_{∞}) satisfies a certain natural torsion condition. Then, in [14], Fukaya and Kato formulated a natural main conjecture for any compact p-adic Lie extension of F and any critical motive M which is defined over F and has good ordinary reduction at all places above p.

non-commutative p-adic L-functions 3

The key feature of [10] is the use of the localization sequence of algebraic K-theory with respect to a canonical Ore set. However, the more general approach of [14] is rather more involved and uses a notion of ‘localized K_{1}-groups’ together with Nekovar’s theory of Selmer complexes and the (conjectural) existence of certain canonical p-adic L-functions. See [37] for a survey. The p-adic L-functions of Fukaya and Kato satisfy an interpolation formula which involves both the ‘non-commutative Tamagawa number conjecture’ (this is a natural refinement of the ‘equivariant Tamagawa number conjecture’ formulated by Flach and the first named author in [6] and hence also implies the ‘main conjecture of non-abelian Iwasawa theory’ discussed by Huber and Kings in [17]) as well as a local analogue of the non-commutative Tamagawa number conjecture. Indeed, by these means, at each continuous finite dimensional padic representation ρ of Gal(F_{∞}/F), the ‘value at ρ’ of the p-adic L-function is explicitly related to the value at the central critical point of the complex L-function associated to the ‘ρ^{∗}-twist’ M(ρ^{∗}) of M. However, if the Selmer module of M(ρ^{∗}) has strictly positive rank (and recent work of Mazur and Rubin [19] shows that this should often be the case), then both sides of the Fukaya-Kato interpolation formula are equal to zero. The main aim of the present article is therefore to extend the formalism of Fukaya and Kato in order to obtain an interesting interpolation formula for all representations ρ as above. To this end we shall introduce a notion of ‘the leading term at ρ’ for elements of suitable localized K_{1}-groups. This notion is defined in terms of the Bockstein homomorphisms that have already played significant roles (either implicitly or explicitly) in work of Perrin-Riou [25, 27], of Schneider [32, 31, 30, 29] and of Greither and the first named author [8, 3] and have been systematically incorporated into Nekovar’s theory of Selmer complexes [22]. We then give two explicit applications of this approach in the setting of extensions F_{∞}/F with F_{cyc} ⊆ F_{∞}. We show first that the ‘p-adic Stark conjecture at s = 1’, as formulated by Serre [33] and interpreted by Tate in [35], can be reinterpreted as providing interpolation formulas for the leading terms of the global Zeta isomorphisms associated to certain Tate motives in terms of the leading terms at s = 1 (in the classical sense) of the p-adic Artin L-functions that are constructed by combining Brauer induction with the fundamental results of Deligne and Ribet and of Cassou-Nogues. We then also prove an interpolation formula for the leading terms of the Fukaya-Kato p-adic L-functions which involves the leading term at the central critical point of the associated complex L-function, the Neron-Tate pairing and Nekovar’s p-adic height pairing. In a subsequent article we shall apply the approach developed here to describe the leading terms of the ‘algebraic p-adic L-functions’ that are introduced by the first named author in [4], and we shall use the resulting description to prove that the main conjecture of Coates et al for an extension F_{∞}/F and an elliptic curve E implies the equivariant Tamagawa number conjecture for the motive h^{1}(E)(1) at each finite degree subextension of F_{∞}/F. We note that this result provides a partial converse to the theorem of Fukaya and Kato which shows that, under a natural torsion hypothesis on Selmer groups, the main conjecture of Fukaya and Kato specialises to recover the main conjecture of Coates et al. The main contents of this article are as follows. In §2 we recall some basic facts regarding (non-commutative) determinant functors and the localized K_{1}- groups of Fukaya and Kato. In §3 we discuss the formalism of Iwasawa theory descent in the setting of localized K_{1}-groups and we introduce a notion of the leading terms at p-adic representations for the elements of such groups. We explain how this formalism applies in the setting of the canonical Ore sets introduced by Coates et al, we show that it can be interpreted as taking values after ‘partial derivation in the cyclotomic direction’, and we use it to extend several well known results concerning Generalized Euler Poincare characteristics. In §4 we recall the ‘global Zeta isomorphisms’ that are conjectured to exist by Fukaya and Kato, and in §5 we prove an interpolation formula for the leading terms of the global Zeta isomorphisms that are associated to certain Tate motives. Finally, in §6, we prove an interpolation formula for the leading terms of the p-adic L-functions that are associated to certain critical motives. We shall use the same notation as in [37].

It is clear that the recent developments in non-commutative Iwasawa theory are due in large part to the energy, encouragement and inspiration of John Coates. It is therefore a particular pleasure for us to dedicate this paper to him on the occasion of his sixtieth birthday. This collaboration was initiated during the conference held in Boston in June 2005 in recognition of the sixtieth birthday of Ralph Greenberg. The authors are very grateful to the organizers of this conference for the opportunity to attend such a stimulating meeting.

2. Preliminaries

2.1. Determinant functors. For any ring R we write B(R) for the category of bounded complexes of (left) R-modules, C(R) for the category of bounded complexes of finitely generated (left) R-modules, P (R) for the category of finitely generated projective (left) R-modules, C^{p}(R) for the category of bounded (cohomological) complexes of finitely generated projective (left) R-modules. By D^{p}(R) we denote the category of perfect complexes as full triangulated subcategory of the derived category D^{b}(R) of the homotopy category of B(R). We write (C^{p}(R), quasi) and (D^{p}(R), is) for the subcategory of quasi-isomorphisms of C^{p}(R) and isomorphisms of D^{p}(R), respectively. For each complex C = (C^{·}, d_{C}^{·}) and each integer r we define the r-fold shift C[r] of C by setting C[r]^{i} = C^{i+r} and d_{C[r]}^{i} = (−1)^{r}d_{C}^{i+r} for each integer i. We first recall that for any (associative unital) ring R there exists a Picard category C_{R} and a determinant functor

d_{R} : (C^{p}(R), quasi) → C_{R}

with the following properties (for objects C, C′ and C′′ of C^{p}(R)) d) ^{1} If 0 → C′ → C → C′′ → 0 is a short exact sequence of complexes,

then there is a canonical isomorphism

d_{R}(C) ≅ d_{R}(C′)d_{R}(C′′)

which we take as an identification.

e) If C is acyclic, then the quasi-isomorphism 0 → C induces a canonical

isomorphism

1_{R} → d_{R}(C).

f) For any integer r one has d_{R}(C[r]) = d_{R}(C)^{(−1)^{r}}.

g) the functor d_{R} factorizes over the image of C^{p}(R) in the category of perfect complexes D^{p}(R), and extends (uniquely up to unique isomorphisms) to (D^{p}(R), is).

h) For each C ∈ D(R) we write H(C) for the complex which has H(C)^{i} = H^{i}(C) in each degree i and in which all differentials are 0. If H(C) belongs to D^{p}(R) (in which case we shall say that C^{·} is cohomologically perfect), then there are canonical isomorphisms

d_{R}(C) ≅ d_{R}(H(C)) ≅ ∏_{i∈ℤ} d_{R}(H^{i}(C))^{(−1)^{i}}.

i) If R′ is any further ring and Y an (R′, R)-bimodule which is both finitely generated and projective as an R′-module, then the functor Y ⊗_{R} − : P(R) → P (R′) extends to a commutative diagram

(D^{p}(R), is) ⟶^{d_{R}} C_{R}
Y ⊗_{R}^{𝕃}−↓  ↓^{Y ⊗_{R}−}
(D^{p}(R′), is) ⟶^{d_{R′}} C_{R′}.

In particular, if R → R′ is a ring homomorphism and C ∈ D^{p}(R), we often write d_{R}(C)_{R′} in place of R′ ⊗_{R} d_{R}(C).

Remark 2.1. Unless R is a regular ring, property d) does not in general extend to arbitrary distinguished triangles. The second displayed isomorphism in h) is induced by properties d) and f). However, whilst a precise description of the first isomorphism in h) is important for the purposes of explicit computations, it is actually rather difficult to find in the literature. Here we use the description given by Knudsen in [18, §3].

Remark 2.2. We have to distinguish between at least two inverses of a map ϕ : d_{R}(C) → d_{R}(D) with C, D ∈ C^{p}(R). The inverse with respect to composition will be denoted by \overline{ϕ} : d_{R}(D) → d_{R}(C) while ϕ^{−1} := \overline{id_{d_{R}(D)^{−1}} · ϕ · id_{d_{R}(C)^{−1}}} : d_{R}(C)^{−1} → d_{R}(D)^{−1} is the unique isomorphism such that ϕ·ϕ^{−1} = id_{1_{R}} under the identification d_{R}(X) · d_{R}(X)^{−1} = 1_{R} for X = C, D. If D = C, then ϕ : d_{R}(C) → d_{R}(C) corresponds uniquely to an element of K_{1}(R) ≅ Aut_{C_{R}}(1_{R}) by the rule ϕ · id_{d_{R}(C)^{−1}} : 1_{R} → 1_{R}. Under this identification \overline{ϕ} and ϕ^{−1} agree in K_{1}(R) and are inverse to ϕ. Furthermore, the following relation between ∘ and · is easily verified: if ϕ : A → B and ψ : B → C are morphisms in C_{R}, then one has ψ ∘ ϕ = ψ · ϕ · id_{B^{−1}}.

1The listing starts with d) to be compatible with the notation of [37] where a)-c) describe properties of the category C_{R}.

We shall use the following

Convention: If ϕ : 1 → A is a morphism and B an object in C_{R}, then we

write \xymatrix{B \ar[r]^{· ϕ} & B · A} for the morphism id_{B} · ϕ. In particular, any morphism

\xymatrix{B \ar[r]^{ϕ} & A} can be written as \xymatrix{B \ar[r]^{· (id_{B^{−1}} · ϕ)} & A} .

Remark 2.3. The determinant of the complex C = [P_{0} →^{ϕ} P_{1}] (in degree 0 and

1) with P_{0} = P_{1} = P is by definition d_{R}(C) ^{def} 1_{R} and is defined even if ϕ is not an isomorphism (in contrast to d_{R}(ϕ)). But if ϕ is an isomorphism, i.e.

if C is acyclic, then by e) there is also a canonical map \xymatrix{1_{R} \ar[r]^{acyc} & d_{R}(C)} , which is in fact nothing else then

[figure: 1_{R}; d_{R}(P_{1})d_{R}(P_{1})^{−1}; d(ϕ)^{−1}·id_{d(P_{1})^{−1}}; d_{R}(P_{0})d_{R}(P_{1})^{−1}; d_{R}(C)]

(and which depends in contrast to the first identification on ϕ). Hence, by

Remark 2.2 the composite [figure: 1_{R}; acyc; d_{R}(C); def; 1_{R}] corresponds to d_{R}(ϕ)^{−1} ∈ K_{1}(R). In order to distinguish between the above identifications between 1_{R} and d_{R}(C) we also say that C is trivialized by the identity when we refer

to d_{R}(C) ^{def} 1_{R} (or its inverse with respect to composition). Obviously, if ϕ = id_{P} , then the above identifications coincide.

Remark 2.4. Let O = O_{L} be the ring of integers of a finite extension L of ℚ_{p} and let A be a finite O-module. For any morphism of the form a : 1_{O} → d_{O}(A), in particular for that induced by some exact sequence of the form \xymatrix{0 \ar[r] & O^{n} \ar[r] & O^{n} \ar[r] & A \ar[r] & 0}, we obtain a canonical element c = c(a) ∈ L^{×} = Aut_{C_{L}}(1_{L}) by means of the composite

[figure: 1_{L}; a_{L}; L ⊗_{O} d_{O}(A); d_{L}(L ⊗_{O} A); acyc; 1_{L}]

where the map ’acyc’ is induced by property e). As an immediate consequence of the elementary divisor theorem one checks that ord_{L}(c) = length_{O}(A).

2.2. The localized K_{1}-group. In [14] a localized K_{1}-group was defined for any full subcategory Σ of C^{p}(R) which satisfies the following four conditions:

(i) 0 ∈ Σ, (ii) if C, C′ are in C^{p}(R) and C is quasi-isomorphic to C′, then C ∈ Σ ⇔

C′ ∈ Σ, (iii) if C ∈ Σ, then also C[n] ∈ Σ for all n ∈ ℤ, (iv) if 0 → C′ → C → C′′ → 0 is an exact sequence in C^{p}(R) with C′, C′′ ∈

Σ then also C ∈ Σ. Since we want to apply the same construction to a subcategory which is not necessarily closed under extensions, we weaken the last condition to

(iv′) if C′ and C′′ belong to Σ, then C′ ⊕ C′′ belongs to Σ.

Definition 2.5. (Fukaya-Kato) Assume that Σ satisfies (i), (ii), (iii) and (iv′). The localized K_{1}-group K_{1}(R, Σ) is defined to be the (multiplicatively written) abelian group which has as generators symbols of the form [C, a] for each C ∈ Σ and morphism a : 1_{R} → d_{R}(C) in C_{R} and relations

(0) [0, id_{1_{R}}] = 1,

(1) [C′, d_{R}(f) ∘ a] = [C, a] if f : C → C′ is an quasi-isomorphism with C (and thus C′) in Σ,

(2) if 0 → C′ → C → C′′ → 0 is an exact sequence in Σ, then

[C, a] = [C′, a′] · [C′′, a′′]

where a is the composite of a′ · a′′ with the isomorphism induced by property d),

(3) [C[1], a^{−1}] = [C, a]^{−1}.

Remark 2.6. Relation (3) is a simple consequence of the relations (0)-(2). Note also that this definition of K_{1}(R, Σ) makes no use of the conditions (iii) and (iv′) that the category Σ is assumed to satisfy. In particular, if Σ satisfies (iv) (rather than only (iv′)), then the above definition coincides with that given in [14, §1]. We shall often refer to a morphism in C_{R} of the form a : 1_{R} → d_{R}(C) or a : d_{R}(C) → 1_{R} as a trivialization (of C).

We now assume given a left denominator set S of R and we let R_{S} := S^{−1}R denote the corresponding localization and Σ_{S} the full subcategory of C^{p}(R) consisting of all complexes C such that R_{S} ⊗_{R}C is acyclic. For any C ∈ Σ_{S} and any morphism a : 1_{R} → d_{R}(C) in C_{R} we write θ_{C,a} for the element of K_{1}(R_{S}) which corresponds under the canonical isomorphism K_{1}(R_{S}) ≅ Aut_{C_{R_{S}}} (1_{R_{S}}) to the composite

1_{R_{S}} ⟶ d_{R_{S}}(R_{S} ⊗_{R} C) → 1_{R_{S}}

where the first arrow is induced by a and the second by the fact that R_{S} ⊗_{R} C is acyclic. Then it can be shown that the assignment [C, a] ↦ θ_{C,a} induces an isomorphism of groups

ch_{R,Σ_{S}} : K_{1}(R, Σ_{S}) ≅ K_{1}(R_{S})

(cf. [14, prop. 1.3.7]). Hence, if Σ is any subcategory of Σ_{S} we also obtain a composite homomorphism

ch_{R,Σ} : K_{1}(R, Σ) → K_{1}(R, Σ_{S}) ≅ K_{1}(R_{S}).

In particular, we shall often use this construction in the following case: A^{·} ∈ Σ_{S} and Σ is equal to smallest full subcategory Σ_{A^{·}} of C^{p}(R) that contains A^{·} and also satisfies the conditions (i), (ii), (iii) and (iv) that are described above. (With this definition, it is easily seen that Σ_{A^{·}} ⊂ Σ_{ss}).

3. Leading terms

In this section we define a notion of the leading term at a continuous finite dimensional p-adic representation of elements of suitable localized K_{1}-groups. To do this we introduce an appropriate ‘semisimplicity’ hypothesis and use a natural construction of Bockstein homomorphisms. We also discuss several alternative characterizations of this notion. We explain how this formalism applies in the context of the canonical localizations introduced in [10] and we use it to extend several well known results concerning Generalized Euler Poincare characteristics.

