The GL_{2} main conjecture for elliptic curves without complex multiplication

J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob

April 15, 2004

1 Introduction

The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex L-functions, typified by the conjecture of Birch and Swinnerton-Dyer and its generalizations. Our goal in the present paper is to develop algebraic techniques which enable us to formulate a precise version of such a main conjecture for motives over a large class of p-adic Lie extensions of number fields. The methods which we develop in general were inspired by the Heidelberg Habilitation thesis of one of us (Venjakob [33]).

Let G be a compact p-adic Lie group with no element of order p and write Λ(G) for the Iwasawa algebra of G (see §2). Let M be a finitely generated torsion Λ(G)-module and write χ(G, M) for its G-Euler characteristic (see §3, (34)). How can we define a characteristic element of M and in particular relate this to the Euler characteristic of M and its twists? Let us quickly recall how such characteristic elements are defined in classical commutative Iwasawa theory when G = ℤ_{p}^{d} for some integer d ≥ 1. In this case, the structure theory for finitely generated torsion Λ(G)-modules [3] shows that there exist a finite number of non-zero elements f_{1}, ⋯, f_{r} such that we have an exact sequence of Λ(G)-modules

0 → ⊕_{i=1}^{r}Λ(G)/Λ(G)f_{i} → M → D → 0,

where D is a pseudo-null Λ(G)-module. We then define a characteristic element of M to be

f_{M} := ∏_{i=1}^{r} f_{i}(T)

which is uniquely determined up to multiplication by a unit in Λ(G). The classical theory (see [26, V16]) shows that if H_{0}(G, M) is finite, then χ(G, M) and χ(G, D) are both finite, and we have

χ(G, M) =| f_{M}(0) |_{p}^{−1}, χ(G, D) = 1

where f_{M}(0) denotes the image of f_{M} under the augmentation map from Λ(G) to ℤ_{p}.

In the non-commutative case, assuming G is p-valued, it is shown in [7] that there is an exact sequence

0 → ⊕_{i=1}^{r}Λ(G)/L_{i} → M/M_{0} → D → 0,

where the L_{i} are non-zero reflexive left ideals of Λ(G), M_{0} is the maximal pseudo-null submodule of M, and D is some pseudo-null Λ(G)-module. The whole approach of the commutative case to define characteristic elements now seems to break down irretrievably. Firstly, it is no longer true (see [6]) that χ(G, D) is finite implies that χ(G, D) = 1 for D pseudo-null. Secondly, it is also not true in general (see the appendix to [32]) that a reflexive left ideal in Λ(G) is always principal.

The goal of this paper is to provide a way out of this dilemma via localisation techniques for an important class of G. Namely, we assume that G has a closed normal subgroup H such that Γ = G/H ≃ ℤ_{p}. For example, this is automatically true when G is the Galois group of a p-adic Lie extension of a number field F, which contains the cyclotomic ℤ_{p}-extension of F. We prove in §2 that the Iwasawa algebra Λ(G) contains a canonical Ore set S^{∗}, enabling us to define the localised algebra Λ(G)_{S^{∗}}. Write 𝔐_{H}(G) for the category consisting of all finitely generated Λ(G)-modules which are annihilated by S^{∗}. We prove that a finitely generated module M belongs to 𝔐_{H}(G) if and only if M/M(p) is finitely generated over Λ(H) where M(p) denotes the p-primary submodule of M. We optimistically believe that 𝔐_{H}(G) contains all the torsion Λ(G)-modules which are of interest in arithmetic applications (for a precise statement see Conjecture 5.1). We then exploit the well known localization sequence of K-theory for the Ore set S^{∗} (see [28]) to define a characteristic element ξ_{M} in K_{1}(Λ(G)_{S^{∗}}) for any module M in 𝔐_{H}(G). We also show that one can relate ξ_{M} to the Euler charcateristics of twists of M by arbitrary continuous representations of G, with values in GL_{n}(O), where O is the ring of integers of a finite extension of ℚ_{p}. This uses the Akashi series of M which was introduced in [6].

The paper ends by formulating and briefly discussing the main conjecture for an elliptic curve E over ℚ over the field generated by the coordinates of its p-power division points, where p is a prime ≥ 5 of good ordinary reduction for E. We also give some numerical evidence in support of this main conjecture based on the remarkable calculations of [12]. We have striven to keep the technical discussions to a minimum in the present paper. A forthcoming paper (Fukaya and Kato, [14]) by two of us will consider quite generally the Iwasawa theory of motives over arbitrary p-adic Lie extensions of number fields and its connexion with the Tamagawa number conjecture. In particular, that work applies to any p-adic Lie extension with Galois group G, and does not need either of the two basic hypotheses made in the present paper (namely that G has no element of order p, and that G has a quotient isomorphic to ℤ_{p}). Acknowledgements: This work was greatly assisted by the generous hospitality provided by the Department of Mathematics of Kyoto University for JC, the Harishchandra Research Institute, Allahabad for JC and RS, and the Tata Institute of Fundamental Research, Mumbai for JC and OV. Finally, we would like to thank Peter Schneider for pointing out to us the alternative characterization of the Ore set S given at the end of §2.

2 The canonical Ore set.

Let G be a compact p-adic Lie group. We define

Λ(G) = lim_{U^{⟵}} ℤ_{p}[G/U], Ω(G) = lim_{U^{⟵}} 𝔽_{p}[G/U],  (1)

where U runs over all open normal subgroups of G. By a module over these algebras, we shall always mean, unless specified otherwise, a left module. It is well known that Λ(G) and Ω(G) are left and right Noetherian. Moreover, if Λ(G) has no zero divisors (e.g. if G is pro-p and has no element of order p), we shall always write Q(G) for the skew field of fractions of Λ(G).

We assume throughout this paper that G has a closed normal subgroup H such that

Γ = G/H →^{≃} ℤ_{p}.  (2)

Inspired by the results in Venjakob [33], our aim is to construct a certain canonical Ore set S in Λ(G).

Definition. Let S be the set of all f in Λ(G) such that Λ(G)/Λ(G)f is a finitely generated Λ(H)-module.

Let H′ be any open subgroup of H. Then Λ(H) is a free left or right module of rank [H : H′] over Λ(H′). Hence, if M is a Λ(H)-module, M will be finitely generated over Λ(H) if and only if it is finitely generated over Λ(H′), and similarly if M is a right Λ(H)- module. To exploit this observation, we will choose H′ = J, where J denotes any open normal subgroup of H which is pro-p. For such a pro-p J, Λ(J) is a local ring, whose maximal ideal is the kernel of the augmentation map from Λ(J) to 𝔽_{p}, and we can then use Nakayama’s lemma for Λ(J)-modules (see the proof of Lemma 2.1). Let

φ_{J} : Λ(G) → Λ(G/J), ψ_{J} : Λ(G) → Ω(G/J)  (3)

be the natural surjections. For J as above, let us note that we can always find an open subgroup J′ of J which is normal in G because the set of G conjugates of such a J is finite.

Lemma 2.1. Let J be any pro-p open subgroup of H, which is normal in G. Then (i) S is the set of all f in Λ(G) such that Λ(G/J)/Λ(G/J)φ_{J}(f) is a finitely generated ℤ_{p}-module, (ii) S is the set of all f in Λ(G) such that Ω(G/J)/Ω(G/J)ψ_{J}(f) is finite, and (iii) S is the set of all f in Λ(G) such that right multiplication by ψ_{J}(f) on Ω(G/J) is injective.

Proof. As remarked above, we can replace H by J in the definition of S. If f is any element of Λ(G), and if we put M = Λ(G)/Λ(G)f, then we clearly have

(M)_{J} = Λ(G/J)/Λ(G/J)φ_{J}(f), M/𝔪_{J}M = Ω(G/J)/Ω(G/J)ψ_{J}(f),  (4)

where 𝔪_{J} denotes the maximal ideal of Λ(J). Thus assertions (i) and (ii) are immediate from Nakayama’s lemma. To prove (iii), we use the important fact that we can always find a subgroup Π of G/J satisfying

Π →^{≃} ℤ_{p}, Π is in the centre of G/J.  (5)

To establish this, let us write Γ′ for some lifting of Γ to G/J. Then we can write G/J as the semi-direct product of H/J and Γ′, where Γ′ acts on H/J via conjugation. But, as H/J is finite, an open subgroup of Γ′ must clearly act trivially on H/J by conjugation, and we take Π to be this open subgroup. Now Ω(Π) is isomorphic to the ring of formal power series 𝔽_{p}[[T]] in an indeterminate T with coefficients in 𝔽_{p}. Thus the quotient field of Ω(Π), which we denote by R(Π), is a commutative field. Consider

V (G/J) = R(Π) ⊗_{Ω(Π)} Ω(G/J) = Ω(G/J) ⊗_{Ω(Π)} R(Π).  (6)

It is a finite dimensional algebra over the commutative field R(Π), which lies in its centre. If f is any element of Λ(G), let us write α_{J}(f) for the element of V (G/J) defined by ψ_{J}(f). By linear algebra, right multiplication by α_{J}(f) is surjective if and only if it is injective. Note also that, as Ω(Π) is a discrete valuation ring with residue field 𝔽_{p}, a finitely generated Ω(Π)-module is Ω(Π)-torsion if and only if it is finite. It follows easily that, for any f in Ω(G), Ω(G/J)/Ω(G/J)ψ_{J}(f) is finite if and only if right multiplication by ψ_{J}(f) on Ω(G/J) has finite kernel. But Ω(G/J) clearly has no finite Ω(G/J)-submodule, and so the equivalence of (ii) and (iii) is now clear.

Lemma 2.2. Let J be any pro-p open subgroup of H, which is normal in G. Then (i) S is the set of all f in Λ(G) such that Λ(G)/fΛ(G) is a finitely generated right Λ(H)-module; (ii) S is the set of all f in Λ(G) such that Λ(G/J)/φ_{J}(f)Λ(G/J) is a finitely generated ℤ_{p}-module, (iii) S is the set of all f in Λ(G) such that Ω(G/J)/ψ_{J}(f)Ω(G/J) is finite, and (iv) S is the set of all f in Λ(G) such that the left multiplication by ψ_{J}(f) on Ω(G/J) is injective.

Proof. We first note that it suffices only to prove (iv). Indeed, once (iv) is established, we can simply reverse the arguments of the proof of Lemma 2.1, but carrying them out for right modules, to deduce (i), (ii), and (iii). To prove (iv), we simply note the following. It was shown at the end of the proof of Lemma 2.1 that S consists of all f in Λ(G) such that there exists z in the algebra V (G/J) with z.α_{J}(f) = 1. But, as multiplication on the left or on the right in V (G/J) is R(Π)-linear, and V (G/J) is finite dimensional over R(Π), it follows from linear algebra that z.α_{J}(f) = 1 if and only if α_{J}(f).z = 1. This completes the proof of lemma 2.2.

Let M be a left or right Λ(G)-module. We say that M is S-torsion if, for each x in M, there exists s in S such that s.x = 0 or x.s = 0, according as the action is on the left or right.

Proposition 2.3. Let M be a finitely generated left or right Λ(G)-module. Then M is finitely generated over Λ(H) if and only if M is S-torsion.

Proof. We give the argument for left Λ(G)-modules. Suppose first that M is S-torsion. Take a finite family (x_{i})_{1≤i≤r} of elements of M which generates M as a Λ(G)-module, and choose f_{i} in S such that f_{i}.x_{i} = 0 for each i. Thus we get a Λ(G)-surjection

θ : ⨁_{i=1}^{r} Λ(G)/Λ(G)f_{i} → M  (7)

by mapping 1 in the i-th direct summand to x_{i}. By the definition of S, each module occurring in the finite direct sum is finitely generated over Λ(H), and hence M is also.

Conversely, assume that M is finitely generated over Λ(H). We must show that every element x of M is annihilated by an element of S. As before, fix any pro-p open subgroup J of H, which is normal in G. Again, we choose a subgroup Π of G/J satisfying (5), and we identify Λ(Π) with ℤ_{p}[[T]] by fixing a topological generator of Π. Now M is finitely generated over Λ(J). Let τ denote any lifting of T to Λ(G). For each integer n ≥ 1, define the Λ(J)-submodule

U_{n} = Λ(J)x + Λ(J)τx + ⋯ + Λ(J)τ^{n}x.  (8)

Since Λ(J) is Noetherian, it follows that there must exist an integer n ≥ 1, and a_{0}, …, a_{n−1} in Λ(J) such that

τ^{n}x = (a_{0} + a_{1}τ + ⋯ + a_{n−1}τ^{n−1})x.

Hence, if we define

s_{n} = τ^{n} − a_{n−1}τ^{n−1} − ⋯ − a_{0},

we have s_{n}.x = 0. But clearly ψ_{J}(s_{n}) is a non-zero element of Ω(Π), and hence, by (iii) of lemma 2.1, s_{n} belongs to S. This completes the proof of Proposition 2.3.

Theorem 2.4. The set S is multiplicatively closed, and is a left and right Ore set in Λ(G). The elements of S are non-zero divisors in Λ(G).

Proof. Take s_{1} and s_{2} in S. Then we have the exact sequence of Λ(G)-modules

0 → W → Λ(G)/Λ(G)s_{1}s_{2} → Λ(G)/Λ(G)s_{2} → 0,  (9)

where W = Λ(G)s_{2}/Λ(G)s_{1}s_{2}. As W is a homomorphic image of Λ(G)/Λ(G)s_{1}, it follows that the Λ(H)-module in the middle of (9) is finitely generated over Λ(H), and thus s_{1}s_{2} belongs to S. Now take f to be any element of S, and x to be any element of Λ(G). By Proposition 2.3, the left module Λ(G)/Λ(G)f and the right module Λ(G)/fΛ(G) are both S-torsion. Hence there exist s and s′ in S such that sx ∈ Λ(G)f and xs′ ∈ fΛ(G). This proves that S is a left and right Ore set in Λ(G).

To prove the final assertion of the theorem, take f to be any element of S. Now

Λ(G) = lim_{J^{⟵}} Λ(G/J),

where the projective limit is taken over all pro-p open subgroups J of H, which are normal in G. Thus, fixing any such J, it suffices to show that φ_{J}(f) is not a zero divisor in Λ(G/J). As before, we choose a subgroup Π of G/J satisfying (5), and let Q(Π) denote the quotient field of Λ(Π). Consider the algebra

W (G/J) = Q(Π) ⊗_{Λ(Π)} Λ(G/J) = Λ(G/J) ⊗_{Λ(Π)} Q(Π),  (10)

which is finite dimensional over the commutative field Q(Π), which lies in its centre. By virtue of (i) of Lemma 2.1 and (ii) of Lemma 2.2, we see that the image of φ_{J}(f) in W (G/J) has a right and left inverse. Hence φ_{J}(f) cannot be a divisor of zero in Λ(G/J), and the proof of Theorem 2.4 is complete.

