On main conjectures in

non-commutative Iwasawa theory

and related conjectures

David Burns

Abstract. The main conjecture of non-commutative Iwasawa theory is shown to be equivalent, modulo an appropriate torsion hypothesis, to a family of main conjectures over cyclotomic ℤ_{p}-extensions together with suitable compatibility relations between the associated abelian p-adic L-functions. This result combines with techniques of Ritter and Weiss and of Kakde to give a concrete strategy for deducing main conjectures for (non-abelian) compact p-adic Lie extensions of arbitrarily large rank from main conjectures in the classical sense. As a first application of this approach we combine it with a recent result of Ritter and Weiss to prove, modulo an appropriate vanishing hypothesis on μ-invariants, the main conjecture of non-commutative Iwasawa theory for Tate motives over compact p-adic Lie extensions of totally real number fields of arbitrarily large rank. We then also deduce several interesting consequences of this result concerning a range of related conjectures including special cases of the equivariant Tamagawa number conjecture.

1991 Mathematics Subject Classification: Primary 11S40; Secondary 11R29 20C05

Introduction

The ‘main conjecture of non-commutative Iwasawa theory’ was formulated by Coates, Fukaya, Kato, Sujatha and Venjakob in [19] and has excited much interest. For a rational prime p and any suitable motive M defined over a number field k, the conjecture is formulated for any compact p-adic Lie extension K of k that contains the cyclotomic ℤ_{p}-extension k^{∞} of k and is such that Gal(K/k) has no element of order p (for more details see §1.3.1). Even though for this conjecture the group Gal(K/k) can have arbitrarily large (finite) rank as a p-adic Lie group, in this article we shall prove in Theorem 2.2 and Corollary 2.3 that if one assumes, as in [19], that the relevant Selmer group is annihilated by a certain Ore localisation of the Iwasawa algebra of Gal(K/k), then the central conjecture of [19] is valid if and only if for every finite extension F of k in K the main conjecture for the extension F^{∞}/F is valid for a certain family of Dirichlet twists of M over F and, in addition, the associated abelian p-adic L-functions satisfy an appropriate compatibility condition as F and the Dirichlet twists vary within this family (cf. Remark 2.4). Whilst such a reduction is perhaps by itself quite striking our approach also combines with recent results of Ritter and Weiss and of Kakde to give a concrete strategy for deducing main conjectures of non-commutative Iwasawa theory for extensions of arbitrarily large rank from main conjectures in the classical sense. This strategy is described in Remark 2.5 and, with future applications in mind, we take care to ensure that it applies to main conjectures that involve Selmer groups that are only ‘S^{∗}-torsion’ rather than ‘S-torsion’ in the terminology of [19] (which is in general necessary in the context of abelian varieties) and also use interpolation conditions related to the leading terms rather than values of elements of Whitehead groups at Artin representations (which is necessary to make descent computations). As a first explicit application of our methods we combine them with results of Wiles and of Ritter and Weiss to prove, modulo the vanishing of appropriate μ-invariants, the main conjecture for Tate motives over compact p-adic Lie extensions of totally real fields of arbitrarily large rank. When combined with the descent formalism of Venjakob and the present author this result in turn gives new evidence in support of the equivariant Tamagawa number conjecture and hence, via the results of [8] and [9], in support of a wide range of more explicit conjectures due to Chinburg, to Chinburg, Kolster, Pappas and Snaith, to Breuning and the present author and to the present author (see, for example, Corollaries 2.8 and 2.10 and Remark 2.9). Further arithmetic applications of the approach developed here will be given elsewhere including proofs of natural non-commutative generalisations of the main conjectures of geometric Iwasawa theory that have been proved by Crew in [20] and which follow as a consequence of the result of Emerton and Kisin in [22, Cor. 1.8]. In the course of obtaining the main results of this article we shall also prove several algebraic results which are of independent interest including natural generalisations of the Weierstrass Preparation Theorem in non-commutative Iwasawa theory of Venjakob (see Corollary 5.5) and of K-theoretical results of Kato, of Hara, of Kakde (see Remark 6.4) and of Ritter and Weiss and we also give a more elementary proof of the main K-theoretical result of Ritter and Weiss in [36] (see Remark 6.2). The main contents of this article is as follows. In §1 we axiomatise the formulation of main conjectures in non-commutative Iwasawa theory and in §2 we state our main results. In §3 we use reduced norms to reinterpret the values and leading terms of elements of Whitehead groups at Artin representations. In §4 we prove a general reduction result (Theorem 2.1) relating main conjectures for non-abelian compact padic Lie groups of rank greater than one to main conjectures for groups of rank one. In §5 we develop techniques to deal with non-abelian groups of rank one and use them to prove a ‘Weierstrass Preparation Theorem’ for the Whitehead groups of the canonical Ore localisation of Iwasawa algebras of pro-p compact p-adic Lie groups. In §6 we use the results of §5 to prove a precise relationship between the Whitehead group of the localised Iwasawa algebra of a compact p-adic Lie group G and the corresponding Whitehead groups for a certain family of rank one abelian subquotients of G (it is this result that underlies our general strategy for investigating the main conjectures of non-commutative Iwasawa theory). In §7 we combine the results of §4, §5 and §6 to prove several consequences of Theorem 2.1 including Theorem 2.2 and Corollary 2.3. In §8 we use techniques of Ritter and Weiss to reinterpret the interpolation conditions that occur in Theorem 2.2 and Corollary 2.3 in terms of the existence of suitable elements of the centre of the total quotient ring of Iwasawa algebras for groups of rank one. Finally in §9 we combine Theorem 2.2 with results of Wiles and of Ritter and Weiss to prove the main conjecture for Tate motives over compact p-adic Lie extensions of totally real fields (or arbitrarily large rank) and we then deduce several interesting consequences of this result. With the exception of the arithmetic results discussed in §2.3 and §9 this article is a long overdue update of a preprint first circulated in 2007. The general approach described here was presented in a seminar at the University of Kyoto in early 2006 and the author is extremely grateful to Kazuya Kato for his invitation to visit the University of Kyoto, for his interest and generous encouragement and for several very insightful remarks. He is also grateful to Jan Nekovář and Otmar Venjakob for many stimulating discussions and to the referee for numerous helpful comments. Some of this article was written when the author held a Visiting Professorship at the University of Paris 6 in June 2007 and he is very grateful to the Institute of Mathematics for this wonderful opportunity. The results described in §2.3 and §9 were obtained just after Ritter and Weiss made [37] available and shortly after we circulated (in April 2010) a version of this article including these arithmetic results Kakde obtained a more elegant proof of Theorem 9.1(ii). (Kakde’s proof is now given in [29] and uses our Theorem 2.1 as well as a more explicit variant of Theorem 6.1 and the general approach that we introduce in Remark 2.5 but does not rely on the main result of [37].)

1. Main Conjectures

In this section we axiomatise the formulation of main conjectures in non-commutative Iwasawa theory (in the spirit of [19]).

1.1. Preliminaries and notation.

1.1.1. Groups. We consider triples of the form

(1)  \underline{G} := (G, π\underline{G}, γ\underline{G})

where G is a compact p-adic Lie group, π\underline{G} : G → Γ(\underline{G}) is a continuous surjective homomorphism to a group that is topologically isomorphic to ℤ_{p} and γ\underline{G} is a topological generator of Γ(\underline{G}). We write H(\underline{G}) for the closed normal subgroup ker(π\underline{G}) of G. When \underline{G} is clear from context we usually abbreviate π\underline{G}, Γ(\underline{G}), γ\underline{G} and H(\underline{G}) to π_{G}, Γ(G), γ_{G} and H(G) (or even π, Γ, γ and H) respectively. For any Galois extension of fields F/E we set G_{F/E} := Gal(F/E).

1.1.2. Iwasawa algebras and K-theory. We fix an algebraic closure ℚ_{p}^{c} of ℚ_{p}. If O is the valuation ring of a finite extension of ℚ_{p} in ℚ_{p}^{c} and G denotes either G, H or Γ, then we write Λ_{O}(G) for the O-Iwasawa algebra of G and Q_{O}(G) for the total quotient ring of Λ_{O}(G) (if O = ℤ_{p}, then we omit the subscripts). We write S and S^{∗} for the multiplicatively closed left and right Ore sets of non-zero divisors of Λ_{O}(G) introduced in [19, §2] and Λ_{O}(G)_{S} and Λ_{O}(G)_{S^{∗}} for the corresponding localisations of Λ_{O}(G) (note that this notation does not explicitly indicate that S and S^{∗} depend upon both O and G). We recall that if G has rank one, then Λ_{O}(G)_{S^{∗}} = Q_{O}(G). In the remainder of §1.1 we let S′ denote either S or S^{∗}. We write 𝔐_{S′}(Λ_{O}(G)) for the category of finitely generated left Λ_{O}(G)-modules M with the property that Λ_{O}(G)_{S′} ⊗_{Λ_{O}(G)} M vanishes. We also write K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}) for the relative algebraic K_{0}-group of the canonical homomorphism Λ_{O}(G) → Λ_{O}(G)_{S′} and recall that there is a canonical exact sequence of abelian groups of the form

(2)  K_{1}(Λ_{O}(G))⟶^{α_{O,G,S′}}K_{1}(Λ_{O}(G)_{S′})⟶^{∂_{O,G,S′}}K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}).

In the sequel we often write α_{G,S′} and ∂_{G,S′} (or even α_{G} and ∂_{G}) in place of α_{O,G,S′} and ∂_{O,G,S′}, believing that O and S′ will be clear from context. We recall that if G has no element of order p, then the homomorphism ∂_{O,G,S′} is surjective by Coates et al [19, Prop. 3.4] (and see also Remark 1.1 below in this regard) and the group K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}) is naturally isomorphic to the Grothendieck group G_{0}(𝔐_{S′}(Λ_{O}(G))) of the category 𝔐_{S′}(Λ_{O}(G)). One can find an explicit description of the latter isomorphism in, for example, [15, §1.2]. We let D^{p}(Λ_{O}(G)) denote the derived category of perfect complexes of left Λ_{O}(G)- modules and write D_{S′}^{p}(Λ_{O}(G)) for the full subcategory of D^{p}(Λ_{O}(G)) comprising complexes C^{•} such that Λ_{O}(G)_{S′} ⊗_{Λ_{O}(G)}^{𝕃} C^{•} is acyclic (or equivalently, such that H^{i}(C^{•}) belongs to 𝔐_{S′}(Λ_{O}(G)) for each degree i). Each object C^{•} in D_{S′}^{p}(Λ_{O}(G)) has a natural ‘refined’ Euler characteristic χ_{O,G}^{ref}(C^{•}) in K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}) and we use the same conventions in this regard as in [15, §1.4] (where χ_{O,G}^{ref}(C^{•}) is written as χ(C^{•})). In fact, since O and G are usually clear from context we often abbreviate χ_{O,G}^{ref}(C^{•}) to χ^{ref}(C^{•}).

Remark 1.1. The homomorphism ∂_{O,G,S′} in (2) has also recently been shown to be surjective in the case that G has an element of order p. For S′ = S this is proved by Kakde in [29] by using the same approach as in [19, Prop. 3.4]; with S′ equal to either S or S^{∗}, it is proved by Witte in the preprint [44] by using the methods of Waldhausen K-theory. (We are grateful to the referee for bringing these results of Witte to our attention.)

1.1.3. Leading terms and values. If O′ is the valuation ring of a finite extension of ℚ_{p} in ℚ_{p}^{c} and O ⊆ O′, then each continuous representation ρ : G → GL_{n}(O′) gives rise to a ring homomorphism

Λ_{O}(G)_{S′} → M_{n}(O′) ⊗_{O′} Q_{O′}(Γ) ≅ M_{n}(Q_{O′}(Γ))

that sends every element g of G to ρ(g)⊗π\underline{G}(g). This homomorphism in turn induces a group homomorphism

(3) Φ_{O,\underline{G},S′,ρ} : K_{1}(Λ_{O}(G)_{S′}) → K_{1}(M_{n}(Q_{O′}(Γ)))

≅ K_{1}(Q_{O′}(Γ)) ≅ Q_{O′}(Γ)^{×} ≅ Q(O′[[T]])^{×}

where the first isomorphism is induced by Morita equivalence, the second by taking determinants (over Q_{O′}(Γ)) and the third by the identification γ − 1 = T. We often abbreviate Φ_{O,\underline{G},S′,ρ} to Φ_{\underline{G},ρ}, believing that O and S′ will be clear from context. The ‘value’ ξ\underline{G}(ρ), resp. ‘leading term’ ξ∗ ; \underline{G}(ρ), at ρ of an element ξ of K_{1}(Λ_{O}(G)_{S′}) is then defined to be the value, resp. leading term, at T = 0 of Φ_{O,\underline{G},S′,ρ}(ξ). In particular, ξ\underline{G}(ρ) is equal to ‘∞’ if the algebraic order of Φ_{O,\underline{G},S′,ρ}(ξ) at T = 0 is strictly negative. These constructions apply to any element ρ of the set A(G) of Artin representations of G. (For any p-adic Lie group H a representation H → GL_{n}(ℚ_{p}^{c}) is said to be an ‘Artin representation’ if it is continuous and has finite image.) For each ρ in A(G) we write ρ̌ for the corresponding contragredient representation and note that ρ̌ ∈ A(G). For any p-adic Lie group H we write H^{∨} for the subset of A(H) comprising all (continuous open kernel) homomorphisms H → ℚ_{p}^{c×}.

Remark 1.2. The above definitions imply that for all ξ in K_{1}(Λ_{O}(G)_{S′}) and all

ρ and ρ′ in A(G) one has ξ\underline{G}(ρ + ρ′) = ξ\underline{G}(ρ)ξ\underline{G}(ρ′) and ξ∗ ; \underline{G}(ρ + ρ′) = ξ∗ ; \underline{G}(ρ)ξ∗ ; \underline{G}(ρ′)

and if ρ is isomorphic to ρ′ (which will be written ρ ≅ ρ′) also ξ\underline{G}(ρ) = ξ\underline{G}(ρ′) and ξ∗ ; \underline{G}(ρ) = ξ∗ ; \underline{G}(ρ′), where any equalities involving ‘∞’ are to be interpreted in the obvious way. This allows one to extend the definition of the value and leading term of ξ to isomorphism classes of virtual Artin representations.

1.2. Abstract main conjectures.

1.2.1. Interpolation data. With Remark 1.2 in mind we introduce suitable notions of abstract interpolation data. We assume to be given for each ρ ∈ A(G) elements ξ_{ρ} and ξ_{ρ}^{∗} of ℚ_{p}^{c}. We say that the collection (ξ_{ρ})_{ρ} is ‘interpolation data’ if the following conditions are satisfied:

• ξ_{ρ} = ξ_{ρ′} if ρ is isomorphic to ρ′;

• ξ_{ρ+ρ′} = ξ_{ρ}ξ_{ρ′} for all ρ and ρ′ in A(G);
• for all ρ ∈ A(G) one has ξ_{ρ⊗ω} ≠ 0 for almost all ω ∈ Γ^{∨} ⊂ A(G).

We say that the collection (ξ_{ρ}^{∗})_{ρ} is ‘leading term interpolation data’ if the following conditions are satisfied:

• ξ_{ρ}^{∗} = ξ_{ρ′}^{∗} if ρ is isomorphic to ρ′;

• ξ_{ρ+ρ′}^{∗} = ξ_{ρ}^{∗}ξ_{ρ′}^{∗} for all ρ and ρ′ in A(G);

• ξ_{ρ}^{∗} ≠ 0 for all ρ ∈ A(G).

1.2.2. The abstract formulation. For any set of interpolation data (ξ_{ρ})_{ρ} and any complex C^{•} in D_{S^{∗}}^{p}(Λ_{O}(G)) we can now formulate an abstract ‘main conjecture of noncommutative Iwasawa theory’ as follows.

MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}): There exists an element ξ of K_{1}(Λ_{O}(G)_{S^{∗}}) such that both (i) for all ρ ∈ A(G) one has ξ\underline{G}(ρ ⊗ ω) = ξ_{ρ⊗ω} for almost all ω ∈ Γ^{∨}, and

(ii) ∂_{O,G,S^{∗}}(ξ) = χ_{O,G}^{ref}(C^{•}).

The results of [15] show that it is also important to consider (not least for the purposes of descent computations) a variant of this statement. To state this variant we fix a complex C^{•} as above and a set of leading term interpolation data (ξ_{ρ}^{∗})_{ρ}.

MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}): There exists an element ξ of K_{1}(Λ_{O}(G)_{S^{∗}}) such that both (i) for all ρ ∈ A(G) one has ξ∗ ; \underline{G}(ρ) = ξ_{ρ}^{∗}, and

(ii) ∂_{O,G,S^{∗}}(ξ) = χ_{O,G}^{ref}(C^{•}).

For an explicit comparison of the interpolation conditions in these statements see §3.2. For simplicity we usually abbreviate MC(\underline{G}, ℤ_{p}, C^{•}, (ξ_{ρ})_{ρ}) and MC^{∗}(\underline{G}, ℤ_{p}, C^{•}, (ξ_{ρ}^{∗})_{ρ}) to MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) and MC^{∗}(\underline{G}, C^{•}, (ξ_{ρ}^{∗})_{ρ}) respectively.

Remark 1.3. In neither MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}) or MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}) do we assert that the element ξ of K_{1}(Λ_{O}(G)_{S^{∗}}) is unique. (See however Remark 4.4 for more in this regard.)

1.3. Arithmetic examples. We describe some important arithmetic applications of the formalism of §1.2.2.

1.3.1. Elliptic curves. Let E be an elliptic curve that is defined over ℚ and does not have complex multiplication and p a prime, with p ≥ 5, at which E has good ordinary reduction. Then the Galois extension K of ℚ generated by all p-power torsion points of E contains the cyclotomic ℤ_{p}-extension ℚ^{∞} of ℚ and G_{K/ℚ} is isomorphic to an open subgroup of GL_{2}(ℤ_{p}) that has no element of order p. Hence there exists data \underline{G} as in (1) with G := G_{K/ℚ} and π\underline{G} the restriction map G → G_{ℚ^{∞}/ℚ}. With X(E/K) denoting the Pontryagin dual of the Selmer module of E over K, Coates et al [19, Conj. 5.1] conjecture that X(E/K) belongs to 𝔐_{S^{∗}}(Λ(G)), or equivalently that C^{•} := X(E/K)[−2] belongs to D_{S^{∗}}^{p}(Λ(G)). Then, with O a suitable unramified extension of ℤ_{p} and interpolation data ξ_{ρ} specified by a formula involving the value at s = 1 of the Hasse-Weil L-function L(E, ρ̌, s) of the ‘ρ̌-twist’ of E, the assertion MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}) recovers the ‘main conjecture of non-commutative Iwasawa theory’ of [19, Conj. 5.8]. The refinement of [19, Conj. 5.8] that is formulated by Venjakob and the present author in [15, Conj. 7.4] is also equivalent to the assertion MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}) where C^{•} is constructed from the Selmer complex of the p-adic realisation of a suitable critical motive M and each term ξ_{ρ}^{∗} is specified by an explicit formula involving the leading term at s = 1 of the complex L-function of the ‘ρ̌-twist’ of M.

1.3.2. Tate motives. Let k be a totally real number field and K a totally real compact p-adic Lie extension of k that is unramified outside a finite set of places Σ of k and contains the cyclotomic ℤ_{p}-extension k^{∞} of k (which implies, in particular, that Σ contains all places of k above p). Let \underline{G} denote the triple (G, π\underline{G}, γ\underline{G}) where G := G_{K/k}, π_{G} is the natural restriction map G → G_{k^{∞}/k} and γ\underline{G} is a choice of topological generator of G_{k^{∞}/k}. In §9 we shall study an analogue of [19, Conj. 5.8] concerning the Galois group X_{Σ}(K) of the maximal abelian pro-p-extension of K that is unramified outside Σ and with interpolation data specified in terms of the leading terms at s = 1 of suitable p-adic Artin L-functions. This is equivalent to the conjecture considered by Kato [31], by Hara [26, 27], by Kakde [28] and by Venjakob and the present author [15] and can be phrased in the form MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}).

Remark 1.4. If p is odd and K/k is any compact p-adic Lie extension of number fields that is ramified at only finitely many places, then there exists a compact p-adic Lie extension K′ of k which contains K, is ramified at only finitely many places, is totally real if K is totally real and is such that G_{K′/k} has no element of order p (see [15, Lem. 6.1]). This means that for the examples discussed in both §1.3.1 and §1.3.2 it is in principle enough to study main conjectures under the simplifying assumption that G has no element of order p.

1.3.3. Geometric analogues. One can also apply the formalism of §1.2.2 in the setting of geometric Iwasawa theory to formulate natural non-commutative generalisations of the main conjectures that are proved by Crew in [20] and which follow as a consequence of the result of Emerton and Kisin in [22, Cor. 1.8]. We use the approach of this article to give a proof of these generalisations in [10].

2. Statement of the main results

2.1. Further preliminaries and notation. Before stating the main results of this article we quickly review some necessary background material. Throughout we use S′ to denote either S or S^{∗}.

2.1.1. Rank one subquotients. With \underline{G} as in (1) we write SQ_{1}(G) for the set of rank one subquotients of G that are of the form G = U/J with U an open subgroup of G and J an open subgroup of H that is contained, and normal, in U. (Strictly speaking, we should define SQ_{1}(G) to be the set of all such pairs (U, J) but, for notational simplicity, we have preferred not to do this.) For each such subquotient we define a triple \underline{G} := (G, π\underline{G}, γ\underline{G}) as in (1) in the following way: π\underline{G} is the natural projection G → HU/H =: Γ(\underline{G}) (so H(\underline{G}) = (H ∩ U)/J is finite) and γ\underline{G} := γd_{G} ; \underline{G} where we write d_{G} for the index of HU in G. We set T_{G} := (1 + T)^{d_{G}} − 1. Then for each finite extension O of ℤ_{p} in ℚ_{p}^{c} and each Artin representation ψ : G → GL_{n}(O′) with O ⊆ O′, the correspondence γ^{d_{G}} −1 = T_{G} gives rise to a homomorphism

Φ_{O,\underline{G},ψ} : K_{1}(Q_{O}(G)) = K_{1}(Λ_{O}(G)_{S^{∗}}) → Q(O′[[T_{G}]])^{×}

as in (3). (When O is clear from context we often abbreviate Φ_{O,\underline{G},ψ} to Φ_{\underline{G},ψ}.) In particular, for each element x of K_{1}(Q_{O}(G)) there is a unique integer r_{ψ}(x) such that Φ_{O,\underline{G},ψ}(x) = T_{G}^{r_{ψ}(x)}G_{x}(T_{G}) with G_{x}(T_{G}) ∈ Q(O′[[T_{G}]])^{×} and G_{x}(0) ∈ ℚ_{p}^{c} \ \{0\}. This integer r_{ψ}(x) depends only upon ψ and ∂_{O,G,S^{∗}}(x). Hence, after recalling Remark 1.1, for any object C^{•} of D_{S^{∗}}^{p}(Λ_{O}(G)) we can set r_{ψ}(C^{•}) := r_{ψ}(x) for any pre-image x of χ_{O,G}^{ref}(C^{•}) under ∂_{O,G,S^{∗}}. For G = U/J in SQ_{1}(G) we use I_{G}^{G} (rather than the more precise notation I_{U,J}^{G}) to denote the composite map

I_{G}^{G} : A(G) → A(U) → A(G)

where the first arrow is inflation and the second is induction. Further, if G is a quotient of G, then we usually regard A(G) as a subset of A(G) and so omit explicit reference to the inflation map I_{G}^{G}. We also write

q_{O,G,G,S′} : K_{1}(Λ_{O}(G)_{S′}) → K_{1}(Λ_{O}(U)_{S′}) → K_{1}(Λ_{O}(G)_{S′}),

or more simply q_{G} whenever O, G and S′ are clear from context, for the composite of the natural restriction and projection homomorphisms. For each C^{•} in D^{p}(Λ_{O}(G)) we write res_{U}^{G}C^{•} for the object of D^{p}(Λ_{O}(U)) obtained by restricting scalars and regard C_{G}^{•} := Λ_{O}(G)⊗_{Λ_{O}(U)}^{𝕃} res_{U}^{G}C^{•} as an object of D^{p}(Λ_{O}(G)) in the natural way. We use the commutative diagram

(4)  \begin{matrix}K_{1}(Λ_{O}(G)_{S′}) & ⟶^{∂_{O,G,S′}} & K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}) \\ ^{q_{O,G,G,S′}} ↓ & & ↓ \\ K_{1}(Λ_{O}(G)_{S′}) & ⟶^{∂_{O,G,S′}} & K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}). \end{matrix}

Here the right hand vertical arrow denotes the homomorphism induced by the natural commutative diagram of scalar extensions

Λ_{O}(G) ⟶ Λ_{O}(G)_{S′}
↓  ↓
Λ_{O}(G) ⟶ Λ_{O}(G)_{S′}

and therefore sends each element of the form χ_{O,G}^{ref}(C^{•}) to χ_{O,G}^{ref}(C_{G}^{•}).

2.1.2. Abelian subquotients. We write SQ_{1}^{ab}(G) for the subset of SQ_{1}(G) comprising subquotients A = U/J that are abelian. For each such A the group H(A) is finite and Γ(A) is torsion-free and so we may fix an isomorphism of topological groups of the form

(5)  A ≅ H(A) × Γ(A).

It is important to note that, unless H(A) is trivial, such an isomorphism is not canonical and many of the constructions and results that are to be discussed in the sequel will depend upon the choices made here. Via the isomorphism (5) we identify Λ_{O}(A) with the group ring Λ_{O}(Γ(A))[H(A)], and hence Λ_{O}(A)_{S′} with Λ_{O}(Γ(A))_{S′}[H(A)], and also regard H(A)^{∨} as a subset of A^{∨}. For each ϕ ∈ H(A)^{∨} we write O[ϕ] for the extension of O generated by {ϕ(h) : h ∈ H(A)} and note that the ring homomorphism Q_{O}(A) = Q_{O}(Γ(A))[H(A)] → Q_{O[ϕ]}(Γ(A)) sending each ∑_{h∈H(A)} x_{h}h to ∑_{h∈H(A)} x_{h}ϕ(h) induces a homomorphism of groups

q_{O,A,ϕ} : K_{1}(Q_{O}(A)) ≅ Q_{O}(A)^{×} → Q_{O[ϕ]}(Γ(A))^{×}.

