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

Received: September 30, 2005 Revised: April 24, 2006

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.

2000 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 [11], 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 ordinary reduction at all places above p and whose Selmer group (over F_{∞}) satisfies a certain natural torsion condition. Then, in [16], 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. The key feature of [11] is the use of the localization sequence of algebraic K-theory with respect to a canonical Ore set. However, the more general approach of [16] is rather more involved and uses a notion of ‘localized K_{1}-groups’ together with Nekovář’s theory of Selmer complexes and the (conjectural) existence of certain canonical p-adic L-functions. See [39] 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 [7] and hence also implies the ‘main conjecture of non-abelian Iwasawa theory’ discussed by Huber and Kings in [19]) 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, where ρ^{∗} denotes the contragredient of the representation ρ. However, if the Selmer module of M(ρ^{∗}) has strictly positive rank (and by a recent result of Mazur and Rubin [21], which is itself equivalent to a special case of an earlier result of Nekovář [24, Th. 10.7.17], 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 [27, 29], of Schneider [34, 33, 32, 31] and of Greither and the first named author [9, 4] and have been systematically incorporated into Nekovář’s theory of Selmer complexes [24]. 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 [35] and interpreted by Tate in [37], 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-Nogués. 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 Nekovář’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 [5], 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.

Non-Commutative p-Adic L-Functions 167

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-Poincaré 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 [39]. 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 associative unital ring R we write B(R) for the category of bounded (cohomological) complexes of (left) Rmodules, C(R) for the category of bounded (cohomological) complexes of finitely generated (left) R-modules, P (R) for the category of finitely generated projective (left) R-modules and C^{p}(R) for the category of bounded (cohomological) complexes of finitely generated projective (left) R-modules. We also write D^{p}(R) for the category of perfect complexes as full triangulated subcategory of the bounded derived category D^{b}(R) of (left) R-modules. We write (P (R), is), (C^{p}(R), quasi) and (D^{p}(R), is) for the subcategories of isomorphisms in P (R), quasi-isomorphisms in C^{p}(R) and isomorphisms in 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 recall that in [16, §1.2] Fukaya and Kato construct an explicit alternative to the category of virtual objects that is used in [7]. Indeed, they construct explicitly a category C_{R} and a ‘determinant functor’

d_{R} : (P (R), is) → C_{R}

which possess the following properties: a) C_{R} has an associative and commutative product structure (M, N) ↦

M · N (which we often write more simply as MN) with canonical unit object 1_{R} = d_{R}(0). If P is any object of P (R), then in C_{R} the object d_{R}(P) has a canonical inverse d_{R}(P)^{−1}. Every object of C_{R} is of the form d_{R}(P) · d_{R}(Q)^{−1} for suitable objects P and Q of P (R);

b) All morphisms in C_{R} are isomorphisms and elements of the form d_{R}(P) and d_{R}(Q) are isomorphic in C_{R} if and only if P and Q correspond to the same element of the Grothendieck group K_{0}(R). There is a natural identification Aut_{C_{R}}(1_{R}) ≅ K_{1}(R) and if Mor_{C_{R}}(M, N) is non-empty, then it is a K_{1}(R)-torsor where each element α of K_{1}(R) ≅ Aut_{C_{R}}(1_{R}) acts on ϕ ∈ Mor_{C_{R}}(M, N) to give αϕ : M = 1_{R} · M ⟶^{α·ϕ} 1_{R} · N = N;

c) d_{R} preserves the product structure: specifically, for each P and Q in

P (R) one has d_{R}(P ⊕ Q) = d_{R}(P) · d_{R}(Q). The functor d_{R} can be extended to give a functor

d_{R} : (C^{p}(R), quasi) → C_{R}

in the following way: for each C ∈ C^{p}(R) one sets

d_{R}(C) := d_{R}(⨁_{i∈ℤ} C^{2i})d_{R}(⨁_{i∈ℤ} C^{2i+1})^{−1}.

This extended functor then has the following properties for all objects C, C′ and C′′ of C^{p}(R):

d) If 0 → C′ → C → C′′ → 0 is a short exact sequence in C^{p}(R), then there exists a canonical morphism in C_{R} of the form

d_{R}(C) ≅ d_{R}(C′)d_{R}(C′′),

which we take to be an identification;

e) If C is acyclic, then the quasi-isomorphism 0 → C induces a canonical morphism in C_{R} of the form

1_{R} = d_{R}(0) → d_{R}(C);

f) For any integer r there exists a canonical morphism d_{R}(C[r]) ≅ d_{R}(C)^{(−1)^{r}} in C_{R} which we take to be an identification;

g) The functor d_{R} factorizes through the image of C^{p}(R) in D^{p}(R) and extends (uniquely up to unique isomorphism) to give a functor

d_{R} : (D^{p}(R), is) → C_{R}.

h) For each C ∈ D^{b}(R) we write H(C) for the complex with 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 morphisms in C_{R} of the form

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

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

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

which commutes (up to canonical isomorphism). In particular, if R → R′ is any ring homomorphism and C ∈ D^{p}(R), then 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 extend to arbitrary exact triangles in D^{p}(R). In general therefore all constructions in the sequel which involve complexes must be made in such a way to avoid this problem (nevertheless, we suppress any explicit discussion of this issue in the present manuscript and simply refer the reader to [7] for details as to how this problem can be overcome). The second displayed morphism in h) is induced by the properties d) and f). However, whilst a precise description of the first morphism 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 [20, §3].

Remark 2.2. In the sequel we will have to distinguish between two inverses of a morphism ϕ : C → D with C, D ∈ C_{R}. The inverse with respect to composition will be denoted by \overline{ϕ} : D → C while

ϕ^{−1} := \overline{id_{D^{−1}} · ϕ · id_{C^{−1}}} : C^{−1} → D^{−1}

is the unique isomorphism such that ϕ · ϕ^{−1} = id_{1_{R}} under the identification X · X^{−1} = 1_{R} for both X = C and X = D. If D = C, then ϕ : C → C corresponds uniquely to an element of K_{1}(R) ≅ Aut_{C_{R}}(1_{R}) by the rule ϕ · id_{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}}.

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. In this remark we let C denote the complex P_{0} →^{ϕ} P_{1}, in which the first term is placed in degree 0 and P_{0} = P_{1} = P. Then, by definition, one

has d_{R}(C) ^{def} 1_{R} . However, if ϕ is an isomorphism (so C is acyclic), then by property e) there is also a canonical morphism \xymatrix{1_{R} \ar[r]^{acyc} & d_{R}(C)} . This latter morphism coincides with the composite

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

and thus depends on ϕ. Indeed, Remark 2.2 shows that the composite morphism

[figure: 1_{R}; acyc; d_{R}(C); def; 1_{R}]

corresponds to the element d_{R}(ϕ)^{−1} of K_{1}(R). Thus, in order to distinguish between the above identifications of 1_{R} with d_{R}(C), we shall say that C is

trivialized by the identity when using either d_{R}(C) ^{def} 1_{R} or its inverse with respect to composition.

Remark 2.4. Let O = O_{L} be the valuation ring of a finite extension L of ℚ_{p} and A a finite O-module. Then for any morphism in C_{O} of the form a : 1_{O} → d_{O}(A), and in particular therefore for that induced by any exact sequence of O-modules 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 [16, §1.3] a localized K_{1}-group is 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 C[n] ∈ Σ for all n ∈ ℤ, (iv) if 0 → C′ → C → C′′ → 0 is an exact sequence in C^{p}(R) with both

C′ ∈ Σ and C′′ ∈ Σ, then 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 and Kato) Let Σ be any full subcategory of C^{p}(R) which satisfies the conditions (i), (ii), (iii) and (iv′). Then the localized K_{1}- group K_{1}(R, Σ) is defined to be the (multiplicatively written) abelian group which has as generators all symbols of the form [C, a] where C ∈ Σ and a is a morphism 1_{R} → d_{R}(C) in C_{R}, and as relations

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

(1) [C′, d_{R}(f) ∘ a] = [C, a] if f : C → C′ is a quasi-isomorphism with C (and thus also 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), (1) and (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 by Fukaya and Kato. 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 to be 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)  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. [16, 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: C is a fixed object of D^{p}(R) which is such that R_{S} ⊗_{R} C is acyclic and Σ denotes the smallest full subcategory Σ_{C} of C^{p}(R) which contains all objects of C^{p}(R) that are isomorphic in D^{p}(R) to C and also satisfies the conditions (i), (ii), (iii) and (iv) that are described above. (With this definition, it is easily seen that Σ_{C} ⊂ Σ_{S}).

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 [11] and we use it to extend several well known results concerning Generalized Euler-Poincaré 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}. 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. Then, since H^{1}(G, ℤ_{p}) ≅ Ext_{Λ(G)}^{1}(ℤ_{p}, ℤ_{p}) by [25, Prop. 5.2.14], the element θ corresponds to a canonical extension of Λ(G)- modules of the form

(2)  0 → ℤ_{p} → E_{θ} → ℤ_{p} → 0.

Indeed, one has E_{θ} = ℤ_{p}^{2} upon which G acts via the matrix (\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 (2) induces an exact sequence in B(Λ(G)) of the form

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

This sequence in turn induces a ‘cup-product’ morphism in D^{b}(Λ(G)) of the form

(3)  A^{•} ⟶^{θ} A^{•}[1].