We are grateful to P. Schneider (Lemma 2.5 and Proposition 2.6 below are due to him) for kindly pointing out to us the following alternative characterisation of the set S.

Lemma 2.5. For any two pro-p open subgroups J ⊆ J′ of H which are normal in G, the kernel of the natural map Ω(G/J) → Ω(G/J′) is a nilpotent ideal.

Proof. We recall that, quite generally, the prime radical N (A) in a Noetherian ring A is nilpotent and contains any other nilpotent ideal [19, 0.2.6, 2.3.7]. The assertion therefore is the claim that

Ker (Ω(G/J) → Ω(G/J′) ⊆ N (Ω(G/J)).

The subgroup Δ := J′/J is a finite normal p-subgroup of G/J. The elements δ − 1 for δ ∈ Δ generate the above left hand side as a left ideal. Let Π ⊆ G/J be a central subgroup as in the proof of Lemma 2.1; it maps isomorphically onto a central subgroup in G/J′. Hence we may view Ω(G/J) → Ω(G/J′) as a map of Ω(Π)-algebras. Since

N (Ω(G/J)) = Ω(G/J) ∩ N (V (G/J))

it suffices to prove that

Ker (V (G/J) → V (G/J′)) ⊆ N (V (G/J)).

Again, the left hand side is generated, as a left ideal, by the elements δ − 1 for δ ∈ Δ. The right hand side is the intersection of the annihilator (prime) ideals of all the simple modules over the finite dimensional R(Π)-algebra V (G/J). Hence we have to show that Δ acts trivially on any simple V (G/J)-module E. Since Δ is a p-group and E is a vector space over a field of characteristic p, the fixed vectors E^{Δ} under Δ in E certainly are non-zero. Since Δ is normal in G/J, these fixed vectors E^{Δ} form a V (G/J)-submodule of E. Hence we must have E^{Δ} = E.

The lemma implies that the two sided ideal

N := preimage of N (Ω(G/J)) in Λ(G)

is independent of the choice of a pro-p open subgroup J in H which is normal in G.

Proposition 2.6. The set S is equal to the set of all elements in Λ(G) which are regular modulo N.

Proof. Choose a pro-p open subgroup J ⊆ H which is normal in G. Since Λ(G)/N = Ω(G/J)/N (Ω(G/J)) an element f ∈ Λ(G) is regular modulo N if and only if ψ_{J}(f) is regular modulo N (Ω(G/J)), i.e. becomes a unit in the finite dimensional algebra V (G/J)/N (V (G/J)). But units modulo nilpotent ideals are units. Hence f is regular modulo N if and only if Ψ_{J}(f) is a non-zero divisor in Ω(G/J). By Lemma 2.1(iii) and 2.2(iii), the latter is equivalent to f belonging to S.

3 Akashi series and Euler characteristics.

Most of the results in this section are already established in [33], but we reprove them here in a slightly different fashion. As always, G will denote a compact p-adic Lie group with a closed normal subgroup H such that G/H = Γ is isomorphic to ℤ_{p}. We fix from now on a topological generator of Γ, and identify Λ(Γ) with the formal power series ring ℤ_{p}[[T]] by mapping this topological generator to 1 + T. We write Q(Γ) for the fraction field of Λ(Γ). Similarly, if O denotes the ring of integers of some finite extension of ℚ_{p}, we write Λ_{O}(Γ) (which we identify with O[[T]]) for the O-Iwasawa algebra of Γ, and Q_{O}(Γ) for the quotient field of Λ_{O}(Γ). Let S be the Ore set in Λ(G) which is defined in §2. Since p ∉ S, we shall also need to consider

Definition.

S^{∗} = ⋃_{n≥0} p^{n}S.

As p lies in the centre of Λ(G), S^{∗} is again a multiplicatively closed left and right Ore set in Λ(G), all of whose elements are non-zero divisors. We write Λ(G)_{S}, Λ(G)_{S^{∗}} for the localizations of Λ(G) at S and S^{∗}, so that

Λ(G)_{S^{∗}} = Λ(G)_{S}[\frac{1}{p}]  (11)

If M is a Λ(G)-module, we write M(p) for the submodule of M consisting of all elements of finite order. It is clear from Proposition 2.3 that M will be S^{∗}-torsion if and only if M/M(p) is finitely generated over Λ(H). We write 𝔐_{H}(G) for the category of all finitely generated Λ(G)-modules, which are S^{∗}-torsion. Note that in the special case in which H = 1 and G = Γ, 𝔐_{H}(G) is the category of all finitely generated torsion Λ(G)-modules. Both for motivation, and because we shall need it later in this section, we prove the following lemma (see [6]).

Lemma 3.1. For each M in 𝔐_{H}(G), the homology groups H_{i}(H, M) (i ≥ 0) are all finitely generated torsion Λ(Γ)-modules. If G has no element of order p, H_{i}(H, M) = 0 for i ≥ d, where d is the dimension of G as a p-adic Lie group.

Proof. We first observe that the H_{i}(H, M) are finitely generated Λ(Γ)-modules, because

H_{i}(H, M) = Tor_{i}^{Λ(G)}(Λ(G/H), M) (i ≥ 0),

and the modules on the right are finitely generated over Λ(Γ) since M is finitely generated over Λ(G). Put M_{f} = M/M(p). Now, for all i ≥ 0, H_{i}(H, M(p)) is killed by p^{t}, where t is any integer ≥ 0 such that p^{t}.M(p) = 0. Also, H_{i}(H, M_{f}) is a finitely generated ℤ_{p}-module since M_{f} is finitely generated over Λ(H). It now follows from the long exact sequence of H-homology that H_{i}(H, M) is a torsion Λ(Γ)-module for all i ≥ 0. The final assertion of the lemma is true because H has p-cohomological dimension d − 1 when H has no elements of order p [24].

As above, let O denote the ring of integers of some finite extension L of ℚ_{p}, and let us assume that we are given a continuous homomorphism

ρ : G → GL_{n}(O),  (12)

where n is some integer ≥ 1. If M is a finitely generated Λ(G)-module, put M_{O} = M⊗_{ℤ_{p}}O, and define

tw_{ρ}(M) = M_{O} ⊗_{O} O^{n}.  (13)

We endow tw_{ρ}(M) with the diagonal action of G i.e. if σ is in G, σ(m⊗z) = (σm)⊗(σz), where it is understood that G acts on O^{n} on the left via the homomorphism ρ. By compactness, this left action of G extends to an action of the whole Iwasawa algebra Λ(G).

Lemma 3.2. If M ∈ 𝔐_{H}(G), then, for all continuous representations ρ of the form (12), tw_{ρ}(M) ∈ 𝔐_{H}(G).

Proof. Assume that M ∈ 𝔐_{H}(G). Since O is a free module of finite rank over ℤ_{p}, it follows easily that M_{O} ∈ 𝔐_{H}(G). Since tw_{ρ}(M_{O}(p)) is killed by the same power of p which kills M_{O}(p), it suffices to prove that tw_{ρ}(R) is finitely generated over Λ(H), where R = M_{O}/M_{O}(p). Now we have a surjection Λ(H)^{m} → R, which plainly induces a surjection of Λ(H)-modules tw_{ρ_{H}}(Λ(H)^{m}) → tw_{ρ}(R), where ρ_{H} denotes the restriction of ρ to the subgroup H of G. But it is well known [33] that tw_{ρ_{H}}(Λ(H)^{m}) is again a free Λ(H)-module of finite rank. This completes the proof of Lemma 3.2.

If R is a ring with unit element, we write R^{×} for the group of units of R, and M_{n}(R) for the ring of n × n-matrices with entries in R. By continuity, the group homomorphism ρ induces a ring homomorphism

ρ : Λ(G) → M_{n}(O).  (14)

A second ring homomorphism

Φ_{ρ} : Λ(G) → M_{n}(Λ_{O}(Γ)),  (15)

which we now define, will play an important role in our work. If σ is in G, we write σ̄ for its image in Γ = G/H. We define a continuous group homomorphism

G → (M_{n}(O) ⊗_{ℤ_{p}} Λ(Γ))^{×}  (16)

by mapping σ to ρ(σ) ⊗ σ̄. Noting that

M_{n}(O) ⊗_{ℤ_{p}} Λ(Γ) = M_{n}(Λ_{O}(Γ))

because O is a free ℤ_{p}-module of finite rank, we obtain (15) by extending (16) to the whole of Λ(G). Note that Φ_{ρ}(p) = pI_{n}, where I_{n} is the unit matrix. Also it is easily seen that (14) is the composition of (15) with the map from M_{n}(Λ_{O}(Γ)) to M_{n}(O) induced by the augmentation map from Λ_{O}(Γ) to O.

Lemma 3.3. The map (15) extends to a ring homomorphism, which we also denote by Φ_{ρ},

Φ_{ρ} : Λ(G)_{S^{∗}} → M_{n}(Q_{O}(Γ)).  (17)

Proof. We must show that, for all s in S^{∗}, Φ_{ρ}(s) is invertible in M_{n}(Q_{O}(Γ)), or equivalently that Φ_{ρ}(s) has non-zero determinant. Since this is clearly true for s = p, we can assume that s is in S. Let k denote the residue field of O, and let

Φ̃_{ρ} : Λ(G) → M_{n}(k[[T]])

denote the composition of Φ_{ρ} with the canonical map from M_{n}(Λ_{O}(Γ)) to M_{n}(k[[T]]). It suffices to show that Φ̃_{ρ}(s) has non-zero determinant. Note that Φ̃_{ρ} is induced by the map

G → (M_{n}(k) ⊗_{k} k[[T]])^{×}  (18)

given by σ ↦ ρ̃(σ) ⊗ σ̄, where ρ̃(σ) denotes the image of ρ(σ) in M_{n}(k). We now exploit the fact that GL_{n}(k) is a finite group. Thus we can find a pro-p open subgroup J of H, which is normal in G, and which is contained in Ker (ρ̃). Clearly Φ̃_{ρ} can be factored through a map

δ : Ω(G/J) → M_{n}(k[[T]]).  (19)

Let ψ_{J} be given by (3). Then we must show that δ(ψ_{J}(s)) has non-zero determinant. As in §2, we can find a central subgroup Π of G/J with Π →^{≃} ℤ_{p}. Replacing Π by an open subgroup if necessary, we may also assume that Π lies in the kernel of the homomorphism from G/J to GL_{n}(k) induced by ρ̃. Hence it is clear from (18) that δ(Ω(Π)) must be contained in k[[T]].I_{n}. Let R(Π) (resp. Q(k[[T]])) denote the quotient field of Ω(Π) (resp. k[[T]]). Since the natural map from Π to Γ is injective, Q(k[[T]]) can be viewed as a finite extension of R(Π). Hence (19) induces a ring homomorphism

α : R(Π) ⊗_{Ω(Π)} Ω(G/J) → M_{n}(Q(k[[T]])).  (20)

But, as proven in §2, ψ_{J}(s) is a unit in the algebra on the left in (20). Hence α(ψ_{J}(s)) is invertible in M_{n}(Q(k[[T]]), and so has non-zero determinant. This completes the proof of Lemma 3.3.

If R is a ring, we write K_{m}R for the m-th K-group of R (we shall only need the cases m = 0, 1). Clearly (14) induces a homomorphism

K_{1}(Λ(G)) → K_{1}(M_{n}(O)) = O^{×}.  (21)

This can be extended to a map from K_{1}(Λ(G)_{S^{∗}}) to L ∪ {∞}, where L is the fraction field of O, in the following manner. Firstly, (17) induces a homomorphism

Φ′_{ρ} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(M_{n}(Q_{O}(Γ)))) = Q_{O}(Γ)^{×}.  (22)

Let φ : Λ_{O}(Γ) → O be the augmentation map, and write 𝔭 = Ker (φ). Of course, writing Λ_{O}(Γ)_{𝔭} ⊂ Q_{O}(Γ) for the localization of Λ_{O}(Γ) at 𝔭, it is clear that φ extends naturally to a homomorphism

φ : Λ_{O}(Γ)_{𝔭} → L.

Let ξ be any element of K_{1}(Λ(G)_{S^{∗}}). If Φ′_{ρ}(ξ) belongs to Λ_{O}(Γ)_{𝔭}, we define ξ(ρ) to be φ(Φ′_{ρ}(ξ)). However, if Φ′_{ρ}(ξ) does not belong to Λ_{O}(Γ)_{𝔭}, we define ξ(ρ) = ∞. This gives us the desired extension of (21).

We now use a well known localization theorem in K-theory to define the notion of a characteristic element for any module M in the category 𝔐_{H}(G). To ensure that we can work with modules rather than complexes, we assume for the rest of this section the following

Hypothesis on G. G has no element of order p.

It is well known [4] that this implies that Λ(G) has finite global dimension equal to d + 1, where d is the dimension of G as a p-adic Lie group. Since 𝔐_{H}(G) is the category of all finitely generated Λ(G)-modules which are S^{∗}-torsion, there is a connecting homomorphism (see [28])

∂_{G} : K_{1}(Λ(G)_{S^{∗}}) → K_{0}(𝔐_{H}(G)),  (23)

where K_{0}(𝔐_{H}(G)) denotes the Grothendieck group of the category 𝔐_{H}(G), such that we have, as part of a larger exact sequence of localization, the exact sequence

⋯ → K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}}) →^{∂_{G}} K_{0}(𝔐_{H}(G)) → K_{0}(Λ(G)) → K_{0}(Λ(G)_{S^{∗}}) → 0. (24)

Proposition 3.4. Assume that G has no element of order p. Then ∂_{G} is surjective.

Before giving the proof of Proposition 3.4, we need the following preliminary lemma.

Lemma 3.5. Let P be any pro-p open normal subgroup of G. Then the canonical map

K_{0}(Λ(G)) → K_{0}(ℤ_{p}[G/P])

is injective.

