Links Between Cyclotomic

and GL_{2} Iwasawa Theory

John Coates, Peter Schneider, Ramdorai Sujatha

Received: September 26, 2002 Revised: February 26, 2003

Vying with the light Of the heaven-coursing sun, Oh, let me search, That I find it once again, The Way that was so pure.

Dedicated to Kazuya Kato

Abstract. We study, in the case of ordinary primes, some connections between the GL_{2} and cyclotomic Iwasawa theory of an elliptic curve without complex multiplication.

2000 Mathematics Subject Classification: 11G05; 11R23. Keywords and Phrases: Iwasawa theory, elliptic curves.

1 Introduction

Let F be a finite extension of ℚ, E an elliptic curve defined over F, and p a prime number such that E has good ordinary reduction at all primes v of F dividing p. Let F^{cyc} denote the cyclotomic ℤ_{p}-extension of F, and put Γ = G(F^{cyc}/F). A very well known conjecture due to Mazur [15] asserts that the dual of the Selmer group of E over F^{cyc} is a torsion module over the Iwasawa algebra Λ(Γ) of Γ (the best result in the direction of this conjecture is due to Kato [13], who proves it when E is defined over ℚ, and F is an abelian extension of ℚ). Now let F_{∞} denote a Galois extension of F containing F^{cyc} whose Galois group G over F is a p-adic Lie group of dimension > 1. This paper will make some modest observations about the following general problem. To what extent, and in what way, does the arithmetic of E over F^{cyc} influence the arithmetic of E over the much larger p-adic Lie extension F_{∞}? For example, assuming that G is pro-p and has no element of order p, and that Mazur’s conjecture is true for E over F^{cyc}, can one deduce that the dual of the Selmer group of E over F_{∞} is a torsion module over the Iwasawa algebra Λ(G) of G? In their paper [12] in this volume, Hachimori and Venjakob prove the surprising result that the answer is yes for a wide class of non-abelian extensions F_{∞} of F in which G has dimension 2. In the first two sections of this paper, we study the different case in which F_{∞} = F (E_{p^{∞}}) is the field obtained by adjoining to F the coordinates of all p-power division points on E. We assume that E has no complex multiplication, so that G is open in GL_{2}(ℤ_{p}) by a well known theorem of Serre. In [4], it was shown that, in this case, the dual of the Selmer group of E over F_{∞} is Λ(G)-torsion provided Mazur’s conjecture for E over F^{cyc} is true, and, in addition, the μ-invariant of the dual of Selmer of E over F^{cyc} is zero. Although we can do no better than this result as far as showing the dual of E over F_{∞} is Λ(G)-torsion, we do prove some related results which were not known earlier. The main result of this paper is Theorem 3.1, relating the truncated G-Euler characteristic of the Selmer group of E over F_{∞} with the Γ-Euler characteristic of E over F^{cyc} (only a slightly weaker form of this result was shown in [4] under the more restrictive assumption that the Selmer group of E over F is finite). We also establish a relation between the μ-invariant of the dual of the Selmer group of E over F_{∞} and the μ-invariant of the dual of the Selmer group of E overF^{cyc} (see Propositions 3.12 and 3.13). The proof of this relationship between μ-invariants led us to study in §4 a new invariant attached to a wide class of finitely generated torsion modules for the Iwasawa algebra of any pro-p p-adic Lie group G, which has no element of order p, and which has a closed normal subgroup H such that Γ = G/H is isomorphic to ℤ_{p}. This new invariant is a refinement of the G-Euler characteristic of such modules, and, in particular, we investigate its behaviour on pseudo-null modules. Added in proof: Since this paper was written, O. Venjakob, in his Heidelberg Habilitation thesis, has made use of our invariant to prove the existence of an analogue of the characteristic power series of commutative Iwasawa theory for any module in the category 𝔐_{H}(G) which is defined at the beginning of §4.

Notation

Let p be a fixed prime number. If A is an abelian group, A(p) will always denote its p-primary subgroup. Throughout G will denote a compact p-adic Lie group, and we write

Λ(G) = lim_{⟵}_{U} ℤ_{p} [G/U],

Links Between Cyclotomic and GL_{2} Iwasawa Theory 189

where U runs over all open normal subgroups of G, for the Iwasawa algebra of G. All modules we consider will be left modules for Λ(G). If W is a compact Λ(G)-module, we write Ŵ = Hom_{ℤ_{p}}(W, ℚ_{p}/ℤ_{p}) for its Pontrjagin dual. It is a discrete p-primary abelian group, endowed with its natural structure of a left Λ(G)-module. We shall write H_{i}(G, W) for the homology groups of W. If W is a finitely generated Λ(G)-module, then it is well known that, for each i ⩾ 0, H_{i}(G, W) has as its Pontrjagin dual the cohomology group H^{i}(G, Ŵ), which is defined with continuous cochains. When K is a field, \overline{K} will denote a fixed separable closure of K, and G_{K} will denote the Galois group of \overline{K} over K. We write G(L/K) for the Galois group of a Galois extension L over K. If A is a discrete G_{K}-module, H^{i}(K, A) will denote the usual Galois cohomology groups. Throughout, F will denote a finite extension of ℚ, and E an elliptic curve defined over F, which will always be assumed to have End\overline{F} (E) = ℤ. We impose throughout sections 2 and 3 of this paper the hypothesis that E has good ordinary reduction at all primes v of F dividing p. We let S denote any fixed set of places of F such that S contains all primes of F dividing p, and all primes of F where E has bad reduction. We write F_{S} for the maximal extension of F which is unramified outside S and the archimedean primes of F. For each intermediate field L with F ⊂ L ⊂ F_{S}, we put G_{S}(L) = G(F_{S}/L). Finally, we shall always assume that our prime p satisfies p ≥ 5.

2 The fundamental exact sequence

We recall that we always assume that E has good ordinary reduction at all primes v of F dividing p. Let F^{cyc} denote the cyclotomic ℤ_{p}-extension of F, and put Γ = G(F^{cyc}/F). By a basic conjecture of Mazur [15], the dual of the Selmer group of E over F^{cyc} is a torsion Λ(Γ)-module. The aim of this section is to analyse the consequences of this conjecture for the study of the Selmer group of E over the field generated by all the p-power division points on E. If L is any intermediate field with F ⊂ L ⊂ F_{S}, we recall that the Selmer group S(E/L) is defined by

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

where E_{p^{∞}} denotes the Galois module of all p-power division points on E. Here w runs over all non-archimedean valuations of L, and L_{w} denotes the union of the completions at w of all finite extensions of ℚ contained in L. As usual, it is more convenient to view S(E/L) as a subgroup of H^{1}(G_{S}(L), E_{p^{∞}}). For v ∈ S, we define J_{v}(L) = lim_{⟶} J_{v}(K), where the inductive limit is taken with respect to the restriction maps as K ranges over all finite extensions of F contained in L, and where, for such a finite extension K of F, we define

J_{v}(K) = ⨁_{w|v} H^{1}(K_{w}, E(\overline{K}_{w}))(p).

Since L ⊂ F_{S}, we then have S(E/L) = Ker λ_{S}(L), where

(1)  λ_{S}(L) : H^{1}(G_{S}(L), E_{p^{∞}}) → ⨁_{v∈S} J_{v}(L)

denotes the evident localization map. We shall see that, when L is an infinite extension of F, the question of the surjectivity of λ_{S}(L) is a basic one. We write

(2)  X(E/L) = Hom(S(E/L), ℚ_{p}/ℤ_{p})

for the compact Pontrjagin dual of the discrete module S(E/L). We shall be primarily concerned with 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 a finitely generated Λ(G(L/F))-module. We are going to exploit the following well-known lemma (see [17, Lemmas 4 and 5] and also [12, §7] for an account of the proof in a more general setting).

Lemma 2.1 Assume that X(E/F^{cyc}) is Λ(Γ)-torsion. Then the localization map λ_{S}(F^{cyc}) is surjective, i.e. we have the exact sequence of Γ-modules

(3) 0 → S(E/F^{cyc}) → H^{1}(G_{S}(F^{cyc}), E_{p^{∞}}) →^{λ_{S}(F^{cyc})} ⨁_{v∈S} J_{v}(F^{cyc}) → 0.

Moreover, we also have

(4)  H^{2}(G_{S}(F^{cyc}), E_{p^{∞}}) = 0.

We now consider the field F_{∞} = F (E_{p^{∞}}), which always contains F^{cyc} by the Weil pairing. We write

(5)  G = G(F_{∞}/F), H = G(F_{∞}/F^{cyc}),

so that G/H = Γ. By Serre’s theorem, G is an open subgroup of Aut(T_{p}(E)) = GL_{2}(ℤ_{p}), where, as usual, T_{p}(E) = lim_{⟵} E_{p^{n}}. The following is the principal result of this section.

Theorem 2.2 Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all primes v of F dividing p, and (iii) X(E/F^{cyc}) is Λ(Γ)-torsion. Then we have the exact sequence

(6)

0 → S(E/F_{∞})^{G} → W_{∞}^{G} → ⨁_{v∈S} J_{v}(F_{∞})^{G} → H^{1}(G, S(E/F_{∞})) → H^{1}(G, W_{∞}) → 0,

where W_{∞} = H^{1}(G_{S}(F_{∞}), E_{p^{∞}}).