It is clear that this morphism depends upon the choice of γ, but nevertheless we continue to denote it simply by θ. We now let ρ : G → GL_{n}(O) be a (continuous) representation of G on T_{ρ} := O^{n}, where O = O_{L} denotes the valuation ring of a finite extension L of ℚ_{p}. Then in the sequel we are mainly interested in the morphism

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

that is induced by (3), where we consider O^{n} as a right Λ(G)-module via the transpose ρ^{t} of ρ. In particular, in each degree i we shall refer to the induced homomorphism

𝔅_{i} : 𝕋or_{i}^{Λ(G)}(T_{ρ}, A^{•}) → 𝕋or_{i−1}^{Λ(G)}(T_{ρ}, A^{•})

of hyper-tor groups

𝕋or_{i}^{Λ(G)}(T_{ρ}, A^{•}) := H^{−i}(O^{n} ⊗_{Λ(G)}^{𝕃} A^{•})

as the Bockstein homomorphism (in degree i) of (A^{•}, T_{ρ}, γ).

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 ∈ Λ(Γ).

3.2.1. Bockstein homomorphisms. 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^{b}(Λ(Γ)) of the form

(4)  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 morphism 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^{•} is a bounded complex of projective Λ(Γ)-modules, then we set

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

and

A_{0}^{•} := ℤ_{p} ⊗_{Λ(Γ)} A^{•}.

In any such case there exists a morphism of complexes π : cone(A^{•}) → A_{0}^{•} of the form

⟶ 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 ℤ_{p} ⊗_{Λ(Γ)} A^{i} = A_{0}^{i}. It is easy to check that π is a quasi-isomorphism. Now from (4) we obtain short exact sequences

(5)  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 the hyper-homology of A^{•} (with respect to Γ) and 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. Let A^{•} be a bounded complex of projective Λ(Γ)-modules. Then in each degree i the Bockstein homomorphism of the triple (A^{•}, ℤ_{p}, γ) coincides with 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 first and third arrows are as in (5) and κ^{−i+1}(A^{•}) denotes the tautological homomorphism

H^{−i+1}(A^{•})^{Γ} ↪ H^{−i+1}(A^{•}) ↠ H^{−i+1}(A^{•})_{Γ}.

Proof. As is shown by Rapoport and Zink in [30, Lem. 1.2], on the level of complexes the cup product morphism of the triple (A^{•}, ℤ_{p}, γ) is described by the morphism

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 also 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 short exact sequence

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

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

From this description it is clear that for any bounded complex of projective Λ(Γ)-modules A^{•} the pair

(6)  (ℍ_{i}(Γ, A^{•}), 𝔅_{i})

forms a homological complex (which, by re-indexing, we shall consider as cohomological complex whenever convenient). It is also clear that this construction extends in a well-defined fashion to objects A^{•} of D^{p}(Λ(Γ)).

3.2.2. Semisimplicity.

Definition 3.2. (Semisimplicity) For any A^{•} ∈ D^{p}(Λ(Γ)) we set

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

We say that a complex A^{•} ∈ D^{p}(Λ(Γ)) is semisimple if the cohomology of the associated complex (6) 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.

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

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

This implies that a complex A^{•} ∈ D^{p}(Λ(Γ)) is semisimple if and only if the homomorphism 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) For each A^{•} ∈ D^{b}(Λ(Γ)) we write (ℍ_{•}(Γ, A^{•}), 0) for the complex with (ℍ_{•}(Γ, A^{•}), 0)^{i} = ℍ_{i}(Γ, A^{•}) in each degree i and in which all differentials are the zero map. In particular, if A^{•} ∈ Σ_{ss}, then we obtain a canonical composite morphism

(7) t(A^{•}) : d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(Γ)} 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, morphism uses property h) (in §2.1) for the functor d_{ℤ_{p}}, resp. property i) for the natural homomorphism ℤ_{p} → ℚ_{p} and then property e) for the functor d_{ℚ_{p}}.

Remark 3.5. If the complex ℚ_{p} ⊗_{Λ(Γ)} A^{•} is acyclic, then t(A^{•}) coincides with the trivialization obtained by directly applying property e) to ℚ_{p} ⊗_{Λ(Γ)} 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 result shows, the above constructions behave well on short exact sequences of semisimple complexes.

Lemma 3.6. Let A^{•}, B^{•} and C^{•} be objects of Σ_{ss} which together lie in a short exact sequence in C^{p}(Λ(Γ)) of the form

0 → A^{•} → B^{•} → C^{•} → 0.

Then one has

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

and, with respect to the canonical morphism

d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(Γ)} B^{•})_{ℚ_{p}} ≅ d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(Γ)} A^{•})_{ℚ_{p}} · d_{ℤ_{p}}(ℤ_{p} ⊗_{Λ(Γ)} C^{•})_{ℚ_{p}}

that is induced by the given short exact sequence, one has

t(B^{•}) = t(A^{•}) · t(C^{•}).

Proof. We let 𝔭 denote the kernel of the augmentation map Λ(Γ) → ℤ_{p} and R the localization Λ(Γ)_{𝔭} of Λ(Γ) at 𝔭. Then R is a discrete valuation ring with uniformizer T and residue class field R/(T) naturally isomorphic to ℚ_{p}. Further, if a complex K^{•} ∈ D^{p}(Λ(Γ)) is semisimple, then the structure theory of finitely generated Λ(Γ)-modules implies that in each degree i the R-module H^{i}(K_{𝔭}^{•}) is isomorphic to a direct sum of (finitely many) copies of R/(T) and hence also to ℚ_{p} ⊗_{ℤ_{p}} H^{i}(K^{•})_{Γ} ≅ ℚ_{p} ⊗_{ℤ_{p}} H^{i}(K^{•})^{Γ}.

To prove the claimed equality r_{Γ}(B^{•}) = r_{Γ}(A^{•}) + r_{Γ}(C^{•}) it is therefore enough to take dimensions over ℚ_{p} ≅ R/(T) in the long exact cohomology sequence of the following short exact sequence in C^{p}(R)

(8)  0 → A_{𝔭}^{•} → B_{𝔭}^{•} → C_{𝔭}^{•} → 0.

To prove the second claim we note that if K^{•} ∈ C^{p}(Λ(Γ)), then the complex K_{0,𝔭}^{•} := ℚ_{p} ⊗_{Λ(Γ)} K^{•} is isomorphic in D^{p}(ℚ_{p}) to ℚ_{p} ⊗_{R} K_{𝔭}^{•}. Hence, since each term of C_{𝔭}^{•} is a projective R-module, the short exact sequence (8) gives rise to a short exact sequence in C^{p}(ℚ_{p}) of the form

(9)  0 → A_{0,𝔭}^{•} → B_{0,𝔭}^{•} → C_{0,𝔭}^{•} → 0.

Now one has a commutative diagram in C_{ℚ_{p}}

d_{ℚ_{p}}(B_{0,𝔭}^{•}) ⟶  d_{ℚ_{p}}(A_{0,𝔭}^{•})d_{ℚ_{p}}(C_{0,𝔭}^{•})
↓  ↓
d_{ℚ_{p}}(H(B_{0,𝔭}^{•})) ⟶ d_{ℚ_{p}}(H(A_{0,𝔭}^{•}))d_{ℚ_{p}}(H(C_{0,𝔭}^{•}))

in which the upper, resp. lower, horizontal morphism is induced by (9), resp, by the long exact cohomology sequence of (9), and both vertical arrows are induced by applying property h) of d_{ℚ_{p}} in §2.1. (For a proof of the commutativity of the above diagram see [2, Thm. 3.3].) Further, in this situation the exact sequences (5) induce short exact sequences 0 → H^{i}(A_{𝔭}^{•}) → H^{i}(A_{0,𝔭}^{•}) → H^{i+1}(A_{𝔭}^{•}) → 0 (and similarly for B^{•} and C^{•}) which together lie in a short exact sequence of long exact sequences

0  0  0  0
↓  ↓  ↓  ↓
⟶ H^{i}(A_{𝔭}^{•}) ⟶ H^{i}(B_{𝔭}^{•}) ⟶ H^{i}(C_{𝔭}^{•}) ⟶ H^{i+1}(A_{𝔭}^{•}) ⟶
↓  ↓  ↓  ↓
⟶ H^{i}(A_{0,𝔭}^{•}) ⟶ H^{i}(B_{0,𝔭}^{•}) ⟶ H^{i}(C_{0,𝔭}^{•}) ⟶ H^{i+1}(A_{0,𝔭}^{•}) ⟶
↓  ↓  ↓  ↓
⟶ H^{i+1}(A_{𝔭}^{•}) ⟶ H^{i+1}(B_{𝔭}^{•}) ⟶ H^{i+1}(C_{𝔭}^{•}) ⟶ H^{i+2}(A_{𝔭}^{•}) ⟶
↓  ↓  ↓  ↓
0  0  0  0

where the upper and lower, resp. central, row is the exact cohomology sequence of (8), resp. (9). It is now a straightforward exercise to derive the required equality t(B^{•}) = t(A^{•}) · t(C^{•}) from the commutativity of both of the above diagrams. □

3.2.3. Leading terms. 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 canonical 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 Lemma 3.6 into account, it can be shown that 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}).

In particular therefore, (property g) of the functor d_{Λ(Γ)} combines with relation (1) in the definition of K_{1}(Λ(Γ), Σ_{ss}) to imply that) the notation [A^{•}, a]^{∗}(ρ_{triv}) extends in a well-defined fashion to pairs of the form (A^{•}, a) where A^{•} ∈ D^{p}(Λ(Γ)) is semisimple and a is a morphism in C_{Λ(Γ)} of the form 1_{Λ(Γ)} → d_{Λ(Γ)}(A^{•}).

