Elliptic curves and module structure over Hopf orders

and

The conjecture of Chinburg-Stark for abelian extensions of a quadratic imaginary field

Werner Bley Institut für Mathematik Universität Augsburg 86135 Augsburg Germany email: bley@math.uni-augsburg.de

December 22, 1997 2

Preface

This habilitation thesis consists of two parts, both originating in the field of Galois module structure.

The first part is concerned with the study of the module structure of rings of integers. My investigations are on the one hand based on methods and results of M. J. Taylor, Ph. Cassou-Noguès and R. Schertz concerning classical Galois module structure of rings of integers over associated orders in the spirit of H. W. Leopoldt, and on the other hand on a more recent approach of M. J. Taylor dealing with the module structure of so-called Kummer orders over explicitly given Hopf orders. It was one of my main motivations to illuminate the relation between both approaches.

The second part is in the first place a contribution to the theory of Artin Lseries. More precisely, we consider a strong version of Stark’s conjecture, due to T. Chinburg, for abelian extensions of a quadratic imaginary number field. The relation to Galois module theory stems from the fact that the validity of this conjecture provides important information about an invariant attached to the Galois structure of the group of S-units, the so-called third Chinburg invariant.

There is still another common feature which connects both parts: the source of the main results is always the analytic theory of elliptic units. Only the interplay between algebraic and analytic methods leads to nice theories with valuable results.

The first part is not written in the style of a research paper. On the contrary, the subject is introduced in a very detailed and essentially self-contained way. For many results, although already in the literature, I give complete proofs; hopefully always with appropriate references. The second part is still detailed, but not nearly as self-contained as the first.

I gratefully acknowledge the help of many people in producing this habilitation thesis. First I wish to thank J. Ritter and R. Schertz for their constant support during the last seven years. I am also grateful to J. Ritter and A. Weiss for suggesting to me to work on the conjecture of Chinburg-Stark. Special thanks go to R. Boltje for many fruitful discussions and also for his consent to include a joint paper in the first part.

Contents

I Elliptic curves and module structure over Hopf orders 6

1 Introduction 6

2 Algebra 12 2.1 Forms of group algebras . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Indices and Hopf structures . . . . . . . . . . . . . . . . . . . . . 18 2.4 Some A-modules and resolvents . . . . . . . . . . . . . . . . . . . 26

3 The cyclotomic case 32 3.1 First proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Second proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Composite results . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Relative Lubin-Tate formal groups 39

5 Elliptic curves 50 5.1 A special class of elliptic curves . . . . . . . . . . . . . . . . . . . 50 5.2 Modules associated with elliptic curves . . . . . . . . . . . . . . . 56 5.3 An analytic resolvent formula à la Schertz . . . . . . . . . . . . . 58 5.4 Special values of the φ-function . . . . . . . . . . . . . . . . . . . 61 5.5 Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.6 Elliptic curves and Lubin-Tate theory . . . . . . . . . . . . . . . 70 5.7 Module structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Kummer orders and Taylor’s conjecture 82

II The conjecture of Chinburg-Stark for abelian extensions of a quadratic imaginary field 87

1 Introduction 87

2 Preliminary results 91

3 Some reductions 94

4 Elliptic units 96

5 The Ramachandra unit lattice 104

6 A unit lattice à la Hasse 111 7 Proofs of the main results 117 8 Galois module structure 125 References 128

Part I

Elliptic curves and module structure over Hopf orders

1 Introduction

In [T2], M. J. Taylor initiated the study of the Galois module structure of certain Kummer orders associated to torsion points on an abelian variety. In [BB] we axiomatized the situation considered by Taylor and applied our results to investigate the module structure of the full ring of integers of certain algebras obtained by dividing torsion points with respect to the group law of a relative Lubin-Tate formal group.

Besides the examples arising from Lubin-Tate theory, there is a variety of natural examples for the general situation considered in [BB]. These notes are intended to discuss some of them, in particular we will deal with examples associated to elliptic curves.

This paper is also an attempt to understand the connections between earlier work of M. J. Taylor and Ph. Cassou-Noguès [CT], R. Schertz [Sch2] and others concerning the Galois module structure of rings of integers over their associated orders for certain wildly ramified relative ray class field extensions of a quadratic imaginary number field, and the more recent work of M. J. Taylor and A. Srivastav [T2], [ST] on the module structure of Kummer orders over certain explicitly given Hopf orders.

We assume the following situation:

• O is a Dedekind domain of characteristic 0, K its field of fractions, K̄ an algebraic closure of K, and Ω := Gal(K̄/K) the absolute Galois group of K.

• G is a finite group on which Ω acts from the left.

• Γ is a finite set on which G and Ω act from the left such that G acts simply transitively (thus |G| = |Γ|) and ^{ω}(^{g}γ) = ^{(^{ω}g)}(^{ω}γ) for all ω ∈ Ω, g ∈ G, and γ ∈ Γ (the last condition just means that the semidirect product G⋊Ω acts on Γ).

We always write the action of ω ∈ Ω on G and Γ exponentially from the left (as in ^{ω}g and ^{ω}γ for g ∈ G and γ ∈ Γ) and its action on K̄ as ω(x) for x ∈ K̄. For any intermediate field K ⊆ L ⊆ K̄, we set Ω_{L} := Gal(K̄/L) ⩽ Ω, and denote by O_{L} the integral closure of O in L.

The motivation for considering this general algebraic context stems from the observation that many natural examples, which are already discussed in the literature, fit into this framework.

Examples 1.1 For any field of characteristic 0 let μ_{n}, n ∈ ℕ, denote the multiplicative group of n-th roots of unity in its algebraic closure.

(a) Let r, m ∈ ℕ and set K := ℚ(μ_{r}), G := μ_{m}. Fix a primitive r-th root of unity β ∈ μ_{r}, and set Γ := {x ∈ K̄ | x^{m} = β}. Then Γ ⊆ μ_{r+m} and G acts on Γ by multiplication. This very explicit example will be discussed in detail in Section 3. It will serve as motivation and guideline for the more involved applications to elliptic curves.

(b) More generally, let K be a number field, let m ∈ ℕ and G := μ_{m}. For β ∈ K^{×} set Γ := {x ∈ K̄ | x^{m} = β}. Then G acts on Γ by multiplication.

(c) For arbitrary K let L/K be a finite Galois extension with Galois group G, and let Ω act on G by conjugation. Moreover, let γ_{0} ∈ L be a primitive element (L = K(γ_{0})), and let Γ be the set of Galois conjugates of γ_{0}.

(d) Let E/F be an elliptic curve with complex multiplication by O_{k}, where F is a finite extension of a quadratic imaginary number field k. For any integral ideal 𝔞 ⊆ O_{k} write E[𝔞] for the subgroup of points of E(F̄) that are annihilated by all elements a ∈ 𝔞. For x ∈ O_{k} let [x] ∈ End(E) be the corresponding endomorphism of E. Let, for simplicity, (a) = 𝔞 ⊆ O_{k} be a principal ideal and set G := E[𝔞]. For P ∈ E(F̄) define K := F (P) and Γ := {Q ∈ E(F̄) | [a](Q) = P }. Then G acts on Γ by translation.

This kind of example is extensively studied in [Ag], [ST] and [T2]. It is also the main topic of this part (see Section 5).

(e) Let ℚ_{p} ⊆ F be a finite field extension of the field of p-adic numbers, and let 𝔭_{F} = (π) be the maximal ideal of the ring O_{F} of integers in F. Let F be a Lubin-Tate formal group attached to a Lubin-Tate power series f(X) ∈ O_{F} [[X]]. Let [−]: O_{F} → End(F) be the usual ring isomorphism with [π](X) = f(X). For n ∈ ℕ set G_{n} := {x ∈ 𝔭F̄ | [π^{n}](x) = 0}, the subgroup of π^{n}-torsion points in the O_{F} -module 𝔭F̄ endowed with the F-group law and the O_{F} -action ax := [a](x) for a ∈ O_{F} and x ∈ 𝔭F̄, and let F_{n} := F (G_{n}) denote the field obtained by adjoining the elements of G_{n}. For fixed r, m ∈ ℕ_{0}, m ≥ 1, set K := F_{r}, G := G_{m}, choose β ∈ G_{r}, and set Γ := {x ∈ 𝔭F̄ | [π^{m}](x) = β} ⊆ G_{m+r}. Then G acts on Γ by translation.

For more details about this example in the case m ≤ r see [CT, Ch. X] and [T1]. The general case is dealt with in [BB].

We return to the situation described at the beginning and define

A := K̄G , B := Map(G, K̄) , C := Map(Γ, K̄) ,

the group algebra of G over K̄, the set of maps from G and from Γ to K̄, respectively. Then A is a K̄-Hopf algebra, B is its K̄-dual Hopf algebra, C is a K̄-algebra and a right A-module by the K̄-linear extension of the action (f · g)(γ) := f(^{g}γ) of G on C, where g ∈ G, f ∈ C, and γ ∈ Γ.

Note that Ω acts on A, B, and C by

ω(∑_{g∈G} λ_{g}g) := ∑_{g∈G} ω(λ_{g})^{ω}g , (^{ω}b)(g) := ω(b(^{ω^{−1}}g)) , (^{ω}f)(γ) := ω(f(^{ω^{−1}}γ)) ,

for ω ∈ Ω, λ_{g} ∈ K̄, b ∈ B, g ∈ G, f ∈ C, and γ ∈ Γ. This is an action via K-Hopf algebra automorphisms on A and B, and via K-algebra automorphisms on C. Hence, we may take fixed points with respect to the subgroup Ω_{L} = Gal(K̄/L) for any intermediate field K ⊆ L ⊆ K̄:

A_{L} := (K̄G)^{Ω_{L}} , B_{L} := Map(G, K̄)^{Ω_{L}} , C_{L} := Map(Γ, K̄)^{Ω_{L}} .

We omit the index L for L = K. Then A_{L} and B_{L} are L-Hopf subalgebras of A and B, and they are L-dual to each other. Moreover, C_{L} is an L-algebra and an A_{L}-module by restriction of the A-action on C. In Proposition 2.12 we show that C_{L} together with its A_{L}-module structure is a Galois object in the sense of [CS], and, if G is abelian, that C_{L} is a free A_{L}-module of rank 1. In the abelian case, we also define a resolvent (f, χ) ∈ K̄, for f ∈ C and χ ∈ Ĝ := Hom(G, K̄^{×}), and show that, for given f ∈ C_{L}, one has f · A_{L} = C_{L} if and only if (f, χ) ≠ 0 for all χ ∈ Ĝ, see Proposition 2.13.

For any intermediate field K ⊆ L ⊆ K̄, the L-algebras B_{L} and C_{L} are commutative. Let us assume that G is abelian. Then also A_{L} is commutative, and we may define the maximal O_{L}-orders

A_{L} , B_{L} = Map(G, OK̄)^{Ω_{L}}, C_{L} := Map(Γ, OK̄)^{Ω_{L}} ,

of A_{L}, B_{L}, and C_{L}, respectively. Moreover, for arbitrary G, set

A_{L}^{∘} := (OK̄G)^{Ω_{L}} ,

which is an O_{L}-order of A_{L}. Then, C_{L} is an A_{L}^{∘}-module and we may define the associated order

A_{L}^{ass} := {a ∈ A_{L} | C_{L} · a ⊆ C_{L}}

of C_{L} in A_{L}, which contains A_{L}^{∘}. If G is abelian, then we have inclusions

A_{L}^{∘} ⊆ A_{L}^{ass} ⊆ A_{L} .

Again , we omit the index L in A_{L}, B_{L}, C_{L}, A_{L}^{∘}, A_{L}^{ass}, if L = K.

We henceforth assume that G is abelian. Then C_{L} is a free A_{L}-module of rank one (see Proposition 2.12). This can be viewed as the analogue of the classical Normal Basis Theorem, which states that for a Galois extension N/M the MGal(N/M)-module N is free of rank one. As in the classical situation we may then ask for an integral analogue of the Normal Basis Theorem. As already mentioned, A_{L}^{∘} acts on C_{L}; but if our aim is to prove that C_{L} is free (or at least locally free) over some order Ã_{L} in A_{L}, then we have to study C_{L} as a module over A_{L}^{ass}. In fact, if C_{L} = f ·Ã_{L} for f ∈ C_{L}, then the definition of the associated order immediately implies A_{L}^{ass} = Ã_{L}. Since associated orders are determined locally, we can only hope for local freeness, if we consider C_{L} over A_{L}^{ass}.

To emphasize the analogy to classical Galois module structure theory we recall some well-known results in this context. Let, for the moment, N/M be a Galois extension of number fields with group G. The Normal Basis Theorem suggests one seek an integral normal basis. But by a theorem of E. Noether we know that O_{N} is locally free over O_{M}G, if and only if N/M is at most tamely ramified. If M = ℚ and N/M is abelian and tame, then the Theorem of Hilbert-Speiser asserts that O_{N} is free of rank one over ℤG. By work of Leopoldt [Leo] the following generalization of the Hilbert-Speiser Theorem holds: for an abelian extension N/ℚ the ring of integers O_{N} is free over its associated order in ℚG.

In the relative case, when ℚ is replaced by some number field M, there is a beautiful theory due to A. Fröhlich and M. J. Taylor for tamely ramified extensions N/M (see [Fr2]). If we allow wild ramification the situation is very different. Even the local situation is far from being well understood. In particular there is no analogue of E. Noether’s theorem and there are very few criteria for determining whether O_{N} is locally free over its associated order.

It is therefore important to look for interesting classes of wildly ramified extensions with known Galois structure. Some of the most interesting examples in this context are certain relative ray class field extensions N/M of a quadratic imaginary number field k. There are results due to Cassou-Noguès, Schertz, Taylor and others, stating that O_{N} is free over its associated order in MG, see e.g. [CT, Ch. XI], [Sch2]. We shall see that these examples are essentially special cases of Example 1.1(d).

In order to sketch the connection of our approach and classical Galois module theory we return to the situation described at the beginning.

Remarks 1.2 (a) Suppose that the action of Ω on Γ is transitive. Fix an element γ_{0} and denote its stabilizer by Ω_{L}, corresponding to an intermediate field K ⊆ L ⊆ K̄. Then one has a bijection Ω/Ω_{L} ≃ G, where ωΩ_{L} is mapped to g ∈ G, if ^{ω}γ_{0} = ^{g}γ_{0}. Moreover, associating to f ∈ C the value f(γ_{0}) ∈ K̄, defines isomorphisms C ≃ L and C ≃ O_{L} of L- (resp. O_{L}-) algebras.

(b) The absolute Galois group Ω acts trivially on G if and only if A = KG.

If one assumes suitable Kummer conditions in Examples 1.1 (a), (d) and (e), then A = KG. Moreover, in many interesting examples Ω acts transitively on Γ and, in the notation of Remark 1.2 (a), the fixed field L of the stabilizer stab_{Ω}(γ_{0}) is an abelian Galois extension of K such that the bijection Gal(L/K) = Ω/Ω_{L} ≃ G is a group isomorphism (see e.g. Lemma 4.1). In this case, the A-module structure of C corresponds to the KG-module structure of L, and we are in the classical situation of Galois module structure theory.

To conclude this introductory section we outline how the rest of this part is arranged and we describe the main results. Section 2 is taken almost unchanged from [BB]. Since it describes the basic algebraic set-up and develops results which are needed in all of the applications, it had to be included for the reader’s convenience. The sections 2.1 and 2.2 summarize properties of the Hopf algebras A_{L} and B_{L}, respectively, for intermediate fields K ⊆ L ⊆ K̄. Most of these properties can already be found in [T2] without proofs. For the reader’s convenience and for later use we provide proofs of all assertions. In Section 2.3 we show that A_{L}^{∘} (resp. A_{L} if G is abelian) is an O_{L}-Hopf order in A_{L} if and only if the discriminant ideals d_{B_{L}} of B_{L} (resp. d_{A_{L}} of A_{L}) over O_{L} are trivial, see Proposition 2.8 (resp.Proposition 2.10). Thus, the structural information of being a Hopf order is completely described by an arithmetic invariant, the discriminant. Moreover, we compute the index ideal [A_{L} : A_{L}^{∘}]_{O_{L}} in terms of d_{A_{L}} and d_{B_{L}}, see Lemma 2.6. Section 2.4 is concerned with the L-algebra structure and the A_{L}-module structure of C_{L}. If G is abelian we determine, for any f ∈ C_{L} with f · A_{L} = C_{L}, the index ideal [C_{L} : f · A_{L}^{∘}]_{O_{L}} in terms of d_{B_{L}}, d_{C_{L}}, and the resolvents (f, χ), χ ∈ Ĝ, see Proposition 2.13.

In Section 3 we discuss the cyclotomic example of 1.1(a) in some detail. We show that C is free of rank one over its associated order (see Theorem 3.6). The proof is achieved prime by prime, and so the main work has to be done in the case K = ℚ(ζ_{p^{r}}), G = μ_{p^{r}}, with a rational prime p and m, r ∈ ℕ_{0}, m ≥ 1. For r = 0 the associated order is given by A^{∘} and the proof of Theorem 3.6 is more or less trivial. In the more interesting case r ≥ 1 the associated order A^{ass} is equal to the unique maximal order A in A (Theorem 3.1), but with regard to further applications it is more important to note that A^{ass} = A also coincides with the Cartier dual A^{gs} of the Hopf order which represents the group scheme of p^{m}-th roots of unity (Theorem 3.5).

In Section 4 we study a slight generalization of Example 1.1(e), namely relative Lubin-Tate extensions. This section is also part of [BB] (except for the trivial case r = 0) and only included for the sake of completeness. These local results will be needed for our treatment of the elliptic curve examples 1.1(d). As in the cyclotomic example we show that for r ≥ 1 one has A^{ass} = A^{gs} (Corollary 4.10), where here A^{gs} denotes the Cartier dual of the O_{K}-Hopf algebra which represents the O_{K}-group scheme of 𝔭_{F}^{m}-torsion on F. Furthermore we prove that C ≃ A^{ass} (see Theorem 4.12). The main tool for our proof of Theorem 4.12 will be the factorization of a suitable resolvent function which takes values in OF̄[[X]] (see Theorem 4.11).

Finally Section 5 contains the main results of these notes. We apply the results of the previous sections to derive analogous results for Example 1.1(d). To that end we introduce in Section 5.1 a special class of elliptic curves. Let k be a quadratic imaginary number field. For an integral ideal 𝔣 of k we denote the ray class field of conductor 𝔣 by k(𝔣). Let F/k be an abelian number field extension and E/F an elliptic curve with complex multiplication by O_{k}. In addition we assume that F (E_{tor})/k is abelian. This is equivalent to the existence of a Groessencharacter φ of k such that ψ_{E/F} = φ ∘ N_{F/k}, where ψ_{E/F} denotes the Groessencharacter associated to E/F and N_{F/k} the norm map from F to k. We fix φ and set 𝔣 = lcm(𝔣_{φ}, 𝔣_{F/k}), where 𝔣_{φ} and 𝔣_{F/k} denote the conductors of φ and F/k, respectively. Note that 𝔣 is an invariant of E/F and does not depend on the choice of φ.

As in the cyclotomic example the major problem is to derive a result in the prime power case. Let 𝔭 be a prime ideal of k such that (𝔣, 𝔭) = 1. We set G = E[𝔭^{m}] and fix a primitive 𝔭^{r+m}-torsion point Q_{0} ∈ E(F̄), where r, m ∈ ℕ_{0}, m ≥ 1. We set K = k(𝔣𝔭^{r}) and Γ = {Q_{0} +_{E} P | P ∈ G}. Then K(Q_{0}) = k(𝔣𝔭^{r+m}) and Ω = Ω_{K} acts on Γ (Lemma 5.9). Again the case r = 0 is easily established (Theorem 5.20). Hence we assume r ≥ 1 in the following. In this case we obtain a K-algebra isomorphism

τ : C ⟶ L := k(𝔣𝔭^{r+m}),
f ⟼ f(Q_{0})

and a bijection φ : G → Gal(L/K) by Remark 1.2. Via τ we endow L with the structure of an A-module. We remark that under the Kummer condition r ≥ m ≥ 1 one has A = KG, φ is a group isomorphism, and identifying G and Gal(L/K) the map τ is actually an isomorphism of A = KG-modules. In other words, we are in the classical situation of Galois module structure, studying the Galois structure of the ring of integers in the wildly ramified extension k(𝔣𝔭^{r+m})/k(𝔣𝔭^{r}).