3.1. Bockstein homomorphisms. Let G be a compact p-adic Lie group which contains a closed normal subgroup H such that the quotient group Γ := G/H is topologically isomorphic to ℤ_{p}. Then we fix a topological generator γ of Γ and denote by

θ ∈ H^{1}(G, ℤ_{p}) = Hom_{cont}(G, ℤ_{p})

the unique homomorphism G ↠ Γ → ℤ_{p} which sends γ to 1. We write Λ(G) for the Iwasawa algebra of G. Since H^{1}(G, ℤ_{p}) ≅ Ext_{Λ(G)}^{1}(ℤ_{p}, ℤ_{p}) by [23, prop. 5.2.14], the element θ corresponds to a canonical extension of Λ(G)-modules

(1)  0 → ℤ_{p} → E_{θ} → ℤ_{p} → 0.

Indeed, one has E_{θ} = ℤ_{p}^{2} upon which G acts by (\begin{matrix}1 & θ \\ 0 & 1 \end{matrix}) .

For any A^{·} in B(Λ(G)) we endow the complex A^{·} ⊗_{ℤ_{p}} E_{θ} with the natural diagonal G-action. Then (1) induces an exact sequence in B(Λ(G)) of the form

0 → A^{·} → A^{·} ⊗_{ℤ_{p}} E_{θ} → A^{·} → 0.

In D^{b}(Λ(G)) we thus obtain a morphism, the cup product by θ (depending on the choice of γ),

A^{·} ⟶^{θ∪−} A^{·}[1]

which we also denote just by θ. Now let ρ : G → GL_{n}(O) be a (continuous) representation of G on T_{ρ} = O^{n}, where O = O_{L} denotes the ring of integers of a finite extension L of ℚ_{p}. We will be mainly interested in the morphism induced by θ after forming the derived tensor product with O^{n} considered as right Λ(G)-module via the transpose ρ^{t} of ρ

O^{n} ⊗_{Λ(G)}^{𝕃} A^{·} ⟶^{θ_{∗}} O^{n} ⊗_{Λ(G)}^{𝕃} A^{·}[1].

In particular, we have homomorphisms

𝕋or_{i}^{Λ(G)}(T_{ρ}, A^{·}) ⟶^{H^{−i}(θ_{∗})} 𝕋or_{i−1}^{Λ(G)}(T_{ρ}, A^{·})

of the hypertor groups 𝕋or_{i}^{Λ(G)}(T_{ρ}, A^{·}) := H^{−i}(O^{n} ⊗_{Λ(G)}^{𝕃} A^{·}). We refer to the map

𝔅_{i} := H^{−i}(θ_{∗})

as the Bockstein morphism (in degree i).

3.2. The case G = Γ. In this section we consider the case G = Γ and take the trivial Γ-module ℤ_{p} for ρ. We set T := γ − 1. For any complex A^{·} ∈ B(Λ(Γ)) it is clear that the canonical short exact sequence

0 → Λ(Γ) ⟶^{×T} Λ(Γ) → ℤ_{p} → 0

induces an exact triangle in D^{p}(Λ(Γ)) of the form

(2)  A^{·} ⟶^{×T} A^{·} → ℤ_{p} ⊗_{Λ(Γ)}^{𝕃} A^{·} → A^{·}[1].

However, in order to be as concrete as possible, we choose to describe this result on the level of complexes. To this end we fix the following definition of the mapping cone of a map f : A^{·} → B^{·} of complexes:

cone(f) := B^{·} ⊕ A^{·}[1],

with differential in degree i equal to

d_{cone(f)}^{i} := (\begin{matrix}d_{B^{·}}^{i} & f^{i} \\ 0 & −d_{A^{·}}^{i+1} \end{matrix}) : B^{i} ⊕ A^{i+1} → B^{i+1} ⊕ A^{i+2}.

If A^{·} belongs to C^{p}(Λ(Γ)), then we set

cone(A^{·}) := cone(A^{·} ⟶^{T} A^{·})

and

A_{0}^{·} := ℤ_{p} ⊗_{Λ(G)}^{𝕃} A^{·}.

Then we have the following morphism of complexes π : cone(A^{·}) → A_{0}^{·}

⟶ A^{i−1} ⊕ A^{i} ⟶^{d_{cone}^{i−1}} A^{i} ⊕ A^{i+1} ⟶^{d_{cone}^{i}} A^{i+1} ⊕ A^{i+2} ⟶^{d_{cone}^{i+1}}
π^{i−1}↓  _{π^{i}}↓  _{π^{i+1}}↓
⟶ A_{0}^{i−1}  ⟶^{d_{A^{·}_{0}}^{i−1}}  A_{0}^{i}  ⟶^{d_{A^{·}_{0}}^{i}} A_{0}^{i+1}  ⟶^{d_{A^{·}_{0}}^{i+1}}

where, in each degree i, π^{i} sends (a, b) ∈ A^{i} ⊕ A^{i+1} to the image of a in A^{i}/T A^{i} ≅ A_{0}^{i}. It is easy to check that π is a quasi-isomorphism. Now from (2) we obtain short exact sequences

(3)  0 → H^{i}(A^{·})_{Γ} → ℍ_{−i}(Γ, A^{·}) → H^{i+1}(A^{·})^{Γ} → 0

where ℍ_{i}(Γ, A^{·}) := 𝕋or_{i}^{Λ(Γ)}(ℤ_{p}, A^{·}) denotes hyper group homology of A^{·} with respect to Γ. Furthermore, for any Λ(Γ)-module M we write M_{Γ} = M/T M and M^{Γ} = _{T} M (kernel of multiplication by T) for the maximal quotient module, resp. submodule, of M upon which Γ acts trivially.

Lemma 3.1. In each degree i the homomorphism 𝔅_{i} : ℍ_{i}(Γ, A^{·}) → ℍ_{i−1}(Γ, A^{·}) can be computed as the composite

ℍ_{i}(Γ, A^{·}) → H^{−i+1}(A^{·})^{Γ} ⟶^{κ^{−i+1}(A^{·})} H^{−i+1}(A^{·})_{Γ} → ℍ_{i−1}(Γ, A^{·})

where the map

κ^{i} : H^{i}(A^{·})^{Γ} ↪ H^{i}(A^{·}) ↠ H^{i}(A^{·})_{Γ}

is induced by the identity.

Proof. As shown in [28, lem. 1.2], on the level of complexes θ is given by the map

θ : cone(A^{·}) → cone(A^{·})[1]

which sends (a, b) ∈ A^{i} ⊕ A^{i+1} to (b, 0) ∈ A^{i+1} ⊕ A^{i+2}. Now let ā be in ker d_{A_{0}^{·}}^{−i} representing a class in ℍ_{i}(Γ, A^{·}). Then there exists (a, b) ∈ ker(d_{cone}^{−i}) with π^{−i}((a, b)) = ā. Since (a, b) ∈ ker(d_{cone}^{−i}) one has b ∈ ker(d_{A^{·}}^{i+1}) and T b = −d_{A^{·}}^{i}(a). This implies that d_{A^{·}}^{i}(a) is divisible by T (in A^{i+1}) and that b = −T^{−1}d_{A^{·}}^{i}(a) ∈ A^{i+1}. Thus θ maps (a, b) to (−T^{−1}d_{A^{·}}^{i}(a), 0) and the class in ℍ_{i−1}(Γ, A^{·}) is represented by −\overline{T^{−1}d_{A^{·}}^{i}(a)} ∈ ker(d_{A_{0}^{·}}^{−i+1}). By using the canonical exact sequence

0 → A^{·} → cone(A^{·}) → A^{·}[1] → 0

one immediately verifies that 𝔅_{i} coincides with the map described in the lemma. □

From this description it follows immediately that the pair

(4)  (ℍ_{i}(Γ, A^{·}), 𝔅_{i})

forms a (homological) complex, which by reindexing we consider as cohomological complex if necessary. We refer to the morphism 𝔅_{i} defined here as the Bockstein morphism in degree i of the pair (A^{·}, T).

Definition 3.2. (Semisimplicity) We shall say that a complex A^{·} ∈ C^{p}(Λ(Γ)) is semisimple if the cohomology of the associated complex (4) is ℤ_{p}-torsion (and hence finite) in all degrees. We let Σ_{ss} denote the full subcategory of C^{p}(Λ(Γ)) consisting of those complexes that are semisimple. For any A^{·} ∈ Σ_{ss} we set

r_{Γ}(A^{·}) := ∑_{i∈ℤ}(−1)^{i+1} dim_{ℚ_{p}}(H^{i}(A^{·})^{Γ} ⊗_{ℤ_{p}} ℚ_{p}) ∈ ℤ.

Remark 3.3. (i) If A^{·} ∈ Σ_{ss}, then the cohomology of A^{·} is a torsion Λ(Γ)- module in all degrees. (ii) In each degree i Lemma 3.1 gives a canonical exact sequence

0 → cok(κ^{−i}) → ker(𝔅_{i})/im(𝔅_{i+1}) → ker(κ^{−i+1}) → 0.

This implies that a complex A^{·} ∈ C^{p}(Λ(Γ)) belongs to Σ_{ss} if and only if the map κ^{i}(A^{·})⊗_{ℤ_{p}} ℚ_{p} is bijective in each degree i, and hence also that in any such case one has

r_{Γ}(A^{·}) = ∑_{i∈ℤ}(−1)^{i+1} dim_{ℚ_{p}}(H^{i}(A^{·})_{Γ} ⊗_{ℤ_{p}} ℚ_{p}).

Definition 3.4. (The canonical trivialization) We write (ℍ_{·}(Γ, A^{·}), 0) for the complex which has (ℍ_{·}(Γ, A^{·}), 0)^{i} = ℍ_{i}(Γ, A^{·}) in each degree i and in which all differentials are the zero map. If A^{·} ∈ Σ_{ss}, then we obtain a canonical trivialization of ℤ_{p} ⊗_{Λ(Γ)}^{𝕃} A^{·} by taking the composite

(5) t(A^{·}) : d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(Γ)}^{𝕃} A^{·})_{ℚ_{p}} ≅ d_{ℤ_{p}}((ℍ_{·}(Γ, A^{·}), 0))_{ℚ_{p}}

= d_{ℤ_{p}}((ℍ_{·}(Γ, A^{·}), 𝔅_{·}))_{ℚ_{p}} ≅ 1_{ℚ_{p}}

where the first, resp. last, isomorphism uses property h), resp. e), of the functor d_{ℤ_{p}}.

Remark 3.5. If ℚ_{p} ⊗_{Λ(Γ)}^{𝕃} A^{·} is acyclic, then t(A^{·}) coincides with the trivialization obtained by applying property e) directly to ℚ_{p} ⊗_{Λ(Γ)}^{𝕃} A^{·}.

The category Σ_{ss} satisfies the conditions (i), (ii), (iii) and (iv′) that are described in §2 (but does not satisfy condition (iv)). In addition, as the following lemma shows, the above constructions behave well on exact triangles of semisimple complexes.

Lemma 3.6. Let A^{·}, B^{·} and C^{·} be objects of Σ_{ss} which together lie in an exact triangle in D^{p}(Λ(Γ)) of the form

A^{·} → B^{·} → C^{·} → A^{·}[1].

Then one has

r_{Γ}(B^{·}) = r_{Γ}(A^{·}) + r_{Γ}(C^{·})

and, with respect to the canonical isomorphism

d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(G)}^{𝕃} B^{·})_{ℚ_{p}} = d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(G)}^{𝕃} A^{·})_{ℚ_{p}}d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(G)}^{𝕃} C^{·})_{ℚ_{p}}

that is induced by the given exact triangle, one has

t(B^{·}) = t(A^{·})t(C^{·}).

Proof. Easy and left to the reader. □

We write ρ_{triv} for the trivial representation of Γ.

Definition 3.7. (The leading term) For each A^{·} ∈ Σ_{ss} and each morphism a : 1_{Λ(Γ)} → d_{Λ(Γ)}(A^{·}) in C_{Λ(Γ)} we define the leading term (A^{·}, a)^{∗}(ρ_{triv}) of the pair (A^{·}, a) at ρ_{triv} to be equal to (−1)^{r_{Γ}(A^{·})} times the element of ℚ_{p} \ {0} which corresponds via the isomorphisms ℚ_{p}^{×} ≅ K_{1}(ℚ_{p}) ≅ Aut_{C_{ℚp}} (1_{ℚ_{p}}) to the composite morphism

1_{ℚ_{p}} ⟶^{ℚ_{p}⊗_{Λ(Γ)}a} d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(Γ)} A^{·})_{ℚ_{p}} ⟶^{t(A^{·})} 1_{ℚ_{p}}.

After taking into account Lemma 3.6, this construction induces a well defined homomorphism of groups

(−)^{∗}(ρ_{triv}) : K_{1}(Λ(Γ), Σ_{ss}) → ℚ_{p}^{×}
[A^{·}, a] ↦ [A^{·}, a]^{∗}(ρ_{triv}) := (A^{·}, a)^{∗}(ρ_{triv}).

The reason for the occurrence of ρ_{triv} in the above definition will become clear in the next subsection. In the remainder of the current section we justify the name ‘leading term’ by explaining the connection between (A^{·}, a)^{∗}(ρ_{triv}) and the leading term of an appropriate characteristic power series. To this end we note that Σ_{ss} is a subcategory of the full subcategory of C^{p}(Λ(Γ)) consisting of those complexes C for which Q(Γ) ⊗_{Λ(Γ)} C is acyclic (by Remark 3.3(i)), and hence that there exists a homomorphism

ch_{Γ} := ch_{Λ(Γ),Σ_{ss}} : K_{1}(Λ(Γ), Σ_{ss}) → K_{1}(Q(Γ)) ≅ Q(Γ)^{×}.

If L is any field extension of ℚ_{p}, and O the valuation ring in L, then the identification Λ_{O}(Γ) ≅ O[[T]] (which, of course, depends on the choice of T = γ − 1) allows any element F ∈ Q_{O}(Γ)^{×} to be written uniquely as

(6)  F (T) = T^{r}G(T)

with r = r(F) ∈ ℤ and G(T) ∈ Q_{O}(Γ) such that G(0) ∈ L^{×}. The leading coefficient of F with respect to its Taylor series expansion in the Laurent series ring L{{T }} is thus equal to F^{∗}(0) := G(0). Let 𝔭 be the kernel of the augmentation map Λ(Γ) → ℤ_{p} and denote by R the localisation Λ(Γ)_{𝔭}. This is a discrete valuation ring with uniformizer T and the residue class field R/(T) is isomorphic to ℚ_{p}. The notation of semi-simplicity extends naturally to R via the exact functor R ⊗_{Λ(Γ)} − : indeed, if A^{·} ∈ D^{p}(R), then the analogue of (2) induces a Bockstein map for 𝕋or_{·}^{R}(R/(T), A^{·}) and we say that A^{·} is semisimple if the associated complex (𝕋or_{·}^{R}(R/(T), A^{·}), 𝔅_{·}) is acyclic.

Proposition 3.8. Let a : 1_{Λ(Γ)} → d_{Λ(Γ)}(A^{·}) be a morphism in C_{Λ(Γ)} with A^{·} in Σ_{ss}.

(i) (Order of vanishing) For L := [A^{·}, a] one has r(ch_{Γ}(L)) = r_{Γ}(A^{·}). (ii) (Leading terms) One has a commutative diagram of abelian groups

K_{1}(Λ(Γ), Σ_{ss}) ⟶^{ch_{Γ}} K_{1}(Q(Γ))
(−)^{∗}(ρ_{triv})↓  ↓^{(−)^{∗}(0)}
ℚ_{p}^{×}  ℚ_{p}^{×}.