In fact, we expect λ_{S}(F_{∞}) to be surjective for all prime numbers p. If p ≥ 5, it is shown in [4] that H^{i}(G, W_{∞}) = 0 for all i ≥ 2, and that H^{i}(G, J_{v}(F_{∞})) = 0 for all i ≥ 1 and all v ∈ S. Thus if λ_{S}(F_{∞}) is surjective for a prime number p ≥ 5, the exact sequence (6) follows. However, what is surprising about Theorem 2.2 is that we can establish it without knowing the surjectivity of λ_{S}(F_{∞}) (in our present state of knowledge [4], to prove the surjectivity of λ_{S}(F_{∞}) we must assume hypotheses (i) and (ii) of Theorem 2.2, replace (iii) by the stronger hypothesis that X(E/F^{cyc}) is a finitely generated ℤ_{p}-module, and, in addition, assume that G is pro-p). Finally, we mention that if hypotheses (i) and (ii) of Theorem 2.2 hold, and if also G is pro-p, then X(E/F_{∞}) is Λ(G)-torsion if and only if λ_{S}(F_{∞}) is surjective. We now proceed to establish Theorem 2.2 via a series of lemmas. For these lemmas, we assume that the hypotheses (i), (ii) and (iii) of Theorem 2.2 are valid.

Lemma 2.3 We have the exact sequence

0 → S(E/F_{∞})^{H} → H^{1}(G_{S}(F_{∞}), E_{p^{∞}})^{H} →^{ρ_{S}(F_{∞})} ⨁_{v∈S} J_{v}(F_{∞})^{H} → 0,

where ρ_{S}(F_{∞}) is induced by the localization map λ_{S}(F_{∞}).

Proof All we have to show is that ρ_{S}(F_{∞}) is surjective. We clearly have the commutative diagram

\xymatrix{0 \ar[r] & S(E/F_{∞})^{H} \ar[r] & H^{1}(G_{S}(F_{∞}), E_{p^{∞}})^{H} \ar[r]_{ρ_{S}(F_{∞})} & ⨁_{v∈S} J_{v}(F_{∞})^{H} \\ 0 \ar[r] & S(E/F^{cyc}) \ar[u] \ar[r] & H^{1}(G_{S}(F^{cyc}), E_{p^{∞}}) \ar[u] \ar[r]_{λ_{S}(F^{cyc})} & ⨁_{v∈S} J_{v}(F^{cyc}) \ar[u]_{γ_{S}(F^{cyc})} \ar[r] & 0,}

where γ_{S}(F^{cyc}) is induced by restriction, and where the surjectivity of λ_{S}(F^{cyc}) is given by Lemma 2.1, thanks to our assumption that X(E/F^{cyc}) is Λ(Γ)- torsion. Hence it suffices to prove the surjectivity of γ_{S}(F^{cyc}). This is essentially contained in the proof of [4, Lemma 6.7], but we give the detailed proof as it is only shown there that γ_{S}(F^{cyc}) has finite cokernel. As is well known and easy to see, there are only finitely many primes of F^{cyc} above each non-archimedean prime of F. Hence we have

Coker(γ_{S}(F^{cyc})) = ⨁_{w|S} Coker(γ_{w}(F^{cyc})),

where w runs over all primes of F^{cyc} lying above primes in S, and where, as H^{2}(F_{∞,w}, E) = 0,

Coker(γ_{w}(F^{cyc})) = H^{2}(Ω_{w}, E(F_{∞,w}))(p);

here Ω_{w} denotes the decomposition group in H of some fixed prime of F_{∞} lying above w. Now Ω_{w} is a p-adic Lie group with no elements of order p as p ≥ 5, and so Ω_{w} has finite p-cohomological dimension equal to its dimension as a p-adic Lie group. Moreover, a simple local analysis (see [2] or [4]) shows that Ω_{w} has dimension at most 1 when w does not divide p, and dimension 2 when w does divide p. We claim that we always have

(7)  H^{2}(Ω_{w}, E_{p^{∞}}) = 0.

This is plain from the above remarks when v does not divide p. When v divides p, Ω_{w} has p-cohomological dimension equal to 2, and, thus H^{2}(Ω_{w}, E_{p^{∞}}) is a divisible group. On the other hand, as E has good reduction at w, it is well-known (see [4] for a direct argument, or [8] for a general result) that H^{2}(Ω_{w}, E_{p^{∞}}) is finite, whence (7) follows. We now finish the proof using (7). Assume first that w does not divide p. Then (see [2, Lemma 3.7]) we have

Coker(γ_{w}(F^{cyc})) = H^{2}(Ω_{w}, E_{p^{∞}}),

and so we obtain the surjectivity of γ_{w}(F^{cyc}) from (7). Suppose now that w does divide p. Then it follows from the results of [3] that

Coker(γ_{w}(F^{cyc})) = H^{2}(Ω_{w}, Ẽ_{w,p^{∞}}),

where Ẽ_{w,p^{∞}} denotes the image of E_{p^{∞}} under reduction modulo w. But as Ω_{w} has p-cohomological dimension equal to 2, the vanishing of H^{2}(Ω_{w}, Ẽ_{w,p^{∞}}) is an immediate consequence of (7). This completes the proof of Lemma 2.3. □

Lemma 2.4 We have H^{i}(H, H^{1}(G_{S}(F_{∞}), E_{p^{∞}})) = 0 for all i ≥ 1.

Proof We have

(8)  H^{m}(G_{S}(F_{∞}), E_{p^{∞}}) = 0, (m ≥ 2).

Indeed, (8) is obvious for m > 2 as G_{S}(F_{∞}) has p-cohomological dimension equal to 2, and it is a consequence of Iwasawa’s work on the cyclotomic ℤ_{p}- extension of number fields when m = 2 (see [2, Theorem 2.10]). In view of (8), the Hochschild-Serre spectral sequence gives

(9) H^{i+1}(G_{S}(F^{cyc}), E_{p^{∞}}) → H^{i}(H, H^{1}(G_{S}(F_{∞}), E_{p^{∞}})) → H^{i+2}(H, E_{p^{∞}}).

The group on the left of (9) vanishes (for i = 1, we use Lemma 2.1). Now H has p-cohomological dimension 3, and so H^{i+2}(H, E_{p^{∞}}) is zero for i > 1, and divisible for i = 1. On the other hand, it is known [6] that H^{k}(H, E_{p^{∞}}) is finite for all k ≥ 0, whence, in particular, we must have H^{3}(H, E_{p^{∞}}) = 0. Thus (9) gives Lemma 2.4 as required. □

Lemma 2.5 We have H^{1}(H, S(E/F_{∞})) = 0.

Proof Let A_{∞} denote the image of λ_{S}(F_{∞}). Hence, in view of Lemma 2.4 with i = 1, we have the exact sequence

0 → S(E/F_{∞})^{H} → H^{1}(G_{S}(F_{∞}), E_{p^{∞}})^{H} → A_{∞}^{H} → H^{1}(H, S(E/F_{∞})) → 0.

But the surjectivity of ρ_{S}(F_{∞}) in Lemma 2.3 shows that

A_{∞}^{H} = ⨁_{v∈S} J_{v}(F_{∞})^{H},

whence it is clear that H^{1}(H, S(E/F_{∞})) = 0, as required. □

Remark 2.6 By a similar argument to that given in the proof of [2, Theorem 3.2], we see that H^{i}(H, J_{v}(F_{∞})) = 0 for all i ≥ 1, for all primes p ≥ 5, and all finite places v of F. Thus we deduce from Lemma 2.4 that the surjectivity of λ_{S}(F_{∞}) implies that H^{i}(H, S(E/F_{∞})) = 0 for all i ≥ 1. Unfortunately, we cannot at present prove the surjectivity of λ_{S}(F_{∞}) assuming only the hypotheses (i), (ii) and (iii) of Theorem 2.2.

Lemma 2.7 We have isomorphisms

\begin{matrix}H^{1}(Γ, H^{1}(G_{S}(F_{∞}), E_{p^{∞}})^{H}) ≃ H^{1}(G, H^{1}(G_{S}(F_{∞}), E_{p^{∞}})) \\ H^{1}(Γ, S(E/F_{∞})^{H}) ≃ H^{1}(G, S(E/F_{∞})). \end{matrix}

Proof This is immediate from Lemmas 2.4 and 2.5, and the usual inflationrestriction exact sequence for H^{1}. □

Lemma 2.8 We have H^{1}(Γ, ⨁_{v∈S} J_{v}(F_{∞})^{H}) = 0.

Proof To simplify notation, let us put K = F^{cyc}. Since the map

γ_{S}(K) : ⨁_{v∈S} J_{v}(K) → ⨁_{v∈S} J_{v}(F_{∞})^{H}

is surjective by Lemma 2.3, and since Γ has p-cohomological dimension equal to 1, it suffices to show that

(10)  H^{1}(Γ, ⨁_{v∈S} J_{v}(K)) = 0.

It is well-known that (10) is valid, but we sketch a proof now for completeness. For each place v of F, let w be a fixed place of K above v, and let Γ_{v} ⊂ Γ denote the decomposition group of w over v, which is an open subgroup of Γ. As usual, it follows from Shapiro’s lemma that

H^{1}(Γ, J_{v}(K)) ≃ H^{1}(Γ_{v}, H^{1}(K_{w}, E)(p)),

and so we must prove that

(11)  H^{1}(Γ_{v}, H^{1}(K_{w}, E)(p)) = 0.