In [Sch2], R. Schertz constructed certain Galois resolvents for extensions of the form k(𝔣𝔭^{r+m})/k(𝔣𝔭^{r}), r ≥ m ≥ 1, and computed under some restrictions on 𝔣 and 𝔭 their prime ideal factorization. Via τ^{−1} his resolvents are mapped to the kind of resolvents we define in (12) and we can use his techniques and results to obtain useful resolvents for r, m ≥ 1. This is the content of Sections 5.3, 5.4 and 5.5.

For a prime 𝔓 of F above 𝔭 the formal group Ê associated with the kernel of reduction modulo 𝔓 is a relative Lubin-Tate formal group (Lemma 5.8).This fact provides the link from the global theory to the local Lubin-Tate theory of Section 4. Section 5.6 is devoted to the investigation of this connection.

In Section 5.7 we derive our main results on the module structure of C. Again the case r = 0 is easily established. The associated order A^{ass} is in this case given by A^{∘} and C is free of rank one over A^{ass}. The case r ≥ 1 is much more interesting. In order to control the ramification of primes 𝔔 of K not dividing 𝔭 we impose the following hypothesis:

E attains good reduction over K = k(𝔣𝔭^{r}).

This is in fact a very mild condition, since for a fixed elliptic curve E/F this hypothesis is satisfied for all but a finite number of pairs 𝔭, r (see Lemma 5.26).

We will then prove that the associated order A^{ass} coincides with the Cartier dual A^{gs} of the O_{K}-Hopf order which represents the O_{K}-group scheme of 𝔭^{m}- torsion points on E. Furthermore we show that C is a locally free, rank one A^{ass}-module (Theorem 5.24).

Due to the lack of appropriate resolvents we are not able to prove a complete result for the question of global freeness of C over A^{ass}. However, if 𝔣 is composite and (𝔣, 𝔭𝔭̄) = 1 we can show that C is free over its associated order (Theorem 5.27). If 𝔣 is the power of an O_{k}-prime ideal we make use of the fact that the associated orders behave well under a change of the base field. This provides restriction homomorphisms between the relevant locally free classgroups. Using this tool we prove that C is free over its associated order if (𝔣, 𝔭𝔭̄) = 1 and (𝔭, w_{k}) = 1, where w_{k} denotes the number of roots of unity in k (Theorem 5.29). As a corollary we obtain results for the classical Galois module structure of rings of integers: let r, m ≥ 1 be integers with 1 ≤ m ≤ r, let 𝔣 be an integral O_{k}- ideal with |{ε ∈ O_{k}^{∗} | ε ≡ 1 (mod 𝔣)}| = 1 and let 𝔭 be a prime with (𝔣, 𝔭𝔭̄) = 1 and (𝔭, w_{k}) = 1. Set K = k(𝔣𝔭^{r}) and L = k(𝔣𝔭^{r+m}). Then O_{L} is free over its associated order in KGal(L/K) (Corollary 5.30). Finally we use the results of the prime power case to derive results in the composite case (Theorem 5.32).

In the last Section 6 we relate our results to the theory of Kummer orders introduced by M.J. Taylor in [T2]. In particular, our results imply that for the special class of elliptic curves described in Section 5.1 the Kummer order Õ_{P} is essentially the full ring of integers (Theorem 6.2). Our approach also provides infinitely many examples of elliptic curves defined over a number field M with complex multiplication by O_{k} and everywhere good reduction for which a conjecture of M. J. Taylor [T2, §1] holds in full generality (Theorem 6.4).

2 Algebra

2.1 Forms of group algebras

In this section we study the properties of the K̄-Hopf algebra A and its subalgebras A_{L} = A^{Ω_{L}}, for intermediate fields K ⊆ L ⊆ K̄.

For a K-Hopf algebra we write as usual Δ_{H} : H → H ⊗_{K} H for the diagonal, S_{H} : H → H for the antipode and ε_{H} : H → K for the augmentation. If there is no danger of confusion we omit the index.

We view A as a Hopf algebra with Δ(g) = g ⊗ g, S(g) = g^{−1} and ε(g) = 1 for g ∈ G.

Lemma 2.1 Let K ⊆ L ⊆ K̄ be an intermediate field, and let g_{1}, …, g_{r} ∈ G be a set of representatives for the Ω_{L}-orbits of G. For each i ∈ {1, …, r} let L_{i} be the fixed field of stab_{Ω_{L}}(g_{i}) ⩽ Ω_{L}, and let x_{i,1}, …, x_{i,r_{i}} be an L-basis of L_{i}. Finally, for 1 ⩽ i ⩽ r and 1 ⩽ j ⩽ r_{i}, set

a_{i,j} := ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(x_{i,j}) ^{ω}g_{i} ∈ A ,

and assume that L ⊆ M ⊆ N ⊆ K̄ are further intermediate fields. Then the following assertions hold:

(i) The elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form an L-basis of A_{L}. If, for each i ∈ {1, …, r} the elements x_{i,1}, …, x_{i,r_{i}} form an O_{L}-basis of O_{L_{i}} then the elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form an O_{L}-basis of A_{L}^{0}.

(ii) One has dim_{L}(A_{L}) = |G|.

(iii) The elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form also an M-basis of A_{M}; in particular, they form a K̄-basis of A.

(iv) The map

i_{L}^{M}: A_{L} ⊗_{L} A_{L} → A_{M} ⊗_{M} A_{M} , a ⊗_{L} a′ ↦ a ⊗_{M} a′ ,

is an injective L-algebra map such that i_{M}^{N} ∘ i_{L}^{M} = i_{L}^{N}. In particular, A_{L} ⊗_{L} A_{L} can be regarded as an L-subalgebra of A ⊗K̄ A.

(v) Under the identification A_{L} ⊗_{L} A_{L} ⊆ A ⊗K̄ A of (iv), A_{L} is an L-Hopf subalgebra of A.

(vi) The map

ϕ_{L}^{M}: M ⊗_{L} A_{L} → A_{M} , λ ⊗_{L} a ↦ λa ,

is an isomorphism of M-Hopf algebras such that the diagrams

\begin{matrix}M ⊗_{L} A_{L} & ⟶^{ϕ_{L}^{M}} & A_{M} \\ ^{ω⊗ω}↓ & & ↓^{ω} \\ ω(M) ⊗_{ω(L)} A_{ω(L)} & ⟶^{ϕ_{ω(L)}^{ω(M)}} & A_{ω(M)} \end{matrix}  \begin{matrix}N ⊗_{M} M ⊗_{L} A_{L} & ⟶^{N⊗ϕ_{L}^{M}} & N ⊗_{M} A_{M} \\ ^{can⊗A_{L}}↓ & & ↓^{ϕ_{M}^{N}} \\ N ⊗_{L} A_{L} & ⟶^{ϕ_{L}^{N}} & A_{N} \end{matrix}

commute for each ω ∈ Ω, where can: N ⊗_{M} M → N is the multiplication map.

Proof (i) It is easy to see that the elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, lie in A_{L}. On the other hand, for arbitrary a = ∑_{g∈G} λ_{g}g ∈ A_{L}, the coefficient λ_{g} is fixed under stab_{Ω_{L}}(g), for each g ∈ G, and so λ_{g_{i}} ∈ L_{i} for each i ∈ {1, …, r}. Moreover, for each i ∈ {1, …, r} and each ω ∈ Ω_{L}, one has λ_{^{ω}g_{i}} = ω(λ_{g_{i}}). Now it follows easily that the elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form an L-basis of A_{L}. The second assertions is shown in a similar way.

(ii) Indeed, by (i) we have

dim_{L}(A_{L}) = ∑_{i=1}^{r} r_{i} = ∑_{r=1}^{r}[L_{i} : L] = ∑_{i=1}^{r}[Ω_{L} : stab_{Ω_{L}}(g_{i})] = |G| .

(iii) First we show that the elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form a K̄-basis of A. Expressing the elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, by the basis G of A produces a matrix which is a block diagonal matrix with r blocks indexed by the Ω_{L}-orbits of G, where the i-th block is given by

(ω(x_{i,j}))_{{j∈\{1,... ,r_{i}\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})}}

whose determinant does not vanish, since L_{i}/L is separable. This shows the result for M = K̄. For arbitrary M, it suffices by (ii) to show that the elements a_{i,j} (1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}) are M-linearly independent. But this follows from the case M = K̄.

(iv) This follows immediately from (iii), since i_{L}^{M} maps an L-basis to an M-basis.

(v) It is easy to see that S(A_{L}) ⊆ A_{L}, since taking inverses in G commutes with the Ω-action on A. Let a ∈ A_{L}. Then, by (iii), we may write

Δ(a) = ∑_{i,j,i′,j′} λ_{i,j,i′,j′} a_{i,j} ⊗K̄ a_{i′,j′} ∈ A ⊗K̄ A

with uniquely determined coefficients λ_{i,j,i′,j′} ∈ K̄. We apply an arbitrary element ω ∈ Ω_{L} on both sides. Since Δ respects the Ω-action, we obtain

Δ(a) = ∑_{i,j,i′,j′} ω(λ_{i,j,i′,j′}) a_{i,j} ⊗K̄ a_{i′,j′} .

Thus, by their uniqueness, the coefficients λ_{i,j,i′,j′} are contained in L, and

Δ(a) ∈ iK̄ ; L(A_{L} ⊗_{L} A_{L}).

(vi) It is easy to see that ϕ_{L}^{M} is a homomorphism of M-Hopf algebras and that the two diagrams commute. Moreover, by (iii), ϕ_{L}^{M} is an isomorphism.

For the rest of this section we assume that G is abelian. Let L be a subextension of K̄/K. In order to understand the L-algebra structure of A_{L} we work with the Wedderburn decomposition of A. Let Ĝ := Hom(G, K̄^{×}) denote the abelian group of K̄-characters of G. Then Ω acts on Ĝ by

(^{ω}χ)(g) := ω(χ(^{ω^{−1}}g))

for χ ∈ Ĝ, ω ∈ Ω, and g ∈ G. It is well-known that the map

ρ: A = K̄G → ∏_{χ∈Ĝ} K̄ , ∑_{g∈G} λ_{g}g ↦ (∑_{g∈G} λ_{g}χ(g))_{χ∈Ĝ}

is an isomorphism of K̄-algebras. For χ ∈ Ĝ, we denote by e_{χ} ∈ A the element corresponding to the primitive idempotent ε_{χ} of ∏_{χ∈Ĝ} K̄ which has entry 1 in the component χ and 0 everywhere else. Then

e_{χ} = \frac{1}{|G|} ∑_{g∈G} χ(g^{−1})g .

Note that ^{ω}(e_{χ}) = e_{(^{ω}χ)} for each ω ∈ Ω and χ ∈ Ĝ. The Ω-action on A is transported via ρ to the action

ω((λ_{χ})_{χ∈Ĝ}) = (ω(λ_{(^{ω^{−1}}χ)}))_{χ∈Ĝ}

for (λ_{χ})_{χ∈Ĝ} ∈ ∏_{χ∈Ĝ} K̄ and ω ∈ Ω. Thus, the application of ω moves the χ-component λ_{χ} to the ^{ω}χ-component while simultaneously applying ω to λ_{χ}. This implies that ρ restricts to an isomorphism

ρ_{L}: A_{L} ≃ {(λ_{χ}) ∈ ∏_{χ∈Ĝ} K̄ | ω(λ_{χ}) = λ_{(^{ω}χ)} for all ω ∈ Ω_{L}} .  (1)

Lemma 2.2 Assume that G is abelian and let K ⊆ L ⊆ K̄ be an intermediate field. Let χ_{1}, …, χ_{s} ∈ Ĝ be a set of representatives for the Ω_{L}-orbits of Ĝ. For each k ∈ {1, …, s}, let L̂_{k} denote the fixed field of stab_{Ω_{L}}(χ_{k}) ⩽ Ω_{L}, and let y_{k,1}, …, y_{k,s_{k}} be an L-basis of L̂_{k}. For 1 ⩽ k ⩽ s and 1 ⩽ l ⩽ s_{k}, set

â_{k,l} := ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ω(y_{k,l}) ^{ω}e_{χ_{k}} ∈ A .

Then the following assertions hold:

(i) The composition

ρ̃: A ⟶^{ρ} ∏_{χ∈Ĝ} K̄ ⟶^{p} ∏_{k=1}^{s} K̄ ,

where p denotes the projection to the χ_{1}, …, χ_{s}-components, restricts to an L-algebra isomorphism

ρ̃_{L}: A_{L} → ∏_{k=1}^{s} L̂_{k} .

In particular, the maximal order A_{L} of A_{L} is given by ρ̃_{L}^{−1}(∏_{k=1}^{s} OL̂_{k}).

(ii) The elements â_{k,l}, 1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k}, form an L-basis of A_{L}. If, for each k ∈ {1, …, s}, the elements y_{k,1}, …, y_{k,s_{k}} form an O_{L}-basis of OL̂_{K} then the elements â_{k,l}, 1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k}, form an O_{L}-basis of A_{L}.

Proof (i) This follows immediately from (1).

(ii) The elements â_{k,l} are contained in A_{L} and ρ̃_{L}(â_{k,l}), 1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k} form an L-basis of ∏_{k=1}^{s} L̂_{k}. Thus (ii) follows from (i). The integral version can be seen in the same way.

2.2 Duality

It is well-known that the Hopf algebra dual (also called Cartier dual) of A is the K̄-Hopf algebra B := Map(G, K̄) consisting of all set maps from G to K̄. The duality is given by the non-degenerate bilinear form

(−, −): A ⊗K̄ B → K̄, (∑_{g∈G} λ_{g}g, f) ↦ ∑_{g∈G} λ_{g}f(g) .

The K̄-linear structure of B is obvious. Multiplication is given by (f_{1}f_{2})(g) = f_{1}(g)f_{2}(g), for f_{1}, f_{2} ∈ B and g ∈ G, with the constant map with value 1 as unity. The diagonal Δ, augmentation ϵ, and antipode S are given by

Δ(f)(g_{1}, g_{2}) = f(g_{1}g_{2}) ,
ϵ(f) = f(1) ,
S(f)(g) = f(g^{−1}) ,

respectively, for f ∈ B, g, g_{1}, g_{2} ∈ G, where in the definition of Δ we identify B ⊗K̄ B with Map(G × G, K̄) in the obvious way. For the K̄-basis elements l_{g}, g ∈ G, with l_{g}(h) := δ_{g,h} for h ∈ G, we have

Δ(l_{g}) = ∑_{{g_{1},g_{2}∈G \atop g_{1}g_{2}=g}} l_{g_{1}} ⊗K̄ l_{g_{2}} .

Note that Ω acts on B by

(^{ω}f)(g) := ω(f(^{ω^{−1}}g))

for ω ∈ Ω, f ∈ B, and g ∈ G. This action respects the K̄-Hopf algebra structure, as is easily verified. For a ∈ A, f ∈ B, and ω ∈ Ω one has

(^{ω}a, ^{ω}f) = ω((a, f)) .  (2)

For each intermediate field K ⊆ L ⊆ K̄ we set

B_{L} := B^{Ω_{L}} = Map(G, K̄)^{Ω_{L}} = {f: G → K̄ | f(^{ω}g) = ω(f(g)) for all ω ∈ Ω_{L}} .

Then B_{L} is an L-subalgebra of B.

The next lemma shows that B_{L} is an L-Hopf subalgebra of B and is the L-dual of A_{L} with respect to the restricted bilinear form (−, −). By B_{L} we denote the maximal order in B_{L} which is given by

B_{L} = Map(G, OK̄)^{Ω_{L}} .

Lemma 2.3 Let K ⊆ L ⊆ K̄ be an intermediate field, and let g_{1}, …, g_{r} ∈ G, L_{1}, …, L_{r}, and x_{i,1}, …, x_{i,r_{i}} ∈ L_{i} (1 ⩽ i ⩽ r) be given as in Lemma 2.1. Moreover, for 1 ⩽ i ⩽ r and 1 ⩽ j ⩽ r_{i}, let b_{i,j} ∈ B be defined by

b_{i,j}(g) := \begin{cases}ω(x_{i,j}), & if g = ^{ω}g_{i} for some ω ∈ Ω_{L}, \\ 0, & otherwise. \end{cases}

Assume that L ⊆ M ⊆ N ⊆ K̄ are further intermediate fields. Then the following assertions hold:

(i) The map

σ_{L}: B_{L} → ∏_{i=1}^{r} L_{i} , b ↦ b(g_{i}) ,

is an L-algebra isomorphism. In particular, σ_{L} restricts to an isomorphism

σ_{L}: B_{L} → ∏_{i=1}^{r} O_{L_{i}} .

(ii) The elements b_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form an L-basis of B_{L}. If, for each i ∈ {1, …, r}, the elements x_{i,1}, …, x_{i,r_{i}} form an O_{L}-basis of O_{L_{i}} then the elements b_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form an O_{L}-basis of B_{L}.

(iii) One has dim_{L}(B_{L}) = |G|.

(iv) The elements b_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form also an M-basis of B_{M}.

(v) The map

j_{L}^{M}: B_{L} ⊗_{L} B_{L} → B_{M} ⊗_{M} B_{M} , b ⊗_{L} b′ ↦ b ⊗_{M} b′ ,

is an injective L-algebra homomorphism such that j_{M}^{N} ∘ j_{L}^{M} = j_{L}^{N}. In particular, B_{L} ⊗_{L} B_{L} can be regarded as L-subalgebra of B ⊗K̄ B.

(vi) Under the identification B_{L} ⊗_{L} B_{L} ⊆ B ⊗K̄ B of (v), B_{L} is an L-Hopf subalgebra of B.

(vii) The map

ψ_{L}^{M}: M ⊗_{L} B_{L} → B_{M} , λ ⊗_{L} b ↦ λb ,

is an isomorphism of M-Hopf algebras such that the diagrams

\begin{matrix}M ⊗_{L} B_{L} & ⟶^{ψ_{L}^{M}} & B_{M} \\ ^{ω⊗ω}↓ & & ↓^{ω} \\ ω(M) ⊗_{ω(L)} B_{ω(L)} & ⟶^{ψ_{ω(L)}^{ω(M)}} & B_{ω(M)} \end{matrix}  \begin{matrix}N ⊗_{M} M ⊗_{L} B_{L} & ⟶^{N⊗ψ_{L}^{M}} & N ⊗_{M} B_{M} \\ ^{can⊗B_{L}}↓ & & ↓^{ψ_{M}^{N}} \\ N ⊗_{L} A_{L} & ⟶^{ψ_{L}^{N}} & B_{N} \end{matrix}

commute for all ω ∈ Ω.

(viii) The map A_{L} ⊗_{L} B_{L} → A_{M} ⊗_{M} B_{M}, a ⊗_{L} b ↦ a ⊗_{M} b, is injective; in particular, A_{L} ⊗_{L} B_{L} can be regarded as an L-subspace of A ⊗K̄ B. Morover, the restriction of (−, −) to A_{L} ⊗_{L} B_{L} takes values in L and is non-degenerate.

(ix) The L-Hopf algebras A_{L} and B_{L} are dual to each other with respect to (−, −): A_{L} ⊗_{L} B_{L} → L.

Proof (i) For b ∈ B_{L} and i ∈ {1, …, r}, the elements b(^{ω}g_{i}) = ω(b(g_{i})), ω ∈ Ω_{L}, are determined by b(g_{i}). Hence, σ_{L} is injective. On the other hand, for given elements λ_{i} ∈ L_{i}, i = 1, …, r, the map b: G → K̄, defined by b(^{ω}g_{i}) := ω(λ_{i}), for i = 1, …, r and ω ∈ Ω_{L}, is obviously in B_{L}.

(ii) This follows immediately from (i).

(iii) This follows from (ii) and the equation ∑_{i=1}^{r}[Ω_{L} : stab_{Ω_{L}}(g_{i})] = |G|. (iv) This is proved in a similar way as Lemma 2.1 (iii) by reduction to the

case M = K̄ and using the basis l_{g}, g ∈ G, of B which leads to the same transition matrix as in the proof of Lemma 2.1 (iii).

(v) This follows from (iv).

(vi) This is proved in a similar way as Lemma 2.1 (v) using the basis b_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, of B_{L}.

(vii) By (iv), ψ_{L}^{M} is an isomorphism of M-spaces. The remaining assertions are easily verified.

(viii) The injectivity of the map A_{L} ⊗_{L} B_{L} → A_{M} ⊗_{M} B_{M} follows from Lemma 2.1 (iii) and part (iv). Moreover, (A_{L}, B_{L}) ⊆ L by Equation (2). Using the bases a_{i,j} of A_{L} from Lemma 2.1 and b_{i,j} of B_{L}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, with their property from Lemma 2.1 (iii) and from (iv), we see that (A_{L}, B_{L}) = L and that the restricted pairing A_{L} ⊗_{L} B_{L} → L is non-degenerate.