Proof. It is easy to see that both maps (−)^{∗}(ρ_{triv}) and ch_{Γ} factor via flat base change R ⊗_{Λ(Γ)} − through K_{1}(R, Ξ), where Ξ denotes the full subcategory of C^{p}(R) consisting of those complexes which are semisimple. Thus it suffices to show the commutativity of the above diagram with K_{1}(Λ(Γ), Σ_{ss}) replaced by K_{1}(R, Ξ). Moreover, by Lemma 3.9 below this is reduced to the case where A^{·} is a complex of the form R ⟶^{a} R where R occurs in degrees −1 and 0 and a denotes multiplication by either T or 1. Further, since the complex R ⟶^{×1} R is acyclic we shall therefore assume that a denotes multiplication by T. Now Mor_{C_{R}}(1_{R}, d(A^{·})) is a K_{1}(R)-torsor and so all possible trivialization arise in the following way where ϵ ∈ R^{×} : The module homomorphisms R → A_{i} which send 1 to 1 ∈ R = A_{−1} and to ϵ ∈ R = A_{0}, respectively, induce maps can_{1} : d_{R}(R) → d_{R}(A_{−1}) and can_{ϵ} : d_{R}(R) → d_{R}(A_{0}) which give rise to the trivialization a_{ϵ} := (can_{1})^{−1} · can_{ϵ} : 1_{R} → d_{R}(A^{·}). Setting L_{ϵ} := [A^{·}, a_{ϵ}] one checks easily that ch_{Γ}(L_{ϵ}) = T^{−1}ϵ and thus ch_{Γ}(L_{ϵ})^{∗}(0) = ϵ(0). On the other hand, the bockstein map 𝔅_{1} is just ℤ_{p} ⟶^{−1} ℤ_{p} as one checks using the description of 𝔅 in the proof of Lemma 3.1. Thus L_{ϵ}^{∗}(ρ_{triv}) is equal to (−1)^{r_{Γ}(A^{·})} times the determinant of

ℚ_{p} ⟶^{ϵ(0)} ℚ_{p} ⟶^{(𝔅_{1})_{ℚp}^{−1}=−1} ℚ_{p} ⟶^{1} ℚ_{p}.

Hence, observing that r_{Γ}(A^{·}) = −1 = r(ch_{Γ}(L_{ϵ})), we have L_{ϵ}^{∗}(ρ_{triv}) = ϵ(0) which proves the Proposition. □

Lemma 3.9. Let R be a discrete valuation ring with uniformizer T and assume that A^{·} ∈ C^{p}(R) is semisimple. Then A^{·} is isomorphic in C^{p}(R) to the direct sum of finitely many complexes of the form R → R where the differential is equal to multiplication by either 1 or T.

Proof. Assume that m is the maximal degree such that A^{m} ≠ 0 and fix an isomorphism D : R^{d} ≅ A^{m}. Let (e_{1}, …, e_{d}) be the standard basis of R^{d}. The semisimplicity of A^{·} is equivalent to the property that T · H^{i}(A^{·}) = 0 for all i ∈ ℤ as can be seen easily by using an analogue of Lemma 3.1; indeed, both conditions are equivalent to the property that the analogues of the maps κ^{i} in that Lemma are isomorphisms for all i. Thus, for 1 ≤ i ≤ d, T e_{i} is in the image of D^{−1}∘d^{m−1}. We set h_{i} = 1 if e_{i} is already in this image and h_{i} = T, otherwise. By H we denote the diagonal matrix with the elements h_{1}, …, h_{d}. Since the image of the map R^{d} ⟶^{H} R^{d} coincides by definition with that of D^{−1} ∘d^{m−1} we obtain a retraction E : R^{d} ⟶ A^{m−1} (i.e. with left inverse ”H^{−1} ∘ D^{−1} ∘ d^{m−1}”) making the following diagram commutative

⟶ 0 ⟶ R^{d} ⟶^{H} R^{d} ⟶ 0 ⟶
↓  ^{E}↓  ^{D}↓  ↓
⟶ A^{m−2} ⟶^{d^{m−2}} A^{m−1} ⟶^{d^{m−1}} A^{m} ⟶^{d^{m}} 0 ⟶ .

If B^{·} denotes the upper row of this diagram and C^{·} := A^{·}/B^{·} the associated quotient complex (not the mapping cone!), one checks readily that there exists a split exact sequence 0 → B^{·} → A^{·} → C^{·} → 0. Since C^{·} is again semisimple and has a strictly shorter length than A^{·} the proof is accomplished by induction. □

Remark 3.10. It will be clear to the reader that analogous statements hold for all results of this subsection if we replace ℤ_{p} by O, ℚ_{p} by L, Λ(Γ) by Λ_{O}(Γ) := O[[Γ]], and ℚ(Γ) by Q_{O}(Γ), the quotient field of Λ_{O}(Γ).

3.3. The general case. For any continuous representation ρ : G → GL_{n}(O) we regard the complex

A^{·}(ρ^{∗}) := A^{·} ⊗_{ℤ_{p}} O^{n}

as a complex of Λ(G)-modules by means of the following G-action: g(a ⊗ o) := ga ⊗ ρ^{∗}(g)o for g ∈ G, a ∈ A^{i} and o ∈ O^{n}.

Definition 3.11. (Semisimplicity at ρ) We shall say that a complex A^{·} ∈ D^{p}(Λ(G)) is semisimple at ρ if the cohomology of the associated complex (ℍ_{·}(G, A^{·}(ρ^{∗})), 𝔅_{·}) is ℤ_{p}-torsion in each degree. We let Σ_{ss−ρ} denote the full subcategory of C^{p}(Λ(G)) consisting of all complexes that are semisimple at ρ, and we note that Σ_{ss−ρ} satisfies the conditions (i), (ii), (iii) and (iv′) that are described in §2. For each A^{·} ∈ Σ_{ss−ρ} we set

r_{G}(A^{·})(ρ) := ∑_{i∈ℤ}(−1)^{i+1} dim_{L} (ℍ_{i}(H, A^{·}(ρ^{∗}))^{Γ} ⊗_{O} L) ∈ ℤ,

where ℍ_{i}(H, −) := 𝕋or_{i}^{Λ(H)}(O, −) denotes hyper H-group homology.

Definition 3.12. (Finiteness at ρ) Similarly we shall say that a complex A^{·} ∈ D^{p}(Λ(G)) is finite at ρ if the cohomology groups ℍ_{i}(G, A^{·}(ρ^{∗})) are ℤ_{p}-torsion in each degree. We let Σ_{fin−ρ} denote the full subcategory of C^{p}(Λ(G)) consisting of all complexes that are finite at ρ, and we note that Σ_{fin−ρ} satisfies the conditions (i), (ii), (iii) and (iv) that are described in §2. In particular we have Σ_{fin−ρ} ⊆ Σ_{ss−ρ}.

In the next result we consider the tensor product Λ_{O}(Γ) ⊗_{O} O^{n} as a natural (Λ_{O}(Γ), Λ(G))-bimodule where Λ_{O}(Γ) acts by multiplication on the left and Λ(G) acts on the right via the rule (τ ⊗ o)g := τḡ⊗ ρ(g)^{t}o for each g ∈ G (with image ḡ in Γ), o ∈ O^{n} and τ ∈ Λ_{O}(Γ). For a complex A^{·} ∈ Σ_{ss−ρ}, we write t(A^{·}(ρ^{∗})) for the trivialization t(A^{·}_{ρ^{∗}}) : d_{O}(O ⊗_{Λ_{O}(Γ)}^{𝕃} A^{·}_{ρ^{∗}})_{L} → 1_{L} defined in (5). By the following Lemma this amounts to the same as taking the composite

(7) t(A^{·}(ρ^{∗})) : d_{O}(O ⊗_{Λ_{O}(G)}^{𝕃} A^{·}(ρ^{∗}))_{L} ≅ d_{O}((ℍ_{·}(G, A^{·}(ρ^{∗})), 0))_{L}

= d_{O}((ℍ_{·}(G, A^{·}(ρ^{∗})), 𝔅_{·}))_{L} ≅ 1_{L}

where the first, resp. last, isomorphism uses property h), resp. e), of the functor d_{O}.

Lemma 3.13. Fix A^{·} ∈ C^{p}(Λ(G)).

(i) One has A^{·} ∈ Σ_{ss−ρ} if and only if

A_{ρ}^{·} := (Λ_{O}(Γ) ⊗_{O} O^{n}) ⊗_{Λ(G)}^{𝕃} A^{·} ≅ Λ_{O}(Γ) ⊗_{Λ_{O}(G)}^{𝕃} A^{·}(ρ^{∗})

belongs to Σ_{ss} (when considered as an object of C^{p}(Λ_{O}(Γ))). In addition, in any such case the Bockstein map of ℍ(G, A^{·}(ρ^{∗})) is equal to the Bockstein map of the pair (A^{·}_{ρ}, T) and one has natural isomorphisms in C^{p}(O)

O ⊗_{Λ_{O}(Γ)}^{𝕃} A_{ρ}^{·} ≅ O^{n} ⊗_{Λ(G)}^{𝕃} A^{·}

and in B(O).

Λ_{O}(Γ) ⊗_{Λ(G)}^{𝕃} A^{·} ≅ O ⊗_{Λ(H)}^{𝕃} A^{·}.
(ii) If A^{·} ∈ Σ_{ss−ρ}, then r_{G}(A^{·})(ρ) = r_{Γ}(A_{ρ}^{·}).

(iii) If A^{·}, B^{·} and C^{·} are objects of Σ_{ss−ρ} which together lie in an exact triangle in D^{p}(Λ(G)) of the form

A^{·} → B^{·} → C^{·} → A^{·}[1],

then one has

r_{G}(B^{·})(ρ) = r_{G}(A^{·})(ρ) + r_{G}(C^{·})(ρ)

and, with respect to the canonical isomorphism

d_{O}(O ⊗_{Λ_{O}(G)}^{𝕃} B^{·}(ρ^{∗}))_{L} = d_{O}(O ⊗_{Λ_{O}(G)}^{𝕃} A^{·}(ρ^{∗}))_{L}d_{O}(O ⊗_{Λ_{O}(G)}^{𝕃} C^{·}(ρ^{∗}))_{L}

that is induced by the given exact triangle, one has

t(B^{·}(ρ^{∗})) = t(A^{·}(ρ^{∗})) · t(C^{·}(ρ^{∗})).

Proof. Claim (i) follows from the fact that the specified actions induce natural isomorphisms in D^{p}(O) of the form

O^{n} ⊗_{Λ(G)}^{𝕃} A^{·} ≅ O ⊗_{Λ_{O}(G)}^{𝕃} A^{·}(ρ^{∗})
≅ O ⊗_{Λ_{O}(Γ)}^{𝕃} (Λ_{O}(Γ) ⊗_{Λ_{O}(G)}^{𝕃} A^{·}(ρ^{∗}))
≅ O ⊗_{Λ_{O}(Γ)}^{𝕃} ((Λ_{O}(Γ) ⊗_{O} O^{n}) ⊗_{Λ(G)}^{𝕃} A^{·})

and from the functorial construction of the Bockstein map. Claim (ii) follows directly by using claim (i) to compare the definitions of the terms r_{G}(A^{·})(ρ) and r_{Γ}(A_{ρ}^{·}). To prove claim (iii) we observe that, by claim (i), the given triangle induces an exact triangle of semisimple complexes in D^{p}(Λ(Γ)) of the form

A_{ρ}^{·} → B_{ρ}^{·} → C_{ρ}^{·} → A_{ρ}^{·}[1].

The equalities of claim (iii) thus follow from claim (ii) and the results of Lemma 3.6 as applied to the above triangle. □

Definition 3.14. (The leading term at ρ) For each complex A^{·} ∈ Σ_{ss−ρ} and each morphism a : 1_{Λ(G)} → d_{Λ(G)}(A^{·}) we define the leading term (A^{·}, a)^{∗}(ρ) of the pair (A^{·}, a) at ρ to be equal to (−1)^{r_{G}(A^{·})(ρ)} times the element of L \ {0} which corresponds via the isomorphisms L^{×} ≅ K_{1}(L) ≅ Aut_{C_{L}}(1_{L}) to the composite morphism

1_{L} ⟶^{L^{n}⊗_{Λ(G)}a} d_{L}(L^{n} ⊗_{Λ(G)}^{𝕃} A^{·}) ⟶^{t(A^{·}(ρ^{∗}))} 1_{L}.

Since Σ_{A^{·}} ⊂ Σ_{ss−ρ} Lemma 3.13 shows that the above construction induces a well-defined homomorphism of groups

(−)^{∗}(ρ) : K_{1}(Λ(G), Σ_{A^{·}}) → L^{×}
[A^{·}, a] ↦ [A^{·}, a]^{∗}(ρ) := (A^{·}, a)^{∗}(ρ).

If A^{·} is clear from context, then we often write a^{∗}(ρ) in place of [A^{·}, a]^{∗}(ρ). It is easily checked that (in the case G = Γ and ρ = ρ_{triv}) these definitions are compatible with those given in §3.2. Further, in §3.4.3 we shall reinterpret the expression [A^{·}, a]^{∗}(ρ) defined above as the leading term at s = 0 of a natural p-adic meromorphic function.

Remark 3.15. If A^{·} ∈ Σ_{fin−ρ}, then we set [A^{·}, a](ρ) := [A^{·}, a]^{∗}(ρ) and call this the value of [A^{·}, a] at ρ. Taking into account Remark 3.5 it is clear that this definition coincides with that given in [14, 4.1.5].

3.4. Canonical localizations. We apply the constructions of §3.3 in the setting of the canonical localizations of Λ(G) that were introduced in [10].

3.4.1. The canonical Ore sets. We recall from [10, §2-§3] that there are canonical left and right denominator sets S and S^{∗} of Λ(G) where

S := {λ ∈ Λ(G) : Λ(G)/Λ(G) · λ is a finitely generated Λ(H)-module}

and

S^{∗} := ⋃_{i≥0} p^{i}S.

We write S^{∗}-tor for the category of finitely generated Λ(G)-modules M which satisfy Λ(G)_{S^{∗}} ⊗_{Λ(G)} M = 0. We further recall from loc. cit. that a finitely generated Λ(G)-module M belongs to S^{∗}-tor, if and only if M/M(p) is finitely generated when considered as a Λ(H)-module (by restriction) where M(p) is the submodule of M consisting of those elements which are annihilated by some power of p.

3.4.2. Leading terms. We use the notation of Definition 3.14. If ρ : G → GL_{n}(O) is any continuous representation and A^{·} any object of Σ_{S^{∗}}, then Σ_{A^{·}} ⊂ Σ_{S^{∗}} and so there exists a canonical homomorphism

ch_{G,A^{·}} := ch_{Λ(G),Σ_{A^{·}}} : K_{1}(Λ(G), Σ_{A^{·}}) → K_{1}(Λ(G)_{S^{∗}}).

In addition, the ring homomorphism Λ(G)_{S^{∗}} → M_{n}(Q(Γ)) which sends each element g ∈ G to ρ(g)ḡ where ḡ is the image in Γ, induces a homomorphism

ρ_{∗} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(M_{n}(Q_{O}(Γ))) ≅ K_{1}(Q_{O}(Γ)) ≅ Q_{O}(Γ)^{×}.

Proposition 3.16. Let A^{·} be a complex which belongs to Σ_{S^{∗}} ∩ Σ_{ss−ρ}. (i) (Order of vanishing) One has r_{G}(A^{·})(ρ) = r_{Γ}(A_{ρ}^{·}) = r(ρ_{∗} ∘ch_{G,A^{·}}(A^{·})). (ii) (Leading terms) The following diagram of abelian groups commutes

K_{1}(Λ(G), Σ_{A^{·}}) ⟶^{ch_{G,A^{·}}} K_{1}(Λ(G)_{S^{∗}})
(−)^{∗}(ρ)↓  ↓^{(ρ_{∗}(−))^{∗}(0)}
L^{×}  L^{×},

where (−)^{∗}(0) denotes the ‘leading term’ homomorphism K_{1}(Q_{O}(Γ)) → L^{×} according to Proposition 3.8.

Proof. By Lemma 3.13(i) we have ℍ_{i}(H, A^{·}(ρ^{∗})) = H^{−i}(O ⊗_{Λ(H)}^{𝕃} A^{·}(ρ^{∗})) = H^{−i}(A_{ρ}^{·}). Taking also into account Proposition 3.8, (i) follows from Definition 3.2 and 3.11. Claim (ii) is proved by the same arguments as used in [14, lem.