We begin by noting that, for each algebraic extension L of F_{v}, we have

(12)  H^{2}(L, E_{p^{∞}}) = 0.

When L is a finite extension of F_{v}, Tate local duality shows that H^{2}(L, E_{p^{∞}}) is dual to H^{0}(L, T_{p}(E)), and this latter group is zero because the torsion subgroup of E(L) is finite. Clearly (12) is now true for all algebraic extensions L of F_{v} by passing to the inductive limit over all finite extensions of F_{v} contained in L. Suppose now that v does not divide p. Then

H^{1}(K_{w}, E_{p^{∞}}) ≃ H^{1}(K_{w}, E)(p).

In view of (12), the Hochschild-Serre spectral sequence for K_{w}/F_{v} shows that we have the exact sequence

H^{2}(F_{v}, E_{p^{∞}}) → H^{1}(Γ_{v}, H^{1}(K_{w}, E_{p^{∞}})) → H^{3}(Γ_{v}, E_{p^{∞}}(K_{v})).

But the group on the left is zero again by (12), and the group on the right is zero because Γ_{v} has p-cohomological dimension equal to 1. This proves (11) in this case. Suppose next that v divides p. As K_{w} is a deeply ramified p-adic field, it follows from [3] that

H^{1}(K_{w}, E)(p) ≃ H^{1}(K_{w}, Ẽ_{w,p^{∞}}),

where Ẽ_{w,p^{∞}} denotes the image of E_{p^{∞}} under reduction modulo w. As Ẽ_{w,p^{∞}} is a quotient of E_{p^{∞}}, and as the Galois group of \overline{F_{v}} over K_{w} has p-cohomological dimension at most 2, we conclude from (12) that

(13)  H^{2}(K_{w}, Ẽ_{w,p^{∞}}) = 0.

In fact, the Galois group of \overline{F}_{v} over K_{w} has p-cohomological dimension 1, so that (13) also follows directly from this fact. In view of (13), the Hochschild-Serre spectral sequence for K_{w}/F_{v} yields the exact sequence

H^{2}(F_{v}, Ẽ_{w,p^{∞}}) → H^{1}(Γ_{v}, H^{1}(K_{w}, Ẽ_{w,p^{∞}}) → H^{3}(Γ_{v}, Ẽ_{w,p^{∞}}).

The group on the right is zero because Γ_{v} has p-cohomological dimension equal to 1. By Tate local duality, the dual of the group on the left is H^{0}(F_{v}, T_{p}(Ê_{w})), where T_{p}(Ê_{w}) lim_{⟵} Ê_{w,p^{n}}, and Ê_{w,p^{n}} denotes the kernel of multiplication by p^{n}

on the formal group Ê_{w} of E at w. But again H^{0}(F_{v}, T_{p}(Ê_{w})) = 0 because the torsion subgroup of E(F_{v}) is finite, and so we have proven (11) in this case. This completes the proof of Lemma 2.8. □

Proof of Theorem 2.2 We can now prove Theorem 2.2. Put W_{∞} = H^{1}(G_{S}(F_{∞}), E_{p^{∞}}). Taking Γ-cohomology of the exact sequence of Lemma 2.3, and using Lemma 2.8, we obtain the long exact sequence

0 → S(E/F_{∞})^{G} → W_{∞}^{G} → ⨁_{v∈S} J_{v}(F_{∞})^{G} →

→ H^{1}(Γ, S(E/F_{∞})^{H}) → H^{1}(Γ, W_{∞}^{H}) → 0.

Theorem 2.2 now follows immediately from Lemma 2.7. □

The following is a curious consequence of the arguments in this section, and we have included it because a parallel result has a striking application to the work of Hachimori and Venjakob [12].

Proposition 2.9 Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all primes v of F dividing p, (iii) X(E/F^{cyc}) is Λ(Γ) -torsion, and (iv) G is pro-p. Then X(E/F_{∞}) is Λ(G)-torsion if and only if H^{2}(H, S(E/F_{∞})) = 0.

Proof Since G is pro-p, it is well-known (see for example, Theorem 4.12 of [2]) that X(E/F_{∞}) is Λ(G)-torsion if and only if λ_{S}(F_{∞}) is surjective. We have already observed in Remark 2.6 that the surjectivity of λ_{S}(F_{∞}) implies that H^{i}(H, S(E/F_{∞})) = 0 for all i ≥ 1. Conversely, assume that H^{2}(H, S(E/F_{∞})) = 0. As in the proof of Lemma 2.5, let A_{∞} denote the image of λ_{S}(F_{∞}). Taking the H-cohomology of the exact sequence

0 → S(E/F_{∞}) → H^{1}(G_{S}(F_{∞}), E_{p^{∞}}) → A_{∞} → 0,

we conclude from Lemma 2.4 that

H^{1}(H, A_{∞}) ≃ H^{2}(H, S(E/F_{∞})).

Hence our hypothesis implies that H^{1}(H, A_{∞}) = 0. Now let B_{∞} = Coker(λ_{S}(F_{∞})). Taking H-cohomology of the exact sequence

0 → A_{∞} → ⨁_{v∈S} J_{v}(F_{∞}) → B_{∞} → 0,

and using Lemma 2.3 and the fact mentioned in Remark 2.6 that H^{1}(H, J_{v}(F_{∞})) = 0 for all v ∈ S, we conclude that

B_{∞}^{H} = H^{1}(H, A_{∞}).

Hence B_{∞}^{H} = 0. But as H is pro-p and B_{∞} is a p-primary discrete H-module, it follows that B_{∞} = 0. Thus λ_{S}(F_{∞}) is surjective, and this completes the proof of Proposition 2.5. □

We conclude this section by proving a result relating the so-called μ-invariants of the Λ(G)-module X(E/F_{∞}) and the Λ(Γ)-module X(E/F^{cyc}). Let us assume for the rest of this section that G is pro-p. Let W be any finitely generated Λ(G)-module. We write W (p) for the submodule of all elements of W which are annihilated by some power of p, and we then define

(14)  W_{f} = W/W (p).

We recall that the homology groups H_{i}(G, W) are the Pontrjagin duals of the cohomology groups H^{i}(G, Ŵ), where Ŵ = Hom_{ℤ_{p}}(W, ℚ_{p}/ℤ_{p}) is the discrete pprimary Pontrjagin dual of W (see [11]). As is explained in [11], the H^{i}(G, W) are finitely generated ℤ_{p}-modules, and thus the H_{i}(G, W (p)) are finite groups for all i ≥ 0. Now the μ-invariant of W, which we shall denote by μ_{G}(W), can be defined in various equivalent fashions (see [22], [11]) in terms of the structure theory of the Λ(G)-module W (p). However, for us it will be more convenient to use the description of μ_{G}(W) in terms of the Euler characteristics which is proven in [11], namely

(15)  p^{μ_{G}(W)} = ∏_{i≥0} #(H_{i}(G, W (p)))^{(−1)^{i}}.

As usual, we shall denote the right hand side of (15) by χ(G, W (p)). We shall use the analogous notation and results for the μ-invariants of finitely generated Λ(Γ)-modules. For the remainder of this section, we always assume the hypotheses (i)-(iv) of Proposition 2.9.

Lemma 2.10 Both H_{0}(H, X(E/F_{∞})_{f}) and H_{1}(H, X(E/F_{∞})_{f}) are finitely generated torsion Λ(Γ)-modules, and H_{1}(H, X(E/F_{∞})_{f}) is annihilated by some power of p.

Proof For simplicity, put X = X(E/F_{∞}) . By duality, the restriction map on cohomology induces a Γ-homomorphism

(16)  α : X_{H} = H_{0}(H, X) → X(E/F^{cyc}).

Thanks to the basic results of [5], it is shown in [4, Lemma 6.7], that Ker(α) is a finitely generated ℤ_{p}-module, and that Coker(α) is finite. As X(E/F^{cyc}) is assumed to be Λ(Γ)-torsion, it follows that H_{0}(H, X) is a finitely generated torsion Λ(Γ)-module. Now if we take H-cohomology of the exact sequence

(17)  0 → X(p) → X → X_{f} → 0,

and recall that H_{1}(H, X) = 0 by Lemma 2.5, we obtain the exact sequence of Λ(Γ)-modules

(18)  0 → H_{1}(H, X_{f}) → H_{0}(H, X(p)) → H_{0}(H, X) → H_{0}(H, X_{f}) → 0.

The right hand end of (18) shows that H_{0}(H, X_{f}) is Λ(Γ)-torsion, and the left hand end shows that H_{1}(H, X_{f}) is finitely generated over Λ(Γ) and annihilated by a power of p, because these two properties clearly hold for H_{0}(H, X(p)). This completes the proof of Lemma 2.10. □

Remark 2.11 Put X = X(E/F_{∞}), and continue to assume that hypotheses (i)-(iv) of Proposition 2.9 hold. As H is pro-p, Nakayama’s lemma also shows that X_{f} = X/X(p) is finitely generated as a Λ(H)-module if and only if (X_{f})_{H} = H_{0}(H, X_{f}) is a finitely generated ℤ_{p}-module. As H_{0}(H, X_{f}) is a finitely generated torsion Λ(Γ)-module, it follows that X_{f} is finitely generated as a Λ(H)-module if and only if μ_{Γ}(H_{0}(H, X_{f})) = 0.