Proof. Let Δ = G/P, and let I denote the kernel of the natural map from Λ(G) to ℤ_{p}[Δ]. Let M be any finitely generated projective Λ(G)-module such that the class of M/IM in K_{0}(ℤ_{p}[Δ]) is zero, i.e. we have a ℤ_{p}[Δ]-isomorphism

α : M/IM ⊕ ℤ_{p}[Δ]^{r} →^{≃} ℤ_{p}[Δ]^{r},

for some integer r ≥ 0. Since M is a projective Λ(G)-module, we can find a Λ(G)- homomorphism

β : M ⊕ Λ(G)^{r} → Λ(G)^{r}  (25)

which lifts α. In particular, it is then clear that (Coker (β))_{P} = 0, whence, as P is pro-p, it follows from Nakayama’s lemma that Coker (β) = 0. Taking P -homology of the exact sequence

0 → Ker (β) → M ⊕ Λ(G)^{r} → Λ(G)^{r} → 0,

and noting that H_{1}(P, Λ(G)^{r}) = 0 because Λ(G) is a free Λ(P)-module of finite rank, we conclude that (Ker (β))_{P} = 0, and so also Ker (β) = 0. Thus β is an isomorphism, and the class of M in K_{0}(Λ(G)) is zero. This completes the proof of Lemma 3.5.

We now prove Proposition 3.4. For the proof, we fix a pro-p open normal subgroup P of G, and put Δ = G/P. We write V = V(Δ) for the set of irreducible representations of the finite group Δ over ℚ̄_{p}, and we take L to be some fixed finite extension of ℚ_{p} such that all representations in V can be realized over L. Thus we have an isomorphism of rings

γ : L[Δ] →^{≃} ∏_{ρ∈V} M_{n_{ρ}}(L),  (26)

where n_{ρ} denotes the dimension of ρ. The proof proceeds by constructing a canonical homomorphism

λ : K_{0}(Λ(G)) → ∏_{ρ∈V} K_{0}(L),  (27)

which will be the composition λ = λ_{4} ∘ λ_{3} ∘ λ_{2} ∘ λ_{1} of four natural maps λ_{i} (i = 1, …, 4), defined as follows. Firstly, λ_{1} is the canonical map appearing in Lemma 3.5. Secondly, we take

λ_{2} : K_{0}(ℤ_{p}[Δ]) → K_{0}(ℚ_{p}[Δ])

and

λ_{3} : K_{0}(ℚ_{p}[Δ]) → K_{0}(L[Δ])

to be the maps induced by the evident inclusions of rings. Finally, we take λ_{4} to be the isomorphism

λ_{4} : K_{0}(L[Δ]) →^{≃} ∏_{ρ∈V} K_{0}(M_{n_{ρ}}(L)) →^{≃} ∏_{ρ∈V} K_{0}(L),

where the first map is induced by γ, and the second is given by Morita equivalence. Now λ_{1} is injective by Lemma 3.5. Moreover, it is well known from the representation theory of finite groups that λ_{2} is injective (see [27], Chap. 16, Theorem 34, Corollary 2) and that λ_{3} is injective (see [27], Chap. 14, §14.6). Hence we conclude that the homomorphism λ is always injective.

To complete the proof of Proposition 3.4, we shall employ an alternative description of the map λ. Consider the map

τ : K_{0}(Λ_{O}(G)) → K_{0}(O)  (28)

induced by the augmentation map from Λ_{O}(G) to O. Recall that, since G has no element of order p, Λ_{O}(G) has finite global dimension, and we can identify K_{0}(Λ_{O}(G)) with the Grothendieck group of the category of all finitely generated Λ_{O}(G)-modules (see [19], §12.4.8). Let U (resp. A) be a finitely generated Λ_{O}(G)-module (resp. O-module), and write [U] (resp. [A]) for the class of U (resp. A) in K_{0}(Λ_{O}(G)) (resp. K_{0}(O)). Then it is easily seen that τ is given explicitly by

τ([U]) = ∑_{i≥0}(−1)^{i}[H_{i}(G, U)].  (29)

Let us also note that τ clearly factors through the map

ε : K_{0}(Λ_{O}(G)) → K_{0}(Λ_{O}(Γ))  (30)

by the natural surjection from Λ_{O}(G) to Λ_{O}(Γ). Moreover, ε is given explicitly by

ε([U]) = ∑_{i≥0}(−1)^{i}[H_{i}(H, U)].  (31)

Let j : K_{0}(O) → K_{0}(L) be the isomorphism induced by the inclusion of O in L. For each ρ in V, let tw_{ρ}(M) be the Λ_{O}(G)-module defined by (13). For each finitely generated Λ(G)-module M, it can be readily verified that

λ([M]) = ∏_{ρ∈V} j(τ([tw_{ρ}(M)])).  (32)

We can now finish the proof of Proposition 3.4. Suppose that M belongs to 𝔐_{H}(G). By Lemmas 3.1 and 3.2, we conclude that H_{i}(H, tw_{ρ}(M)) is Λ_{O}(Γ)-torsion for all i ≥ 0, and thus their class in K_{0}(Λ_{O}(Γ)) vanishes since the latter is isomorphic to ℤ, via the map which assigns to a finitely generated Λ_{O}(Γ)-module its rank. Hence it follows from (31) that ε([tw_{ρ}(M)]) = 0, whence certainly τ([tw_{ρ}(M)]) = 0, for all ρ in V. But then (32) implies that λ([M]) = 0, and so we must have [M] = 0, because λ is injective. It now follows from the exactness of (24) that ∂_{G} is surjective, and the proof of Proposition 3.4 is complete.

In view of Proposition 3.4, we can now define the notion of a characteristic element for each M in 𝔐_{H}(G).

Definition. For each M in 𝔐_{H}(G), a characteristic element of M is any ξ_{M} in K_{1}(Λ(G)_{S^{∗}}) such that

∂_{G}(ξ_{M}) = [M].  (33)

We recall that we say an Λ(G)-module M has finite G-Euler characteristic if H_{i}(G, M) is finite for all i ≥ 0. If M has finite G-Euler characteristic, we define

χ(G, M) = ∏_{i≥0} ♯(H_{i}(G, M))^{(−1)^{i}},  (34)

the product on the right is finite beause of our hypothesis that G has no element of order p. The principal result of this section, which is directly inspired by a parallel result in [33], is the following relation between characteristic elements and G-Euler characteristics. We write | |_{p} for the valuation of ℚ̄_{p}, normalized so that |p|_{p} = 1/p. For our continuous homomorphism

ρ : G → GL_{n}(O),

we write m_{ρ} = [L : ℚ_{p}], where L is the quotient field of O. Let ρ̂ denote the contragradient representation of G, i.e. ρ̂(g) = ρ(g^{−1})^{t} for g in G, where ′t′ denotes the transpose matrix.

Theorem 3.6. Assume that G has no element of order p. Take M in 𝔐_{H}(G), and let ξ_{M} be a characteristic element of M. Then, for every continuous homomorphism ρ : G → GL_{n}(O) such that χ(G, twρ̂(M)) is finite, we have

ξ_{M}(ρ) ≠ 0, ∞,  (35)

and

χ(G, twρ̂(M)) = |ξ_{M}(ρ)|_{p}^{−m_{ρ}},  (36)

where m_{ρ} = [L : ℚ_{p}], and L is the quotient field of O.

Before giving the proof of Theorem 3.6, we recall an important ingredient in it, which was first introduced in [6]. Assume M lies in 𝔐_{H}(G). By Lemma 3.1, the H_{i}(H, M) (i ≥ 0) are finitely generated torsion Λ(Γ)-modules, which are zero for i ≥ d. Let f_{i,M} denote a characterisitic power series for H_{i}(H, M) as a Λ(Γ)-module. We then define the Akashi series Ak(M) by

Ak(M) = ∏_{i≥0} f_{i,M}^{(−1)^{i}} mod Λ(Γ)^{×}.  (37)

As is explained in [6], Ak induces in the evident fashion a homomorphism

Ak : K_{0}(𝔐_{H}(G)) → Q(Γ)^{×}/Λ(Γ)^{×}.  (38)

Suppose now that M is also an O-module, so that we can regard the H_{i}(H, M) as finitely generated torsion Λ_{O}(Γ)-modules. Let g_{i,M} denote a characteristic power series of H_{i}(H, M) as a Λ_{O}(Γ)-module. We then define

Ak_{O}(M) = ∏_{i≥0} g_{i,M}^{(−1)^{i}} mod Λ_{O}(Γ)^{×}.  (39)

Now Λ_{O}(Γ) is a free Λ(Γ)-module of rank m_{ρ} = [L : ℚ_{p}], and we let N denote the norm map from Λ_{O}(Γ) to Λ(Γ). It follows from [3], Chap.VII, §4.8, Prop. 18 that N(g_{i,M}) = f_{i,M} mod Λ(Γ)^{×} for all i ≥ 0, and so we conclude that

N(Ak_{O}(M)) = Ak(M).  (40)

We now begin the proof of Theorem 3.6. Since the map M ↦ twρ̂(M) preserves exact sequences, we can define a map

Δ_{ρ} : K_{0}(𝔐_{H}(G)) → Q_{O}(Γ)^{×}/Λ_{O}(Γ)^{×}  (41)

by

Δ_{ρ}([M]) = Ak_{O}(twρ̂(M)).  (42)

Let

∂_{Γ} : Q_{O}(Γ)^{×} → Q_{O}(Γ)^{×}/Λ_{O}(Γ)^{×}

be the natural surjection. Consider the diagram

K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{G}} K_{0}(𝔐_{H}(G))
Φ′_{ρ}↓  ↓^{Δ_{ρ}}
Q_{O}(Γ)^{×}  ⟶^{∂_{Γ}} Q_{O}(Γ)^{×}/Λ_{O}(Γ)^{×}.

(43)

Lemma 3.7. The diagram (43) is commutative.

Let us now assume Lemma 3.7, and show that Theorem 3.6 follows. Let M and ξ_{M} be as in Theorem 3.6, and assume that χ(G, twρ̂(M)) is finite. As before, let φ : Λ_{O}(Γ) → O be the augmentation map, and 𝔭 its kernel. We again write

φ : Λ_{O}(Γ)_{𝔭} → L

for the homomorphism induced by φ. Since χ(G, twρ̂(M)) is finite, it is proven in [6] that φ(f_{i,twρ̂(M)}) ≠ 0 for all i ≥ 0, that φ(Ak(twρ̂(M))) ≠ 0, ∞, and that

χ(G, twρ̂(M)) = |φ(Ak(twρ̂(M)))|_{p}^{−1}.  (44)

Using (40) and the obvious fact that N commutes with the augumentation map, it follows that φ(Ak_{O}(twρ̂(M))) ≠ 0, ∞, and that

χ(G, twρ̂(M)) = |φ(Ak_{O}(twρ̂(M)))|_{p}^{−m_{ρ}}.  (45)

But the commutativity of (43) shows that

∂_{Γ}(Φ′_{ρ}(ξ_{M})) = Ak_{O}(twρ̂(M)).  (46)

Hence assertions (35) and (36) are clear from (45) and (46).

We now prove the commutativity of the diagram (43), which will complete the proof of Theorem 3.6. Let K_{0}^{tor}(M_{n}(Λ_{O}(Γ))) be the Grothendieck group of the category of all finitely generated torsion M_{n}(Λ_{O}(Γ))-modules. Now Morita equivalence and the functoriality of the localization sequence in K-theory shows that we have a commutative diagram

K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{G}} K_{0}(𝔐_{H}(G))
u_{1}↓  ↓^{u_{0}}
K_{1}(M_{n}(Q_{O}(Γ))) ⟶^{∂_{n}} K_{0}^{tor}(M_{n}(Λ_{O}(Γ)))
v_{1}↓≅  ≅↓^{v_{0}}
K_{1}(Q_{O}(Γ)) ⟶^{∂_{1}} K_{0}^{tor}(Λ_{O}(Γ)),

(47)

where u_{0} is induced by the ring homomorphism Φ_{ρ} given in (15), u_{1} is induced by the extension (17) of Φ_{ρ}, ∂_{n} and ∂_{1} are connecting homomorphisms of localization, and v_{0} and v_{1} are given by Morita equivalence. Granted the canonical identifications

K_{1}(Q_{O}(Γ)) = Q_{O}(Γ)^{×}, K_{0}^{tor}(Λ_{O}(Γ)) = Q_{O}(Γ)^{×}/Λ_{O}(Γ)^{×},

it therefore suffices to compute v_{0} ∘ u_{0}, and show that it is indeed Δρ̂ given by ρ̂(g) = ρ(g^{−1})^{t}. Take M in 𝔐_{H}(G). We can find a finite resolution

0 → P_{r} → P_{r−1} → ⋯ → P_{0} → M → 0,  (48)

where each P_{i} is a finitely generated projective Λ(G)-module. Thus, viewing M_{n}(Λ_{O}(Γ)) as a right Λ(G)-module via the homomorphism Φ_{ρ}, we have, by definition,

u_{0}([M]) = ∑_{i≥0}(−1)^{i}[M_{n}(Λ_{O}(Γ)) ⊗_{Λ(G)} P_{i}].  (49)

On the other hand, for an arbitrary ring R, the Morita equivalence between the category of M_{n}(R)-modules and the category of R-modules is given by N ↦ (R^{n})^{t} ⊗_{M_{n}(R)} N where (R^{n})^{t} means the space of row vectors over R with n entries (′t′ denotes the transpose) on which M_{n}(R) acts from the right in the evident way. Put R = Λ_{O}(Γ). We endow R ⊗_{O} (O^{n})^{t} with the right G-action given by (τ ⊗ x^{t})g = τḡ ⊗ x^{t}ρ(g)^{t}, where τ is in R, x is in O^{n}, g is in G, and ḡ denotes the image of g in Γ. Recalling that the left G-action on twρ̂(P_{i}) = P_{i} ⊗_{ℤ_{p}} O^{n} is given by g(y ⊗ x) = gy ⊗ ρ̂(g)x for y in P_{i}, we obtain an isomorphism of R-modules

(R ⊗_{O} (O^{n})^{t}) ⊗_{Λ(G)} P_{i} →^{≃} R ⊗_{Λ_{O}(G)} twρ̂(P_{i})

by mapping τ ⊗ x^{t} ⊗ y to τ ⊗ y ⊗ x. Combining this isomorphism with the natural identification

((R^{n})^{t} ⊗_{M_{n}(R)} M_{n}(R)) ⊗_{Λ(G)} P_{i} = (R ⊗_{O} (O^{n})^{t}) ⊗_{Λ(G)} P_{i},

we deduce finally that we have an isomorphism of R-modules

((R^{n})^{t} ⊗_{M_{n}(R)} M_{n}(R)) ⊗_{Λ(G)} P_{i} →^{≃} R ⊗_{Λ_{O}(G)} twρ̂(P_{i}).