4.3.10]. Indeed, one need only observe that the above diagram arises as the following composite commutative diagram

K_{1}(Λ(G), Σ_{A^{·}}) ⟶^{ch_{Λ(G),Σ_{A^{·}}}} K_{1}(Λ(G)_{S^{∗}})
(Λ_{O}(Γ)⊗_{O}O^{n})⊗_{Λ(G)}−↓  ↓^{ρ_{∗}}
K_{1}(Λ_{O}(Γ), Σ_{ss}) ⟶^{ch_{Λ_{O} (Γ),Σss}} K_{1}(Q_{O}(Γ))
(−)^{∗}(ρ_{triv})↓  ↓^{(−)^{∗}(0)}
L^{×}  L^{×}

where the lower square is as in Proposition 3.8. □

Thus for any F ∈ K_{1}(Λ(G)_{S^{∗}}) we write also F^{∗}(ρ) for the leading term ρ_{∗}(F)(0) of F at ρ. We note that, by Proposition 3.16 , this notation is consistent with that of Definition 3.14 in the case that F belongs to the image of ch_{G,A^{·}}. Similarly, we shall use the notation F (ρ) := F^{∗}(ρ) if r(ρ_{∗}(F)) = 0.

3.4.3. Partial derivatives. We now observe that the constructions of the previous section allow an interpretation of the expression (A^{·}, a)^{∗}(ρ) defined in §3.3 as the leading term at s = 0 of a natural p-adic meromorphic function. At the outset we fix a representation of G of the form χ : G ↠ Γ → ℤ_{p}^{×} of infinite order and we set

c_{χ,γ} := log_{p}(χ(γ)) ∈ ℚ_{p}^{×}.

We also fix A^{·} ∈ Σ_{S^{∗}} and a morphism a : 1_{Λ(G)} → d_{Λ(G)}(A^{·}), put L := [A^{·}, a], and for any continuous representation ρ : G → GL_{n}(O) we set

f_{ρ}(T) := ρ_{∗}(ch_{G,A^{·}}(L)) ∈ K_{1}(Q_{O}(Γ)) ≅ Q_{O}(Γ)^{×}.

Then, since the zeros and poles of elements in Q_{O}(Γ) are discrete, the function

s ↦ f_{L}(ρχ^{s}) := f_{ρ}(χ(γ)^{s} − 1)

is a p-adic meromorphic function on ℤ_{p}.

Lemma 3.17. Let A^{·} and a be as above and set r := r_{G}(A^{·})(ρ). Then, (i) in any sufficiently small neighbourhood of 0 in ℤ_{p} one has

L^{∗}(ρχ^{s}) = L(ρχ^{s}) = f_{L}(ρχ^{s})

for all s ≠ 0, (ii) c_{χ,γ}^{r}L^{∗}(ρ) is the leading coefficient at s = 0 of f_{L}(ρχ^{s}), and (iii) if r ≥ 0, one has

c_{χ,γ}^{r}L^{∗}(ρ) = \frac{1}{r!} \frac{d^{r}}{ds^{r}} f_{L}(ρχ^{s})|_{s=0}.

Proof. In any sufficiently small neighbourhood of 0 in ℤ_{p} one has f_{ρχ^{s}}(0) ∈ L^{×} for all s ≠ 0. Since f_{ρχ^{s}}(T) = f_{ρ}(χ(γ)^{s}(T + 1) − 1) we may therefore deduce from Proposition 3.16 that L^{∗}(ρχ^{s}) = f_{ρχ^{s}}(0) = f_{ρ}(χ(γ)^{s} − 1) = f_{L}(ρχ^{s}) for any such value of s.

In addition, if r ≥ 0 and we factorize f_{ρ}(T) as T^{r}G_{ρ}(T) with G_{ρ}(T) ∈ Q_{O}(Γ), then G_{ρ}(0) = f_{ρ}^{∗}(0) and

\frac{1}{r!} \frac{d^{r}}{ds^{r}} f_{L}(ρχ^{s})|_{s=0} = lim_{0≠s→0} \frac{f_{ρ}(χ(γ)^{s} − 1)}{s^{r}}
= lim_{0≠s→0} (\frac{(χ(γ)^{s} − 1)^{r}}{s^{r}} G_{ρ}(χ(γ)^{s} − 1))
= ( lim_{0≠s→0} \frac{χ(γ)^{s} − 1}{s})^{r}G_{ρ}(0)
= (log_{p}(χ(γ)))^{r}f_{ρ}^{∗}(0)
= c_{χ,γ}^{r}L^{∗}(ρ),

where the last equality follows from Proposition 3.16. If r < 0 we do not have the interpretation as partial derivative but the same arguments prove the statement concerning the leading coefficient at s = 0. □

Remark 3.18. Of particular interest is the case χ = χ_{cyc} in which the above calculus can be interpreted as partial derivation in the ‘cyclotomic’ direction (cf. Remark 5.6) .

3.5. Generalized Euler-Poincare characteristics. In this section we observe that the constructions made in §3.3 give rise to a natural extension of results from [10, 14, 36]. We fix a continuous representation ρ : G → GL_{n}(O) and a complex A^{·} ∈ Σ_{ss−ρ} and in each degree i we set H_{𝔅}^{i}(G, −) := H^{i}((ℍ_{−·}(G, −), 𝔅_{−·})). We then define the (generalized) additive, resp. multiplicative, Euler-Poincare characteristic of the complex A^{·}(ρ^{∗}) by setting

χ_{add}(G, A^{·}(ρ^{∗})) := ∑_{i∈ℤ}(−1)^{i}length_{O}(H_{𝔅}^{i}(G, A^{·}(ρ^{∗}))),

resp.

χ_{mult}(G, A^{·}(ρ^{∗})) := (#κ_{L})^{χ_{add}(G,A^{·}(ρ^{∗}))}

where κ_{L} is the residue class field of L. We recall that for single Λ(G)-module M or rather its Pontryagin-dual D similar Euler characteristics have also been studied by other authors (cf. [11, 40, 16]). Indeed, they use the Hochschild-Serre spectral sequence to construct differentials

d^{i} : H^{i}(G, D) → H^{i}(H, D)^{Γ} → H^{i}(H, D)_{Γ} → H^{i+1}(G, D)

where the middle map is induced by the identity; then their generalized Euler characteristic is defined similar as above using the complex (H^{·}(G, D), d^{·}) instead of (ℍ_{−·}(G, −), 𝔅_{−·}). But by Lemma 3.13 (i) one sees immediately that the Pontryagin dual of d_{i} coincides with the Bockstein map 𝔅_{i+1} : ℍ_{i+1}(G, P^{·}) → ℍ_{i}(G, P^{·}) as computed in Lemma 3.1 with A^{·} = Λ(Γ) ⊗_{Λ(G)}^{𝕃} P^{·}, where P^{·} denotes a projective solution of M. More precisely, in [11] a truncated version of this generalized Euler characteristic is used.

Proposition 3.19. Let ord_{L} denote the valuation of L which takes the value 1 on any uniformizing parameter and | − |_{p} the p-adic absolute value, normalized such that |p|_{p} = p^{−1}. If A^{·} ∈ Σ_{ss−ρ} and a : 1_{Λ(G)} → d_{Λ(G)}(A^{·}) is any morphism, then for L := [A^{·}, a] one has

χ_{add}(G, A^{·}(ρ^{∗})) = ord_{L}(L^{∗}(ρ))

and

χ_{mult}(G, A^{·}(ρ^{∗})) = |L^{∗}(ρ)|_{p}^{−[L:ℚ_{p}]}.

Proof. First observe that using Lemma 3.13, property h) and the fact that O is regular we obtain canonical isomorphisms

\xymatrix{1_{O} \ar[r]^{O^{n}⊗_{Λ(G)}a} & d_{O}(O^{n} ⊗_{Λ(G)}^{𝕃} A^{·})} ≅ d_{O}(O ⊗_{Λ_{O}(G)}^{𝕃} A^{·}(ρ^{∗})))
≅ d_{O}(H_{𝔅}^{·}(G, A^{·}(ρ^{∗})))
≅ ∏_{i∈ℤ} d_{O}(H_{𝔅}^{i}(G, A^{·}(ρ^{∗})))^{(−1)^{i}}.

After tensoring with L and identifying then the factors at the end with the unit object by acyclicity we recover the definition of the leading term (A^{·}, a)^{∗}(ρ). From this description it becomes clear that the latter can also be written modulo O^{×} as the product over i of the following maps

\xymatrix{(1_{O})_{L} \ar[r]^{def} & d_{O}(H_{𝔅}^{i}(A^{·}(ρ^{∗})))_{L}^{(−1)^{i}} \ar[r]^{acyc} & 1_{L}} .

The result follows now from Remark 2.4. □

Remark 3.20. In the special case that the complex A^{·}(ρ^{∗}) ⊗_{ℤ_{p}} ℚ_{p} is acyclic, the leading term L^{∗}(ρ) is just the value of L at ρ (in the sense of Remark 3.15) and so the result of Proposition 3.19 recovers the results of [10, thm. 3.6], [36, prop. 6.3 ] and [14, rem. 4.1.13].

4. Global Zeta isomorphisms

In this section we recall the Tamagawa Number Conjecture as formulated by Fukaya and Kato in [14].

4.1. Galois cohomology. The main reference for this section is [14, §1.6], but see also [6], here we use the same notation as in [37]. For simplicity we assume that p is odd throughout this section. Let U = spec(ℤ[\frac{1}{S}]) be a dense open subset of spec(ℤ) where S contains S_{p} := {p} and S_{∞} := {∞} (by abuse of notation). We write G_{S} for the Galois group of the maximal outside S unramified extension of ℚ. Let X be a topological abelian group with a continuous action of G_{S}. As usual we write RΓ(U, X) (RΓ_{c}(U, X)) for global Galois cohomology with restricted ramification (and compact support), for any place v of ℚ we denote by RΓ(ℚ_{v}, X) local Galois cohomology. Let L be a finite extension of ℚ_{p} with ring of integers O and V a finite dimensional Lvector space with continuous G_{ℚ_{v}}- and G_{ℚ}-action, respectively. Then the finite parts of global and local cohomology are written as RΓ_{f}(ℚ, V) and RΓ_{f}(ℚ_{v}, V), respectively. There is a canonical exact triangle

(8)  \xymatrix{RΓ_{c}(U, V) \ar[r] & RΓ_{f}(ℚ, V) \ar[r] & ⨁_{S} RΓ_{f}(ℚ_{v}, V) \ar[r] & .}

For any prime ℓ set t_{ℓ}(V) := D_{dR}(V)/D_{dR}^{0}(V) if ℓ = p and t_{ℓ}(V) = 0 otherwise. For the definition of the canonical isomorphism in C_{L}

(9)  η_{ℓ}(V) : 1_{L} → d_{L}(RΓ_{f}(ℚ_{l}, V))d_{L}(t_{ℓ}(V)).

we refer the reader to [14, §2.4.4] or the appendix of [37].

4.2. K-Motives over ℚ. For the background on this material we refer the reader to [14, §2.2, 2.4], [6, §3] or the survey [37, §2]. Let K be a finite extension of ℚ. As usual we write M_{B}, M_{dR} and M_{λ} for the Betti, de Rham and λ-adic realizations of a K-motive M where λ varies over all finite places of K. Also, we write t_{M} = M_{dR}/M_{dR}^{0} for the tangent space of M. Henceforth, for any commutative ring R and R[G(ℂ/ℝ)]-module X we denote by X^{+} and X^{−} the R-submodule of X on which complex conjugation acts by +1 and −1, respectively. In our later calculations we will use the following isomorphisms

• The comparison between the Betti- and the λ-adic realization induces canonical isomorphisms of K_{λ} and K_{l}-modules

(10)  g_{λ}^{+} : K_{λ} ⊗_{K} M_{B}^{+} ≅ M_{λ}^{+} and g_{ℓ}^{+} : K_{ℓ} ⊗_{K} M_{B}^{+} ≅ M_{ℓ}^{+}.

• The comparison between de Rahm and Betti-cohomology induces the the ℝ-linear period map

(11)  \xymatrix{ℝ ⊗_{ℚ} M_{B}^{+} \ar[r]^{α_{M}} & ℝ ⊗_{ℚ} t_{M}.}

• The comparison between p-adic and de Rham cohomology induces an isomorphism of K_{λ}-vector spaces

(12)  \xymatrix{t_{p}(M_{λ}) = D_{dR}(M_{λ})/D_{dR}^{0}(M_{λ}) \ar[r]^{g_{dR}^{t}}_{≅} & K_{λ} ⊗_{K} t_{M}.}

The motivic cohomology K-vector spaces H_{f}^{0}(M) := H^{0}(M) and H_{f}^{1}(M) may be defined by algebraic K-theory or motivic cohomology a la Voevodsky. They are conjectured to be finite dimensional. For the definition we refer the reader to [6].

4.3. The Tamagawa Number Conjecture. Let us first recall from [14, §2.2.2] that, for each embedding K → ℂ, the complex L-function attached to a K-motive M is defined as Euler product

L_{K}(M, s) = ∏_{p} P_{p}(M, p^{−s})^{−1}

for the real part of s large enough. We assume meromorphic continuation and write as usual L_{K}^{∗}(M) ∈ ℝ^{×} and r(M) ∈ ℤ for its leading coefficient and order of vanishing at s = 0, respectively.

To establish a link between L^{∗}(M) and Galois cohomology one uses the fundamental line:

Δ_{K}(M) : = d_{K}(H_{f}^{0}(M))^{−1}d_{K}(H_{f}^{1}(M))d_{K}(H_{f}^{0}(M^{∗}(1))^{∗})d_{K}(H_{f}^{1}(M^{∗}(1))^{∗})^{−1}
d_{K}(M_{B}^{+})d_{K}(t_{M})^{−1}.

The conjecture formulated in [14, §2.2.7] induces a canonical isomorphism in C_{K_{ℝ}} (period-regulator map)

(13)  ϑ_{∞} : K_{ℝ} ⊗_{K} Δ_{K}(M) ≅ 1_{K_{ℝ}}.

In addition, a standard conjecture on Cycle class and Chern class maps induces, for any place λ above p, the p-adic period-regulator isomorphism in C_{K_{λ}} (involving the maps η_{l})

(14)  ϑ_{λ} : Δ_{K}(M)_{K_{λ}} ≅ d_{K_{λ}}(RΓ_{c}(U, M_{λ}))^{−1}.

Now let F_{∞} be a p-adic Lie extension of ℚ with Galois group G = G(F_{∞}/ℚ). By Λ = Λ(G) we denote its Iwasawa algebra. For a ℚ-motive M over ℚ we fix a G_{ℚ}-stable ℤ_{p}-lattice T_{p} of M_{p} and define a left Λ-module

𝕋 := Λ ⊗_{ℤ_{p}} T_{p}

on which Λ acts via multiplication on the left factor from the left while G_{ℚ} acts diagonally via g(x ⊗ y) = xḡ^{−1} ⊗ g(y), where ḡ denotes the image of g ∈ G_{ℚ} in G. Let λ a finite place of K, O_{λ} the ring of integers of the completion K_{λ} of K at λ and assume that ρ : G → GL_{n}(O_{λ}) is a continuous representation of G which, for some suitable choice of a basis, is the λ-adic realization N_{λ} of a K-motive N. We also write ρ for the induced ring homomorphism Λ(G) → M_{n}(O_{λ}) and we consider O_{λ}^{n} as a right Λ(G)-module via action by ρ^{t} on the left, viewing O_{λ}^{n} as set of column vectors (contained in K_{λ}^{n}.) Note that, setting M(ρ^{∗}) := N^{∗} ⊗ M, we obtain an isomorphism of Galois representations

O_{λ}^{n} ⊗_{Λ(G)} 𝕋 ≅ T_{λ}(M(ρ^{∗})),

where T_{λ}(ρ^{∗}) is the O_{λ}-lattice ρ^{∗} ⊗ T_{p} of M(ρ^{∗})_{λ} and ρ^{∗} denotes the contragredient(=dual) representation of ρ.