Proposition 2.12 Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all primes v of F dividing p (iii) G is pro-p, and (iv) X(E/F^{cyc}) is Λ(Γ)- torsion. Then we have

(19)  μ_{G}(X(E/F_{∞})) = μ_{Γ}(X(E/F^{cyc})) + δ + ϵ,

where, writing X = X(E/F_{∞}),

(20) δ = Σ_{i=0}^{1} (−1)^{i+1}μ_{Γ}(H_{i}(H, X_{f})), ϵ = Σ_{i=1}^{3}(−1)^{i}μ_{Γ}(H_{i}(H, X(p))).

Proof As each module in the exact sequence (18) is Λ(Γ)-torsion, it follows (see [11, Prop. 1.9]) that the alternating sum of the μ_{Γ}-invariants taken along (18) is zero. Moreover, the μ_{Γ}-invariants of the two middle terms in (18) can be calculated as follows. Firstly, as Ker(α) and Coker(α) are finitely generated ℤ_{p}-modules, it follows from (16) that

(21)  μ_{Γ}(H_{0}(H, X)) = μ_{Γ}(X(E/F^{cyc})).

Secondly, for i = 1, 2, 3, 4, the Hochschild-Serre spectral sequence yields the short exact sequence

(22) 0 → H_{0}(Γ, H_{i}(H, X(p))) → H_{i}(G, X(p)) → H_{1}(Γ, H_{i−1}(H, X(p))) → 0.

Also, we have H_{4}(H, X(p)) = 0 because H has p-homological dimension equal to 3. It follows easily that

(23)  χ(G, X(p)) = ∏_{i=0}^{3} χ(Γ, H_{i}(H, X(p)))^{(−1)^{i}},

whence by (15) for both the group G and the group Γ, we obtain

(24)  μ_{G}(X) = Σ_{i=0}^{3}(−1)^{i}μ_{Γ}(H_{i}(H, X(p))).

Proposition 2.12 now follows immediately (21) and (24) and from the fact that the alternating sum of the μ_{Γ}-invariants along (18) is 0. □

We now give a stronger form of (19) when we impose the additional hypothesis that X(E/F_{∞}) is Λ(G)-torsion.

Proposition 2.13 Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all primes v of F dividing p, (iii) G is pro-p, (iv) X(E/F^{cyc}) is Λ(Γ)- torsion, and (v) X(E/F_{∞}) is Λ(G)-torsion. Then H_{i}(H, X_{f}) (i = 1, 2), where X_{f} = X(E/F_{∞})/X(E/F_{∞})(p), is a finitely generated Λ(Γ)-module which is killed by a power of p. Moreover,

(25) μ_{G}(X(E/F_{∞})) = μ_{Γ}(X(E/F^{cyc})) + Σ_{i=0}^{2} (−1)^{i+1}μ_{Γ}(H_{i}(H, X_{f})).

Corollary 2.14 Assume hypotheses (i)-(iv) of Proposition 2.13 and replace (v) by the hypothesis that X(E/F_{∞})/X(E/F_{∞})(p) is finitely generated over Λ(H). Then

(26)  μ_{G}(X(E/F_{∞})) = μ_{Γ}(X(E/F^{cyc})).

To deduce the corollary, we first note that the hypothesis that X_{f} is finitely generated over Λ(H) implies that X_{f} is Λ(G)-torsion, whence X is also Λ(G)-torsion. Secondly, the H_{i}(H, X_{f}) are finitely generated ℤ_{p}-modules once X_{f} is finitely generated over Λ(H), and hence their μ_{Γ}-invariants are zero. Thus (26) then follows from (25). We remark that, in all cases known to date in which we can prove X(E/F_{∞}) is Λ(G)-torsion, one can show that X_{f} is finitely generated over Λ(H), but we have no idea at present of how to prove this latter assertion in general.

Proof of Proposition 2.13 Again put X = X(E/F_{∞}). Since we are now assuming that X is Λ(G)-torsion, or equivalently that λ_{S}(F_{∞}) is surjective, we have already remarked (see Remark 2.6) that

(27)  H_{i}(H, X) = 0 for all i ≥ 1.

We next claim that

(28)  H_{3}(H, X_{f}) = 0.

Indeed, as H has p-cohomological dimension 3, and multiplication by p is injective on X_{f}, it follows that multiplication by p must also be injective on H_{3}(H, X_{f}). On the other hand, taking H-homology of the exact sequence (17), we see that H_{3}(H, X_{f}) injects into the torsion group H_{2}(H, X(p)) because H_{3}(H, X) = 0. Thus (28) follows. Moreover, using (27) and (28), we conclude from the long exact sequence of H-cohomology of (17) that

(29)  H_{1}(H, X(p)) = H_{2}(H, X_{f}), H_{i}(H, X(p)) = 0 (i = 2, 3).

Thus (25) now follows from (19), completing the proof of Proposition 2.13. □

3 The truncated Euler characteristic

We assume throughout this section our three standard hypotheses: (i) p ≥ 5, (ii) E has good ordinary reduction at all primes v of F dividing p, and (iii) X(E/F^{cyc}) is Λ(Γ)-torsion. Since G has dimension 4 as a p-adic Lie group and has no element of order p, G has p-cohomological dimension equal to 4. We begin by defining the notion of the truncated G-Euler characteristic χ_{t}(G, A) of a discrete p-primary G-module A. The main aim of this section is to compute the truncated G-Euler characteristic of S(E/F_{∞}) in terms of the Γ-Euler characteristic of S(E/F^{cyc}). A Birch-Swinnerton-Dyer type formula for the latter Euler characteristic is well-known (see Schneider [19], Perrin-Riou [18]), and we shall recall this at the end of this section. If D is a discrete p-primary Γ-module, we have

H^{0}(Γ, D) = D^{Γ}, H^{1}(Γ, D) ≅ D_{Γ},

and hence there is the obvious map

(30)  ϕ_{D} : H^{0}(Γ, D) → H^{1}(Γ, D)

given by ϕ_{D}(x) = residue class of x in D_{Γ}. If f is any homomorphism of abelian groups, we define

(31)  q(f) = #(Ker(f))/#(Coker(f)),

saying q(f) is finite if both Ker(f) and Coker(f) are finite. We say D has finite Γ-Euler characteristic if q(ϕ_{D}) is finite, and we then define χ(Γ, D) = q(ϕ_{D}). Suppose now that A is a discrete p-primary G-module. As in the previous section, we write H = G(F_{∞}/F^{cyc}), so that H is a closed normal subgroup of G with G/H = Γ. Parallel to (30), we define a map

(32)  ξ_{A} : H^{0}(G, A) → H^{1}(G, A)

by ξ_{A} = η ∘ ϕ_{A^{H}} , where η is the inflation map from H^{1}(Γ, A^{H}) to H^{1}(G, A). We say that A has finite truncated G-Euler characteristic if q(ξ_{A}) is finite, and we then define the truncated G-Euler characteristic of A by

(33)  χ_{t}(G, A) = q(ξ_{A}).

Of course, if the H^{i}(G, A) are finite for all i ⩾ 0, and zero for i ⩾ 2, then the truncated G-Euler characteristic of A coincides with the usual G-Euler characteristic of A. However, we shall be interested in G-modules A, which arise naturally in arithmetic, and for which we can often show that the truncated Euler characteristic is finite and compute it, without being able to prove anything about the H^{i}(G, A) for i ⩾ 2. In fact (see the remarks immediately after Theorem 2.2), it is conjectured that in the arithmetic example we consider when A = S(E/F_{∞}), we always have H^{i}(G, A) = 0 for i ⩾ 2. If v is a finite prime of F, we write L_{v}(E, s) for the Euler factor at v of the complex L-function of E over F. In particular, when E has bad reduction (the only case we shall need) L_{v}(E, s) is 1, (1 − (Nv)^{−s})^{−1}, and (1 + (Nv)^{−s})^{−1}, according as E has additive, split multiplicative, or non-split multiplicative reduction at v. Also, if u and v are non-zero elements of ℚ_{p}, u ∼ v means that u/v is a p-adic unit. The following is the main result of this section. As usual, j_{E} denotes the j-invariant of the elliptic curve E.

Theorem 3.1 Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all places v of F dividing p, and (iii) X(E/F^{cyc}) is Λ(Γ)-torsion. Then χ_{t}(G, S(E/F_{∞})) is finite if and only if χ(Γ, S(E/F^{cyc})) is finite. Moreover, when χ(Γ, S(E/F^{cyc})) is finite, we have

χ_{t}(G, S(E/F_{∞})) ∼ χ(Γ, S(E/F^{cyc})) × ∏_{v∈M} L_{v}(E, 1)^{−1},

where M consists of all places v of F where the j-invariant of E is non-integral,

The following is a special case of Theorem 3.1 (see [2, Theorem 1.15] for a weaker result in this direction). Assuming hypotheses (i) and (ii) of Theorem 3.1, it is well-known (see [6] or Greenberg [10]) that both X(E/F^{cyc}) is Λ(Γ)- torsion and χ(Γ, S(E/F^{cyc})) is finite when S(E/F) is finite.

Corollary 3.2 Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all places v of F dividing p, and (iii) S(E/F) is finite. Then χ_{t}(G, S(E/F_{∞})) is finite and

χ_{t}(G, S(E/F_{∞})) ∼ χ(Γ, S(E/F^{cyc})) × ∏_{v∈M} L_{v}(E, 1)^{−1},

We now give the proof of Theorem 3.1 via a series of lemmas. For these lemmas, we assume that hypotheses (i), (ii) and (iii) of Theorem 3.1 are valid. We let S′ be the subset of S consisting of all places v of F such that ord_{v}(j_{E}) < 0. We then define

(34)  S′(E/F^{cyc}) = Ker(H^{1}(G_{S}(F^{cyc}), E_{p^{∞}}) → ⨁_{v∈S\S′} J_{v}(F^{cyc})).