Hence

v_{0}([M_{n}(Λ_{O}(Γ)) ⊗_{Λ(G)} P_{i}]) = Λ_{O}(Γ) ⊗_{Λ_{O}(G)} twρ̂(P_{i}).  (50)

But it has already been remarked that, if N is any free Λ_{O}(G)-module, then twρ̂(N) is again free. Hence it follows easily that twρ̂(P_{i}) is again a projective Λ_{O}(G)-module, and the exact sequence

0 → twρ̂(P_{r}) → twρ̂(P_{r−1}) → ⋯ → twρ̂(P_{0}) → twρ̂(M) → 0

is a projective resolution of twρ̂(M). Tensoring this exact sequence with Λ_{O}(Γ) over Λ_{O}(G), we get a complex whose cohomology groups are the H_{i}(H, twρ̂(M)) (i ≥ 0). Thus it follows that the image of v_{0} ∘ u_{0}([M]) under the canonical isomorphism from K_{0}^{tor}(Λ_{O}(Γ)) to Q_{O}(Γ)^{×}/Λ_{O}(Γ)^{×} is precisely Ak_{O}(twρ̂(M)). This completes the proof of Theorem 3.6.

We say our continuous representation ρ : G → GL_{n}(O) is an Artin representation if Ker (ρ) is open in G, or equivalently if ρ factors through a finite quotient of G. The following stronger form of Theorem 3.6, but for a much more restricted class of modules, will be needed in the last section to study some of the consequences of the ”main conjecture”.

Theorem 3.8. Assume that G has no element of order p. Let M be a module in 𝔐_{H}(G), which, in addition, satisfies

H_{i}(H′, M) = 0 for all i ≥ 1,  (51)

and for all open subgroups H′ of H, which are normal in G. Let ξ_{M} denote a characteristic element of M. Then ξ_{M}(ρ) ≠ ∞ for every Artin representation ρ of G. Moreover, for each Artin representation ρ of G, ξ_{M}(ρ) ≠ 0 if and only if χ(G, twρ̂(M)) is finite.

The following lemma is the essential ingredient in the proof of Theorem 3.8.

Lemma 3.9. Let M be a module in 𝔐_{H}(G), satisfying (51). Then, if ρ is any Artin representation of G,

Ak_{O}(tw_{ρ}(M)) ∈ Λ_{O}(Γ)[\frac{1}{p}] modulo Λ_{O}(Γ)^{×}.  (52)

Moreover, if φ : Λ_{O}(Γ)_{𝔭} → L is the homomorphism induced by the augmentation map, then φ(Ak_{O}(tw_{ρ}(M))) ≠ 0 if and only if χ(G, tw_{ρ}(M)) is finite.

Proof. Let ρ be any Artin representation of G, and put W = tw_{ρ}(M). Let g_{i,W} denote the characteristic power series of H_{i}(H, W) as a Λ_{O}(Γ)-module (i ≥ 0). Let π denote a local parameter of O. We now prove that we can take

g_{i,W} = π^{μ_{i}} (i ≥ 1)  (53)

for some integer μ_{i} ≥ 0. Take H′ = Ker (ρ) ∩ H. Hence (51) holds for H′, whence it follows easily that we have H_{i}(H′, M_{O}) = 0 for all i ≥ 1. But since H′ ⊂ Ker (ρ), we have W = M_{O}^{n} as H′-modules, and so we conclude that

H_{i}(H′, W) = 0 (i ≥ 1).  (54)

Put Δ = H/H′. It follows from (53) and the Hochschild-Serre spectral sequence that

H_{i}(H, W) = H_{i}(Δ, W_{H′}) (i ≥ 1).  (55)

But the group on the right is clearly annihilated by the order of the finite group Δ, whence it is plain that the characteristic power series of the module on the left must be a power of π. This proves (53). Hence

Ak_{O}(W) = g_{0,W} ∏_{i≥1} π^{(−1)^{i}μ_{i}} mod Λ_{O}(Γ)^{×},  (56)

and so assertion (52) is proven. It also follows from (56) that φ(Ak_{O}(W)) ≠ 0 if and only if φ(g_{0,W}) ≠ 0. Now a standard argument with the Hochschild-Serre spectral sequence (see [6]) shows that χ(G, M) is finite if and only if φ(g_{i,W}) ≠ 0 for all i ≥ 0. But, in view of (53), it is clear that φ(g_{i,W}) ≠ 0 for all i ≥ 0 if and only if φ(g_{0,W}) ≠ 0. This completes the proof of Lemma 3.9.

Theorem 3.8 is an almost immediate consequence of Lemmas 3.7 and 3.9. By definition, for any Artin representation ρ of G, ξ_{M}(ρ) ≠ ∞ means that Φ′_{ρ}(ξ_{M}) belongs to Λ_{O}(Γ)_{𝔭}. But, by Lemma 3.7,

∂_{Γ}(Φ′_{ρ}(ξ_{M})) = Ak_{O}(twρ̂(M)),  (57)

whence it is clear from (52) that Φ′_{ρ}(ξ_{M}) does belong to Λ_{O}(Γ)_{𝔭}. The final assertion of Theorem 3.8 is also clear from (57) and the final assertion of Lemma 3.9. This completes the proof of Theorem 3.8.

We next establish an analogue of the classical Artin formalism for G-Euler characteristics. The result is of interest in its own right, but we shall also use it to study the numerical example at the end of this section. In fact, as we shall show, the Artin formalism holds for all compact p-adic Lie groups G with no element of order p, and we do not need for this result the existence of a closed normal subgroup H of G with G/H ≅ ℤ_{p}. We use similar notation to earlier. Let G′ be an arbitrary open normal subgroup of G, and let Δ = G/G′. Let V = V(Δ) denote the set of irreducible representations of Δ over ℚ̄_{p}. We take L to be any finite extension of ℚ_{p} such that all representations in V can be realized over L, and we write O for the ring of integers of L.

Theorem 3.10. Let G be any compact p-adic Lie group with no element of order p, G′ an open normal subgroup of G, and let Δ = G/G′. Let M be a finitely generated Λ(G)-module. Then χ(G′, M) is finite if and only if χ(G, tw_{ρ}(M)) is finite for all ρ in V = V(Δ). When χ(G′, M) if finite, we have

χ(G′, M)^{[L:ℚ_{p}]} = ∏_{ρ∈V} χ(G, tw_{ρ}(M))^{n_{ρ}},  (58)

where n_{ρ} is the dimension of ρ.

We now prove Theorem 3.10. Put R = O[Δ]. If ρ is in V, we let L_{ρ} denote a free O-module of rank n_{ρ}, endowed with a left action of R which realizes ρ. We then have an exact sequence of left R-modules

0 → R → ⨁_{ρ∈V} L_{ρ}^{n_{ρ}} → W → 0,  (59)

where W is finite. If U is any left R-module, we can view U as a right R-module by defining u.δ to be δ^{−1}.u for δ in Δ and u in U. In particular, we can view all the modules appearing in (59) to be right R-modules or Λ_{O}(G)-modules in this fashion. Now take M to be the module appearing in Theorem 3.10, and put M_{O} = M ⊗_{ℤ_{p}} O. We take a resolution

0 → P_{n} → P_{n−1} → ⋯ → P_{0} → M_{O} → 0,  (60)

where P_{i} (0 ≤ i ≤ n) is a finitely generated projective Λ_{O}(G)-module. Tensoring (59) on the right over Λ_{O}(G) with P_{i}, we obtain an exact sequence

0 → R ⊗_{Λ_{O}(G)} P_{i} → ⨁_{ρ∈V}(L_{ρ} ⊗_{Λ_{O}(G)} P_{i})^{n_{ρ}} → W ⊗_{Λ_{O}(G)} P_{i} → 0,  (61)

the injectivity on the left is valid because P_{i} is projective. Hence we obtain an exact sequence of complexes

0 → C → ⨁_{ρ∈V} D_{ρ}^{n_{ρ}} → K → 0,  (62)

where C = (C_{i}), D_{ρ} = (D_{ρ,i}), K = (K_{i}), with i running from 0 to n, are given by

C_{i} = R ⊗_{Λ_{O}(G)} P_{i}, D_{ρ,i} = L_{ρ} ⊗_{Λ_{O}(G)} P_{i}, K_{i} = W ⊗_{Λ_{O}(G)} P_{i}.  (63)

Now

R ⊗_{Λ_{O}(G)} M_{O} = (M_{O})_{G′}, L_{ρ} ⊗_{Λ_{O}(G)} M_{O} →^{≃} (tw_{ρ}(M)) ⊗_{Λ_{O}(G)} O = (tw_{ρ}(M))_{G}. (64)

Writing H_{i}(C), H_{i}(D_{ρ}), H_{i}(K) for the i-th homology groups of the complexes C, D_{ρ}, K, we therefore have, for 0 ≤ i ≤ n,

H_{i}(C) = H_{i}(G′, M_{O}), H_{i}(D_{ρ}) = H_{i}(G, tw_{ρ}(M)).  (65)

Now, as O is flat over ℤ_{p},

H_{i}(G′, M_{O}) = H_{i}(G′, M) ⊗_{ℤ_{p}} O.  (66)

The long exact sequence for the cohomology of the exact sequence of complexes (62) shows that the conclusions of Theorem 3.10 will all follow provided we can prove that

H_{i}(K) is finite for 0 ≤ i ≤ n and ∏_{i=0}^{n} ♯(H_{i}(K))^{(−1)^{i}} = 1.  (67)

To establish (67), we note that each C_{i} is a projective finitely generated R-module, and hence C defines a class in K_{0}(R), which we denote by [C]. In fact, we claim that [C] = 0. Indeed, the natural map from K_{0}(R) to K_{0}(L[Δ]) is injective (see [27], Chap. 16, Theorem 34, Corollary 2), and [C] must be sent to zero under this map because of the fact that H_{i}(C) is finite for 0 ≤ i ≤ n. As [C] maps to zero under an injective map, we must have [C] = 0. We now define a canonical homomorphism

θ : K_{0}(R) → ℚ_{p}^{×}.  (68)

Note that, if A is any finitely generated projective R-module, then W ⊗_{R} A is clearly finite. Moreover, as A is projective, Tor_{1}^{A}(W, A) = 0, and we therefore can define θ by sending [A] to ♯(W ⊗_{R} A). In particular, as [C] = 0, we conclude that θ([C]) = 1. But

K_{i} = W ⊗_{R} C_{i} (i = 0, ⋯, n),

and so K_{i} is finite, and

θ([C]) = ∏_{i=0}^{n} ♯(K_{i})^{(−1)^{i}} = ∏_{i=0}^{n} ♯(H_{i}(K))^{(−1)^{i}}.  (69)

Since θ([C]) = 1, (67) follows from (69). This completes the proof of Theorem 3.10.

Example. We end this section by discussing a numerical example of our algebraic theory, arising from the arithmetic of elliptic curves. Let E = X_{1}(11) be the elliptic curve over ℚ of conductor 11

y^{2} + y = x^{3} − x^{2}.  (70)

Take p = 5, and let E_{5^{∞}} denote the group of all 5-power division points on E. Define

F_{∞} = ℚ(E_{5^{∞}}).  (71)

By the Weil pairing, F_{∞} contains ℚ(μ_{5^{∞}}), where μ_{5^{∞}} denotes the group of all 5-power roots of unity. In particular, F_{∞} contains the cyclotomic ℤ_{5}-extension of ℚ, which we denote by ℚ^{cyc}. Define the Galois groups

G = G(F_{∞}/ℚ), H = G(F_{∞}/ℚ^{cyc}), Γ = G(ℚ^{cyc}/ℚ),  (72)

providing an example of compact 5-adic Lie groups to which our general theory applies. We remark in passing that not only is G open in GL_{2}(ℤ_{5}), but in fact it is well known ([13], Prop. 1.5) to be isomorphic to the subgroup of GL_{2}(ℤ_{5}) consisting of all elements

(\begin{matrix}a & b \\ c & d \end{matrix}) with

(\begin{matrix}a & b \\ c & d \end{matrix}) ≡ (\begin{matrix}1 & 0 \\ 0 & ∗ \end{matrix}) mod 5.

We shall be interested in the following two irreducible Artin representations ρ_{i} (i = 1, 2) of G. Let E_{2} denote the unique elliptic curve over ℚ of conductor 11 with E_{2}(ℚ) = 0 (thus E_{2} is the curve A_{2} of [13]). If we write E_{5} and E_{2,5} for the groups of 5-division points on E, E_{2}, respectively, and define

K_{1} = ℚ(E_{5}), K_{2} = ℚ(E_{2,5}),  (73)

then K_{1} and K_{2} are both cyclic extensions of degree 5 of the field ℚ(μ_{5}), and they are both contained in F_{∞}. Let χ_{i} (i = 1, 2) denote any non-trivial character of the Galois group of K_{i} over ℚ(μ_{5}). Then we define ρ_{i} to be the character of the Galois group of K_{i} over ℚ which is induced by χ_{i}, and we can then view ρ_{i} as an Artin representation of G. It is easily seen that ρ_{i} is irreducible of degree 4, and, in fact, ρ_{i} can be realized as a 4-dimensional representation even over ℚ. Also ρ̂_{i} = ρ_{i} for i = 1, 2.

We write X(E/F_{∞}) for the compact Pontrjagin dual of the Selmer group of E over F_{∞} (see §4 for a fuller discussion of Selmer groups). Then X(E/F_{∞}) is a finitely generated Λ(G)-module, and it is proven in [5] that X(E/F_{∞}) is finitely generated over Λ(H). Hence X(E/F_{∞}) belongs to our category 𝔐_{H}(G). We write ξ_{E,5} for any choice of a characteristic element of X(E/F_{∞}).