(ix) This follows from part (viii) and from the existence of bases a_{i,j} of A_{L} and b_{i,j} of B_{L} with the properties from Lemma 2.1 (iii) and from (iv).

2.3 Indices and Hopf structures

Let K ⊆ L ⊆ K̄ be an intermediate field. We may take duals of O_{L}-lattices in A_{L} and B_{L} with respect to the non-degenerate pairing

A_{L} ⊗_{L} B_{L} → L , (∑_{g∈G} λ_{g}g, f) ↦ ∑_{g∈G} λ_{g}f(g) .

More precisely, if R ⊆ A_{L} and S ⊆ B_{L} are O_{L}-lattices then

R^{∗} := {f ∈ B_{L} | (r, f) ∈ O_{L} for all r ∈ R}

and

S^{∗} := {a ∈ A_{L} | (a, s) ∈ O_{L} for all s ∈ S}

are O_{L}-lattices in B_{L} and A_{L} respectively. Note that R is an O_{L}-order (resp. O_{L}-subcoalgebra) of A_{L} if and only if R^{∗} is an O_{L}-subcoalgebra (resp. O_{L}- order) in B_{L}. A similar statement holds for S. Moreover, R is an O_{L}-Hopf order in A_{L} if and only if R^{∗} is an O_{L}-Hopf order in B_{L}, and similarly for S.

Recall that for O_{L}-lattices X ⊆ Y of equal O_{L}-rank the O_{L}-order ideal [Y : X]_{O_{L}} is defined as the product 𝔭_{1} ⋯ 𝔭_{t} of non-zero prime ideals 𝔭_{1}, …, 𝔭_{t} of O_{L} if R/𝔭_{1}, …, R/𝔭_{t} are the O_{L}-composition factors of Y/X. More generally, for O_{L}-lattices X and Y in a finite dimensional L-vector space the order ideal [Y : X]_{O_{L}} is defined as the fractional ideal [Y : X ∩ Y]_{O_{L}}[X : X ∩ Y]_{O_{L}}^{−1}. For the following properties of order ideals see for example [R, §4]. If L ⊆ L′ ⊆ K̄ is a finite extension field of L then

[O_{L′} ⊗_{O_{L}} Y : O_{L′} ⊗_{O_{L}} X]_{O_{L′}} = [Y : X]_{O_{L}}O_{L′} .

If 𝔭 is a non-zero prime ideal of O_{L} then the following localization property holds:

[Y_{𝔭} : X_{𝔭}]_{(O_{L})_{𝔭}} = ([Y : X]_{O_{L}})_{𝔭} .

If Y and X are free O_{L}-modules and M is the matrix of coefficients arising from expressing an O_{L}-basis of X by an O_{L}-basis of Y, then

[Y : X]_{O_{L}} = det(M)O_{L} .

Finally, if L ⊆ L′ ⊆ K̄ is as above and X′ ⊆ Y′ are O_{L′}-lattices then

[Y′ : X′]_{O_{L}} = N_{L′/L}([Y′ : X′]_{O_{L′}} ) .

If G is abelian we denote by d_{A_{L}} the discriminant ideal of the maximal order A_{L} over O_{L}, i.e. d_{A_{L}} = ∏_{k=1}^{s} d_{L̂_{k}/L} in the notation of Lemma 2.2. Similarly, for arbitrary G, we denote by d_{B_{L}} the discriminant ideal of the maximal order B_{L} over O_{L}, i.e. d_{B_{L}} = ∏_{i=1}^{r} d_{L_{i}/L} in the notation of Lemma 2.1 and Lemma 2.3. If L ⊆ L′ ⊆ K̄ is as above then we write D_{L′/L} for the different of O_{L′} over O_{L}.

Lemma 2.4 With the notation of Lemma 2.1 and Lemma 2.3 one has

B_{L}^{∗} = {∑_{i=1}^{r} ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(λ_{i}) ^{ω}g_{i} | λ_{i} ∈ D_{L_{i}/L}^{−1} for i = 1, …, r} .

Moreover, B_{L}^{∗} is an O_{L}-order in A_{L} if and only if d_{B_{L}} = O_{L}.

Proof Each element a ∈ A_{L} can be written in the form

a = ∑_{i=1}^{r} ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(λ_{i}) ^{ω}g_{i}

with uniquely determined λ_{i} ∈ L_{i}, i = 1, …, r, by Lemma 2.1 (i). Then a ∈ B_{L}^{∗} if and only if (a, f) ∈ O_{L} for all f ∈ B_{L}. By Lemma 2.3 (i), each f ∈ B_{L}^{∗} is a sum of elements of the form ∑_{ω} ω(x_{i}) l_{^{ω}g_{i}}, x_{i} ∈ O_{L_{i}}, i = 1, …, r, where the sum runs over coset representatives of Ω_{L}/stab_{Ω_{L}}(g_{i}). Hence, a ∈ B_{L}^{∗} if and only if

∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(λ_{i}) ω(x_{i}) = Tr_{L_{i}/L}(λ_{i}x_{i}) ∈ O_{L}

for all i ∈ {1, …, r} and all x_{i} ∈ O_{L_{i}}, which is equivalent to λ_{i} ∈ D_{L_{i}/L}^{−1} for all i ∈ {1, …, r}.

If d_{B_{L}} = O_{L} then D_{L_{i}/L} = O_{L_{i}} for all i = 1, …, r, and therefore B_{L}^{∗} = A_{L}^{∘} is an O_{L}-order. Suppose, conversely, that B_{L}^{∗} is an O_{L}-order and let i ∈ {1, …, r} be arbitrary. Let i′ ∈ {1, …, r} and ν ∈ Ω_{L} be such that g_{i}^{−1} = ^{ν}g_{i′}. Note that stab_{Ω_{L}}(g_{i}) = stab_{Ω_{L}}(g_{i}^{−1}) and that ν : L_{i′} → L_{i} is a L-isomorphism. Let λ_{i} ∈ D_{L_{i}/L}^{−1} and λ′_{i′} ∈ D_{L_{i′} /L}^{−1} be arbitrary. Then

∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(λ_{i}) ^{ω}g_{i} , ∑_{ω′∈Ω_{L}/stab_{Ω_{L}} (g_{i′})} ω′(λ′_{i′}) ^{ω′}g_{i′}

are elements of B_{L}^{∗}. Hence, also their product lies in B_{L}^{∗}. In particular the coefficient at 1 ∈ G of this product is an element of O_{L}:

∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(λ_{i}) ων(λ′_{i′}) = Tr_{L_{i}/L}(λ_{i}ν(λ′_{i′})) ∈ O_{L} .  (3)

If λ′_{i′} runs through D_{L_{i′} /L}^{−1} then ν(λ′_{i′}) runs through D_{L_{i}/L}^{−1} and (3) implies that Tr_{L_{i}/L}(D_{L_{i}/L}^{−2}) ⊆ O_{L}. But this is only possible if d_{L_{i}/L} = O_{L}. Since this holds for all i ∈ {1, …, r} we obtain d_{B_{L}} = O_{L}.

Corollary 2.5 One has A_{L}^{∘} ⊆ B_{L}^{∗} and [B_{L}^{∗} : A_{L}^{∘}]_{O_{L}} = d_{B_{L}}.

Proof The inclusion A_{L}^{∘} ⊆ B_{L}^{∗} is clear from Lemma 2.4. Applying ρ̃_{L} from Lemma 2.2 (i) we have

[B_{L}^{∗} : A_{L}^{∘}]_{O_{L}} = [∏_{i=1}^{r} D_{L_{i}/L}^{−1} : ∏_{i=1}^{r} O_{L_{i}}]_{O_{L}} = ∏_{i=1}^{r}[O_{L_{i}} : D_{L_{i}/L}]_{O_{L}}

= ∏_{i=1}^{r} d_{L_{i}/L} = d_{B_{L}} .

We remark that, even for G abelian, B_{L}^{∗} is not necessarily contained in A_{L}.

For the following lemma we assume that G is abelian. Let L ⊆ L′ ⊆ K̄ be a finite extension field of L containing L_{1}, …, L_{r} and L̂_{1}, …, L̂_{s} in the notation of Lemma 2.1 and Lemma 2.2. Then A_{L′} = L′G, B_{L′} = Map(G, L′), and A_{L′} ≅ ∏_{χ∈Ĝ} L′ via ρ_{L′} as L′-algebras. Such a finite extension L′ will be called a splitting field for A_{L} and B_{L}.

Lemma 2.6 Let G be abelian. Then

[A_{L} : A_{L}^{∘}]_{O_{L}}^{2} = |G|^{|G|}d_{B_{L}}d_{A_{L}}^{−1}

for each intermediate field K ⊆ L ⊆ K̄.

Proof Let 𝔭 be a non-zero prime ideal of O_{L}. Since (A_{L})_{𝔭} is the maximal (O_{L})_{𝔭}-order in A_{L} and (A_{L}^{∘})_{𝔭} is the set of elements in A_{L} whose coefficients with respect to G are integral over (O_{L})_{𝔭}, we may as well assume that O_{L} is local with maximal ideal 𝔭. Let L′ be a splitting field for A_{L} and B_{L}. We determine

[A_{L} : A_{L}^{∘}]_{O_{L}}^{2}O_{L′} = [O_{L′} ⊗_{O_{L}} A_{L} : O_{L′} ⊗_{O_{L}} A_{L}^{∘}]_{O_{L′}}^{2}

using the isomorphism ϕ_{L}^{L′}: L′⊗_{L}A_{L} → A_{L′}, the maximal O_{L′}-order A_{L′} of A_{L′}, and the isomorphism ρ_{L′}: A_{L′} → ∏_{χ∈Ĝ} L′. More precisely, [A_{L} : A_{L}^{∘}]_{O_{L}}O_{L′} is the quotient of the order ideals

[ρ_{L′}(A_{L′}) : (ρ_{L′} ∘ ϕ_{L}^{L′})(O_{L′} ⊗_{O_{L}} A_{L}^{∘})]_{O_{L′}}^{2}  (4)

and

[ρ_{L′}(A_{L′}) : (ρ_{L′} ∘ ϕ_{L}^{L′})(O_{L′} ⊗_{O_{L}} A_{L})]_{O_{L′}}^{2} .  (5)

For the computation of the order ideal (5) let χ_{k}, L̂_{k}, y_{k,l}, and â_{k,l}, 1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k}, be given as in Lemma 2.2 such that y_{k,1}, …, y_{k,s_{k}} is an O_{L}-basis of OL̂_{k} for each k = 1, …, s. Then the elements

â_{k,l} = ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ω(y_{k,l}) e_{^{ω}χ_{k}}  (1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k})

form an O_{L}-basis of A_{L} and (ρ_{L′} ∘ ϕ_{L}^{L′})(O_{L′} ⊗_{O_{L}} A_{L}) has as O_{L′}-basis the elements

(χ(â_{k,l}))_{χ∈Ĝ} (1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k}) .

Expressing this basis by the O_{L′}-basis ε_{χ}, χ ∈ Ĝ, of ρ_{L′}(A_{L′}) = ∏_{χ∈Ĝ} O_{L′}, we obtain a block diagonal transition matrix with blocks indexed by Ĝ/Ω_{L}. The block belonging to χ_{k} is given by

(ω(y_{k,l}))_{{l∈\{1,... ,s_{k}\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})}} .

The square of the determinant of this block generates the O_{L}-ideal d_{L̂_{k}/L}. Thus, the squared order ideal in (5) is given by

∏_{k=1}^{s} d_{L̂_{k}/L}O_{L′} = d_{A_{L}/L}O_{L′} .  (6)

For the computation of the squared order ideal (4) let g_{i}, L_{i}, x_{i,j}, and a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, be given as in Lemma 2.1 such that x_{i,1}, …, x_{i,r_{i}} is an O_{L}-basis of O_{L_{i}} for each i ∈ {1, …, r}. Then, the elements

a_{i,j} = ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(x_{i,j}) ^{ω}g_{i}  (1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i})

form an O_{L}-basis of A_{L}^{∘}. Thus, the elements

(  ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(x_{i,j}) χ(^{ω}g_{i}))_{χ∈Ĝ}  (1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i})

form an O_{L′}-basis of (ρ_{L′} ∘ϕ_{L}^{L′})(O_{L′} ⊗_{O_{L}} A_{L}^{∘}). Expressing this basis by the basis ε_{χ}, χ ∈ Ĝ, of ρ_{L′}(A_{L′}) = ∏_{χ∈Ĝ} O_{L′}, we obtain the transition matrix

M = ( ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(x_{i,j}) χ(^{ω}g_{i}))_{{(i,j) \atop χ∈Ĝ}}

which is the product M = M_{1}M_{2} of the block diagonal matrix M_{1} with blocks indexed by i = 1, …, r, the i-th block given by

M_{1,i} = (ω(x_{i,j}))_{{j∈\{1,... ,r_{i}\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})}} ,

and the matrix

M_{2} = (χ(g))_{{g∈G \atop χ∈Ĝ}} .

Now, det(M_{1,i})^{2} generates the O_{L}-ideal d_{L_{i}/L}; thus det(M_{1})^{2}O_{L} = d_{B_{L}}. Moreover, since

∑_{g∈G} χ_{1}(g) χ_{2}(g) = ∑_{g∈G}(χ_{1}χ_{2})(g) = \begin{cases}|G|, & if χ_{1} = χ_{2}^{−1}, \\ 0, & otherwise, \end{cases}

for χ_{1}, χ_{2} ∈ Ĝ, we have

det(M_{2})^{2} = det(M_{2}^{t}M_{2}) = ±|G|^{|G|} .  (7)

Altogether this yields

det(M)^{2}O_{L′} = |G|^{|G|}d_{B_{L}}O_{L′}  (8)

which expresses the squared order ideal in (4). Now, dividing the ideal in (8) by the ideal in (6), the result follows, since extending O_{L}-ideals to O_{L′}-ideals is injective.

Next we investigate under which circumstances A_{L}^{∘} and (in the abelian case) A_{L} are Hopf orders over O_{L} in A_{L}. The crucial point is to get hold of the image of the diagonal.

Since the following might be of general interest we place ourselves for the moment in a more general setting. Let H be a finite group, E/F a field extension in characteristic zero, and R ⊆ F a subring such that F is the field of fractions of R. Suppose that A ⊆ EH is an R-subalgebra of EH with R-basis a_{1}, …, a_{n} which is also an E-basis of EH. Then the multiplication map E ⊗_{R} A → EH, λ⊗_{R} a ↦ λa, is an E-algebra isomorphism and the map A⊗_{R} A → EH ⊗_{E} EH, a⊗_{R}a′ ↦ a⊗_{E}a′, is injective, so that A⊗_{R}A can be regarded as an R-subalgebra in EH ⊗_{E} EH. We would like to decide whether Δ(A) ⊆ A ⊗_{R} A or not. We write

a_{i} = ∑_{h∈H} α_{i,h}h

for i = 1, …, n with α_{i,h} ∈ E, and we consider the matrix

S := (α_{i,h})_{{i∈\{1,... ,n\} \atop h∈H}} ∈ GL_{n}(E) ,

together with its inverse

T := S^{−1} = (β_{h,i})_{{h∈H \atop i∈\{1,... ,n\}}} .

Then we have the following criterion:

Lemma 2.7 Keeping the above notation, the following assertions are equivalent:

(i) Δ(A) ⊆ A ⊗_{R} A.

(ii) ∑_{h∈H} α_{i,h}β_{h,j}β_{h,j′} ∈ R for all i, j, j′ ∈ {1, …, n}.

Proof We write

Δ(a_{i}) = ∑_{j,j′=1}^{n} γ_{j,j′}^{(i)}a_{j} ⊗_{E} a_{j′}

with uniquely determined coefficients γ_{j,j′}^{(i)} ∈ E. Then, for i ∈ {1, …, n}, we

have Δ(a_{i}) ∈ A ⊗_{R} A if and only if γ_{j,j′}^{(i)} ∈ R for all j, j′ ∈ {1, …, n}. Using the expansion Δ(a_{i}) = ∑_{h∈H} α_{i,h}h ⊗ h, we obtain the equivalent equation

∑_{h∈H} α_{i,h}h ⊗ h = ∑_{j,j′=1}^{n} ∑_{h_{1},h_{2}∈H} γ_{j,j′}^{(i)}α_{j,h_{1}}α_{j′,h_{2}}h_{1} ⊗ h_{2} .

Hence, the coefficients γ_{j,j′}^{(i)} are uniquely determined by the system of linear equations

∑_{j,j′} γ_{j,j′}^{(i)}α_{j,h_{1}}α_{j′,h_{2}} = \begin{cases}α_{i,h}, & if h_{1} = h_{2} =: h, \\ 0, & if h_{1} ≠ h_{2}, \end{cases}

one equation for each i ∈ {1, …, n} and each pair (h_{1}, h_{2}) ∈ H × H. Now it is easy to verify that these equations are satisfied for γ_{j,j′}^{(i)} = ∑_{h∈H} α_{i,h}β_{h,j}β_{h,j′}, and the result follows.

In the following proposition we apply Lemma 2.7 to the O_{L}-order A_{L}^{∘} in A_{L}.

Proposition 2.8 The O_{L}-order A_{L}^{∘} is a Hopf order in A_{L} if and only if d_{B_{L}} = O_{L}.

Proof Clearly, A_{L}^{∘} is stable under the antipode of A_{L}. The inclusion Δ(A_{L}^{∘}) ⊆ A_{L}^{∘} ⊗_{O_{L}} A_{L}^{∘} can be tested by localization. Thus we may assume that O_{L} is a local ring. In this case we choose g_{i}, L_{i}, x_{i,j}, and a_{i,j} as in Lemma 2.1 such that, for each i ∈ {1, …, r}, the elements x_{i,1}, …, x_{i,r_{i}}, form an O_{L}-basis of O_{L_{i}}. Then the elements a_{i,j}, 1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i}, form an O_{L}-basis of A_{L}^{∘}, and, in the notation introduced before Lemma 2.7, the coefficient matrix S is a block diagonal matrix, the blocks indexed by i = 1, …, r, and the i-th block given by

(ω(x_{i,j}))_{{i∈\{1,... ,r_{i}\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})}} .

If x_{i,1}^{∗}, …, x_{i,r_{i}}^{∗} ∈ L_{i} is a dual basis of x_{i_{1}}, …, x_{i,r_{i}} with respect to the trace form Tr_{L_{i}/L}, then the inverse T of S is given by the block diagonal matrix with the i-th block

(ω(x_{i,j}^{∗}))_{{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i}) \atop i∈\{1,... ,r_{i}\}}} .

Now, taking into account the block diagonal structure of S and T, Lemma 2.7 states that Δ(A_{L}^{∘}) ⊆ A_{L}^{∘} ⊗_{O_{L}} A_{L}^{∘} if and only if

∑_{ω∈Ω_{L}/stab_{Ω_{L}} (g_{i})} ω(x_{i,j}) ω(x_{i,j′}^{∗}) ω(x_{i,j′′}^{∗}) = Tr_{L_{i}/L}(x_{i,j}x_{i,j′}^{∗}x_{i,j′′}^{∗}) ∈ O_{L} (9)

for all i ∈ {1, …, r} and all j′, j′′ ∈ {1, …, r_{i}}. But since x_{i,j}^{∗}, j = 1, …, r_{i}, form an O_{L}-basis of D_{L_{i}/L}^{−1}, the condition in (9) holds for given i ∈ {1, …, r} and all j′, j′′ ∈ {1, …, r_{i}} if and only if Tr_{L_{i}/L}(D_{L_{i}/L}^{−2}) ⊆ O_{L}. But this is equivalent to D_{L_{i}/L} = O_{L_{i}} and to d_{L_{i}/L} = O_{L}. Now the result follows.

Corollary 2.9 The following statements are equivalent:

(i) The order A_{L}^{∘} is a Hopf order in A_{L}.

(ii) The discriminant ideal d_{B_{L}} is trivial.

(iii) One has A_{L}^{∘} = B_{L}^{∗}.

(iv) One has (A_{L}^{∘})^{∗} = B_{L}.

(v) The maximal order B_{L} of B_{L} is an O_{L}-Hopf order.

If (i)–(v) hold then A_{L}^{∘} is the smallest O_{L}-Hopf order of A_{L}.

Proof The equivalence of (i) and (ii) is the content of Proposition 2.8. The statements (ii) and (iii) are equivalent by Corollary 2.5 which states [B_{L}^{∗} : A_{L}^{∘}] = d_{B_{L}}. Obviously, (iii) and (iv) are equivalent, since O_{L} is a Dedekind domain. Moreover, (i) and (iv) imply (v). Finally, (v) implies that B_{L}^{∗} is an order in A_{L}, and then Lemma 2.4 implies (ii).