(When O is clear from context we often abbreviate q_{O,A,ϕ} to q_{A,ϕ}.) It is easy to check that if one identifies Q_{O[ϕ]}(Γ(A))^{×} with Q(O[ϕ][[T_{A}]])^{×} via γ^{d_{A}} − 1 = T_{A}, then q_{O,A,ϕ} coincides with the homomorphism Φ_{O,\underline{A},ϕ} defined in §2.1.1. For each ϕ ∈ H(A)^{∨} we define a right action of Λ_{O}(A) on O[ϕ] by setting (x)a := xϕ(a)^{−1} for all x ∈ O[ϕ] and a ∈ A. Via this action we obtain for each C^{•} in D_{S′}^{p}(Λ_{O}(A)) an object of D_{S′}^{p}(Λ_{O[ϕ]}(Γ(A))) by setting

C_{ϕ}^{•} := O[ϕ] ⊗_{O[H(A)]}^{𝕃} C^{•}.

2.1.3. Generalised μ-invariants. For each A ∈ SQ_{1}^{ab}(G) there is a canonical direct sum decomposition of finite abelian groups H(A) = H(A)_{p} ×H(A)_{p′} with H(A)_{p} the Sylow p-subgroup of H(A) and H(A)_{p′} of order prime to p. For each homomorphism ϕ ∈ H(A)^{∨} we write ϕ_{(p′)} for the element of H(A)^{∨} that agrees with ϕ on H(A)_{p′} and is trivial on H(A)_{p}. If C^{•} belongs to D_{S^{∗}}^{p}(Λ(A)), then for each ϕ ∈ H(A)^{∨} and each integer j the cohomology group H^{j}(C_{ϕ_{(p′)}}^{•}) is a finitely generated torsion Λ(Γ(A))-module. Since H^{j}(C_{ϕ_{(p′)}}^{•}) vanishes for almost all j we may therefore define

μ_{ϕ}(C^{•}) := rank_{ℤ_{p}}(ℤ_{p}[ϕ_{(p′)}])^{−1} ∑_{j∈ℤ}(−1)^{j}μ_{Γ(A)}(H^{j}(C_{ϕ_{(p′)}}^{•}))

where we write μ_{Γ(A)}(N) for the μ-invariant of a finitely generated Λ(Γ(A))-module N. The proof of Lemma 7.1(iii) below will show that each such sum μ_{ϕ}(C^{•}) is in fact an integer.

2.2. The main algebraic results. To state the main algebraic results of this article we fix a complex C^{•} in D_{S^{∗}}^{p}(Λ_{O}(G)) and sets (ξ_{ρ})_{ρ} and (ξ_{ρ}^{∗})_{ρ} of interpolation and leading term interpolation data respectively. The first result we state is proved in §4 and shows that main conjectures for any compact p-adic Lie group G (of arbitrary finite rank) as in (1) are equivalent to a family of main conjectures for groups of rank one. To state this result we write Q_{1}(G) for the set of quotients of G of the form G = G/U with U an open subgroup of H that is normal in G (so Γ(G) = Γ and A(G) ⊆ A(G)).

Theorem 2.1.

(i) MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}) is valid if and only if MC(\underline{G}, O, C_{G}^{•}, (ξ_{ρ})_{ρ}) is valid for all

G in Q_{1}(G).

(ii) MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}) is valid if and only if MC^{∗}(\underline{G}, O, C_{G}^{•}, (ξ_{ρ}^{∗})_{ρ}) is valid for

all G in Q_{1}(G).

The following consequence of Theorem 2.1 will be proved in §7 and shows that main conjectures for groups of arbitrary (finite) rank are equivalent to the validity of main conjectures for a family of ℤ_{p}-extensions together with the existence of suitable ‘analytic p-adic L-functions’ for a family of rank one extensions. To state this result we define an Ore set

S̃ := \begin{cases}S, & if G has an element of order p \\ S^{∗}, & otherwise \end{cases}

(and in this regard we also recall Remark 1.4). We also note that the conditions (i)(a) and (ii)(a) in this result depend upon the choice of isomorphism (5) for each abelian subquotient A in SQ_{1}^{ab}(G).

Theorem 2.2. We assume that p is odd and fix a complex C^{•} in Dp ; S̃(Λ(G)). For every subquotient A in SQ_{1}^{ab}(G) and homomorphism ϕ in H(A)^{∨} we then write MC_{A,ϕ} and MC_{A,ϕ}^{∗} in place of the assertions MC(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{A,ϕ}^{•}, (ξ_{I_{A}^{G}(ϕ⊗ω)})_{ω}) and

MC^{∗}(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{A,ϕ}^{•}, (d_{A}^{−r_{ϕ⊗ω}(C_{A,ϕ}^{•})}ξ_{I_{A}^{G}(ϕ⊗ω)}^{∗})_{ω}) respectively.

(i) MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) is valid if and only if the following conditions are satisfied.

(a) MC_{A,ϕ} is valid for all A ∈ SQ_{1}^{ab}(G) and all ϕ ∈ H(A)^{∨}.

(b) For all G ∈ Q_{1}(G) there exists an element ξ_{G} of K_{1}(Λ(G)_{S}) with the following property: for all ρ ∈ A(G) one has (ξ_{G})\underline{G}(ρ⊗ω) = p^{−m_{ρ}(C^{•})}ξ_{ρ⊗ω} for almost all ω ∈ Γ^{∨}, where m_{ρ}(C^{•}) is the integer defined in §7.1. (ii) MC^{∗}(\underline{G}, C^{•}, (ξ_{ρ}^{∗})_{ρ}) is valid if and only if the following conditions are satisfied.

(a) MC_{A,ϕ}^{∗} is valid for all A ∈ SQ_{1}^{ab}(G) and all ϕ ∈ H(A)^{∨}.

(b) For all G ∈ Q_{1}(G) there exists an element ξ_{G} of K_{1}(Λ(G)_{S}) with

(ξ_{G})∗ ; \underline{G}(ρ) = p^{−m_{ρ}(C^{•})}ξ_{ρ}^{∗} for all ρ ∈ A(G).

In the next result we use the same notation as in Theorem 2.2. We also assume that the condition of Theorem 2.2(i)(a), resp. Theorem 2.2(ii)(a), is satisfied and for every A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} we write ξ_{A,ϕ}, resp. ξ_{A,ϕ}^{∗}, for the unique element of Q_{ℤ_{p}[ϕ]}(Γ(A))^{×} that validates MC_{A,ϕ}, resp. MC_{A,ϕ}^{∗} (such elements ξ_{A,ϕ} and ξ_{A,ϕ}^{∗} are necessarily unique as a consequence of Remark 3.5 below). We also write

(6)  Δ_{G,S} : K_{1}(Λ(G)_{S}) → ∏_{{A∈SQ{ab \atop 1}(G) \atop ϕ∈H(A)^{∨}}} Q_{ℤ_{p}[ϕ]}(Γ(A))^{×}

for the diagonal map ∏_{A,ϕ}(q_{A,ϕ} ∘ q_{A}). The following result will be proved in §7.4.

Corollary 2.3. Assume that p is odd.

(i) MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) is valid if and only if (p^{−μ_{ϕ}(C_{A}^{•})}ξ_{A,ϕ})_{A,ϕ} ∈ im(Δ_{G,S}).

(ii) MC^{∗}(\underline{G}, C^{•}, (ξ_{ρ}^{∗})_{ρ}) is valid if and only if (p^{−μ_{ϕ}(C_{A}^{•})}ξ_{A,ϕ}^{∗})_{A,ϕ} ∈ im(Δ_{G,S}).

Remark 2.4. Let k be a number field and K a Galois extension of k with k^{∞} ⊆ K and G := G_{K/k} a compact p-adic Lie group. Write F for the set of intermediate fields of K/k that are of finite degree over k and let M be a motive defined over k. Fix A := U/J ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} and set F := K^{U} ∈ F. If MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) corresponds (in the sense, for example, of either §1.3.1 or §1.3.2) to the main conjecture of non-commutative Iwasawa theory for K/k and M, then it can be shown that MC(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{A,ϕ}^{•}, (ξ_{I_{A}^{G}(ϕ⊗ω)})_{ω}) corresponds to the main conjecture of Iwasawa theory for F^{∞}/F and the ϕ̌-twist of the base change of M to F. In any such case Theorem 2.2(i) therefore shows that the non-commutative main conjecture is valid if and only if the cyclotomic main conjectures are valid for a family of Dirichlet twists of base changes of M to fields in F and in addition there exist ‘non-abelian p-adic Lfunctions’ with suitable interpolation properties. The result of Corollary 2.3(i) shows further that the existence of non-abelian p-adic L-functions amounts to a compatibility relation between the cyclotomic p-adic L-functions of Dirichlet twists of M over fields in F. The results of Theorem 2.2(ii) and Corollary 2.3(ii) can be interpreted in a similar way.

Remark 2.5. Our proof of Corollary 2.3 (via Theorem 6.1) will also suggest the following concrete strategy for proving statements of the form MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) or MC^{∗}(\underline{G}, C^{•}, (ξ_{ρ}^{∗})_{ρ}):

• Verify MC_{A,ϕ}, resp. MC_{A,ϕ}^{∗}, for all A in SQ_{1}^{ab}(G) and all ϕ in H(A)^{∨} and obtain an explicit description of the validating elements ξ_{A,ϕ}, resp. ξ_{A,ϕ}^{∗}.

• Describe explicitly a subgroup Φ of ∏_{A,ϕ} Q_{ℤ_{p}[ϕ]}(Γ(A))^{×} which both contains im(Δ_{G,S}) and satisfies Φ ∩ ∏_{A,ϕ} Λ_{ℤ_{p}[ϕ]}(Γ(A))^{×} = im(Δ_{G,S} ∘ α_{G,S}) and then use this description to show that the element (p^{−μ_{ϕ}(C_{A}^{•})}ξ_{A,ϕ})_{A,ϕ}, resp. (p^{−μ_{ϕ}(C_{A}^{•})}ξ_{A,ϕ}^{∗})_{A,ϕ}, belongs to Φ.

After hearing of the ideas underlying this general approach, Kato improved it considerably in [31] by suggesting how to describe explicit subgroups Φ which make the approach workable for specific groups G, and these results of Kato were subsequently generalised by Hara [26, 27] and by Kakde [28]. By using this strategy Kato, Hara and Kakde were able to prove (in the indicated references) results that are equivalent to special cases of Theorem 2.6 below. More recently, Kakde was able in [29] to complete the development of this approach by giving an explicit description of a suitable subgroup Φ for all (rank one) pro-p groups G and then to use this subgroup to give an alternative proof of Theorem 2.6. In the article [10] the present author then combined this strategy with the K-theoretical results of Kakde in [29] to prove the results that are mentioned in §1.3.3.

2.3. Arithmetic applications. To describe a first arithmetic application of the results in §2.2 we fix notation as in §1.3.2. For each ρ ∈ A(G) we also write L_{p,Σ}(s, ρ) for the associated Σ-truncated p-adic Artin L-function (for more details see §9.1). For each such ρ we write ⟨ρ, 1_{G}⟩ for the multiplicity with which the trivial representation 1_{G} of G occurs in ρ and recall that the function

(7)  L′_{p,Σ}(s, ρ) := (1 − s)^{⟨ρ,1_{G}⟩}L_{p,Σ}(s, ρ)

is p-adic holomorphic at s = 1. If ρ is irreducible we set

c_{ρ} := \begin{cases}1 − ρ(γ−1 ; \underline{G}), & if G_{K/k^{∞}} ⊆ ker(ρ) and ρ ≠ 1_{G} \\ 1, & otherwise \end{cases}

and then extend this definition by linearity to all ρ ∈ A(G). With ς_{p} denoting the group of p-th roots of unity in an algebraic closure of k^{∞} we write ω_{k} for the cyclotomic character G_{k^{∞}/k} ≅ G_{k^{∞}(ς_{p})/k(ς_{p})} → ℤ_{p}^{×}. For any number field E we write μ_{p}(E) for the p-adic Iwasawa μ-invariant of the extension E^{∞}/E and recall that Iwasawa has conjectured that μ_{p}(E) = 0. Before stating the next result we finally recall Remark 1.4.

Theorem 2.6. Assume that p is odd, that G has no element of order p and that there exists a finite subextension k′ of K/k such that K/k′ is a pro-p extension and μ_{p}(k′) = 0. Then the group X_{Σ}(K) belongs to 𝔐_{S}(Λ(G)) and there exists an element ξ of K_{1}(Λ(G)_{S}) which satisfies both of the following conditions:

(i) for all ρ ∈ A(G) one has

(8)  ξ\underline{G}(ρ ⊗ ω) = c_{ρ⊗ω}^{−1}log_{p}(ω_{k}(γ_{G}))^{−⟨ρ⊗ω,1_{G}⟩}L′_{p,Σ}(1, ρ ⊗ ω)

for almost all ω ∈ Γ^{∨}; (ii) ∂_{G,S}(ξ) = χ^{ref}(X_{Σ}(K)[0]).

Remark 2.7. We will prove Theorem 2.6 as a special case of a more general result (Theorem 9.1) that allows the group G to have non-trivial elements of p-power order. Our proof of Theorem 2.6 will also show that if p is odd and the ‘p-adic Stark conjecture at s = 1’ is valid for all ρ ∈ A(G), then for any ξ as in Theorem 2.6 and all ρ ∈ A(G) one has ξ\underline{G}(ρ) = ξ∗ ; \underline{G}(ρ) ≠ 0 and the equality (8) is valid for all ω ∈ Γ^{∨}. We recall that the p-adic Stark conjecture at s = 1 is first stated precisely in [14, Rem. 5.3] but originates with Serre in [40] and is discussed by Tate in [41, Chap. VI, §5].

In the rest of this subsection we record some explicit consequences of Theorem 2.6. For any finite Galois extension of number fields F/k and any integer r we write ℚ(r)_{F/k} for the motive h^{0}(Spec(F))(r), regarded in the natural way as defined over k and with coefficients ℚ[G_{F/k}]. For each algebra R we write Cl(R) for the reduced projective class group of R.

Corollary 2.8. Let F/k be a finite Galois extension of totally real number fields and set G := G_{F/k}. Assume that for each odd prime divisor p of |G| there exists an intermediate field k′ of F/k such that F/k′ is of p-power degree and μ_{p}(k′) = 0. Assume also that the p-adic Stark conjecture at s = 1 is valid for all odd primes p. Then the following conjectures are valid for F/k:

• the equivariant Tamagawa number conjecture of [11, Conj. 4.1] for the pair (ℚ(1)_{F/k}, ℤ[\frac{1}{2}][G]); • the image under K_{0}(ℤ[G], ℝ[G]) → K_{0}(ℤ[\frac{1}{2}][G], ℝ[G]) of the leading term conjecture formulated by Breuning and the present author in [4, Conj. 3.3];

• the image under Cl(ℤ[G]) → Cl(ℤ[\frac{1}{2}][G]) of the ‘Ω(1)-Conjecture’ of Chinburg

[16, Question 3.2]. If, in addition, for each place w of F that has odd residue characteristic and ramifies wildly in F/k the w-adic completion of F is absolutely abelian, then the following conjectures are also valid for F/k:

• the image under K_{0}(ℤ[G], ℝ[G]) → K_{0}(ℤ[\frac{1}{2}][G], ℝ[G]) of the leading term conjecture formulated by Breuning and the present author in [4, Conj. 4.1];

• the image under Cl(ℤ[G]) → Cl(ℤ[\frac{1}{2}][G]) of the ‘Ω(3)-Conjecture’ of Chinburg [16, Question 3.2].

Remark 2.9. The equivariant Tamagawa number conjecture of Flach and the present author is a natural refinement of the seminal Tamagawa number conjecture of Bloch and Kato [2] and, when specialised to pairs of the form (ℚ(a)_{F/k}, ℤ[G]) with a ∈ {0, 1}, is known to imply a variety of explicit congruences between the values at s = 0 of derivatives of Artin L-functions (for details see [9, Th. 4.1.1, Cor. 4.1.6 and §9.1]). The leading term conjectures of Breuning and the present author [4, Conj. 3.3 and Conj. 4.1] are refinements of explicit conjectures of Tate and Stark concerning the leading terms of Artin L-functions at s = 1 and s = 0 respectively. The ‘Ω-conjectures’ of Chinburg are the central conjectures of Galois module theory. However, aside from the results of Flach and the present author in [12, Th. 4.1 and Rem. 11], where [11, Conj. 4.1] is verified for (ℚ(a)_{F/ℚ}, ℤ[G_{F/ℚ}]) with a ∈ {0, 1} and F belonging to a certain infinite family of quaternionic extensions of ℚ, we are not aware of any other previous (conditional) results that establish the validity for any non-abelian extensions of any of the conjectures listed in Corollary 2.8.

For a precise formulation of the conjecture of Chinburg, Kolster, Pappas and Snaith that occurs in the following result see [17, Question 1.3] and [18].

Corollary 2.10. Let F be a finite CM Galois extension of a totally real number field k and set G := G_{F/k}. Write τ for complex conjugation in G and for each integer r let e_{r} denote the central idempotent \frac{1}{2}(1 + (−1)^{r+1}τ) of ℤ[\frac{1}{2}][G]. Assume that for each odd prime divisor p of |G| there exists an intermediate field k′ of F/k such that F/k′ is of p-power degree and μ_{p}(k′) = 0. Then for each strictly negative integer r the following conjectures are valid for F/k:

• the equivariant Tamagawa number conjecture of [11, Conj. 4.1] for the pair (ℚ(r)_{F/k}, ℤ[\frac{1}{2}][G]e_{r});

• the image under Cl(ℤ[G]) → Cl(ℤ[\frac{1}{2}][G]e_{r}) of the conjecture of Chinburg, Kolster, Pappas and Snaith concerning the invariant Ω_{−r}(F/k).

Example 2.11. To give an explicit example in which Corollary 2.10 implies the unconditional validity of the listed conjectures we fix a finite totally real abelian extension k of ℚ. If F′ is any Galois extension of k of degree a power of an odd prime p, then for any natural number n the field F := F′(\sqrt{−n}) is a CM Galois extension of k, the field k′ := k(\sqrt{−n}) is abelian over ℚ so μ_{p}(k′) = 0 (by Ferrero and Washington [23]) and the degree of F/k′ is a power of p and so F/k validates all hypotheses that occur in Corollary 2.10. This gives the first unconditional evidence in favour of both [11, Conj. 4.1] for the pair (ℚ(r)_{F/k}, ℤ[\frac{1}{2}][G]e_{r}) and of the above conjecture of Chinburg, Kolster, Pappas and Snaith for a class of non-abelian extensions F/k and all strictly negative integers r. (Prior to this, the only evidence in favour of these cases of [11, Conj. 4.1] in any non-abelian setting was provided by Hara in [27] where he considered the ‘p-part’ of [11, Conj. 4.1] for (ℚ(r)_{F/k}, ℤ[G_{F/k}]) for extensions of totally real fields F/k of exponent p and integers r = 1 − m with m a natural number divisible by p − 1.)

3. Values, leading terms and reduced norms

In this section we let S′ denote either S or S^{∗} and, as preparation for our proof of the results in §2.2, we analyse the homomorphisms Φ_{O,\underline{G},S′,ρ} defined in (3). For any ρ in A(G) the subgroup U := H ∩ ker(ρ) is open in H and normal in G. The quotient G_{ρ} := G/U therefore belongs to Q_{1}(G) and so gives a triple \underline{G_{ρ}} as in (1) in which Γ(\underline{G_{ρ}}) = Γ(\underline{G}), γ\underline{G_{ρ}} = γ\underline{G} and π\underline{G_{ρ}} is induced by π\underline{G}. Identifying ρ with an element of A(G_{ρ}) in the obvious way one also has

(9)  Φ_{O,\underline{G},S′,ρ}(ξ) = Φ_{O,\underline{G_{ρ}},S′,ρ}(q_{G_{ρ}}(ξ))

for all ξ in K_{1}(Λ_{O}(G)_{S′}). This allows us to study elements of the form Φ_{O,\underline{G},S′,ρ}(ξ) via an analysis of the corresponding elements for the rank one group G_{ρ}.

3.1. The reduced norm. In this subsection we assume that, unless explicitly stated otherwise, G has rank one. The centre ζ(G) of G is thus open and so Λ_{O}(G)_{S} = Λ_{O}(G)_{S^{†}} with S^{†} := Λ_{O}(ζ(G)) ∩ S. Further, Λ_{O}(G)_{S^{∗}} is the total quotient ring Q_{O}(G) of Λ_{O}(G) and is a semisimple algebra. We write ζ(Q_{O}(G)) for the centre of Q_{O}(G) and Nrd_{Λ_{O}(G)_{S′}} for the reduced norm homomorphism

K_{1}(Λ_{O}(G)_{S′}) → K_{1}(Q_{O}(G)) → ζ(Q_{O}(G))^{×}

(cf. [21, §45A]) and then set

SK_{1}(Λ_{O}(G)_{S′}) := ker(Nrd_{Λ_{O}(G)_{S′}}).

For each element x of Λ_{O}(G)_{S′} we write [x] for its image under the natural homomorphism Λ_{O}(G)_{S′}^{×} → K_{1}(Λ_{O}(G)_{S′}). We also set Q^{c}(Γ) := ℚ_{p}^{c} ⊗_{E} Q_{O}(Γ) where E is the fraction field of O. We use an explicit description of Nrd_{Λ_{O}(G)_{S′}} given by Ritter and Weiss. To be specific, we use the fact that for each Artin representation ρ : G → GL_{n}(O′) with O ⊆ O′ there exists a homomorphism of groups

j_{ρ} : ζ(Q_{O}(G))^{×} → Q^{c}(Γ)^{×}

which has all of the following properties:

(10)  ⋂_{ρ∈A(G)} ker(j_{ρ}) = {1};

if V_{ρ} is a ℚ_{p}^{c}-space which realises ρ and x belongs to Λ_{O}(G) ∩ Λ_{O}(G)_{S′}^{×}, then

(11)  j_{ρ}(Nrd_{Λ_{O}(G)_{S′}} ([x])) = det_{Q^{c}(Γ)}(r_{ρ}(x))

where r_{ρ}(x) is the endomorphism of Q_{O}(Γ) ⊗_{Λ_{O}(Γ)} Hom_{ℚ_{p}^{c}[H]}(V_{ρ}, ℚ_{p}^{c} ⊗_{O} Λ_{O}(G)) that is induced by right multiplication by x on Λ_{O}(G); for ξ ∈ K_{1}(Λ_{O}(G)_{S′}), ω ∈ Γ^{∨} and ρ ∈ A(G) one has

(12)  j_{ρ⊗ω}(Nrd_{Λ_{O}(G)_{S′}} (ξ)) = tw_{ω}(j_{ρ}(Nrd_{Λ_{O}(G)_{S′}} (ξ)))

where tw_{ω} is the automorphism of Q^{c}(Γ) that sends γ to ω(γ)γ, or equivalently, T to ω(γ)T + ω(γ) − 1. (Note that (10) follows from the definition of j_{ρ} in [34, Prop. 6(3)], whilst (11) and (12) are a consequence of [loc. cit., Th. 8].)

Lemma 3.1. For ξ ∈K_{1}(Λ_{O}(G)_{S′})and ρ∈A(G)one has j_{ρ}(Nrd_{Λ_{O}(G)_{S′}} (ξ))=Φ_{\underline{G},ρ}(ξ).

Proof. We may assume that ξ = [x] with x ∈ Λ_{O}(G) ∩ Λ_{O}(G)_{S′}^{×} and so compute j_{ρ}(Nrd_{Λ_{O}(G)_{S′}} (ξ)) via (11). To do this we use the natural isomorphisms

Hom_{ℚ_{p}^{c}}(V_{ρ}, ℚ_{p}^{c} ⊗_{O} Λ_{O}(G)) ≅ Vρ̌ ⊗_{O} Λ_{O}(G) ≅ Λ_{O}(G) ⊗_{Λ_{O}(H)} (Vρ̌ ⊗_{O} Λ_{O}(H))

where each g ∈ G, resp. g ∈ H, acts on Vρ̌ ⊗_{O} Λ_{O}(G), resp. on Vρ̌ ⊗_{O} Λ_{O}(H), via v ⊗ λ ↦ g(v) ⊗ gλ, and G acts by left multiplication on the third module (cf. [6, p. 71, Ex. 2]). Upon passing to H-coinvariants these maps induce an isomorphism of Q^{c}(Γ)-modules

(13)  Q_{O}(Γ) ⊗_{Λ_{O}(Γ)} Hom_{ℚ_{p}^{c}[H]}(V_{ρ}, ℚ_{p}^{c} ⊗_{O} Λ_{O}(G)) ≅ Q^{c}(Γ) ⊗_{ℚ_{p}^{c}} Vρ̌

under which r_{ρ}(x) corresponds to the image of x under the continuous O-linear homomorphism

Δ_{ρ} : Λ_{O}(G) → End_{ℚ_{p}^{c}}(Q^{c}(Γ) ⊗_{ℚ_{p}^{c}} Vρ̌)

that satisfies Δ_{ρ}(g)(λ ⊗ v) = λπ\underline{G}(g) ⊗ g^{−1}(v) for all g ∈ G, λ ∈ Q^{c}(Γ) and v ∈ Vρ̌. Now for each g in G the matrix of the action of g^{−1} on Vρ̌ is the transpose of the matrix of the action of g on V_{ρ}. Taking this into account, an explication of the role of Morita equivalence in the definition of Φ_{\underline{G},ρ} shows that Φ_{\underline{G},ρ}(x) = det_{Q^{c}(Γ)}(Δ_{ρ}(x)), as required to complete the proof. □

Remark 3.2. Lemma 3.1 implies that for all ξ ∈ K_{1}(Λ_{O}(G)_{S′}) and ρ ∈ A(G) one has Φ_{\underline{G},ρ}(ξ) = Det_{Q_{O}(G)}(ξ)(ρ) where Det_{Q_{O}(G)}(ξ) is the function on A(G) defined by Ritter and Weiss in [34].

Remark 3.3. If now G has any finite rank, then (9) combines with Lemma 3.1 and the equality (12) (with G and ξ replaced by G_{ρ} and q_{G_{ρ}}(ξ) respectively) to imply that Φ_{\underline{G},ρ⊗ω}(ξ) = tw_{ω}(Φ_{\underline{G},ρ}(ξ)) for all ξ ∈ K_{1}(Λ_{O}(G)_{S′}), ρ ∈ A(G) and ω ∈ Γ^{∨}.