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 (in the usual sense) of an appropriate characteristic power series. To this end we note that Remark 3.3(i) implies 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, where we write Q(Γ) for the quotient field of Λ(Γ). Hence there exists a homomorphism

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

Now the identification between Λ(Γ) and the power series ring ℤ_{p}[[T]] (which, of course, depends on the choice of T = γ − 1) allows any element F of Q(Γ)^{×} to be written uniquely as

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

with r = r(F) ∈ ℤ and G(T) ∈ Q(Γ) such that G(0) ∈ ℚ_{p}^{×}. The leading coefficient of F with respect to its expansion in the Laurent series ring ℚ_{p}{{T }} is therefore equal to F^{∗}(0) := G(0).

Proposition 3.8. Let A^{•} be any object of D^{p}(Λ(Γ)) which is semisimple and a any morphism in C_{Λ(Γ)} of the form 1_{Λ(Γ)} → d_{Λ(Γ)}(A^{•}).

(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. We use the localization R of Λ(Γ) that was introduced in the proof of Lemma 3.6. It is easy to see that both of the homomorphisms (−)^{∗}(ρ_{triv}) and ch_{Γ} factor via the 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 K^{•} with the property that in each degree i the R-module H^{i}(K^{•}) is isomorphic to a direct sum of (finitely many) copies of R/(T). 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 ⟶^{d} R where R occurs in degrees −1 and 0 and d denotes multiplication by either T or 1. Further, since the complex R ⟶^{×1} R is acyclic we shall therefore assume that d denotes multiplication by T. Now Mor_{C_{R}}(1_{R}, d_{R}(A^{•})) is a K_{1}(R)-torsor and so all possible trivializations arise in the following way: if ϵ is any fixed element of R^{×}, then the R-module homomorphism R → A^{−1}, resp. R → A^{0}, that sends 1 ∈ R to 1 ∈ R, resp. to ϵ ∈ R, induces a morphism can_{1} : d_{R}(R) → d_{R}(A^{−1}), resp. can_{ϵ} : d_{R}(R) → d_{R}(A^{0}), in C_{R}, and hence also a morphism a_{ϵ} := (can_{1})^{−1}·can_{ϵ} : 1_{R} → d_{R}(A^{•}). Setting L_{ϵ} := [A^{•}, a_{ϵ}] ∈ K_{1}(R, Ξ), 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 homomorphism 𝔅_{1} of the triple (A^{•}, R/(T), γ) is equal to ℚ_{p} ⟶^{−1} ℚ_{p} as one checks by using the description given in the proof of Lemma 3.1. Thus L_{ϵ}^{∗}(ρ_{triv}) is, by definition, equal to (−1)^{r_{Γ}(A^{•})} times the determinant of

ℚ_{p} ⟶^{ϵ(0)} ℚ_{p} ⟶^{(𝔅_{1})^{−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) = ch_{Γ}(L_{ϵ})^{∗}(0). This proves both claims of the Proposition. □

Lemma 3.9. Let R be a discrete valuation ring with uniformizer T and assume that A^{•} ∈ C^{p}(R) is such that in each degree i the R-module H^{i}(A^{•}) is annihilated by T. 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}. Then, by assumption, for each integer i with 1 ≤ i ≤ d, one has T e_{i} ∈ im(D^{−1}∘d^{m−1}). For each such i we set h_{i} := 1 if e_{i} ∈ im(D^{−1} ∘ d^{m−1}) and, otherwise, we set h_{i} := T. We write H for the diagonal d × d-matrix with entries h_{1}, …, h_{d}.

Then, since the image of the map R^{d} ⟶^{H} R^{d} is equal to im(D^{−1} ∘ d^{m−1}), there exists a retraction E : R^{d} ⟶ A^{m−1} (i.e. with left inverse ‘H^{−1} ∘ D^{−1} ∘ d^{m−1}’) that makes 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 ⟶ .

Now if B^{•} denotes the upper row of this diagram and C^{•} := A^{•}/B^{•} the associated quotient complex (not the mapping cone!), then one checks readily that there exists a split exact sequence 0 → B^{•} → A^{•} → C^{•} → 0. This implies that C^{•} belongs to C^{p}(R) and has cohomology annihilated by T (in all degrees). Thus, since the length of C^{•} is strictly shorter than the length of A^{•}, the proof can be completed 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 the quotient field Q_{O}(Γ) of Λ_{O}(Γ).

3.3. The general case. We extend the constructions of §3.2 to the setting of the Bockstein homomorphisms that are discussed at the end of §3.1. If A^{•} ∈ C^{p}(Λ(G)), then for any continuous representation of G of the form ρ : G → GL_{n}(O) we regard the complex

A^{•}(ρ^{∗}) := O^{n} ⊗_{ℤ_{p}} A^{•}

as a complex of (left) Λ_{O}(G)-modules by means of the following G-action: g(x ⊗_{ℤ_{p}} a) := ρ^{∗}(g)(x) ⊗_{ℤ_{p}} g(a) for each g ∈ G, x ∈ O^{n} and a ∈ A^{i}. With this action, there exists a natural isomorphism in C^{p}(ℤ_{p}) between ℤ_{p} ⊗_{Λ(G)} A^{•}(ρ^{∗}) and the complex O^{n} ⊗_{Λ(G)} A^{•} that occurs in §3.1. Further, it can be shown that the Bockstein homomorphisms 𝔅_{•} of the triple (A^{•}, T_{ρ}, γ) give rise to a complex of the form (ℍ_{•}(G, A^{•}(ρ^{∗})), 𝔅_{•}) where for each integer i and each normal closed subgroup J of G we set

ℍ_{i}(J, A^{•}(ρ^{∗}))) := H^{−i}(ℤ_{p} ⊗_{Λ(J)} A^{•}(ρ^{∗})) ≅ 𝕋or_{i}^{Λ(J)}(T_{ρ}, A^{•})

(see, for example, the proof of Lemma 3.13 below).

Definition 3.11. (Semisimplicity at ρ) For each A^{•} ∈ D^{p}(Λ(G)) we set

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

where L is the fraction field of O. We 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 those 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.

Definition 3.12. (Finiteness at ρ) We say that a complex A^{•} ∈ D^{p}(Λ(G)) is finite at ρ if the groups ℍ_{i}(G, A^{•}(ρ^{∗})) are ℤ_{p}-torsion in all degrees i. We let Σ_{fin−ρ} denote the full subcategory of C^{p}(Λ(G)) consisting of those 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 an (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} x)g := τḡ ⊗_{O} ρ(g)^{t}(x) for each g ∈ G (with image ḡ in Γ), x ∈ O^{n} and τ ∈ Λ_{O}(Γ). For each complex A^{•} ∈ Σ_{ss−ρ} we then set

A_{ρ}^{•} := (Λ_{O}(Γ) ⊗_{O} O^{n}) ⊗_{Λ(G)} A^{•} ∈ C^{p}(Λ_{O}(Γ)).

Lemma 3.13. Fix A^{•} ∈ C^{p}(Λ(G)).

(i) There are natural quasi-isomorphisms in C^{p}(Λ_{O}(Γ)) of the form

A_{ρ}^{•} ≅ Λ_{O}(Γ) ⊗_{Λ_{O}(G)} A^{•}(ρ^{∗}) ≅ O ⊗_{Λ_{O}(H)} A^{•}(ρ^{∗}).

(ii) One has r_{G}(A^{•})(ρ) = r_{Γ}(A_{ρ}^{•}).

(iii) The Bockstein homomorphism in any given degree of (A^{•}, T_{ρ}, γ) (as defined in §3.1) coincides with the Bockstein homomorphism in the same degree of (A_{ρ}^{•}, ℤ_{p}, γ).

(iv) One has A^{•} ∈ Σ_{ss−ρ} if and only if A_{ρ}^{•} ∈ Σ_{ss} (when considered as an object of C^{p}(Λ_{O}(Γ))). Further, if this is the case, then the trivialization

t(A_{ρ}^{•}) : d_{O}(O ⊗_{Λ_{O}(Γ)} A_{ρ}^{•})_{L} → 1_{L}

that is defined as in (7) coincides with the composite morphism

(11) t(A^{•}(ρ^{∗})) : d_{O}(O ⊗_{Λ_{O}(G)} 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, morphism uses property h) (in §2.1) for the functor d_{O}, resp. property i) for the homomorphism O → L and then property e) for the functor d_{L}.

(v) If A^{•}, B^{•} and C^{•} are objects of Σ_{ss−ρ} which together lie in a short exact sequence in C^{p}(Λ(G)) of the form

0 → A^{•} → B^{•} → C^{•} → 0,

then one has

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

and, with respect to the canonical morphism

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

that is induced by the given short exact sequence, one has

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

Proof. Claim (i) is clear (given the specified actions). Claim (ii) then follows by using the isomorphisms of claim (i) to directly compare the definitions of r_{G}(A^{•})(ρ) and r_{Γ}(A_{ρ}^{•}). In a similar way, claims (iii) and (iv) follow from the functorial construction of Bockstein homomorphisms and the fact that there are natural isomorphisms in C^{p}(O) of the form

O^{n} ⊗_{Λ(G)} A^{•} ≅ O ⊗_{Λ_{O}(G)} A^{•}(ρ^{∗})
≅ O ⊗_{Λ_{O}(Γ)} (Λ_{O}(Γ) ⊗_{Λ_{O}(G)} A^{•}(ρ^{∗}))
≅ ℤ_{p} ⊗_{Λ(Γ)} A_{ρ}^{•}.

Finally, to prove claim (v) we observe that, by claim (i), the given short exact sequence gives rise to a short exact sequence of semisimple complexes in C^{p}(Λ_{O}(Γ)) of the form

0 → A_{ρ}^{•} → B_{ρ}^{•} → C_{ρ}^{•} → 0.