Conjecture 4.1 (Fukaya/Kato [14, conj. 2.3.2] or [37, conj. 4.1 ]). There exists a (unique) isomorphism in C_{Λ}

ζ_{Λ}(M) := ζ_{Λ}(𝕋) : 1_{Λ} → d_{Λ}(RΓ_{c}(U, 𝕋))^{−1}

with the following property: for all K, λ and ρ as above the (generalized) base change K_{λ}^{n} ⊗_{Λ} − sends ζ_{Λ}(M) to

\xymatrix{1_{K_{λ}} \ar[r]^{ζ_{K}(M(ρ^{∗}))_{K_{λ}}} & Δ_{K}(M(ρ^{∗}))_{K_{λ}} \ar[r]^{ϑ_{λ}} & d_{K_{λ}}(RΓ_{c}(U, M(ρ^{∗})_{λ}))^{−1},}

where ζ_{K}(M(ρ^{∗})) : 1_{K} → Δ_{K}(M(ρ^{∗})) is the unique isomorphism such that, for every embedding K → ℂ, the leading coefficient L_{K}^{∗}(M(ρ^{∗})) is equal to the composite

\xymatrix{1_{ℂ} \ar[r]^{ζ_{K}(M(ρ^{∗}))_{ℂ}} & Δ_{K}(M(ρ^{∗}))_{ℂ} \ar[r]^{(ϑ_{∞})_{ℂ}} & 1_{ℂ}.}

It is easily shown that this conjecture implies the Equivariant Tamagawa Number Conjecture formulated by Flach and the first named author in [6, conj. 4(iv)] and hence also implies the ‘main conjecture of non-abelian Iwasawa theory’ discussed by Huber and Kings in [17].

5. The interpolation formula for Tate motives

In this section we shall give a first explicit application of the formalism developed in §3. More precisely, we show that the ‘p-adic Stark conjecture at s = 1’, as formulated by Serre in [33] and discussed by Tate in [35, chap. VI, §5], can be naturally interpreted as an interpolation formula for the leading term (in the sense of Definition 3.14) of certain global Zeta isomorphisms that are predicted to exist by Conjecture 4.1 in terms of the leading terms (in the classical sense) of suitable p-adic Artin L-functions. Interested readers can find further explicit results concerning Conjecture 4.1 in the special case that we consider here in both [2] and [7]. Throughout this section we fix an odd prime p and a totally real Galois extension F_{∞} of ℚ which contains the cyclotomic ℤ_{p}-extension ℚ_{cyc} of ℚ and is such that G := G(F_{∞}/ℚ) is a compact p-adic Lie group. We assume further that F_{∞}/ℚ is unramified outside a finite set of prime numbers S (which therefore contains p). We set H := G(F_{∞}/ℚ_{cyc}) and Γ := G(ℚ_{cyc}/ℚ) ≅ G/H. We fix a subfield E of F_{∞} which has finite degree over ℚ and, for simplicity, we assume throughout that the following condition is satisfied:

(15)  E ∩ ℚ_{cyc} = ℚ and E_{w} ∩ ℚ_{p,cyc} = ℚ_{p} for all w ∈ S_{p}(E).

We let 𝕋 denote the Λ(G)-module Λ(G) endowed with the following action of G(\overline{ℚ}/ℚ): each σ ∈ G(\overline{ℚ}/ℚ) acts on 𝕋 as multiplication by the element χ_{cyc}(σ̄)σ̄^{−1} where σ̄ denotes the image of σ in G and χ_{cyc} is the cyclotomic character G → Γ → ℤ_{p}^{×}. For each subfield F of F_{∞} which is Galois over ℚ we let 𝕋_{F} denote the Λ(G(F/ℚ))-module Λ(G(F/ℚ)) ⊗_{Λ(G)} 𝕋. We also set U := Spec(ℤ[\frac{1}{S}]) and note that for each such field F there is a natural isomorphism in D^{p}(Λ(G(F/ℚ))) of the form

(16)  Λ(G(F/ℚ)) ⊗_{Λ(G)}^{𝕃} RΓ_{c}(U, 𝕋) ≅ RΓ_{c}(U, 𝕋_{F}).

We regard each character of Ḡ := G(E/ℚ) as a character of G via the natural projection G → Ḡ. For any field C we write R_{C}(Ḡ) for the ring of finite dimensional C-valued characters of Ḡ, and for each C-valued character ρ we fix a representation space V_{ρ} of character ρ. For any ℚ_{p}[Ḡ]-module N, resp. endomorphism α of a ℚ_{p}[Ḡ]-module N, we write N^{ρ} for the ℂ_{p}-module

HomḠ(V_{ρ}, ℂ_{p} ⊗_{ℚ_{p}} N) ≅ ((V_{ρ^{∗}})_{ℂ_{p}} ⊗_{ℚ_{p}} N)Ḡ,

resp. α^{ρ} for the induced endomorphism of N^{ρ}. We use similar notation for complex characters ρ and ℚ[Ḡ]-modules N.

For any abelian group A we write A ˆ⊗ℤ_{p} for its p-adic completion \underleftarrow{lim}_{n} A/p^{n}A.

5.1. Leopoldt’s Conjecture. We recall that Leopoldt’s conjecture (for the field E at the prime p) is equivalent to the injectivity of the natural localisation map

λ_{p} : O_{E} [\frac{1}{p}]^{×} ⊗ ℤ_{p} → ∏_{w∈S_{p}(E)} E_{w}^{×} ˆ⊗ℤ_{p},

where S_{p}(E) denotes the set of places of E which lie above p. If ρ ∈ R_{ℂ_{p}}(Ḡ), then we say that Leopoldt’s conjecture ‘is valid at ρ’ if (ℚ_{p} ⊗_{ℤ_{p}} ker(λ_{p}))^{ρ} = 0. We set c_{γ} := c_{χ_{cyc},γ} ∈ ℚ_{p}^{×} and for each ρ ∈ R_{ℂ_{p}}(Ḡ) we define

⟨ρ, 1⟩ := dim_{ℂ_{p}}(H^{0}(Ḡ, V_{ρ})) = dim_{ℂ_{p}}((ℚ_{p})^{ρ}).

Lemma 5.1. We fix ρ ∈ R_{ℂ_{p}}(Ḡ) and assume that Leopoldt’s conjecture is valid at ρ.

(i) There are canonical isomorphisms

(H_{c}^{i}(U, 𝕋_{E}) ⊗_{ℤ_{p}} ℚ_{p})^{ρ} ≅ \begin{cases}(cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p})^{ρ}, & if i = 2 \\ (ℚ_{p})^{ρ}, & if i = 3 \\ 0, & otherwise. \end{cases}

(ii) RΓ_{c}(U, 𝕋) ∈ Σ_{ss−ρ} and r_{G}(RΓ_{c}(U, 𝕋))(ρ) = ⟨ρ, 1⟩.

(iii) For each w ∈ S_{p}(E) we write N_{E_{w}/ℚ_{p}} for the morphism E_{w}^{×} ˆ⊗ℤ_{p} → ℚ_{p}^{×} ˆ⊗ℤ_{p} that is induced by the field theoretic norm map. Then, with respect to the identifications given in claim (i), the morphism

(𝔅^{2})^{ρ} : (H_{c}^{2}(U, 𝕋_{E}) ⊗_{ℤ_{p}} ℚ_{p})^{ρ} → (H_{c}^{3}(U, 𝕋_{E}) ⊗_{ℤ_{p}} ℚ_{p})^{ρ},

is induced by the map

log_{p,γ,E} : ∏_{w∈S_{p}(E)} E_{w}^{×} ˆ⊗ℤ_{p} → ℤ_{p}

which sends each element (e_{w})_{w} to − ∑_{w} c_{γ}^{−1} log_{p}(N_{E_{w}/ℚ_{p}}(e_{w})).

Proof. Claim (i) can be verified by combining the exact cohomology sequence of the tautological exact triangle

(17)  RΓ_{c}(U, 𝕋_{E}) → RΓ(U, 𝕋_{E}) → ⨁_{ℓ∈S} RΓ(ℚ_{ℓ}, 𝕋_{E}) → RΓ_{c}(U, 𝕋_{E})[1]

together with the canonical identifications H^{i}(U, 𝕋_{E}) ≅ H^{i}(O_{E}[\frac{1}{S}], ℤ_{p}(1)) and H^{i}(ℚ_{ℓ}, 𝕋_{E}) ≅ ⨁_{w∈S_{ℓ}(E)} H^{i}(E_{w}, ℤ_{p}(1)) and an explicit computation of each of the groups H^{i}(O_{E}[\frac{1}{S}], ℤ_{p}(1)) and H^{i}(E_{w}, ℤ_{p}(1)). As this is routine we leave explicit details to the reader except to note that H_{c}^{2}(U, 𝕋_{E})⊗_{ℤ_{p}}ℚ_{p} is canonically isomorphic to cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p} (independently of Leopoldt’s conjecture), whilst the fact that E is totally real implies the vanishing of (H_{c}^{1}(U, 𝕋_{E}) ⊗_{ℤ_{p}} ℚ_{p})^{ρ} is equivalent to that of (ker(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p})^{ρ}. To prove claim (ii) and (iii) we write E_{cyc}, E_{w,cyc} for each w ∈ S_{p}(E) and ℚ_{p,cyc} for the cyclotomic ℤ_{p}-extensions of E, E_{w} and ℚ_{p}. Then (15) induces a direct product decomposition G(E_{cyc}/ℚ) ≅ Γ × Ḡ and hence allows us to identify Γ with each of G(E_{cyc}/E), G(E_{w,cyc}/E_{w}) and G(ℚ_{p,cyc}/ℚ_{p}). We note also that, in terms of the notation of Lemma 3.13, the isomorphism (16) (with F = E_{cyc}) induces a canonical isomorphism in D^{p}(Λ(Γ)) of the form

(18)  RΓ_{c}(U, 𝕋)_{ρ} ≅ O^{n} ⊗_{ℤ_{p}[Ḡ]} RΓ_{c}(U, 𝕋_{E_{cyc}}),

where Γ acts naturally on the right hand factor in the tensor product. From Lemma 3.13(i), we may therefore deduce that RΓ_{c}(U, 𝕋) ∈ Σ_{ss−ρ} if and only if O^{n} ⊗_{ℤ_{p}[Ḡ]} RΓ_{c}(U, 𝕋_{E_{cyc}}) ∈ Σ_{ss}. But the latter containment can be easily verified by using the criterion of Remark 3.3(ii): indeed, one need only note that H_{c}^{i}(U, 𝕋_{E_{cyc}}) is finite if i ∉ {2, 3}, that H_{c}^{3}(U, 𝕋_{E_{cyc}}) identifies with ℤ_{p} (as a Γ-module) and that the exact sequences of (3) combine with the descriptions of claim (i) to imply that ((ℚ_{p}⊗_{ℤ_{p}} H_{c}^{1}(U, 𝕋_{E_{cyc}}))^{ρ})^{Γ} and (ℚ_{p}⊗_{ℤ_{p}} H_{c}^{1}(U, 𝕋_{E_{cyc}}))_{Γ}^{ρ} both vanish. In addition, the same observations combine with Lemma 3.13(ii) to imply that r_{G}(RΓ_{c}(U, 𝕋))(ρ) = dim_{ℂ_{p}}((ℚ_{p})^{ρ}). Regarding claim (iii), the isomorphism (18) combines with Lemma 3.13(i) and Lemma 3.1 to imply that (𝔅^{2})^{ρ} = ˆ(𝔅^{2})^{ρ} where 𝔅̂^{2} is the Bockstein morphism of the pair (RΓ_{c}(U, 𝕋_{E_{cyc}}), T). Also, the cohomology sequence of the triangle (17) gives rise to a commutative diagram

⨁_{w∈S_{p}(E)} ℚ_{p} ⊗_{ℤ_{p}} H^{1}(E_{w}, ℤ_{p}(1)) ⟶ ℚ_{p} ⊗_{ℤ_{p}} H_{c}^{2}(U, 𝕋_{E})
(−1)×(ℚ_{p}⊗_{ℤp} 𝔅_{w}^{1})_{w}↓  ↓^{ℚ_{p}⊗_{ℤp} 𝔅̂^{2}}
⨁_{w∈S_{p}(E)} ℚ_{p} ⊗_{ℤ_{p}} H^{2}(E_{w}, ℤ_{p}(1)) ⟶ ℚ_{p} ⊗_{ℤ_{p}} H_{c}^{3}(U, 𝕋_{E})

where the upper row is the (tautological) surjection induced by the canonical identifications H^{1}(E_{w}, ℤ_{p}(1)) ≅ E_{w}^{×} ˆ⊗ℤ_{p} and ℚ_{p} ⊗_{ℤ_{p}} H_{c}^{2}(U, 𝕋_{E}) ≅ ℚ_{p} ⊗_{ℤ_{p}} cok(λ_{p}), the lower row is the surjection induced by the canonical identifications H^{2}(E_{w}, ℤ_{p}(1)) ≅ ℤ_{p} and H_{c}^{3}(U, 𝕋_{E}) ≅ ℤ_{p} together with the identity map on ℤ_{p}, 𝔅_{w}^{1} is the Bockstein morphism of the pair (RΓ(E_{w,cyc}, ℤ_{p}(1)), T) and the factor −1 occurs on the left hand vertical arrow because the diagram compares Bockstein morphisms in degrees 1 and 2. Further, for each w ∈ S_{p}(E) the natural isomorphism (in D^{p}(ℤ_{p}))

ℤ_{p} ⊗_{ℤ_{p}[G(E_{w}/ℚ_{p})]}^{𝕃} RΓ(E_{w}, ℤ_{p}(1)) ≅ RΓ(ℚ_{p}, ℤ_{p}(1))

induces a commutative diagram

H^{1}(E_{w}, ℤ_{p}(1)) ⟶ H^{1}(ℚ_{p}, ℤ_{p}(1))
𝔅_{w}^{1}↓  ↓^{𝔅_{p}^{1}}
H^{2}(E_{w}, ℤ_{p}(1)) ⟶ H^{2}(ℚ_{p}, ℤ_{p}(1))

where the upper horizontal arrow is induced by the canonical identifications H^{1}(E_{w}, ℤ_{p}(1)) ≅ E_{w}^{×} ˆ⊗ℤ_{p} and H^{1}(ℚ_{p}, ℤ_{p}(1)) ≅ ℚ_{p}^{×} ˆ⊗ℤ_{p} together with the map N_{E_{w}/ℚ_{p}}, the lower horizontal arrow is induced by the canonical identifications H^{2}(E_{w}, ℤ_{p}(1)) ≅ ℤ_{p} and H^{2}(ℚ_{p}, ℤ_{p}(1)) ≅ ℤ_{p} together with the identity map on ℤ_{p}, and 𝔅_{p}^{1} is the Bockstein morphism of the pair (RΓ(ℚ_{p,cyc}, ℤ_{p}(1))), T).