3.2. The evaluation kernel. For any data \underline{G} = (G, π, γ) as in (1), and with G of any finite rank, we define the ‘evaluation kernel’ of K_{1}(Λ_{O}(G)_{S′}) to be the subgroup

EK_{1}(Λ_{O}(G)_{S′}) := ⋂_{ρ∈A(G)} ker(Φ_{O,\underline{G},S′,ρ}).

The next result describes several useful characterisations of this subgroup. This result implies in particular that the interpolation property of either MC(\underline{G}, O, C^{•}, \{ξ_{ρ}\}_{ρ})(i) or MC^{∗}(\underline{G}, O, C^{•}, \{ξ_{ρ}^{∗}\}_{ρ})(i) uniquely determines both the value ξ\underline{G}(ρ) and leading term ξ∗ ; \underline{G}(ρ) of ξ at all ρ in A(G).

Lemma 3.4. Fix an element ξ of K_{1}(Λ_{O}(G)_{S′}). Then the following conditions are equivalent.

(i) ξ ∈ EK_{1}(Λ_{O}(G)_{S′}).

(ii) q_{G}(ξ) ∈ SK_{1}(Λ_{O}(G)_{S′}) for all G ∈ Q_{1}(G).

(iii) ξ∗ ; \underline{G}(ρ) = 1 for all ρ ∈ A(G).

(iv) ξ\underline{G}(ρ) = 1 for all ρ ∈ A(G).

(v) For all ρ ∈ A(G) one has ξ\underline{G}(ρ ⊗ ω) = 1 for almost all ω ∈ Γ^{∨}.

Proof. When considering the conditions (i), (iii), (iv) and, in view of Remark 3.3, also (v) we can clearly restrict attention to representations ρ in A(G) for which Φ_{\underline{G},ρ}(ξ) ≠ 1 and this is what we do in the sequel. We thus fix such a representation ρ : G → GL_{n}(O′) with O ⊆ O′. Then for almost all ω ∈ Γ^{∨} neither of the elements Φ_{\underline{G},ρ}(ξ) or Φ_{\underline{G},ρ}(ξ)′ := Φ_{\underline{G},ρ}(ξ) − 1 of Q(O′[[T]]) has either a zero or a pole at T = ω(γ) − 1. For all such ω one thus has

Φ_{\underline{G},ρ}(ξ)(ω(γ) − 1) = tw_{ω}(Φ_{\underline{G},ρ}(ξ))(0) = Φ_{\underline{G},ρ⊗ω}(ξ)(0) =: ξ\underline{G}(ρ ⊗ ω) = ξ∗ ; \underline{G}(ρ ⊗ ω),

where the fourth equality follows from our assumption on ω, and hence also Φ_{\underline{G},ρ}(ξ)′(ω(γ) − 1) = ξ\underline{G}(ρ ⊗ ω) − 1 = ξ∗ ; \underline{G}(ρ ⊗ ω) − 1. One also knows that the element Φ_{\underline{G},ρ}(ξ)′ is identically zero if and only if it has infinitely many distinct zeroes in ℤ_{p}^{c} \ ℤ_{p}^{c×} (by the Weierstrass Preparation Theorem). Finally we note that each element ω(γ) − 1 belongs to ℤ_{p}^{c} \ ℤ_{p}^{c×} and that ω′(γ) ≠ ω(γ) for all ω′ ∈ Γ^{∨} \ {ω}. Taken together these observations make clear that if either (iii), (iv) or (v) is valid, then Φ_{\underline{G},ρ}(ξ) = 1 for all ρ ∈ A(G). Since the reverse implications are obvious we have at this stage shown the equivalence of the conditions (iii), (iv), (v) and (i). In addition, the conditions (i) and (ii) are equivalent because the equality (9) for each ρ ∈ A(G) implies that ξ ∈ EK_{1}(Λ_{O}(G)_{S′}) if and only if for all G ∈ Q_{1}(G) one has q_{G}(ξ) ∈ ker(Φ_{O,\underline{G},S′,ψ}) for all ψ ∈ A(G), or equivalently, by Lemma 3.1 and (10), that q_{G}(ξ) ∈ SK_{1}(Λ_{O}(G)_{S′}). □

Remark 3.5. If G has rank one, then Q_{O}(G) is a (finite) product of central simple algebras over finite extensions of fields of the form Q_{O}(Π) with Π a group that is topologically isomorphic to ℤ_{p}. Since the cohomological dimension of any such field Q_{O}(Π) is equal to three (see [39]) a conjecture of Suslin implies that SK_{1}(Q_{O}(G)) vanishes (see [32, p. 250] and note that Suslin’s conjecture is known to be valid if G is abelian [21, Prop. (45.12)]). Hence, if this is true, then Lemma 3.4 implies that if G has rank one, then at most one element of K_{1}(Λ_{O}(G)_{S^{∗}}) can satisfy the interpolation property of either MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ})(i) or MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ})(i).

3.3. Functoriality. We now record several functoriality results that will be useful in subsequent sections. The key fact in this regard is that if G belongs to SQ_{1}(G) and G′ = U′/J′ to SQ_{1}(G), then (11) combines with [34, Lem. 9] to imply that there is a commutative diagram of group homomorphisms

(14)

K_{1}(Λ_{O}(G)_{S′}) ⟶^{Nrd_{Λ_{O} (G)_{S′}}} ζ(Q_{O}(G))^{×}
q_{G,G′} ↓  ^{q′_{G,G′}} ↓
K_{1}(Λ_{O}(G′)_{S′}) ⟶^{Nrd_{Λ_{O} (G′)_{S′}}} ζ(Q_{O}(G′))^{×}

where q_{G,G′} is the composite of the natural restriction K_{1}(Λ_{O}(G)_{S′}) → K_{1}(Λ_{O}(U′)_{S′}) and projection K_{1}(Λ_{O}(U′)_{S′}) → K_{1}(Λ_{O}(G′)_{S′}) maps and q′_{G,G′} is uniquely characterised (because of (10) with G = G′) by the following property: for each y in ζ(Q_{O}(G))^{×} and each ψ in A(G′) one has j_{ψ}(q′_{G,G′}(y)) = j_{I_{G′}^{G} (ψ)}(y).

Lemma 3.6. Fix an element ξ of K_{1}(Λ_{O}(G)_{S^{∗}}), a subquotient G′ in SQ_{1}(G) and an Artin representation ψ : G′ → GL_{n}(O′) with O ⊆ O′. Recall the integers d_{G′}, r_{I_{G′}^{G} (ψ)}(ξ) and r_{ψ}(q_{G′}(ξ)) defined in §2.1.1.

(i) Φ_{\underline{G},I_{G′}^{G} (ψ)}(ξ) = Φ_{\underline{G′},ψ}(q_{G′}(ξ)) in Q(O′[[T_{G′}]])^{×} ⊂ Q(O′[[T]])^{×}.

(ii) ξ\underline{G}(I_{G′}^{G}(ψ)) = q_{G′}(ξ)\underline{G′}(ψ).

(iii) ξ_{\underline{G}′}^{∗}(I_{G′}^{G}(ψ)) = d_{G′}^{r_{ψ}(q_{G′} (ξ))}q_{G′}(ξ)∗ ; \underline{G′}(ψ).

(iv) r_{I_{G′}^{G} (ψ)}(ξ) = r_{ψ}(q_{G′}(ξ)).

Proof. To prove claim (i) we write G′ = U/J′ with U open in G and J′ open in H and normal in U, and set J := ⋂_{τ} τ^{−1}J′τ where τ runs over a (finite) set of coset representatives of U in G. Then J is open in H and normal in G and G′ ∈ SQ_{1}(G) with G := G/J ∈ Q_{1}(G). Also J ⊆ ker(ind_{G′}^{G}ψ) and so (9) implies that Φ_{\underline{G},I_{G′}^{G} (ψ)}(ξ) = Φ_{\underline{G},I_{G′}^{G} (ψ)}(q_{G}(ξ)). Claim (i) follows upon combining this equality with the description of Lemma 3.1, the commutativity of (14) and the fact that q_{G,G′} ∘ q_{G} = q_{G′}. Following claim (i) we set

f := Φ_{\underline{G},I_{G′}^{G} (ψ)}(ξ) = Φ_{\underline{G′},ψ}(q_{G′}(ξ)) ∈ Q(O′[[T_{G′}]])^{×}.

Now T_{G′} := (1 + T)^{d_{G′}} − 1 and hence T_{G′} = 0 if T = 0 (which implies claim (ii)), the algebraic order at 0 of f as a function of T_{G′} is equal to the algebraic order at 0 of f as a function of T (proving claim (iv)) and the leading term of f as a power series in T is equal to d_{G′}^{r_{ψ}(q_{G′} (ξ))} times the leading term of f as a power series in T_{G′} (which proves claim (iii)). □

4. The reduction to groups of rank one

4.1. The proof of Theorem 2.1. If ξ is an element of K_{1}(Λ_{O}(G)_{S^{∗}}) that validates MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}), resp. MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}), then for all G ∈ Q_{1}(G) the element q_{G}(ξ) validates MC(\underline{G}, O, C_{G}^{•}, (ξ_{ρ})_{ρ}), resp. MC^{∗}(\underline{G}, O, C_{G}^{•}, (ξ_{ρ}^{∗})_{ρ}). Indeed this is a direct consequence of (9) and of the commutative diagram (4) with S′ = S^{∗}. In the remainder of the argument we thus fix for each G ∈ Q_{1}(G) an element ξ_{G} of K_{1}(Q_{O}(G)) that validates MC(\underline{G}, O, C_{G}^{•}, (ξ_{ρ})_{ρ}), resp. MC^{∗}(\underline{G}, O, C_{G}^{•}, (ξ_{ρ}^{∗})_{ρ}). We also fix an element ξ^{alg} ∈ K_{1}(Λ_{O}(G)_{S^{∗}}) with ∂_{O,G,S^{∗}}(ξ^{alg}) = χ_{O,G}^{ref}(C^{•}) and for each G ∈ Q_{1}(G) set ξ̃_{G} := q_{G}(ξ^{alg})^{−1}ξ_{G} ∈ K_{1}(Q_{O}(G)). Before stating the next result we set Nrd_{G} := Nrd_{Λ_{O}(G)_{S^{∗}}} for each G ∈ Q_{1}(G). We note that if G and G′ belong to Q_{1}(G) and G′ is a quotient of G then the maps q_{G,G′} and q′_{G,G′} that occur in (14) are just the natural projection homomorphisms. Finally, if G denotes either G or an element of Q_{1}(G), then we abbreviate the homomorphisms α_{O,G,S^{∗}} and ∂_{O,G,S^{∗}} in the sequence (2) to α_{G} and ∂_{G} respectively.

Lemma 4.1. One has (Nrd_{G}(ξ̃_{G}))_{G∈Q_{1}(G)} ∈ \underleftarrow{lim}_{G∈Q_{1}(G)} im(Nrd_{G} ∘ α_{G}).

Proof. For each G ∈ Q_{1}(G) our choice of ξ_{G} implies ∂_{G}(ξ_{G}) = χ^{ref}(C_{G}^{•}) whilst our choice of ξ^{alg} combines with (4) to imply ∂_{G}(q_{G}(ξ^{alg})) = χ^{ref}(C_{G}^{•}). The element ξ̃_{G} = q_{G}(ξ^{alg})^{−1}ξ_{G} therefore belongs to ker(∂_{G}) = im(α_{G}) and so Nrd_{G}(ξ̃_{G}) ∈ im(Nrd_{G} ∘α_{G}). It is thus enough to prove that if G and G′ belong to Q_{1}(G) and G′ is a quotient of G, then q′_{G,G′}(Nrd_{G}(ξ̃_{G})) = Nrd_{G′}(ξ̃_{G′}). But q_{G,G′}(q_{G}(ξ^{alg})) = q_{G′}(ξ^{alg}) so (14) implies

q′_{G,G′}(Nrd_{G}(ξ̃_{G})) = Nrd_{G′}(q_{G′}(ξ^{alg})^{−1}q_{G,G′}(ξ_{G})) = Nrd_{G′}(ξ̃_{G′})Nrd_{G′}(ξ_{G′}^{−1}q_{G,G′}(ξ_{G})).

It therefore suffices to prove that ξ_{G′}^{−1}q_{G,G′}(ξ_{G}) belongs to SK_{1}(Q_{O}(G′)), or equivalently (by Lemma 3.4 with G = G′) that for all ρ ∈ A(G′) one has (ξ_{G′})\underline{G′}(ρ ⊗ ω) = (q_{G,G′}(ξ_{G}))\underline{G′}(ρ ⊗ ω) for almost all ω ∈ Γ^{∨} ⊆ A(G′). But it is clear from the definition (3) that one has Φ_{\underline{G′},ψ}(q_{G,G′}(ξ_{G})) = Φ_{\underline{G},ψ}(ξ_{G}) for all ψ ∈ A(G′) ⊆ A(G) and so (q_{G,G′}(ξ_{G}))\underline{G′}(ρ ⊗ ω) = (ξ_{G})\underline{G}(ρ ⊗ ω). The required equalities are thus equivalent to asking that (ξ_{G′})\underline{G′}(ρ ⊗ ω) = (ξ_{G})\underline{G}(ρ ⊗ ω) for almost all ω ∈ Γ^{∨} ⊆ A(G′) and this is obviously true because of the interpolation properties that ξ_{G′} and ξ_{G} are assumed to satisfy. □

Upon combining Lemma 4.1 with the result of Proposition 4.2 below we may fix an element u of K_{1}(Λ_{O}(G)) with Nrd_{G}(q_{G}(α_{G}(u))) = Nrd_{G}(ξ̃_{G}) for all G ∈ Q_{1}(G). We now claim that the element ξ′ := α_{G}(u)ξ^{alg} validates MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}), resp. MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}). To verify the necessary interpolation properties we fix a representation ρ ∈ A(G) and set G := G/(H ∩ ker(ρ)) ∈ Q_{1}(G). Then one has

Φ_{\underline{G},ρ}(ξ′) = Φ_{\underline{G},ρ}(q_{G}(ξ′)) = j_{ρ}(Nrd_{G}(q_{G}(ξ′))) = j_{ρ}(Nrd_{G}(q_{G}(α_{G}(u))q_{G}(ξ^{alg})))

= j_{ρ}(Nrd_{G}(q_{G}(ξ^{alg})ξ̃_{G})) = j_{ρ}(Nrd_{G}(ξ_{G})) = Φ_{\underline{G},ρ}(ξ_{G}),

where the first displayed equality follows from (9), the second and last from Lemma 3.1, the third from the definition of ξ′, the fourth from our choice of u and the fifth from the definition of ξ̃_{G}. Thus, since ξ_{G} satisfies MC(\underline{G}, O, C_{G}^{•}, (ξ_{ρ})_{ρ})(i), resp. MC^{∗}(\underline{G}, O, C_{G}^{•}, (ξ_{ρ}^{∗})_{ρ})(i), it is obvious from the displayed equality that ξ′ satisfies MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ})(i), resp. MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ})(i). It therefore only remains to verify that ξ′ satisfies MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ})(ii), resp. MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ})(ii). This is true since

∂_{G}(ξ′) = ∂_{G}(α_{G}(u)ξ^{alg}) = ∂_{G}(α_{G}(u)) + ∂_{G}(ξ^{alg}) = ∂_{G}(ξ^{alg}) = χ^{ref}(C^{•}),

where the third equality is valid because α_{G}(u) ∈ im(α_{G}) = ker(∂_{G}) and the last because of our choice of ξ^{alg}. This completes the proof of Theorem 2.1.

Proposition 4.2. The natural composite homomorphism

K_{1}(Λ_{O}(G)) → \underleftarrow{lim}_{G} im(α_{G}) → \underleftarrow{lim}_{G} im(Nrd_{G} ∘ α_{G})

is surjective, where the inverse limits are taken over Q_{1}(G).

Proof. For each G ∈ Q_{1}(G) there is a tautological exact sequence of abelian groups

0 → ker(Nrd_{G} ∘ α_{G}) → K_{1}(Λ_{O}(G)) → im(Nrd_{G} ∘ α_{G}) → 0.

These sequences are compatible with the natural transition homomorphisms as G varies over Q_{1}(G) and so induce upon passage to the inverse limit an exact sequence

(15) \underleftarrow{lim}_{G∈Q_{1}(G)} K_{1}(Λ_{O}(G)) → \underleftarrow{lim}_{G∈Q_{1}(G)} im(Nrd_{G} ∘ α_{G}) → \underleftarrow{lim}_{G∈Q_{1}(G)}^{1} ker(Nrd_{G} ∘ α_{G}).

Next we recall that Fukaya and Kato [24, Prop. 1.5.1] have shown the natural homomorphism K_{1}(Λ_{O}(G)) → \underleftarrow{lim}_{n≥0} K_{1}(Λ_{O}(G)/J^{n}) to be bijective, where J denotes the radical of Λ_{O}(G). Since the J-topology on Λ_{O}(G) coincides with the canonical topology this result implies that the natural homomorphisms

(16)  K_{1}(Λ_{O}(G)) → \underleftarrow{lim}_{J ∈Q_{1}(G)} K_{1}(Λ_{O}(J)) → \underleftarrow{lim}_{Δ∈Q_{0}(G)} K_{1}(O[Δ])

are both bijective, where Q_{0}(G) is the set of finite quotients of G. Upon combining this isomorphism with the exact sequence (15) it is clear that the claimed surjectivity will follow if \underleftarrow{lim}_{G}^{1} ker(Nrd_{G} ∘ α_{G}) vanishes. But, since Q_{1}(G) contains a cofinal subset indexed by the natural numbers, the derived limit will vanish provided that each group ker(Nrd_{G} ∘ α_{G}) has a natural compact topology. To prove this we show that if G is any element of Q_{1}(G) then, with respect to the isomorphism (16) with G = G, the subgroup ker(Nrd_{G} ∘ α_{G}) is equal to \underleftarrow{lim}_{Δ∈Q_{0}(G)} SK_{1}(O[Δ]) where SK_{1}(O[Δ]) is the kernel of the natural map ι_{Δ} : K_{1}(O[Δ]) → K_{1}(ℚ_{p}^{c}[Δ]). This implies the claimed result since, by a result of Bass, each group SK_{1}(O[Δ]) is finite (cf. [21, Th. (45.20)]). For each Δ in Q_{0}(G) we write Ir_{ℚ_{p}^{c}}(Δ) for the set of irreducible representations of Δ over ℚ_{p}^{c}. We regard Ir_{ℚ_{p}^{c}}(Δ) as a subset of A(G) in the obvious way and then note that for each ψ ∈ Ir_{ℚ_{p}^{c}}(Δ) the equality (11) implies the following diagram commutes

K_{1}(Λ_{O}(G)) ⟶^{α_{G}} im(α_{G}) ⟶^{j_{ψ}∘Nrd_{G}} (ℚ_{p}^{c} ⊗_{O} Λ_{O}(Γ))^{×}
π_{Δ}↓  ↓^{ϵ}
K_{1}(O[Δ]) ⟶^{ι_{Δ}} K_{1}(ℚ_{p}^{c}[Δ]) ⟶^{λ_{ψ}}  ℚ_{p}^{c,×}.

Here π_{Δ} and ϵ are the natural projections and λ_{ψ} is the composite of the reduced norm map K_{1}(ℚ_{p}^{c}[Δ]) → ζ(ℚ_{p}^{c}[Δ])^{×}, the natural identification ζ(ℚ_{p}^{c}[Δ])^{×} = ∏_{Ir_{ℚ^{c}_{p}} (Δ)} ℚ_{p}^{c,×}

and the ‘projection to the ψ-component’ map. Now if x ∈ ker(Nrd_{G} ∘ α_{G}), then this diagram implies that x ∈ ker(λ_{ψ} ∘ ι_{Δ} ∘ π_{Δ}) for all ψ ∈ Ir_{ℚ_{p}^{c}}(Δ). Since the map ∏_{ψ∈Ir_{ℚ^{c}_{p}} (Δ)} λ_{ψ} is injective it thus follows that π_{Δ}(x) ∈ ker(ι_{Δ}) = SK_{1}(O[Δ]), as required. To prove the converse, we assume x is an element of K_{1}(Λ_{O}(G)) with x ∈ ker(ι_{Δ} ∘π_{Δ}) for all Δ ∈ Q_{0}(G). Then Lemma 3.1 combines with the above diagram to imply that

(α_{G}(x))\underline{G}(ψ) := Φ_{\underline{G},ψ}(α_{G}(x))(0) = ϵ(j_{ψ} ∘Nrd_{G}(α_{G}(x))) = λ_{ψ}(ι_{Δ} ∘π_{Δ}(x)) = λ_{ψ}(1) = 1

for all ψ ∈ Ir_{ℚ_{p}^{c}}(Δ). Since any ρ ∈ A(G) is a direct sum of finitely many such ψ (with Δ := G/ ker(ρ)) one therefore has (α_{G}(x))\underline{G}(ρ) = 1 and so Lemma 3.4 implies α_{G}(x) ∈ SK_{1}(Q_{O}(G)), or equivalently that x belongs to ker(Nrd_{G} ∘ α_{G}), as required. □

4.2. Some remarks. We end this section with some remarks about Proposition 4.2.

Remark 4.3. For any finite group Δ and natural number n the group GL_{n}(O[Δ]) can be topologised by choosing as a basis of open neighbourhoods of I_{n} the subgroups {I_{n} +π^{k}M_{n}(O[Δ]) : k ≥ 1} where π is a uniformizing parameter for O. This induces a natural compact topology on K_{1}(O[Δ]) (cf. [21, p. 155]) and hence, via the bijection (16), a compact topology on K_{1}(Λ_{O}(G)). The argument of Proposition 4.2 shows that, with respect to this topology, the subgroup \underleftarrow{lim}_{G∈Q_{1}(G)} ker(Nrd_{G} ∘ α_{G}) is closed.

Remark 4.4. From Lemma 3.4 it is clear that any element ξ that validates either MC(\underline{G}, O, C^{•}, (ξ_{ρ})_{ρ}) or MC^{∗}(\underline{G}, O, C^{•}, (ξ_{ρ}^{∗})_{ρ}) is unique modulo the subgroup im(α_{O,G,S^{∗}}) ∩ EK_{1}(Λ_{O}(G)_{S^{∗}}) of K_{1}(Λ_{O}(G)_{S^{∗}}). The argument of Proposition 4.2 implies that α_{O,G,S^{∗}} restricts to give a surjection from \underleftarrow{lim}_{Δ∈Q_{0}(G)} SK_{1}(O[Δ]) to im(α_{O,G,S^{∗}}) ∩ EK_{1}(Λ_{O}(G)_{S^{∗}}).

Remark 4.5. Proposition 4.2 also implies that the kernel of the natural diagonal homomorphism

K_{1}(Λ(G)_{S^{∗}}) ⟶^{(∂_{G}∘q_{G})_{G}} ∏_{G∈Q_{1}(G)} K_{0}(Λ(G), Λ(G)_{S^{∗}})

is contained in ker(∂_{G,S^{∗}}) + EK_{1}(Λ(G)_{S^{∗}}). One can show that this fact has the following consequence for the ‘algebraic p-adic L-functions’ L_{G,γ}^{alg}(C^{•}) that are discussed in [7] and [15]: if C^{•} belongs to D_{S}^{p}(Λ(G)), then one has ∂_{G,S^{∗}}(vL_{G,γ}^{alg}(C^{•})) = χ^{ref}(C^{•}) for some v ∈ EK_{1}(Λ(G)_{S^{∗}}). This observation generalises the results of both [7, Th. 4.1] and [15, Prop. 4.7].

5. Weierstrass Preparation Theorems

Theorem 2.1 makes clear the importance of main conjectures in the rank one case. We now prepare to make a detailed analysis of Whitehead groups in this case by discussing the Weierstrass Preparation Theorem proved by Venjakob in [42].

5.1. The rank one case. In this subsection we assume that G is a pro-p group of rank one. In this case G is isomorphic to a semi-direct product H ⋊ Γ for a finite p-group H. Hence, if we fix an element γ̃ of G with π\underline{G}(γ̃) = γ, then for any finite extension O of ℤ_{p} the algebra A := Λ_{O}(G) identifies with the skew power series ring R[[T; σ, δ]] in the sense of Venjakob [42, §2] with R := O[H], T := γ̃ − 1, σ the O-linear ring endomorphism of R sending each element h of H to γ̃hγ̃^{−1} and δ the σ-derivation of R given by σ − id. The ring R is local with maximal ideal 𝔪 equal to the kernel of the natural homomorphism R → O/℘, with ℘ the maximal ideal of O, and is both separated and complete with respect to its 𝔪-adic topology. It is clear that δ(R) ⊆ 𝔪 and also easy to check that δ(𝔪) ⊆ 𝔪^{2}. The Weierstrass Preparation Theorem of Venjakob [42, Cor. 3.2] can therefore be applied to the algebra A. Following [42, §3] we say that an element s = T^{d} + ∑_{i=0}^{i=d−1} a_{i}T^{i} of Λ_{O}(G) is a distinguished polynomial (of degree d) in Λ_{O}(G) if a_{i} ∈ 𝔪 for all i with 0 ≤ i < d.

Lemma 5.1. Let Π be any central subgroup of G that is isomorphic to ℤ_{p}. Then each element of Λ_{O}(G)_{S}^{×} can be written uniquely as a product of the form ut with u ∈ Λ_{O}(G)^{×} and t = ss̃^{−1} where s is a distinguished polynomial in Λ_{O}(G) and s̃ is a distinguished polynomial in Λ_{O}(Π).