We remark that we could also define S′(E/F_{∞}) analogously, but in fact S′(E/F_{∞}) = S(E/F_{∞}) as J_{v}(F_{∞}) = 0 for v in S′ (see [2, Lemma 3.3]).

Lemma 3.3 We have the exact sequence of Γ-modules

0 → S(E/F^{cyc}) → S′(E/F^{cyc}) → ⨁_{v∈S′} J_{v}(F^{cyc}) → 0.

Proof This is clear from the commutative diagram with exact rows (cf. Lemma 2.1)

\xymatrix{0 \ar[r] & S(E/F^{cyc}) \ar[r] \ar[d] & H^{1}(G_{S}(F^{cyc}), E_{p^{∞}}) \ar[r]^{λ_{S}(F^{cyc})} \ar[d] & ⨁_{v∈S} J_{v}(F^{cyc}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & S′(E/F^{cyc}) \ar[r] & H^{1}(G_{S}(F^{cyc}), E_{p^{∞}}) \ar[r] & ⨁_{v∈S\S′} J_{v}(F^{cyc}) \ar[r] & 0}

where all the vertical arrows are the natural maps, the first is the natural inclusion, the middle map is the identity and the right vertical map is the natural projection. □

Lemma 3.4 We have that χ(Γ, S′(E/F^{cyc})) is finite if and only if χ(Γ, S(E/F^{cyc})) is finite. Moreover, when χ(Γ, S(E/F^{cyc})) is finite, we have

χ(Γ, S′(E/F^{cyc})) ∼ χ(Γ, S(E/F^{cyc})) × ∏_{v∈M} L_{v}(E, 1)^{−1}

and M is the set of places with non-integral j-invariant as before.

Proof The lemma is very well known (see, for example, [4, Lemma 5.6 and Corollary 5.8] for a special case), and so we only sketch the proof. Indeed, by the multiplicativity of the Euler characteristic in exact sequences, we see that all follows from Lemma 3.3 and the fact that

χ(Γ, ⨁_{v∈S′} J_{v}(F^{cyc})) ∼ ∏_{v∈M} L_{v}(E, 1)^{−1},

(see [4, Lemmas 5.6 and 5.11]). □

Lemma 3.5 Let f : A → B be a homomorphism of p-primary Γ-modules with both Ker(f) and Coker(f) finite. If

g : A^{Γ} → B^{Γ}, h : A_{Γ} → B_{Γ}

denote the two maps induced by f, then q(g) and q(h) are finite, and q(g) = q(h).

Proof This follows easily from breaking up the commutative diagram

0 ⟶ A^{Γ} ⟶ A ⟶^{γ−1} A ⟶ A_{Γ} ⟶ 0
g↓  ^{f}↓  ^{f}↓  ^{h}↓
0 ⟶ B^{Γ} ⟶ B ⟶^{γ−1} B ⟶ B_{Γ} ⟶ 0,

where γ is a topological generator of Γ, into two commutative diagrams of short exact sequences and applying the snake lemma to each of these diagrams. □

The heart of the proof of Theorem 3.1 is to apply Lemma 3.5 to the map

(35)  f : S′(E/F^{cyc}) → S(E/F_{∞})^{H}

given by the restriction. We therefore need

Lemma 3.6 If f is given by (35), both Ker(f) and Coker(f) are finite.

Proof Put S′′ = S \ S′. As J_{v}(F_{∞}) = 0 for v ∈ S′, we have the commutative diagram with exact rows

\xymatrix{0 \ar[r] & S(E/F_{∞})^{H} \ar[r] & H^{1}(G_{S}(F_{∞}), E_{p^{∞}})^{H} \ar[r] & ⨁_{v∈S′′} J_{v}(F_{∞})^{H} \\ 0 \ar[r] & S′(E/F^{cyc}) \ar[r] \ar[u]^{f} & H^{1}(G_{S}(F^{cyc}), E_{p^{∞}}) \ar[r] \ar[u] & ⨁_{v∈S′′} J_{v}(F^{cyc}) \ar[r] \ar[u]^{γ′′_{S}(F^{cyc})} & 0.}

Now it is known that H^{i}(H, E_{p^{∞}}) is finite for all i ≥ 0 (see [6, Appendix A.2.6]). Hence Lemma 3.6 will follow from this diagram provided we show that the kernel of γ_{S′′}(F^{cyc}) is finite (it is here that we will use the fact that S′′ contains no place of potential multiplicative reduction for E). Now

Ker(γ_{S′′}(F^{cyc})) = ⨁_{w} Ker(γ_{w}(F^{cyc})),

where

Ker(γ_{w}(F^{cyc})) = H^{1}(Ω_{w}, E(F_{∞,w}))(p);

here w runs over all primes of F^{cyc} lying above primes in S′′, and Ω_{w} denotes the decomposition group in H of some fixed prime of F_{∞} above w. Then (see [2, Lemma 3.7]) we have

(36)  Ker(γ_{w}(F^{cyc})) = H^{1}(Ω_{w}, E_{p^{∞}}).

But E has potential good reduction at w, and F_{w}^{cyc} contains the unique unramified ℤ_{p} extension of F_{w}. Hence, as p ≥ 5, it follows from Serre-Tate [20] that Ω_{w} must be a finite group of order prime to p, and thus (36) shows that γ_{w}(F^{cyc}) is injective in this case. Suppose next that w does divide p. Then it follows from the results of [3] that

(37)  Ker(γ_{w}(F^{cyc})) = H^{1}(Ω_{w}, Ẽ_{w,p^{∞}}),

where Ẽ_{w,p^{∞}} denotes the image of E_{p^{∞}} under reduction modulo w. But it is known (see either [8] or [4, Lemma 5.25]) that the cohomology groups H^{i}(Ω_{w}, Ẽ_{w,p^{∞}}) are finite for all i ≥ 0. Hence Ker(γ_{w}(F^{cyc})) is finite, and the proof of Lemma 3.6 is complete. □

We can now finish the proof of Theorem 3.1. Consider the Γ-modules

A = S′(E/F^{cyc}), B = S(E/F_{∞})^{H},

and the map given by (35). By Lemma 2.7, we have

H^{0}(Γ, B) = H^{0}(G, S(E/F_{∞})), H^{1}(Γ, B) = H^{1}(G, S(E/F_{∞})),

and we clearly have the commutative diagram

H^{0}(Γ, A) ⟶^{g} H^{0}(Γ, B) = H^{0}(G, S(E/F_{∞}))
ϕ_{A}↓  ^{ξ_{B}} ↓
H^{1}(Γ, A) ⟶^{h} H^{1}(Γ, B) = H^{1}(G, S(E/F_{∞})),

here g and h are induced by f, and ϕ_{A} and ξ_{B} are defined by (30) and (32), respectively. By Lemmas 3.5 and 3.6, we know that q(g) and q(h) are finite, whence it is plain from the diagram that q(ϕ_{A}) is finite if and only if q(ξ_{B}) is finite. Now it is a basic property of the q-function, that q of a composition of two maps is the product of the q′s of the individual maps. Hence, as ξ_{B} ∘g = ϕ_{A} ∘h, we conclude that q(ξ_{B})q(g) = q(ϕ_{A})q(h), when q(ϕ_{A}) or equivalently q(ξ_{B}) is finite. But q(g) = q(h) by Lemma 3.5, and so q(ϕ_{A}) = q(ξ_{B}). Applying Lemma 3.4 to compute q(ϕ_{A}) in terms of χ(Γ, S(E/F^{cyc})), the proof of Theorem 3.1 is now complete. □

We now briefly recall the value of χ(Γ, S(E/F^{cyc})) determined by Perrin-Riou [18] and Schneider [19], and use Theorem 3.1 to discuss several numerical examples. We need the following notation. If A is an abelian group, A(p) will denote its p-primary subgroup. If w is any finite prime of F, we put c_{w} = [E(F_{w}) : E_{0}(F_{w})], where E_{0}(F_{w}) denotes the subgroup of points in E(F_{w}) with non-singular reduction modulo w. Put

τ_{p}(E) = |Π_{w}c_{w}|_{p}^{−1},

where the p-adic valuation is normalized so that |p|_{p} = p^{−1}. For each place v of F dividing p, let k_{v} be the residue field of v, and let Ẽ_{v} over k_{v} be the reduction of E modulo v. We say that a prime v of F dividing p is anomalous if Ẽ_{v}(k_{v})(p) ≠ 0. We always continue to assume that p ⩾ 5, and that E has good ordinary reduction at all primes v of F dividing p. Write � (E/F) for the Tate-Shafarevich group of E over F.

Case 1. We assume that E(F) is finite, and � (E/F)(p) is finite, or equivalently S(E/F) is finite. Then it is shown in [18], [19] that H^{i}(Γ, S(E/F^{cyc})) (i = 0, 1) is finite, and

χ(Γ, S(E/F^{cyc})) = \frac{#(� (E/F)(p))}{#(E(F)(p))^{2}} × τ_{p}(E) × ∏_{v|p} #(Ẽ_{v} (k_{v})(p))^{2}.