The equalities of claim (v) thus follow from claims (ii), (iii) and (iv) and the results of Lemma 3.6 as applied to the last displayed short exact sequence. □

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^{•}) in C_{Λ(G)} 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 canonical 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)} A^{•}) ⟶^{t(A^{•}(ρ^{∗}))} 1_{L}.

Then, since Σ_{A^{•}} ⊂ Σ_{ss−ρ}, Lemma 3.13(v) can be used to show that this construction induces a well-defined homomorphism of groups

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

In particular, (property g) of the functor d_{Λ(G)} combines with relation (1) in the definition of K_{1}(Λ(G), Σ_{A^{•}}) to imply that) the notation [A^{•}, a]^{∗}(ρ) extends in a well-defined fashion to pairs of the form (A^{•}, a) where A^{•} ∈ D^{p}(Λ(G)) is semisimple at ρ and a is a morphism in C_{Λ(G)} of the form 1_{Λ(G)} → d_{Λ(G)}(A^{•}).

If A^{•} is clear from the 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^{•} ∈ D^{p}(Λ(G)) is both semisimple at ρ and such that r_{G}(A^{•})(ρ) = 0 (which is the case, for example, if A^{•} is finite at ρ), then we set [A^{•}, a](ρ) := [A^{•}, a]^{∗}(ρ) and refer to this as the value of [A^{•}, a] at ρ. In particular, after taking account of Remark 3.5, it is clear that this definition coincides with that given in [16, 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 [11].

3.4.1. The canonical Ore sets. We recall from [11, §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) denotes the submodule of M consisting of those elements that are annihilated by some power of p.

3.4.2. Leading terms. In this subsection we use the notation of Definition 3.14 and the isomorphism K_{1}(Λ(G), Σ_{S^{∗}}) ≅ K_{1}(Λ(G)_{S^{∗}}) described at the end of §2.2. 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), Σ_{S^{∗}}) ≅ 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 denotes the image of g in Γ, induces a homomorphism of groups

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 both Σ_{S^{∗}} and 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^{×} which occurs in Proposition 3.8 (and Remark 3.10).

Proof. By Lemma 3.13(i) one has ℍ_{i}(H, A^{•}(ρ^{∗})) = H^{−i}(O ⊗_{Λ_{O}(H)} A^{•}(ρ^{∗})) = H^{−i}(A_{ρ}^{•}) in each degree i. Thus, after taking account of Proposition 3.8 (and Remark 3.10), claim (i) follows directly from Definitions 3.2 and 3.11.

Claim (ii) is proved by the same argument as used in [16, 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. □

For any element F of K_{1}(Λ(G)_{S^{∗}}) we write F^{∗}(ρ) for the leading term (ρ_{∗}(F))^{∗}(0) of F at ρ. 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^{•}}. In a similar way, if r(ρ_{∗}(F)) = 0, then we shall use the notation F (ρ) := F^{∗}(ρ).

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 (in the usual sense) 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}^{×} which has infinite order and set

c_{χ,γ} := log_{p}(χ(γ)) ∈ ℚ_{p}^{×}.

We also fix an object A^{•} of Σ_{S^{∗}} and a morphism a : 1_{Λ(G)} → d_{Λ(G)}(A^{•}) in C_{Λ(G)}, we set L := [A^{•}, a] ∈ K_{1}(Λ(G), Σ_{A^{•}}) and for any continuous representation ρ : G → GL_{n}(O) we define

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

Then, since the zeros and poles of elements of 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 U of 0 in ℤ_{p} one has

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

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

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

Proof. If U is any sufficiently small neighbourhood of 0 in ℤ_{p}, then one has f_{ρχ^{s}}(0) ∈ L^{×} for all s ∈ U \ {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}) = L(ρχ^{s}) = f_{ρχ^{s}}(0) = f_{ρ}(χ(γ)^{s} − 1) = f_{L}(ρχ^{s}) for any s ∈ U \ {0}. This proves claim (i). 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. This proves claim (iii). Also, if r < 0, then (whilst we no longer have the interpretation of the limit as a partial derivative) the same arguments prove the statement concerning the leading coefficient at s = 0 that is made in claim (ii). □

Remark 3.18. Lemma 3.17 is of particular interest in the case that χ is equal to the cyclotomic character of G when the above calculus can be interpreted as partial derivation in the ‘cyclotomic’ direction (cf. Remark 5.6).

3.5. Generalized Euler-Poincaré characteristics. In this subsection we show that the constructions made in §3.3 give rise to a natural extension of certain results from [11, 16, 38]. To do this 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, A^{•}(ρ^{∗})) := H^{i}((ℍ_{−•}(G, A^{•}(ρ^{∗})), 𝔅_{−•})).

We then define the (generalized) additive, respectively multiplicative, Euler-Poincaré 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^{∗}))),

respectively

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

where κ_{L} denotes the residue class field of L. We recall that for a single Λ(G)- module M, or rather its Pontryagin-dual D, similar Euler characteristics have already been studied by several other authors (cf. [12, 42, 18]). 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 second arrow is induced by the identity map on H^{i}(H, D); then the generalized Euler characteristics studied in loc. cit. are defined just as above but by using the complex (H^{•}(G, D), d^{•}) in place of (ℍ_{−•}(G, −), 𝔅_{−•}). However, Lemma 3.13(i) implies that the Pontryagin dual of d^{i} is equal to the Bockstein homomorphism 𝔅_{i+1} : ℍ_{i+1}(G, P^{•}) → ℍ_{i}(G, P^{•}) where P^{•} is a projective resolution of M.

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 so that |p|_{p} = p^{−1}. If A^{•} ∈ Σ_{ss−ρ} and a : 1_{Λ(G)} → d_{Λ(G)}(A^{•}) is any morphism in C_{Λ(G)}, 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. We observe first that by combining Lemma 3.13 with property h) in §2.1 (with R = O) we obtain canonical morphisms

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

After applying L⊗_{O} − to this composite morphism and then identifying all factors in the product expression with 1_{L} by acyclicity we recover the definition of the leading term L^{∗}(ρ) := (A^{•}, a)^{∗}(ρ). On the other hand, if we take the product

over all i of any arbitrarily chosen maps f_{i} : 1_{O} → d_{O}(H_{𝔅}^{i}(G, A^{•}(ρ^{∗})))^{(−1)^{i}}, this will coincide with the above map modulo O^{×}. Thus the product over all i of the maps

\xymatrix{(1_{O})_{L} \ar[r]^{(f_{i})_{L}} & d_{O}(H_{𝔅}^{i}(G, A^{•}(ρ^{∗})))_{L}^{(−1)^{i}} \ar[r]^{acyc} & 1_{L}} ,

which calculate the length of H_{𝔅}^{i}(G, A^{•}(ρ^{∗})) by Remark 2.4, differs from L^{∗}(ρ) only by a unit in O and hence the claimed result follows. □

Remark 3.20. If the complex ℚ_{p} ⊗_{ℤ_{p}} A^{•}(ρ^{∗}) is acyclic, then the leading term L^{∗}(ρ) is equal to the value of L at ρ (in the sense of Remark 3.15). This implies that Proposition 3.19 recovers the results of [11, Thm. 3.6], [38, Prop. 6.3 ] and [16, Rem. 4.1.13].

4. Global Zeta isomorphisms

In this section we recall the non-commutative Tamagawa Number Conjecture that has been formulated by Fukaya and Kato.

4.1. Galois cohomology. The main reference for this section is [16, §1.6], but see also [7]; here we use the same notation as in the survey article [39]. For simplicity we assume throughout this section that p is odd. We fix a finite set S of places of ℚ which contains both S_{p} := {p} and S_{∞} := \{∞\} and let U denote the corresponding dense open subset Spec(ℤ[\frac{1}{S}]) of Spec(ℤ). We fix an algebraic closure ℚ̄ of ℚ and, for each place v of ℚ, an algebraic closure ℚ̄_{v} of ℚ_{v}. We then set G_{ℚ} := Gal(ℚ̄/ℚ) and G_{ℚ_{v}} := Gal(ℚ̄_{v}/ℚ_{v}) and write G_{S} for the Galois group of the maximal extension of ℚ inside ℚ̄ which is unramified outside S. If X is any topological abelian group which is endowed with a continuous action of G_{S}, then we write RΓ(U, X) (RΓ_{c}(U, X)) for global Galois cohomology with restricted ramification (and compact support) and for any place v of ℚ we denote by RΓ(ℚ_{v}, X) the corresponding local Galois cohomology complex. We let L denote a finite extension of ℚ_{p}, we write O for the valuation ring of L and we let V denote a finite dimensional L-vector space which is endowed with a continuous action of G_{ℚ}. Then the ‘finite parts’ of global and local Galois cohomology are written as RΓ_{f}(ℚ, V) and RΓ_{f}(ℚ_{v}, V) respectively, and there exists a canonical exact triangle of the form (12)

\xymatrix{RΓ_{c}(U, V) \ar[r] & RΓ_{f}(ℚ, V) \ar[r] & ⨁_{v∈S} RΓ_{f}(ℚ_{v}, V) \ar[r] & RΓ_{c}(U, V)[1].}

We set t_{p}(V) := D_{dR}(V)/D_{dR}^{0}(V) and also t_{ℓ}(V) := 0 for each prime number ℓ ≠ p. Then, for each prime ℓ, Fukaya and Kato define a canonical morphism in C_{L} of the form

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

For the explicit definition of this morphism we refer the reader either to the original reference [16, §2.4.4] or to the survey article [39, Appendix].