To prove claim (iii) it thus suffices to recall that, with respect to the identifications H^{1}(ℚ_{p}, ℤ_{p}(1)) ≅ ℚ_{p}^{×} ˆ⊗ℤ_{p} and H^{2}(ℚ_{p}, ℤ_{p}(1)) ≅ ℤ_{p}, the map 𝔅_{p}^{1} is equal to c_{γ}^{−1} · log_{p} (see, for example, [8, p. 352]). □

5.2. The p-adic Stark conjecture at s = 1. For each character χ ∈ R_{ℂ}(Ḡ) we write L_{S}(s, χ) for the Artin L-function of χ that is truncated by removing the Euler factors attached to primes in S (cf. [35, Chap. 0, §4]). Then, for each character ρ ∈ R_{ℂ_{p}}(G) there exists a unique p-adic meromorphic function L_{p,S}(·, ρ) : ℤ_{p} → ℂ_{p} such that for each strictly negative integer n and each isomorphism j : ℂ_{p} ≅ ℂ one has

L_{p,S}(n, ρ)^{j} = L_{S}(n, (ρ · ω^{n−1})^{j})

where ω : G(\overline{ℚ}/ℚ) → ℤ_{p}^{×} is the Teichmüller character. Henceforth we will fix such an isomorphism j and often omit it from the notation. This function is the ‘(S-truncated) p-adic Artin L-function’ of ρ that is constructed by Greenberg in [15] by combining techniques of Brauer induction with the fundamental results of Deligne and Ribet [13] and Cassou-Noguès [9]. In this section we recall a conjecture of Serre regarding the ‘leading term at s = 1’ of L_{p,S}(s, ρ). To this end we set E_{∞} := ℝ ⊗_{ℚ} E ≅ ∏_{Hom(E,ℂ)} ℝ and write log_{∞}(O_{E}^{×}) for the inverse image of O_{E}^{×} ↪ E_{∞}^{×} under the (componentwise) exponential map exp_{∞} : E_{∞} → E_{∞}^{×}. We set E_{0} := {x ∈ E : Tr_{E/ℚ}(x) = 0}. Then log_{∞}(O_{E}^{×}) is a lattice in ℝ⊗_{ℚ} E_{0} and so there is a canonical isomorphism of ℂ[Ḡ]-modules μ_{∞} : ℂ ⊗_{ℤ} log_{∞}(O_{E}^{×}) ≅ ℂ ⊗_{ℚ} E_{0}. This implies that the ℚ[Ḡ]- modules E_{0} and O_{E}^{×} ⊗_{ℤ} ℚ are (non-canonically) isomorphic. We note also that the composite morphism

(19) log_{∞}(O_{E}^{×}) ⟶^{exp_{∞}} O_{E}^{×} ⟶^{λ_{p}} ∏_{w∈S_{p}(E)} U_{E_{w}}^{1}

⟶^{(u_{w})_{w}↦(log_{p}(u_{w}))_{w}} ∏_{w∈S_{p}(E)} E_{w} ≅ ℚ_{p} ⊗_{ℚ} E,

factors through the inclusion ℚ_{p} ⊗_{ℚ} E_{0} ⊂ ℚ_{p} ⊗_{ℚ} E and hence induces an isomorphism of ℚ_{p}[Ḡ]-modules μ_{p} : ℚ_{p} ⊗_{ℤ} log_{∞}(O_{E}^{×}) ≅ ℚ_{p} ⊗_{ℚ} E_{0}.

Conjecture 5.2. For each ρ ∈ R_{ℂ_{p}}(Ḡ) we set

L_{p,S}^{∗}(1, ρ) := lim_{s→1}(s − 1)^{⟨ρ,1⟩} · L_{p,S}(s, ρ).

Then L_{p,S}^{∗}(1, ρ) is equal to the leading term of L_{p,S}(s, ρ) at s = 1, and for each choice of isomorphism of ℚ[Ḡ]-modules g : E_{0} → O_{E}^{×} ⊗_{ℤ} ℚ one has

\frac{L_{p,S}^{∗}(1, ρ)}{det_{ℂ_{p}}((ℂ_{p} ⊗_{ℚ_{p}} μ_{p}) ∘ (ℂ_{p} ⊗_{ℚ} g))^{ρ}} = \frac{L_{S}^{∗}(1, ρ)}{det_{ℂ}(μ_{∞} ∘ (ℂ ⊗_{ℚ} g))^{ρ}} .

Remark 5.3. This conjecture is the ‘p-adic Stark conjecture at s = 1’ as discussed by Tate in [35, Chap. VI, §5] (where it is attributed to Serre [33]). More precisely, there are some imprecisions in the statement of [35, Chap. VI, §5] (for example, and as already noted by Solomon in [34, §3.3], the intended meaning of the symbols ‘log U’ and ‘μ_{p}’ in [35, p. 137] is unclear) and Conjecture 5.2 represents a natural clarification of the presentation given in loc. cit.

Remark 5.4. We fix a subgroup J of Ḡ and write 1_{J} for the trivial character of J. If ρ = IndḠ ; J 1_{J}, then the inductive behaviour of L-functions combines with the analytic class number formula for E^{J} to show that Conjecture 5.2 is valid for ρ if and only if the residue at s = 1 of the p-adic zeta function of the field E^{J} is non-zero and equal to 2^{[E^{J}:ℚ]−1}hR_{p}e_{p}/\sqrt{|d|} where h, R_{p} and d are the class number, p-adic regulator and absolute discriminant of E^{J} respectively and e_{p} := ∏_{v∈S_{p}(E^{J})}(1 − Nv^{−1}) (cf. [35, rem., p. 138]). From the main result of Colmez in [12] one may thus deduce that Conjecture 5.2 is valid for ρ = Ind_{J}^{G} 1_{J} if and only if Leopoldt’s conjecture is valid for E^{J}. We note also that if Leopoldt’s conjecture is valid for E, then it is valid for all such intermediate fields E^{J}.

5.3. The interpolation formula. We now reinterpret the equality of Conjecture 5.2 as an interpolation formula for the Zeta isomorphism ζ_{Λ(G)}(𝕋).

Theorem 5.5. If Conjecture 5.2 is valid, then for each ρ ∈ R_{ℂ_{p}}(Ḡ) one has RΓ_{c}(U, 𝕋) ∈ Σ_{ss−ρ}, r_{G}(RΓ_{c}(U, 𝕋))(ρ) = ⟨ρ, 1⟩ and

(20)  c_{γ}^{⟨ρ,1⟩} · ζ_{Λ(G)}(𝕋)^{∗}(ρ) = L_{p,S}^{∗}(1, ρ).

Remark 5.6. One can naturally interpret (20) as an equality of leading terms of p-adic meromorphic functions. Indeed, whilst Conjecture 5.2 predicts that L_{p,S}^{∗}(1, ρ) is the leading term at s = 1 of L_{p,S}(s, ρ), Lemma 3.17 interprets the left hand side of (20) as the leading term at s = 0 of the function f_{L}(ρχ_{cyc}^{s}) with L := [RΓ_{c}(U, 𝕋), ζ_{Λ(G)}(𝕋)] ∈ K_{1}(Λ(G), Σ_{ss−ρ}).

Proof. We note first that if Conjecture 5.2 is valid, then Remark 5.4 implies that Leopoldt’s conjecture is valid for E and so Lemma 5.1(ii) implies that RΓ_{c}(U, 𝕋) ∈ Σ_{ss−ρ} and r_{G}(RΓ_{c}(U, 𝕋))(ρ) = ⟨ρ, 1⟩ for each ρ ∈ R_{ℂ_{p}}(Ḡ). We now fix ρ ∈ R_{ℂ_{p}}(Ḡ) and a number field K over which the character ρ can be realised. We fix an embedding K ↪ ℂ and write λ for the place of K which is induced by j. We set M := h^{0}(Spec E)(1) and note that M([ρ]^{∗}) := M ⊗[ρ]^{∗} is a K-motive, where [ρ]^{∗} denotes the dual of the Artin motive corresponding to ρ. To evaluate ζ_{Λ(G)}(𝕋)^{∗}(ρ) we need to make Definition 3.14 explicit. To do this we use the observations of [5, §1.1, §1.3] to explicate the isomorphism ζ_{K}(M([ρ]^{∗}))_{K_{λ}} which occurs in Conjecture (4.1). Indeed one has H_{f}^{1}(M) = O_{E}^{×} ⊗_{ℤ} ℚ, H_{f}^{0}(M^{∗}(1)) = ℚ, t_{M} := E and H_{f}^{0}(M) = H_{f}^{1}(M^{∗}(1)) = M_{B}^{+} = 0 (the latter since E is totally real) so that

ℂ ⊗_{K} Δ_{K}(M([ρ]^{∗})) = d_{ℂ}((ℚ ⊗_{ℤ} O_{E}^{×})_{ρ})d_{ℂ}((ℚ)_{ρ})d_{ℂ}((E)_{ρ})^{−1}

and ζ_{K}(M([ρ]^{∗}))_{K_{λ}} is equal to the composite morphism

(21) 1_{ℂ_{p}} →1_{ℂ_{p}}
→d_{ℂ_{p}}((ℚ_{p} ⊗_{ℤ} O_{E}^{×})^{ρ})d_{ℂ_{p}}((ℚ_{p})^{ρ})d_{ℂ_{p}}((ℚ_{p} ⊗_{ℚ} E)^{ρ})^{−1}
→d_{ℂ_{p}}(ℂ_{p} ⊗_{K_{λ}} H_{c}^{2}(U, M([ρ]^{∗})_{λ}))^{−1}d_{ℂ_{p}}(ℂ_{p} ⊗_{K_{λ}} H_{c}^{3}(U, M([ρ]^{∗})_{λ})
→d_{ℂ_{p}}(ℂ_{p} ⊗_{K_{λ}} RΓ_{c}(U, M([ρ]^{∗})_{λ}))^{−1}.

In this displayed formula we have used the following notation: the first map corresponds to multiplication by L_{S}^{∗}(1, ρ); the second map is induced by applying (ℂ_{p} ⊗_{ℝ,j^{−1}} −)^{ρ} to both the natural isomorphism E ⊗_{ℚ} ℝ ≅ ∏_{Hom(E,ℂ)} ℝ and also the exact sequence

(22)  0 → O_{E}^{×} ⊗_{ℤ} ℝ ⟶^{(log ∘σ)_{σ}} ∏_{σ∈Hom(E,ℂ)} ℝ \underrightarrow{(x_{σ})_{σ}↦∑_{σ} x_{σ}} ℝ → 0;

the third map is induced by Lemma 5.1(i) and the inverse of the isomorphism

(23)  ∏_{w∈S_{p}(E)} U_{E_{w}}^{1} ⊗_{ℤ_{p}} ℚ_{p} ⟶^{(u_{w})_{w}↦(log_{p}(u_{w}))_{w}} ∏_{w∈S_{p}(E)} E_{w} ≅ ℚ_{p} ⊗_{ℚ} E;

the last map is induced by property h) as described in §2.1 (with R = ℂ_{p}). Also, from Lemma 5.1(iii) we know that ℂ_{p} ⊗_{K_{λ}} t(RΓ_{c}(U, 𝕋)(ρ^{∗})) is equal to the composite

d_{ℂ_{p}}(ℂ_{p} ⊗_{K_{λ}} RΓ_{c}(U, M([ρ]^{∗})_{λ})^{−1} (24)

→d_{ℂ_{p}}(H_{c}^{2}(U, M([ρ]^{∗})_{λ})^{−1}d_{ℂ_{p}}(H_{c}^{3}(U, M([ρ]^{∗})_{λ})
→d_{ℂ_{p}}((ℚ_{p})^{ρ})^{−1}d_{ℂ_{p}}((ℚ_{p})^{ρ}) = 1_{ℂ_{p}}

where the first arrow is induced by property h) and the second by Lemma 5.1(i) and the map −c_{γ}^{−1} log_{p,γ,E}. Now, after taking account of Lemma 5.1(ii), ζ_{Λ(G)}(𝕋)^{∗}(ρ) is defined to be (−1)^{⟨ρ,1⟩} times the element of ℂ_{p}^{×} which corresponds to the composite of (21) and (24) (cf. Definition 3.14). Thus, after noting that there is a commutative diagram of the form

∏_{w∈S_{p}(E)} U_{E_{w}}^{1} ⊗_{ℤ_{p}} ℚ_{p} ⟶ cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p}
(23)↓  ↓^{(−1)×log_{p,γ,E}}
E ⊗_{ℚ} ℚ_{p}  ⟶^{Tr_{E/ℚ}}  ℚ_{p}

where the upper horizontal arrow is the tautological projection, the observations made above imply that

(25)  c_{γ}^{⟨ρ,1⟩} · ζ_{Λ(G)}(𝕋)^{∗}(ρ) = L_{S}^{∗}(1, ρ) · ξ

where ξ is the element of ℂ_{p}^{×} which corresponds to the composite

(26)  1_{ℂ_{p}} =d_{ℂ_{p}}((ℚ_{p} ⊗_{ℤ} O_{E}^{×})^{ρ})d_{ℂ_{p}}((ℚ_{p} ⊗_{ℤ} O_{E}^{×})^{ρ})^{−1}
→d_{ℂ_{p}}((ℚ_{p} ⊗_{ℚ} E_{0})^{ρ})d_{ℂ_{p}}((ℚ_{p} ⊗_{ℤ} O_{E}^{×})^{ρ})^{−1}
→d_{ℂ_{p}}((ℚ_{p} ⊗_{ℚ} E_{0})^{ρ})d_{ℂ_{p}}((ℚ_{p} ⊗_{ℚ} E_{0})^{ρ})^{−1} = 1_{ℂ_{p}}

where the first arrow is induced by applying ℂ_{p} ⊗_{ℝ,j^{−1}} − to the isomorphism ℝ ⊗_{ℤ} O_{E}^{×} ≅ ℝ ⊗_{ℚ} E_{0} coming from the map (log ∘σ)_{σ} in (22) and the second by the isomorphism ℚ_{p} ⊗_{ℤ} O_{E}^{×} ≅ ℚ_{p} ⊗_{ℚ} E_{0} coming from the second and third arrows in (19). (Note that the factor (−1)^{⟨ρ,1⟩} in the definition of ζ_{Λ(G)}(𝕋)^{∗}(ρ) cancels against the factor −1 which occurs in the commutative diagram above, and hence does not occur in the formula (25)). But, upon comparing the definitions of μ_{∞} and μ_{p} in §5.2 with the maps involved in (26), one finds that ξ is equal to

det_{ℂ_{p}}((ℂ_{p} ⊗_{ℚ_{p}} μ_{p}) ∘ (ℂ_{p} ⊗_{ℂ,j^{−1}} μ_{∞})^{−1})^{ρ} = \frac{det_{ℂ_{p}}((ℂ_{p} ⊗_{ℚ_{p}} μ_{p}) ∘ (ℂ_{p} ⊗_{ℚ} g))^{ρ}}{(det_{ℂ}(μ_{∞} ∘ (ℂ ⊗_{ℚ} g))^{ρ})}

and hence (25) implies that

\frac{c_{γ}^{⟨ρ,1⟩} · ζ_{Λ(G)}(𝕋)^{∗}(ρ)}{det_{ℂ_{p}}((ℂ_{p} ⊗_{ℚ_{p}} μ_{p}) ∘ (ℂ_{p} ⊗_{ℚ} g))^{ρ}} = \frac{L_{S}^{∗}(1, ρ)}{det_{ℂ}(μ_{∞} ∘ (ℂ ⊗_{ℚ} g))^{ρ}} .

The claimed equality (20) now follows immediately upon comparing this equality to that of Conjecture 5.2. □

Corollary 5.7. If Leopoldt’s conjecture is valid for E at p, then for every ℚ_{p}-rational character ρ of Ḡ there exists a natural number n_{ρ} such that

(c_{γ}^{⟨ρ,1⟩} · ζ_{Λ(G)}(𝕋)^{∗}(ρ))^{n_{ρ}} = L_{p,S}^{∗}(1, ρ)^{n_{ρ}}.

Further, if ρ is a permutation character, then one can take n_{ρ} = 1.

Proof. If ρ is ℚ_{p}-rational, then Artin’s Induction Theorem implies the existence of a natural number n_{ρ} such that in R_{ℂ_{p}}(G) one has n_{ρ} · ρ = ∑_{H} n_{H} · ind_{H}^{G} 1_{H} where H runs over the set of subgroups of Ḡ and each n_{H} is an integer (cf. [35, chap. II, thm. 1.2]). Further, ρ is said to be a permutation character if and only if there exists such a formula with n_{ρ} = 1. The stated result thus follows by combining Theorem 5.5 with Remark 5.4 and the fact that each side of (20) is both additive and inductive in ρ. □

6. The interpolation formula for critical motives

As a second application of the formalism introduced in §3, in this section we prove an interpolation formula for the leading terms (in the sense of Definition 3.14) of the p-adic L-functions that Fukaya and Kato conjecture to exist for any critical motive which has good ordinary reduction at all places above p. (We recall that a motive M is said to be ‘critical’ if the map (11) is bijective). To study these p-adic L-functions we must combine Conjecture 4.1 together with a local analogue of this conjecture (which is also due to Fukaya and Kato, and is recalled as Conjecture 6.1 below) and aspects of Nekovar’s theory of Selmer complexes and of the theory of p-adic height pairings.