Proof. Every element of Λ_{O}(G)_{S}^{×} can be written in the form xy^{−1} with x and y in S. Further, since Π is open in G, one can assume y belongs to Λ_{O}(Π). By comparing [19, Lem. 2.1] to [42, Th. 3.1] one finds that an element of Λ_{O}(G) belongs to S if and only if it has ‘finite reduced order’ (in the terminology of [42, §2]). From [42, Cor. 3.2] it follows that the elements x and y above decompose as x = vs and y = ṽs̃ with v ∈ Λ_{O}(G)^{×}, ṽ ∈ Λ_{O}(Π)^{×} and s and s̃ distinguished polynomials in Λ_{O}(G) and Λ_{O}(Π) respectively. Since xy^{−1} = vs(ṽs̃)^{−1} = vṽ^{−1}ss̃^{−1} (as ṽ is central) we thus obtain a decomposition xy^{−1} = ut of the required form by setting u := vṽ^{−1} and t := ss̃^{−1}. To check uniqueness we assume u_{1}s_{1}s̃_{1}^{−1} = u_{2}s_{2}s̃_{2}^{−1} where for i = 1, 2 one has u_{i} ∈ Λ_{O}(G)^{×}, s_{i} is a distinguished polynomial in Λ_{O}(G) and s̃_{i} is a distinguished polynomial in Λ_{O}(Π). Then u_{1}s_{1}s̃_{2} = u_{2}s_{2}s̃_{1} (since s̃_{1} and s̃_{2} are central). But s_{1}s̃_{2} and s_{2}s̃_{1} are distinguished polynomials in Λ_{O}(G) and so by the uniqueness of the Weierstrass decomposition proved in [42, Cor. 3.2] one has u_{1} = u_{2} and s_{1}s̃_{2} = s_{2}s̃_{1} (so s_{1}s̃_{1}^{−1} = s_{2}s̃_{2}^{−1}). □

For each ρ in A(G) we fix a finite extension O_{ρ} of O in ℚ_{p}^{c} over which Vρ̌ can be realised.

Lemma 5.2. Let s be a distinguished polynomial in Λ_{O}(G). Then for each representation ρ in A(G) there exists a natural number m_{ρ,s} such that Φ_{\underline{G},ρ}([s])^{p^{m}} is a distinguished polynomial in O_{ρ}[[T]] for all integers m ≥ m_{ρ,s}.

Proof. From the argument in the proof of Lemma 3.1 one has

Φ_{\underline{G},ρ}([s])^{p^{m}} = Φ_{\underline{G},ρ}([s^{p^{m}}]) = det_{Q^{c}(Γ)}(Δ_{ρ}(s^{p^{m}}))

where Δ_{ρ}(s^{p^{m}}) is an explicit endomorphism of the module Q^{c}(Γ) ⊗_{ℚ_{p}^{c}} Vρ̌ that occurs in (13). We fix a (finitely generated) O_{ρ}[G]-module Lρ̌ which is free over O_{ρ} and such that ℚ_{p}^{c} ⊗_{O_{ρ}} Lρ̌ = Vρ̌. We then set n := dim_{ℚ_{p}^{c}}(Vρ̌), fix an O_{ρ}-basis {b_{i}}_{1≤i≤n} of Lρ̌ and write M(s^{p^{m}}) for the matrix in M_{n}(O_{ρ}[[T]]) that represents, with respect to the associated O_{ρ}[[T]]-basis {1 ⊗_{O_{ρ}} b_{i}}_{1≤i≤n}, the restriction of Δ_{ρ}(s^{p^{m}}) to the module Λ_{O_{ρ}}(Γ) ⊗_{O_{ρ}} Lρ̌ = O_{ρ}[[T]] ⊗_{O_{ρ}} Lρ̌. There exists a natural number m_{0} such that 𝔪^{p^{m}} ⊂ pR for all integers m ≥ m_{0}. For each element r of 𝔪 one also has (rT)^{p^{m}} = (∏_{i=0}^{i=p^{m}} σ^{i}(r))T^{p^{m}} ∈ 𝔪^{p^{m}}T^{p^{m}}. Thus, if s is a distinguished polynomial of degree d in Λ_{O}(G), then the binomial theorem implies that

(17)  s^{p^{m}} ≡ T^{dp^{m}} ≡ (T + 1)^{dp^{m}} − 1 (mod pΛ_{O}(G))

for all integers m ≥ m_{0}. In addition, the representation ρ̌ has open kernel and so there is a natural number m_{1} such that

(18)  (T + 1)^{dp^{m}} = γ̃^{dp^{m}} ∈ ker(ρ̌)

for all integers m ≥ m_{1}. We now set m_{ρ,s} := m_{0} + m_{1} and consider the matrix M(s^{p^{m}}) for any integer m with m ≥ m_{ρ,s}. For any such m the containment (18) implies that the matrix in M_{n}(O_{ρ}[[T]]) that represents (with respect to any choice of O_{ρ}[[T]]-basis) the action of (T + 1)^{dp^{m}} − 1 = γ̃^{dp^{m}} − 1 on O_{ρ}[[T]] ⊗_{O_{ρ}} Lρ̌ is equal to (T + 1)^{dp^{m}} − 1 times the n × n identity matrix. Thus, since s^{p^{m}} is a monic polynomial of degree dp^{m}, the congruence (17) combines with our chosen basis {1 ⊗_{O_{ρ}} b_{i}}_{1≤i≤n} of O_{ρ}[[T]] ⊗_{O_{ρ}} Lρ̌ to imply that each diagonal entry of M(s^{p^{m}}) is a monic polynomial of degree dp^{m} that is congruent modulo pO_{ρ}[[T]] to T^{dp^{m}} and in addition that each off-diagonal entry of M(s^{p^{m}}) is a polynomial of degree less than dp^{m} that belongs to pO_{ρ}[[T]]. From this explicit description it is then clear that det(M(s^{p^{m}})) is a distinguished polynomial in O_{ρ}[[T]] of degree ndp^{m}, as required. □

We can now prove the main result of this section.

Theorem 5.3. Let s_{1} and s_{2} be distinguished polynomials in Λ_{O}(G).

Then the product x := s_{1}s_{2}^{−1} belongs to Λ_{O}(G)_{S}^{×}. Further, if p is odd, O/ℤ_{p} is unramified and Φ_{\underline{G},ρ}([x]) belongs to Λ_{O_{ρ}}(Γ)^{×} for every irreducible representation ρ in A(G), then the element [x] belongs to SK_{1}(Q_{O}(G)).

Proof. We let {ρ_{i} : i ∈ I} be a (finite) set of representatives of the irreducible representations in A(G) under the following equivalence relation: ρ is equivalent to ρ′ if and only if ρ = ρ′ ⊗ ω for some ω ∈ Γ^{∨}. We then set m := ∑_{i∈I}(m_{ρ_{i},s_{1}} + m_{ρ_{i},s_{2}}) where each m_{ρ_{i},s_{j}} is the natural number that occurs in Lemma 5.2. For this choice of m Lemma 5.2 implies that for each index i the series Φ_{\underline{G},ρ_{i}}([x])^{p^{m}} = Φ_{\underline{G},ρ_{i}}([s_{1}])^{p^{m}}Φ_{\underline{G},ρ_{i}}([s_{2}])^{−p^{m}} is a quotient of distinguished polynomials in Q_{O_{ρ_{i}}} (Γ). Hence, if Φ_{\underline{G},ρ_{i}}([x]), and thus also Φ_{\underline{G},ρ_{i}}([x])^{p^{m}}, belongs to Λ_{O_{ρ_{i}}} (Γ)^{×}, then the (classical) Weierstrass Preparation Theorem for Λ_{O_{ρ_{i}}} (Γ) implies that Φ_{\underline{G},ρ_{i}}([x])^{p^{m}} = 1. Since this is true for all i in I Remark 3.3 implies that

(19)  Φ_{\underline{G},ρ}([x])^{p^{m}} = 1 for all ρ ∈ A(G).

We now write \overline{G} for the abelianisation of G and \overline{q} for the natural homomorphism Λ_{O}(G) → Λ_{O}(\overline{G}). We claim that

(20)  \overline{q}(x) = 1.

To prove this we note that \overline{G} = Δ × Γ for a suitable quotient Δ of H and that for both j = 1 and j = 2 the element

\overline{q}(s_{j}) := T^{n_{j}} + ∑_{i=0}^{i=n_{j}−1} c_{ji}T^{i}

is a distinguished polynomial in Λ_{O}(\overline{G}) = O[Δ][[T]]. We fix an irreducible (degree one) representation \overline{ρ} in A(\overline{G}) and write ρ for the associated element of A(G). Then

Φ_{\underline{G},ρ}([s_{j}]) = Φ_{\underline{\overline{G}},\overline{ρ}}([\overline{q}(s_{j})]) = T^{n_{j}} + ∑_{i=0}^{i=n_{j}−1} \overline{ρ}(c_{ji})T^{i}

is a distinguished polynomial in Λ_{O_{ρ}}(Γ) for j = 1 and j = 2. But, by assumption, one has

Φ_{\underline{\overline{G}},\overline{ρ}}([\overline{q}(s_{1})])Φ_{\underline{\overline{G}},\overline{ρ}}([\overline{q}(s_{2})])^{−1} = Φ_{\underline{G},ρ}([x]) ∈ Λ_{O_{ρ}}(Γ)^{×},

and so the Weierstrass Preparation Theorem for Λ_{O_{ρ}}(Γ) implies that Φ_{\underline{\overline{G}},\overline{ρ}}([\overline{q}(s_{1})]) = Φ_{\underline{\overline{G}},\overline{ρ}}([\overline{q}(s_{2})]). Thus one has n_{1} = n_{2} and \overline{ρ}(c_{1i}) = \overline{ρ}(c_{2i}) for all \overline{ρ} and all integers i with 0 ≤ i < n_{1}. This in turn implies that c_{1i} = c_{2i} for all i and hence that \overline{q}(s_{1}) is equal to \overline{q}(s_{2}), as required to prove (20). Now from [19, Lem. 2.1(iii)] we know that Λ_{O}(G)_{S} is equal to the localisation Λ_{O}(G)_{•} of Λ_{O}(G) defined by Ritter and Weiss in [36, Introduction]. Hence, if p is odd and O/ℤ_{p} is unramified, then we can apply Wall’s Theorem on torsion determinants (as extended to Iwasawa algebras in [36, Lem. 12]) to deduce from (19) and (20) that Φ_{\underline{G},ρ}([x]) = 1 for all ρ in A(G). (In making this deduction note Remark 3.2 implies that Φ_{\underline{G},ρ}([x]) = Det_{Q_{O}(G)}([x])(ρ) in the notation of loc. cit.) This fact in turn implies that [x] belongs to EK_{1}(Q_{O}(G)) = SK_{1}(Q_{O}(G)), as required. □

Remark 5.4. The result of [36, Lem. 12], and hence also the argument used to prove Theorem 5.3, does not extend to the case of (finite) ramified extensions O of ℤ_{p}.

5.2. The higher rank case. The Weierstrass Preparation Theorem of Venjakob [42, Cor. 3.2] is only valid for certain pro-p compact p-adic Lie groups as in (1) and gives rise to a difficult technical problem: there are ‘left-hand’ and ‘right-hand’ versions of the result and the connection between them is extremely difficult to describe explicitly (for an example see [13, §2, Ex. 1]). By using Theorem 5.3 we now prove a ‘Weierstrass Preparation Theorem’ that is more general than Venjakob’s and also intrinsically commutative in nature.

Corollary 5.5. Let G be a pro-p compact p-adic Lie group as in (1) and O a finite extension of ℤ_{p} in ℚ_{p}^{c}. Then every element ξ in the image of the composite map S ⊂ Λ_{O}(G)_{S^{∗}}^{×} → K_{1}(Λ_{O}(G)_{S^{∗}}) has a decomposition of the form

(21)  ξ = ξ^{U}ξ^{D} = ξ^{D}ξ^{U}

that satisfies all of the following conditions.

(i) ξ^{U} ∈ im(α_{O,G,S^{∗}}).

(ii) For all G in Q_{1}(G) the element q_{G}(ξ^{D}) is the image of a distinguished poly-

nomial under the composite map S ⊂ Λ_{O}(G)_{S^{∗}}^{×} → K_{1}(Λ_{O}(G)_{S^{∗}}). (iii) If p is odd and O/ℤ_{p} is unramified, then the elements ξ^{U} and ξ^{D} are unique

up to multiplication by an element of im(α_{O,G,S^{∗}}) ∩ EK_{1}(Λ_{O}(G)_{S^{∗}}).

Proof. We first establish the existence of a decomposition (21). To do this we fix an element s of S which maps to ξ in K_{1}(Λ_{O}(G)_{S^{∗}}). Then for each G in Q_{1}(G) the Weierstrass Preparation Theorem of Venjakob [42, Cor. 3.2] applies to the image s_{G} of s under the natural map Λ_{O}(G)_{S^{∗}}^{×} → Λ_{O}(G)_{S^{∗}}^{×} and implies that there is a unique element u_{G} of Λ_{O}(G)^{×} and a unique distinguished polynomial d_{G} in Λ_{O}(G) with s_{G} = u_{G}d_{G}. Since (s_{G})_{G} ∈ \underleftarrow{lim}_{G} Λ_{O}(G) the uniqueness of these decompositions implies that (u_{G})_{G} belongs to \underleftarrow{lim}_{G} Λ_{O}(G)^{×} and hence identifies with an element u of Λ_{O}(G)^{×}. We define ξ^{U} and ξ^{D} to be the images of u and u^{−1}s under the natural map Λ_{O}(G)_{S^{∗}}^{×} → K_{1}(Λ_{O}(G)_{S^{∗}}). It is then clear that ξ = ξ^{U}ξ^{D} = ξ^{D}ξ^{U}, that ξ^{U} ∈ im(α_{O,G,S^{∗}}) as required by (i), and that for all G ∈ Q_{1}(G) one has q_{G}(ξ^{D}) = [u_{G}^{−1}][s_{G}] = [d_{G}] so that ξ^{D} satisfies (ii). To prove (iii) we assume that also ξ = ξ̃^{U} ξ̃^{D} where ξ̃^{U} and ξ̃^{D} satisfy conditions (i) and (ii) respectively. Then it is clear that η := ξ̃^{D}(ξ^{D})^{−1} = (ξ̃^{U})^{−1}ξ^{U} belongs to im(α_{O,G,S^{∗}}). For each G in Q_{1}(G) it is also clear that q_{G}(η) = q_{G}(ξ̃^{D})q_{G}(ξ^{D})^{−1} belongs to im(α_{O,G,S^{∗}}) and is represented by a quotient of distinguished polynomials in Λ_{O}(G). Thus, since O/ℤ_{p} is unramified and Φ_{\underline{G},ρ}(im(α_{O,G,S^{∗}})) ⊆ Λ_{O_{ρ}}(Γ)^{×} for all ρ ∈ A(G), Theorem 5.3 (with G replaced by G) implies q_{G}(η) ∈ SK_{1}(Q_{O}(G)). From Lemma 3.4 it then follows that η belongs to EK_{1}(Λ_{O}(G)_{S^{∗}}), and hence to EK_{1}(Λ_{O}(G)_{S^{∗}}) ∩ im(α_{O,G,S^{∗}}), as required. □

Remark 5.6. The group im(α_{O,G,S^{∗}}) ∩ EK_{1}(Λ_{O}(G)_{S^{∗}}) that occurs in Corollary 5.5(iii) also measures the ambiguity in the choice of a potential validating element for the main conjectures discussed in §1.2.2 (cf. Remark 4.4).

Remark 5.7. A compact p-adic Lie group G is said to be ‘p-elementary’ if it is a direct product of a pro-p group P and a finite abelian group of order prime to p. (So, in particular, every abelian compact p-adic Lie group is p-elementary). In this case Λ(G) is a finite product of algebras Λ_{O}(P) for suitable finite unramified extensions O of ℤ_{p} and P has the structure (1) if G does. Via this decomposition of Λ(G) all of the results of this section extend directly to the case of p-elementary compact p-adic Lie groups G with the structure (1).

6. Relations between abelian subquotients

In this section we investigate the homomorphism

Δ_{O,G,S} : K_{1}(Λ_{O}(G)_{S}) → ∏_{A,ϕ} Q_{O[ϕ]}(Γ(A))^{×},

as defined in (6), where we recall A runs over SQ_{1}^{ab}(G) and ϕ over H(A)^{∨}. Our main result concerning this homomorphism will subsequently play a key role in the proof of Corollary 2.3 and to state it we set Δ_{O,G} := Δ_{O,G,S} ∘α_{O,G,S} with α_{O,G,S} the homomorphism K_{1}(Λ_{O}(G)) → K_{1}(Λ_{O}(G)_{S}) that occurs in (2).

Theorem 6.1.

(i) ker(Δ_{O,G,S}) = EK_{1}(Λ_{O}(G)_{S}).

(ii) If p is odd and O/ℤ_{p} is unramified, then one has

im(Δ_{O,G,S}) ∩ ∏_{A,ϕ} Λ_{O[ϕ]}(Γ(A))^{×} = im(Δ_{O,G}),

where in the product A runs over SQ_{1}^{ab}(G) and ϕ over H(A)^{∨}.

Remark 6.2. In this remark we assume that G has rank one. Then Remark 3.2 combines with the equality in Theorem 6.1(ii) to imply that if an element ξ of K_{1}(Λ_{O}(G)_{S}) is such that Det_{Q_{O}(G)}(ξ)(ρ) belongs to Λ_{O_{ρ}}(Γ(\underline{G}))^{×} for all ρ in A(G), then Det_{Q_{O}(G)}(ξ) belongs to Det_{Q_{O}(G)}(ker(∂_{O,G,S})). This observation gives a new proof of the main K-theoretical result of Ritter and Weiss in [36] (this result is stated as Theorem B in loc. cit. and assumes that G is a pro-p group): we note, in particular, that this proof avoids any use of the theory of ‘integral logarithms’ which is the key tool in the proof given in [36]. However, if one removes the hypothesis that G is a pro-p group, then our proof of Theorem 6.1(ii) proceeds via a reduction to the case of p-elementary groups and it is not clear to us whether such a reduction step can also be made without using integral logarithms (for further details see Remark 6.8).

Remark 6.3. By dealing with p-adic Lie groups of arbitrarily high rank our methods can also be used to extend the main arithmetic results of Ritter and Weiss in [35] and [36]. Indeed, the latter results assume, in terms of the notation used in loc. cit., that the Galois group X_{Σ}(K_{∞}) belongs to the category 𝔐_{S}(Λ(G)). However, our approach can be used to prove the statements of [35, Th. A] and [36, Th. A] under the presumably weaker hypothesis that there exists a totally real compact p-adic Lie extension K′ of k with K_{∞} ⊆ K′ and such that both G_{K′/k} has no element of order p, (which, we recall, is always possible by Remark 1.4) and in addition the group X_{Σ}(K′) belongs to the category 𝔐_{S^{∗}}(Λ(G_{K′/k})).

Remark 6.4. For certain special classes of p-adic Lie groups G the equality of Theorem 6.1(ii) has already been proved by Kato in [30, 31], by Hara in [26, 27] and by Kakde in [28]. In addition, the referee has informed us that a more general version of Theorem 6.1(ii) (in which, in particular, there is no hypothesis that p is odd and O/ℤ_{p} is unramified) has recently been proved by Witte in [44, Cor. 3.5]. It is also interesting to note that the methods that are used in loc. cit. are very different from those used here and rely on a systematic use of Waldhausen K-theory.

6.1. The proof of Theorem 6.1(i). We regard each ρ ∈ A(G) as a representation \overline{ρ} of the finite group G(ρ) := G/ ker(ρ). Then, by applying Brauer’s induction theorem to G(ρ), we deduce the existence of a finite set {U_{i}/ ker(ρ) : i ∈ I} of subgroups of G(ρ) and an isomorphism of virtual representations \overline{ρ} ≅ ∑_{i∈I} m_{i} · I_{U_{i}/ ker(ρ)}^{G(ρ)}(ψ_{i}) with m_{i} ∈ ℤ and ψ_{i} ∈ (U_{i}/ ker(ρ))^{∨} for all i ∈ I. By inflation we regard each ψ_{i} as an an element of A_{i}^{∨} with A_{i} the abelianisation of U_{i}/(ker(ρ) ∩ H) ∈ SQ_{1}(G) and then decompose it as a product ψ_{i} = ϕ_{i} ×ω_{i} with ϕ_{i} ∈ H(A_{i})^{∨} and ω_{i} ∈ Γ(A_{i})^{∨}. It follows that ρ ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ϕ_{i} ⊗ ω_{i}) and hence for each ξ in K_{1}(Λ_{O}(G)_{S}) that

(22) Φ_{\underline{G},ρ}(ξ) = ∏_{i∈I} Φ_{\underline{G},I_{A_{i}}^{G} (ϕ_{i}⊗ω_{i})}(ξ)^{m_{i}} = ∏_{i∈I} Φ_{\underline{A_{i}},ϕ_{i}⊗ω_{i}}(q_{A_{i}}(ξ))^{m_{i}}

= ∏_{i∈I} tw_{ω_{i}}(Φ_{\underline{A_{i}},ϕ_{i}}(q_{A_{i}}(ξ)))^{m_{i}} = ∏_{i∈I} tw_{ω_{i}}(q_{A_{i},ϕ_{i}} ∘ q_{A_{i}}(ξ))^{m_{i}}

where the first equality is clear, the second is by Lemma 3.6(i), the third by Remark 3.3 and the last because Φ_{\underline{A_{i}},ϕ_{i}} identifies with q_{A_{i},ϕ_{i}} (cf. §2.1.2). Since each function tw_{ω_{i}} is an automorphism of Q_{O[ϕ_{i}]}(Γ(A_{i}))^{×} this equality implies that Φ_{\underline{G},ρ}(ξ) = 1 for all ρ ∈ A(G) if and only if q_{A,ϕ} ∘q_{A}(ξ) = 1 for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨}. This implies that EK_{1}(Λ_{O}(G)_{S}) = ker(Δ_{O,G,S}) and so completes the proof of Theorem 6.1(i).

6.2. The proof of Theorem 6.1(ii). We assume in the rest of §6 that p is odd and O/ℤ_{p} is unramified and write Δ′_{O,G,S} for the composite homomorphism

K_{1}(Λ_{O}(G)_{S}) ⟶^{Δ_{O,G,S}} ∏_{A,ϕ} Q_{O[ϕ]}(Γ(A))^{×} → ∏_{A,ϕ}(Q_{O[ϕ]}(Γ(A))^{×}/Λ_{O[ϕ]}(Γ(A))^{×})

where the second arrow is the product of the natural projection maps. Then Theorem 6.1(ii) is true if and only if one has

ker(Δ′_{O,G,S}) = im(α_{O,G,S}) + EK_{1}(Λ_{O}(G)_{S}).

Now it is clear that im(α_{O,G,S}) ⊆ ker(Δ′_{O,G,S}) and that Theorem 6.1(i) implies EK_{1}(Λ_{O}(G)_{S}) ⊆ ker(Δ′_{O,G,S}) and so we are reduced to proving that every element of ker(Δ′_{O,G,S}) belongs to im(α_{O,G,S}) + EK_{1}(Λ_{O}(G)_{S}). We therefore fix an element ξ of ker(Δ′_{O,G,S}) for the remainder of §6. Our proof of Theorem 6.1(ii) will proceed by showing that q_{G}(ξ) belongs to im(α_{O,G,S})+EK_{1}(Λ_{O}(G)_{S}) for all G ∈ SQ_{1}^{ab}(G) in §6.2.1, for all ‘p-elementary sections’ on G in §6.2.2, for all G ∈ Q_{1}(G) in §6.2.3 and then finally for G = G in §6.2.4.

6.2.1. The abelian rank one case. For each abelian subquotient A in SQ_{1}^{ab}(G) the following result can be applied to the element ξ_{A} = q_{A}(ξ) and shows that it belongs to im(α_{O,A,S}).

Lemma 6.5. Assume that p is odd and let A be an abelian group of rank one as in (1). Then an element ξ_{A} of K_{1}(Q_{O}(A)) ≅ Q_{O}(A)^{×} belongs to im(α_{O,A,S}) ≅ Λ_{O}(A)^{×} if and only if it belongs to K_{1}(Λ_{O}(A)_{S}) = Λ_{O}(A)_{S}^{×} and for all ϕ ∈ H(A)^{∨} the element ξ_{A,ϕ} := q_{A,ϕ}(ξ_{A}) belongs to Λ_{O[ϕ]}(Γ(A))^{×}.

Proof. Necessity of the given conditions is clear. To prove sufficiency we use Lemma 5.1 (and Remark 5.7) to note that, after multiplying by an element of Λ_{O}(A)^{×}, we may assume ξ_{A} = ss̃^{−1} where s and s̃ are distinguished polynomials in Λ_{O}(A). It suffices to prove that in this case the stated conditions imply ξ_{A} = 1. We regard each ϕ ∈ H(A)^{∨} as an Artin representation of A as in §2.1.2. Then ξ_{A,ϕ} = Φ_{\underline{A},ϕ}([ξ_{A}]) and so for all ω ∈ Γ(A)^{∨} the stated conditions combine with Remark 3.3 to imply that

Φ_{\underline{A},ϕ⊗ω}([ξ_{A}]) = tw_{ω}(Φ_{\underline{A},ϕ}([ξ_{A}])) = tw_{ω}(ξ_{A,ϕ}) ∈ Λ_{O[ϕ]}(Γ(A))^{×}.

Thus, as every irreducible Artin representation of A is of the form ϕ⊗ω, Theorem 5.3 (and Remark 5.7) implies ξ_{A} ∈ SK_{1}(Q_{O}(A)) and hence ξ_{A} = 1 (by Remark 3.5). □

6.2.2. The p-elementary rank one case. We recall that a ‘p-elementary section’ of G is an element J of SQ_{1}(G) that is a p-elementary group in the sense of Remark 5.7. We write SQ_{1}^{p−el}(G) for the set of such subquotients. For each J in SQ_{1}^{p−el}(G) one has SQ_{1}^{ab}(J) ⊆ SQ_{1}^{ab}(G) and there is an associated triple \underline{J} as defined in §2.1.1. Thus, if A is in SQ_{1}^{ab}(J), then Lemma 6.5 implies that q_{A}(ξ) belongs to im(α_{O,A,S}). Since also q_{J} (ξ) belongs to K_{1}(Λ_{O}(J)_{S}) and one has q_{A} = q_{J ,A} ∘ q_{J}, the following result (with ξ_{J} = q_{J} (ξ)) therefore implies that q_{J} (ξ) belongs to the subgroup im(α_{O,J ,S}) + SK_{1}(Λ_{O}(J)_{S}), as required.

Lemma 6.6. Assume that p is odd and fix J in SQ_{1}^{p−el}(G) and an element ξ_{J} in K_{1}(Λ_{O}(J)_{S}). If q_{J ,A}(ξ_{J}) belongs to the subgroup im(α_{O,A,S}) for all abelian subquotients A in SQ_{1}^{ab}(J), then ξ_{J} belongs to the subgroup im(α_{O,J ,S}) + SK_{1}(Λ_{O}(J)_{S}).