4.2. K-Motives over ℚ. For further background on this (standard) material we refer the reader to either [16, §2.2, 2.4], [7, §3] or [39, §2]. We fix a finite extension K of ℚ and a motive M that is defined over ℚ and has coefficients K. As usual we write M_{B}, M_{dR}, M_{ℓ} and M_{λ} for the Betti, de Rham, ℓ-adic and λ-adic realizations of M, where ℓ ranges over rational primes and λ over non-archimedean places of K. We also let t_{M} denote the tangent space M_{dR}/M_{dR}^{0} of M. For any ring R and R[Gal(ℂ/ℝ)]-module X we denote by X^{+} and X^{−} the R-submodule of X upon which complex conjugation acts as multiplication by +1 and −1 respectively. In our later calculations we will use each of the following isomorphisms:

• The comparison isomorphisms between the Betti and λ-adic realizations of M induce canonical isomorphisms of K_{λ}-modules, respectively K_{ℓ}-modules, of the form

(14) g_{λ}^{+} : K_{λ} ⊗_{K} M_{B}^{+} ≅ M_{λ}^{+}, respectively g_{ℓ}^{+} : K_{ℓ} ⊗_{K} M_{B}^{+} ≅ M_{ℓ}^{+}.

• We set K_{ℝ} := ℝ ⊗_{ℚ} K. Then the comparison isomorphism between the de Rham and Betti realizations of M induces a canonical K_{ℝ}- equivariant period map

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

• For each p-adic place λ of K, the comparison isomorphism between the p-adic and de Rham realizations of M induces a canonical isomorphism of K_{λ}-modules of the form

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

We further recall that the ‘motivic cohomology groups’ H_{f}^{0}(M) := H^{0}(M) and H_{f}^{1}(M) of M are K-modules that can be defined either in terms of algebraic K-theory or motivic cohomology in the sense of Voevodsky (cf. [7]). They are both conjectured to be finite dimensional.

4.3. The Tamagawa Number Conjecture. For each embedding K → ℂ the complex L-function that is associated to a K-motive M is defined (for the real part of s large enough) as an Euler product

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

over all rational primes ℓ. We assume meromorphic continuation of this function and write L_{K}^{∗}(M) ∈ ℂ^{×} and r(M) ∈ ℤ for its leading coefficient and order of vanishing at s = 0 respectively. To establish a link between L_{K}^{∗}(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}.

Indeed, as described in [16, §2.2.7], it is conjectured that archimedean regulators and height pairings combine with the period map α_{M} to induce a canonical morphism in C_{K_{ℝ}} (the ‘period-regulator isomorphism’) of the form

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

In addition, a standard conjecture on cycle class maps and Chern class maps induces, for each non-archimedean place λ of K, a canonical ‘λ-adic periodregulator isomorphism’ in C_{K_{λ}} (which involves the morphism in (13))

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

We now fix a compact p-adic Lie extension F_{∞} of ℚ which is unramified outside S. We set G := Gal(F_{∞}/ℚ) and write Λ(G) for the associated Iwasawa algebra. For any motive M over ℚ we fix a G_{ℚ}-stable full ℤ_{p}-sublattice T_{p} of M_{p} and define a (left) Λ-module by setting

𝕋 := Λ(G) ⊗_{ℤ_{p}} T_{p}

on which Λ(G) acts via left multiplication (on the left hand factor) and each element g of G_{ℚ} acts diagonally via g(x⊗_{ℤ_{p}} y) = xḡ^{−1}⊗_{ℤ_{p}} g(y), where ḡ denotes the image of g in G ⊂ Λ(G). For any non-archimedean place λ of K we write O_{λ} for the valuation ring of K_{λ}. We consider a continuous representation ρ : G → GL_{n}(O_{λ}) of G which, with respect to a suitable choice of basis, is the λ-adic realization N_{λ} of a K-motive N. We continue to denote by ρ 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 the transpose 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}(M(ρ^{∗})) is the O_{λ}-lattice O_{λ}^{n} ⊗ T_{p} of M(ρ^{∗})_{λ}, on which g ∈ G_{ℚ} acts diagonally: g(x ⊗ t) = ρ^{∗}(g)x ⊗ g · t denoting by ρ^{∗} the contragredient representation of ρ.

Conjecture 4.1 (Fukaya and Kato, [16, Conj. 2.3.2]). Set Λ := Λ(G). Then there exists a canonical morphism 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 the composite morphism

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

where

ζ_{K}(M(ρ^{∗})) : 1_{K} → Δ_{K}(M(ρ^{∗}))

denotes the unique morphism which is 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_{∞}(N))_{ℂ}} & 1_{ℂ}.}

Fukaya and Kato refer to the (conjectural) morphism ‘ζ_{Λ}(M)’ in Conjecture 4.1 as a global Zeta isomorphism. We note also that it is straightforward to show that Conjecture 4.1 implies the ‘p-primary component’ of the Equivariant Tamagawa Number Conjecture that is formulated by Flach and the first named author in [7, Conj. 4(iv)] and hence also implies the ‘main conjecture of nonabelian Iwasawa theory’ that is discussed by Huber and Kings in [19]. For a further discussion of Conjecture 4.1 see [39, §4].

5. The interpolation formula for Tate motives

In this section we 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 [35] and discussed by Tate in [37, 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, for example, both [3] and [8]. Throughout this section we set G(F/E) := Gal(F/E) for any Galois extension of fields F/E. We also 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 is both Galois and of finite degree over ℚ, we set Ḡ := G(E/ℚ) and we write S_{p}(E) for the set of p-adic places of E and 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} respectively. For simplicity, we always assume that the following condition is satisfied

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

We note that this condition implies that there is a direct product decomposition G(E_{cyc}/ℚ) ≅ Γ × Ḡ and hence allows us to regard γ as a topological generator of each of the groups Γ, G(E_{cyc}/E), G(E_{w,cyc}/E_{w}) for w ∈ S_{p}(E) and G(ℚ_{p,cyc}/ℚ_{p}). We let 𝕋 denote the (left) Λ(G)-module Λ(G) endowed with the following (left) action of G_{ℚ}: each σ ∈ G_{ℚ} acts on 𝕋 as right 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 (left) 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

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

We regard each character of Ḡ as a character of G via the natural projection G ↠ Ḡ. For any field C we write R_{C}^{+}(\overline{G}) and R_{C}(\overline{G}) for the set of finite dimensional C-valued characters of \overline{G} and for the ring of finite dimensional C-valued virtual characters of \overline{G}, respectively. For each ρ ∈ R_{C}^{+}(\overline{G}) we fix a representation space V_{ρ} of character ρ and for any ℚ_{p}[\overline{G}]-module N, respectively 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,

respectively 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} ℤ_{p} → ∏_{w∈S_{p}(E)} E_{w}^{×} ˆ⊗ℤ_{p}.

If ρ ∈ R_{ℂ_{p}}^{+}(Ḡ), then in the sequel we say that Leopoldt’s Conjecture ‘is valid at ρ’ if one has (ℚ_{p} ⊗_{ℤ_{p}} ker(λ_{p}))^{ρ} = 0. We set c_{γ} := c_{χ_{cyc},γ} ∈ ℚ_{p}^{×} (see §3.4.3) 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

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

(ii) RΓ_{c}(U, 𝕋) is semisimple at ρ and one has 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 homomorphism

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 Bockstein homomorphism in degree −2 of (RΓ_{c}(U, 𝕋), T_{ρ}, γ) is equal to −c_{γ}^{−1} times the homomorphism

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

that is induced by the homomorphism

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

which sends each element (e_{w})_{w} to ∑_{w} 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

(21)  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 ℚ_{p}⊗_{ℤ_{p}}H_{c}^{2}(U, 𝕋_{E}) is canonically isomorphic to ℚ_{p} ⊗_{ℤ_{p}} cok(λ_{p}) (independently of Leopoldt’s Conjecture), whilst the fact that E is totally real implies that the vanishing of (ℚ_{p} ⊗_{ℤ_{p}} H_{c}^{1}(U, 𝕋_{E}))^{ρ} is equivalent to that of (ℚ_{p} ⊗_{ℤ_{p}} ker(λ_{p}))^{ρ}.

To prove claims (ii) and (iii) we note first that, in terms of the notation used in §3.3, the isomorphism (20) (with F = E_{cyc}) induces a canonical isomorphism in D^{p}(Λ_{O}(Γ)) of the form

(22)  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(iv) we may therefore deduce that RΓ_{c}(U, 𝕋) is semisimple at ρ if and only if the complex O^{n} ⊗_{ℤ_{p}[Ḡ]} RΓ_{c}(U, 𝕋_{E_{cyc}}) ∈ D^{p}(Λ_{O}(Γ)) is semisimple. But the latter condition is easy to check 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 (5) 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 (22) combines with Lemma 3.13(iii) to imply that (𝔅_{−2})^{ρ} = (𝔅̂_{−2})^{ρ} where 𝔅̂_{−2} is the Bockstein homomorphism in degree −2 of (RΓ_{c}(U, 𝕋_{E_{cyc}}), ℤ_{p}, γ), with γ regarded as a topological generator of G(E_{cyc}/E). Also, by comparing (21) to the corresponding exact triangle with E_{cyc} in place of E, we obtain a morphism of exact triangles of the form

RΓ(ℚ_{p}, 𝕋_{E_{cyc}}) ⟶^{γ−1} RΓ(ℚ_{p}, 𝕋_{E_{cyc}}) ⟶ RΓ(ℚ_{p}, 𝕋_{E}) ⟶
↓  ↓  ↓
RΓ_{c}(U, 𝕋_{E_{cyc}})[1] ⟶^{γ−1} RΓ_{c}(U, 𝕋_{E_{cyc}})[1] ⟶ RΓ_{c}(U, 𝕋_{E})[1] ⟶ .