6.1. Local epsilon isomorphisms. We fix once and for all the p-adic period t = ”2πi”, i.e. a generator of ℤ_{p}(1). Further we write ϵ_{p}(V) := ϵ(D_{pst}(V)) for Deligne’s ϵ-factor at p where D_{pst}(V) is endowed with the linearized action of the Weil-group and thereby considered as a representation of the Weil-Deligne group, see [14, §3.2] or [27, appendix C] (Here we suppress the dependence of the choice of a Haar measure and of t = 2πi in the notation. The above choice of t = (t_{n}) ∈ ℤ_{p}(1) determines a homomorphism ψ_{p} : ℚ_{p} → ℚ̄_{p}^{×} with ker(ψ_{p}) = ℤ_{p} sending \frac{1}{p^{n}} to t_{n} ∈ μ_{p^{n}}). Note that (B_{dR})^{I_{p}} = ℚ̂_{p}^{nr}, the completion of the maximal unramified extension ℚ_{p}^{nr} of ℚ_{p}. Let L be any finite extension of ℚ_{p} and V any finite dimensional L-vector space with continuous G_{ℚ_{p}}-action. We set L̃ := ℚ̂_{p}^{nr} ⊗_{ℚ_{p}} L,

Γ^{∗}(−j) := \begin{cases}Γ(j) = (j − 1)!, & if j > 0, \\ lim_{s→j}(s − j)Γ(s) = (−1)^{j}((−j)!)^{−1}, & if j ≤ 0, \end{cases}

and Γ_{L}(V) := ∏_{j∈ℤ} Γ^{∗}(j)^{−h(−j)}, where h(j) := dim_{L} gr^{j}D_{dR}(V). In [14, prop. 3.3.5] the existence of a map

ϵ_{dR}(V) : 1L̃ → dL̃(V)dL̃(D_{dR}(V))^{−1}

with certain properties was shown. Using this we define an isomorphism

ϵ_{p,L}(V) : 1L̃ → (d_{L}(RΓ(ℚ_{p}, V))d_{L}(V))L̃

as product of Γ_{L}(V), η_{l}(V) · \overline{(η_{l}(V^{∗}(1))^{∗})} and ϵ_{dR}(V). Now let T be a Galois stable O := O_{L}-lattice of V and set Õ := W (\overline{𝔽_{p}}) ⊗_{ℤ_{p}} O, where W (\overline{𝔽_{p}}) denotes the Wittring of \overline{𝔽_{p}}. We define

Λ̃ := W (\overline{𝔽_{p}})[[G]] = lim ; ⟵_{n}(W (\overline{𝔽_{p}}) ⊗_{ℤ_{p}} ℤ_{p}[G/G_{n}]),

we assume L = ℚ_{p} and set as before 𝕋 := Λ ⊗_{ℤ_{p}} T. We write T (ρ^{∗}) for the O-lattice ρ^{∗} ⊗ T of ρ^{∗} ⊗ V, which we assume to be de Rham. The following conjecture is a local integrality statement

Conjecture 6.1 (Fukaya/Kato [14, conj. 3.4.3], see also [37, conj. 5.9]). There exists a (unique) isomorphism in CΛ̃

ϵ_{p,Λ}(𝕋) : 1Λ̃ → (d_{Λ}(RΓ(ℚ_{p}, 𝕋))d_{Λ}(𝕋))Λ̃

such that for all ρ : G → GL_{n}(O) ⊆ GL_{n}(L), L a finite extension of ℚ_{p} with ring of integers O, we have

L^{n} ⊗_{Λ} ϵ_{p,Λ}(𝕋) = ϵ_{p,L}(T (ρ^{∗})).

6.2. p-adic height pairings. To prepare for our derivation of the interpolation formula in this section we discuss certain preliminaries regarding height pairings. We fix a continuous finite-dimensional L-linear representation W of G_{ℚ} which satisfies the following ‘condition of Dabrowski-Panchiskin’: (DP) W is de Rham and there exist a G_{ℚ_{p}}-subrepresentation Ŵ of W (re-

stricted to G_{ℚ_{p}}) such that D_{dR}^{0}(Ŵ) = t_{p}(W) := D_{dR}(W)/D_{dR}^{0}(W). Thus we have an exact sequence of G_{ℚ_{p}}-representations

0 → Ŵ → W → W̃ → 0

such that D_{dR}^{0}(Ŵ) = t_{p}(W̃) = 0 (cf. [21, prop. 1.28]). Setting Z := W^{∗}(1), Ẑ := W̃^{∗}(1), and Z̃ := Ŵ^{∗}(1), we obtain by Kummer duality the analogous exact sequence

0 → Ẑ → Z → Z̃ → 0

and Z satisfies the condition (DP). Let S be a finite set of places of ℚ containing the places S_{∞} := {∞} as well as S_{p} := {p} and such that W (and thus Z) are also representations of G_{S}(ℚ), the Galois group of the maximal outside S unramified extension of ℚ. Furthermore, we set S_{f} := S \ S_{∞}. The Selmer complex SC_{U}(Ŵ , W) is by definition the mapping fibre (27)

[figure: SC_{U}(Ŵ , W); RΓ(U, W); RΓ(ℚ_{p}, W/Ŵ) ⊕ ⨁_{S_{f} \S_{p}} RΓ(ℚ_{ℓ}, W)]

while SC(Ŵ , W) is the mapping fibre (28)

[figure: SC(Ŵ , W); RΓ(U, W); RΓ(ℚ_{p}, W/Ŵ) ⊕ ⨁_{S_{f} \S_{p}} RΓ_{/f}(ℚ_{ℓ}, W)].

Here RΓ_{/f}(ℚ_{ℓ}, W) is defined as mapping cone

(29)  [figure: RΓ_{f}(ℚ_{ℓ}, W); RΓ(ℚ_{ℓ}, W); RΓ_{/f}(ℚ_{ℓ}, W)]

For any G_{ℚ_{p}}-representation V and any prime number ℓ we define an element of L[u] by setting

P_{ℓ}(V, u) := P_{L,ℓ}(V, u) := \begin{cases}det_{L}(1 − φ_{ℓ}u|V^{I_{ℓ}}), & if ℓ ≠ p \\ det_{L}(1 − φ_{p}u|D_{cris}(V)), & if ℓ = p, \end{cases}

where φ_{l} denotes the geometric Frobenius automorphism at l. The following three conditions are easily seen to be equivalent

(A_{1}) P_{l}(W, 1)P_{l}(Z, 1) ≠ 0 for all l ∈ S_{f} \ S_{p}, (A_{2}) H^{0}(ℚ_{ℓ}, W) = H^{0}(ℚ_{ℓ}, Z) = 0 for all l ∈ S_{f} \ S_{p}, (A_{3}) RΓ_{f}(ℚ_{ℓ}, W) is quasi-null for all l ∈ S_{f} \ S_{p}.

We also consider the following group of conditions (B_{1}) P_{p}(W, 1)P_{p}(Z, 1) ≠ 0 (B_{2}) D_{cris}(W)^{φ_{p}−1} = D_{cris}(Z)^{φ_{p}−1} = 0

(B_{3}) H^{0}(ℚ_{p}, W) = H^{0}(ℚ_{p}, Z) = 0 Then (B_{1}) is equivalent to (B_{2}) and (B_{3}) implies (B_{2}) by [21, thm. 1.15]. Finally we consider the following conditions

(C_{1}) P_{p}(W̃ , 1)P_{p}(Z̃, 1) ≠ 0

(C_{2}) D_{cris}(W̃)^{φ_{p}−1} = D_{cris}(Z̃)^{φ_{p}−1} = 0 (C_{3}) H^{0}(ℚ_{p}, W̃) = H^{0}(ℚ_{p}, Z̃) = 0.

These conditions are again easily seen to be equivalent (to see that (C_{2}) is equivalent to (C_{3}) use (loc. cit.) and the fact that t_{p}(W̃) = t_{p}(Z̃) = 0).

Lemma 6.2. Let X denote either W or Z.

(i) If condition (A_{1}) holds, then for every ℓ ∈ S_{f} \ S_{p} the following complexes are quasi-null

RΓ(ℚ_{ℓ}, X) ≅ RΓ_{f}(ℚ_{ℓ}, X) ≅ RΓ_{/f}(ℚ_{ℓ}, X) ≅ 0.

(ii) If condition (C_{1}) is satisfied, then there are isomorphisms

RΓ_{/f}(ℚ_{p}, X) ≅ RΓ(ℚ_{p}, X̃) and RΓ_{f}(ℚ_{p}, X) ≅ RΓ(ℚ_{p}, X̂)

in D^{p}(L). (iii) If conditions (A_{1}) and (C_{1}) are both satisfied, then we have a quasi-

isomorphism in D^{p}(L)

SC_{U}(Ŵ , W) ≅ RΓ_{f}(ℚ, W).

Proof. We assume (A_{1}). Then by local duality and the local Euler characteristic formula it follows immediately that RΓ(ℚ_{ℓ}, X) is quasi-null. The other statements in (i) are obvious. To prove (ii) we assume (C_{1}). Then, since every complex of vector spaces is quasi-isomorphic to its cohomology, considered as complex with zero differential, we have RΓ(ℚ_{p}, X̂) ≅ RΓ_{f}(ℚ_{p}, X̂) ≅ RΓ_{f}(ℚ_{p}, X) by [14, lem. 4.1.7]. Thus the exact triangles

RΓ(ℚ_{p}, X̂) → RΓ(ℚ_{p}, X) → RΓ(ℚ_{p}, X̃) →

and

RΓ_{f}(ℚ_{p}, X) → RΓ(ℚ_{p}, X) → RΓ_{/f}(ℚ_{p}, X) →

are naturally isomorphic in D^{p}(L). Finally, we note that claim (iii) follows immediately from claims (i) and (ii) and the definitions of SC_{U}(Ŵ , W) and RΓ_{f}(ℚ, W). □

Let M be any motive over ℚ, V = M_{p} its p-adic realization, ρ an Artin representation defined over the number field K and [ρ] its corresponding Artin motive. Then, for any place λ of K above p, the λ-adic realisation

(30)  W := N_{λ} = V ⊗_{ℚ_{p}} [ρ]_{λ}^{∗}

of the motive N := M(ρ^{∗}) := M ⊗ [ρ]^{∗} is an L := K_{λ}-adic representation. We assume that V (and hence, since [ρ]^{∗} is pure of weight zero, also W) satisfies the condition (DP). We fix a Galois stable lattice T of V and set T_{ρ} := T ⊗_{ℤ_{p}} O^{n}, a Galois stable lattice in W (where we assume that without loss of generality [ρ]_{λ}^{∗} is given as ρ^{∗} : G_{ℚ} → GL_{n}(O)). Similarly we fix a G_{ℚ_{p}}-stable lattice T̂ of V̂ and take as T̃ the lattice in Ṽ induced from T. Finally we set T̂_{ρ} := T̂⊗_{ℤ_{p}} O^{n} and T̃_{ρ} := T̃⊗_{ℤ_{p}} O^{n}, they are Galois stable O-lattices of Ŵ and W̃ , respectively.

Example 6.3. Let A be an abelian variety that is defined over ℚ and set M := h^{1}(A)(1). If A has good ordinary reduction at p, then W := N_{λ} satisfies the conditions (DP), (A_{1}), (B_{1}) and (C_{1}) (the last three for weight reasons - Weil conjectures, while more general, the condition (DP) holds for any motive which has good ordinary reduction at p (see [26])). However, if, for example, A is an elliptic curve with (split) multiplicative reduction at p, then the condition (B_{1}) is not satisfied.

Now we define a G_{ℚ_{p}}-stable ℤ_{p}-lattice

T̂ := T ∩ V̂ ,

of V̂ . As before let 𝕋 denote the big Galois representation Λ ⊗_{ℤ_{p}} T and put 𝕋̂ := Λ ⊗_{ℤ_{p}} T̂ similarly. Then 𝕋̂ is G_{ℚ_{p}}-stable sub-Λ-module of 𝕋. In fact, it is a direct summand of 𝕋 and we have an isomorphism in CΛ̃

(31)  β : d_{Λ}(𝕋^{+})Λ̃ ≅ d_{Λ}(𝕋̂)Λ̃.

Now the Selmer complexes SC_{U}(𝕋̂, 𝕋) and SC(𝕋̂, 𝕋) are defined analogously as for W above. We point out that SC_{U}(X̂, X) coincides with the Selmer complex R̃Γ_{f}(X) in [22, (11.3.1.5)] for X ∈ {W, Z}. More generally, if we define

𝕋_{cyc,ρ} := Λ(Γ) ⊗ T_{ρ}

and similarly 𝕋̂_{cyc,ρ} and 𝕋̃_{cyc,ρ}, then SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ}) identifies with the Selmer complex { \atop RΓ_{f,IW}} (ℚ_{cyc}/ℚ, T_{ρ}) defined in [22, (8.8.5)] (with Nekovar’s local conditions induced by T_{ℓ}^{+} = 𝕋̂_{cyc}(ρ), if ℓ = p, and 0 otherwise, and taking S_{f} for his Σ). Here Γ is the Galois group of the cyclotomic ℤ_{p}-extension ℚ_{cyc} of ℚ. Thus we obtain a pairing

h_{p}(W) : H_{f}^{1}(ℚ, W) × H_{f}^{1}(ℚ, Z) → L

from [22, §11] where h_{p}(W) is called h̃_{π,1,1}. By [22, thm. 11.3.9] the pairing h_{p}(W) coincides up to sign with the height-pairing constructed by Schneider [30] (in the case of abelian varieties), Perrin-Riou [24] (for semi-stable representations) and those constructed earlier by Nekovar [21]: see also §8.1 in (loc. cite.), Mazur-Tate [20] and Zarhin [39] for alternative definitions of related height pairings. It follows from the construction of Nekovar’s height pairing (cf. [22, the sentence after (11.1.3.2)]) that the induced map

(32)  ad(h_{p}(W)) : H_{f}^{1}(ℚ, W) → H_{f}^{1}(ℚ, Z)^{∗}

is equal to the composite

(33) H_{f}^{1}(ℚ, W) ≅ H^{1}(SC_{U}(Ŵ , W)) ⟶^{𝔅} H^{2}(SC_{U}(Ŵ , W))

≅ H_{f}^{2}(ℚ, W) ≅ H_{f}^{1}(ℚ, Z)^{∗} where the first and third maps are by Lemma 6.2(iii), 𝔅 denotes the Bockstein morphism for SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ}) and the last map is by global duality. For the evaluation at representations in the next subsection we need the following descent properties. Let Υ be the set of all l ≠ p such that the ramification index of l in F_{∞}/ℚ is infinite. Note that Υ is empty if G has a commutative open subgroup.

Proposition 6.4. [14, prop. 1.6.5] With the above notation we have canonical isomorphisms (for all l)

L^{n} ⊗_{Λ,ρ} RΓ(U, 𝕋) ≅ RΓ(U, W),  L^{n} ⊗_{Λ,ρ} RΓ_{c}(U, 𝕋) ≅ RΓ_{c}(U, W),
L^{n} ⊗_{Λ,ρ} RΓ(ℚ_{ℓ}, 𝕋) ≅ RΓ(ℚ_{ℓ}, W), L^{n} ⊗_{Λ,ρ} SC_{U}(𝕋̂, 𝕋) ≅ SC_{U}(Ŵ , W).

For l ∉ Υ ∪ S_{p} we also have: L^{n} ⊗_{Λ,ρ} RΓ_{f}(ℚ_{ℓ}, 𝕋) ≅ RΓ_{f}(ℚ_{ℓ}, W).

But note that the complex RΓ_{f}(ℚ_{ℓ}, 𝕋̂) for l ∈ Υ and thus SC(𝕋̂, 𝕋) does not descend like this in general. Instead, according to [14, prop. 4.2.17] one has a distinguished triangle (34)

[figure: L^{n} ⊗_{Λ,ρ} SC(𝕋̂, 𝕋); SC(Ŵ , W); ⨁_{l∈Υ} RΓ_{f}(ℚ_{ℓ}, W)].

6.3. The interpolation formula. In this section we assume that the motive N := M(ρ^{∗}) is critical. Then, assuming [37, conj. 3.3] of Fontaine and Perrin-Riou it follows that the motivic cohomology groups

(D_{1}) H_{f}^{0}(N) = H_{f}^{0}(N^{∗}(1)) = 0 both vanish. Assuming also the conjecture [37, conj. 3.6] on p-adic regulator maps, this is equivalent to the condition

(D_{2}) H_{f}^{0}(ℚ, W) = H_{f}^{0}(ℚ, Z) = 0. We also consider the condition

(F) The pairing h_{p}(W) is non-degenerate.