Proof. Let x be a pre-image of ξ_{J} under the surjection ϖ : Λ_{O}(J)_{S}^{×} → K_{1}(Λ_{O}(J)_{S}). Then, as ϖ(Λ_{O}(J)^{×}) = im(α_{O,J ,S}), Lemma 5.1 allows us to assume x = ss̃^{−1} where s and s̃ are distinguished polynomials in Λ_{O}(J). For each ρ ∈ A(J) we now use the expression for Φ_{\underline{J} ,ρ}(ξ_{J}) given by (22) (with G = J and ξ = ξ_{J}). Since, by assumption, each element q_{J ,A_{i}}(ξ_{J}) belongs to im(α_{O,A_{i},S}) ≅ Λ_{O}(A_{i})^{×} this expression implies that Φ_{\underline{J} ,ρ}(ξ_{J}) belongs to Λ_{O_{ρ}}(Γ)^{×}. We may thus apply Theorem 5.3 (and Remark 5.7) to deduce that ξ_{J} belongs to SK_{1}(Λ_{O}(J)_{S}), as required. □

6.2.3. The general rank one case. For our chosen element ξ in ker(Δ′_{O,G,S}) the argument in §6.2.2 has shown that q_{J} (ξ) belongs to im(α_{O,J ,S}) + SK_{1}(Λ_{O}(J)_{S}) for all subquotients J in SQ_{1}^{p−el}(G). The commutativity of diagram (14) then implies that for every subquotient G in Q_{1}(G) we can apply the result of Lemma 6.7 below with y = Nrd_{Λ_{O}(G)_{S}}(q_{G}(ξ)) to deduce that the element Nrd_{Λ_{O}(G)_{S}}(q_{G}(ξ)) belongs to Nrd_{Λ_{O}(G)_{S}}(im(α_{O,G,S})). This in turn implies that q_{G}(ξ) belongs to im(α_{O,G,S}) + SK_{1}(Λ_{O}(G)_{S}), as required. Before stating the next result we recall (from [34, the proof of Th. 8]) that there is a canonical group isomorphism of the form

(23)  ι_{G} : ζ(Q_{O}(G))^{×} ≅ Hom^{E}(R(G), Q^{c}(Γ(G))^{×}).

Here E denotes the field of fractions of O and R(G) the additive group that is generated by the set A(G) and we write Hom^{E} for the collection of homomorphisms f : R(G) → Q^{c}(Γ(G))^{×} which for all ρ ∈ A(G) satisfy both f(ρ^{α}) = f(ρ)^{α} for all α ∈ G_{ℚ_{p}^{c}/E} and f(ρ ⊗ ω) = tw_{ω}(f(ρ)) for all ω ∈ Γ(G)^{∨}. Under this isomorphism one has ι_{G}(z)(ρ) = j_{ρ}(z) for each z ∈ ζ(Q_{O}(G))^{×} and ρ ∈ A(G), where j_{ρ} is the homomorphism discussed in §3.1. In particular, from Remark 3.2 one has

(24)  ι_{G}(Nrd_{Q_{O}(G)}(y)) = Det_{Q_{O}(G)}(y)

for all y in K_{1}(Λ_{O}(G)_{S}). We further recall that if G is any rank one group and J any element of SQ_{1}(G), then there is a canonical ‘restriction’ homomorphism of abelian groups

q′_{G,J} : ζ(Q_{O}(G))^{×} → ζ(Q_{O}(J))^{×}

which fits into a commutative diagram as in (14). The following result is due to Ritter and Weiss.

Lemma 6.7. Let G be any rank one group with the structure (1). Then an element y of im(Nrd_{Λ_{O}(G)_{S}}) belongs to the subgroup Nrd_{Λ_{O}(G)_{S}}(im(α_{O,G,S})) if and only if for all J in SQ_{1}^{p−el}(G) the element q′_{G,J} (y) belongs to Nrd_{Λ_{O}(J)_{S}}(im(α_{O,J ,S})).

Proof. For each group G we set α_{G} := α_{O,G,S}. If y = Nrd_{Λ_{O}(G)_{S}}(α_{G}(y′)) for an element y′ in K_{1}(Λ_{O}(G)), then for every subquotient J = U/J in SQ_{1}^{p−el}(G) one has

q′_{G,J} (y) = Nrd_{Λ_{O}(J)_{S}}(q_{G,J} (α_{G}(y′))) = Nrd_{Λ_{O}(J)_{S}}(α_{J} (q′′_{G,J} (y′)))

where the first equality follows from the commutativity of (14) and in the second we write q′′_{G,J} for the composite K_{1}(Λ_{O}(G)) → K_{1}(Λ_{O}(U)) → K_{1}(Λ_{O}(J)) of the natural restriction and projection maps and use the obvious fact that q_{G,J} ∘ α_{G} = α_{J} ∘ q′′_{G,J}. The above formula for q′_{G,J} (y) shows that the stated conditions are necessary. To prove the converse we write y = Nrd_{Λ_{O}(G)_{S}}(y′) with y′ in K_{1}(Λ_{O}(G)_{S}). Then, since SQ_{1}^{ab}(G) ⊆ SQ_{1}^{p−el}(G), the stated assumptions combine with the commutativity of (14) to imply that q_{G,A}(y′) belongs to Λ_{O}(A)^{×} for all A in SQ_{1}^{ab}(G). By using Brauer induction as in (22) this then shows that for all ρ in A(G) the series Φ_{\underline{G},ρ}(y′) = Det_{Q_{O}(G)}(y′)(ρ) is a unit in Λ^{c}(Γ(G)) := ℤ_{p}^{c} ⊗_{ℤ_{p}} Λ(Γ(G)), where we write ℤ_{p}^{c} for the integral closure of ℤ_{p} in ℚ_{p}^{c}. Since any such function Det_{Q_{O}(G)}(y′) automatically satisfies the remaining conditions which define the subgroup HOM^{E}(R(G), Λ^{c}(Γ(G))^{×}) of Hom^{E}(R(G), Λ^{c}(Γ(G))^{×}), we may therefore deduce from the proof of [35, Th. B] that there exists an element y′′ of K_{1}(Λ_{O}(G)) with Det_{Q_{O}(G)}(y′) = Det_{Q_{O}(G)}(α_{G}(y′′)). From the equality (24) it then follows that y = Nrd_{Λ_{O}(G)_{S}}(y′) = Nrd_{Λ_{O}(G)_{S}}(α_{G}(y′′)), as required. □

Remark 6.8. The result of Ritter and Weiss in [35, Th. B] plays the key role in the proof of Lemma 6.7 given above. The proof of this result in loc. cit. proceeds by means of a reduction to the p-elementary case that was discussed in Remark 6.2 and this reduction argument combines certain standard induction techniques together with the natural character action on K_{1}-groups and also makes important use of the theory of integral logarithms.

6.2.4. The general case. The argument given in §6.2.3 above has shown that, for all subquotients G in Q_{1}(G), the image under q_{G} of our chosen element ξ belongs to im(α_{O,G,S}) + SK_{1}(Λ_{O}(G)_{S}). Theorem 6.1(ii) therefore follows directly by applying the following result with ξ′ = ξ.

Lemma 6.9. Fix an element ξ′ in K_{1}(Λ_{O}(G)_{S}). If q_{G}(ξ′) belongs to im(α_{O,G,S}) + SK_{1}(Λ_{O}(G)_{S}) for all G ∈ Q_{1}(G), then ξ′ belongs to im(α_{O,G,S}) + EK_{1}(Λ_{O}(G)_{S}).

Proof. In this argument all inverse limits are over Q_{1}(G). For each G ∈ Q_{1}(G) we set α_{G} := α_{O,G,S}, D(G) := im(α_{G}), Nrd_{G} := Nrd_{Λ_{O}(G)_{S}}, SK(G) := SK_{1}(Λ_{O}(G)_{S})(= ker(Nrd_{G})) and B(G) := D(G) ∩ SK(G). Then there is a short exact sequence

(25)  0 → ker(α_{G}) → ker(Nrd_{G} ∘ α_{G}) → B(G) → 0

in which the third arrow is induced by α_{G}. By taking inverse limit of these sequences we obtain an exact sequence

\underleftarrow{lim}_{G}^{1} ker(Nrd_{G} ∘ α_{G}) → \underleftarrow{lim}_{G}^{1}B(G) → \underleftarrow{lim}_{G}^{2} ker(α_{G}).

But \underleftarrow{lim}_{G}^{2} ker(α_{G}) vanishes (by Roos [38]) whilst the argument of Proposition 4.2 implies \underleftarrow{lim}_{G}^{1} ker(Nrd_{G} ∘ α_{G}) vanishes and so \underleftarrow{lim}_{G}^{1}B(G) must also vanish. Hence upon taking the inverse limit over the obvious exact commutative diagram

0 ⟶ B(G) ⟶  SK(G)  ⟶  SK(G)/B(G)  ⟶ 0
⊆↓  ^{⊆}↓  ≅↓
0 ⟶ D(G) ⟶ D(G) + SK(G) ⟶ (D(G) + SK(G))/D(G) ⟶ 0

we obtain an exact commutative diagram of the form

\underleftarrow{lim}_{G}SK(G)  ⟶  \underleftarrow{lim}_{G}(SK(G)/B(G))  ⟶ 0
⊆↓  ≅↓
\underleftarrow{lim}_{G} D(G) ⟶^{θ} \underleftarrow{lim}_{G}(D(G) + SK(G)) ⟶ \underleftarrow{lim}_{G}(D(G) + SK(G))/D(G).

Now since, by assumption, (q_{G}(ξ′))_{G} ∈ \underleftarrow{lim}_{G}(D(G) + SK(G)) an easy diagram chase shows that we may choose (z_{G})_{G} ∈ \underleftarrow{lim}_{G}SK(G) with (q_{G}(ξ′)z_{G}^{−1})_{G} = θ((y_{G})_{G}) for some (y_{G})_{G} ∈ \underleftarrow{lim}_{G}D(G). Next we compare the sequences obtained by taking the inverse limits of (25) and of the tautological exact sequences 0 → ker(α_{G}) → K_{1}(Λ_{O}(G)) → D(G) → 0. Since \underleftarrow{lim}_{G}^{1} ker(Nrd_{G} ∘ α_{G}) vanishes, we obtain in this way an exact commutative diagram

\underleftarrow{lim}_{G} ker(Nrd_{G} ∘ α_{G}) ⟶ \underleftarrow{lim}_{G}B(G) ⟶ \underleftarrow{lim}_{G}^{1} ker(α_{G}) ⟶ 0
⊆↓  ^{⊆}↓  ∥
\underleftarrow{lim}_{G}K_{1}(Λ_{O}(G)) ⟶^{ϕ} \underleftarrow{lim}_{G}D(G) ⟶ \underleftarrow{lim}_{G}^{1} ker(α_{G}).

This diagram implies there exists an element (z̃_{G})_{G} of \underleftarrow{lim}_{G}B(G) ⊆ \underleftarrow{lim}_{G}SK(G) with (y_{G}z̃_{G}^{−1})_{G} ∈ im(ϕ). Recalling the first isomorphism in (16) we may therefore choose u ∈ K_{1}(Λ_{O}(G)) with (q_{G}(α_{G}(u)))_{G} = (y_{G}z̃_{G}^{−1})_{G} where α_{G} := α_{O,G,S}, and hence

q_{G}(α_{G}(u)^{−1}ξ′) = y_{G}^{−1}z̃_{G}q_{G}(ξ′) = z_{G}z̃_{G} ∈ SK(G)

for all G ∈ Q_{1}(G).

Now every ρ ∈ A(G) factors through a representation \overline{ρ} of G_{ρ} := G/(H ∩ ker(ρ)) ∈ Q_{1}(G). From (9) and Lemma 3.4, it therefore follows that

(α_{G}(u)^{−1}ξ′)∗ ; \underline{G}(ρ) = (q_{G_{ρ}}(α_{G}(u)^{−1}ξ′))∗ ; \underline{G_{ρ}}(\overline{ρ}) = (z_{G_{ρ}} z̃_{G_{ρ}})∗ ; \underline{G_{ρ}}(\overline{ρ}) = 1

for all ρ ∈ A(G) and hence α_{G}(u)^{−1}ξ′ ∈ EK_{1}(Λ_{O}(G)_{S}). It is thus clear that ξ′ = α_{G}(u)(α_{G}(u)^{−1}ξ′) belongs to im(α_{G}) + EK_{1}(Λ_{O}(G)_{S}), as claimed. □

7. The proofs of Theorem 2.2 and Corollary 2.3

In this section we prove Theorem 2.2 and Corollary 2.3. At the outset we fix data \underline{G}, C^{•}, (ξ_{ρ})_{ρ} and (ξ_{ρ}^{∗})_{ρ} as in Theorem 2.2.

7.1. The integers m_{ρ}(C^{•}). We define the integers m_{ρ}(C^{•}) that occur in Theorem 2.2(b) in the following way. For each C^{•} in D_{S^{∗}}^{p}(Λ(G)) we write χ_{G}^{μ}(C^{•}) for the ‘equivariant multiplicative μ-invariant’ that is defined by Venjakob and the present author in [15, §4.1] (this is an element of K_{1}(Λ(G)_{S^{∗}})). We recall that, by the very definition of χ_{G}^{μ}(C^{•}), for each ρ ∈ A(G) there is a (unique) integer m_{ρ}(C^{•}) with

(26)  Φ_{\underline{G},ρ}(χ_{G}^{μ}(C^{•})) = p^{m_{ρ}(C^{•})} ∈ Q(Γ)^{×}.

We shall use the following properties of these integers.

Lemma 7.1. Fix C^{•} in D_{S^{∗}}^{p}(Λ(G)).

(i) For all ρ ∈ A(G) and ω ∈ Γ^{∨} one has m_{ρ⊗ω}(C^{•}) = m_{ρ}(C^{•}).

(ii) For all A ∈ SQ_{1}^{ab}(G) one has q_{A}(χ_{G}^{μ}(C^{•})) = χ_{A}^{μ}(C_{A}^{•}).

(iii) For all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} one has m_{I_{A}^{G}(ϕ)}(C^{•}) = m_{ϕ}(C_{A}^{•}) = μ_{ϕ}(C_{A}^{•}), where the latter integer is as defined in §2.1.3.

Proof. Claim (i) is true since Remark 3.3 combines with (26) to imply that

p^{m_{ρ⊗ω}(C^{•})} = Φ_{\underline{G},ρ⊗ω}(χ_{G}^{μ}(C^{•})) = tw_{ω}(Φ_{\underline{G},ρ}(χ_{G}^{μ}(C^{•}))) = tw_{ω}(p^{m_{ρ}(C^{•})}) = p^{m_{ρ}(C^{•})}.

Claim (ii) is proved in [15, Lem. 4.5 and Lem. 4.6]. To prove claim (iii) we note Lemma 3.6(i) combines with claim (ii) to imply that

p^{m_{I^{G}_{A}(ϕ)}(C^{•})} = Φ_{\underline{G},I_{A}^{G}(ϕ)}(χ_{G}^{μ}(C^{•})) = Φ_{\underline{A},ϕ}(q_{A}(χ_{G}^{μ}(C^{•}))) = Φ_{\underline{A},ϕ}(χ_{A}^{μ}(C_{A}^{•})) = p^{m_{ϕ}(C_{A}^{•})}

and hence that m_{I_{A}^{G}(ϕ)}(C^{•}) = m_{ϕ}(C_{A}^{•}). It is thus enough to prove m_{ϕ}(C_{A}^{•}) = μ_{ϕ}(C_{A}^{•}) and this follows directly from (26) and the explicit definitions of χ_{A}^{μ}(C_{A}^{•}) and μ_{ϕ}(C_{A}^{•}). Indeed, the key point in this regard is simply the following: if (as in [15, §3.3]) one labels the primitive idempotents of the semisimple ℤ_{p}-algebra ℤ_{p}[H(A)_{p′}] as {e_{i} : i ∈ I} and writes ℤ_{p}[H(A)_{p′}]e_{i(ϕ)} for the (unique) simple component through which the restriction of ϕ to H(A)_{p′} factors, then μ_{ϕ}(C_{A}^{•}) is equal to the integer μ_{Λ(A)}^{i(ϕ)}(C_{A}^{•}) that is defined at the end of [15, §3.4.1]. □

7.2. A first reduction step. We now reduce the proof of Theorem 2.2 to consideration of complexes in D_{S}^{p}(Λ(G)). The proof of the next result uses the following general facts about refined Euler characteristics (for more details of which see [15] and the references indicated therein). Let G be any group as in (1) and C^{•} a complex in D_{S^{∗}}^{p}(Λ_{O}(G)).

• For any exact triangle C_{1}^{•} → C^{•} → C_{2}^{•} → C_{1}^{•}[1] in D_{S^{∗}}^{p}(Λ_{O}(G)) one has χ^{ref}(C^{•}) = χ^{ref}(C_{1}^{•}) + χ^{ref}(C_{2}^{•});

• If G has no element of order p, then the complex C_{1}^{•} := ⨁_{i∈ℤ} H^{i}(C^{•})[−i] belongs to D_{S^{∗}}^{p}(Λ_{O}(G)) and one has χ^{ref}(C^{•}) = χ^{ref}(C_{1}^{•});

• Fix A = U/J ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨}. Then for both S′ = S and S′ = S^{∗} there is a commutative diagram

(27)

K_{1}(Λ_{O}(G)_{S′}) ⟶^{∂_{G,S′}}  K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′})
q_{A,ϕ}∘q_{G,A}↓  ↓^{q′_{G,A,ϕ}}
Λ_{O[ϕ]}(Γ(A))_{S′}^{×} ⟶^{∂_{Γ(A),S′}} K_{0}(Λ_{O[ϕ]}(Γ(A)), Λ_{O[ϕ]}(Γ(A))_{S′}).

Here we write q′_{G,A,ϕ} for the composite homomorphism

K_{0}(Λ_{O}(G), Λ_{O}(G)_{S′}) → K_{0}(Λ_{O}(U), Λ_{O}(U)_{S′}) → K_{0}(Λ_{O}(A), Λ_{O}(A)_{S′})

→ K_{0}(Λ_{O[ϕ]}(Γ(A)), Λ_{O[ϕ]}(Γ(A))_{S′})

where the first, second and third arrows denote the natural restriction, projection and scalar extension maps and so for all C^{•} in D_{S′}^{p}(Λ_{O}(G)) one has q′_{G,A,ϕ}(χ^{ref}(C^{•})) = χ^{ref}(C_{A,ϕ}^{•}).

For each finitely generated Λ(G)-module N we write N_{tor} for its ℤ_{p}-torsion submodule and set N_{tf} := N/N_{tor}.

Lemma 7.2. We assume that G has no element of order p. We fix C^{•} in D_{S^{∗}}^{p}(Λ(G)), write C̃^{•} for the complex ⨁_{i∈ℤ} H^{i}(C^{•})_{tf}[−i] in D_{S}^{p}(Λ(G)) and for all ρ ∈ A(G) set ξ̃_{ρ} := p^{−m_{ρ}(C^{•})}ξ_{ρ} ∈ ℚ_{p}^{c} and ξ̃_{ρ}^{∗} := p^{−m_{ρ}(C^{•})}ξ_{ρ}^{∗} ∈ ℚ_{p}^{c} \ {0}.

(i) An element ξ of K_{1}(Λ(G)_{S^{∗}}) validates MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) if and only if χ_{G}^{μ}(C^{•})^{−1}ξ validates MC(\underline{G}, C̃^{•}, (ξ̃_{ρ})_{ρ}).

(ii) Fix A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨}. An element ξ_{A,ϕ} of Q_{ℤ_{p}[ϕ]}(Γ(A))^{×} validates MC(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{A,ϕ}^{•}, (ξ_{I_{A}^{G}(ϕ⊗ω)})_{ω}) if and only if ξ̃_{A,ϕ} := p^{−μ_{ϕ}(C_{A}^{•})}ξ_{A,ϕ} validates MC(\underline{Γ(A)}, ℤ_{p}[ϕ], C̃_{A,ϕ}^{•}, (ξ̃_{I_{A}^{G}(ϕ⊗ω)})_{ω}).

(iii) For every A and ϕ as in claim (ii) one has p^{−μ_{ϕ}(C̃_{A}^{•})}ξ̃_{A,ϕ} = p^{−μ_{ϕ}(C_{A}^{•})}ξ_{A,ϕ}. (iv) Claims (i), (ii) and (iii) remain valid if one simultaneously replaces ‘MC’

by ‘MC^{∗}’, the terms (ξ_{ρ})_{ρ} and (ξ̃_{ρ})_{ρ} in claim (i) by (ξ_{ρ}^{∗})_{ρ} and (ξ̃_{ρ}^{∗})_{ρ} respectively and the terms (ξ_{I_{A}^{G}(ϕ⊗ω)})_{ω} and (ξ̃_{I_{A}^{G}(ϕ⊗ω)})_{ω} in claim (ii) by

(d_{A}^{−r_{ϕ⊗ω}(C_{A,ϕ}^{•})}ξ_{I_{A}^{G}(ϕ⊗ω)}^{∗})_{ω} and (d_{A}^{−r_{ϕ⊗ω}(C̃_{A,ϕ}^{•})}ξ̃_{I_{A}^{G}(ϕ⊗ω)}^{∗})_{ω} respectively.

Proof. Set C_{1}^{•} := ⨁_{i∈ℤ} H^{i}(C^{•})[−i] and C_{2}^{•} := ⨁_{i∈ℤ} H^{i}(C^{•})_{tor}[−i]. Then, as G has no element of order p, one has

χ^{ref}(C̃^{•}) = χ^{ref}(C_{1}^{•}) − χ^{ref}(C_{2}^{•}) = χ^{ref}(C^{•}) − χ^{ref}(C_{2}^{•}) = χ^{ref}(C^{•}) − ∂_{G,S^{∗}}(χ_{G}^{μ}(C^{•}))

where the first equality is a consequence of the obvious exact triangle

C_{2}^{•} → C_{1}^{•} → C̃^{•} → C_{2}^{•}[1]

in D_{S^{∗}}^{p}(Λ(G)) and the third follows from the equality χ^{ref}(C_{2}^{•}) = ∂_{G,S^{∗}}(χ_{G}^{μ}(C^{•})) proved in [15, Prop. 4.7(i)]. Claim (i) follows directly from the last displayed equality and the fact that (26) implies χ_{G}^{μ}(C^{•})\underline{G}(ρ) = p^{m_{ρ}(C^{•})} for all ρ ∈ A(G).

We now fix A and ϕ as in claim (ii). Then the above displayed equality combines with the commutativity of diagram (27) to imply that

χ^{ref}(C̃_{A,ϕ}^{•}) = q′_{G,A,ϕ}(χ^{ref}(C̃^{•}))
= q′_{G,A,ϕ}(χ^{ref}(C^{•})) − q′_{G,A,ϕ}(∂_{G,S^{∗}}(χ_{G}^{μ}(C^{•})))
= χ^{ref}(C_{A,ϕ}^{•}) − ∂_{Γ(A),S^{∗}}(q_{A,ϕ}(q_{A}(χ_{G}^{μ}(C^{•}))))
= χ^{ref}(C_{A,ϕ}^{•}) − ∂_{Γ(A),S^{∗}}(q_{A,ϕ}(χ_{A}^{μ}(C_{A}^{•})))
= χ^{ref}(C_{A,ϕ}^{•}) − ∂_{Γ(A),S^{∗}}(p^{μ_{ϕ}(C_{A}^{•})})

where the penultimate equality follows from Lemma 7.1(ii) and the last is true because (26) (with G, ρ and C^{•} replaced by A, ϕ and C_{A}^{•}) combines with Lemma 7.1(iii) to imply q_{A,ϕ}(χ_{A}^{μ}(C_{A}^{•})) = Φ_{\underline{A},ϕ}(χ_{A}^{μ}(C_{A}^{•})) = p^{μ_{ϕ}(C_{A}^{•})}. To deduce claim (ii) from the last displayed equation it suffices to note that for all ω ∈ Γ(A)^{∨} one has

(p^{μ_{ϕ}(C_{A}^{•})})\underline{Γ(A)}(ω) = p^{μ_{ϕ}(C_{A}^{•})} = p^{m_{ϕ}(C_{A}^{•})} = p^{m_{ϕ⊗ω}(C_{A}^{•})} = p^{m_{I^{G}_{A}(ϕ⊗ω)}(C^{•})},

where the second and last equalities follow from Lemma 7.1(iii) and the third from Lemma 7.1(i). Claim (iii) simply asserts that μ_{ϕ}(C̃_{A}^{•}) = 0 and since ℤ_{p}[ϕ_{(p′)}] ⊗_{ℤ_{p}[H(A)]}^{𝕃} C̃_{A}^{•} belongs to D_{S}^{p}(Λ(Γ(A))) this follows directly from [15, Lem. 3.1(iii)] and our definition of μ_{ϕ}(C̃_{A}^{•}). Claim (iv) is now also straightforward to verify. Indeed, given the above arguments, the only point worth mentioning is that in proving the analogue of claim (ii) one uses the fact that r_{ϕ⊗ω}(C_{A,ϕ}^{•}) = r_{ϕ⊗ω}(C̃_{A,ϕ}^{•}). □

Lemma 7.2 implies that if G has no element of order p, then Theorem 2.2 and Corollary 2.3 are valid as stated if and only if they are valid with C^{•} replaced by C̃^{•} (which is an object of D_{S}^{p}(Λ(G))) and the terms ξ_{ρ}, ξ_{ρ}^{∗} and ξ_{A,ϕ} replaced by ξ̃_{ρ}, ξ̃_{ρ}^{∗} and ξ̃_{A,ϕ} respectively. If, on the other hand, G has an element of order p, then S̃ = S and so Theorem 2.2 and Corollary 2.3 both assume that C^{•} belongs to D_{S}^{p}(Λ(G)). In view of Theorem 2.1 we will therefore assume (without further comment) that for the rest of §7 the following conditions are satisfied:

• G has rank one;

• C^{•} belongs to D_{S}^{p}(Λ(G)) (so m_{ρ}(C^{•}) = 0 for all ρ ∈ A(G) and μ_{ϕ}(C_{A}^{•}) = 0

for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨}). We abbreviate MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}) and MC^{∗}(\underline{G}, C^{•}, (ξ_{ρ}^{∗})_{ρ}) to MC and MC^{∗}. For all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} we also abbreviate MC(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{A,ϕ}^{•}, (ξ_{I_{A}^{G}(ϕ⊗ω)})_{ω}) and

MC^{∗}(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{A,ϕ}^{•}, (d_{A}^{−r_{ϕ⊗ω}(C_{A,ϕ}^{•})}ξ_{I_{A}^{G}(ϕ⊗ω)}^{∗})_{ω}) to MC_{A,ϕ} and MC_{A,ϕ}^{∗} respectively. We use the following exact commutative diagram (where both rows are as in (2) and ι and ι′ denote the homomorphisms induced by the inclusion Λ(G)_{S} ⊂ Λ(G)_{S^{∗}})

K_{1}(Λ(G)) ⟶ K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{G,S^{∗}}} K_{0}(Λ(G), Λ(G)_{S^{∗}})
∥  ↑_{ι′}  ↑^{ι}
K_{1}(Λ(G)) ⟶ K_{1}(Λ(G)_{S}) ⟶^{∂_{G,S}} K_{0}(Λ(G), Λ(G)_{S}).