If (i)–(v) hold then B_{L} is certainly the largest O_{L}-Hopf order in B_{L}. Therefore its dual A_{L}^{∘} is the smallest O_{L}-Hopf order in A_{L}.

Proposition 2.10 Let G be abelian. Then the maximal O_{L}-order A_{L} of A_{L} is a Hopf order if and only if d_{A_{L}} = O_{L}.

Proof Since the antipode S is an L-algebra automorphism of A_{L}, the maximal O_{L}-order of A_{L} is stable under S. As in the proof of Proposition 2.8 we may assume that O_{L} is local. Now let χ_{k}, L̂_{k}, y_{k,l}, and â_{k,l} be given as in Lemma 2.2 such that, for each k ∈ {1, …, s}, the elements y_{k,1}, …, y_{k,s_{k}} form an O_{L}-basis of OL̂_{k}. Then the elements â_{k,l}, 1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k}, form an O_{L}-basis of A_{L}. The matrix S in the notation prededing Lemma 2.7 is given by

\frac{1}{|G|}( ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ω(y_{k,l}) (^{ω}χ_{k})(g^{−1}))_{{(k,l) \atop g∈G}} .

We can write S = S_{1}S_{2}, where S_{1} is the block diagonal matrix with blocks indexed by k = 1, …, s and whose k-th block is given by

S_{1,k} = (ω(y_{k,l}))_{{l∈\{1,... ,s_{k}\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})}} ,

and where

S_{2} = \frac{1}{|G|}((^{ω}χ_{k})(g^{−1}))_{{(k,ω) \atop g∈G}} .

If, for each k ∈ {1, …, s}, we denote by y_{k,1}^{∗}, …, y_{k,s_{k}}^{∗} ∈ L̂_{k} the dual basis of y_{k_{1}}, …, y_{k,s_{k}} with respect to Tr_{L̂_{k}/L}, then

T_{1,k} = (ω(y_{k,l}^{∗}))_{{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k}) \atop l∈\{1,... ,s_{k}\}}}

is the inverse of S_{1,k}. Moreover, by the orthogonality relations of irreducible characters, the inverse of S_{2} is given by

T_{2} = ((^{ω}χ_{k})(g))_{{g∈G \atop (k,ω)}} .

Thus, the inverse of S is given by

T = ( ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ω(y_{k,l}^{∗}) (^{ω}χ_{k})(g))_{{g∈G \atop (k,l)}} .

Now, Lemma 2.7 states that Δ(A_{L}) ⊆ A_{L} ⊗_{O_{L}} A_{L} if and only if

\frac{1}{|G|} ∑_{g∈G} ∑_{ω,ω′,ω′′} ω(y_{k,l}) (^{ω}χ_{k})(g) ω′(y_{k′,l′}^{∗}) (^{ω′}χ_{k′})(g) ω′′(y_{k′′,l′′}^{∗}) (^{ω′′}χ_{k′′})(g) =

= \frac{1}{|G|} ∑_{ω,ω′,ω′′} ω(y_{k,l}) ω′(y_{k′,l′}^{∗}) ω′′(y_{k′′,l′′}^{∗}) ∑_{g∈G}(^{ω}χ_{k})(^{ω′}χ_{k′})(^{ω′′}χ_{k′′})(g) ∈ O_{L}

for all pairs (k, l), (k′, l′), (k′′, l′′), where the triple sum runs independently over all ω ∈ Ω_{L}/stab_{Ω_{L}}(χ_{k}), ω′ ∈ Ω_{L}/stab_{Ω_{L}}(χ_{k′}), and ω′′ ∈ Ω_{L}/stab_{Ω_{L}}(χ_{k′′}). By the orthogonality relations of irreducible characters the last sum over g ∈ G reduces to |G| if (^{ω}χ_{k})(^{ω′}χ_{k′})(^{ω′′}χ_{k′′}) = 1 and vanishes otherwise. If d_{A_{L}} = O_{L} then y_{k,l}^{∗} ∈ OL̂_{k} for all pairs (k, l), and the above condition is certainly satisfied. Conversely, if the above condition is satisfied, then we choose k′ ∈ {1, …, s} arbitrarily and remark that stab_{Ω_{L}}(χ_{k′}) = stab_{Ω_{L}}(χ_{k′}^{−1}). Let k′′ ∈ {1, …, s} and κ ∈ Ω_{L} be such that χ_{k′}^{−1} = ^{κ}χ_{k′′}. Moreover let k ∈ {1, …, s} be such that χ_{k} = 1. Then L̂_{k} = L and the above sum reduces further to

∑_{ω′,ω′′} ω′(y_{k′,l′}^{∗}) ω′′(y_{k′′,l′′}^{∗}) δ_{^{ω′}χ_{k′} ^{ω′′}χ_{k′′} ,1} =

= ∑_{ω′} ω′(y_{k′,l′}^{∗}) ω′κ(y_{k′′,l′′}^{∗}) = Tr_{L̂_{k′} /L}(y_{k′,l′}^{∗} κ(y_{k′′,l′′}^{∗})) ,

since ^{ω′}χ_{k′} ^{ω′′}χ_{k′′} = 1 if and only if ω′′ = ω′κ. The elements κ(y_{k′′,1}), …, κ(y_{k′′,s_{k′′}}) form an O_{L}-basis of κ(D_{L̂_{k′′} /L}^{−1}) = D_{L̂_{k′} /L}^{−1}. Since the last term in the above equation lies in O_{L} for all l′ and l′′, we have Tr_{L̂_{k′} /L}(D_{L̂_{k′} /L}^{−2}) ⊆ O_{L}. But this implies d_{L̂_{k′} /L} = O_{L}. This holds for all

k′ ∈ {1, …, s}. Thus d_{A_{L}} = O_{L}.

2.4 Some A-modules and resolvents

Let C := Map(Γ, K̄) be the K̄-algebra with pointwise multiplication. Then, Ω acts on C via K-algebra automorphisms by

(^{ω}f)(γ) := ω(f(^{ω^{−1}}γ)) ,

for ω ∈ Ω, f ∈ C, γ ∈ Γ. For an intermediate field K ⊆ L ⊆ K̄ let

C_{L} := C^{Ω_{L}} = Map(Γ, K̄)^{Ω_{L}}

be th L-algebra of Ω_{L}-fixed points of C. Moreover, let C_{L} be the maximal O_{L}-order of C_{L}. Thus,

C_{L} = Map(Γ, OK̄)^{Ω_{L}} .

Note that C is a right A-module via the G-action on Γ:

(f · (∑_{g∈G} λ_{g}g))(γ) := ∑_{g∈G} λ_{g}f(^{g}γ)  (10)

for λ_{g} ∈ K̄, f ∈ C, ω ∈ Ω. This action of A on C satisfies

ω(f · a) = (^{ω}f) · (^{ω}a)

for all a ∈ A, f ∈ C, ω ∈ Ω. Thus, the A-module structure on C restricts to an A_{L}-module structure on C_{L} for any intermediate field K ⊆ L ⊆ K̄. Moreover, as apparent from (10), C_{L} is an A_{L}^{∘}-module by restriction.

Similar to Lemma 2.3 for the algebra B we have the following lemma for C.

Lemma 2.11 Let K ⊆ L ⊆ K̄ be an intermediate field, and let γ_{1}, …, γ_{t} ∈ Γ be a set of representatives for the Ω_{L}-orbits of Γ. For each m ∈ {1, …, t} let L̃_{m} denote the fixed field of stab_{Ω_{L}}(γ_{m}) ⩽ Ω_{L}, and let z_{m,1}, …, z_{m,t_{m}} be an L-basis of L̃_{m}. For 1 ⩽ m ⩽ t and 1 ⩽ n ⩽ t_{m}, let c_{m,n} ∈ C be defined by

c_{m,n}(γ) := \begin{cases}ω(z_{m,n}), & if γ = ^{ω}γ_{m} for some ω ∈ Ω_{L}, \\ 0, & otherwise, \end{cases}

for γ ∈ Γ. Assume that L ⊆ M ⊆ N ⊆ K̄ are further intermediate fields. Then the following assertions hold:

2.4 Some A-modules and resolvents 27

(i) The map

τ: C_{L} ≃ ∏_{m=1}^{t} L̃_{m} , f ↦ (f(γ_{m})) ,

is an L-algebra isomorphism. In particular, τ restrict to an isomorphism

τ_{L}: C_{L} ≃ ∏_{m=1}^{t} OL̃_{m} .

(ii) The elements c_{m,n}, 1 ⩽ m ⩽ t, 1 ⩽ n ⩽ t_{m}, form an L-basis of C_{L}. If, for each m ∈ {1, …, t}, the elements z_{m,1}, …, z_{m,t_{m}} form an O_{L}-basis of OL̃_{m}, then the elements c_{m,n}, 1 ⩽ m ⩽ t, 1 ⩽ n ⩽ t_{m}, form an O_{L}-basis of C_{L}.

(iii) One has dim_{L}(C_{L}) = |G|.

(iv) The elements c_{m,n}, 1 ⩽ m ⩽ t, 1 ⩽ n ⩽ t_{m}, form an M-basis of C_{M}.

(v) The map

π_{L}^{M}: M ⊗_{L} C_{L} → C_{M} , λ ⊗_{L} c ↦ λc ,

is an isomorphism of M-algebras such that the diagrams

\begin{matrix}M ⊗_{L} C_{L} & ⟶^{π_{L}^{M}} & C_{M} \\ ^{ω⊗ω}↓ & & ↓^{ω} \\ ω(M) ⊗_{ω(L)} C_{ω(L)} & ⟶^{π_{ω(L)}^{ω(M)}} & C_{ω(M)} \end{matrix}  \begin{matrix}N ⊗_{M} M ⊗_{L} C_{L} & ⟶^{N⊗π_{L}^{M}} & N ⊗_{M} C_{M} \\ ^{can⊗C_{L}}↓ & & ↓^{π_{M}^{N}} \\ N ⊗_{L} C_{L} & ⟶^{π_{L}^{N}} & C_{N} \end{matrix}

commute for all ω ∈ Ω.

Proof All assertions are proved in a similar way as the analogous assertions of Lemma 2.3.

Next we show that the L-algebra C_{L} is an A_{L}-Galois extension in the sense of [CS]. Let us shortly recall the relevant notions in a general setting.

Let R be a commutative ring, let H be an R-Hopf algebra which is finitely generated and projective as R-module. Furthermore, let S be a commutative R-algebra, finitely generated and projective as R-module, which is also a right H-module. Then S is called an H-Galois extension of R if and only if the following conditions are satisfied:

(G1) (i) (st) · h = ∑_{(h)}(s · h_{(1)})(t · h_{(2)}) , (ii) 1_{S} · h = ϵ(h)1_{S} , for all h ∈ H, s, t ∈ S, where Δ(h) = ∑_{(h)} h_{(1)} ⊗ h_{(2)} is the Sweedler notation and the module structure of S over H is denoted by a dot.

(G2) The map

H ⊗_{R} S → Hom_{R}(S, S) , h ⊗_{R} s ↦ (t ↦ (t · h)s) ,

is an isomorphism.

In fact, it is well-known that (G1) is equivalent to S being an H^{∗}-object, and that (G2) is equivalent to the condition that the H^{∗}-object S is a Galois H^{∗}-object in the terminology of [CS, §7].

Proposition 2.12 Let K ⊆ L ⊆ K̄ be an intermediate field. Then the Lalgebra C_{L} is an A_{L}-Galois extension of L. Moreover, if G is abelian, then C_{L} is a free A_{L}-module of rank 1.

Proof It suffices to verify (G1) in the case L = K̄. So let g ∈ G, f, f′ ∈ C, and γ ∈ Γ. Then

((ff′) · g)(γ) = (ff′(^{g}γ) = f(^{g}γ)f′(^{g}γ) = (f · g)(γ)(f′ · g)(γ)
= ((f · g)(f′ · g))(γ) ,

thus, (ff′) · g = (f · g)(f′ · g) which is the statement in (G1) (i). Moreover, (1_{C} · g)(γ) = 1_{C}(^{g}γ) = 1 = 1_{C}(γ), for all γ ∈ Γ. Hence, also (G1) (ii) holds.

In order to prove (G2), we use Lemma 2.1 (vi) and Lemma 2.11 (v) to reduce the assertion to the case L = K̄. Moreover, by Lemma 2.1 (ii) and Lemma 2.11 (iii), it suffices to show that the map in (G2) is injective. So let ∑_{g∈G} g ⊗ f_{g} ∈ A ⊗K̄ C with arbitrarily chosen f_{g} ∈ C for g ∈ G such that it vanishes under the map in (G2). Then

∑_{g∈G} f(^{g}γ)f_{g}(γ) = 0 , for all f ∈ C and all γ ∈ Γ .  (11)

Let g_{0} ∈ G and γ ∈ Γ. Then, choosing f ∈ C in such a way that f(^{g_{0}}γ) = 1 and f(γ′) = 0 for γ′ ≠ ^{g_{0}}γ, the equation in (11) implies f_{g_{0}}(γ) = 0. Thus f_{g_{0}} = 0 for all g_{0} ∈ G. This shows that C_{L} is an A_{L}-Galois extension of L.

If G is abelian, A_{L} is commutative and B_{L} is cocommutative. By [CH, Prop. 2.3, Thm. 3.1], C_{L} and B_{L} are isomorphic as A_{L}-modules, where the A_{L}-module structure of B_{L} is given by (b · ∑_{g∈G} λ_{g}g)(g′) := ∑_{g∈G} λ_{g}b(gg′) for all b ∈ B_{L}, g′ ∈ G. Moreover, A_{L} and B_{L} are isomorphic as A_{L}-modules by sending 1_{A_{L}} to the element l_{1} ∈ B_{L} with l_{1}(g) = δ_{g,1}.

Let G be abelian. Since C_{L} is free over A_{L} for any intermediate field K ⊂ L ⊂ K̄, the following question arises naturally: Is C_{L} free over some O_{L}-order in A_{L}. It is well-known that, if this is the case, it is only possible for the associated order

A_{L}^{ass} := {a ∈ A_{L} | C_{L} · a ⊆ C_{L}}

of C_{L} in A_{L}. Obviously, A_{L}^{∘} ⊆ A_{L}^{ass} ⊆ A_{L}. We will prove in the following chapters that in many of the examples described in 1.1 the module C_{L} is free over A_{L}^{ass}. In the proofs we make use of the resolvent

(f, χ)_{γ_{0}} := ∑_{g∈G} f(^{g}γ_{0})χ(g^{−1}) ∈ K̄  (12)

attached to f ∈ C and χ ∈ Ĝ for fixed γ_{0} ∈ Γ.

For an intermediate field K ⊆ L ⊆ K̄, let L′ ⊆ K̄ now denote a finite extension of L such that Ω_{L′} acts trivially on Ĝ and Γ. We call such a field a splitting field for A_{L} and C_{L}. Note that, since

ω((f, χ)_{γ_{0}}) = (^{ω}f, ^{ω}χ)_{^{ω}γ_{0}} ,

for all ω ∈ Ω, f ∈ C, χ ∈ Ĝ, and γ_{0} ∈ Γ, we then have (f, χ)_{γ_{0}} ∈ L′ for all f ∈ C_{L}, χ ∈ Ĝ, and γ_{0} ∈ Γ. We denote by d_{C_{L}} the discriminant ideal of C_{L} over O_{L}. Thus, in the notation of Proposition 2.11 we have d_{C_{L}} = ∏_{m=1}^{t} d_{L̃_{m}/L}.

Proposition 2.13 Assume that G is abelian and fix γ_{0} ∈ Γ. Let K ⊆ L ⊆ K̄ be an intermediate field, let L ⊆ L′ ⊆ K̄ be a splitting field for A_{L} and C_{L}, and let f ∈ C_{L}. Then f · A_{L} = C_{L} if and only if (f, χ)_{γ_{0}} ≠ 0 for all χ ∈ Ĝ. Moreover, if f · A_{L} = C_{L} then

[C_{L} : f · A_{L}^{∘}]_{O_{L}}^{2}O_{L′} = d_{B_{L}}d_{C_{L}}^{−1} ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2}O_{L′} .

Proof (a) Let χ_{k}, L̂_{k}, y_{k,l}, and the L-basis

â_{k,l} = ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ω(y_{k,l}) ^{ω}e_{χ}  (1 ⩽ k ⩽ s, 1 ⩽ l ⩽ s_{k})  (13)

of A_{L} be given as in Lemma 2.2. Then the elements

((f · â_{k,l})(γ))_{γ∈Γ} ∈ ∏_{γ∈Γ} L′  (14)

generate (τ_{L′} ∘π_{L}^{L′})(L′⊗_{L}f ·A_{L}) ⊆ ∏_{γ∈Γ} L′ over L′. We express these generators by the primitive idempotents ε_{γ}, γ ∈ Γ, of ∏_{γ∈Γ} L′and obtain a transition matrix

M = ((f · â_{k,l})(γ))_{{(k,l) \atop γ∈Γ}} = ((f · â_{k,l})(^{g}γ_{0}))_{{(k,l) \atop g∈G}} ,

so that f · A_{L} = C_{L} if and only if det(M) ≠ 0. We determine det(M). For the entry of M we have

(f · â_{k,l})(^{g}γ_{0}) = \frac{1}{|G|} ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ∑_{g′∈G} ω(y_{k,l}) (^{ω}χ_{k})(g′^{−1})f(^{g′g}γ_{0})

= \frac{1}{|G|} ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ∑_{h∈G} ω(y_{k,l}) (^{ω}χ_{k})(gh^{−1})f(^{h}γ_{0})

= \frac{1}{|G|} ∑_{ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})} ω(y_{k,l}) (f, ^{ω}χ_{k})_{γ_{0}}(^{ω}χ_{k})(g) .

We can write M as M = M_{1}M_{2}, where M_{1} is a block diagonal matrix, the blocks M_{1,k} indexed by k = 1, …, s, with

M_{1,k} = (ω(y_{k,l}))_{{l∈\{1,... ,s_{k}\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k})}} ,

and where

M_{2} = \frac{1}{|G|}((f, ^{ω}χ_{k})_{γ_{0}}(^{ω}χ_{k})(g))_{{k,ω∈Ω_{L}/stab_{Ω_{L}} (χ_{k}) \atop g∈G}} = \frac{1}{|G|}((f, χ)_{γ_{0}}χ(g))_{{χ∈Ĝ \atop g∈G}} .

Now, det(M_{1,k})^{2}O_{L} = d_{L̂_{k}/L}, hence det(M_{1})^{2}O_{L} = d_{A_{L}/L}. Moreover,

det(M_{2})^{2} = ( ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2} ) |G|^{−2|G|} det(χ(g))2_{{χ∈Ĝ \atop g∈G}} = ± ( ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2} ) |G|^{−|G|}

by (7). Thus,

det(M)^{2}O_{L′} = d_{A_{L}}|G|^{−|G|} ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2}O_{L′} .  (15)

This shows that f · A_{L} = C_{L} if and only if (f, χ)_{γ_{0}} ≠ 0 for all χ ∈ Ĝ.

(b) Next we show the assertion about [C_{L} : f · A_{L}^{∘}]_{O_{L}}. Since both sides of the equation behave well under localization, we may assume that O_{L} is local. We transport the two O_{L′}-lattices O_{L′} ⊗_{O_{L}} C_{L} and O_{L′} ⊗_{O_{L}} f · A_{L}^{∘} by the isomorphism τ_{L′} ∘ π_{L}^{L′} into ∏_{γ∈Γ} L′.

Retracing the calculations in (a) one observes that, if y_{k,1}, …, y_{k,s_{k}} is an O_{L}-basis of OL̂_{k}, for each k = 1, …, s, then the elements (13) form an O_{L}-basis of A_{L}, and the elements in (14) form an O_{L′}-basis of (τ_{L′} ∘π_{L}^{L′})(O_{L′} ⊗_{O_{L}} f ·A_{L}). Thus, the calculation in (a) shows that

[τ_{L′}(C_{L′}) : (τ_{L′} ∘ π_{L}^{L′})(O_{L′} ⊗_{O_{L}} f · A_{L})]_{O_{L′}}^{2} = d_{A_{L}}|G|^{−|G|} ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2}O_{L′} . (16)

Moreover, we already know from Proposition 2.6 that

[f · A_{L} : f · A_{L}^{∘}]_{O_{L}}^{2} = [A_{L} : A_{L}^{∘}]_{O_{L}}^{2} = d_{B_{L}}d_{A_{L}}^{−1}|G|^{|G|} .  (17)

Now, it suffices to show that

[τ(C_{L′}) : (τ_{L′} ∘ π_{L}^{L′})(O_{L′} ⊗_{O_{L}} C_{L})]_{O_{L′}}^{2} = d_{C_{L}}O_{L′} ,  (18)

since then, dividing the product of (16) and (17) by (18) yields the result.