Thus, by combining the description of Lemma 3.1 with consideration of the long exact cohomology sequences of this diagram we obtain 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})
(ℚ_{p}⊗_{ℤp} 𝔅_{−1,w})_{w}↓  ↓^{(−1)×(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}).

Here the upper row is the (tautological) surjection that is 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_{−1,w} is the Bockstein homomorphism in degree −1 of (RΓ(E_{w,cyc}, ℤ_{p}(1)), ℤ_{p}, γ) where γ is considered as a topological generator of G(E_{w,cyc}/E_{w}), and the factor −1 occurs on the right hand vertical arrow because of the 1-shift in the lower row of the previous diagram. 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_{−1,w}↓  ↓^{𝔅_{−1,p}}
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_{−1,p} is the Bockstein homomorphism in degree −1 of (RΓ(ℚ_{p,cyc}, ℤ_{p}(1)), ℤ_{p}, γ). To prove claim (iii) it thus suffices to recall that, with respect to the natural 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_{−1,p} is equal to c_{γ}^{−1} ·log_{p} (see, for example, [9, 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. [37, 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 ι : ℂ_{p} ≅ ℂ one has

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

where ω : G_{ℚ} → ℤ_{p}^{×} is the Teichmüller character (cf. [37, Chap. V., Thm. 2.2]). Indeed, this function is the ‘S-truncated p-adic Artin L-function’ of ρ that is constructed by Greenberg in [17] by combining techniques of Brauer induction with the fundamental results of Deligne and Ribet [15] and Cassou-Noguès [10]. For typographical simplicity in the sequel, we fix an isomorphism ι : ℂ_{p} ≅ ℂ as above and hence often omit it from the notation. 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}.

By a standard argument (cf. [14, §6, Exer. 6]) this implies that the ℚ[Ḡ]- modules E_{0} and ℚ ⊗_{ℤ} log_{∞}(O_{E}^{×}) are (non-canonically) isomorphic. We also note that the composite homomorphism

(23) 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 (Serre). 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} → ℚ ⊗_{ℤ} log_{∞}(O_{E}^{×}) 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 [37, Chap. VI, §5], where it is attributed to Serre [35]. More precisely, there are some slight imprecisions in the discussion of [37, Chap. VI, §5] (for example, and as already noted by Solomon in [36, §3.3], the intended meaning of the symbols ‘log U’ and ‘μ_{p}’ in [37, 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 p-adic zeta function of the field E^{J} has a simple pole at s = 1 with residue 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. [37, Rem., p. 138]). From the main result (§5, Thm.) of Colmez in [13] one may thus deduce that Conjecture 5.2 is valid for ρ = IndḠ ; J 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) that is predicted to exist by Conjecture 4.1.

Theorem 5.5. If Conjecture 5.2 is valid, then for each ρ ∈ R_{ℂ_{p}}^{+}(Ḡ) the complex RΓ_{c}(U, 𝕋) is semisimple at ρ and one has both r_{G}(RΓ_{c}(U, 𝕋))(ρ) = ⟨ρ, 1⟩ and

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

Remark 5.6. One can naturally interpret (24) 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 (24) 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 r_{G}(RΓ_{c}(U, 𝕋))(ρ) = ⟨ρ, 1⟩ for each ρ ∈ R_{ℂ_{p}}^{+}(Ḡ) and also that RΓ_{c}(U, 𝕋) is semisimple at each such ρ. 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 the fixed isomorphism ι : ℂ_{p} ≅ ℂ. 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 [6, §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). This implies that

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

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

(25) 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,ι^{−1}} −)^{ρ} to both the natural isomorphism ℝ ⊗_{ℚ} E ≅ ∏_{Hom(E,ℂ)} ℝ and also the exact sequence

(26)  0 → ℝ ⊗_{ℤ} O_{E}^{×} ⟶^{(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

(27)  ∏_{w∈S_{p}(E)} ℚ_{p} ⊗_{ℤ_{p}} U_{E_{w}}^{1} ⟶^{(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} (28)

→ 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) in §2.1 (with R = ℂ_{p}) and the second by Lemma 5.1(i) and the homomorphism −c_{γ}^{−1} log_{p,E} described in Lemma 5.1(iii). Now, after taking account of Lemma 5.1(ii), the leading term ζ_{Λ(G)}(𝕋)^{∗}(ρ) is defined (in Definition 3.14) to be equal to (−1)^{⟨ρ,1⟩} times the element of ℂ_{p}^{×} which corresponds to the composite of (25) and (28). Thus, after noting that there is a commutative diagram of the form

∏_{w∈S_{p}(E)} ℚ_{p} ⊗_{ℤ_{p}} U_{E_{w}}^{1} ⟶ ℚ_{p} ⊗_{ℤ_{p}} cok(λ_{p})
(27)↓  ↓^{log}_{p,E}
ℚ_{p} ⊗_{ℚ} E  ⟶^{Tr_{E/ℚ}}  ℚ_{p}

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

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

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

(30)  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}}.

Here the first arrow is induced by applying ℂ_{p} ⊗_{ℝ,ι^{−1}} − to the isomorphism ℝ⊗_{ℤ}O_{E}^{×} ≅ ℝ⊗_{ℚ}E_{0} coming from the map (log ∘σ)_{σ} in (26) and the second by the isomorphism ℚ_{p} ⊗_{ℤ} O_{E}^{×} ≅ ℚ_{p} ⊗_{ℚ} E_{0} coming from the second and third arrows in (23). (Note also that the factor (−1)^{⟨ρ,1⟩} in the definition of ζ_{Λ(G)}(𝕋)^{∗}(ρ) cancels against the factor −1 in the term −c_{γ}^{−1} which occurs in the morphism (28) and hence does not occur in the formula (29)). But, upon comparing the definitions of μ_{∞} and μ_{p} in §5.2 with the maps involved in (30), one finds that ξ is equal to

det_{ℂ_{p}}((ℂ_{p} ⊗_{ℚ_{p}} μ_{p}) ∘ (ℂ_{p} ⊗_{ℂ,ι^{−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 (29) 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}} .

Documenta Mathematica · Extra Volume Coates (2006) 165–209

The claimed equality (24) 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 finite dimensional ℚ-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-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. [37, 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 (24) 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 (15) 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 Nekovář’s theory of Selmer complexes and of the theory of p-adic height pairings.

6.1. Local epsilon isomorphisms. At the outset we fix a ‘p-adic period’ t (that is, a topological generator of ℤ_{p}(1)). Let L be any finite extension of ℚ_{p} and V any finite-dimensional L-vector space with continuous G_{ℚ_{p}}-action. Then we write ϵ_{p}(V) := ϵ(D_{pst}(V)) for Deligne’s epsilon-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 [16, §3.2] or [29, App. C]. (Note that this notation hides dependence on the choice of a Haar measure and p-adic period. Note also that the choice of t = (t_{n}) ∈ ℤ_{p}(1) determines a homomorphism ψ_{p} : ℚ_{p} → \overline{ℚ_{p}}^{×} with ker(ψ_{p}) = ℤ_{p} by sending \frac{1}{p^{n}} to t_{n} ∈ μ_{p^{n}}). The subfield of inertial invariants (B_{dR})^{I_{p}} of B_{dR} identifies with the completion ℚ̂_{p}^{nr} of the maximal unramified extension ℚ_{p}^{nr} of ℚ_{p} in \overline{ℚ_{p}}. For L and V as above we set L̃ := ℚ̂_{p}^{nr} ⊗_{ℚ_{p}} L and

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). We let

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

denote the morphism that is obtained by taking the product of Γ_{L}(V) with the morphisms η_{ℓ}(V) and \overline{(η_{ℓ}(V^{∗}(1))^{∗})} from (13) and the morphism

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

that is constructed by Fukaya and Kato in [16, Prop. 3.3.5]. We set Λ := Λ(G) and define

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

where U runs over all open normal subgroups of G and W (\overline{𝔽_{p}}) denotes the Witt ring of \overline{𝔽_{p}}. Now we fix a finite-dimensional ℚ_{p}-linear representation V of G_{ℚ_{p}}, a full Galois stable ℤ_{p}-sublattice T of V, set 𝕋 := Λ ⊗_{ℤ_{p}} T and we write O for the valuation ring of L. For any continuous representations ρ : G → GL_{n}(O) we denote by V (ρ^{∗}) the Galois representation ρ^{∗} ⊗ V := O^{n} ⊗_{ℤ_{p}} V, on which G_{ℚ_{p}} acts diagonally, via ρ^{∗} on the first factor. The following conjecture will play a key role in the sequel (for further discussion of this conjecture see [39, Conj. 5.9]).

Conjecture 6.1 (Fukaya and Kato, [16, Conj. 3.4.3]). There exists a canonical morphism in CΛ̃ of the form

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

which is such that for all finite degree extensions L of ℚ_{p}, with valuation ring O, and all continuous representations ρ : G → GL_{n}(O) ⊆ GL_{n}(L) such that V (ρ^{∗}) is de Rham one has

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

6.2. Selmer complexes. We fix a continuous finite-dimensional L-linear representation W of G_{ℚ} which satisfies the following ‘condition of Da̧browski-Panchishkin’: (DP) W is de Rham and there exists 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. [23, 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 we note that Z also satisfies the condition (DP). We now fix a finite set S of places of ℚ which contains both S_{∞} := {∞} and S_{p} := {p} and is such that W (and hence also Z) is a representation of G_{S}, and we set U := Spec(ℤ[\frac{1}{S}]). Then the Selmer complex SC_{U}(Ŵ , W) is defined to be the natural mapping fibre (31)

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

while the modified Selmer complex SC(Ŵ , W) is defined to be the natural mapping fibre (32)

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

where in both cases ℓ runs over all prime numbers that are distinct from p. Also, for each such ℓ, the complex RΓ_{/f}(ℚ_{ℓ}, W) is defined as the natural mapping cone

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

For any G_{ℚ_{p}}-representation V and prime number ℓ we define an element of the polynomial ring 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 of ℓ. Then the following three conditions are easily seen to be equivalent:

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

We also consider the following 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.