Since χ^{ref}(C^{•}) is assumed to belong to im(ι), and the map ∂_{G,S} is surjective (see Remark 1.1), we can fix an element ξ^{alg} of K_{1}(Λ(G)_{S}) with

∂_{G,S^{∗}}(ι′(ξ^{alg})) = χ^{ref}(C^{•}).

An easy diagram chase then shows that any element of K_{1}(Λ(G)_{S^{∗}}) that validates MC, resp. MC^{∗}, must also belong to im(ι′).

7.3. The proof of Theorem 2.2. Following the observation just made above we assume first that there exists an element ξ′ of K_{1}(Λ(G)_{S}) such that ι′(ξ′) validates MC, resp. MC^{∗}. Then (9) implies the respective condition (b) in Theorem 2.2 is satisfied by ξ_{G} := q_{G}(ξ′). In a similar way, Lemma 3.6(ii), resp. Lemma 3.6(iii), implies that for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} the element ξ_{A,ϕ} := q_{A,ϕ} ∘ q_{A}(ξ′) validates MC_{A,ϕ}(i), resp. MC_{A,ϕ}^{∗}(i). Finally we note that the commutative diagram (27) implies ξ_{A,ϕ} also validates MC_{A,ϕ}(ii), resp. MC_{A,ϕ}^{∗}(ii). This proves necessity of the conditions in Theorem 2.2. To prove the converse we note that, since G has rank one, the respective condition (b) in Theorem 2.2 implies the existence of an element ξ′ of K_{1}(Λ(G)_{S}) such that for all ρ ∈ A(G) one has (ξ′)\underline{G}(ρ ⊗ ω) = ξ_{ρ⊗ω} for almost all ω ∈ Γ^{∨}, resp. (ξ′)∗ ; \underline{G}(ρ) = ξ_{ρ}^{∗}. This combines with Lemma 3.6 to imply, in particular, that for all A and ϕ as above q_{A,ϕ} ∘ q_{A}(ξ′) validates MC_{A,ϕ}(i), resp. MC_{A,ϕ}^{∗}(i). Now the respective condition (a) implies that MC_{A,ϕ}, resp. MC_{A,ϕ}^{∗}, is valid for all A and ϕ and so Remark 3.5 implies q_{A,ϕ} ∘ q_{A}(ξ′) must also validate MC_{A,ϕ}(ii), resp. MC_{A,ϕ}^{∗}(ii). From the commutativity of diagram (27) it therefore follows that

∂_{ℤ_{p}[ϕ],Γ(A),S}(q_{A,ϕ} ∘ q_{A}(ξ^{alg})) = χ^{ref}(C_{A,ϕ}^{•}) = ∂_{ℤ_{p}[ϕ],Γ(A),S}(q_{A,ϕ} ∘ q_{A}(ξ′))

and hence

q_{A,ϕ} ∘ q_{A}(ξ′(ξ^{alg})^{−1}) ∈ ker(∂_{ℤ_{p}[ϕ],Γ(A),S}) = Λ_{ℤ_{p}[ϕ]}(Γ(A))^{×} ⊂ Q_{ℤ_{p}[ϕ]}(Γ(A))^{×}.

The element Δ_{G,S}(ξ′(ξ^{alg})^{−1}) therefore belongs to im(Δ_{G,S}) ∩ ∏_{A,ϕ} Λ_{ℤ_{p}[ϕ]}(Γ(A))^{×} and so Theorem 6.1(ii) implies that Δ_{G,S}(ξ′(ξ^{alg})^{−1}) = Δ_{G,S}(u) for some element u of im(α_{G,S}) = ker(∂_{G,S}). Applying Theorem 6.1(i) we can now deduce that the product v := u^{−1}ξ′(ξ^{alg})^{−1} belongs to ker(Δ_{G,S}) = EK_{1}(Λ(G)_{S}) and it is then straightforward to check that the element ξ′v^{−1} = ξ^{alg}u validates both conditions of MC, resp. MC^{∗}, as required. This completes the proof of Theorem 2.2.

7.4. The proof of Corollary 2.3. We assume first that there is an element ξ′ of K_{1}(Λ(G)_{S}) such that ι′(ξ′) validates MC, resp. MC^{∗}. Then, just as in §7.3, for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} the element q_{A,ϕ} ∘ q_{A}(ξ′) validates MC_{A,ϕ}, resp. MC_{A,ϕ}^{∗}. Remark 3.5 therefore implies that q_{A,ϕ} ∘ q_{A}(ξ′) is equal to ξ_{A,ϕ}, resp. ξ_{A,ϕ}^{∗}, and hence that (ξ_{A,ϕ})_{A,ϕ}, resp. (ξ_{A,ϕ}^{∗})_{A,ϕ}, is equal to Δ_{G,S}(ξ′). This proves necessity of the conditions stated in Corollary 2.3. We assume next that there is an element ξ′ of K_{1}(Λ(G)_{S}) with Δ_{G,S}(ξ′) equal to (ξ_{A,ϕ})_{A,ϕ}, resp. (ξ_{A,ϕ}^{∗})_{A,ϕ}. To deduce Corollary 2.3 from Theorem 2.2 it suffices to show that for all G ∈ Q_{1}(G) the element ξ_{G} := q_{G}(ξ′) has the interpolation properties described in Theorem 2.2(i)(b), resp. 2.2(ii)(b). This is in turn a direct consequence of the following result.

Lemma 7.3. Fix G in Q_{1}(G) and ξ in K_{1}(Λ(G)_{S}).

(i) The following conditions are equivalent: (a) for all ρ ∈ A(G) one has ξ\underline{G}(ρ ⊗ ω) = ξ_{ρ⊗ω} for almost all ω ∈ Γ^{∨};

(b) for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} one has q_{A}(ξ)\underline{A}(ϕ ⊗ ω) = ξ_{I_{A}^{G}(ϕ⊗ω)}

for almost all ω ∈ Γ(A)^{∨}. (ii) If for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} one has r_{ϕ}(q_{A}(ξ)) = r_{ϕ}(C_{A}^{•}), then

the following conditions are equivalent:

(a) for all ρ ∈ A(G) one has ξ∗ ; \underline{G}(ρ) = ξ_{ρ}^{∗};

(b) for all A ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨} one has q_{A}(ξ)∗ ; \underline{A}(ϕ ⊗ ω) =

d−r_{ϕ}(C_{A}^{•}) ; \underline{A}ξ_{I_{A}^{G}(ϕ⊗ω)}^{∗} for all ω ∈ Γ(A)^{∨}.

Proof. The fact that (i)(a) implies (i)(b) and (ii)(a) implies (ii)(b) is a direct consequence of Lemma 3.6(ii) and (iii). To prove the converse we note that, just as in §6.1, for each ρ ∈ A(G) there exists a finite subset {A_{i} : i ∈ I} of SQ_{1}^{ab}(G) for which ρ ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ψ_{i}) with m_{i} ∈ ℤ and ψ_{i} ∈ A_{i}^{∨} for each i. Now each group Γ(A_{i}) identifies with a subgroup of Γ and if for each ω ∈ Γ^{∨} we write ω_{i} for the restriction of ω to Γ(A_{i}), then ρ ⊗ ω ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ψ_{i} ⊗ ω_{i}). Hence, since each ψ_{i} ⊗ ω_{i} can be written in the form ϕ_{i} ⊗ ω′_{i} with ϕ_{i} ∈ H(A_{i})^{∨} and ω′_{i} ∈ Γ(A_{i})^{∨}, the hypothesis (i)(b) combines with Lemma 3.6(ii) to imply that for each i ∈ I one has ξ\underline{G}(I_{A_{i}}^{G}(ψ_{i} ⊗ ω_{i})) = ξ_{I_{A_{i}}^{G} (ψ_{i})⊗ω} ≠ 0 for almost all ω ∈ Γ^{∨}. For almost all ω ∈ Γ^{∨} one

therefore has ξ\underline{G}(I_{A_{i}}^{G}(ψ_{i} ⊗ ω_{i})) ≠ 0 for all i ∈ I and also

ξ\underline{G}(ρ ⊗ ω) = ∏_{i∈I} ξ\underline{G}(I_{A_{i}}^{G}(ψ_{i} ⊗ ω_{i}))^{m_{i}} = ∏_{i∈I}(ξ_{I_{A_{i}}^{G} (ψ_{i}⊗ω_{i})})^{m_{i}} = ξ_{∑_{i∈I} m_{i}·I_{A_{i}}^{G} (ψ_{i}⊗ω_{i})} = ξ_{ρ⊗ω}

where the third equality is because ξ_{ρ′+ρ′′} = ξ_{ρ′}ξ_{ρ′′} for all ρ′ and ρ′′. This shows that (i)(b) implies (i)(a). In a similar way, Lemma 3.6(iii) implies that under hypothesis (ii)(b) one has ξ∗ ; \underline{G}(ρ) = ξ_{ρ}^{∗} for all ρ ∈ A(G), as required to prove (ii)(a). □

8. p-adic L-functions and reduced norms

In this section we reinterpret the interpolation conditions of Theorem 2.2 in terms of the approach developed by Ritter and Weiss in [34, 35, 36]. To do this we fix a complex C^{•} and interpolation data (ξ_{ρ})_{ρ}, resp. (ξ_{ρ}^{∗})_{ρ}, as in Theorem 2.2 and for each subquotient G in SQ_{1}(G) we write MC_{G} and MC_{G}^{∗} in place of MC(\underline{G}, C_{G}^{•}, (ξ_{I_{G}^{G}(ψ)})_{ψ})

and MC^{∗}(\underline{G}, C_{G}^{•}, (d_{G}^{−r_{ψ}(C_{G}^{•})}ξ_{I_{G}^{G}(ψ)}^{∗})_{ψ}) respectively.

8.1. p-adic L-functions. Lemma 7.2 allows us to assume throughout this subsection (and without further comment) that the following conditions are satisfied:

• G has rank one;

• C^{•} belongs to D_{S}^{p}(Λ(G)) (so m_{ρ}(C^{•}) = 0 for all ρ ∈ A(G)). We write ℤ_{p}^{c} for the integral closure of ℤ_{p} in ℚ_{p}^{c} and set ℤ_{p}^{c}[[T]] := ℤ_{p}^{c} ⊗_{ℤ_{p}} ℤ_{p}[[T]]. For any group G as in (1) we set Λ_{ℤ_{p}^{c}}(G) := ℤ_{p}^{c} ⊗_{ℤ_{p}} Λ(G) and Q^{c}(G) := ℚ_{p}^{c} ⊗_{ℚ_{p}} Q(G). We also set Γ := Γ(G) and identify Q^{c}(Γ) with Q^{c}(ℤ_{p}[[T]]) := ℚ_{p}^{c} ⊗_{ℚ_{p}} Q(ℤ_{p}[[T]]) via γ\underline{G} − 1 = T. For G in SQ_{1}(G) we similarly identify Q^{c}(Γ(G)) with Q^{c}(ℤ_{p}[[T_{G}]]) ⊆ Q^{c}(ℤ_{p}[[T]]). For each ρ ∈ A(G) we recall the homomorphism j_{ρ} : ζ(Q(G))^{×} → Q^{c}(Γ(G))^{×} discussed in §3.1.

Theorem 8.1. Assume that MC_{A}, resp. MC_{A}^{∗}, is valid for all A ∈ SQ_{1}^{ab}(G). Then for all G ∈ SQ_{1}(G) there exists a unique element L_{G} of ζ(Q(G))^{×} which satisfies all of the following properties.

(i) For all ρ ∈ A(G) one has j_{ρ⊗ω}(L_{G})(0) = ξ_{I_{G}^{G}(ρ⊗ω)} for almost all ω ∈ Γ(G)^{∨},

resp. j_{ρ}(L_{G})^{∗}(0) = d_{G}^{−r_{ρ}(C_{G}^{•})}ξ_{I_{G}^{G}(ρ)}^{∗}. (ii) q′_{G,G}(L_{G}) = L_{G} with q′_{G,G} the homomorphism in diagram (14). (iii) j_{ρ}(L_{G}) ∈ Λ_{ℤ_{p}^{c}}(Γ(G))_{S}^{×} for all ρ ∈ A(G). (iv) L_{G} belongs to im(Nrd_{Λ(G)_{S}}) if and only if there exists an element ξ_{G} in

K_{1}(Λ(G)_{S}) for which for all ρ ∈ A(G) one has (ξ_{G})\underline{G}(ρ ⊗ ω) = ξ_{I_{G}^{G}(ρ⊗ω)} for

almost all ω ∈ Γ(G)^{∨}, resp. (ξ_{G})∗ ; \underline{G}(ρ) = d_{G}^{−r_{ρ}(C_{G}^{•})}ξ_{I_{G}^{G}(ρ)}^{∗}. If such an element ξ_{G} exists, then one has L_{G} = Nrd_{Λ(G)_{S}}(ξ_{G}).

Proof. For each A in SQ_{1}^{ab}(G) we write ξ_{A} for the element of Q(A)^{×} ≅ K_{1}(Q(A)) which validates the assertion MC_{A}, resp. MC_{A}^{∗}. (Such an element ξ_{A} is unique by Remark 3.5). For ρ ∈ A(G) we also set ξ̃_{ρ} := ξ_{ρ}, resp. ξ̃_{ρ} := ξ_{ρ}^{∗}. For each ρ ∈ A(G) the argument of §6.1 shows that there is a finite subset {A_{i} : i ∈ I} of SQ_{1}^{ab}(G) such that ρ ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ρ_{i}) with m_{i} ∈ ℤ and ρ_{i} ∈ A_{i}^{∨} for all i ∈ I. We set

L_{G,ρ} := ∏_{i∈I} j_{ρ_{i}}(ξ_{A_{i}})^{m_{i}} ∈ Q^{c}(ℤ_{p}[[T_{G}]])^{×}.

Then, for almost all ω ∈ Γ^{∨}, each series j_{ρ_{i}}(ξ_{A_{i}}) has neither a zero or a pole at ω(γ_{G}) − 1 and also r_{ρ_{i}⊗ω_{i}}(C_{A}^{•}) = 0 where ω_{i} is the restriction of ω to Γ(A_{i}) ⊆ Γ, and so the interpolation conditions of MC_{A_{i}}, resp. MC_{A_{i}}^{∗}, imply

(28) L_{G,ρ}(ω(γ_{G}) − 1) = ∏_{i∈I} j_{ρ_{i}}(ξ_{A_{i}})(ω(γ_{G}) − 1)^{m_{i}} = ∏_{i∈I} j_{ρ_{i}⊗ω_{i}}(ξ_{A_{i}})(0)^{m_{i}}

= ∏_{i∈I} ξ̃_{I_{A_{i}}^{G} (ρ_{i}⊗ω_{i})}^{m_{i}} = ξ̃_{∑_{i∈I} m_{i}·I_{A_{i}}^{G} (ρ_{i}⊗ω_{i})} = ξ̃_{I_{G}^{G}(ρ)⊗ω}.

The second equality here is by (12) (with G = A_{i} so Nrd_{Q(G)}(ξ_{A_{i}}) = ξ_{A_{i}}), the third is because j_{ρ_{i}⊗ω_{i}}(ξ_{A_{i}}) = Φ_{\underline{A_{i}},ρ_{i}⊗ω_{i}}(ξ_{A_{i}}) (by Lemma 3.1) and the fourth because ξ̃_{ρ_{1}+ρ_{2}} = ξ̃_{ρ_{1}} ξ̃_{ρ_{2}} for all ρ_{1}, ρ_{2} in A(G). The equality (28) shows in particular that the value L_{G,ρ}(ω(γ_{G}) − 1) for infinitely many ω, and hence also the element L_{G,ρ} itself, is independent of the choice of decomposition ρ ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ρ_{i}). In addition, if ξ_{A_{i}} validates MC_{A_{i}}^{∗} for each i ∈ I, then one has

(29) L_{G,ρ}^{∗}(0) = ∏_{i∈I} j_{ρ_{i}}(ξ_{A_{i}})^{∗}(0)^{m_{i}} = ∏_{i∈I}((d_{A_{i}}d_{G}^{−1})^{r_{ρ_{i}} (ξ_{A_{i}})}Φ_{\underline{A_{i}},ρ_{i}}(ξ_{A_{i}})^{∗}(0))^{m_{i}}

= d_{G}^{− ∑_{i∈I} m_{i}r_{ρ_{i}} (ξ_{A_{i}})} ∏_{i∈I}(ξ_{I_{A_{i}}^{G} (ρ_{i})}^{∗})^{m_{i}} = d_{G}^{−r_{ρ}(C_{G}^{•})}ξ_{∑_{i∈I} m_{i}·I_{A_{i}}^{G} (ρ_{i})}^{∗} = d_{G}^{−r_{ρ}(C_{G}^{•})}ξ_{I_{G}^{G}(ρ)}^{∗}.

Here the second equality is because the leading coefficient at 0 of j_{ρ_{i}}(ξ_{A_{i}}) = Φ_{\underline{A_{i}},ρ_{i}}(ξ_{A_{i}}) as a function of T_{G} = (1 + T)^{d_{G}} − 1 is equal to (d_{A_{i}}d_{G}^{−1})^{r_{ρ_{i}} (ξ_{A_{i}})} times its leading coefficient at 0 as a function of T_{A} = (1 + T)^{d_{A}} − 1, the third is because

Φ_{\underline{A_{i}},ρ_{i}}(ξ_{A_{i}})^{∗}(0) =: ξ_{\underline{A_{i}},ρ_{i}}^{∗}(0) = d_{A_{i}}^{−r_{ρ_{i}} (C_{A_{i}}^{•})}ξ_{I_{A_{i}}^{G} (ρ_{i})}^{∗}

and r_{ρ_{i}}(ξ_{A_{i}}) = r_{ρ_{i}}(C_{A_{i}}^{•}) and the fourth follows from the equality r_{ρ}(C_{G}^{•}) = ∑_{i∈I} m_{i}r_{ρ_{i}}(ξ_{A_{i}}) that is proved in Lemma 8.2 below. Now for each A ∈ SQ_{1}^{ab}(G) and ψ ∈ A^{∨} the map j_{ψ} : Q(A)^{×} → Q^{c}(Γ(A))^{×} in §3.1 is induced by the ℤ_{p}-linear homomorphism Q(A) → Q^{c}(Γ(A)) that sends each a ∈ A to ψ(a)π\underline{A}(a). For ω ∈ Γ(A)^{∨}, α ∈ G_{ℚ_{p}^{c}/ℚ_{p}} and x ∈ Q(A)^{×} one thus has j_{ψ⊗ω}(x) = tw_{ω}(j_{ψ}(x)) and j_{ψ^{α}}(x) = j_{ψ}(x)^{α}, where G_{ℚ_{p}^{c}/ℚ_{p}} acts on Q^{c}(Γ(A))^{×} via its natural action on the coefficients. For each ρ ∈ A(G) these facts combine with the above definition of L_{G,ρ} to imply that for all ω ∈ Γ(G)^{∨} and α ∈ G_{ℚ_{p}^{c}/ℚ_{p}} one has

(30)  L_{G,ρ⊗ω} = tw_{ω}(L_{G,ρ}) and L_{G,ρ^{α}} = (L_{G,ρ})^{α}

where G_{ℚ_{p}^{c}/ℚ_{p}} acts on Q^{c}(ℤ_{p}[[T]]) in the obvious way. In view of (30), the isomorphism (23) allows us to define L_{G} to be the unique element of ζ(Q(G))^{×} with j_{ρ}(L_{G}) = L_{G,ρ} for all ρ ∈ A(G). Then for all ω ∈ Γ(G)^{∨} one has

j_{ρ⊗ω}(L_{G})(0) = L_{G,ρ⊗ω}(0) = tw_{ω}(L_{G,ρ})(0) = L_{G,ρ}(ω(γ_{G}) − 1)

and similarly j_{ρ⊗ω}(L_{G})^{∗}(0) = L_{G,ρ}^{∗}(ω(γ_{G})−1). Hence, from (28), resp. (29), it follows that for all ρ ∈ A(G) one has j_{ρ⊗ω}(L_{G})(0) = ξ_{ρ⊗ω} for almost all ω ∈ Γ(G)^{∨}, resp. j_{ρ}(L_{G})^{∗}(0) = d_{G}^{−r_{ρ}(C_{G}^{•})}ξ_{ρ}^{∗}, as required by claim (i). To prove claim (ii) we fix ρ ∈ A(G) and a decomposition ρ ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ρ_{i}) as above. Then I_{G}^{G}(ρ) ≅ ∑_{i∈I} m_{i}·I_{A_{i}}^{G}(ρ_{i}) and so j_{I_{G}^{G}(ρ)}(L_{G}) = ∏_{i∈I} j_{ρ_{i}}(ξ_{A_{i}})^{m_{i}} = j_{ρ}(L_{G}). The equality of claim (ii) now follows from the explicit definition of q′_{G,G}. The definition of L_{G} combines with claim (ii) to show it is enough to prove claim (iii) in the case G = A ∈ SQ_{1}^{ab}(G). In this case L_{A} = ξ_{A} and so, since ∂_{A,S}(ξ_{A}) = χ_{A}^{ref}(C_{A}^{•}) and C_{A}^{•} belongs to D_{S}^{p}(Λ(A)), one has L_{A} ∈ Λ(A)_{S}^{×}. It thus suffices to prove that j_{ρ}(x) ∈ Λ_{ℤ_{p}^{c}}(Γ(A))_{S} for all x ∈ Λ(A)_{S}^{×} and ρ ∈ A^{∨}. We now fix an isomorphism of topological groups A ≅ H(A) × Γ(A) as in (5) and so regard Γ(A) as a subgroup of A. For all ρ ∈ A^{∨} the continuous ℤ_{p}^{c}-linear endomorphism tw_{ρ} of Q^{c}(Γ(A)) that sends each g in Γ(A) to ρ(g)g restricts to give an endomorphism of Λ_{ℤ_{p}^{c}}(Γ(A))_{S}. (This is because if O is any finite extension of ℤ_{p} in ℚ_{p}^{c} that contains ℤ_{p}[ρ], π is a uniformizing parameter for O and x an element of Λ_{O}(Γ(A)) \ πΛ_{O}(Γ(A)), then tw_{ρ}(x) ∈ Λ_{O}(Γ(A)) \ π · Λ_{O}(Γ(A)).) Claim (iii) thus follows from the fact that j_{ρ} sends each element ∑_{h∈H(A)} x_{h}h of Λ(A)_{S}^{×} = Λ(Γ(A))_{S}[H(A)]^{×} to ∑_{h∈H(A)} tw_{ρ}(x_{h})ρ(h). We note next that if ξ_{G} is any element with the properties described in claim (iv), then Lemma 3.1 (and (12)) combines with claim (i) to imply that for all ρ ∈ A(G) and infinitely many ω ∈ Γ(G)^{∨} one has

j_{ρ}(L_{G})(ω(γ_{G}) − 1) = j_{ρ⊗ω}(L_{G})(0) = ξ̃_{I_{G}^{G}(ρ⊗ω)}

= j_{ρ⊗ω}(Nrd_{Λ(G)_{S}}(ξ_{G}))(0) = j_{ρ}(Nrd_{Λ(G)_{S}}(ξ_{G}))(ω(γ_{G}) − 1)

and hence j_{ρ}(L_{G}) = j_{ρ}(Nrd_{Λ(G)_{S}}(ξ_{G})) ∈ Q^{c}(ℤ_{p}[[T]]). Since this is true for all ρ one has L_{G} = Nrd_{Λ(G)_{S}}(ξ_{G}) (by (10) with G replaced by G) and so L_{G} ∈ im(Nrd_{Λ(G)_{S}}), as required. In a similar way, if L_{G} ∈ im(Nrd_{Λ(G)_{S}}), then Lemma 3.1 combines with claim (i) to show that any ξ′ ∈ K_{1}(Λ(G)_{S}) with Nrd_{Λ(G)_{S}}(ξ′) = L_{G} has the interpolation property described in claim (iv). This proves claim (iv). □

In the next result we use the same notation as in the proof of Theorem 8.1.

Lemma 8.2. r_{ρ}(C_{G}^{•}) = ∑_{i∈I} m_{i}r_{ρ_{i}}(ξ_{A_{i}}).