Let γ_{m}, L̃_{m}, z_{m,n}, and c_{m,n} be given as in Lemma 2.11 such that, for each m ∈ {1, …, t}, the elements z_{m,1}, …, z_{m,t_{m}} form an O_{L}-basis of OL̃_{m}. Then, the elements c_{m,n}, 1 ⩽ m ⩽ t, 1 ⩽ n ⩽ t_{m}, form an O_{L}-basis of C_{L}. Thus, the elements (τ_{L′} ∘ π_{L}^{L′})(c_{m,n}) form an O_{L′}-basis of (τ_{L′} ∘ π_{L}^{L′})(O_{L′} ⊗_{O_{L}} C_{L}). We express this O_{L′}-basis by the canonical O_{L′}-basis of primitive idempotents ε_{γ}, γ ∈ Γ, of ∏_{γ∈Γ} O_{L′} and obtain a transition matrix M. This is a block diagonal matrix with blocks M_{1}, …, M_{t}, where

M_{m} = (ω(z_{m,n}))_{{n∈\{1,... ,tm\} \atop ω∈Ω_{L}/stab_{Ω_{L}} (γm)}}

for m ∈ {1, …, t}. Since det(M_{m})^{2}O_{L} = d_{L̃_{m}/L}, we obtain (18), and the proof is complete.

As an immediate consequence of Proposition 2.13 we obtain:

Corollary 2.14 In the situation of Proposition 2.13 with f ∈ C_{L} such that (f, χ)_{γ_{0}} ≠ 0 for all χ ∈ Ĝ one has

[C_{L} : f · Ã_{L}]_{O_{L}}^{2}O_{L′} = d_{B_{L}}d_{C_{L}}^{−1}[Ã_{L} : A_{L}^{∘}]_{O_{L}}^{−2} ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2}O_{L′}

for each O_{L}-order Ã_{L} of A_{L} contained in A_{L}^{ass}. In particular, if f ∈ C_{L} is such that

[Ã_{L} : A_{L}^{∘}]_{O_{L}}^{2}d_{C_{L}}O_{L′} = d_{B_{L}} ∏_{χ∈Ĝ}(f, χ)_{γ_{0}}^{2}O_{L′} ,

then C_{L} = f · Ã_{L} is a free Ã_{L}-module of rank one with basis {f}, and consequently, Ã_{L} = A_{L}^{ass}.

To conclude this chapter we consider the case G = Γ. Then C_{L} = B_{L} for any intermediate field K ⊆ L ⊆ K̄. We define a map l ∈ C_{L} by

l(g) = \begin{cases}1, & if g = 1, \\ 0, & if g ≠ 1 \end{cases}  (19)

Corollary 2.15 Let G = Γ. Then A_{L}^{ass} = A_{L}^{∘} and C_{L} is a free A_{L}^{∘}-module of rank one with basis {l}, where l is defined in (19).

Proof We fix γ_{0} = 1. Then (l, χ)_{γ_{0}} = 1 for all χ ∈ Ĝ. Thus the corollary is an immediate consequence of the second part of Corollary 2.14.

3 The cyclotomic case

This section mainly serves as a motivation and a guide-line for the applications to Lubin-Tate formal groups and elliptic curves.

3.1 First proof

Let r, m ∈ ℕ_{0}, m ≥ 1, and let p be a rational prime number. Fix a primitive p^{r}-th root of unity ζ_{p^{r}}, let G = μ_{p^{m}} be the group of p^{m}-th roots of unity and let Γ be the set of p^{m}-th roots of ζ_{p^{r}},

Γ = {γ ∈ ℚ̄ | γ^{p^{m}} = ζ_{p^{r}}}.

We consider ℚ(ζ_{p^{r}}) as our base field and set

K = ℚ(ζ_{p^{r}}), Ω = Ω_{K}, L = ℚ(ζ_{p^{r+m}}),

where ζ_{p^{r+m}} is a primitive p^{r+m}-th root of unity such that ζ_{p^{r+m}}^{p^{m}} = ζ_{p^{r}}. Then G acts simply transitive on Γ,

Γ = {g · ζ_{p^{r+m}} | g ∈ G},

and Ω acts on G and Γ such that ^{ω}(g · γ) = ^{ω}g^{ω}γ for all ω ∈ Ω, g ∈ G and γ ∈ Γ. Following Section 1 we set

A = (K̄G)^{Ω}, B = Map(G, K̄)^{Ω}, C = Map(Γ, K̄)^{Ω}.

As an immediate consequence of Lemma 2.11 we obtain an isomorphism τ,

\begin{matrix}τ : C & ⟶ & ℚ ⊕ ℚ(ζ_{p}) ⊕ … ℚ(ζ_{p^{m}}), \\ f & ⟼ & (f(1), f(ζ_{p}), …, f(ζ_{p^{m}})), \end{matrix} if r = 0,  (20)

and

\begin{matrix}τ : C & ⟶ & L = ℚ(ζ_{p^{r+m}}), \\ f & ⟼ & f(ζ_{p^{r+m}}), \end{matrix} if r ≥ 1.  (21)

We first have a closer look at the more interesting case r ≥ 1. Via τ we endow L with the structure of an A-module. In the sequel we will always write G = {g_{k} = ζ_{p^{m}}^{k} | k ∈ ℤ/p^{m}ℤ}. Let a = ∑_{k=0}^{p^{m}−1} a_{k}g_{k} ∈ A and f ∈ C. Then

τ (f · ∑_{k=0}^{p^{m}−1} a_{k}g_{k}) = ∑_{k=0}^{p^{m}−1} a_{k}f (ζ_{p^{r+m}}^{1+p^{r}k})

= ∑_{k=0}^{p^{m}−1} a_{k}f (ζ_{p^{r+m}})^{σ(1+p^{r}k)} ,

where σ(1 + p^{r}k) denotes the Artin symbol. Thus the A-module structure of L is explicitly given by

α · ∑_{k=0}^{p^{m}−1} a_{k}g_{k} = ∑_{k=0}^{p^{m}−1} a_{k}α^{σ(1+p^{r}k)}  (22)

for α ∈ L, ∑_{k=0}^{p^{m}−1} a_{k}g_{k} ∈ A.

Note that the map

ℤ/p^{m}ℤ ⟶ Gal(L/K),
k + p^{m}ℤ ⟼ σ(1 + p^{r}k)

is a bijection. In addition, if m ≤ r, then it is also a homomorphism of groups, since then σ(1 + p^{r}k_{1})σ(1 + p^{r}k_{2}) = σ(1 + p^{r}(k_{1} + k_{2})) for k_{1}, k_{2} ∈ ℤ. Moreover, A = KG in this case, and identifying G and Gal(L/K) via g_{k} ↦ σ(1 + p^{r}k) the new module structure defined in (22) is nothing else but the usual KGal(L/K)- module structure. However, if m > r, then the new structure does not coincide with the classical Galois module structure. But from a geometric point of view it is nevertheless very natural.

For r ≥ 1 we shall always identify C and L via τ; in particular we write c_{x} = τ^{−1}(x) for x ∈ L. For r = 0 we define a map l ∈ C by

l(γ) = \begin{cases}1, & if γ = 1, \\ 0, & if γ ≠ 1. \end{cases}  (23)

Recall also that A^{∘} = (OK̄G)^{Ω}.

Our aim is to study the integral closure C of O_{K} in C as a module over its associated order A^{ass} = {a ∈ A | C ·a ⊆ C}.The main result of this Section reads as follows:

Theorem 3.1 (i) Let r = 0. Then A^{ass} = A^{∘}and C is free of rank one over A^{ass}. An explicit generator is given by the map l defined in (23).

(ii) Let r ≥ 1. Then A^{ass} is equal to the unique maximal order A ⊆ A and C is free of rank one over A^{ass}. An explicit generator is given by the map c_{θ}, where θ = \frac{1−ζ_{p^{r}}}{1−ζ_{p^{r+m}}}.

Since this section should serve as a motivation on the one hand and a guideline for further applications on the other hand we will give two proofs of Theorem 3.1. The first one is quite short and completely elementary, but has the disadvantage that it does not carry over to applications to Lubin-Tate formal groups or elliptic curves. The second one may appear to be artificial, but it is more structural and based on geometric properties which are shared by the cyclotomic example and the examples concerning Lubin-Tate groups and elliptic curves.

First proof of Theorem 3.1 (i) This is exactly the assertion of Corollary 2.15.

(ii) We claim that Ω acts trivially on Ĝ. The action of Ω obviously factors through Ω_{ℚ(ζ_{p^{m}})}. Thus the claim is certainly true for m ≤ r. For m > r every Galois automorphism ω of ℚ(ζ_{p^{m}})/ℚ(ζ_{p^{r}}) is of the form ω = σ(1 + p^{r}x) with x ∈ ℤ. Then ω^{−1} = σ(1 + p^{r}y) with y ∈ ℤ such that

(1 + p^{r}x)(1 + p^{r}y) ≡ 1(mod p^{m}).

Let χ ∈ Ĝ be defined by χ(g_{1}) = ζ_{p^{m}}^{s}, s ∈ ℤ. Then ^{ω}χ = χ because of

(^{ω}χ) (g_{1}) = ω(χ(ω^{−1}(g_{1})) = σ(1 + p^{r}x)(χ(σ(1 + p^{r}y)ζ_{p^{m}})
= σ(1 + p^{r}x) (χ(ζ_{p^{m}}^{1+p^{r}y})) = σ(1 + p^{r}x) (ζ_{p^{m}}^{(1+p^{r}y)s})
= ζ_{p^{m}}^{(1+p^{r}x)(1+p^{r}y)s} = ζ_{p^{m}}^{s}.

As a consequence of Lemma 2.2 the K-algebra A splits completely over K and the idempotents e_{χ}, χ ∈ Ĝ, form a ℤ-basis of the maximal order A.

Via the ismorphism τ of (21) the order C identifies with O_{L} = O_{K}[ζ_{p^{r+m}}]. Let χ ∈ Ĝ be uniquely defined by χ(g_{1}) = ζ_{p^{m}}^{s}, 0 ≤ s ≤ p^{m} − 1. Using (22) we derive for 0 ≤ j ≤ p^{m} − 1

ζ_{p^{r+m}}^{j} · e_{χ} = \frac{1}{p^{m}} ∑_{k=0}^{p^{m}−1} χ(g_{k}^{−1})ζ_{p^{r+m}}^{jσ(1+p^{r}k)}

= \frac{1}{p^{m}} ∑_{k=0}^{p^{m}−1} ζ_{p^{m}}^{k(j−s)}ζ_{p^{r+m}}^{j} = \begin{cases}ζ_{p^{r+m}}^{j}, & if j = s, \\ 0, & otherwise. \end{cases}

Thus for θ = 1+ζ_{p^{r+m}} +… ζ_{p^{r+m}}^{p^{m}−1} we obtain τ(C) = O_{L} = θ·A and the assertion of the Theorem follows.

Remarks 3.2 (i) In the case r ≥ 1 the associated order A^{ass} is a Hopf order by Proposition 2.10. This raises the following natural question: what is the affine O_{K}-group scheme which is represented by A^{ass}? This question is somehow the starting point for the second proof of Theorem 3.1 in the case r ≥ 1.

(ii) In the case r = 0 the associated order A^{ass} is not a Hopf order by Proposition 2.8, except for p = 2 and m = 1.

3.2 Second proof

We assume the situation described in Section 1. Since the case r = 0 is settled in full generality by Corollary 2.15, we also assume r ≥ 1. Combining ideas of [T1] and [T2] the strategy for our second proof is as follows:

1. Compute the Cartier dual A^{gs} of the O_{K}-Hopf order B^{gs} ⊆ B which represents the affine O_{K}-group scheme μ_{p^{m}} of p^{m}-th roots of unity. A^{gs} is a O_{K}-Hopf order in A.

2. Prove that A^{gs} ⊆ A^{ass}.

3. Apply the theory of tame objects over Hopf algebras of [CH], in particular [CH, Th,(5.4)], to derive that C is locally free over A^{gs}. This implies A^{gs} = A^{ass}.

4. Define “suitable” resolvents (c_{θ}, χ)_{γ_{0}} as in (12), where γ_{0} = ζ_{p^{r+m}}, and compute their prime ideal factorization.

5. Compute [A^{ass} : A^{∘}]_{O_{K}}, d_{B} and d_{C} and apply Corollary 2.14 to conclude that C = c_{θ} · A^{ass}.

The affine group scheme μ_{p^{m}} of p^{m}-th roots of unity is represented by the O_{K}-Hopf order

B^{gs} = O_{K}[X]/ (X^{p^{m}} − 1) .

We view B^{gs} as a order in B via the rule b(X)(g) = b(g) for b(X) ∈ O_{K}[X] and g ∈ G.

Let Tr_{B/K} : B → K denote the trace map Tr_{B/K}(b(X)) = ∑_{g∈G} b(g) and define, as usual, the inverse different of B^{gs} by

D^{−1}(B^{gs}) = {b ∈ B | Tr_{B/K}(bB^{gs}) ⊆ O_{K}}.

In order to make this chapter as self-contained as possible, we will provide proofs of the following two results, although they are already stated in [T2].

Lemma 3.3 [T2, Lemma 1] The inverse different of B^{gs} is given by

D^{−1}(B^{gs}) = \frac{1}{p^{m}} B^{gs}.

Proof We view the set {1, X̄, …, X̄^{p^{m}−1}} ⊆ B^{gs} as a K-basis of B. Let b = ∑_{i=0}^{p^{m}−1} a_{i}X̄^{i} ∈ B, a_{i} ∈ K. Then

b ∈ D^{−1}(B^{gs}) ⟺ ∑_{i=0}^{p^{m}−1} a_{i} ∑_{g∈G} g^{i−j} ∈ O_{K} for j = 0, …, p^{m} − 1.  (24)

From

∑_{g∈G} g^{i−j} = ∑_{k=0}^{p^{m}−1} ζ_{p^{m}}^{k(i−j)} = \begin{cases}p^{m}, & if i − j ≡ 0 (mod p^{m}), \\ 0, & otherwise, \end{cases}

and (24) we conclude that

b ∈ D^{−1}(B^{gs}) ⟺ p^{m}a_{i} ∈ O_{K} for i = 0, …, p^{m} − 1
⟺ b ∈ \frac{1}{p^{m}} B^{gs}.

Proposition 3.4 [T2, Prop. 1] The Cartier dual A^{gs} of B^{gs} is given by

A^{gs} = \{\frac{1}{p^{m}} ∑_{g∈G} f(g)g | f ∈ B^{gs}\} .

Proof We follow very closely the proof of [T2, Prop. 1]. Since the trace pairing

( , ) : B × B → K, (b_{1}, b_{2}) = Tr_{B/K}(b_{1}b_{2})

is non-degenerate we have a natural isomorphism

ξ : D^{−1}(B^{gs}) ⟶ Hom_{O_{K}} (B^{gs}, O_{K}),
d ⟼ (b ↦ ξ(d)(b) = Tr_{B/K}(db)).

By the definition of the Cartier dual we get a natural identification

η : Hom_{O_{K}} (B^{gs}, O_{K}) ⟶ A^{gs},
h ⟼ ∑_{g∈G} a_{g}g, if h(b) = ∑_{g∈G} a_{g}b(g), ∀b ∈ B^{gs}.

Thus for d ∈ D^{−1}(B^{gs}) we obtain

(η ∘ ξ)(d) = ∑_{g∈G} d(g)g

and the equality in the proposition follows from Lemma 3.3.

Since {1, X̄, …, X̄^{p^{m}−1}} constitutes an O_{K}-basis of B^{gs}, the elements

e_{s} = \frac{1}{p^{m}} ∑_{k=0}^{p^{m}−1} ζ_{p^{m}}^{ks}g_{k}, s = 0, …, p^{m} − 1,

form an O_{K}-basis of A^{gs} by Proposition 3.4. Obviously e_{s} = e_{χ} for the abelian character χ ∈ Ĝ defined by χ(g_{1}) = ζ_{p^{m}}^{−s} and thus A^{gs} = A.

So far we have worked out the first step of our stratgy for the second proof of Theorem 3.1. In order to show that A^{gs} ⊆ A^{ass} we proceed as in our first proof. The A^{gs}-module C is certainly locally free, since A^{gs} = A = A^{ass} is a maximal order. In further applications this will not be true, but the theory of tame objects over Hopf orders of Childs and Hurley [CH] provides an easily tested criterion to prove local freeness.

We now compute the resolvent (c_{θ}, χ)_{γ_{0}} for θ = 1 + ζ_{p^{r+m}} + … + ζ_{p^{r+m}}^{p^{m}−1}, γ_{0} = ζ_{p^{r+m}} and χ ∈ Ĝ uniquely defined by χ(g_{1}) = ζ_{p^{m}}^{s}, 0 ≤ s ≤ p^{m} − 1. We obtain

(c_{θ}, χ)_{γ_{0}} = ∑_{g∈G} χ(g^{−1})c_{θ}(gγ_{0})

= ∑_{k=0}^{p^{m}−1} ζ_{p^{m}}^{−ks}c_{θ} (ζ_{p^{r+m}}^{1+p^{r}k})

= ∑_{k=0}^{p^{m}−1} ζ_{p^{m}}^{−ks}c_{θ} (ζ_{p^{r+m}})^{σ(1+p^{r}k)}

= ∑_{k=0}^{p^{m}−1} ζ_{p^{m}}^{−ks}θ^{σ(1+p^{r}k)}
= ζ_{p^{r+m}}^{s} · p^{m}.

From Lemma 2.6 we know that [A^{ass} : A^{∘}]_{O_{K}}^{2} = p^{mp^{m}}d_{B} (note that d_{A} = (1)) and [Wa, Ch. 2, Prop. 2.1] implies that d_{C} = d_{L/K} = p^{mp^{m}}. We are now in position to apply Corollary 2.14:

([A^{ass} : A^{∘}]_{O_{K}}^{2} d_{C}) O_{ℚ(ζ_{p^{m}})} = (p^{mp^{m}}d_{B}d_{C}) O_{ℚ(ζ_{p^{m}})}
= (p^{2mp^{m}}d_{B}) O_{ℚ(ζ_{p^{m}})}

= (d_{B} ∏_{χ∈Ĝ}(c_{θ}, χ)_{γ_{0}}^{2} ) O_{ℚ(ζ_{p^{m}})}.

Hence C = c_{θ} · A^{ass}. In fact we have proved a slightly stronger result, namely:

Theorem 3.5 If r ≥ 1, then the A^{ass}-module C is free of rank one on any map c ∈ C with the property

(c, χ)_{γ_{0}} ∼ p^{m}

for all χ ∈ Ĝ. Moreover, A^{ass} = A^{gs}.

3.3 Composite results

So far we have only considered the prime power case. In this section we will use Theorem 3.1 to derive a complete result in the composite case.

Let n, f^{∗} ∈ ℕ and set f = nf^{∗}. Let G = μ_{n} and Γ = {γ ∈ ℚ̄ | γ^{n} = ζ_{f^{∗}}}. We write

n = ∏_{i=1}^{s} p_{i}^{m_{i}}, f^{∗} = ∏_{i=1}^{s} p_{i}^{r_{i}}, f^{∗} = ∏_{i=1}^{s} p_{i}^{r_{i}+m_{i}}, 0 ≤ r_{i}, m_{i}, r_{i} + m_{i} ≥ 1,

with mutually distinct rational primes p_{i}. Then

G = ∏_{i=1}^{s} G_{i}, Γ = ∏_{i=1}^{s} Γ_{i},

where G_{i} = μ_{p_{i}^{m_{i}}} and Γ_{i} = {γ ∈ ℚ̄ | γ^{p_{i}^{m_{i}}} = ζ_{p_{i}^{r_{i}}} }, i = 1, …, s. We write

K = ℚ(ζ_{f^{∗}}), L = ℚ(ζ_{f}), Ω = Ω_{K},

and define

A = (K̄G)^{Ω}, B = Map(G, K̄)^{Ω}, C = Map(Γ, K̄)^{Ω}.

Let B^{gs} ⊆ B denote the O_{K}-Hopf order which represents the affine O_{K}-group scheme μ_{n} of n-th roots of unity. As before we write A^{gs} for its Cartier dual in A.

Theorem 3.6 Assume the notation as above. Then C is free of rank one over its associated order A^{ass}. If, in addition, r_{i} ≥ 1 for i = 1, …, s, then A^{ass} = A^{gs}.