Proposition 3.11. Put M = X(E/F_{∞}). For i = 1, 2, χ(G, tw_{ρ_{i}}(M)) is finite, and, in fact,

χ(G, tw_{ρ_{1}}(M)) = 5^{3}, χ(G, tw_{ρ_{2}}(M)) = 5.  (74)

In view of Proposition 3.11, we deduce from Theorem 3.6 that ξ_{E,5}(ρ_{i}) ≠ 0, ∞ (i = 1, 2) and

|ξ_{E,5}(ρ_{1})|_{5}^{−1} = 5^{3}, |ξ_{E,5}(ρ_{2})|_{5}^{−1} = 5.  (75)

We now prove Proposition 3.11, by combining Theorem 3.10 with the following explicit Euler characteristic formula for X(E/F_{∞}) (which is Theorem 1.1 of [5] applied to this example for the prime p = 5). Put F = ℚ(μ_{5}), and let K be any finite extension of F, which is contained in F_{∞}. Write

G_{K} = G(F_{∞}/K).

For each finite place v of K, we write d_{v} for the degree of K completed at v over the completion of ℚ at v, k_{v} for the residue field, and Ẽ_{v} for the reduction of E modulo v. Let � (E/K) be the Tate-Shafarevich group of E over K. Assume now that

E(K) and � (E/K)(5) are both finite.  (76)

Since X(E/F_{∞}) is finitely generated over Λ(H), it follows that the hypotheses of Theorem 1.1 of [5] hold for E over K and the prime p = 5. Hence Theorem 1.1 of [5] shows that, putting X = X(E/F_{∞}), we have χ(G_{K}, X) is finite, and is given by the arithmetic formula

χ(G_{K}, X) = \frac{♯(� (E/K)(5))}{♯(E(K)(5))^{2}} × ∏_{v|11}(5|d_{v}|_{5}^{−1}) × ∏_{v|5} ♯(Ẽ_{v}(k_{v})(5))^{2} ;  (77)

here the first product on the right of (77) is taken over the primes v of K dividing 11, and the second over primes v of K dividing 5. When K = F, it is well known (see [9]) that

E(F)(5) = ℤ/5ℤ, � (E/F)(5) = 0.

Since there are four primes of F above 11 with d_{v} = 1, and since there is a unique prime above 5 with residue field 𝔽_{5}, we conclude from (77) that

χ(G_{F} , X) = 5^{4}.  (78)

Taking next K = K_{1}, it is proven in [13] that

E(K_{1}) = (ℤ/5ℤ)^{2}, � (E/K_{1})(5) = (ℤ/5ℤ)^{2},

and that there are four primes of K_{1} above 11 with d_{v} = 5, and five primes of K_{1} above 5 with residue field 𝔽_{5}. Hence we conclude from (77) that

χ(G_{K_{1}}, X) = 5^{16}.  (79)

Applying Theorem 3.10 to both of the open subgroups G′ = G_{F} and G′ = G_{K_{1}}, we deduce that χ(G, tw_{ρ_{1}}(X)) is finite, and

χ(G, tw_{ρ_{1}}(X))^{4} = 5^{12},

which proves the first assertion of Proposition 3.11. Now take K = K_{2}. It is proven in [13] that

E(K_{2}) = ℤ/5ℤ, � (E/K_{2})(5) = 0,

and that there are four primes of K_{2} above 11 with d_{v} = 5, and one prime of K_{2} above 5 with residue field 𝔽_{5}. Hence we conclude from (77) that

χ(G_{K_{2}}, X) = 5^{8}.  (80)

Applying Theorem 3.10 to both of the open subgroups G′ = G_{F} and G′ = G_{K_{2}}, we deduce that χ(G, tw_{ρ_{2}}(X)) is finite, and

χ(G, tw_{ρ_{2}}(X))^{4} = 5^{4},

which proves the second assertion of Proposition 3.11.

4 Additional properties of characteristic elements.

We now establish some additional properties of the characteristic elements of modules in the category 𝔐_{H}(G), and of the group K_{1}(Λ(G)_{S^{∗}}) in which they lie. We end this section with some conjectures about the integrality properties of characteristic elements.

We begin with some general remarks. Let R be a ring with unit element, which is left and right Noetherian. We recall that the Jacobson radical of R, which we denote by Jac(R), is the intersection of all maximal left ideals of R, or equivalently the intersection of all maximal right ideals of R. We say that R is semi-local if the ring R/Jac(R) is both left and right Artinian. The following lemma is well known (see [1], Chap.III, Prop. 2.12, and Chap.IX, Prop. 1.3).

Lemma 4.1. Let I be a two sided ideal of a ring R, and assume that R is I-adically complete. If I is contained in Jac(R), then the natural map from K_{0}(R) to K_{0}(R/I) is an isomorphism.

If G is an arbitrary compact p-adic Lie group, it is well known (see [20], Chap.V, Prop. 5.2.16) that Λ(G) is a semi-local ring. We assume for the rest of this section that again G has a closed normal subgroup H such that G/H is isomorphic to ℤ_{p}, and we let S be the Ore set in Λ(G) defined in §2.

Proposition 4.2. The ring Λ(G)_{S} is semi-local.

We first establish the following lemma.

Lemma 4.3. Let J be any pro-p open subgroup of H which is normal in G, and let ψ_{J} be the homomorphism (3). Define S_{J} = ψ_{J}(S). Then S_{J} is an Ore set of non-zero divisors in Ω(G/J), and the ring Ω(G/J)_{S_{J}} is Artinian.

Proof. Lemmas 2.1 and 2.2 show that S_{J} consists of all non-zero divisors in Ω(G/J). Moreover, as S is an Ore set and ψ_{J} is surjective, it follows that S_{J} satisfies the Ore condition. Thus Ω(G/J)_{S_{J}} is the total ring of quotients of Ω(G/J). Now choose a subgroup Π of G/J satisfying (2), and let Ω(Π) = 𝔽_{p}[[T]] and its quotient field R(Π) be as in §2. Define Σ to be the set of non-zero elements of Ω(Π). If α is any element of Σ, it is clear that multiplication by α on the right or the left induces an automorphism of the R(Π)-vector space V (G/J) defined by (6). Hence, by the proof of Lemmas 2.1 and 2.2, α belongs to S_{J}; in particular, α is not a divisor of zero in Ω(G/J). As Ω(Π) is in the centre of Ω(G/J), it follows that Σ is an Ore set in Ω(G/J). We claim that

Ω(G/J)_{S_{J}} = Ω(G/J)_{Σ}.  (81)

Note that (81) proves the assertion of Lemma 4.3, because the fact that Ω(G/J) is a finitely generated Ω(Π)-module implies that Ω(G/J)_{Σ} is a finite dimensional vector space over R(Π), and so Ω(G/J)_{Σ} is Artinian. As Σ ⊂ S_{J}, to prove (81) we must show that every element of S_{J} is invertible in Ω(G/J)_{Σ}. Take any s in S_{J}. As Ω(Π) lies in the centre of Ω(G/J), which is finitely generated over Ω(Π), s satisfies an equation (see [19], Lemma 5.3.2) of the form

s^{n} + a_{1}s^{n−1} + ⋯ + a_{n} = 0 (a_{i} ∈ Ω(Π)).  (82)

Moreover, since s is not a zero divisor in Ω(G/J), we can assume that a_{n} ≠ 0. Now this equation can be rewritten as

(b_{0}s^{n−1} + b_{1}s^{n−1} + ⋯ + b_{n−1})s = 1,

where b_{0} = −1/a_{n}, and b_{i} = −a_{i}/a_{n} (i = 1, ⋯, n − 1), proving that s has an inverse in Ω(G/J)_{Σ}, as required. This completes the proof of Lemma 4.3.

We now can prove Proposition 4.2. If M is any left Λ(G)-module, we define

M_{S} = Λ(G)_{S} ⊗_{Λ(G)} M.

Let J be any subgroup of G as in Lemma 4.3, and let I denote the kernel of ψ_{J}. Since Λ(G)_{S} is flat over Λ(G), we have the exact sequence

0 → I_{S} → Λ(G)_{S} → Ω(G/J)_{S} → 0.

But

Ω(G/J)_{S} = Ω(G/J)_{S_{J}} .

Hence, by lemma 4.3, it suffices to show that I_{S} is contained in the Jacobson radical of Λ(G)_{S}. In view of [19], Theorem 0.3.8, we must prove that 1 − x is a unit in Λ(G)_{S} for every x in I_{S}. Write x = s^{−1}θ with θ in I and s in S. Since θ is in I, ψ_{J}(s − θ) = ψ_{J}(s). Hence, by Lemma 2.1, s − θ belongs to S, But then

1 − x = s^{−1}(s − θ)

is clearly a unit in Λ(G)_{S}, and the proof of Proposition 4.2 is complete.

We remark that Λ(G)_{S^{∗}} is not, in general, a semi-local ring, as is shown by the following example. Take G = H × K, where both H and K are isomorphic to ℤ_{p}. Then we can identify Λ(G) with R = ℤ_{p}[[U, V]] by mapping fixed topological generators of H and K to U + 1 and V + 1, respectively. Then the set S of §2 in this case is the complement of the prime ideal (p, U) in R. Thus R_{S} is a local ring of dimension 2, whose maximal ideal is generated by p and U. Hence R_{S^{∗}} = R_{S}[1/p] is a ring of dimension 1. But R_{S^{∗}} has infinitely many prime ideals, and is certainly not semi-local. Indeed, if g is any irreducible distinguished polynomial in ℤ_{p}[U], then gR is a prime ideal whence gR_{S^{∗}} is also prime because S^{∗} ∩ gR is empty.

Theorem 4.4. Assume that G has no element of order p. Then the natural maps Λ(G)_{S}^{×} → K_{1}(Λ(G)_{S}) and Λ(G)_{S^{∗}}^{×} → K_{1}(Λ(G)_{S^{∗}}) are both surjective.

If R is any semi-local ring, it is well known that the natural map R^{×} → K_{1}R is surjective (see [29], [30]). In view of Proposition 4.2, this establishes the first assertion of Theorem 4.4, and we now give the proof of the second assertion. We establish two preliminary lemmas. Note that, if P is a direct summand of Λ(G) viewed as a left Λ(G)- module, then P is a finitely generated projective Λ(G)-module, and P/pP is a finitely generated projective Ω(G)-module.

Lemma 4.5. The group K_{0}(Ω(G)) is generated by the classes [P/pP], where P runs over all direct summands of Λ(G) viewed as a left Λ(G)-module.

Proof. Let I = pΛ(G). Then I is contained in the Jacobson radical of Λ(G) since, for each λ in I, 1−λ is invertible with inverse ∑_{n=0}^{∞} λ^{n}. Moreover, Λ(G) is I-adically complete, and so, by Lemma 4.1, the canonical map from K_{0}(Λ(G)) to K_{0}(Ω(G)) is an isomorphism. In particular, K_{0}(Ω(G)) is generated by the classes [P/pP], where P ranges over all finitely generated projective Λ(G)-modules. To complete the proof of lemma, we must show that we still get a set of generators if we allow P to only run over the direct summands of Λ(G). Pick a pro-p open normal subgroup U of G, and put Δ = G/U. Let D denote the kernel of the natural map from Λ(G) to 𝔽_{p}[Δ], so that D is generated as a Λ(G)-module by the set W_{U} consisting of p and 1 − u, where u ranges over U. Now Λ(U) is a local ring, whose maximal ideal Jac(Λ(U)) is clearly generated by W_{U}. But Λ(G) is finitely generated over Λ(U), and so it is well known that ([19], Prop. 9.1.3)

Jac(Λ(G)) ∩ Λ(U) ⊆ Jac(Λ(U)).

This proves that D is contained in Jac(Λ(G)), and, as Λ(G) is plainly D-adically complete, we conclude from Lemma 4.1 that the natural map from K_{0}(Λ(G)) to K_{0}(𝔽_{p}[Δ]) is an isomorphism. Put

R = 𝔽_{p}[Δ]/Jac(𝔽_{p}[Δ]).

Since 𝔽_{p}[Δ] is Artinian, Jac(𝔽_{p}[Δ]) is nilpotent, and so Lemma 4.1 proves that the canonical map from K_{0}(𝔽_{p}[Δ]) to K_{0}(R) is an isomorphism. However, R is a semisimple ring, and thus K_{0}(R) is generated by the simple projective R-modules, and these are given precisely by the direct summands of R viewed as a left R-module. It is then well known (see [1], Chap.III, Prop. 2.12) that one can lift these generators of K_{0}(R) successively back to K_{0}(𝔽_{p}[Δ]) and to K_{0}(Λ(G)), so that they remain direct summands of 𝔽_{p}[Δ] and Λ(G), respectively. This completes the proof of Lemma 4.5.

Lemma 4.6. Put R = Λ(G)[1/p], and assume that G has no element of order p. Then the natural map R^{×} → K_{1}R is surjective.

Proof. Let C denote the category of all finitely generated Λ(G)-modules, which are annihilated by some power of p. Then, by dévissage, it is well known (see, for example, [19], Theorem 12.4.7) that we can naturally identify K_{0}(C) with K_{0}(D), where D is the category of all finitely generated Ω(G)-modules. As G has no element of order p, Ω(G) has finite global dimension, an so we can identify K_{0}(D) with K_{0}(Ω(G)). Thus the localization sequence of K-theory for the multiplicative Ore set consisting of the powers of p lying in ℤ gives the commutative diagram

\begin{matrix}Λ(G)^{×} & ⟶^{↪} & R^{×} & ⟶^{g} & K_{0}(Ω(G)) \\ ↓ & & ^{f}↓ & & ∥ \\ K_{1}(Λ(G)) & ⟶ & K_{1}(R) & ⟶^{∂} & K_{0}(Ω(G)) \end{matrix}  (83)

where R = Λ(G)[1/p], f is the natural map, and the bottom row is exact. Also ∂ is the connecting homomorphism, and g = ∂ ∘ f. Since Λ(G) is semi-local, the left vertical map is surjective (see [29], [30]). We conclude from (83) that f will be surjective provided we can prove g is surjective. We now proceed to show this.

By Lemma 4.5, K_{0}(Ω(G)) is generated by the classes [P/pP], where P runs over the direct summands of Λ(G), viewed as a left Λ(G)-module. Hence it suffices to show that each such class belongs to the image of g. Take such a P, and let P′ be the Λ(G)-module such that

P ⊕ P′ = Λ(G).  (84)

In terms of the decomposition (84), we can define endomorphisms of Λ(G), viewed as a left Λ(G)-module, by

α(x + y) = px + y, β(x + y) = x + py (x ∈ P, y ∈ P′).