Example 1. Take F = ℚ and let E be the elliptic curve

X_{1}(11) : y^{2} + y = x^{3} − x^{2}.

Kolyvagin’s theorem tells us that both E(ℚ) and � (E/ℚ) are finite, since the complex L-function of E does not vanish at s1. The conjecture of Birch and Swinnerton-Dyer predicts that � (E/ℚ) = 0, but the numerical verification of this via Heegner points does not seem to exist in the literature. Now it is well known that S(E/ℚ^{cyc}) = 0 for p = 5, and that S(E/ℚ^{cyc}) = 0 for a good ordinary prime p > 5 provided � (E/ℚ)(p) = 0 (see [6]). Hence, assuming � (E/ℚ)(p) = 0 when p > 5, we conclude that

χ(Γ, S(E/ℚ^{cyc})) = 1

for all good ordinary primes p ⩾ 5. Our knowledge of X(E/F_{∞}) is extremely limited. We know (see [2]) that X(E/F_{∞})⨂_{ℤ_{p}} ℚ_{p} has infinite dimension over ℚ_{p}

for all primes p ⩾ 5. Take S{p, 11}, and note that 11 is the only prime where E has non-integral j-invariant and L_{11}(E, S) = (1 − 11^{−s})^{−1}. First, assume that p = 5. Theorem 3.1 then tells us that

χ_{t}(G, S(E/F_{∞})) = 5.

But it is shown in [4] that the map λ_{S}(G_{∞}) is surjective when p = 5. Hence we have also H^{i}(G, S(E/F_{∞})) = 0 for i = 2, 3, 4 and so we have the stronger result that S(E/F_{∞}) has finite G-Euler characteristic, and χ(G, S(E/F_{∞})) = 5. Next assume that p is a good ordinary prime > 5, and that � (E/ℚ)(p) = 0. Then Theorem 3.1 gives

χ_{t}(G, S(E/F_{∞})) = 1.

However, it has not been proven yet that λ_{S}(F_{∞}) is surjective for a single good ordinary prime p > 5, and so we know nothing about the H^{i}(G, S(E/F_{∞})) for i = 2, 3, 4. But we can deduce a little more from the above evaluation of the truncated Euler characteristic. For E = X_{1}(11), Serre has shown that GGL_{2}(ℤ_{p}) for all p ⩾ 7, whence it follows by a well known argument that H^{i}(G, E_{p^{∞}}) = 0 for all i ⩾ 1. But it is proven in [2] (see Lemmas 4.8 and 4.9) that the order of H^{1}(G, S(E/F_{∞})) must divide the order of H^{3}(G, E_{p^{∞}}). Hence finally we conclude that

H^{0}(G, S(E/F_{∞})) = H^{1}(G, S(E/F_{∞})) = 0

for all good ordinary primes p > 5 such that � (E/ℚ)(p) = 0.

Case 2. We assume now that E(F) has rank g ≥ 1, and that � (E/F)(p) is finite. We write

< , >_{F,p}: E(F) × E(F) → ℚ_{p}

for the canonical p-adic height pairing (see [16], [18], [19]), which exists because of our hypotheses that E has good ordinary reduction at all places v of F dividing p. If P_{1}, ⋯, P_{g} denote a basis of E(F) modulo torsion, we define

R_{p}(E/F) = det < P_{i}, P_{j} > .

It is conjectured that we always have R_{p}(E/F) ≠ 0, but this is unknown. Recalling that we are assuming that � (E/F)(p) is finite, the principal result of [18], [19] is that firstly χ(Γ, S(E/F^{cyc})) is finite if and only if R_{p}(E/F) ≠ 0, and secondly, when R_{p}(E/F) ≠ 0, we have

χ(Γ, S(E/F^{cyc})) = p^{−g} | ρ_{p} |_{p}^{−1} .

where

ρ_{p} = \frac{R_{p}(E/F) × #(� (E/F)(p))}{♯(E(F)(p))^{2}} × τ_{p}(E) × ∏_{v|p} #(Ẽ_{v}(k_{v})(p))^{2}.

Documenta Mathematica · Extra Volume Kato (2003) 187–215

The work of Mazur and Tate [16] gives a very efficient numerical method for calculating R_{p}(E/F) up to a p-adic unit, and so the right hand side of this formula can be easily computed in simple examples, provided we know the order of � (E/F)(p).

Example 2. Take F = ℚ, and let E be the elliptic curve of conductor 37.

E : y^{2} + y = x^{3} − x.

It is well-known that E(ℚ) is a free abelian group of rank 1 generated by P = (0, 0). Moreover, the complex L-function of E over ℚ has a simple zero at s = 1, and so � (E/ℚ) is finite by Kolyvagin’s theorem. In fact, the conjecture of Birch and Swinnerton-Dyer predicts that � (E/ℚ) = 0, but again the numerical verification of this via Heegner points does not seem to exist in the literature. We put h_{p}(P) =< P, P >_{ℚ,p}. The supersingular primes for E less than 500 are 2, 3, 17, 19, 257, 311. The ordinary primes for E less than 500 which are anamolous are 53, 127, 443. The following values for h_{p}(P) for p ordinary with p < 500 have been calculated by C. Wuthrich. We have

| h_{p}(P) |_{p}= p^{−2} for p = 13, 67 and | h_{p}(P) |_{p}= p for p = 53, 127, 443,

and | h_{p}(P) |_{p}= p^{−1} for the remaining primes. As c_{37} = 1, we conclude from the above formula that χ(Γ, S(E/ℚ^{cyc})) = 1 for all ordinary primes p < 500, with the exception of p = 13, 67, where we have χ(Γ, S(E/ℚ^{cyc})) = p; here we are assuming that � (E/ℚ)(p) = 0. Now 37 is the only prime where the jinvariant of E is non-integral, and L_{37}(E, s) = (1+37^{−s})^{−1}. Hence we conclude from Theorem 3.1 that χ_{t}(G, S(E/F_{∞})) is finite and

χ_{t}(G, S(E/F_{∞})) = χ(Γ, S(E/ℚ^{cyc})).

We point out that we do not know that λ_{S}(F_{∞}) is surjective for a single ordinary prime p for this curve.

4 An algebraic invariant

In this last section, we attach a new invariant to a wide class of modules over the Iwasawa algebra of a compact p-adic Lie group G satisfying the following conditions: (i) G is pro-p, (ii) G has no element of order p, and (iii) G has a closed normal subgroup H such that G/H is isomorphic to ℤ_{p}. We always put

(38)  Γ = G/H,

and write Q(Γ) for the quotient field of the Iwasawa algebra Λ(Γ) of Γ. We assume that G satisfies these hypotheses for the rest of this section. By [14], G has finite p-cohomological dimension, which is equal to the dimension d of G as a p-adic Lie group. We are interested in the following full subcategory of the category of all finitely generated torsion Λ(G)-modules. Let 𝔐_{H}(G) denote the category whose objects are all Λ(G)-modules which are finitely generated over Λ(H) (such modules are automatically torsion Λ(G)-modules, because Λ(G) is not finitely generated as a Λ(H)-module). Perhaps somewhat unexpectedly, it is shown in [4] that many Λ(G)-modules which arise in the GL_{2}-Iwasawa theory of elliptic curves belong to 𝔐_{H}(G) (specifically, in the notation of §2, X(E/F_{∞}) is a Λ(G)-module in 𝔐_{H}(G) when we assume that (i) p ⩾ 5, (ii) E has good ordinary reduction at all primes v of F dividing p, (iii) G is pro-p, and (iv) X(E/F^{cyc}) is a finitely generated ℤ_{p}-module; here H = G(F_{∞}/F^{cyc})).

If f and g are any two non-zero elements of Q(Γ), we write f ∼ g if fg^{−1} is a unit in Λ(Γ). To each M is 𝔐_{H}(G), we now attach a non-zero element f_{M} of Q(Γ), which is canonical in the sense that it is well-defined up to the equivalence relation ∼. As M is a finitely generated Λ(H)-module, all of the homology groups

(39)  H_{i}(H, M)  (i ⩾ 0)

are finitely generated ℤ_{p}-modules. On the other hand, as M is a Λ(G)-module, these homology groups have a natural structure as Λ(Γ)-modules, and they must therefore be torsion Λ(Γ)-modules. Let g_{i} in Λ(Γ) be a characteristic power series for the Λ(Γ)-module H_{i}(H, M), and define

(40)  f_{M} = ∏_{i⩾0} g_{i}^{(−1)^{i}}.

This product is, of course, finite because H_{i}(H, M) = 0 for i ⩾ d, and it is well defined up to ∼, because each g_{i} is well defined up to multiplication by a unit in Λ(Γ).

Lemma 4.1 (i) If M(p) denotes the submodule of M in 𝔐_{H}(G) consisting of all elements of p-power order, then f_{M} ∼ f_{M/M(p)}; (ii) If we have an exact sequence of modules in 𝔐_{H}(G),

(41)  0 → M_{1} → M_{2} → M_{3} → 0,

then f_{M_{2}} ∼ f_{M_{1}} · f_{M_{3}}.