Proof. Fix ξ_{G} ∈ K_{1}(Λ(G)_{S}) with ∂_{G,S}(ξ_{G}) = χ_{G}^{ref}(C_{G}^{•}). Then for each i ∈ I one has (C_{G}^{•})_{A_{i}} = C_{A_{i}}^{•} and so the commutative diagram (4) with O, G, G and S′ replaced by ℤ_{p}, G, A_{i}, and S implies that

∂_{A_{i},S}(q_{G,A_{i}}(ξ_{G})) = χ_{A_{i}}^{ref}((C_{G}^{•})_{A_{i}}) = χ_{A_{i}}^{ref}(C_{A_{i}}^{•}) = ∂_{A_{i},S}(ξ_{A_{i}})

and hence that q_{G,A_{i}}(ξ_{G}) = u_{i}ξ_{A_{i}} with u_{i} ∈ ker(∂_{A_{i},S}) = Λ(A_{i})^{×}. Since ρ ≅ ∑_{i∈I} m_{i} · I_{A_{i}}^{G}(ρ_{i}) Lemma 3.6(i) therefore implies that

Φ_{\underline{G},ρ}(ξ_{G}) = ∏_{i∈I} Φ_{\underline{A}_{i},ρ_{i}}(q_{G,A_{i}}(ξ_{G}))^{m_{i}} = v ∏_{i∈I} Φ_{\underline{A}_{i},ρ_{i}}(ξ_{A_{i}})^{m_{i}}

with v := ∏_{i∈I} Φ_{\underline{A}_{i},ρ_{i}}(u_{i})^{m_{i}}. The claimed formula now follows by combining the last displayed equality together with the facts that v belongs to ℤ_{p}^{c}[[T]]^{×} and that r_{ρ}(C_{G}^{•}) and r_{ρ_{i}}(ξ_{A_{i}}) are equal to the algebraic orders at T = 0 of the functions Φ_{\underline{G},ρ}(ξ_{G}) and Φ_{\underline{A}_{i},ρ_{i}}(ξ_{A_{i}}) respectively. □

8.2. Reduction to p-elementary groups. We next record a useful consequence of Theorem 8.1. In this result we fix a complex C^{•} in Dp ; S̃(Λ(G)) and sets (ξ_{ρ})_{ρ} and (ξ_{ρ}^{∗})_{ρ} of interpolation and leading term interpolation data respectively, but we no longer assume that G has rank one.

Corollary 8.3. Assume p is odd. Then MC(\underline{G}, C^{•}, (ξ_{ρ})_{ρ}), resp. MC^{∗}(\underline{G}, C^{•}, (ξ_{ρ}^{∗})_{ρ}), is valid if and only if MC_{J}, resp. MC_{J}^{∗}, is valid for all J ∈ SQ_{1}^{p−el}(G).

Proof. If C^{•} belongs to Dp ; S̃(Λ(G)) but not D_{S}^{p}(Λ(G)), then G has no element of order p and the argument of Lemma 7.2 implies that Corollary 8.3 is valid as stated if and only if it is valid with C^{•} replaced by a complex C̃^{•} in D_{S}^{p}(Λ(G)) (and with appropriate changes in the data (ξ_{ρ})_{ρ}, resp. (ξ_{ρ}^{∗})_{ρ}). In the sequel we shall thus assume that C^{•} belongs to D_{S}^{p}(Λ(G)). Following Theorem 2.1 we also assume G has rank one. Then, in view of Theorems 2.2 and 8.1(ii) it is enough to show that the validity of MC_{J}, resp. MC_{J}^{∗}, for every J ∈ SQ_{1}^{p−el}(G) implies that L_{G} belongs to im(Nrd_{Λ(G)_{S}}). To do this we recall that for each such subquotient J the isomorphisms (23) fit into a commutative diagram

ζ(Q(G))^{×} ⟶^{ι_{G}} Hom^{ℚ_{p}}(R(G), Q^{c}(Γ)^{×})
q′_{G,J} ↓  ↓^{q′′_{G,J}}
ζ(Q(J))^{×} ⟶^{ι_{J}} Hom^{ℚ_{p}}(R(J), Q^{c}(Γ(J))^{×})

where for each homomorphism f one has q′′_{G,J}(f)(ψ) = f(I_{J}^{G}(ψ)) for all ψ in A(J). Set f_{anal} := ι_{G}(L_{G}). Then the above diagram combines with Theorem 8.1(ii) and (iv) (with G = J) and our assumption that MC_{J}, resp. MC_{J}^{∗}, is valid to imply that

(31)  q′′_{G,J}(f_{anal}) ∈ ι_{J}(im(Nrd_{Λ(J)_{S}})).

We now write Λ(G)_{∧} for the pro-p-completion of the ring Λ(G)_{S} and set Λ^{c}(Γ)_{∧} := ℤ_{p}^{c} ⊗_{ℤ_{p}} Λ(Γ)_{∧}. Then the containments (31) for each J in SQ_{1}^{p−el}(G) combine with the result of [35, Th. C and §4, Rem. 2] to imply that f_{anal} belongs to the subgroup Det(K_{1}(Λ(G)_{∧})) of HOM^{ℚ_{p}}(R(G), Λ^{c}(Γ)_{∧}^{×}), where the homomorphism ‘Det’ and notation ‘HOM’ is as defined in loc. cit.

We next fix an element ξ^{alg} of K_{1}(Λ(G)_{S}) as described at the end of §7.2 and set f_{alg} := ι_{G}(Nrd_{Λ(G)_{S}}(ξ^{alg})). Then (24) implies that f_{alg} = Det_{Q(G)}(ξ^{alg}) and so the quotient f_{anal}/f_{alg} belongs to Det(K_{1}(Λ(G)_{∧})). In addition, since we are assuming that MC_{J}, resp. MC_{J}^{∗}, is valid for all J in SQ_{1}^{p−el}(G) the argument used in the proof of Lemma 6.7 shows that for all ρ in A(G) the value f_{anal}(ρ)/f_{alg}(ρ) of f_{anal}/f_{alg} at ρ belongs to Λ^{c}(Γ)^{×}. Applying the result of [35, Th. B] we can therefore deduce that the function ι_{G}(L_{G} ·Nrd_{Λ(G)_{S}}(ξ^{alg})^{−1}) = f_{anal}/f_{alg} belongs to Det(K_{1}(Λ(G))) = ι_{G}(Nrd_{Λ(G)_{S}}(im(α_{G,S}))). This in turn implies that L_{G} · Nrd_{Λ(G)_{S}}(ξ^{alg})^{−1} belongs to Nrd_{Λ(G)_{S}}(im(α_{G,S})) ⊂ im(Nrd_{Λ(G)_{S}}) and hence that L_{G} belongs to im(Nrd_{Λ(G)_{S}}), as required to complete the proof. □

Example 8.4. Let G denote the p-adic Lie group (\begin{matrix}ℤ_{p}^{×} & ℤ_{p} \\ 0 & 1 \end{matrix}) considered by Kato in

[30] (this is the Galois group of ‘false Tate curve extensions’). This group has rank two, contains no element of order p and has an associated triple (1) with H(\underline{G}) =

(\begin{matrix}(ℤ_{p}^{×})_{tor} & ℤ_{p} \\ 0 & 1 \end{matrix}). We claim that every element G of SQ_{1}^{p−el}(G) is either abelian or

pro-p. To prove this it suffices to show that if G = H × P with H a non-trivial finite abelian group of order t coprime to p and P a pro-p group, then P is abelian. To

do this we note G is an open subgroup of (\begin{matrix}ℤ_{p}^{×} & ℤ/p^{n} \\ 0 & 1 \end{matrix}) for some integer n and fix a

non-trivial element x = (\begin{matrix}η & r \\ 0 & 1 \end{matrix}) of H. Then x^{t} = (\begin{matrix}η^{t} & (∑_{i=0}^{i=t−1} η^{i})r \\ 0 & 1 \end{matrix}) is the identity

matrix so η ≢ 1(mod p). Now if y_{i} = (\begin{matrix}η_{i} & r_{i} \\ 0 & 1 \end{matrix}) are any elements of P for i = 1, 2 then

the equality xy_{i} = y_{i}x implies r_{i} = (η − 1)^{−1}(η_{i} − 1)r in ℤ/p^{n}. This in turn implies (η_{1} − 1)r_{2} = (η_{2} − 1)r_{1} and hence that y_{1}y_{2} = y_{2}y_{1}, as required.

9. Tate motives

In this section we use Theorem 2.2 to prove Theorem 2.6 and Corollaries 2.8 and 2.10. We fix an algebraic closure ℚ^{c} of ℚ and regard all number fields as subfields of ℚ^{c}. For any number field k we set G_{k} := G_{ℚ^{c}/k} and write χ_{k} and κ_{k} for the cyclotomic and Teichmüller characters G_{k} → ℤ_{p}^{×}. We also set ω_{k} := χ_{k}κ_{k}^{−1} and note that this homomorphism factors through the restriction map G_{k} → G_{k^{∞}/k}.

9.1. p-adic Artin L-functions. We first recall (from Greenberg [25]) some basic facts about p-adic Artin L-functions. To do this we fix an odd prime p, a totally real number field F and a compact p-adic Lie extension E of F that is unramified outside a finite set of places Σ of F. We set G := G_{E/F} and write A^{+}(G) for the subset of A(G) comprising representations that are trivial on G_{E/E′} for some intermediate field E′ of E/F that is totally real. For any finite dimensional complex representation ϱ of any finite quotient of G and any finite set of places Σ′ of F we write L_{Σ′}(s, ϱ) for the Artin L-function of ϱ truncated by removing the Euler factors of all non-archimedean places in Σ′. Then for each ρ ∈ A^{+}(G) the Σ-truncated p-adic Artin L-function L_{p,Σ}(s, ρ) is the unique p-adic meromorphic function on ℤ_{p} with the property that for all strictly negative integers n and all field isomorphisms j : ℂ_{p} ≅ ℂ one has

(32)  j(L_{p,Σ}(n, ρ)) = L_{Σ_{p}}(n, (ρ ⊗ κ_{k}^{n−1})^{j})

where Σ_{p} denotes the union of Σ and all places of F that lie above p. We now assume that F^{∞} ⊆ E, set Γ := G_{F^{∞}/F} and fix a topological generator γ_{G} of Γ. Then for each ρ ∈ A^{+}(G) there exists a unique element f_{Σ,ρ}(T) of Q^{c}(ℤ_{p}[[T]]) := ℚ_{p}^{c} ⊗_{ℚ_{p}} Q(ℤ_{p}[[T]]) with all of the following properties:

(33)  L_{p,Σ}(1 − s, ρ) = f_{Σ,ρ}(ω_{F} (γ_{G})^{s} − 1);

f_{Σ,ρ+ρ′}(T) = f_{Σ,ρ}(T)f_{Σ,ρ′}(T) for all ρ′ ∈ A^{+}(G);

(34)  f_{Σ,ρ⊗ω}(T) = f_{Σ,ρ}(ω(γ_{G})(1 + T) − 1)

for all ω ∈ Γ^{∨}; with respect to the natural action of G_{ℚ_{p}^{c}/ℚ_{p}} on Q^{c}(ℤ_{p}[[T]]) one has f_{Σ,ρ^{α}}(T) = α(f_{Σ,ρ}(T)) for all α ∈ G_{ℚ_{p}^{c}/ℚ_{p}}; if ρ has rank one, then f_{Σ,ρ}(T) = G_{Σ,ρ}(T)/H_{ρ}(T) where G_{Σ,ρ}(T) and H_{ρ}(T) are the Deligne-Ribet power series used by Wiles in [43].

9.2. The proof of Theorem 2.6. If E/F is any extension as above, then for each integer a we write Λ(G)^{#}(a) for the (left) Λ(G)-module Λ(G) endowed with the following commuting (left) action of G_{F}: each σ in G_{F} acts on Λ(G)^{#}(a) as right multiplication by χ_{F} (σ)^{a}σ̄^{−1} where σ̄ is the image of σ in G. Then, following Fukaya and Kato [24, §2.1.1] and Nekovář [33], one obtains an object of D^{p}(Λ(G)) by setting

C_{E/F,Σ}^{•} := RΓ_{c,ét}(Spec(O_{F,Σ_{p}}), Λ(G)^{#}(1))

where the subscript ‘c’ denotes cohomology with compact support and O_{F,Σ_{p}} the subring of F comprising elements that are integral at all non-archimedean places outside Σ_{p}. After recalling the function L′_{p,Σ}(s, ρ) defined by (7) we can now state the main result of this subsection.

Theorem 9.1. Assume that p is odd. Let k be a totally real number field and K a totally real compact p-adic Lie extension of k that contains k^{∞}, is unramified outside a finite set of places Σ of k and is such that there exists a finite subextension k′ of K/k for which K/k′ is a pro-p-extension and μ_{p}(k′) = 0. Set G := G_{K/k} and Γ := G_{k^{∞}/k} and write \underline{G} for the corresponding triple (G, π_{G}, γ_{G}) as in (1).

(i) Then C_{K/k,Σ}^{•} belongs to D_{S}^{p}(Λ(G)).

(ii) There exists an element ξ̃ of K_{1}(Λ(G)_{S}) that satisfies both of the following conditions:

(a) for all ρ ∈ A(G) one has

ξ̃\underline{G}(ρ ⊗ ω) = log_{p}(ω_{k}(γ_{G}))^{−⟨ρ⊗ω,1_{G}⟩}L′_{p,Σ}(1, ρ ⊗ ω)

for almost all ω ∈ Γ^{∨};

(b) ∂_{G,S}(ξ̃) = χ^{ref}(C_{K/k,Σ}^{•}). (iii) If the ‘p-adic Stark conjecture at s = 1’ is valid for all ρ ∈ A(G), then for any

element ξ̃ as in claim (ii) one has ξ̃∗ ; \underline{G}(ρ ⊗ ω) = ξ̃\underline{G}(ρ ⊗ ω) and the displayed formula in claim (ii)(a) is valid for all ω ∈ Γ^{∨}.

Corollary 9.2. Theorem 2.6 (and the assertion made in Remark 2.7) is valid.

Proof. It is enough to show that if G has no element of order p, then Theorem 9.1(ii), resp. Theorem 9.1(iii), is equivalent to Theorem 2.6, resp. Remark 2.7. This equivalence is proved by Venjakob and the present author in [15, §8.1]. □

9.3. The proof of Theorem 9.1. At the outset we fix an odd prime p and recall that if F′ is a finite extension of F with E′ := (F′)^{∞} ⊆ E and we set G := G_{E/F} and G′ := G_{E′/F′}, then there is a descent isomorphism in D^{p}(Λ(G′)) of the form

(35)  Λ(G′) ⊗_{Λ(G_{E/F′})}^{𝕃} res_{G_{E/F′}}^{G} (C_{E/F,Σ}^{•}) ≅ C_{E′/F′,Σ}^{•}.

In addition Σ = Σ_{p} (since F^{∞} ⊆ E so all places of F above p ramify in E/F) and if E is totally real there are canonical isomorphisms

(36)  H^{i}(C_{E/F,Σ}^{•}) ≅ \begin{cases}X_{Σ}(E), & if i = 2, \\ ℤ_{p}, & if i = 3, \\ 0, & otherwise \end{cases}

(for further details regarding these well-known isomorphisms see, for example, [13, the proof of Lem. 3]). In particular, if E is totally real, then (36) implies that C_{E/F,Σ}^{•} belongs to D_{S}^{p}(Λ(G)) if and only if X_{Σ}(E) is a finitely generated Λ(H)-module with H := G_{E/E′}. Now, after taking account of the natural isomorphism of algebras Λ(G′) ≅ ℤ_{p}⊗_{Λ(H)}Λ(G_{E/F′}), the hypertor spectral sequence combines with the isomorphism (35) and the explicit descriptions (36) to give an exact sequence of ℤ_{p}-modules of the form

Tor_{Λ(H)}^{2}(ℤ_{p}, ℤ_{p}) → ℤ_{p} ⊗_{Λ(H)} X_{Σ}(E) → X_{Σ}(E′) → Tor_{Λ(H)}^{1}(ℤ_{p}, ℤ_{p}).

Thus, since Tor_{Λ(H)}^{i}(ℤ_{p}, ℤ_{p}) is a finitely generated ℤ_{p}-module for each i > 0, Nakayama’s lemma implies that if H is pro-p, then X_{Σ}(E) is a finitely generated Λ(H)-module if and only if X_{Σ}(E′) is a finitely generated ℤ_{p}-module (or equivalently, μ_{p}(F′) = 0). In particular, under the conditions of Theorem 9.1, the existence of k′ implies that C_{K/k,Σ}^{•} belongs to D_{S}^{p}(Λ(G)), as claimed by Theorem 9.1(i). We now assume that G = G_{E/F} denotes either G (so E = K and F = k) or an element of SQ_{1}(G). For each ψ ∈ A(G) we then define

L̃′_{p,Σ}(1, ψ) := log_{p}(ω_{F} (γ_{G}^{d_{G}}))^{−⟨ψ,1_{G}⟩}L′_{p,Σ}(1, ψ)

with d_{G} := [Γ : Γ(G)]. The set {L̃′_{p,Σ}(1, ψ)}_{ψ∈A(G)} is interpolation data in the sense of §1.2.1 and Theorem 9.1(ii) is equivalent to asserting that MC(\underline{G}, C_{K/k,Σ}^{•}, (L̃′_{p,Σ}(1, ρ))_{ρ}) is valid. It thus suffices to verify that the conditions of Theorem 2.2(i) are satisfied in this case. To consider Theorem 2.2(i)(a) we fix A = U/J ∈ SQ_{1}^{ab}(G) and ϕ ∈ H(A)^{∨}. For each ψ ∈ A(A) one has ⟨I_{A}^{G}(ψ), 1_{G}⟩ = ⟨ψ, 1_{A}⟩ and, because of the inductivity properties of p-adic L-functions, also L′_{p,Σ}(s, I_{A}^{G}(ψ)) = L′_{p,Σ(K^{U})}(s, ψ) where Σ(K^{U}) is the set of places of K^{U} above Σ. This implies, in particular, that L̃′_{p,Σ}(1, I_{A}^{G}(ϕ⊗ω)) = L′_{p,Σ(K^{U})}(1, ϕ ⊗ ω) for almost all ω ∈ Γ(A)^{∨} and so Theorem 2.2(i)(a) (with E = K and F = k) follows from (35) and Lemma 9.3(i) below. We next assume that the ‘p-adic Stark conjecture at s = 1’ is valid for all ρ ∈ A(G). In this case L′_{p,Σ}(1, ρ) ≠ 0 for all ρ ∈ A(G) so {L̃′_{p,Σ}(1, ψ)}_{ψ∈A(G)} is leading term interpolation data and Theorem 9.1(iii) asserts the validity of MC^{∗}(\underline{G}, C_{K/k,Σ}^{•}, (L̃′_{p,Σ}(1, ρ))_{ρ}) and so it suffices to verify that the conditions of Theorem 2.2(ii) are satisfied in this case. Now the p-adic Stark conjecture at s = 1 implies that Leopoldt’s Conjecture is valid at p for every finite subextension of K/k (by [14, Rem. 5.4]) and so the isomorphisms of (35) and (36) combine to imply that for both i = 2, 3 the Λ(A)[\frac{1}{p}]- module ℚ_{p} ⊗_{ℤ_{p}} H^{i}(C_{A}^{•}) is isomorphic to ℚ_{p} (with trivial action). The explicit formula of [15, Lem. 5.5(iv)] then implies that for all ϕ ∈ H(A)^{∨} and ω ∈ Γ(A)^{∨} one has r_{ϕ⊗ω}(C_{A,ϕ}^{•}) = ⟨ϕ ⊗ ω, 1_{A}⟩ and hence

d_{A}^{−r_{ϕ⊗ω}(C_{A,ϕ}^{•})}L̃′_{p,Σ}(1, I_{A}^{G}(ϕ ⊗ ω))=d_{A}^{−⟨ϕ⊗ω,1_{A}⟩}log_{p}(ω_{k}(γ_{G}))^{−⟨ϕ⊗ω,1_{A}⟩}L′_{p,Σ}(1, I_{A}^{G}(ϕ ⊗ ω))
=log_{p}(ω_{K^{U}} (γ_{G}^{d_{A}}))^{−⟨ϕ⊗ω,1_{A}⟩}L′_{p,Σ(K^{U})}(1, ϕ ⊗ ω).

Given this equality, the validity of Theorem 2.2(ii)(a) follows from ((35) and) Lemma 9.3(ii) below. At this stage our proof of Theorem 9.1(ii), resp. Theorem 9.1(iii), will be completed if we prove that the respective condition (b) of Theorem 2.2 is satisfied. To do this we use the approach of §8 and so need to show that for all G ∈ Q_{1}(G) the containment L_{G} ∈ im(Nrd_{Λ(G)_{S}}) required by Theorem 8.1(iv) is valid. But, given the interpolation properties of MC(\underline{G}, C_{K/k,Σ}^{•}, (L̃′_{p,Σ}(1, ρ))_{ρ}), resp. MC^{∗}(\underline{G}, C_{K/k,Σ}^{•}, (L̃′_{p,Σ}(1, ρ))_{ρ}), in this case the element L_{G} is such that if G = G/J and we set E := K^{J}, then for all ρ ∈ A(G) there are infinitely many ω ∈ Γ^{∨} with

(37) j_{ρ}(L_{G})(ω(γ) − 1) = j_{ρ⊗ω}(L_{G})(0) = L_{p,Σ}(1, ρ ⊗ ω)

= j_{ρ⊗ω}(L_{E/k,Σ})(0) = j_{ρ}(L_{E/k,Σ})(ω(γ) − 1).

Here L_{E/k,Σ} is the element of ζ(Q(G))^{×} constructed in [34] to satisfy

j_{ρ⊗ω}(L_{E/k,Σ}) = ι(L_{E/k,Σ})(ρ ⊗ ω) = f_{Σ,ρ⊗ω}(T)

for all ρ and ω, where ι is the isomorphism from (23), and so the third equality is a consequence of (33). Now, since for any ρ the equalities (37) are valid for infinitely many ω they imply that j_{ρ}(L_{G}) = j_{ρ}(L_{E/k,Σ}) for all ρ and hence, by (10), that L_{G} = L_{E/k,Σ}. The required containment L_{G} ∈ im(Nrd_{Λ(G)_{S}}) is therefore equivalent in this case to the main result of Ritter and Weiss in [37] (to see this compare the statement of [37, Theorem] with the argument given in the proof of Corollary 8.3). Modulo the next result, this completes the proof of Theorem 9.1. Note that the following result does not assume the vanishing of any μ-invariants.

Lemma 9.3. Let F′ be a totally real number field and E′ a totally real rank one p-adic Lie extension of F′ that contains F^{′∞} and is unramified outside a finite set of places Σ of F′. Set A := G_{E′/F′} and write \underline{A} for the triple (1) with H(A) := G_{E′/F^{′∞}} and Γ(A) := G_{F^{′∞}/F′}. Then C^{•} := C_{E′/F′,Σ}^{•} belongs to D_{S^{∗}}^{p}(Λ(A)). Assume now that A is abelian.

(i) Then MC(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{ϕ}^{•}, (L′_{p,Σ}(1, ϕ ⊗ ω))_{ω}) is valid for all ϕ ∈ H(A)^{∨}. (ii) If the p-adic Stark conjecture at s = 1 is valid for every ρ ∈ A(A), then MC^{∗}(\underline{Γ(A)}, ℤ_{p}[ϕ], C_{ϕ}^{•}, (log_{p}(ω_{F′}(γ_{A}))^{−⟨ϕ⊗ω,1_{A}⟩}L′_{p,Σ}(1, ϕ⊗ω))_{ω}) is valid for all ϕ ∈ H(A)^{∨}.