Thus α and β are given, respectively, by multiplication on the right by a = α(1), b = β(1). Since ab = p, it follows that a and b are both units in R. On the other hand, the explicit description of ∂ in the localization sequence (see [28]) shows that

∂(f(a)) = [Coker α] = [P/pP].

This completes the proof of Lemma 4.6.

We can now prove the second assertion of Theorem 4.4. As before, we let R = Λ(G)[1/p], Ω(G) = Λ(G)/pΛ(G), and put

R′ = Λ(G)_{S}[\frac{1}{p}] = Λ(G)_{S^{∗}}, Ω′(G) = Λ(G)_{S}/pΛ(G)_{S}.

As in the proof of Lemma 4.6, we can identify the Grothendieck group of the category of all finitely generated Λ(G)_{S}-modules, which are annihilated by some power of p, with K_{0}(Ω′(G)). Thus, parallel to (83), we have the exact sequence of localization

K_{1}(Λ(G)_{S}) → K_{1}(R′) →^{∂′} K_{0}(Ω′(G)).  (85)

Let f′ denote the natural map from (R′)^{×} to K_{1}(R′), and put g′ = ∂′ ∘ f′. As Λ(G)_{S} is semi-local, the natural map from Λ(G)_{S}^{×} to K_{1}(Λ(G)_{S}) is surjective. In view of this last remark, we conclude from the exact sequence (85) that f′ is surjective provided g′ is surjective. To establish the surjectivity of g′, we note that we have a commutative diagram

R^{×} ⟶^{g} K_{0}(Ω(G))
↓  ↓^{h}
(R′)^{×} ⟶^{g′} K_{0}(Ω′(G)),

where the vertical maps are the natural ones. We claim that both g and h are surjective, which will clearly imply that g′ is surjective. But the surjectivity of g was proven in the course of the proof of Lemma 4.6. Let S′ denote the image of S in Ω(G). By (iii) of Lemmas 2.1 and 2.2, S′ consists of non-zero divisors in Ω(G), and it is plain that S′ is an Ore set in Ω(G). As localization with respect to S is an exact functor, we see easily that

Ω′(G) = Ω(G)_{S′}.

But then it is well known ([19], Theorem 12.4.9) that the natural map h from K_{0}(Ω(G)) to K_{0}(Ω(G)_{S′}) is surjective. This completes the proof of Theorem 4.4.

We next show that any module M in the category 𝔐_{H}(G) has many twists ρ by continuous representations of G of the form (12) with finite G-Euler characteristics. Indeed, consider continuous representations

η : Γ → O^{×}.  (86)

Since Γ = G/H, we can always view such η as a continuous representation of G.

Lemma 4.7. Assume that G has no element of order p, and let M ∈ 𝔐_{H}(G). Then, for all but finitely many Artin characters η of Γ, χ(G, tw_{η}(M)) is finite. Moreover, if η is a fixed character of Γ of infinite order, χ(G, tw_{η^{n}}(M)) is finite for all but finitely many n ∈ ℤ.

Proof. We pick a topological generator γ_{0} of Γ, and identify Λ_{O}(Γ) with O[[T]] by mapping γ_{0} to 1 + T. As earlier, let M_{O} = M ⊗_{ℤ_{p}} O. Since M belongs to 𝔐_{H}(G), H_{i}(H, M_{O}) is a torsion Λ_{O}(Γ)-module for all i ≥ 0, and we write f_{i,M} for a characteristic series of H_{i}(H, M_{O}). Since η(H) = 1, we have

H_{i}(H, tw_{η}(M)) = H_{i}(H, M_{O}) ⊗_{O} A_{η},

where A_{η} denotes a free O -module of rank 1 on which Γ acts via η, i.e. γ.a = η(γ)a for γ in Γ and a in A_{η}. It follows easily that

f_{i,tw_{η}(M)}(T) = f_{i,M}(η(γ_{0})^{−1}(1 + T) − 1).  (87)

Now a standard argument with the Hochschild-Serre spectral sequence (see, for example, [6]) shows that χ(G, tw_{η}(M)) is finite provided f_{i,tw_{η}(M)}(0) ≠ 0, or equivalently

f_{i,M}(η(γ_{0})^{−1} − 1) ≠ 0 (0 ≤ i ≤ d − 1) ;  (88)

here d denotes the dimension of G as a p-adic Lie group. Let ℨ denote the set of all zeros of the f_{i,M}(T) 0 ≤ i ≤ d − 1) lying in the maximal ideal of the ring of integers of ℚ̄_{p}. By the Weierstrass Preparation Theorem, ℨ is a finite set. Hence, as η runs over all the Artin characters of Γ, only finitely many of the η(γ_{0})^{−1} − 1 can lie in ℨ, and the first assertion of the lemma is proven. Now suppose that η has infinite order, so that the characters η^{n} (n ∈ ℤ) are all distinct. Thus, for fixed η of infinite order, only finitely many of the η(γ_{0})^{−n} −1, as n runs over ℤ, can lie in ℨ, and the second assertion of Lemma 4.7 follows.

The following two examples illustrate what differences occur between the commutative and the non-commutative theory, especially when one considers modules which lie outside the category 𝔐_{H}(G) in the non-commutative case.

Example. Suppose now that G is commutative and has no element of order p, and let M be any finitely generated torsion Λ(G)-module. A slight generalization of a lemma of Greenberg [15] then shows that we can always find a closed subgroup H of G such that Γ = G/H is isomorphic to ℤ_{p}, and M belongs to the category 𝔐_{H}(G). Thus, in this case, one can always find, by Lemma 4.7, continuous representations ρ of G such that χ(G, tw_{ρ}(M)) is finite. The following example shows that this last assertion is false, in general, when G is not commutative. Take G = GL_{3}(ℤ_{p}), and assume that p ≥ 5 to ensure that G has no element of order p. Let

u = (\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , v = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) , f = u − v.

Consider the module

M = Λ(G)/Λ(G)f.

We claim that, for every continuous representation ρ of G of the form (12), the Euler characteristic χ(G, tw_{ρ}(M)) is never finite. Indeed, a free resolution of tw_{ρ}(M) by Λ_{O}(G)- modules is given by

0 → tw_{ρ}(Λ(G)) → tw_{ρ}(Λ(G)) → tw_{ρ}(M) → 0,

since, as was remarked earlier (see [33]), tw_{ρ}(Λ(G)) is a free Λ_{O}(G)-module of rank n. Taking G-homology of this complex, it follows that H_{i}(G, tw_{ρ}(M)) = 0 for i ≥ 2, and that

H_{1}(G, tw_{ρ}(M)) = Ker (ρ(f)), H_{0}(G, tw_{ρ}(M)) = Coker (ρ(f)).

But we now prove that 0 is an eigenvalue of ρ(f) for every continuous ρ. We first note that for a ∈ ℤ_{p}, we have a-th powers u^{a} and v^{a} defined as

u^{a} = (\begin{matrix}1 & 0 & a \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , v^{a} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix}) .

Taking a in ℤ_{p}^{×}, we define

σ_{a} = (\begin{matrix}a & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , τ_{a} = (\begin{matrix}1 & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & 1 \end{matrix}) ,

and one verifies immediately that

σ_{a}uσ_{a}^{−1} = u^{a}, τ_{a}vτ_{a}^{−1} = v^{a}.

Hence the eigenvalues of ρ(u) and ρ(v) in ℚ̄_{p} must be stable under exponentiation by any a in ℤ_{p}^{×}. Since there are only finitely many of these eigenvalues, they must therefore all be p-power roots of unity. Moreover, ρ(u) and ρ(v) commute because uv = vu. Hence ρ(u) and ρ(v) must have a common eigenvector z, with respective eigenvalues α and β. Since α and β are both p-power roots of unity, we have either

β = α^{w} or α = β^{w}  (89)

for some element w in ℤ. Define

g = (\begin{matrix}1 & w − 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , h = (\begin{matrix}1 & 0 & 0 \\ w − 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

One verifies that

g^{−1}ug = u, g^{−1}vg = u^{1−w}v, h^{−1}uh = uv^{1−w}, h^{−1}vh = v.  (90)

If the first option of (89) holds, one uses the first two equations of (90) to verify that ρ(g)z is an eigenvector of both ρ(u) and ρ(v) with the same eigenvalue α. Similarly, if the second option of (89) is true, one uses the second two equations of (90) to conclude that ρ(h)z is an eigenvector for both ρ(u) and ρ(v) with the same eigenvalue. In either case, we see that 0 is an eigenvalue of ρ(f), completing the proof that χ(G, tw_{ρ}(M)) is never finite.

Example. We give a second example to illustrate the differences between the commutative and non-commutative theory. Assume that G is p-valued in the sense of Lazard [17] (for example, provided p > n + 1, every pro-p closed subgroup of GL_{n}(ℤ_{p}) is p-valued). Then it follows from [17] that both Λ(G) and Ω(G) = Λ(G)/pΛ(G) have no zero divisors. We consider any Λ(G)-module of the form M = Λ(G)/Λ(G)w, where w is an element of Λ(G), which is not a unit, and which does not belong to pΛ(G). Since Ω(G) has no zero divisors, we see that M has no p-torsion. Moreover, it is easily seen that M is not pseudo-null as a Λ(G)-module in the sense of [33]. Further, in a similar manner to that of the previous example, we see that for every continuous representation ρ of G, we have H_{i}(G, tw_{ρ}(M)) = 0 for i ≥ 2, and

H_{1}(G, tw_{ρ}(M)) = Ker (ρ(w)), H_{0}(G, tw_{ρ}(M)) = Coker (ρ(w)).

Suppose first that G is commutative. Thus G is isomorphic to ℤ_{p}^{d} for some integer d ≥ 1, and Λ(G) is isomorphic to the ring ℤ_{p}[[T_{1}, ⋯, T_{d}]] of formal power series in d variables with coefficients in ℤ_{p}. Identifying w with a formal power series w(T_{1}, ⋯, T_{d}), and recalling that w is not a unit, and is not divisible by p, a well known argument with the Weierstrass preparation theorem shows that there always exist α_{1}, ⋯, α_{d} in the maximal ideal of the ring of integers of ℚ̄_{p} such that w(α_{1}, ⋯, α_{d}) = 0. Let O be the ring of integers of any finite extension of ℚ_{p} containing α_{1}, ⋯, α_{d}. We can define ρ : G → O^{×} by specifying ρ(γ_{i}) = α_{i} + 1 (1 ≤ i ≤ d), where γ_{1}, ⋯, γ_{d} denote the ℤ_{p}-basis of G with γ_{i} = T_{i} + 1 (1 ≤ i ≤ d). Clearly ρ(w) = 0, and thus tw_{ρ}(M) does not have finite G-Euler characteristic. On the other hand, take G to be kernel of the reduction map from GL_{3}(ℤ_{p}) to GL_{3}(𝔽_{p}), and assume that p ≥ 5 to ensure that G is p-valued. Define

x = (\begin{matrix}1 & 0 & p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , w = x − 1 − p.

An entirely analogous argument to that given for u and v in the previous example shows that, for every continuous representation ρ of G, the eigenvalues of ρ(x) in ℚ̄_{p} are all ppower roots of unity. Thus the eigenvalues of ρ(w) in ℚ̄_{p} are never zero, and so it follows that χ(G, tw_{ρ}(M)) is finite for every continuous representation ρ of G.

We end this section by making some conjectures about the integrality properties of characteristic elements of modules in the category 𝔐_{H}(G). For brevity, let us write

R_{1} = Λ(G), R_{2} = Λ(G)[\frac{1}{p}], R_{3} = Λ(G)_{S^{∗}}.  (91)

Since S^{∗} consists of non-zero divisors in Λ(G), we have natural inclusions R_{1} ⊂ R_{2} ⊂ R_{3}. We also write

A_{1} = Λ_{O}(Γ), A_{2} = Λ_{O}(Γ)[\frac{1}{p}].  (92)

It is also convenient to give a name α to the canonical map

α : R_{3}^{×} → K_{1}(R_{3}),  (93)

which we know is surjective by Theorem 4.4. Finally, we write ℜ(G) for the set of all continuous representations ρ of G of the form (11), where O is allowed to run over the rings of integers of all finite extensions of ℚ_{p}, and 𝔄(G) for the subset of ℜ(G) consisting of the Artin representations. In the next conjecture, the statements for all ρ in ℜ(G), and for all ρ in 𝔄(G) will just be abbreviated to the respective assertions for all ρ, and for all Artin ρ.

Conjecture 4.8. Assume that G has no element of order p. In each of the following four cases, the assertions (a), (b), (c) and (d) are equivalent for any element ξ of K_{1}(R_{3}):-

Case 1. (a) ξ ∈ α(R_{2} ∩ R_{3}^{×}); (b) ξ(ρ) ≠ ∞ for all ρ; (c) Φ′_{ρ}(ξ) ∈ A_{2} for all ρ; (d) Φ′_{ρ}(ξ) ∈ A_{2} for all Artin ρ.

Case 2. (a) ξ ∈ α(R_{1} ∩ R_{3}^{×}); (b) ξ(ρ) is finite and in O for all ρ; (c) Φ′_{ρ}(ξ) ∈ A_{1} for all ρ; (d) Φ′_{ρ}(ξ) ∈ A_{1} for all Artin ρ.

Case 3. (a) ξ ∈ α(R_{2}^{×}); (b) ξ(ρ) ≠ 0, ∞ for all ρ; (c) Φ′_{ρ}(ξ) ∈ A_{2}^{×} for all ρ; (d) Φ′_{ρ}(ξ) ∈ A_{2}^{×} for all Artin ρ.

Case 4. (a) ξ ∈ α(R_{1}^{×}); (b) ξ(ρ) is finite and in O^{×} for all ρ; (c) Φ′_{ρ}(ξ) ∈ A_{1}^{×} for all ρ; (d) Φ′_{ρ}(ξ) ∈ A_{1}^{×} for all Artin ρ.

We remark that Case 1 of Conjecture 4.8 has an interesting consequence for the modules M in 𝔐_{H}(G) which, in addition, satisfy condition (51). Let M be any such module, and let ξ_{M} be a characteristic element of M. It follows from (52) and (57) that Φ′_{ρ}(ξ_{M}) belongs to A_{2} for all ρ in 𝔄(G). Hence, if Conjecture 4.8 is valid, the assertions (a), (b), and (c) of Case 1 would be true for ξ_{M}, and, in particular, we would have ξ_{M} ∈ α(R_{2}∩R_{3}^{×}). But we have to confess that at present we are unable to prove (a), (b), and (c) of Case 1 for the characteristic elements of such modules M.