We note that (B_{1}) is equivalent to (B_{2}) and that [23, Thm. 1.15] shows that (B_{3}) implies (B_{2}). Finally we consider the following mutually equivalent conditions (to see that (C_{2}) is equivalent to (C_{3}) one uses loc. cit. and the fact that t_{p}(W̃) = t_{p}(Z̃) = 0) :

(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.

Lemma 6.2. Let X denote either W or Z.

(i) If condition (A_{1}) is satisfied, then for every prime ℓ ≠ p all of 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 in D^{p}(L) of

the form

RΓ_{/f}(ℚ_{p}, X) ≅ RΓ(ℚ_{p}, X̃)

and

RΓ_{f}(ℚ_{p}, X) ≅ RΓ(ℚ_{p}, X̂) ≅ RΓ_{f}(ℚ_{p}, X̂).

(iii) If conditions (A_{1}) and (C_{1}) are both satisfied, then there exists an isomorphism in D^{p}(L) of the form

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 claim (i) are then obvious. To prove claim (ii) we assume (C_{1}). Then, since every bounded complex of finitely generated L-modules is canonically isomorphic in D^{b}(L) to its cohomology, considered as a complex with zero differentials, we have RΓ(ℚ_{p}, X̂) ≅ RΓ_{f}(ℚ_{p}, X̂) ≅ RΓ_{f}(ℚ_{p}, X) by [16, 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 respective definitions of SC_{U}(Ŵ , W) and RΓ_{f}(ℚ, W). □

6.3. p-adic height pairings. To prepare for our derivation of the interpolation formula in §6.4 we now discuss certain preliminaries regarding p-adic height pairings. We 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] the corresponding Artin motive. We fix a p-adic place λ of K, set L := K_{λ} and write O for the valuation ring of L. Then the λ-adic realisation

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

of the motive N := M(ρ^{∗}) := M ⊗ [ρ]^{∗} is an L-adic representation. We assume that V (and hence, since [ρ]^{∗} is pure of weight zero, also W) satisfies the condition (DP). We fix a full Galois stable ℤ_{p}-sublattice 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 full G_{ℚ_{p}}- stable ℤ_{p}-sublattice T̂ of V̂ and we define T̃ to be the lattice in Ṽ that is induced from T. Finally we set T̂_{ρ} := T̂ ⊗_{ℤ_{p}} O^{n} and T̃_{ρ} := T̃ ⊗_{ℤ_{p}} O^{n} (which are Galois stable O-sublattices 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}). Indeed, the last three conditions are valid for weight reasons, and more generally, condition (DP) is known to be valid for any motive which has good ordinary reduction at p (see [28]). More precisely, for A (still in the good ordinary case) we have Ŵ = V̂ ⊗ [ρ]_{λ}^{∗} where V̂ = V_{p}(Â^{∨}) denotes the p-adic Tate-module of the formal group of the dual abelian variety A^{∨} of A. However, if, for example, A is an elliptic curve with (split) multiplicative reduction at p, then M does not satisfy the condition (B_{1}).

Now we define a G_{ℚ_{p}}-stable ℤ_{p}-sublattice of V̂ by setting

T̂ := T ∩ V̂ .

As before we let 𝕋 denote the Galois representation Λ⊗_{ℤ_{p}}T and set 𝕋̂ := Λ⊗_{ℤ_{p}} T̂ similarly. Then 𝕋̂ is a G_{ℚ_{p}}-stable Λ-submodule of 𝕋. It is in fact a direct summand of 𝕋 and there exists a morphism in CΛ̃ of the form

(35)  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. Then SC_{U}(X̂, X) coincides with the Selmer complex R̃Γ_{f}(X) that occurs in [24, (11.3.1.5)] for X ∈ {W, Z}. More generally, we set Γ := Gal(ℚ_{cyc}/ℚ) and define

𝕋_{cyc,ρ} := Λ(Γ) ⊗_{ℤ_{p}} T_{ρ}

and similarly also 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}) that is defined in [24, (8.8.5)] (with Nekovář’s local conditions induced by setting T_{ℓ}^{+} := 𝕋̂_{cyc}(ρ) if ℓ = p and T_{ℓ}^{+} := 0 otherwise, and with Nekovář’s set Σ taken to be the set of all rational primes). Thus we obtain a pairing

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

from [24, §11] where h_{p}(W) is denoted h̃_{π,1,1}. Now, by [24, Thm. 11.3.9], the pairing h_{p}(W) coincides up to sign with the height pairings constructed by Schneider [32] (in the case of abelian varieties) and Perrin-Riou [26] (for semistable representations) and also those constructed earlier by Nekovář [23]: see also [loc. cit., §8.1] and the papers of Mazur and Tate [22] and Zarhin [41] for alternative definitions of related height pairings. It follows from the construction of Nekovář’s height pairing (cf. [24, the sentence after (11.1.3.2)]) that the induced map

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

is equal to the composite

(37) 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 homomorphism for SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ}) and the last map comes from global duality.

6.4. The interpolation formula. In this section we assume that the motive N := M(ρ^{∗}) is critical. Then, assuming the conjecture [39, Conj. 3.3] of Fontaine and Perrin-Riou to be valid, the motivic cohomology groups

(D_{1}) H_{f}^{0}(N) = H_{f}^{0}(N^{∗}(1)) = 0 both vanish. In fact, if we also assume the validity of a well-known conjecture [39, Conj. 3.6] on p-adic regulator maps, this last condition is equivalent to the condition

(D_{2}) H_{f}^{0}(ℚ, W) = H_{f}^{0}(ℚ, Z) = 0 where W is defined in (34) and Z := W^{∗}(1). We also consider the condition

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

Example 6.4. If A is an abelian variety over ℚ, then the motive M = h^{1}(A)(1) satisfies the conditions (D_{1}) and (D_{2}). 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 a result of Bertrand [1] that for an elliptic curve with complex multiplication, the height of a point of infinite order is non-zero (but even this is unknown in the non CM case). Computationally, however, there has been a lot of work done recently by Stein and Wuthrich [40]. We are grateful to J. Coates, P. Schneider and C. Wuthrich for providing us with these examples.

We now fix a compact p-adic Lie extension F_{∞} of ℚ which contains ℚ_{cyc} and is unramified outside S. We let G denote the group Gal(F_{∞}/ℚ), with quotient Γ := Gal(ℚ_{cyc}/ℚ), and we set Λ := Λ(G). In [16] Fukaya and Kato use the morphisms ζ_{Λ}(M) and ϵ_{p,Λ}(𝕋̂) that are predicted to exist by Conjecture 4.1 and Conjecture 6.1 to construct canonical ‘p-adic L-function’ morphisms in C_{Λ} of the form

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

and

(39)  L_{β} := L_{β}(M) : 1_{Λ} → d_{Λ}(SC(𝕋̂, 𝕋))

both depending on the isomorphism β in (35). We set SC_{U} := SC_{U}(𝕋̂, 𝕋) and SC := SC(𝕋̂, 𝕋). Then the morphisms L_{U,β} and L_{β} give rise to elements [SC_{U}, L_{U,β}] and [SC, L_{β}] of K_{1}(Λ(G), Σ_{SC_{U}}) and K_{1}(Λ(G), Σ_{SC}) respectively (where we use the notation Σ_{C} introduced at the end of §2.2), and for simplicity we continue to denote these elements by L_{U,β} and L_{β} respectively. We write Υ for the set of all primes ℓ ≠ p with the property that the ramification index of ℓ in F_{∞}/ℚ is infinite. We note that Υ is empty if G has a commutative open subgroup.

Theorem 6.5. We assume that the motive M(ρ^{∗}) is critical, that the representation W defined in (34) satisfies the conditions (DP), (A_{1}), (B_{1}), (C_{1}), (D_{2}) and (F) and that the morphisms ζ_{Λ}(M) and ϵ_{p,Λ}(𝕋̂) that are described in Conjecture 4.1 and Conjecture 6.1 exist. Then both SC_{U}(𝕋̂, 𝕋) and SC(𝕋̂, 𝕋) are semisimple at ρ, one has 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_{β}^{∗}(ρ) (respectively L_{U,β}^{∗}(ρ)) is equal to the product

(40) (−1)^{r} \frac{L_{K,B}^{∗}(M(ρ^{∗}))}{Ω_{∞}(M(ρ^{∗}))R_{∞}(M(ρ^{∗}))} · Ω_{p,β}(M(ρ^{∗}))R_{p}(M(ρ^{∗}))

· Γ_{ℚ_{p}}(V̂)^{−1} · \frac{P_{L,p}(Ŵ^{∗}(1), 1)}{P_{L,p}(Ŵ , 1)},

where L_{K,B}^{∗}(M(ρ^{∗})) denotes the leading term at s = 0 of the B-truncated complex L-function of M(ρ^{∗}) with B := Υ∪S_{p} (respectively B := S \S_{∞}). Further, the regulator terms R_{∞}(M(ρ^{∗})) and R_{p}(M(ρ^{∗})) and period terms Ω_{∞}(M(ρ^{∗})) and Ω_{p,β}(M(ρ^{∗})) that occur in the above formula are as defined in the course of the proof given below.