Proof. Since A has rank one it has a central open subgroup C and so C^{•} belongs to D_{S^{∗}}^{p}(Λ(A)) if each Λ(C)-module H^{i}(C^{•}) is torsion. But, given the description (36), this follows immediately from the well-known fact that X_{Σ}(E′) is a torsion Λ(C)- module. We assume henceforth that A is abelian, fix ϕ ∈ H(A)^{∨} and write R for the ring Λ_{ℤ_{p}[ϕ]}(Γ(A)) ≅ ℤ_{p}[ϕ][[T]] and char_{R}(M) for the characteristic ideal of any finitely generated torsion R-module M. The groups K_{1}(Q(R)) and K_{0}(R, Q(R)) identify with Q(R)^{×} and the multiplicative group of invertible R-lattices in Q(R) respectively and with respect to this identification one has χ_{ℤ_{p}[ϕ],Γ(A)}^{ref}(C_{ϕ}^{•}) = ∏_{i∈ℤ} char_{R}(H^{i}(C_{ϕ}^{•}))^{(−1)^{i}}. For any Λ(A)-module M there are R-modules M^{(ϕ)} := H^{0}(H(A), ℤ_{p}[ϕ] ⊗_{ℤ_{p}} M) and M_{(ϕ)} := ℤ_{p}[ϕ] ⊗_{ℤ_{p}[H(A)]} M ≅ H_{0}(H(A), ℤ_{p}[ϕ] ⊗_{ℤ_{p}} M). Now since C^{•} belongs to D^{p}(Λ(A)) and is acyclic outside degrees 2 and 3 (by (36)) it is isomorphic to a complex Ψ^{2} → Ψ^{3} where both terms are of finite projective dimension as ℤ_{p}[H(A)]-modules and Ψ^{2} occurs in degree two. Since the functors M ↦ M^{(ϕ)} and M ↦ M_{(ϕ)} are respectively left and right exact there is therefore a natural exact commutative diagram of R-modules

Ψ_{(ϕ)}^{2} ⟶ Ψ_{(ϕ)}^{3} ⟶ H^{3}(C^{•})_{(ϕ)} ⟶ 0
↓  ↓
0 ⟶ H^{2}(C^{•})^{(ϕ)} ⟶ Ψ^{2,(ϕ)} ⟶ Ψ^{3,(ϕ)}

in which both vertical arrows are induced by ∑_{h∈H(A)} h and are therefore bijective (because the ℤ_{p}[H(A)]-modules Ψ^{2} and Ψ^{3} are of finite projective dimension). Since C_{ϕ}^{•} is isomorphic to Ψ_{(ϕ)}^{2} → Ψ_{(ϕ)}^{3} this diagram combines with (36) to imply C_{ϕ}^{•} is acyclic outside degrees 2 and 3 and that there are canonical isomorphisms H^{2}(C_{ϕ}^{•}) ≅ X_{Σ}(E′)^{(ϕ)} and H^{3}(C_{ϕ}^{•}) ≅ (ℤ_{p})_{(ϕ)}. But if ϕ is trivial, resp. non-trivial, then (ℤ_{p})_{(ϕ)} is ℤ_{p}, resp. is finite, and so char_{R}(H^{3}(C_{ϕ}^{•})) is equal to (T), resp. R. Claim (i), resp. claim (ii), therefore asserts the existence of an element ξ_{ϕ} of Q(R) such that

(38)  char_{R}(X_{Σ}(E′)^{(ϕ)}) = \begin{cases}(T ξ_{ϕ}), & if ϕ is trivial \\ (ξ_{ϕ}), & otherwise \end{cases}

and, in addition, for almost all ω ∈ Γ(A)^{∨} the value at T = 0 of tw_{ω}(ξ_{ϕ}) is defined and equal to L_{p,Σ}(1, ϕ ⊗ ω), resp. for all ω ∈ Γ(A)^{∨} the leading term at T = 0 of tw_{ω}(ξ_{ϕ}) is equal to log_{p}(ω_{F′}(γ_{A}))^{−⟨ϕ⊗ω,1_{A}⟩}L′_{p,Σ}(1, ϕ ⊗ ω). We claim that these conditions are satisfied by the element ξ_{ϕ} := f_{Σ,ϕ}(T). Indeed, with this definition, it is a straightforward exercise to show that (38) follows from the main conjecture of Iwasawa theory for totally real fields proved by Wiles [43, p. 496] (for brevity however we simply note that (38) is equivalent to the containment (∗∗) that is deduced from the results of Wiles by Ritter and Weiss in the proof of [34, Th. 16]). From (33) it is also clear that tw_{ω}(ξ_{ϕ})(0) is defined and equal to L_{p,Σ}(1, ϕ ⊗ ω) for almost all ω ∈ Γ(A)^{∨}. In addition, if the p-adic Stark conjecture at s = 1 is valid for ϕ ⊗ ω, then the algebraic order of L_{p,Σ}(1 − u, ϕ ⊗ ω) at u = 0 is equal to −⟨ϕ ⊗ ω, 1_{A}⟩ and so (33) implies the leading term of tw_{ω}(ξ_{ϕ}) at T = 0 is equal to the value of log_{p}(ω_{F′}(γ_{A}))^{−⟨ϕ⊗ω,1_{A}⟩}u^{⟨ϕ⊗ω,1_{A}⟩}L_{p,Σ}(1 − u, ϕ ⊗ ω) at u = 0, as required. □

9.4. The proof of Corollary 2.8. We set ℤ′ := ℤ[\frac{1}{2}]. We also fix a finite Galois extension F/k of number fields that is unramified outside a finite set of places Σ of k and set G := G_{F/k}. The equivariant Tamagawa number conjecture of [11, Conj. 4.1(iv)] for the pair (ℚ(r)_{F/k}, ℤ′[G]) asserts the vanishing of the element T Ω(ℚ(r)_{F/k}, ℤ′[G]) of K_{0}(ℤ′[G], ℝ[G]) that is defined (unconditionally) in [11, Conj. 4.1(iii)]. Thus, since the product over all odd primes p and all field isomorphisms j : ℂ ≅ ℂ_{p} of the induced composite homomorphism of K-groups

j_{∗} : K_{0}(ℤ′[G], ℝ[G]) → K_{0}(ℤ′[G], ℂ[G]) ⟶^{j} K_{0}(ℤ′[G], ℂ_{p}[G]) → K_{0}(ℤ_{p}[G], ℂ_{p}[G])

is injective (by [4, Lem. 2.1]), it suffices to prove that j_{∗}(T Ω(ℚ(r)_{F/k}, ℤ′[G])) vanishes for every such j. We assume throughout this subsection that F is totally real, that for each odd prime divisor p of |G| there exists an intermediate field k′_{p} of F/k such that F/k′_{p} is of ppower degree and μ_{p}(k′_{p}) = 0, and that the p-adic Stark conjecture at s = 1 is valid for all odd primes p. We fix an odd prime p, set G := G_{F^{∞}/k} and recall from [15, Th. 8.1] that the validity of [15, Conj. 8.3] for F^{∞}/k implies that j_{∗}(T Ω(ℚ(1)_{F/k}, ℤ′[G])) vanishes for every isomorphism j : ℂ ≅ ℂ_{p}. It is this implication that underlies our proof of Corollary 2.8. Now [15, Rem. 7.2] implies that [15, Conj. 8.3] is equivalent, in our terminology, to the assertion MC^{∗}(G, C_{F^{∞}/k,Σ}^{•}, (log_{p}(ω_{k}(γ_{G}))^{−⟨ρ,1_{G}⟩}L′_{p,Σ}(1, ρ))_{ρ}). Thus, if p divides |G|, then since F^{∞}/k′_{p} is pro-p and μ_{p}(k′_{p}) = 0, Theorem 9.1(iii) implies [15, Conj. 8.3] is valid for F^{∞}/k and hence that j_{∗}(T Ω(ℚ(1)_{F/k}, ℤ′[G])) vanishes for every isomorphism of fields j : ℂ ≅ ℂ_{p}. We may therefore assume that p does not divide |G|. In this case G has no element of order p and the further property that any subquotient A in SQ_{1}^{p−el}(G) belongs to SQ_{1}^{ab}(G) and has no element of order p. From Corollary 8.3 it follows that [15, Conj. 8.3] is valid for F^{∞}/k if and only if it is valid for all extensions of the form E^{∞}/E′ where E′/k is finite and E is a finite abelian extension of E′ with E ⊂ F and [E : E′] prime to p. For all such extensions E^{∞}/E′ the validity of [15, Conj. 8.3] follows directly from Lemma 9.3(ii). At this stage we therefore know that j_{∗}(T Ω(ℚ(1)_{F/k}, ℤ′[G])) vanishes for every odd p and every isomorphism of fields j : ℂ ≅ ℂ_{p} and hence that [11, Conj. 4(iv)] is valid for the pair (ℚ(1)_{F/k}, ℤ′[G]). To deduce all of the other results in Corollary 2.8 we use the following convenient notation: if, when applied to the extension F/k, a conjecture [?, Conj. ??] is an equality in K_{0}(ℤ[G], ℝ[G]), then we write [?, Conj. ??]′_{F/k} for the image of this conjectural equality under the natural homomorphism K_{0}(ℤ[G], ℝ[G]) → K_{0}(ℤ′[G], ℝ[G]). Now since the p-adic Stark conjecture at s = 1 is assumed to be valid at all odd p, Leopoldt’s conjecture is also valid at all odd p (by [14, Rem. 5.4]). Hence, since [11, Conj. 4(iv)] is valid for (ℚ(1)_{F/k}, ℤ′[G]), the validity of [4, Conj. 3.3]′_{F/k} follows immediately from [5, Cor. 1.2]. The validity of [4, Conj. 3.3]′_{F/k} then combines with [4, Prop. 3.6(ii)] to imply the validity of the image under Cl(ℤ[G]) → Cl(ℤ′[G]) of Chinburg’s Ω(1)-Conjecture for F/k. Next we recall that if [4, Conj. 3.3]′_{F/k} is valid, then [4, Th. 5.2 and Rem. 5.4] imply [4, Conj. 4.1]′_{F/k} is valid if and only if [1, Conj. 4.1]′_{F/k} is valid. To deduce [4, Conj. 4.1]′_{F/k} it thus suffices to recall that Breuning [3, Prop. 4.6] has proved [1, Conj. 4.1]′_{F/k} is valid provided that for any place w of F which has odd residue characteristic and ramifies wildly in F/k the w-adic completion of F is absolutely abelian (as is assumed at the end of Corollary 2.8). Finally we recall from [4, Prop. 4.4] that [4, Conj. 4.1]′_{F/k} implies the validity of the image under Cl(ℤ[G]) → Cl(ℤ′[G]) of Chinburg’s Ω(3)-Conjecture for F/k. This completes the proof of Corollary 2.8.

9.5. The proof of Corollary 2.10. In this subsection we prove a generalisation of Theorem 9.1 and then use it to prove Corollary 2.10.

9.5.1. Main conjectures Let p be an odd prime, k be a totally real number field and E a CM field that is a compact p-adic Lie extension of k unramified outside a finite set of places Σ of k. Let τ denote the unique complex conjugation in G_{E/k}. For each integer r write A_{r}(G_{E/k}) for the subset of A(G_{E/k}) comprising those (p-adic) Artin representations ρ of G_{E/k} that are trivial on (−1)^{r+1}τ and e_{r} for the central idempotent \frac{1}{2}(1 + (−1)^{r+1}τ) of Λ(G_{E/k}) and then define an object of D^{p}(Λ(G_{E/k})) by setting

(39)  C_{E/k,Σ}^{•}(r) := RΓ_{c,ét}(Spec(O_{k,Σ_{p}}), e_{r}Λ(G_{E/k})^{#}(r)).

We note in particular that if E contains the cyclotomic ℤ_{p}-extension k^{∞} of k then all places of k above p ramify in E/k and so Σ_{p} = Σ.

Corollary 9.4. Let k be a totally real number field and K a CM field that is a compact p-adic Lie extension of k that contains k^{∞} and is unramified outside a finite set of places Σ of k. Set G := G_{K/k} and fix an integer r.

(i) If there exists a finite subextension k′ of K/k for which K/k′ is a pro-p extension and μ_{p}(k′) = 0, then C_{K/k,Σ}^{•}(r) belongs to D_{S}^{p}(Λ(G)).

(ii) If there exists a field k′ as in claim (i), then there also exists an element ξ̃_{r} of K_{1}(Λ(G)_{S}) that satisfies both of the following conditions:

(a) for all irreducible ρ ∈ A(G) one has

ξ̃_{r,\underline{G}}(ρ) = \begin{cases}L_{p,Σ}(r, ρ ⊗ κ_{k}^{1−r}), & if ρ ∈ A_{r}(G) \\ 1, & otherwise; \end{cases}

(b) ∂_{G,S}(ξ̃_{r}) = χ^{ref}(C_{K/k,Σ}^{•}(r)). (iii) If G has rank one and contains no element of order p, then after replacing S

by S^{∗} both claims (i) and (ii) are valid without any hypothesis on μ-invariants.

Proof. For each natural number n we write ς_{p^{n}} for the group of p^{n}-th roots of unity in ℚ^{c}. By replacing K by K(ς_{p}) we can assume (since k^{∞} ⊆ K) that K contains ς_{p^{n}} for all natural numbers n and hence that the cyclotomic character χ_{k} factors through the restriction map G_{k} → G. We write K^{+} for the maximal real subfield of K, set G^{+} := G_{K^{+}/k} and use the restriction map Λ(G) → Λ(G^{+}) to identify e_{1}Λ(G) with Λ(G^{+}). For each integer a we set Λ(G)_{a} := e_{a}Λ(G) and write tw_{a} both for the automorphism of Λ(G) sending each g in G to χ_{k}^{a}(g)g and for the induced isomorphism of algebras Λ(G^{+})_{S} = Λ(G)_{1,S} ≅ Λ(G)_{1−a,S}. We use the fact that the naturality of the long exact sequence of relative K-theory implies that the induced isomorphisms of K-groups tw_{a,∗}^{1} : K_{1}(Λ(G^{+})_{S}) ≅ K_{1}(Λ(G)_{1−a,S}) and tw_{a,∗}^{0} : K_{0}(Λ(G^{+}), Λ(G^{+})_{S}) ≅ K_{0}(Λ(G)_{1−a}, Λ(G)_{1−a,S}) lie in a commutative diagram of connecting homomorphisms

(40)

K_{1}(Λ(G^{+})_{S}) ⟶^{∂_{G^{+},S}} K_{0}(Λ(G^{+}), Λ(G^{+})_{S})
tw_{a,∗}^{1}↓  ↓^{tw}_{a,∗}^{0}
K_{1}(Λ(G)_{1−a,S}) ⟶ K_{0}(Λ(G)_{1−a}, Λ(G)_{1−a,S})

where the lower horizontal arrow denotes the homomorphism obtained by restricting ∂_{G,S} to the direct summand K_{1}(Λ(G)_{1−a,S}) of K_{1}(Λ(G)_{S}). We now consider Λ(G) ⊗_{Λ(G),tw_{a}} Λ(G)^{#}(1) where the tensor product indicates that the first term Λ(G) is regarded a right Λ(G)-module via the homomorphism tw_{a}. This tensor product is endowed with commuting left actions of Λ(G) (via left multiplication on the first factor) and G_{k} (via the action on the second factor specified at the beginning of §9.2) and, with respect to these actions, the map 1 ⊗ g ↦ χ_{k}(g)^{−a}g induces an isomorphism of Λ(G)×ℤ_{p}[G_{k}]-modules Λ(G)⊗_{Λ(G),tw_{a}} Λ(G)^{#}(1) ≅ Λ(G)^{#}(1−a). This isomorphism restricts to give an isomorphism of Λ(G) × ℤ_{p}[G_{k}]-modules

Λ(G) ⊗_{Λ(G),tw_{a}} Λ(G^{+})^{#}(1) = e_{1−a}(Λ(G) ⊗_{Λ(G),tw_{a}} Λ(G)^{#}(1)) ≅ e_{1−a}Λ(G)^{#}(1 − a)

and hence (by [24, Prop. 1.6.5(3)]) an isomorphism in D^{p}(Λ(G))

(41)  Λ(G) ⊗_{Λ(G),tw_{a}} C_{K^{+}/k,Σ}^{•}(1) ≅ C_{K/k,Σ}^{•}(1 − a).

Given this isomorphism (with a = 1−r) Corollary 9.4(i) follows directly from Theorem 9.1(i) (with K and G replaced by K^{+} and G^{+}). With ξ̃ the element of K_{1}(Λ(G^{+})_{S}) described in Theorem 9.1(ii) we next claim that ξ̃_{r} := tw_{1−r}(ξ̃) ∈ K_{1}(Λ(G)_{r,S}) ⊂ K_{1}(Λ(G)_{S}) satisfies the conditions of Corollary 9.4(ii). Indeed, to see that ξ̃_{r} satisfies property (ii)(b) one need only combine Theorem 9.1(ii)(b) with (41) and the commutative diagram (40) (with a = 1 − r). We shall prove that ξ̃_{r} also satisfies property (ii)(a) in Lemma 9.5 below. We now consider claim (iii). In this case G has rank one and so (41) combines with the first paragraph of Lemma 9.3 (with E′ = K^{+} and F′ = k) to imply C_{K/k,Σ}^{•}(r) belongs to D_{S^{∗}}^{p}(Λ(G)). To prove the rest of claim (iii) we use Corollary 8.3. To do this we first extend by linearity the displayed formula in claim (ii)(a) to all ρ ∈ A(G) in order to obtain a set of (leading term) interpolation data. We then note that, since G has no element of order p, any G = U/J ∈ SQ_{1}^{p−el}(G) belongs to SQ_{1}^{ab}(G) and has no element of order p. In addition, since K is CM, the extension K^{J}(ς_{p})/(K^{+})^{U} is a rank one CM abelian extension of a totally real field such that (K^{+})^{U} ⊆ K^{U} ⊂ K^{U,∞} ⊆ K^{J} ⊆ K^{J}(ς_{p}) and G_{K^{J} (ς_{p})/(K^{+})^{U,∞}} has no element of order p. Taking account of Corollary 8.3 with S = S^{∗}, it therefore suffices to prove the remaining assertions of claim (iii) in the case that G is abelian of rank one and contains no element of order p. In this case we set H := H(G^{+}) and

ξ̃:= ∑_{ϕ∈H^{∨}} f_{Σ,ϕ}(T)e_{ϕ} ∈ Q(ℤ_{p}[[T]])[H]^{×} = Q(G^{+})^{×} ≅ K_{1}(Q(G^{+})),

where for each ϕ ∈ H^{∨} we write e_{ϕ} for the idempotent |H|^{−1} ∑_{h∈H} ϕ(h^{−1})h of ℤ_{p}[ϕ][H]. Then the proof of Lemma 9.3 combines with (41) to imply that ∂_{G,S^{∗}}(tw_{1−r}(ξ̃)) = χ^{ref}(C_{K/k,Σ}^{•}(r)) and, by using (33) and (34), it is also straightforward to check that the element tw_{1−r}(ξ̃) has the necessary interpolation properties. This completes the proof of claim (iii). □

Lemma 9.5. Let K/k be as in Corollary 9.4(ii) and recall the element ξ̃ of K_{1}(Λ(G^{+})_{S}) described by Theorem 9.1(ii). Then for all r ∈ ℤ and all ρ ∈ A_{r}(G) one has tw_{1−r}(ξ̃)\underline{G}(ρ) = L_{p,Σ}(r, ρ ⊗ κ_{k}^{1−r}).

Proof. Fix ρ ∈ A_{r}(G) and set ψ := ρ ⊗ κ_{k}^{1−r} ∈ A_{1}(G) = A^{+}(G). Then, in view of (33), the claimed result will follow if one has tw_{1−r}(ξ̃)\underline{G}(ρ) = f_{Σ,ψ}(ω_{k}(γ_{G})^{1−r} − 1) for all r ∈ ℤ. Now Theorem 9.1(ii)(a) implies that there are infinitely many ω ∈ Γ^{∨} with

Φ_{\underline{G},ψ}(ξ̃)(ω(γ_{G}) − 1) = Φ_{\underline{G},ψ⊗ω}(ξ̃)(0) = ξ̃\underline{G}(ψ ⊗ ω)

= L_{p,Σ}(1, ψ ⊗ ω) = f_{Σ,ψ⊗ω}(0) = f_{Σ,ψ}(ω(γ_{G}) − 1),

and hence that Φ_{\underline{G},ψ}(ξ̃) = f_{Σ,ψ}(T). Since tw_{1−r}(ξ̃)\underline{G}(ρ) := Φ_{\underline{G},ρ}(tw_{1−r}(ξ̃))(0) the required equality tw_{1−r}(ξ̃)\underline{G}(ρ) = f_{Σ,ψ}(ω_{k}(γ_{G})^{1−r} − 1) will therefore follow if for all ξ ∈ K_{1}(Λ(G)_{S}) and a ∈ ℤ one has Φ_{\underline{G},ρ}(tw_{a}(ξ))(0) = Φ_{\underline{G},ρ⊗κ_{k}^{a}}(ξ)(ω_{k}(γ_{G})^{a} − 1). To prove this equality we recall that, since Λ(G)_{S} is semi-local (by [19, Prop. 4.2]), the natural homomorphism Λ(G)_{S}^{×} → K_{1}(Λ(G)_{S}) is surjective and so K_{1}(Λ(G)_{S}) is generated by the images of elements in S ⊂ Λ(G). It thus suffices to prove that Φ_{\underline{G},ρ}(tw_{a}(x)) = tw′_{a}(Φ_{\underline{G},ρ⊗κ_{k}^{a}}(x)) for all x ∈ S, where tw′_{a} is the automorphism of Q^{c}(ℤ_{p}[[T]]) that sends T to ω_{k}(γ_{G})^{a}(1+T)−1. Since ρ and κ_{k} are continuous we can also assume that x is a finite sum of the form ∑_{g} c_{g}g with g ∈ G and c_{g} ∈ ℤ_{p}. Then, writing π for the restriction homomorphism π\underline{G} : G → G_{k^{∞}/k}, one has Φ_{\underline{G},ρ}(x) = det(∑_{g} c_{g}ρ(g)π(g)). Since tw_{a}(x) = ∑_{g} c_{g}χ_{k}^{a}(g)g and χ_{k}^{a} = ω_{k}^{a}κ_{k}^{a} and for all g ∈ G both κ_{k}^{a}(g)ρ(g) = (ρ ⊗ κ_{k}^{a})(g) and ω_{k}^{a}(g) = ω_{k}^{a}(π(g)), one therefore has

Φ_{\underline{G},ρ}(tw_{a}(x)) = det(∑_{g} c_{g}χ_{k}^{a}(g)ρ(g)π(g)) = det(∑_{g} c_{g}(ρ ⊗ κ_{k}^{a})(g)ω_{k}^{a}(π(g))π(g))

= tw′_{a}(det(∑_{g} c_{g}(ρ ⊗ κ_{k}^{a})(g)π(g))) = tw′_{a}(Φ_{\underline{G},ρ⊗κ_{k}^{a}}(x)),

as required. □

9.5.2. The deduction of Corollary 2.10 We now fix a finite CM Galois extension F of a totally real number field k that is unramified outside a finite set of places Σ of k and set G := G_{F/k}. We fix a set of representatives Ir(G) of the isomorphism classes of irreducible finite dimensional complex representations of G. We also fix an odd prime p and an isomorphism of fields ℂ ≅ ℂ_{p} and hence regard Ir(G) as a subset of A(G). For each ϱ ∈ Ir(G) we write e_{ϱ} for the primitive central idempotent χ_{ϱ}(1)|G|^{−1} ∑_{g∈G} χ_{ϱ}(g^{−1})g of ℚ_{p}^{c}[G], where χ_{ϱ} is the character of ϱ. For each integer r we set 𝔄_{p,r} := ℤ_{p}[G]e_{r} and A_{p,r} := ℚ_{p}[G]e_{r}. We write D^{p,f}(𝔄_{p,r}) for the subcategory of D^{p}(𝔄_{p,r}) comprising perfect complexes that have finite cohomology groups in all degrees and recall that each C^{•} in D^{p,f}(𝔄_{p,r}) has a canonical Euler characteristic χ^{ref}(C^{•}) in K_{0}(𝔄_{p,r}, A_{p,r}). To be precise, we follow the conventions of [15] and so define χ^{ref}(C^{•}) to be the inverse of the element that corresponds under the isomorphism of [15, (2)] (with R = 𝔄_{p,r} and R′ = A_{p,r}) to the pair (C^{•}, ι_{C^{•}}) with ι_{C^{•}} the canonical isomorphism ℚ_{p} ⊗_{ℤ_{p}} C^{•} ≅ 0 in D^{p}(A_{p,r}). To prove Corollary 2.10 we first recall that, as is shown in [8, §11.2], for each strictly negative integer r the validity of the equivariant Tamagawa number conjecture for (ℚ(r)_{F/k}, ℤ[\frac{1}{2}][G]e_{r}) implies that the elements w_{−r}(F/k) and Ω_{−r}(F/k) of Cl(ℤ[G]) defined by Chinburg et al in [17] coincide after projection to Cl(ℤ[\frac{1}{2}][G]e_{r}) (as asserted by the second claim of Corollary 2.10). To prove Corollary 2.10 it is thus enough to show that Corollary 9.4 implies (modulo the vanishing of appropriate μ-invariants) the validity for each strictly negative integer r of the equivariant Tamagawa number conjecture for (ℚ(r)_{F/k}, ℤ[\frac{1}{2}][G]e_{r}). To give an explicit statement of the latter conjecture we quickly review the approach of [13, §5]. We thus recall that for each odd prime p and strictly negative integer r the complex C_{F/k,Σ}^{•}(r) in (39) is isomorphic in D^{p}(𝔄_{p,r}) to

C_{p}^{•}(r) := e_{r}RΓ_{c,ét}(Spec(O_{F,Σ_{p}}), ℤ_{p}(r))

and that results of Soulé, Dwyer-Friedlander, Quillen and Borel combine to imply that C_{p}^{•}(r) is an object of D^{p,f}(𝔄_{p,r}). In addition, classical results of Klingen and Siegel imply that the element

L_{Σ}(r) := ∑_{ψ∈Ir(G)} L_{Σ}(r, ψ)e_{ψ}

belongs to ζ(A_{p,r})^{×}. The approach of [13, §5] then shows that the equivariant Tamagawa number conjecture for (ℚ(r)_{F/k}, ℤ[\frac{1}{2}][G]e_{r}) is equivalent to asserting that for each odd prime p one has an equality

(42)  δ_{p}(L_{Σ}(r)) = −χ^{ref}(C_{p}^{•}(r)) ∈ K_{0}(𝔄_{p,r}, A_{p,r})

with δ_{p} := ∂_{p} ∘ Nrd_{p}^{−1} where ∂_{p} : K_{1}(A_{p,r}) → K_{0}(𝔄_{p,r}, A_{p,r}) is the connecting homomorphism of relative K-theory and Nrd_{p} : K_{1}(A_{p,r}) ≅ ζ(A_{p,r})^{×} the isomorphism induced by taking reduced norms with respect to the semisimple ℚ_{p}-algebra A_{p,r}. To prove (42) we set K := F^{∞} and G := G_{K/k}. If p divides |G|, then Corollary 2.10 assumes that there exists an intermediate field k′ of F/k such that F/k′ (and hence also K/k′) is pro-p and μ_{p}(k′) = 0. The equalities of Corollary 9.4(ii) and (iii) are therefore valid unconditionally for K/k. Since ℚ_{p} ⊗_{ℤ_{p}} C_{p}^{•}(r) is acyclic the natural ‘descent’ isomorphism in D^{p}(𝔄_{p,r})

𝔄_{p,r} ⊗_{Λ(G)e_{r}}^{𝕃} C_{K/k,Σ}^{•}(r) ≅ C_{F/k,Σ}^{•}(r) ≅ C_{p}^{•}(r)

implies that C_{K/k,Σ}^{•}(r) is ‘semisimple at each element of Ir(G)’ in the sense of [14, Def. 3.11]. By applying the descent formalism of [15, Th. 2.2 and Rem. 2.3] to the equalities of Corollary 9.4(ii), resp. (iii), one therefore obtains an equality

δ_{p}(L_{p,Σ}(r)) = −χ^{ref}(C_{p}^{•}(r))

with

L_{p,Σ}(r) := ∑_{ρ} L_{p,Σ}(r, ρ ⊗ κ_{k}^{1−r})e_{ρ} ∈ ζ(A_{p,r})^{×}

where ρ runs over representations in Ir(G) that are trivial on (−1)^{r+1}τ. To deduce (42) it is thus enough to prove that δ_{p}(L_{p,Σ}(r)) = δ_{p}(L_{Σ}(r)).

46 David Burns

Now (32) implies that L_{p,Σ}(r) = L_{Σ_{p}}(r) and, as each place in Σ_{p} \ Σ is unramified in F/k, one has an equality in ζ(A_{p,r})^{×}

L_{Σ_{p}}(r) = L_{Σ}(r) ∏_{v∈Σ_{p}\Σ} Nrd_{p}([e_{r}(1 − Nv^{−r}σ_{w})])

where we write [u] for the image in K_{1}(A_{p,r}) of each u ∈ A_{p,r}^{×} and σ_{w} for the frobenius automorphism in G of any place w of F above v. It therefore suffices to prove that δ_{p}(Nrd_{p}([e_{r}(1 − Nv^{−r}σ_{w})])) = 0 for all v ∈ Σ_{p} \ Σ. But each such v divides p and so, since r < 0, one has e_{r}(1 − Nv^{−r}σ_{w}) ∈ 𝔄_{p,r}^{×} and thus the exact sequence of relative K-theory implies that

δ_{p}(Nrd_{p}([e_{r}(1 − Nv^{−r}σ_{w})])) = ∂_{p}([e_{r}(1 − Nv^{−r}σ_{w})]) = 0,

as required. This completes our proof of Corollary 2.10.

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King’s College London, Dept. of Mathematics, London WC2R 2LS, United Kingdom david.burns@kcl.ac.uk