Lemma 4.9. In each of the four cases of Conjecture 4.8, (a) implies (b), (b) and (c) are equivalent, and (c) implies (d).

Proof. Plainly (c) implies (d), and it is also clear that (c) implies (b) since, by definition, ξ(ρ) is the image of Φ′_{ρ}(ξ) under the augmentation map once Φ′_{ρ}(ξ) belongs to the localization of A_{1} at the kernel of the augmentation map. The implication (a) implies (b) holds because if g is any element of R_{2}^{×} of the form g = p^{−m}f with f in R_{1} and m an integer ≥ 0, then, putting ξ = α(g), ξ(ρ) = p^{−mn} det ρ(f) for every ρ in ℜ(G). To prove that (b) implies (c), we make use of the following general remark. Recall that we have identified A_{1} with O[[T]] by mapping a fixed topological generator γ_{0} of Γ to 1 + T. Let η : Γ → O^{×} be any continuous homomorphism. Then we claim that, for all ξ in K_{1}(R_{3}) and all ρ in ℜ(G), we have the identity

Φ′_{ρη}(ξ) = Φ′_{ρ}(ξ)(η(γ)^{−1}(1 + T) − 1),  (94)

where both sides of this equation are viewed as elements of the quotient field of O[[T]]. To establish (94), we note that K_{1}(R_{3}) is generated as an abelian group by those ξ which are characteristic elements of modules M in 𝔐_{H}(G), and, for such ξ, (94) follows from (87) and Lemma 3.7. We now use (94) to show that (b) implies (c) in Case 1. Suppose that, for some ρ in ℜ(G), we have Φ′_{ρ}(ξ) = f(T)/g(T), where f(T) and g(T) are relatively prime elements of O[[T]], and g(T) is not a power of the local parameter π of O. Thus g(T) must have a zero T = α in the maximal ideal of the ring of integers of ℚ̄_{p}. Enlarging O if necessary, we can assume that α belongs to O, and define a continuous homomorphism η : Γ → O^{×} by η(γ_{0}) = (1+α)^{−1}. It is then clear from (94) that ξ(ηρ) = ∞, contradicting our assumption that (b) is valid. Similar arguments prove that (b) implies (c) in the remaining three cases of Conjecture 4.8, and the proof of Lemma 4.9 is complete.

The strongest evidence we have for Conjecture 4.8 is that it is true when G is of the form ℤ_{p}^{d} × Δ, where d is any integer ≥ 1 and Δ is a finite group of order prime to p. However, we omit the detailed proof in this case.

5 The main conjecture.

For brevity, we only discuss here the main conjecture for elliptic curves over ℚ over the field generated by the coordinates of all p-power division points on the curve, where p is a prime ≥ 5 of good ordinary reduction. However, it will be clear that a similar conjecture can be formulated in great generality for motives over p-adic Lie extensions of number fields which contain the cyclotomic ℤ_{p}-extension of the base field.

If F is a finite extension of ℚ, we write F^{cyc} for the cyclotomic ℤ_{p}-extension of F, and put Γ_{F} = G(F^{cyc}/F). Let E be an elliptic curve defined over ℚ, and E_{p^{∞}} the group of all p-power division points on E. We define

F_{∞} = ℚ(E_{p^{∞}}).  (95)

By the Weil pairing, F_{∞} ⊃ ℚ(μ_{p^{∞}}), where μ_{p^{∞}} denotes the group of all p-power roots of unity. Hence F_{∞} contains ℚ^{cyc}, and we define

G = G(F_{∞}/ℚ), H = G(F_{∞}/ℚ^{cyc}), Γ = G(ℚ^{cyc}/ℚ).  (96)

Thus G is a compact p-adic Lie group to which the abstract theory developed in the first four sections of this paper applies. In fact, the structure of G is well known. Indeed, G can be identified with a closed subgroup of GL_{2}(ℤ_{p}) = Aut (E_{p^{∞}}). When E admits complex multiplication, G has dimension 2, and when E does not admit complex multiplication, G has dimension 4 [25].

If L is any algebraic extension of ℚ, we recall that the classical Selmer group S(E/L) is defined by

S(E/L) = Ker (H^{1}(L, E_{p^{∞}}) → ∏_{w}(H^{1}(L_{w}, E(L̄_{w}))),

where w runs over all non-archimedean places of L, and L_{w} denotes the union of the completions at w of all finite extensions of ℚ contained in L. We write

X(E/L) = Hom (S(E/L), ℚ_{p}/ℤ_{p})

for the compact Pontrjagin dual of the discrete abelian group S(E/L). We shall mainly be interested in the case in which L is Galois over F, in which case both S(E/L) and X(E/L) have a natural left action of G(L/F), which extends to a left action of the whole Iwasawa algebra Λ(G(L/F)). It is easy to see that X(E/L) is always a finitely generated Λ(G(L/F))-module.

We impose for the rest of the paper the following

Hypotheses on p. p ≥ 5 and E has good ordinary reduction at p.

The condition p ≥ 5 guarantees that G has no element of order p, since G is a closed subgroup of GL_{2}(ℤ_{p}). It is a basic question as to when, assuming our hypotheses on p, the dual Selmer group X(E/F_{∞}) belongs to the category 𝔐_{H}(G), or equivalently is S^{∗}-torsion, where S^{∗} is the Ore set defined in §3.

Conjecture 5.1. Assuming our hypotheses on p, X(E/F_{∞}) belongs to the category 𝔐_{H}(G).

We now briefly discuss what is known at present about Conjecture 5.1. Indeed, all we know is related to the following much older conjecture of Mazur [18].

Conjecture 5.2. Assume E has good ordinary reduction at p. For each finite extension F of ℚ, X(E/F^{cyc}) is Λ(Γ_{F})-torsion.

One might hope that Conjecture 5.1 is equivalent to knowing Conjecture 5.2 for all finite extensions F of ℚ contained in F_{∞}. At present, we can only prove some partial results in this direction, which we now describe. If F is a finite extension of ℚ contained in F_{∞}, we put

G_{F} = G(F_{∞}/F), H_{F} = G(F_{∞}/F^{cyc}).  (98)

Let X be a finitely generated Λ(G)-module. If L is a finite extension of ℚ in F_{∞} such that G_{L} is pro-p, the μ-invariant of X viewed as a Λ(G_{L})-module, is defined in [16] and [31]. Similarly, if Z is any finitely generated Λ(Γ_{F})-module, we write μ_{Γ_{F}} (Z) for its classical μ-invariant. Also, for any field K in F_{∞}, we define

Y (E/K) = X(E/K)/X(E/K)(p).  (99)

Lemma 5.3. Assume that X(E/F_{∞}) belongs to 𝔐_{H}(G). Then, for each finite extension F of ℚ contained in F_{∞}, we have (i) X(E/F^{cyc}) is Λ(Γ_{F})-torsion; (ii) H_{i}(H_{F} , X(E/F_{∞})) = 0 for all i ≥ 1; (iii) H_{i}(H_{F} , Y (E/F_{∞})) is finite for i = 1, 2, and zero for i > 2. Moreover, we have μ_{G_{L}}(X(E/F_{∞})) = μ_{Γ}(X(E/L^{cyc})) for each finite extension L of ℚ in F_{∞} such that G_{L} is pro-p.

Proof. We assume that X(E/F_{∞}) belongs to 𝔐_{H}(G), so that Y (E/F_{∞}) is a finitely generated Λ(H)-module. Take F to be any finite extension of ℚ contained in F_{∞}. Then Y (E/F_{∞})_{H_{F}} is clearly a finitely generated ℤ_{p}-module, because Y (E/F_{∞}) is then a finitely generated Λ(H_{F})-module. Now we have the commutative diagram with exact rows

X(E/F_{∞})_{H_{F}} ⟶ Y (E/F_{∞})_{H_{F}} ⟶ 0
↓^{r_{F}}  ↓^{s_{F}}
X(E/F^{cyc}) ⟶ Y (E/F^{cyc}) ⟶ 0,

where r_{F} is the dual of the restriction map t_{F} from S(E/F^{cyc}) to S(E/F_{∞})^{H_{F}}, and s_{F} is the corresponding induced map. We claim that the cokernel of r_{F} is finite. Indeed, the cokernel of r_{F} is dual to the kernel of the restriction map t_{F}, and Ker (t_{F}) is contained in H^{1}(H_{F} , E_{p^{∞}}). But this latter group is proven to be finite in [8]. Since Coker (r_{F}) is finite, it follows that Coker (s_{F}) is also finite. As was remarked above, Y (E/F_{∞})_{H_{F}} is a finitely generated ℤ_{p}-module, and so we see that Y (E/F^{cyc}) is a finitely generated ℤ_{p}-module. Thus X(E/F^{cyc}) is Λ(Γ_{F})-torsion, proving (i). To prove the remainder of Lemma 5.3, we invoke the arguments of [6], and do not repeat the detailed proofs of these here. Since X(E/F^{cyc}) is Λ(Γ_{F})-torsion, Lemma 2.5 of [6] shows that H_{1}(H_{F} , X(E/F_{∞})) = 0. As we have shown X(E/F^{cyc}) is Λ(Γ_{F})-torsion for all F, Lemma 2.1 and Remark 2.6 of [6] prove that H_{i}(H_{F} , X(E/F_{∞})) = 0 for all i ≥ 2, establishing (ii). Turning to (iii), an entirely similar argument to that given in the proof of Proposition 2.13 of [6] gives that H_{3}(H_{F} , Y (E/F_{∞})) = 0, and that H_{i}(H_{F} , Y (E/F_{∞})) (i = 1, 2) are killed by a power of p. Hence, as the H_{i}(H_{F} , Y (E/F_{∞})) (i ≥ 0) are finitely generated ℤ_{p}-modules, we conclude that H_{i}(H_{F} , Y (E/F_{∞})) must be finite for i = 1, 2. The final assertion of Lemma 5.3 now follows from formula (25) of Proposition 2.13 of [6], on noting that the μ-invariant of the Λ(Γ_{L})-module H_{0}(H_{L}, Y (E/F_{∞})) is zero, because this module is finitely generated over ℤ_{p}. This completes the proof of Lemma 5.3.

Lemma 5.4. Assume that X(E/F^{cyc}) is Λ(Γ_{F})-torsion for all finite extensions F of ℚ in F_{∞}. Suppose, in addition, that there exists a finite extension L of ℚ contained in F_{∞} satisfying: − (i) G_{L} is pro-p, (ii) μ_{G_{L}}(X(E/F_{∞})) = μ_{Γ_{L}}(X(E/L^{cyc})), and (iii) H_{1}(H_{L}, Y (E/F_{∞})) is finite. Then X(E/F_{∞}) belongs to 𝔐_{H}(G).

Proof. We first observe that H_{0}(H_{L}, Y (E/F_{∞})) is a finitely generated torsion Λ(Γ_{L})- module, assuming the hypotheses of the lemma. Indeed, H_{0}(H_{L}, Y (E/F_{∞})) is a quotient of H_{0}(H_{L}, X(E/F_{∞})), and we claim that this latter module is a finitely generated torsion Λ(Γ_{L})-module. This is because we are assuming that the finitely Λ(Γ_{L})-module X(E/L^{cyc}) is Λ(Γ_{L})-torsion, and it is shown in [5] that the kernel of the natural map

r_{L} : X(E/F_{∞})_{H_{L}} → X(E/L^{cyc}),

which is the dual of the restriction map, is a finitely generated ℤ_{p}-modules. Hence Ker (r_{L}) is a finitely generated torsion Λ(Γ_{L})-module, and our claim follows. The delicate part of the proof is to now show that

μ_{Γ_{L}}(H_{0}(H_{L}, Y (E/F_{∞}))) = 0.  (100)

Indeed, (100) implies immediately that Y (E/F_{∞}) is a finitely generated Λ(H)-module, completing the proof of the lemma. This is because (100) shows that H_{0}(H_{L}, Y (E/F_{∞})) is a finitely generated ℤ_{p}-module, and hence, as H_{L} is pro-p, Nakayama’s lemma gives that Y (E/F_{∞}) is finitely generated over Λ(H_{L}). To prove (100), we invoke the full force of Proposition 2.13 of [6]. As remarked earlier, our assumption that X(E/F^{cyc}) is Λ(Γ_{F})- torsion for all finite extensions F of ℚ in F_{∞} implies that all the hypotheses of Proposition 2.13 of [6] are valid. Hence formula (25) of Proposition 2.13 of [6] holds. Inserting conditions (ii) and (iii) in formula (25) of [6], we conclude that the μ_{Γ_{L}}-invariants of H_{0}(H_{L}, Y (E/F_{∞})) and H_{2}(H_{L}, Y (E/F_{∞})) must add up to zero. Thus, as both are nonnegative integers, both of these μ-invariants must be zero, proving (100) in particular.

Corollary 5.5. Assume that p ≥ 5, E has good ordinary reduction at p, and E admits complex multiplication. Then X(E/F_{∞}) belongs to 𝔐_{H}(G) if and only if there exists a finite extension L of ℚ in F_{∞} such that G_{L} is pro-p and μ_{Γ_{L}}(X(E/L^{cyc})) = 0.

Proof. It is a well known consequence of the proof of the two variable main conjecture for E over F_{∞} (see [22]) that X(E/F^{cyc}) is Λ(Γ_{F})-torsion for every finite extension F of ℚ in F_{∞}. Take L to be any finite extension of ℚ contained in F_{∞} such that G_{L} is pro-p. Then it is well known that the thesis of Schneps [23] implies that μ_{G_{L}}(X(E/F_{∞})) = 0. This in turn implies that X(E/F_{∞})(p) = 0 because X(E/F_{∞}) has no non-zero Λ(G_{L})- pseudo-null submodule by [21]. We then have H_{i}(H_{L}, Y (E/F_{∞})) = 0 for all i ≥ 1 because Y (E/F_{∞}) = X(E/F_{∞}). The assertion of the corollary is now clear from Lemmas 5.3 and 5.4.

The following is the one practical criterion we know at present for proving in some concrete examples that X(E/F_{∞}) belongs to 𝔐_{H}(G).

Proposition 5.6. Assume that there exists a finite extension L of ℚ contained in F_{∞}, and an elliptic curve E′ defined over L, as follows: − (i) G_{L} is pro-p, (ii) E′ is isogenous to E over L, and (iii) X(E′/L^{cyc}) is finitely generated ℤ_{p}-module. Then X(E/F_{∞}) belongs to 𝔐_{H}(G).