Remark 6.6. The formulas of Theorem 6.5 represent a natural generalization of the formulas obtained by Perrin-Riou in [29, 4.2.2 and 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 the condition (B_{1}) is not satisfied. Indeed, if condition (B_{1}) fails, then one can have P_{L,p}(W, 0) = 0 and so the order of vanishing at s = 0 of the functions L_{K,B}(M(ρ^{∗}), s) and L_{K}(M(ρ^{∗}), s) may differ. However, to avoid this problem, in formula (40) 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. We first prove all of the assertions concerning SC_{U}(𝕋̂, 𝕋). By [16, 4.1.4(2)] there exists a canonical isomorphism

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

Lemma 3.13 therefore combines with the following result to imply that, under the stated conditions, SC_{U}(𝕋̂, 𝕋) is semisimple at ρ and one has r_{G}(SC_{U}(𝕋̂, 𝕋))(ρ) = dim_{L} H_{f}^{1}(ℚ, W).

Lemma 6.7. We assume that the conditions (A_{1}), (C_{1}) and (D_{2}) are satisfied.

(i) Then SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ}) is semisimple if and only if the condition (F) holds.

(ii) Further, if condition (F) is satisfied, then r_{Γ}(SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ})) =

dim_{L} H_{f}^{1}(ℚ, W).

Proof. By assumption, the condition (D_{2}) can be combined with the isomorphism of Lemma 6.2 (iii) and the global duality isomorphism H_{f}^{3}(ℚ, W) ≅ H_{f}^{0}(ℚ, Z)^{∗} to imply that SC_{U}(Ŵ , W) is acyclic outside degrees 1 and 2. Both claims therefore follow from the fact that the homomorphisms (36) and (37) are known to coincide and that there are canonical isomorphisms L ⊗_{Λ(Γ)} 𝕋_{cyc,ρ} ≅ W, L⊗_{Λ(Γ)}𝕋̂_{cyc,ρ} ≅ Ŵ and thus L⊗_{Λ(Γ)}SC_{U}(𝕋̂_{cyc,ρ}, 𝕋_{cyc,ρ}) ≅ SC_{U}(Ŵ , W). □

We next prove the explicit formula (40) for the leading term L_{U,β}^{∗}(ρ). Our proof of this result is closely modeled on that of [16, Thm. 4.2.26] (as amplified in [39, proof of Thm. 6.4]). At the outset we set N := M(ρ^{∗}), let γ = (γ_{i})_{i} and δ = (δ_{i})_{i} denote a choice of ‘good bases’ (in the sense of [16, 4.2.24(3)]) of M_{B}^{+} and t_{M} for 𝕋̂ and write γ′ and δ′ for the induced K-bases of N_{B}^{+} and t_{N} respectively. Then these choices induce a morphism

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

Furthermore, we 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. Then, letting P^{d} := (P_{1}^{d}, …, P_{d(N)}^{d}) denote the dual basis of P, we obtain a similar morphism

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

Then can := can_{γ′,δ′} · can_{P^{∨},P^{d}} is a morphism (44)

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. We let Ω_{∞}(N) denote the determinant of the canonical isomorphism

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

with respect to the bases γ′ and δ′, and R_{∞}(N) the determinant of the inverse of the canonical isomorphism

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

with respect to the bases P^{d} and P^{∨} respectively. Thus we have morphisms

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^{d}})_{ℂ}} d_{K}(H_{f}^{1}(N))_{ℂ}d_{K}(H_{f}^{1}(N^{∗}(1))^{∗})_{ℂ}^{−1} ⟶^{id·d(h_{∞}(N))^{−1}} 1_{ℂ}

whose product gives

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

Upon comparing this with the leading term

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

we deduce that ζ_{K}(N) : 1_{K} → Δ_{K}(N) is equal to the morphism

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

Before proceeding we recall the relevant descent properties of Selmer complexes.

Lemma 6.8. We use the notation of §6.3.

(i) There exist canonical isomorphisms of the form

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

(ii) There exists an exact triangle of the form

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

Proof. See [16, Prop. 1.6.5 and Prop. 4.2.17]. □

Now, after taking account of Lemma 6.8(i), the leading term L_{U,β}^{∗}(ρ) is defined (in Definition 3.14) to be equal to (−1)^{r} times the morphism

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 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). But Conjecture 4.1 implies that ζ_{Λ}(M)(ρ) is equal to

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

while Conjecture 6.1 implies that

ϵ(𝕋̂)(ρ) = ϵ_{p,L}(Ŵ),

and hence it follows that L_{U,β}^{∗}(ρ) is equal to the product of the following seven terms (47)-(53):

(47)  (−1)^{r} \frac{L_{K}^{∗}(N)}{Ω_{∞}(M(ρ^{∗}))R_{∞}(N)};
(48)  Γ_{L}(Ŵ)^{−1} = Γ_{ℚ_{p}}(V̂)^{−1};

(49) Ω_{p,β}(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 use D_{dR}^{0}(Ŵ) = 0 for the second isomorphism and where we apply Remark 2.2 to regard this as an automorphism of 1L̃;

(50) ∏_{ℓ∈S\{p,∞}} P_{L,ℓ}(W, 1) :

\xymatrix{1_{L} \ar[r]^{∏ η_{ℓ}(W)} & ∏_{ℓ∈S\\{p,∞\}} 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;

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

\xymatrix{1_{L} \ar[r]^{η_{p}(W)·η_{p}(Ŵ)^{−1}} & d_{L}(RΓ_{f}(ℚ_{p}, W))d_{L}(RΓ_{f}(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 described in Lemma 6.2(ii);

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

where we use the fact that t_{p}(Ŵ^{∗}(1)) = D_{dR}^{0}(Ŵ) = 0;

(53) R_{p}(N) : 1_{L} ⟶^{(can_{P^{∨},P^{d}})_{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}

which is equal to the determinant over L of the isomorphism ad(h_{p}(W)) with respect to the chosen bases P^{∨} and P. Indeed, in order to compare L_{U,β}^{∗}(ρ) with the product of the above terms (47)- (53) one just has to verify that after revealing all definitions and identifications, in particular all comparison isomorphisms, the same constituents show up in both expressions (here we rely on Remark 2.2 which implies that all compositions of maps in CL̃ can be interpreted as products and hence are independent of any ordering). Thus we shortly indicate how the constituents of L_{U,β}^{∗}(ρ) give rise to precisely those in the product: As we remarked earlier, ζ_{Λ}(M)(ρ) decomposes up to the comparison isomorphism d(g_{λ}^{+}), which contributes to factor (49), into ζ_{K}(N)_{L} and ϑ_{λ}(N). While ζ_{K}(N)_{L} gives the full factor (47) and contributes with can_{γ,δ} and can_{P^{∨},P^{d}} to the factors (49) and (53), respectively, the second part ϑ_{λ}(N) gives the full factor (50), the half factor (51) in the form of η_{p}(W) and contributes d(g_{dR}^{+}) to factor (49). Further, β(ρ) contributes to factor (49), while according to [16, §3.3] ϵ(𝕋̂)^{−1}(ρ) = ϵ_{p,L}(Ŵ)^{−1} gives the full factors (48) and (52), the other half of (51) in the form of η_{p}(Ŵ)^{−1} and adds ϵ_{dR}(Ŵ) to factor (49). Finally, we had observed at the end of §6.3 that t(SC_{U}(ρ^{∗})) is equal to h_{p}(W). Since L_{U,β}^{∗}(ρ) is equal to the product of the terms (47)-(53), it is therefore enough to show that the product of these terms is also equal to the explicit product expression in (40). But this follows immediately by a direct comparison of the maps involved and then using the fact that

L_{K,B}^{∗}(N) = L_{K}^{∗}(N) · ∏_{ℓ∈S\S_{∞}} P_{L,ℓ}(W, 1) · P_{L,p}(Ŵ , 1)^{−1} · P_{L,p}(Ŵ^{∗}(1), 1).

At this stage we have proved all of the claims in Theorem 6.5 concerning SC_{U}(𝕋̂, 𝕋) and so it only remains to prove the analogous claims for the complex SC(𝕋̂, 𝕋). But these claims can be proved easily by combining the above argument with consideration of the exact triangle

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

(which itself results from comparing the defining exact triangles (31) and (32) firstly with each other and then with the exact triangle in Lemma 6.8(ii)) and the equality

L_{K,Υ′}^{∗}(N) = L_{K,B′}^{∗}(N) ∏_{ℓ∈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. Set M := h^{1}(E)(1) and F_{∞} := ℚ(E(p)) where E(p) denotes the p-power torsion subgroup of E(\overline{ℚ}). Then it is conjectured that SC_{U}(𝕋̂, 𝕋) always belongs to Σ_{S^{∗}} (cf. [11, Conj. 5.1] and [16, 4.3.5 and Prop. 4.3.7]). Further, as was shown in [16], the existence of a morphism L_{β}(M) as in (39) implies the existence of the element L_{E} of K_{1}(Λ(G)_{S^{∗}}) that [11, Conj. 5.7] predicts to exist with a precise interpolation property for all Artin representations ρ such that r_{G}(SC(𝕋̂, 𝕋))(ρ) = 0. More generally, the formula (40) now gives a precise interpolation property for (the leading term of) the element L_{E} at all Artin representations at which the underlying archimedean and p-adic height pairings are non-degenerate.

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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 210

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