Example 6.5. If A is an abelian variety over ℚ, then the conditions (D_{1}) and (D_{2}) are both satisfied for the motive M = h^{1}(A)(1). However, very little is known about the non-degeneracy of the p-adic height pairing in the ordinary case. Indeed, as far as we are aware, the only theoretical evidence for non-degeneracy is an old result of Bertrand [1] that for an elliptic curve with complex multiplication, the height of a point of infinite order is non-zero (even this is unknown in the non CM case). Computationally, there has been a lot of work done recently by Stein and Wuthrich [38]. We are grateful to J. Coates, P. Schneider and C. Wuthrich for providing us with these examples.

Proposition 6.6. Assume that the conditions (A_{1}), (C_{1}) and (D_{2}) are satisfied. Then

(i) SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ}) ∈ Σ_{ss} if and only if the condition (F) holds.

(i) if (F) holds, we have r_{Γ}(SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ})) = dim_{L} H_{f}^{1}(ℚ, W).

Proof. By assumption (D_{2}) implies that H^{i}(SC_{U}(Ŵ , W)) = 0 for i ≠ 1, 2. Thus the claim follows from the fact that (32) and (33) coincide. □

Now let F_{∞} be as before a p-adic Lie extension of ℚ which contains ℚ_{cyc} and G its Galois group (with quotient Γ). We set Λ := Λ(G). Since by [14, 4.1.4(2)] we have a canonical identification

(35)  (Λ_{O}(Γ) ⊗_{O} O^{n}) ⊗_{Λ(G)}^{𝕃} SC_{U}(𝕋̂, 𝕋) ≅ SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ})

we conclude that SC_{U}(𝕋̂, 𝕋) ∈ Σ_{ss−ρ} if and only (F) holds for W. In [14] two p-adic L-functions

(36)  L_{U} := L_{U}(M) : 1_{Λ} → d_{Λ}(SC_{U}(𝕋̂, 𝕋))

and

(37)  L := L(M) : 1_{Λ} → d_{Λ}(SC(𝕋̂, 𝕋))

were defined, modulo the validity of Conjecture 4.1 and Conjecture 6.1. Recall the definition of Σ_{SC_{U}} or Σ_{SC} at the end of section 2 where we abbreviate SC_{U} = SC_{U}(𝕋̂, 𝕋) and SC = SC(𝕋̂, 𝕋). Now L_{U} and L induce classes [SC_{U}, L_{U}] and [SC, L] in K_{1}(Λ(G), Σ_{SC_{U}}) and K_{1}(Λ(G), Σ_{SC}), which for simplicity we again call L_{U} and L, respectively. However, we note that for comparison purposes, it would perhaps be more convenient to define both L and L_{U} as elements of the group K_{1}(Λ(G), Σ′_{SC}) where Σ′_{SC} is the smallest full subcategory of C^{p}(Λ) which satisfies conditions (i)-(iii) and (iv’) from section 2 and contains SC_{U} as well as the Euler factors RΓ_{f}(ℚ_{ℓ}, 𝕋̂) for all l ∈ S_{f} \ S_{p}.

Theorem 6.7. We assume that W satisfies the conditions (A_{1}), (B_{1}), (C_{1}), (D_{1}) and (F) and that the isomorphisms ζ_{Λ}(M) and ϵ_{p,Λ}(𝕋̂) that are described in Conjectures 4.1 and 6.1 both exist. Then both SC_{U}(𝕋̂, 𝕋) and SC(𝕋̂, 𝕋) belong to Σ_{ss}, r := r_{G}(SC_{U}(𝕋̂, 𝕋))(ρ) = r_{G}(SC(𝕋̂, 𝕋))(ρ) = dim_{L} H_{f}^{1}(ℚ, W) and the leading term L^{∗}(ρ) (respectively L_{U}^{∗}(ρ)) is equal to (38)

(−1)^{r} \frac{L_{K,B}^{∗}(M(ρ^{∗}))}{Ω_{∞}(M(ρ^{∗}))R_{∞}(M(ρ^{∗}))} ·Ω_{p}(M(ρ^{∗}))R_{p}(M(ρ^{∗}))·Γ(V̂)^{−1}· \frac{P_{L,p}(Ŵ^{∗}(1), 1)}{P_{L,p}(Ŵ , 1)},

where L_{K,B}^{∗}(M(ρ^{∗})) is the leading coefficient of the complex L-function of N, truncated by removing Euler factors for all primes in B := Υ ∪ S_{p} (respectively B := S \ S_{∞}), and the regulators R_{∞}(M(ρ^{∗})) and R_{p}(M(ρ^{∗})) and periods Ω_{∞}(M(ρ^{∗})) and Ω_{p}(M(ρ^{∗})) are as defined in the course of the proof given below.

Remark 6.8. The formulas of Theorem 6.7 generalize the formulas obtained by Perrin-Riou in [27, 4.2.2,4.3.6]. Further, by slightly altering the definition of the complex L-function an analogous formula can be proved even in the case that (B_{1}) is not satisfied. Indeed, if (B_{1}) fails, then one can have P_{L,p}(W, 0) = 0 and so the order of vanishing of L_{K,B}(M(ρ^{∗}), s) and L_{K}(M(ρ^{∗}), s) may differ. However, to avoid this problem, in formula (38) one need only replace P_{L,p}(Ŵ , 1) by the leading coefficient of P_{L,p}(Ŵ , p^{s}) at s = 0, or equivalently

one can replace the term \frac{L_{K,B}^{∗}(M(ρ^{∗}))}{P_{L,p}(Ŵ ,1)} by \frac{L_{K,B\\{p\}}^{∗}(M(ρ^{∗}))}{\{P_{L,p}(W,u)^{−1}P_{L,p}(Ŵ ,u)\}_{u=1}} .

Proof. This proof is closely modelled on that of [14, thm. 4.2.26] (as amplified in [37, proof of thm. 6.4]). At the outset we let γ = (γ_{i})_{i} and δ = (δ_{i})_{i} be a choice of ‘good bases (in the sense of [14, 4.2.24]) of M_{B}^{+} and t_{M} for 𝕋̂ and let γ′ and δ′ be the induced K-bases of N_{B}^{+} and t_{N}, respectively. This induces a map

(39)  can_{γ′.δ′} : 1_{K} → d_{K}(N_{B}^{+})d_{K}(t_{N})^{−1}.

Furthermore, let P^{∨} = (P_{1}^{∨}, …, P_{d(N)}^{∨}) and P = (P_{1}, …, P_{d(N)}) be K-bases of H_{f}^{1}(N) and H_{f}^{1}(N^{∗}(1)), respectively. Setting P^{d} = (P_{1}^{d}, …, P_{d(N)}^{d}) the dual basis of P we obtain similarly

(40)  can_{P^{∨},P} : 1_{K} → d_{K}(H_{f}^{1}(N))d_{K}(H_{f}^{1}(N^{∗}(1))^{∗})^{−1}.

Then can := can_{γ′.δ′} · can_{P^{∨},P} is an isomorphism (41)

can : 1_{K} → Δ_{K}(N) = d_{K}(N_{B}^{+})d_{K}(t_{N})^{−1}d_{K}(H_{f}^{1}(N))d_{K}(H_{f}^{1}(N^{∗}(1))^{∗})^{−1}.

We fix an embedding of K into ℂ. Now let Ω_{∞}(N) and R_{∞}(N) denote the determinant of the isomorphism

(42)  α_{N} : (N_{B}^{+})_{ℂ} → (t_{N})_{ℂ}

with respect to γ′ and δ′, and the inverse of

(43)  h_{∞} : (H_{f}^{1}(N^{∗}(1))^{∗})_{ℂ} → H_{f}^{1}(N)_{ℂ}

with respect to P^{d} and P, respectively. In other words, we have

Ω_{∞}(N) : 1_{ℂ} ⟶^{(can_{γ′,δ′})_{ℂ}} d_{K}(N_{B}^{+})_{ℂ}d_{K}(t_{N})_{ℂ}^{−1} ⟶^{d(α_{N})·id} 1_{ℂ}

and

R_{∞}(N) : 1_{ℂ} ⟶^{(can_{P^{∨},P})_{ℂ}} d_{K}(H_{f}^{1}(N))_{ℂ}d_{K}(H_{f}^{1}(N^{∗}(1))^{∗})_{ℂ}^{−1} ⟶^{id·d(h_{∞})^{−1}} 1_{ℂ}.

Note that we obtain

Ω_{∞}(N)R_{∞}(N) : 1_{ℂ} ⟶^{can} Δ_{K}(N)_{ℂ} ⟶^{(ϑ_{∞})_{ℂ}} 1_{ℂ}.

Upon comparing this with

L_{K}^{∗}(M) : 1_{ℂ} ⟶^{ζ_{K}(N)_{ℂ}} Δ_{K}(N)_{ℂ} ⟶^{(ϑ_{∞})_{ℂ}} 1_{ℂ}.

we deduce that ζ_{K}(N) : 1_{K} → Δ_{K}(N) coincides with

\frac{L_{K}^{∗}(M)}{Ω_{∞}(N)R_{∞}(N)} · can : 1_{K} → Δ_{K}(N).

Also, L_{U}^{∗}(ρ) is defined (in Definition 3.14) to be the isomorphism

1L̃ ⟶^{ζ_{Λ}(M)(ρ)L̃} d_{L}(RΓ_{c}(U, W))−1 ; L̃ ⟶^{β(ρ)ϵ(𝕋̂)^{−1}(ρ)}

d_{L}(SC_{U}(Ŵ , W))−1 ; L̃ ⟶^{t(SC_{U} (ρ^{∗}))L̃} 1L̃

where we set ζ_{Λ}(M)(ρ) := L^{n} ⊗_{Λ} ζ_{Λ}(N), β(ρ) := L^{n} ⊗_{Λ} β and ϵ(𝕋̂)(ρ) := L^{n} ⊗_{Λ} ϵ_{p,Λ}(𝕋̂). Using that ζ_{Λ}(M)(ρ) equals

1L̃ ⟶^{ζ_{K}(N)L̃} Δ_{K}(N)L̃ ⟶^{ϑ_{λ}} d_{L}(RΓ_{c}(U, W))−1 ; L̃

by [14, conj. 2.3.2] (or [37, conj. 3.7, Conj. 4.1]), one sees easily that L_{U}^{∗}(ρ) is the product of the following terms (44)-(50):

(44)  \frac{L_{K}^{∗}(N)}{Ω_{∞}(M(ρ^{∗}))R_{∞}(N)},

(45)  Γ_{L}(Ŵ)^{−1} = Γ_{ℚ_{p}}(V̂)^{−1},

(46) Ω_{p}(M(ρ^{∗})) : [figure: d_{L}(Ŵ)L̃; · ϵ_{dR}(Ŵ)^{−1}; d_{L}(D_{dR}(Ŵ))L̃; d(g_{dR}^{t}); d_{K}(t_{M(ρ^{∗})})L̃; · can_{γ,δ}]

\xymatrix{d_{K}(M(ρ^{∗})_{B}^{+})L̃ \ar[r]^{d(g_{λ}^{+})} & d_{L}(W^{+})L̃ \ar[r]^{β(ρ)} & d_{L}(Ŵ)L̃}

where we apply Remark 2.2 to obtain an automorphism of 1L̃.

(47) ∏_{S\\{p,∞\}} P_{L,l}(W, 1) : \xymatrix{1_{L} \ar[r]^{∏ η_{l}(W)} & ∏ d_{L}(RΓ_{f}(ℚ_{ℓ}, W)) \ar[r]^{acyc} & 1_{L}}

where the first map comes from the trivialization by the identity and the second from the acyclicity,

(48) {P_{L,p}(W, u)P_{L,p}(Ŵ , u)^{−1}}_{u=1} :

\xymatrix{1_{L} \ar[r]^{η_{W} ·η−1 ; Ŵ} & d_{L}(RΓ_{f}(ℚ_{p}, W))d_{L}(RΓ(ℚ_{p}, Ŵ))^{−1} \ar[r]^{quasi} & 1_{L},}

where we use that t_{p}(W) = D_{dR}(Ŵ) = t_{p}(Ŵ) and the quasi-isomorphism mentioned in Lemma 6.2(ii),

(49) P_{L,p}(Ŵ^{∗}(1), 1) : \xymatrix{1_{L} \ar[r]^{\overline{(η_{Ŵ^{∗}(1)})^{∗}}} & d_{L}(RΓ_{f}(ℚ_{p}, Ŵ^{∗}(1))) \ar[r]^{acyc} & 1_{L},}

where we use that t_{p}(Ŵ^{∗}(1)) = D_{dR}^{0}(Ŵ) = 0, and the determinant over L of the isomorphism ad(h_{p}(W)) with respect to the bases P^{∨} and P:

(50) R_{p}(N) : 1_{L} ⟶^{(can_{P^{∨},P})_{L}} d_{K}(H_{f}^{1}(N))_{L}d_{K}(H_{f}^{1}(N^{∗}(1))^{∗})_{L}^{−1} ⟶^{≅}

d_{K}(H_{f}^{1}(ℚ, W))_{L}d_{K}(H_{f}^{1}(ℚ, Z)^{∗})_{L}^{−1} ⟶^{h_{p}(W)} 1_{L}.

Indeed, as remarked above ζ_{Λ}(M)(ρ) decomposes up to the comparison isomorphism d(g_{λ}^{+}), which contributes to factor (46), into ζ_{K}(N)_{L} and ϑ_{λ}. While ζ_{K}(N)_{L} gives the full factor (44) and contributes with can_{γ,δ} and can_{P^{∨},P} to the factors (46) and (50), respectively, the second part ϑ_{λ} gives the full factor (47), the half factor (48) in form of η(W) and contributes d(g_{dR}^{+}) to factor (46). Further, β(ρ) contributes to factor (46), while according to [14, §3.3] ϵ(𝕋̂)^{−1}(ρ) = ϵ_{p,L}(Ŵ)^{−1} gives the full factors (45) and (49), the other half of (48) in form of η−1 ; Ŵ and adds ϵ_{dR}(Ŵ) to factor (46). Finally, we had observed at the end of §6.2 that t(SC_{U}(ρ^{∗})) equals h_{p}(W). In order to derive the corresponding formula for L^{∗}(ρ) we use the exact triangle

SC_{U}(Ŵ , W) → L^{n} ⊗_{Λ}^{𝕃} SC(𝕋̂, 𝕋) → ⨁_{S_{f} \(S_{p}∪Υ)} RΓ_{f}(ℚ_{ℓ}, W) →

and the equality

L_{K,Υ′}^{∗}(N) = L_{K,B′}^{∗}(N) ∏_{B\Υ} P_{L,l}(W, 1)^{−1}

with Υ′ = Υ ∪ {p} and B′ = S \ S_{∞}. □

Example 6.9. Let E be an elliptic curve defined over ℚ, M = h^{1}(E)(1) and F_{∞} := ℚ(E(p)) where E(p) ⊆ E(\overline{ℚ}] denotes the group of p-power torsion points of E. It is conjectured that SC_{U}(𝕋̂, 𝕋) always belongs to Σ_{S^{∗}} (cf. [10, conj. 5.1] and [14, 4.3.5 and prop. 4.3.7]). As was shown in [14] the existence of L(M) implies the existence of L_{E} ∈ K_{1}(Λ(G)_{S^{∗}}) as conjectured in [10, conj. 5.7] to exist with a prediscribed interpolation property L_{E}(ρ) for r_{G}(SC(𝕋̂, 𝕋))(ρ) = 0. Now the above formula (38) predicts the leading terms L_{E}^{∗}(ρ) for all Artin characters with non-degenerate archimedean and p-adic height pairing.

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David Burns

King’s College London, Dept. of Mathematics, London WC2R 2LS, United Kingdom.

Otmar Venjakob Universität Bonn Mathematisches Institut Beringstraße 1 D-53115 Bonn Germany.