It is very probable that the hypotheses of Proposition 5.6 are valid for most elliptic curves E over ℚ and most primes p of good ordinary reduction. However, in our present state of knowledge, condition (iii) is difficult to verify in numerical examples. Here is an example of one isogeny class of curves to which it can be applied.

Example. Consider the three elliptic curves over ℚ of conductor 11, which we denote by E_{i} (i = 0, 1, 2). Here E_{1} is given by (70), and the other two are given by the equations

E_{0} : y^{2} + y = x^{3} − x^{2} − 10x − 20
E_{2} : y^{2} + y = x^{3} − x^{2} − 7820x − 263580

All three curves are isogenous over ℚ, and so the field F_{∞} is the same for the three curves and all primes p. Taking p = 5, and L = ℚ(μ_{5}), it is well known [22] that G_{L} is pro-5 and X(E_{1}/L^{cyc}) = 0. Hence Proposition 5.6 shows that X(E_{i}/F_{∞}) belongs to 𝔐_{H}(G) for p = 5 and i = 0, 1, 2. At present, we do not know how to prove that X(E_{i}/F_{∞}) belongs to 𝔐_{H}(G) for any good ordinary prime p > 5.

We now prove Proposition 5.6. Pick an isogeny f : E′ → E, which is defined over L. Note that F_{∞} = L(E′_{p^{∞}}). The natural homomorphism

r_{L} : X(E′/F_{∞})_{H_{L}} → X(E′/L^{cyc}),

which is the dual of the restriction map, has a kernel which is a finitely generated ℤ_{p}-module (see [5]). Applying Nakayama’s lemma, we deduce from (i) and (iii) that X(E′/F_{∞}) is finitely generated over Λ(H_{L}). On the other hand, it is easily seen that our isogeny f induces a Λ(G_{L})-homomorphism from X(E/F_{∞}) to X(E′/F_{∞}), whose kernel is killed by a power of p. Thus X(E/F_{∞}) belongs to 𝔐_{H}(G), and the proof of Proposition 5.6 is complete.

We end this paper by conjecturing the existence of a p-adic L-function attached to E over F_{∞}, and formulating a main conjecture which relates this p-adic L-function to X(E/F_{∞}). We refer the reader to [14] for motivation for the definition we have chosen for our p-adic L-function. If q is any prime number, we write Frob_{q} for the Frobenius automorphism of q in Gal(ℚ̄_{q}/ℚ_{q})/I_{q}, where, as usual, I_{q} denotes the inertia subgroup. To fix the definition of our local and global ε-factors, we follow the conventions and normalizations of [10]. In particular, we choose the sign of the local reciprocity map so that q is mapped to Frob_{q}^{−1}. We take the Haar measure on the adèle group of ℚ which is the usual Haar measure on ℝ, and, for each prime number q, gives ℤ_{q} volume 1. We also fix the unique complex character of the adèle group of ℚ whose infinite component is x ↦ exp(2πix), and whose component at a finite prime q is x ↦ exp(−2πix). Now let ρ denote any Artin representation of G. In our earlier p-adic theory, we viewed ρ as being realized over the ring of integers of some finite extension of ℚ_{p}. However, since ρ is an Artin representation of G, finite group theory tells us that there exists a finite extension of ℚ, which we denote by K_{ρ}, such that ρ can be realized in a finite dimensional vector space V_{ρ} over K_{ρ}. We first recall the definition of the complex Artin L-function L(ρ, s). It is given by the Euler product

L(ρ, s) = ∏_{q} P_{q}(ρ, q^{−s})^{−1},

where P_{q}(ρ, T) is the polynomial

P_{q}(ρ, T) = det(1 − Frob_{q}^{−1}.T |V_{ρ}^{I_{q}}).

For each prime number q, we write e_{q}(ρ) for the local ε-factor of ρ at q, normalized as in [10].

The complex L-function which is obtained by twisting E by the Artin representation ρ is defined as follows. For each prime number l, let

T_{l}(E) = lim_{⟵} E_{l^{n}}, V_{l}(E) = T_{l}(E) ⊗_{ℤ_{l}} ℚ_{l}, H_{l}^{1}(E) = Hom (V_{l}(E), ℚ_{l}).  (101)

Moreover, we fix some prime λ of K_{ρ} above l, and put V_{ρ,λ} = V_{ρ} ⊗_{K} K_{ρ,λ}. Then

L(E, ρ, s) = ∏_{q} P_{q}(E, ρ, q^{−s})^{−1},  (102)

where P_{q}(E, ρ, T) is the polynomial

P_{q}(E, ρ, T) = det(1 − Frob_{q}^{−1}.T |(H_{l}^{1}(E) ⊗_{ℚ_{l}} V_{ρ,λ})^{I_{q}});

here l is any prime number different from q. The Euler product L(E, ρ, s) converges only for ℜ(s) > 3/2, and the only thing known about its analytic continuation at present is that it has a meromorphic continuation when ρ factors through a soluble extension of ℚ. We will assume the analytic continuation of L(E, ρ, s) to s = 1 for all Artin characters ρ of G in what follows. The point s = 1 is critical for L(E, ρ, s) for all Artin ρ, in the sense of [11] and the period conjecture of [11] asserts the following in this case. Fix a global minimal Weierstrass equation for E over ℤ, and let ω denote the Néron differential of this equation. Let γ^{+} (resp. γ^{−}) denote a generator for the subspace of H_{1}(E(ℂ), ℤ) fixed by complex conjugation (resp. the subspace on which complex conjugation acts by -1). We define

Ω_{+}(E) = ∫_{γ^{+}} ω, Ω_{−}(E) = ∫_{γ^{−}} ω.

Moreover, let d^{+}(ρ) denote the dimension of the subspace of V_{ρ} on which complex conjugation acts by +1, and d^{−}(ρ) the dimension of the subspace on which complex conjugation acts by -1. According to the period conjecture of [11], we have

\frac{L(E, ρ, 1)}{Ω_{+}(E)^{d^{+}(ρ)}Ω_{−}(E)^{d^{−}(ρ)}} ∈ ℚ̄  (103)

for all Artin representation ρ of G. We shall assume (103) to define our conjectural p-adic L-function. We also tacitly suppose we have fixed embeddings of ℚ̄ into both ℂ and ℚ̄_{p}.

Let j_{E} denote the j-invariant of E, and let R denote the set consisting of the prime p and all prime numbers q with ord_{q}(j_{E}) < 0. We define

L_{R}(E, ρ, s) = ∏_{q∉R} P_{q}(E, ρ, q^{−s})^{−1}.  (104)

Finally, we put

p^{f_{ρ}} = p-part of the conductor of ρ.  (105)

Also, since E is ordinary at p, we have

1 − a_{p}X + pX^{2} = (1 − uX)(1 − wX), u ∈ ℤ_{p}^{×};  (106)

here, as usual, p + 1 − a_{p} = ♯(Ẽ_{p}(𝔽_{p})), where Ẽ_{p} denotes the reduction of E modulo p. We recall that ρ̂ denotes the contragredient representation of ρ.

Conjecture 5.7. Assume that p ≥ 5 and that E has good ordinary reduction at p. Then there exists L_{E} in K_{1}(Λ(G)_{S^{∗}}) such that, for all Artin representations ρ of G, we have L_{E}(ρ) ≠ ∞, and

L_{E}(ρ) = \frac{L_{R}(E, ρ, 1)}{Ω_{+}(E)^{d^{+}(ρ)}Ω_{−}(E)^{d^{−}(ρ)}} · e_{p}(ρ) · \frac{P_{p}(ρ̂, u^{−1})}{P_{p}(ρ, w^{−1})} · u^{−f_{ρ}};  (107)

here e_{p}(ρ) denotes the local ε-factors at p attached to ρ.

We remark that, when E admits complex multiplication, Conjecture 5.7 is true, and can be deduced from the existence of the two variable p-adic L-function of Manin-Vishik, Katz, and Yager attached to E at the ordinary prime p. When E does not admit complex multiplication, the only evidence we have at present to support Conjecture 5.7 is some interesting but still fragmentary numerical data due to Balister [2] (unpublished) and T.Dokchitser-V.Dokchitser [12].

Conjecture 5.8 (The main conjecture). Assume that p ≥ 5, E has good ordinary reduction at p, and X(E/F_{∞}) belongs to 𝔐_{H}(G). Granted Conjecture 5.7, the p-adic L-function L_{E} in K_{1}(Λ(G)_{S^{∗}}) is a characteristic element of X(E/F_{∞}).

It is plain that such a main conjecture will have many deep consequences for the arithmetic of E over F_{∞} and its subfields. We do not enter into a discussion of these here, beyond mentioning the following almost immediate corollaries.

Corollary 5.9. Assume Conjecture 5.8. Then, for each Artin representation ρ of G, χ(G, tw_{ρ}(X(E/F_{∞}))) is finite if and only if L(E, ρ̂, 1) ≠ 0.

Corollary 5.10. Assume Conjecture 5.8. Let ρ be an Artin representation of G such that L(E, ρ̂, 1) ≠ 0. As in Theorem 3.6, let m_{ρ} be the degree of the quotient field of O over ℚ_{p}, where ρ is given by (12). Then χ(G, twρ̂(M)) is equal to the m_{ρ}-th power of the inverse of the p-adic valuation of the right hand side of (107).

We make the following remark about the integrality properties of L_{E} and any characteristic element ξ_{E} of X(E/F_{∞}). On the one hand, the conjectured holomorphy of L(E, ρ, s) at s = 1 gives, via Conjecture 5.7, the assertion that L_{E}(ρ) ≠ ∞ for every Artin representation ρ of G. On the other hand, thanks to Lemma 5.3 and Theorem 3.8, the conjecture that X(E/F_{∞}) belongs to 𝔐_{H}(G) implies that ξ_{E}(ρ) ≠ ∞ for every Artin representation ρ of G. Note that Case 1 of Conjecture 4.8 would then imply that both L_{E} and ξ_{E} belong to the image of the natural map

Λ(G)[\frac{1}{p}]^{×} ∩ Λ(G)_{S^{∗}} ↪ Λ(G)_{S^{∗}} → K_{1}(Λ(G)_{S^{∗}}).

To prove Corollary 5.9, we observe that when X(E/F_{∞}) belongs to 𝔐_{H}(G), Lemma 5.3 and Theorem 3.8 show that ξ_{E}(ρ) ≠ 0 if and only if χ(G, twρ̂(X(E/F_{∞}))) is finite. If we assume Conjecture 5.8, we can take ξ_{E} = L_{E}, and then it is clear from (107) that L_{E}(ρ) ≠ 0 if and only if L(E, ρ, 1) ≠ 0. Corollary 5.10 is then an immediate consequence of (107) and formula (36) of Theorem 3.6.

Needless to say, the evidence in favour of Conjecture 5.8 is still very limited. When E admits complex multiplication, Conjecture 5.8 is true provided we assume that X(E/F_{∞}) belongs to 𝔐_{H}(G) (see Corollary 5.5). Indeed it can be deduced from the proof by Yager [34] and Rubin [22] of what is usually called the two variable main conjecture (recall that in this case G is a p-adic Lie group of dimension 2). When E does not admit complex multiplication, the remarkable calculations of L(E, ρ, 1) made in [12] for certain Artin representations of small degree and small primes p provide some numerical evidence in favour of Conjecture 5.8. We end by giving their results for E = X_{1}(11), p = 5, and the two Artin representations ρ_{1} and ρ_{2} of degree 4 which are discussed at the end of §3.

Example. We use the same notation as at the end of §3. The following data is calculated in [12]. We have

d^{+}(ρ_{i}) = d^{−}(ρ_{i}) = 2 (i = 1, 2)  (108)

f_{ρ_{1}} = 3, f_{ρ_{2}} = 5  (109)

P_{5}(ρ_{1}, X) = 1 − X, P_{5}(ρ_{2}, X) = 1  (110)

P_{5}(E, ρ_{1}, X) = 1 − X + 5X^{2}, P_{5}(E, ρ_{2}, X) = 1  (111)

P_{11}(E, ρ_{1}, X) = 1, P_{11}(E, ρ_{2}, X) = 1  (112)

|e_{5}(ρ_{1})|_{5} = 5^{−3/2}, |e_{5}(ρ_{2})|_{5} = 5^{−5/2}  (113)

Moreover, ♯(Ẽ(𝔽_{5})) = 5, and so

1 − a_{5}X + 5X^{2} = 1 − X + 5X^{2}  (114)

Finally, even though P_{q}(E, ρ_{i}, X) (i = 1, 2) has degree 8 for all q ≠ 5, 11, the remarkable calculations of [12] show that

\frac{L(E, ρ_{1}, 1)}{(Ω_{+}(E)Ω_{−}(E))^{2}} = \frac{−2^{2}}{11^{2}\sqrt{5}}, \frac{L(E, ρ_{2}, 1)}{(Ω_{+}(E)Ω_{−}(E))^{2}} = \frac{−2^{6}}{11^{2}5\sqrt{5}}.  (115)

Thus, since both of these L-values are non-zero, Proposition 3.11 shows that Corollary 5.9 holds for ρ_{1} and ρ_{2}. Finally, one can use the above data to calculate the right hand side of (107), up to a 5-adic unit, for ρ_{1} and ρ_{2}. Using (74) of Proposition 3.11, and recalling that ρ_{1} and ρ_{2} can both be realized over ℤ_{5}, we deduce that Corollary 5.10 is also valid for ρ_{1} and ρ_{2}.

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John Coates Takako Fukaya DPMMS, Faculty of Business and Commerce, University of Cambridge, Keio University, Centre for Mathematical Sciences, Hiyoshi, Kohoku-ku, Wilberforce Road, Yokohama, 223-8521, Japan Cambridge CB3 0WB, England takakof@hc.cc.keio.ac.jp J.H.Coates@dpmms.cam.ac.uk

Kazuya Kato Ramdorai Sujatha Department of Mathematics, School of Mathematics, Kyoto University, Tata Institute of Fundamental Research, Kitashirakawa, Homi Bhabha Road, Kyoto 606-8502, Japan Mumbai 400 005, India kazuya@math.kyoto-u.ac.jp sujatha@math.tifr.res.in

Otmar Venjakob Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany otmar@mathi.uni-heidelberg.de