Proof For i = 1, …, s we set A_{i} = (K̄G_{i})^{Ω} and C_{i} = Map(Γ_{i}, K̄)^{Ω}. Then

A_{1} ⊗_{K} … ⊗_{K} A_{s} → A, a_{1} ⊗ … ⊗ a_{s} ↦ a_{1} ⋯ a_{s}

is an isomorphism of K-algebras. Furthermore,

C_{1} ⊗_{K} … ⊗_{K} C_{s} → C, h_{1} ⊗ … ⊗ h_{s} ↦ (γ_{1} ⋯ γ_{s} ↦ h_{1}(γ_{1}) ⋯ h_{s}(γ_{s}))

is a isomorphism of A-modules.

For i = 1, …, s we set

K_{i} = ℚ(ζ_{p_{i}^{r_{i}}}), A′_{i} = (K̄_{i}G_{i})^{Ω_{K_{i}}} , C′_{i} = Map(Γ_{i}, K̄_{i})^{Ω_{K_{i}}} .

Note that Lemma 2.1(vi) and Lemma 2.11(v) imply

K ⊗_{K_{i}} A′_{i} ≃ A_{i}, K ⊗_{K_{i}} C′_{i} ≃ C_{i}.

Let C′_{i} denote the integral closure of O_{K_{i}} in C′_{i} and denote by A_{i}^{′ass} its associated order. Then, by Theorem 3.1, there exists a map c_{i} ∈ C′_{i} such that

C′_{i} = c_{i} · A_{i}^{′ass}

for i = 1, …, s. Since the discriminants d_{C′_{i}} and d_{K/K_{i}} are relatively prime, we obtain by applying O_{K} ⊗_{O_{K_{i}}} :

C_{i} ≃ O_{K} ⊗_{O_{K_{i}}} C′_{i} = c_{i} (O_{K} ⊗_{O_{K_{i}}} A_{i}^{′ass}) .

Thus A_{i}^{ass} ≃ O_{K} ⊗_{O_{K_{i}}} A_{i}^{′ass}. Again we note that the discriminants d_{C_{i}}, i = 1, …, s, are mutually relative coprime. Hence

C ≃ C_{1} ⊗_{O_{K}} … ⊗_{O_{K}} C_{s} = c_{1} · A_{1}^{ass} ⊗_{O_{K}} … ⊗_{O_{K}} c_{s}A_{s}^{ass},

or in other words,

C = c · A^{ass} with c = c_{1} ⋯ c_{s}, A^{ass} ≃ A_{1}^{ass} ⊗_{O_{K}} … ⊗_{O_{K}} A_{s}^{ass}.

If r_{i} ≥ 1 for i = 1, …, s, then A_{i}^{ass} is the dual of the O_{K}-Hopf order B_{i}^{gs} which represents the O_{K}-group scheme μ_{p_{i}^{m_{i}}}. The affine group scheme μ_{n} is the fibre product of the μ_{p_{i}^{m_{i}}} and therefore represented by B^{gs} = B_{1}^{gs} ⊗_{O_{K}} … ⊗_{O_{K}} B_{s}^{gs}. Now the second assertion of Theorem 3.6 follows from A^{ass} ≃ A_{1}^{ass} ⊗_{O_{K}} … ⊗_{O_{K}} A_{s}^{ass}.

4 Relative Lubin-Tate formal groups

Our main reference for the general theory of relative Lubin-Tate formal groups is [dS].

Throughout this section p denotes a rational prime number. For any extension field ℚ_{p} ⊆ L ⊆ ℚ̄_{p} we write O_{L} for the integral closure of ℤ_{p} in L and 𝔭_{L} for its maximal ideal. We fix a finite extension field F of ℚ_{p} and we set q = |O_{F} /𝔭_{F} |.

Let d > 0 be a fixed integer. We denote by F′/F the unramified extension of degree d > 0 and by ϕ the Frobenius automorphism of F′/F. We fix an element ξ ∈ F such that v_{F} (ξ) = d, where v_{F} denotes the normalized valuation (i.e., v_{F} (F^{∗}) = ℤ) of F.

As in [dS, Ch. I] we set

F_{ξ} = {f ∈ O_{F′}[[X]] | f ≡ π′Xmod X^{2},
N_{F′/F} (π′) = ξ and f ≡ X^{q}mod 𝔭_{F′}[[X]]}

For any power series g over O_{F′} in one or more indeterminates, let g^{ϕ} arise from applying ϕ to the coefficients of g.

From [dS, Ch.I, Th.1.3] we know that for every f ∈ F_{ξ} there exists a unique one-dimensional commutative formal group F_{f} defined over O_{F′} such that f ∈ Hom(F_{f}, F_{f}^{ϕ}). Note that f^{ϕ} ∈ F_{ξ} and F_{f}^{ϕ} = F_{f^{ϕ}}. Of course, classical Lubin-Tate formal groups correspond to the case d = 1.

In order to introduce all the necessary notation we recall

Proposition 4.1 ([dS, Ch.I, (1.5)]) Let f(X) = π_{1}X +…, g(X) = π_{2}X +… be in F_{ξ}. Let a ∈ O_{F′} satisfy a^{ϕ−1} = π_{2}/π_{1}. Then there exists a unique power series [a]_{f,g} ∈ O_{F′}[[X]] such that

(i)  [a]_{f,g} ≡ aXmod X^{2},
(ii)  [a]_{f,g}^{ϕ} ∘ f = g ∘ [a]_{f,g}.

Moreover, [a]_{f,g} ∈ Hom(F_{f}, F_{g}). The map

{a ∈ O_{F′} | a^{ϕ−1} = π_{2}/π_{1}} → Hom(F_{f}, F_{g}), a ↦ [a]_{f,g}

is a group isomorphism and in the case f = g a ring isomorphism O_{F} ≃ End(F_{f}). Furthermore, if h(X) = π_{3}X + … ∈ F_{ξ} and b^{ϕ−1} = π_{3}/π_{2}, then [ab]_{f,h} = [b]_{g,h} ∘ [a]_{f,g}.

In the sequel we shall always write [a]_{f} for [a]_{f,f}. If f(X) = π′X + … ∈ F_{ξ} and i ≥ 0, then we put

f^{(i)} = f^{ϕ^{i−1}} ∘ … ∘ f^{ϕ} ∘ f.  (25)

Then f^{(i)} ∈ Hom(F_{f}, F_{f^{ϕ^{i}}}) and f^{(d)} = [ξ]_{f} ∈ End(F_{f}). Moreover, for α ∈ O_{F′}

we write α^{(i)} = α^{ϕ^{i−1}} ⋯ α^{ϕ} · α. It then follows that f^{(i)}(X) = [π^{′(i)}]_{f,f^{ϕ^{i}}} (X).

As usual we endow the set 𝔭F̄ with the structure of an O_{F} -module, denoted by F_{f}(𝔭F̄), by setting

x +_{f} y = F_{f}(x, y) , a · x = [a]_{f}(x) ,

for x, y ∈ 𝔭F̄ and a ∈ O_{F}.

We now fix ξ ∈ O_{F} with v_{F} (ξ) = d and a power series f(X) = π′X+… ∈ F_{ξ}. Let π ∈ O_{F} be any prime element. For n ∈ ℕ_{0} we define the group of 𝔭_{F}^{n} -torsion points of F_{f} by

G_{f,n} = {x ∈ 𝔭F̄ | [a]_{f}(x) = 0, ∀a ∈ 𝔭_{F}^{n} }
= {x ∈ 𝔭F̄ | [π^{n}]_{f}(x) = 0}
= ker(f^{(n)} : F_{f}(𝔭F̄) → F_{f^{ϕ^{n}}} (𝔭F̄))

(see [dS, Ch. I, 1.7]).

In fact, G_{f,n} is an abelian group under +_{f} of order q^{n}. The field F_{ξ,n} = F′(G_{f,n}) does not depend on the choice of f ∈ F_{ξ}. F_{ξ,n}/F is abelian and F_{ξ,n}/F′ is totally ramified of degree q^{n−1}(q − 1). Every α ∈ G_{f,n} ∖ G_{f,n−1} generates F_{ξ,n} over F′ and, in addition, α is a uniformizer for F_{ξ,n}. This implies that O_{F_{ξ,n}} = O_{F′}[α]. See [dS, Ch. I, Prop. 1.8] for more details.

We are now ready to describe a set-up that fits into the general framework of Section 1. We fix integers r ≥ 0, m ≥ 1 and set G = G_{f,m}. We choose a primitive 𝔭_{F}^{r} -torsion point β (i.e. β ∈ G_{f,r} ∖ G_{f,r−1}) and set

Γ = {γ ∈ 𝔭F̄ | f^{(m)}(γ) = β} .

We consider F_{ξ,r} as our base field and in order to simplify notation we set

K := F_{ξ,r} , Ω := Ω_{K} , L := F_{ξ,r+m} .

The data O_{K}, K, G and Γ now satisfy the axioms postulated at the beginning of Section 1, where Ω := Ω_{K} acts on G and Γ through Galois automorphisms (thus we also write ω(g) and ω(γ) instead of ^{ω}g and ^{ω}γ), and where G acts on Γ by translation:

gγ = g +_{f} γ ,

for g ∈ G and γ ∈ Γ. In particular, |Γ| = |G| = q^{m}.

As in Section 1 we set

A := (K̄G)^{Ω} , B := Map(G, K̄)^{Ω} , C := Map(Γ, K̄)^{Ω} ,

omitting the index K.

For r ≥ 1 we have L = K(γ_{0}). The following lemma reveals the connection to results of Cassou-Noguès and Taylor (see [CT, Ch. X, Thm. 3.3]).

For the rest of this section we fix an element γ_{0} ∈ Γ. Then L = K(γ_{0}).

Lemma 4.2 Suppose that r ⩾ m ⩾ 1. Then the following assertions hold: (a) The map

τ : C ⟶ L, c ⟼ c(γ_{0}),

is an isomorphism of K-algebras.

(b) One has A = KG.

(c) The map

G → Gal(L/K) , g ↦ ω_{g} ,

where ω_{g} is uniquely determined by ^{(ω_{g})}γ_{0} = ^{g}γ_{0} (= g +_{f} γ_{0}), is a group isomorphism.

(d) Identifying G and Gal(L/K) as in (c), the map τ is also an isomorphism of A-modules, where we consider L endowed with the right A-module structure x · a := ax for a ∈ A and x ∈ L.

Proof (a) First we show that Ω acts transitively on Γ. In fact, applying the Weierstrass Preparation theorem we can write f^{(m)}(X) − β = h(X)u(X) with a distinguished polynomial h(X) ∈ O_{K}[[X]] of degree q^{m} and a unit u(X) ∈ O_{K}[[X]]^{∗}. Since β is a uniformizer in K, h(X) is an Eisenstein polynomial and therefore irreducible. Hence the Galois group Ω acts transitively on Γ = {x ∈ F̄ | h(x) = 0}. Now the result follows from K(γ_{0}) = L.

(b) This is immediate from G ⊆ K.

(c) Let g, h ∈ G. Then

ω_{(g+_{f} h)}γ_{0} = (g +_{f} h) +_{f} γ_{0} = g +_{f} ω_{h}(γ_{0}) = ω_{h}(g +_{f} γ_{0})
= ω_{h}ω_{g}(γ_{0}) = (ω_{g}ω_{h})(γ_{0}) ,

since g ∈ K and Ω/Ω_{L} is an abelian group. This implies that the map g ↦ ω_{g} is a group homomorphism, which is bijective, since G and Ω act transitively on Γ.

(d) This follows from

(c · g)(γ_{0}) = c(g +_{f} γ_{0}) = c(^{(ω_{g})}γ_{0}) = ω_{g}(c(γ_{0})) ,

for c ∈ C and g ∈ G.

At this point the same remarks as in the cyclotomic case (following (22)) apply. Of course, the map τ of Lemma 6.2 is an isomorphism of K-algebras for arbitrary r, m ⩾ 1. The map in (c), however, is in general just a bijection and not a homomorphism. Via τ we can endow L with the structure of an A-module, which for m > r does not coincide with the usual Galois module structure.

If r ≥ 1 we shall always identify C and L via τ in the following; in particular we write c_{x} = τ^{−1}(x) for x ∈ L.

We recall the following trace relation which is basic for the rest of this section.

Proposition 4.3 ([Ch, Lemma 3.2]) For i ∈ ℕ_{0} one has

s_{i} = ∑_{g∈G} g^{i} ∈ 𝔭_{F′}^{m}.

Moreover, s_{q^{m}−1} has exact F′-valuation m and for i > q^{m} − 1 the F′-valuation of s_{i} is strictly bigger than m.

Although obvious at this point, we remark that for i = 0 the summand 0^{0} has to be interpreted as 1. We also remark that π^{′(m)} is associated to π^{m}.

We recall that the associated order of C in A is defined by

A^{ass} = {a ∈ A | C · a ⊆ C} .

Motivated by [T1, Theorem 3] and the cyclotomic case of Section 3.2 we will prove that A^{ass} coincides with the Cartier dual of the O_{K}-Hopf order which represents the O_{K}-group scheme of 𝔭_{F}^{m}-torsion on F_{f}. This affine group scheme is represented by the O_{K}-Hopf order

B^{gs} = \frac{O_{K}[[X]]}{(f^{(m)}(X))}.  (26)

We view B^{gs} as an order in B via the rule b(X)(g) = b(g) for b(X) ∈ O_{K}[[X]] and g ∈ G. Let A^{gs} ⊆ A be the Cartier dual of B^{gs}. Then A^{gs} is an O_{K}-Hopf order. For a thorough discussion of these facts the reader is referred to [BT, II, §7].

Let Tr_{B/K} : B → K denote the trace map Tr_{B/K}(b(X)) = ∑_{g∈G} b(g) and write D^{−1}(B^{gs}) for the inverse different of B^{gs}:

D^{−1}(B^{gs}) = {b ∈ B | Tr_{B/K}(bB^{gs}) ⊆ O_{K}}.

The following Lemma is completely analogous to Lemma 3.3 (see also [T2, Lemma 1]).

Lemma 4.4 The inverse different of B^{gs} is given by

D^{−1}(B^{gs}) = \frac{1}{π^{′(m)}} B^{gs}.

Proof Replacing the well-known trace relations for roots of unity by the trace relations of Proposition 4.3 this is proved as Lemma 3.3. For the reader’s convenience we give the details.

The set {¯1, X̄, …, X̄^{q^{m}−1}} constitutes an O_{K}-basis of B^{gs} and also an Kbasis of B. Let b = ∑_{i=0}^{q^{m}−1} a_{i}X̄^{i} ∈ B, a_{i} ∈ K. Then:

b ∈ D^{−1}(B^{gs}) ⟺ ∑_{i=0}^{q^{m}−1} a_{i} ∑_{g∈G} g^{i+j} ∈ O_{K} for j = 0, …, q^{m} − 1.  (27)

From Propostion 4.3 it follows immediately that \frac{1}{π^{′(m)}} B^{gs} ⊆ D^{−1}(B^{gs}). For the converse inclusion we set v_{0} = min{vF̄(a_{i}) | i = 0, …, q^{m} − 1} and i_{0} = min{i | vF̄(a_{i}) = v_{0}}, where vF̄ is the extension to F̄ of the normalized valuation v_{F}. Then the right hand side of (27) equals for j = q^{m} − 1 − i_{0}

∑_{i=0}^{i_{0}−1} a_{i} ∑_{g∈G} g^{q^{m}−1+i−i_{0}} + a_{i_{0}} ∑_{g∈G} g^{q^{m}−1} + ∑_{i=i_{0}+1}^{q^{m}−1} a_{i} ∑_{g∈G} g^{q^{m}−1+i−i_{0}} ∈ O_{K}.

Again from Proposition 4.3 we conclude that the vF̄-valuation of the middle summand is equal to v_{0}+m, whereas the other summands have valuation strictly bigger than v_{0} + m. Thus v_{0} + m ≥ 0, which proves D^{−1}(B^{gs}) ⊆ \frac{1}{π^{′(m)}} B^{gs}.

In what follows, the elements of G play two different roles in the group algebra F̄G: on the one hand they occur as group elements, on the other hand they are field elements of F̄ and occur as coefficients in F̄G. To distinguish these different roles we henceforth write x_{g} instead of g whenever g is considered as a field element. Moreover we write g_{0} for the unit element in G. Following [CT, Ch. X, Def. 3.2] we introduce certain special elements of the algebra A. For i ≥ 0 we set

σ_{i} := \frac{1}{π^{′(m)}} ∑_{g∈G} x_{g}^{i}(g − g_{0}) .  (28)

It is immediate that these elements are Ω-invariant.

The next proposition is the analogue of Proposition 3.4 (see also [T2, Prop. 1]).

Proposition 4.5 The Cartier dual A^{gs} of B^{gs} is given by

A^{gs} = \{\frac{1}{π^{′(m)}} ∑_{g∈G} f(g)g | f ∈ B^{gs}\} = O_{K} · g_{0} + ∑_{i=0}^{q^{m}−2} O_{K} · σ_{i}.

Proof The first equality is proved as Proposition 3.4. In order to prove the second one we first define l′ ∈ B^{gs} by

l′(g) = \begin{cases}π^{′(m)}, & if g = g_{0}, \\ 0, & if g ≠ g_{0}. \end{cases}  (29)

Note that l′ = f^{(m)}(X)/X. Writing f^{(m)}(X) = h(X)u(X) with a distinguished polynomial h(X) ∈ O_{K}[[X]] of degree q^{m} and a unit u(X) = u_{0}+… ∈ O_{K}[[X]]^{∗} we see that

l′ = u_{0} \frac{h(X)}{X} + h(X)\frac{u(X) − u_{0}}{X} ≡ u_{0} \frac{h(X)}{X} (mod (f^{(m)}(X))).

It immediately follows that the set {¯1, X̄, …, X̄^{q^{m}−2}, l′} also forms an O_{K}-basis of B^{gs}. Now the second equality is an immediate consequence of Lemma 4.3.

For later reference we deduce from Lemma 4.4 and Proposition 4.5

Corollary 4.6 [T2, Proposition 2] B^{gs} is a free A^{gs}-module on the map l′ defined in (29).

Proof Consider the map κ = ¯ ∘ η ∘ ξ : D^{−1}(B^{gs}) → A^{gs} defined by

κ(d) = (¯ ∘ η ∘ ξ)(d) = ∑_{g∈G} d(g)g^{−1}, d ∈ D^{−1}(B^{gs})

(for the definition of η ∘ ξ see the proof of Proposition 3.4). It is easily verified that κ is an isomorphism of A^{gs}-modules. Hence we obtain

\frac{1}{π^{′(m)}} B^{gs} = D^{−1}(B^{gs}) = κ^{−1}(A^{gs}) = κ^{−1}(1) · A^{gs} = l · A^{gs}.

In the sequel we assume r ≥ 1, since by Corollary 2.15 the module structure of C = B is well-known for r = 0. Our aim is to show that A^{gs} is equal to the associated order A^{ass}. The following lemma is a first step in this direction.

Lemma 4.7 The elements σ_{i}, i ⩾ 0, are contained in A^{ass}. In particular, A^{gs} ⊆ A^{ass}.

Proof The proof is simply an adaptation of the proof of [CT, Ch. X, Lemma 3.5] to our situation. For the reader’s convenience we give a short translation into our setting and notation. The maximal O_{K}-order C of C identifies via τ with the ring of integers O_{L} in L. Since O_{L} = O_{K}[γ_{0}], it suffices to show that

c_{γ_{0}^{k}} · σ_{i} ∈ C  (30)

for 0 ⩽ i and 0 ⩽ k ⩽ q^{m} − 1. To achieve this we compute τ(c_{γ_{0}^{k}} · σ_{i}):

τ(c_{γ_{0}^{k}} · σ_{i}) = \frac{1}{π^{′(m)}} ∑_{g∈G} x_{g}^{i}(c_{γ_{0}^{k}}(γ_{0} +_{f} g) − c_{γ_{0}^{k}}(γ_{0}))

= \frac{1}{π^{′(m)}} ∑_{g∈G} x_{g}^{i}(^{ω_{g}}(γ_{0}^{k}) − γ_{0}^{k})

= \frac{1}{π^{′(m)}} ∑_{g∈G} x_{g}^{i}(F_{f}(γ_{0}, g)^{k} − γ_{0}^{k})

= \frac{1}{π^{′(m)}} ∑_{g∈G} x_{g}^{i} · gF_{1}(γ_{0}, g) ,

where we have set F_{f}(X, Y)^{k} = X^{k} + Y F_{1}(X, Y). Recall that g = x_{g} and write Y F_{1}(X, Y) = ∑_{s=0}^{∞} X^{s}a_{s}(Y) with a_{s}(Y) ∈ O_{F′} [[Y]]. Then

τ(c_{γ_{0}^{k}} · σ_{i}) = ∑_{s=0}^{∞} γ_{0}^{s} · \frac{1}{π^{′(m)}} ∑_{g∈G} g^{i}a_{s}(g).