Proof (i) is plain from the long exact sequence of H-homology derived from the exact sequence

0 → M(p) → M → M/M(p) → 0,

and the fact that the H_{i}(H, M(p))(i ⩾ 0) are killed by any power of p which annihilates M(p), and are therefore finite. Similarly, (ii) follows from the exact sequence of H-homology derived from (41), and the classical fact that the Λ(Γ)-characteristic series is multiplicative in exact sequences. □

One of the main reasons we are interested in the invariant f_{M} is its link with the G-Euler characteristic of M. If g is any element of Λ(Γ), we write g(0) as usual for the image of g under the augmentation map from Λ(Γ) to ℤ_{p}. Similarly, if g is any non-zero element of Q(Γ), then the fact that Λ(Γ) is a unique factorization domain allows us to write g = h_{1}/h_{2}, where h_{1} and h_{2} are relatively prime elements of Λ(Γ). We say that g(0) is defined if h_{2}(0) ≠ 0, and then put g(0) = h_{1}(0)/h_{2}(0). If M ∈ 𝔐_{H}(G), we say that M has finite G-Euler characteristic if the H_{i}(G, M) are finite for all i ⩾ 0, and, when this is the case, we define

(42)  χ(G, M) = ∏_{i⩾0} #(H_{i}(G, M))^{(−1)^{i}}.

Lemma 4.2 Assume that M in 𝔐_{H}(G) has finite G-Euler characteristic. Then f_{M}(0) is defined and non-zero, and we have

(43)  χ(G, M) = |f_{M}(0)|_{p}^{−1}.

Proof As Γ has cohomological dimension 1, for all i ⩾ 1 the Hochschild-Serre spectral sequence yields an exact sequence

(44)  0 → H_{0}(Γ, H_{i}(H, M)) → H_{i}(G, M) → H_{1}(Γ, H_{i−1}(H, M)) → 0.

The finiteness of the H_{i}(G, M) (i ⩾ 0) therefore implies that, for all i ⩾ 0, we have g_{i}(0) ≠ 0, where, as above, g_{i} denotes a characteristic power series for H_{i}(H, M). Moreover, by a classical formula for Λ(Γ)-modules, we then have

(45)  |g_{i}(0)|_{p}^{−1} = \frac{#(H_{0}(Γ, H_{i}(H, M)))}{#(H_{1}(Γ, H_{i}(H, M)))}

for all i ⩾ 0. The formula (43) is now plain from (44) and (45), and the proof of the lemma is complete. □

Let

(46)  π_{Γ} : Λ(G) → Λ(Γ)

be the natural ring homomorphism.

Lemma 4.3 Let g be a non-zero element of Λ(G) such that N = Λ(G)/Λ(G)g belongs to 𝔐_{H}(G). Then H_{i}(H, N) = 0 for all i > 0. Moreover, f_{N} ∼ π_{Γ}(g) lies in Λ(Γ), and f_{N}(0) ≠ 0 if and only if N has finite G-Euler characteristic.

Proof We have the exact sequence of Λ(G)-modules

(47)  0 → Λ(G) ⟶^{·g} Λ(G) → N → 0,

where ·g denotes multiplication on the right by g. Since H_{i}(G, Λ(G)) = 0 for i ⩾ 1, we conclude that H_{i}(G, N) = 0 for i > 1. But, for any compact Λ(G)-module R, the Hochschild-Serre spectral sequence provides an injection of H_{0}(Γ, H_{i}(H, R)) into H_{i}(G, R), and so the vanishing of H_{i}(G, R) implies the vanishing of the compact Γ-module H_{i}(H, R). It follows that H_{i}(H, Λ(G)) = 0 for i ⩾ 1, and that H_{i}(H, N) = 0 for i > 1. Put h = π_{Γ}(g). Taking H-homology of (47). we obtain the exact sequence of Λ(Γ)-modules

(48)  0 → H_{1}(H, N) → Λ(Γ) ⟶^{·h} Λ(Γ) → H_{0}(H, N) → 0,

where ·h now denotes right multiplication by h. Note that H_{0}(H, N) is Λ(Γ)- torsion as N is in 𝔐_{H}(G), and so we must have h ≠ 0. But then multiplication by h in Λ(Γ) is injective, and so H_{1}(H, N) = 0, and it is then clear than f_{N} ∼ h. Finally, returning to the G-homology of (47), we obtain the exact sequence

(49)  0 → H_{1}(G, N) → ℤ_{p} ⟶^{·h(0)} ℤ_{p} → H_{0}(G, N) → 0,

and so N has finite G-Euler characteristic if and only if h(0) ≠ 0. This completes the proof of Lemma 4.3. □

We recall (see [22]) that a finitely generated torsion Λ(G)-module M is defined to be pseudo-null if Ext_{Λ(G)}^{1}(M, Λ(G)) = 0. If M lies in 𝔐_{H}(G), it is an important fact that M is pseudo-null as a Λ(G)-module if and only if M is Λ(H)-torsion. Since G as an extension of Γ by H necessarily splits, and hence is a semi-direct product, this is proven in [23, Prop. 5.4] provided H is uniform. However, by a well-known argument [22, Prop. 2.7], it then follows in general for our G, since H must always contain an open subgroup which is uniform.

Lemma 4.4 Let M be a module in 𝔐_{H}(G). If G is isomorphic to ℤ_{p}^{r}, for some integer r ⩾ 1, then f_{M} ∼ 1 if and only if M is pseudo-null as a Λ(G)-module.

Proof We assume r ≥ 2, since pseudo-null modules are finite when r = 1. Let M be a Λ(H)-torsion module in 𝔐_{H}(G). We recall that f_{M} = ∏_{i⩾0}g_{i}^{(−1)^{i}},

where g_{i} in Λ(Γ) is a characteristic power series for the torsion Λ(Γ)-module H_{i}(H, M). By (i) of Lemma 4.1, we may assume that M has no p-torsion. Thus the assumptions of Lemma 2 of [9] when applied to M as a Λ(H)-module are satisfied, and we conclude that there exists a closed subgroup J of H such that H/J →^{∼} ℤ_{p} and M is finitely generated as a Λ(J)-module. Analogously to (44), the Hochschild-Serre spectral sequence gives rise to the exact sequences of finitely generated torsion Λ(Γ)-modules, for all i ⩾ 1,

(50) 0 → H_{0}(H/J, H_{i}(J, M)) → H_{i}(H, M) → H_{1}(H/J, H_{i−1}(J, M)) → 0.

Hence, if g_{j,i} denotes the characteristic power series of H_{j}(H/J, H_{i}(J, M)) as a torsion Λ(Γ)-module, we obtain

g_{0} ∼ g_{0,0}, g_{i} ∼ g_{1,i−1} · g_{0,i} (i ⩾ 1).

Thus

f_{M} ∼ g_{0,0} × ∏_{i⩾1}(g_{1,i−1} · g_{0,i})^{(−1)^{i}} ∼ ∏_{i⩾o}(g_{0,i}/g_{1,i})^{(−1)^{i}}.

On the other hand, letting \overline{h} denote a topological generator of H/J, we have the exact sequence of Λ(C)-modules

0 → H_{1}(H/J, H_{i}(J, M)) → H_{i}(J, M) ⟶^{\overline{h}−1} H_{i}(J, M) → H_{0}(H/J, H_{i}(J, M)) → 0.

But H_{i}(J, M) is a finitely generated ℤ_{p}-module, because J was chosen so that M is a finitely generated Λ(J)-module. Hence the above exact sequence is in fact an exact sequence of finitely generated torsion Λ(C)-modules. By the multiplicativity of the characteristic power series along exact sequences, it follows that g_{0,i} ∼ g_{1,i}. Conversely, let f_{M} ∼ 1. Under our hypotheses on G, the Iwasawa algebra Λ(G) is a commutative regular local ring. By the classical structure theorem for finitely generated modules, M is pseudo-isomorphic to 𝔈(M) where

𝔈(M) = ⨁_{i=1}^{m} Λ(G)/Λ(G)g_{i}

with g_{i} non-zero for i = 1, ⋯, m. By Lemma 4.3 and the first part of the proof above, we see that f_{M} ∼ π_{Γ}(g_{1} ⋯ g_{m}). As f_{M} is assumed to be a unit in Λ(Γ), we conclude that π_{Γ}(g_{1} ⋯ g_{m}) is also a unit in Λ(Γ), and hence π_{Γ}(g_{1} ⋯ g_{m}) does not belong to the maximal ideal of Λ(G). But, as the residue field of Λ(G) is 𝔽_{p}, and Λ(G) is local, we see that g_{1} ⋯ g_{m} is a unit in Λ(G) and hence so is each g_{1} (i = 1, ⋯, m). Thus 𝔈(M) is zero and M is pseudo-null, thereby completing the proof of the lemma. □

Lemma 4.5 Assume that G is a direct product C × H, where C is isomorphic to Γ and H has dimension ≥ 1. Suppose that M in 𝔐_{H}(G) is finitely generated as a ℤ_{p}-module, then f_{M} ∼ 1.

Proof We remark that a finitely generated ℤ_{p}-module M in 𝔐_{H}(G) is necessarily pseudo-null since H has dimension ≥ 1. We fix a topological generator c of C end let g_{i}(T) denote the characteristic polynomial of c − 1 on the finite dimensional vector space H_{i}(H, M) ⊗_{ℤ_{p}} ℚ_{p}. We

will show that

∏_{i⩾0} g_{i}^{(−1)^{(i)}} = 1.

Note first of all that, because M is finitely generated as a ℤ_{p}-module, it is Λ(H)-torsion, and so we have (see [11])

∑_{i⩾0}(−1)^{i} deg(g_{i}) = 0.