Since the right-hand term is integral by Proposition 4.3, this establishes (30).

From Lemma 4.7 we deduce the following corollary which may be viewed as a generalization of the trace relations of Proposition 4.3.

Corollary 4.8 Let χ ∈ Ĝ and i ⩾ 0. Then

\frac{1}{π^{′(m)}} ∑_{g∈G} g^{i}χ(g) ∈ OF̄ .

Proof Each χ ∈ Ĝ induces a homomorphism A → F̄ of F -algebras, which we again denote by χ. Together with Lemma 4.7, this implies that

χ(σ_{i}) = \frac{1}{π^{′(m)}} ∑_{g∈G} g^{i}(χ(g) − 1)

is integral over O_{F}. Now we easily deduce the integrality of \frac{1}{π^{′(m)}} ∑_{g∈G} g^{i}χ(g) from Proposition 4.3.

Now that we know that the Hopf order A^{gs} acts on C we can apply the results of [CH] to show that C is a free A^{gs}-module (necessarily of rank one). In this context recall the definition of tamenes of [CH, Def. (2.2)].

The module of integrals I is defined by

I = {a ∈ A^{gs} | a′a = ϵ(a′)a for all a′ ∈ A^{gs}} .

Recall the definition of σ_{i} in (28).

Lemma 4.9 With the above notation one has I = \frac{1}{π^{m}} O_{K} ∑_{g∈G} g.

Proof The inclusion “⊇” is immediate. For the converse let a = ∑_{g∈G} λ_{g}g, λ_{g} ∈ F̄, be an element in I. Since A^{gs} is generated by g_{0}, σ_{0}, …, σ_{q^{m}−1} we obtain the condition

a ∈ I ⟺ σ_{i} · ∑_{g∈G} λ_{g}g = 0 for i = 0, …, q^{m} − 2 .

On multiplying and comparing coefficients we derive

∑_{g∈G\{g_{0}}} x_{g}^{i}(λ_{hg^{−1}} − λ_{h}) = 0 ,

for all i = 0, …, q^{m} − 2 and h ∈ G. The Vandermonde matrix (x_{g}^{i})_{i=0,... ,q^{m}−2, g∈G\{g_{0}}} is obviously invertible. Therefore, λ_{hg^{−1}} = λ_{h}, for all g, h ∈ G, which in turn implies λ = λ · ∑_{g∈G} g with λ ∈ K. Now the result follows, since π^{m} is the highest possible denominator for elements in A^{gs}.

Corollary 4.10 Let r ≥ 1. Then the associated order A^{ass} is equal to A^{gs} and C is free of rank one over A^{ass}.

Proof From [Ch, Lemma 3.1 (b)] we deduce that C ·I = O_{K}, which shows that C is a tame A^{gs}-object. Now [CH, Thm. (5.4)] implies that C is free over A^{gs} of rank one. Hence A^{ass} = A^{gs}.

The disadvantage of the approach we have taken so far is that we do not get an explicit generator. These local results are certainly of interest for themselves, but they also play an important role if we want to derive analogous results in the global situation of Example 1.1 (d). To obtain a link between global and local we will need an explicit generator or, and this will lead to even stronger results, a relation between local and global resolvents.

For χ ∈ Ĝ, we define the resolvent function

R_{χ}(X) := ∑_{g∈G} \frac{f^{(m)}(X)}{g +_{f} X}χ(g^{−1}) ∈ F̄[[X]] .

Theorem 4.11 For each χ ∈ Ĝ one has

R_{χ}(X) = π^{′(m)} · u_{χ}(X)

with a unit u_{χ}(X) ∈ OF̄[[X]]^{×}.

Proof First we note that

R_{χ}(0) = \frac{f^{(m)}(X)}{X}|_{X=0} = π^{′(m)} .

Note that for g ∈ G one has f^{(m)}(g +_{f} X) = f^{(m)}(X). Since f^{(m)}(X) has no constant term, it follows from

R_{χ}(X) = ∑_{g∈G} \frac{f^{(m)}(g +_{f} X)}{g +_{f} X}χ(g^{−1})

that R_{χ}(X) ∈ OF̄[[X]]. Hence it suffices to show that each coefficient of R_{χ}(X) is divisible by π^{′(m)}. However, it is easily seen that we may write

\frac{f^{(m)}(g +_{f} X)}{g +_{f} X} = ∑_{i=0}^{∞} q_{i}(g)X^{i} ,

where q_{i} ∈ O_{F′}[[X]], i ∈ ℕ_{0}, are power series depending only on the formal group F_{f}. Hence we may deduce

R_{χ}(X) = ∑_{i=0}^{∞}[∑_{g∈G} q_{i}(g)χ(g^{−1})]X^{i}.

This completes the proof, since by Corollary 4.8 the terms in brackets are divisible by π^{′(m)}.

Consider the map

c: Γ → F̄ , γ ↦ \frac{f^{(m)}(γ)}{γ} .  (31)

Since f^{(m)}(X) ∈ O_{F′}[[X]], it is Ω-invariant. Thus, c ∈ C, and c = c_{θ} with θ = \frac{f^{(m)}(γ_{0})}{γ_{0}}. Moreover, recalling the definition of the resolvent (c, χ)_{γ_{0}} for χ ∈ Ĝ from Section 2.4, we have the following relation:

(c, χ)_{γ_{0}} = ∑_{g∈G} c(^{g}γ_{0}) χ(g^{−1}) = ∑_{g∈G} \frac{f^{(m)}(g +_{f} γ_{0})}{g +_{f} γ_{0}} χ(g^{−1})

= ∑_{g∈G} \frac{f^{(m)}(γ_{0})}{g +_{f} γ_{0}}χ(g^{−1}) = R_{χ}(γ_{0}) .  (32)

We are now ready to state and prove the main result of this section.

Theorem 4.12

(i) Let r = 0. Then A^{ass} = A^{∘} and C = B is free of rank one with A^{∘}-generator l ∈ B, where l is defined in (19). (ii) Let r ≥ 1. Then one has:

(a) The A^{ass}-module C is free of rank one on any map c ∈ C with the property that

(c, χ)_{γ_{0}} ∼ π^{m}

for all χ ∈ Ĝ.

(b) Any function c_{x} ∈ C with x ∈ L^{×} having 𝔭_{L}-valuation q^{m} − 1 is an A^{ass}- basis of C. In particular,

C = c · A^{ass}  (33)

with c = c_{θ} from (31).

Proof (i) follows from Corollary 2.15, so we only have to consider (ii).

(a) Let L′ be a splitting field for A, B and C. Then, by Corollary 2.14, it suffices to show that

[A^{ass} : A^{∘}]_{O_{K}}^{2} d_{C}O_{L′} = d_{B} ∏_{χ∈Ĝ}(c, χ)_{γ_{0}}^{2}O_{L′} ,  (34)

with A^{∘} = (OF̄G)^{Ω}. By assumption

(c, χ)_{γ_{0}} ∼ π^{m} ,

for each χ ∈ Ĝ. On the other hand [Ch, Lemma 3.1] implies that d_{C} ∼ π^{mq^{m}}. It therefore suffices to show

[A^{ass} : A^{∘}]_{O_{K}}^{2} = d_{B}π^{mq^{m}} .  (35)

In order to prove this equality we split the above index and show

[O_{L′} ⊗_{O_{K}} A^{ass} : O_{L′}G]_{O_{L′}}^{2} = π^{mq^{m}}O_{L′}  (36)

and

[O_{L′}G : O_{L′} ⊗_{O_{K}} A^{∘}]_{O_{L′}}^{2} = d_{B}O_{L′} .  (37)

Equation (37) follows immediately if we recall from Lemma 2.1 that the elements

∑_{ω∈Ω/stab_{Ω}(g_{i})} ω(x_{i,j}) ^{ω}g_{i} (1 ⩽ i ⩽ r, 1 ⩽ j ⩽ r_{i})

constitute an O_{K}-basis of A^{∘}, where {g_{1}, …, g_{r}} ⊆ G is a set of representatives for the Ω-orbits of G and x_{i,1}, …, x_{i,r_{i}} is an O_{K}-basis of O_{K_{i}}, for K_{i} being the fixed field of stab_{Ω}(g_{i}).

The proof of (36) follows very closely the proof of [CT, Ch. X, Thm. 4.1]. The O_{L′}-basis g_{0} ∪ {g − g_{0} | g ∈ G \ {g_{0}}} of O_{L′}G is transformed to the O_{L′}- basis 1 ⊗ g_{0} ∪ {1 ⊗ σ_{i} | i = 0, …, q^{m} − 2} of O_{L′} ⊗_{O_{K}} A^{ass} by means of the matrix

S = ( \begin{matrix}1 & 0 & ⋯ & 0 \\ 0 & π^{′(m)}^{−1} & ⋯ & π^{′(m)}^{−1} \\ 0 & π^{′(m)}^{−1}g_{1} & ⋯ & π^{′(m)}^{−1}g_{N} \\ ⋮ & ⋮ & & ⋮ \\ 0 & π^{′(m)}^{−1}g_{1}^{N−1} & ⋯ & π^{′(m)}^{−1}g_{N}^{N−1} \end{matrix} ) ,

where we have set G = {g_{0}, g_{1}, …, g_{N}} with N = q^{m} − 1. Recall that π^{′(m)} ∼ π^{m}. By the Weierstrass Preparation Theorem we may write f^{(m)}(X) = h_{m}(X)u(X) with a distinguished polynomial h_{m}(X) and a unit u(X) ∈ O_{L}[[X]]^{×}. Since

\frac{h_{m}(X)}{X} = ∏_{j=1}^{N}(X − g_{j}) ,

we conclude by Vandermonde that

det(S)^{2} ∼ ±π^{−2mN} · ∏_{j=1}^{N} (\frac{d}{dX} \frac{h_{m}(X)}{X})|_{X=g_{j}} = ±π^{−2mN} · ∏_{j=1}^{N} \frac{h′_{m}(g_{j})}{g_{j}}.

Adapting the proof of [CT, Ch. X, Lemma 2.5] we can show that h′_{m}(g_{j}) ∼ π^{′(m)} for j = 1, …, N. Furthermore ∏_{j=1}^{N} g_{j} is associated to the leading coefficient of f^{(m)}(X), which is π^{′(m)}. Summing up we obtain

det(S)^{2} ∼ ±π^{−2mN} · π^{mN−m} = π^{−mq^{m}} ,

which proves (36).

(b) It follows from Theorem 4.11 together with (32) that for c = c_{θ} we have (c, χ)_{γ_{0}} ∼ π^{m} for all χ ∈ Ĝ. By (a) we conclude that C = c · A^{ass}. Note that v_{L}(θ) = q^{m} − 1 since γ_{0} (resp. β) is a uniformizing element in L (resp. K).

Let x ∈ N^{×} have 𝔭_{L}-valuation q^{m}−1. Since L/Kis totally ramified of degree q^{m}, there exists a unit u ∈ O_{K}^{×} such that

x ≡ uθ (mod 𝔭_{K}O_{L}) resp. c_{x} ≡ uc_{θ} (mod 𝔭_{K}C) .

Together with (33) this implies

C = c_{x} · A^{ass} + 𝔭_{L}C .

Now the full statement in (b) is a consequence of Nakayama’s Lemma.

We now have a closer look at the associated order when F = ℚ_{p}. The next theorem should be compared to Theorem 3.1 and [CT, Ch. X, Th. 4.1].

Theorem 4.13 Let F = ℚ_{p} and assume that r ≥ 1. Then A^{ass} is the maximal order A in A, and d_{A} = O_{K}.

Proof From Lemma 2.6 we know that

[A : A^{∘}]_{O_{K}}^{2} = d_{B}d_{A}^{−1}p^{mp^{m}} ,

whereas from (35) we obtain

[A^{ass} : A^{∘}]_{O_{K}}^{2} = d_{B}p^{mp^{m}} .

Therefore [A : A^{ass}]_{O_{K}}^{2} = d_{A}^{−1}, which forces A = A^{ass}and d_{A} = O_{K}.

To conclude this Section we prove that B^{gs} and C are naturally isomorphic as A^{gs}-modules after applying the functor O_{L} ⊗_{O_{K}} . In the terminology of [T2] or [BT] this means that C is a principal homogeneous space for B^{gs}. Note that O_{L} ⊗_{O_{K}} B^{gs} = B_{L}^{gs} and O_{L} ⊗_{O_{k}} A^{gs} = A_{L}^{gs}.

Proposition 4.14 Suppose that r ≥ 1. Then the map ξ′ : B → C defined by ξ′(b)(γ) = b(γ −_{f} γ_{0}), b ∈ B, γ ∈ Γ, induces an isomorphism

ξ : O_{L} ⊗_{O_{K}} B^{gs} ⟶ O_{L} ⊗_{O_{K}} C

of A_{L}^{gs}-modules.

Proof By Theorem 4.12(b) it suffices to show that ξ^{−1}(c_{θ}) generates B_{L}^{gs} over A_{L}^{gs}. We write

ξ^{−1}(c_{θ}) = ∑_{g∈G} \frac{1}{π^{′(m)}} c_{θ}(γ_{0} +_{f} g)l′^{g}

with the map l′ defined in (29). By the definition of c_{θ} we further compute

ξ^{−1}(c_{θ}) = ∑_{g∈G} \frac{1}{π^{′(m)}} θ^{ω_{g}}l′^{g}

= l′ · ∑_{g∈G} \frac{1}{π^{′(m)}} θ^{ω_{g}}g.

By Corollary 4.6 it remains to show that

λ := ∑_{g∈G} \frac{1}{π^{′(m)}} θ^{ω_{g}}g ∈ (A_{L}^{gs})^{∗}.

From the definition of θ we deduce

θ^{ω_{g}} = (\frac{f^{(m)}(γ_{0})}{γ_{0}})^{ω_{g}} = \frac{f^{(m)}(γ_{0} +_{g} g)}{γ_{0} +_{f} g}.

Therefore θ^{ω_{g}} = b(X)(g) with b(X) = \frac{f^{(m)}(γ_{0}+_{g}X)}{γ_{0}+_{f} X} ∈ O_{L}[[X]] and we conclude from (26) and Proposition 4.5 that λ ∈ A_{L}^{gs}. Mapping λ to its Wedderburn components (see Lemma 2.2(i)) we derive from (32) and Theorem 4.11 that λ is an invertible element of the maximal order A_{L}. Hence the result follows from (A_{L}^{gs} ∩ A^{∗}) = (A_{L}^{gs})^{∗}.

5 Elliptic curves

5.1 A special class of elliptic curves

The purpose of this section is to introduce a special class of elliptic curves. General references for the theory of elliptic curves are the books of Silverman [Sil1] and [Sil2], in particular [Sil2, Ch. II] for CM-elliptic curves. We will follow very closely the exposition in [GS, §4] and [dS, Ch. II, 1].

Throughout this section k denotes a quadratic imaginary number field. For an integral ideal 𝔣 of k the ray class field of conductor 𝔣 is denoted by k(𝔣). In particular we write k(1) for the Hilbert class field.

Once and for all we fix an algebraic closure ℚ̄ of ℚ and an embedding ℚ̄ ↪ ℂ. Via this embedding we view all number fields as subfields of ℂ.

For an abelian extension N/M of number fields and an integral ideal 𝔞 of M relatively prime to the conductor 𝔣_{N/M} of N/M we denote the Artin symbol by σ(𝔞) or (𝔞, N/M).

Let F be a finite abelian extension of k and let E be an elliptic curve defined over F with complex multiplication by the ring of integers O_{k}. We fix a generalized Weierstrass model for E/F,

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},  (38)

with a_{1}, a_{2}, a_{3}, a_{4}, a_{6} ∈ F and denote by ω = ω_{E} = dx/(2y + a_{1}x + a_{3}) the standard invariant differential associated with (38).

In characteristics ≠ 2, 3 we can replace x and y by

℘ = x + \frac{a_{1}^{2} + 4a_{2}}{12} , ℘′ = 2y + a_{1}x + a_{3},

and (38) is transformed to

(℘′)^{2} = 4℘^{3} − g_{2}℘ − g_{3} with g_{2}, g_{3} ∈ F.  (39)

The pair (E, ω_{E}) determines a complex lattice 𝔴 ⊆ ℂ which is characterized by

g_{2}(𝔴) = g_{2}, g_{3}(𝔴) = g_{3},

where, as usual,

g_{2}(𝔴) = 60 ∑′_{ω∈𝔴} ω^{−4}, g_{3}(𝔴) = 140 ∑′_{ω∈𝔴} ω^{−6}

denote the coefficients in the differential equation of the Weierstass ℘-function.

Since E has complex multiplication by O_{k}, the lattice 𝔴 is of the form Ω𝔞 with Ω ∈ ℂ^{∗} and a fractional ideal 𝔞 of k. Once 𝔞 is fixed, Ω is uniquely determined up to a root of unity in k.

We denote by

ξ( , 𝔴) : ℂ/𝔴 ⟶ E(ℂ),
z + 𝔴 ⟼ \begin{cases}[℘(z | 𝔴), ℘′(z | 𝔴), 1], & if z ∉ 𝔴, \\ [0, 1, 0] , & if z ∈ 𝔴, \end{cases}  (40)

the corresponding Weierstrass isomorphism.

Let θ : O_{k} → End(E) be normalized in the sense of Shimura [Sh, 5.1], i.e. ω ∘ θ(μ) = μω for μ ∈ O_{k}. Then

θ(μ) (ξ(z, 𝔴)) = ξ(μz, 𝔴), z ∈ ℂ, μ ∈ O_{k}.  (41)

For an integral ideal 𝔤 of k we define the group of 𝔤-torsion points of E by

E[𝔤] = {ξ(z, 𝔴) | z ∈ 𝔤^{−1}𝔴/𝔴}.

Furthermore we write E_{tor} for the torsion subgroup of E(ℂ). From the fact that the j-invariant j_{E} of E generates the Hilbert class field k(1) over k we derive k(1) ⊆ F. Therefore [Sil2, Th. 2.3] implies that F (E[𝔤])/F is abelian.

Let ψ = ψ_{E/F} be the Groessencharacter associated to E/F and denote by 𝔣_{ψ} its conductor. Thus ψ is a homomorphism from the group of all fractional ideals of F relatively prime to 𝔣_{ψ} to k^{∗}. For any principal ideal 𝔄 = (a) of F such that a ≡ 1 (mod ^{∗} 𝔣_{ψ}) one has ψ((a)) = ∏_{σ∈Gal(F/k)} a^{σ}.

Let 𝔄 be any integral ideal of F with (𝔄, 𝔣_{ψ}) = 1. Then ψ(𝔄) ∈ O_{k} and we have the formula

ξ(ρ, 𝔴)^{σ(𝔄)} = ξ(ψ(𝔄)ρ, 𝔴)  (42)

for any ρ ∈ 𝔤^{−1}𝔴, (𝔤, N_{F/k}(𝔄)) = 1.

We now impose the following hypothesis on E:

F (E_{tor}) is an abelian extension of k.  (43)

By [Sh, Th. 7.44] this condition is equivalent to the existence of a Groessencharacter φ of k such that ψ = φ ∘ N_{F/k} (see also [GS, Th. 4.1]). Such a Groessencharacter φ is uniquely determined up to multiplication with an abelian character χ of Gal(L/k), where χ is regarded as a Dirichlet character of k ([GS, Remarque (4.2)]). Once and for all we fix such a φ and denote by

𝔣 = lcm(𝔣_{φ}, 𝔣_{F/k})  (44)

the least common multiple of the conductors of φ and F/k. The ideal 𝔣 is in fact independent of the choice of φ (see [GS, p.197]) and only depends on E/F. Note also that φ satisfies φ(η) = η for η ≡ 1 (mod 𝔣), η ∈ O_{k}.

Lemma 5.1 Let φ be a Groessencharacter of k with conductor 𝔣_{φ} and infinity type (1, 0) (i.e., φ((η)) = η for all η ∈ O_{k} with η ≡ 1 (mod 𝔣_{φ})). Then w(𝔣_{φ}) = 1, where w(𝔞) = |{ε ∈ O_{k} | ε ≡ 1 (mod 𝔞)}| for any integral O_{k}-ideal 𝔞.