We now choose a finite extension L/ℚ_{p} which is a splitting field for all the polynomials g_{i}. Then g_{i} also can be viewed as the characteristic polynomial of c − 1 on the L-vector space

H_{i}(H, M) ⊗_{ℤ_{p}} L = H_{i}(H, M ⊗_{ℤ_{p}} L).

Since our assertion is multiplicative in short exact sequences we may argue by induction with respect to the dimension of M ⊗_{ℤ_{p}} L. As the H-action and the

C-action commute, the H-action preserves all C-eigenspaces. This reduces us to the case where c acts on M ⊗_{ℤ_{p}} L by a single eigenvalue λ. Then c certainly

acts on each H_{i}(H, M) ⊗_{ℤ_{p}} L by the same eigenvalue λ, and we obtain

∏_{i⩾0} g_{i}^{(−1)^{(i)}} = (T − λ)^{∑(−1)^{i} deg(g_{i})} = (T − λ)^{0} = 1.

□

When M is a finitely generated ℤ_{p}-module which has finite G-Euler characteristic, then Lemmas 4.4 and 4.5 prove that χ(G, M) = 1. This is a slightly stronger version of the main theorem of [21] for a group G = C × H as in Lemma 4.5.

The previous two results might lead one to believe that the invariant f_{M} is a unit for all pseudo-null modules M in 𝔐_{H}(G). However, the following two examples illustrate that this is not the case.

Example 3 (see [8], §5). Let K be any finite extension of ℚ_{p} which contains μ_{p} if p is odd and μ_{4} if p = 2. Take E to be any Tate elliptic curve over K, and write

G = G(K(E_{p^{∞}})/K), H = G(K(E_{p^{∞}})/K(μ_{p^{∞}})).

Thus G is a p-adic Lie group of dimension 2, H is a closed normal subgroup of G which is isomorphic to ℤ_{p}, and ΓG/H is isomorphic to ℤ_{p}. If M is a ℤ_{p}-module on which G_{K}-acts, M(n), as usual will denote the n-fold Tate twist of M by T_{p}(μ) = lim_{⟵} μ_{p^{n}}. We now consider the G-module

W = T_{p}(E).

As is shown in [8] (see [8], formula (48)), we then have

H_{0}(H, W) = ℤ_{p}, H_{1}(H, W) = ℤ_{p}(−1).

To compute the corresponding f_{W} in Q(Γ), let χ : G_{K} → ℤ_{p}^{∗} be the cyclotomic character of G_{K}, i.e. σ(ζ) = ζ^{χ(σ)} for all σ in G_{k} and all ζ in μ_{p^{∞}}. Fix a topological generator γ of Γ, and identify Λ(Γ) with the ring ℤ_{p}[[T]] of formal power series in T with coefficients in ℤ_{p} by mapping γ to 1 + T. Then we see immediately that

f_{W} = \frac{T}{T + 1 − χ(γ)^{−1}} .

In particular, f_{W} does not belong to Λ(Γ) in this example. We note also that, since G has dimension > 1, W is certainly a pseudo-null Λ(G)-module because W is a finitely generated ℤ_{p}-module (see [24], Prop. 3.6).

Example 4. We now assume that G = C × H, and we take a Λ(H)-module M of the form

(51)  M = Λ(H)/Λ(H)g,

where g is any non-zero element of Λ(H). To make M into a Λ(G)-module, it suffices to give a continuous action of C on M, which commutes with the H-action. Fix a topological generator c of C. If z is any element of Λ(H), we write z(0) for the image of z under the augmentation map in ℤ_{p}. We now take units w and u in Λ(H) satisfying

(52)  gw = ug, w(0) ≡ u(0) ≡ 1 mod p.

We let c act on M by

(53)  c.(z + Λ(H)g) = zw + Λ(H)g.

This is well-defined by the first condition in (52), and extends to a continuous action of C by the second condition in (52). Conversely, let M be any Λ(G)- module which is of the form (51) as a Λ(H)-module with g ≠ 0. It is then easy to see that the action of c on M is necessarily given by (53), where u and w are units in Λ(H) satisfying (52), because Λ(H) is a local ring. As discussed before Lemma 4.4, this Λ(G)-module M is pseudo-null because it is plainly Λ(H)- torsion. To compute the invariant f_{M} of M, we consider the exact sequence of Λ(G)-modules

(54)  0 → Λ(H) ⟶^{·g} Λ(H) → M → 0,

where c acts on the first copy of Λ(H) by right multiplication by u, and on the second by right multiplication by w. Taking H-coinvariants of (54), we obtain that H_{i}(H, M) = 0 for i ≥ 1, and the exact sequence of Λ(C)-modules

(55)  0 → H_{1}(H, M) → ℤ_{p} ⟶^{.g(0)} ℤ_{p} → H_{0}(H, M) → 0;

here c acts on the first copy of ℤ_{p} by multiplication by u(0), and on the second by multiplication by w(0). Thus

(56)  f_{M}(T) = (T − w(0) + 1)/(T − u(0) + 1).

This is not a unit in Λ(Γ) provided w(0) ≠ u(0). To obtain concrete examples of such modules M with w(0) ≠ u(0), we now assume that H contains a subgroup which is a semi-direct product of the following form. Let H_{0} and C_{0} be two closed subgroups of H which are isomorphic to ℤ_{p}, and which are such that

xhx^{−1} = h^{ϕ(x)} (x ∈ C_{0}, h ∈ H_{0}),

where ϕ : C_{0} ↪ Aut(H_{0}) = ℤ_{p}^{∗} is a continuous injective group homomorphism. We note that this hypothesis is valid for any H which is open in SL_{n}(ℤ_{p}) (n ≥ 2), or more generally for any compact H which is open in the group of ℚ_{p}-points of a split semi-simple algebraic group over ℚ_{p}. Now fix a toplogical generator h_{0} of H_{0}, and any element γ of C_{0} with γ ≠ 1. Define

g = h_{0} − 1, w = γ + p^{r},

where r is any integer ≥ 1. We then have

gw = ug, where u = γ · \frac{h_{0}^{ϕ(γ)} − 1}{h_{0} − 1} + p^{r}.

Hence

w(0) = 1 + p^{r}, u(0) = ϕ(γ) + p^{r},

and so u(0) ≠ w(0) because ϕ is injective. Moreover, as w(0) and u(0) are not equal to 1, we see that M has finite G-Euler characteristic, which by Lemma 4.2 is given by

χ(G, M) = | \frac{ϕ(γ) − 1 + p^{r}}{p^{r}} |_{p} .

This therefore gives a new class of pseudo-null Λ(G)-modules, which are not finitely generated over ℤ_{p}, and which have a non-trivial G-Euler characteristic.

Finally, we interpret this example as a statement about K-theory classes. Writing K_{0}(𝔐_{H}(G)) for the Grothendieck group of 𝔐_{H}(G), it is clear from Lemma 4.1 that the map M ↦ f_{M} induces a homomorphism from K_{0}(𝔐_{H}(G)) to Q(Γ)^{∗}/Λ(Γ)^{∗}. Now let M be any Λ(G)-module of the form (51) as a Λ(H)- module with g ≠ 0. It follows from the computation of f_{M} given by (56) that the class of M in K_{0}(𝔐_{H}(G)) is non-zero.

Let C^{o}(G) denote the abelian category of finitely generated torsion Λ(G)- modules which contains 𝔐_{H}(G) as a full subcategory. Thus there is a natural map

i : K_{0}(𝔐_{H}(G)) → K_{0}(C^{0}(G))

on the Grothendieck groups. Again, let M be a Λ(G)-module of the form (51). Writing [M] for the class of M in K_{0}(𝔐_{H}(G)), we have just seen that [M] is not zero provided w(0) ≠ u(0). However, we now show that i[M] = 0. Let Q(G) denote the skew-field of fractions of Λ(G), and let Ω(G) be the abelian group defined by

(57)  Ω(G) := Q(G)^{∗}/Λ(G)^{∗}[Q(G)^{∗}, Q(G)^{∗}],

where for any ring A, A^{∗} denotes the multiplicative group of units in A and [A^{∗}, A^{∗}] is the commutator subgroup. It is well-known [1] that the localisation sequence yields an isomorphism

ϕ : K_{0}(C^{0}(G)) ≃ Ω(G)

which sends the class of any module of the form (Λ(G)/Λ(G)z), z ≠ 0 in Λ(G), to the class of z in Ω(G). Hence (54) shows that ϕ(i([M])) is the coset of (c − u)^{−1}(c − w). We prove that this coset is trivial in Ω(G). Indeed, since c is in the centre of G, by (52), we get

(58)  g(c − w) = (c − u)g.

Hence in Ω(G), we have

g(c − u)^{−1}(c − w) = g(c − w)(c − u)^{−1}
= (c − u)g(c − u)^{−1} by (58)
= (c − u)(c − u)^{−1}g
= g.

Since Ω(G) is a group, it follows that the class of (c − u)^{−1}(c − w) in Ω(G) is trivial, as required.

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John Coates DPMMS, University of Cambridge Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB, England j.h.coates@dpmms.cam.ac.uk Peter Schneider Mathematisches Institut Westfälische Wilhelms-Universität Einsteinstr. 62 D-48149 Münster, Germany pschnei@math.uni-muenster.de

Ramdorai Sujatha School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400 005, India sujatha@math.tifr.res.in 216

Documenta Mathematica · Extra Volume Kato (2003)