Proof By setting φ_{f}(a) := φ((a))/a we obtain a well-defined character φ_{f} : (O_{k}/𝔣_{φ})^{∗} → ℂ^{∗}. Let W = {ε ∈ O_{k}^{∗} | ε ≡ 1 (mod 𝔣_{φ})}. Then we have an embedding

O_{k}^{∗}/W ↪ (O_{k}/𝔣_{φ})^{∗} .

For any η ∈ O_{k}^{∗} we have φ_{f}(η) = 1/η. Since |O_{k}^{∗}/W | = w(1)/w(𝔣_{φ}) we

necessarily have φ_{f}(η)^{w(1)/w(𝔣_{φ})} = 1 = (\frac{1}{η} )^{w(1)/w(𝔣_{φ})}. Since O_{k}^{∗} is cyclic of

order w(1) we conclude that w(𝔣_{φ}) = 1.

Particular instances of this type of elliptic curves are the analytic models of Fueter, Deuring and Legendre in [CT], [Co], [Fl] and [Sch2]. See also the examples (4.5) in [GS].

Let 𝔞 be an integral ideal of k relatively prime to 𝔣. From [GS, (4.3),(4.6)] one deduces the existence of a unique isogeny

λ(𝔞) : E ⟶ E^{σ(𝔞)}  (45)

characterized by the property

P^{σ(𝔞)} = λ(𝔞)(P) for P ∈ E[𝔤], (𝔤, 𝔞) = 1.  (46)

Since ω is a F -basis for the space of differentials of the first kind, we may define the quantity Λ(𝔞) ∈ F by

ω^{σ(𝔞)} ∘ λ(𝔞) = Λ(𝔞)ω.

We write 𝔴_{𝔞} for the lattice which corresponds to the couple (E^{σ(𝔞)}, ω^{σ(𝔞)}).

Proposition 5.2 [GS, Prop. (4.10)]

(i) For an integral ideal 𝔞 of k relatively prime to 𝔣 the map 𝔞 ↦ Λ(𝔞) is characterized by the following commutative diagram:

ℂ/𝔴 ⟶^{ξ(·,𝔴)} E(ℂ)
·Λ(𝔄)↓  ↓^{λ(𝔞)}
ℂ/𝔴_{𝔞} ⟶^{ξ(·,𝔴_{𝔞})} E^{σ(𝔞)}(ℂ)

(ii) For all 𝔞 as above, 𝔴_{𝔞} = Λ(𝔞)𝔞^{−1}𝔴.

(iii) Let 𝔤 be an integral ideal of k such that 𝔣 | 𝔤. Then one has for all ρ ∈ 𝔤^{−1}𝔴

ξ(ρ, 𝔴)^{σ(𝔞)} = ξ(Λ(𝔞)ρ, 𝔴_{𝔞}).

(iv) If 𝔞 and 𝔟 are two integral ideals of k such that (𝔞, 𝔟) = 1, (𝔞𝔟, 𝔣) = 1, then

Λ(𝔞𝔟) = Λ(𝔞)^{σ(𝔟)}Λ(𝔟) = Λ(𝔞)Λ(𝔟)^{σ(𝔞)}  (47)

There is a unique way to extend the definition of Λ to the whole group of fractional ideals relatively prime to 𝔣 such that (47) remains valid. Namely we have to set

Λ(𝔞^{−1})^{σ(𝔞)} = \frac{1}{Λ(𝔞)}.  (48)

This extension will again be called Λ.

The following proposition gives the precise class field theoretical description of the fields generated by the 𝔤-torsion points for 𝔣 | 𝔤.

Proposition 5.3 [GS, Lemme (4.7)] Let 𝔤 be an integral ideal of k divisible by 𝔣. Then F (E[𝔤]) = k(𝔤).

In the rest of this section we will concentrate on relative extensions F (E[𝔣𝔭^{r+m}])/F (E[𝔣𝔭^{r}]) with r, m ≥ 1 and a prime ideal 𝔭 of k. In particular we will work out the values for φ, Λ and λ for 𝔞 such that (𝔞, F/k) = 1 in greater detail.

Lemma 5.4 Let 𝔞 be an integral ideal of k relatively prime to 𝔣 such that (𝔞, F/k) = 1. Then:

(i) Λ(𝔞) ∈ k^{∗}, Λ(𝔞)O_{k} = 𝔞 and 𝔴 = 𝔴_{𝔞}.

(ii) λ(𝔞) = θ(Λ(𝔞)).

(iii) Λ(𝔞) = φ(𝔞), in particular Λ(η) = η for η ≡ 1 (mod 𝔣), η ∈ O_{k}.

Proof Let 𝔟 be any integral ideal of k such that (𝔞𝔣, 𝔟) = 1. From Proposition 5.2 (iv) we obtain

Λ(𝔞𝔟) = Λ(𝔞)^{σ(𝔟)}Λ(𝔟) = Λ(𝔞)Λ(𝔟)^{σ(𝔞)} = Λ(𝔞)Λ(𝔟).

Hence Λ(𝔞)^{σ(𝔟)} = Λ(𝔞), which implies Λ(𝔞) ∈ k^{∗}. From E^{σ(𝔞)} = E and ω^{σ(𝔞)} = ω we derive 𝔴_{𝔞} = 𝔴 and Λ(𝔞)O_{k} = 𝔞.

The commutative diagram of Proposition 5.2 (i) implies that

λ(𝔞) (ξ(z, 𝔴)) = ξ(Λ(𝔞)z, 𝔴) = θ(Λ(𝔞))(ξ(z, 𝔴)).

From this (ii) is immediate.

The cocycle relation (47) implies that Λ is multiplicative on the set of ideals 𝔞 with (𝔞, F/k) = 1. Let φ′ be any extension of Λ to a Groessencharacter of k. For any integral ideal 𝔄 of F we have

ξ(ρ, 𝔴)^{σ(𝔄)} = ξ(ρ, 𝔴)^{σ(N_{F/k}(𝔄))}
= ξ(Λ(N_{F/k}(𝔄))ρ, 𝔴),

where ρ ∈ 𝔤^{−1}𝔴 for any integral O_{k}-ideal 𝔤 with (𝔤, N_{F/k}(𝔄)) = 1.

By the defining property of ψ_{E/F} we obtain Λ ∘ N_{F/k} = ψ_{E/F} and thus also φ′ ∘ N_{F/k} = ψ_{E/F}. Therefore φ′ is of the form φ′ = φχ for a character χ of Gal(F/k). But then Λ(𝔞) = φ′(𝔞) = φ(𝔞)χ(𝔞) = φ(𝔞) for any 𝔞 such that (𝔞, F/k) = 1.

Remark 5.5 From (48) it is clear that (i) and (iii) are also valid for any fractional ideal 𝔞 of k relatively prime to 𝔣 such that (𝔞, F/k) = 1.

Corollary 5.6 Let 𝔤 be an integral ideal of k divisible by 𝔣. Let Q be a primitive 𝔤-torsion point of E. Then F (Q) = k(𝔤).

Proof From Proposition 5.3 we know F (Q) ⊆ k(𝔤). Let 𝔞 be an integral ideal of k such that σ(𝔞) |_{F (Q)}= 1. Let ρ ∈ 𝔤^{−1}𝔴 be such that Q = ξ(ρ, 𝔴). By (46) and Lemma 5.4(ii) we obtain Q^{σ(𝔞)} = ξ(Λ(𝔞)ρ, 𝔴). Hence ξ(Λ(𝔞)ρ, 𝔴) = ξ(ρ, 𝔴) and since Q is primitive, this implies Λ(𝔞) ≡ 1 (mod 𝔤). Let Q_{1} = ξ(ρ_{1}, 𝔴) be an arbitrary 𝔤-torsion point. Then ρ_{1} ∈ 𝔤^{−1}𝔴 and

Q_{1}^{σ(𝔞)} = ξ(Λ(𝔞)ρ_{1}, 𝔴) = ξ(ρ_{1}, 𝔴) = Q_{1}.

Therefore σ(𝔞) fixes F (E[𝔤]) = k(𝔤), and thus F (Q) = k(𝔤).

For a prime ideal 𝔓 of F we denote the 𝔓-adic valuation by v_{𝔓}.

Corollary 5.7 Let 𝔭 be a prime ideal of k such that 𝔭 |̸ 𝔣. Then v_{𝔓}(𝔭O_{F}) = v_{𝔓}(Λ(𝔭)) for all 𝔓 | 𝔭.

Proof Let n ∈ ℕ be such that (𝔭^{n}, F/k) = 1. Then Lemma 5.4 (i) and Proposition 5.2 (iv) impliy

𝔭^{n}O_{F} = Λ(𝔭^{n})O_{F} = Λ(𝔭)Λ(𝔭)^{σ(𝔭)} ⋯ Λ(𝔭)^{σ(𝔭)^{n−1}}.

One deduces that v_{𝔓}(𝔭O_{F}) = v_{𝔓}(Λ(𝔭)), since σ(𝔭) is an element of the decomposition group G_{𝔓} for all 𝔓 above 𝔭.

Let now 𝔭 be a prime ideal of k such that (𝔭, 𝔣) = 1. Let 𝔓 denote a prime of F above 𝔭. Since E has good reduction at 𝔓, we can fix a Weierstrass model (38) over the localization O_{F,𝔓} of O_{F} at 𝔓 such that Δ ∈ O_{F,𝔓}^{∗}.

In the following we will work out the connection to the relative Lubin-Tate theory as it was presented in Section 5.

We view E as an elliptic curve over the completion O_{F_{𝔓}} and denote by Ê the one-parameter formal group of E with respect to the parameter

z = −\frac{x}{y}

(see [Sil1, Ch.IV]).

Lemma 5.8 [dS, CH.II,1.10] Ê is a relative Lubin-Tate group with respect to the unramified extension F_{𝔓}/k_{𝔭} associated with the parameter Λ(𝔭).

Proof Let ϕ = σ(𝔭) be the Frobenius automorphism of F_{𝔓}/k_{𝔭}. The isogeny λ(𝔭) : E → E^{ϕ} induces a homomorphism of formal groups

λ̂(𝔭) : Ê ⟶ Ê^{ϕ}.

We put q = N_{F/ℚ}(𝔓). It follows from (46) that λ(𝔭) gives the q-power Frobenius on the reduction of the curve modulo 𝔓. Therefore one has

λ̂(𝔭)(z) ≡ z^{q} (mod 𝔓).  (49)

The invariant differential ω = dx/(2y + a_{1}x + a_{3}) has an expression

ω(z) = (1 + a_{1}z + O(z^{2}))dz.

Since λ(𝔭) is unramified we can write

λ̂(𝔭)(z) = b_{1}z + O(z^{2}), b_{1} ∈ O_{F_{𝔓}}.

From (ω ∘ λ̂(𝔭))(z) = Λ(𝔭)ω(z) we obtain the equality

(1 + a_{1}b_{1}z + O(z^{2}))d(b_{1}z + O(z^{2})) = (Λ(𝔭) + O(z))dz,

and therefore b_{1} = Λ(𝔭). This implies

λ̂(𝔭)(z) ≡ Λ(𝔭)z (mod deg 2).  (50)

By (49), (50), [dS, ChI, Th. 1.3] and Corollary 5.7 we conclude that Ê is the unique relative Lubin-Tate group with respect to F_{𝔓}/k_{𝔭} associated with the parameter Λ(𝔭).

5.2 Modules associated with elliptic curves

In this section we will use the special class of elliptic curves discussed in the previous section to produce interesting and non-trivial examples that fit into the framework of Section 1. We adopt the notation of Section 5.1. So k is a quadratic imaginary number field and F/k a finite abelian extension. E/F denotes an elliptic curve with complex multiplication by O_{k} and such that F (E_{tor})/k is abelian. Let 𝔭 be a prime ideal of k such that (𝔭, 𝔣) = 1, where 𝔣 is the integral ideal of k defined in (44). The extensions that we will consider are in most cases relative ray class field extensions of the form k(𝔣𝔭^{r+m})/k(𝔣𝔭^{r}) with integers r, m ≥ 1. This kind of extension was already studied by Ph. Cassou-Noguès and M.J. Taylor [CT] and R. Schertz [Sch2] under the additional Kummer condition m ≤ r. In this case the module structure we introduced in Section 1 coincides with classical Galois module structure and we will recover (parts of) their results.

Let r ≥ 0, m ≥ 1 be rational integers. We set G = E[𝔭^{m}]. Fix a primitive 𝔭^{r+m}-torsion point Q_{0} = ξ(ρ_{0}, 𝔴), ρ_{0} ∈ 𝔭^{−r−m}𝔴, of E and put

Γ = {Q_{0} +_{E} P | P ∈ G}.

In the sequel we will use the more convenient notation notation [α] instead of θ(α) for the endomorphism of E corresponding to α ∈ O_{k}.

If 𝔭 = (π) is principal, then Γ is just the set of “π^{m}-th roots” of the primitive π^{r}-torsion point P_{0} = [π^{m}](Q_{0}), namely

Γ = {Q ∈ E(ℚ̄) | [π^{m}](Q) = P_{0}}.

In the following we will write

F_{s} = F (E[𝔣𝔭^{s}]) = k(𝔣𝔭^{s}) for s ∈ ℕ.

The case r = 0 is completely settled by Corollary 2.15. Therefore we henceforth assume r ≥ 1 unless stated otherwise.

We consider F_{r} as our base field and set

K = F_{r}, Ω = Ω_{K}, L = F_{r+m}.

From class field theory we know that [k(𝔣𝔭^{s}) : k(1)] = \frac{w(1)}{w(𝔣𝔭^{s})}Φ(𝔣𝔭^{s}) with the Euler function Φ for the ring O_{k}. Therefore Lemma 5.1 implies

[L : K] = N_{k/ℚ}(𝔭^{m}).

Lemma 5.9 Let r ≥ 1. Then one has

(i)  L = K(Q_{0}).
(ii)  Γ = {Q_{0}^{σ} | σ ∈ Gal(L/K)}.

In particular, Ω acts on Γ.

Proof From Corollary 5.6 and Proposition 5.3 we deduce

L = F_{r+m} = F (E[𝔣], Q_{0}) = k(𝔣)(Q_{0}) = F_{r}(Q_{0}) = K(Q_{0}).

For the proof of (ii) first note that

Gal(L/K) = {σ(1 + η) | η ∈ 𝔣𝔭^{r}/𝔣𝔭^{r+m}}.

For η ∈ 𝔣𝔭^{r} we obtain

\begin{aligned}Q_{0}^{σ(1+η)} & = & ξ(ρ_{0}, 𝔴)^{σ(1+η)} \\ & = & ξ(ρ_{0} + ρ_{0}η, 𝔴) \\ & = & ξ(ρ_{0}, 𝔴) +_{E} ξ(ηρ_{0}, 𝔴) \\ & = & Q_{0} +_{E} ξ(ηρ_{0}, 𝔴). \end{aligned}  (51)

Since ηρ_{0} ∈ 𝔣𝔭^{−m}𝔴 ⊆ 𝔭^{−m}𝔴 we derive ξ(ηρ_{0}, 𝔴) ∈ G. From (i) we already know that the conjugates are mutually distinct, and therefore the map

𝔣𝔭^{r}/𝔣𝔭^{r+m} ⟶ G,
η + 𝔣𝔭^{r+m} ⟼ ξ(ηρ_{0}, 𝔴)

is injective. Now (ii) is implied by |G| = N_{k/ℚ}(𝔭^{m}) = [L : K].

Finally we remark that by Proposition 5.3 the action of Ω on Γ factors through Ω_{L}. Thus the “in particular” statement follows from (ii).

We set

A = (K̄G)^{Ω}, B = Map(G, K̄)^{Ω}, C = Map(Γ, K̄)^{Ω}

and are now exactly in the set-up of Section 1.

We consider the K-algebra homomorphism

τ : C ⟶ L, f ⟼ f(Q_{0}).  (52)

Since Ω acts transitively on Γ, τ is in fact an isomorphism of K-algebras (Lemma 2.11).

From (51) we obtain a bijection

φ : G ⟶ Gal(L/K),
ξ(β, 𝔴) ⟼ σ(1 + \frac{β}{ρ_{0}} ), β ∈ 𝔭^{−m}𝔣𝔴.  (53)

We endow L via the isomorphism τ with the structure of an A-module. Explicitly this means for α ∈ L and ∑_{g∈G} a_{g}g

α · ∑_{g∈G} a_{g}g = ∑_{g∈G} a_{g}α^{σ(1+β_{g}/ρ_{0})},  (54)

where we have set g = ξ(β_{g}, 𝔴), β_{g} ∈ 𝔭^{−m}𝔣𝔴.

Note that for m > r this new module structure does not coincide with the usual Galois module structure. However, under the additional Kummer condition m ≤ r we are in the classical situation studied by [CT] and [Sch2].

Lemma 5.10 Suppose that r ≥ m ≥ 1.

(i) One has A = KG.

(ii) φ is a group isomorphism.

(iii) Identifying G and Gal(L/K) via φ the map τ is an isomorphism of A-modules.

Proof If m ≤ r we have G ⊆ K = F_{r} and therefore (i) is evident. Let β_{1}, β_{2} ∈ 𝔭^{−m}𝔣𝔴. Then β_{1}β_{2}/ρ_{0}^{2} ∈ 𝔣𝔭^{r+m} and we obtain

φ(ξ(β_{1}, 𝔴)) · φ(ξ(β_{2}, 𝔴)) = σ(1 + β_{1}/ρ_{o}) · σ(1 + β_{2}/ρ_{0})
= σ(1 + \frac{β_{1} + β_{2}}{ρ_{0}} + \frac{β_{1}β_{2}}{ρ_{0}^{2}} )
= σ(1 + \frac{β_{1} + β_{2}}{ρ_{0}} )
= φ(ξ(β_{1} + β_{2}, 𝔴)).

Thus φ is a group homomorphism. (iii) is then immediate from (54).

5.3 An analytic resolvent formula à la Schertz

One of the main sources for many results in the field of Galois module structure for rings of integers is the determination of some sort of resolvent (for instance see [CT, Ch. VI], [Fr2] or [Sch2]).

We will use the analytic resolvent function of Schertz [Sch2, §3], since it seems to be best suited for our applications. As already Schertz remarked in [Sch2, Bemerkungen (3), p. 283], his analytic resolvent function produces also some kind of arithmetical resolvent in the case m > r, but it can no longer be interpreted as a Galois resolvent. In fact, this is exactly the resolvent that we need in our context.

Let 𝔴 = ℤω_{1} + ℤω_{2}, Im(\frac{ω_{1}}{ω_{2}} ) > 0, be a complex lattice. We write

σ(z | 𝔴) = z ∏_{ω∈𝔴,ω≠0}(1 − \frac{z}{ω})e^{\frac{z}{ω} + \frac{1}{2}(\frac{z}{ω})^{2}}, z ∈ ℂ,

for the Weierstrass σ-function and define the Dedekind η-function by

η(τ) = q_{τ}^{1/24} ∏_{n=1}^{∞}(1 − q_{τ}^{n}), Im(τ) > 0,

where q_{τ} = e^{2πiτ}. Finally we denote by

ζ(z | 𝔴) = \frac{1}{z} + ∑_{ω∈𝔴,ω≠0} (\frac{1}{z − ω} + \frac{1}{ω} + \frac{z}{ω^{2}} ) , z ∈ ℂ,

the Weierstrass ζ-function and put

η_{1} = 2ζ(\frac{ω_{1}}{2} | 𝔴), η_{2} = 2ζ(\frac{ω_{2}}{2} | 𝔴)

for the basic quasi-periods of ζ.

The resolvents that we are interested in are resolvents of certain elliptic units (or elliptic 𝔭-units). These elliptic units are defined as singular values of the following normalization of the Weierstrass σ-function:

φ (z |{ω_{1} \atop ω_{2}} ) = 2πi exp(−(a_{1}η_{1} + a_{2}η_{2})\frac{z}{2})σ (z |𝔴) η^{2}(\frac{ω_{1}}{ω_{2}} )ω_{2}^{−1},

where the real numbers a_{1}, a_{2} are uniquely determined by z = a_{1}ω_{1} + a_{2}ω_{2}. For later reference we record the transformation formulas

φ ((a_{1} + b_{1})ω_{1} + (a_{2} + b_{2})ω_{2} |{ω_{1} \atop ω_{2}} )

= (−1)^{b_{1}b_{2}+b_{1}+b_{2}}e^{− \frac{2πi}{2} (b_{1}a_{2}−b_{2}a_{1})}φ (a_{1}ω_{1} + a_{2}ω_{2} |{ω_{1} \atop ω_{2}} )  (55)

for a_{1}, a_{2} ∈ ℚ, b_{1}, b_{2} ∈ ℤ, and

φ ((a_{1}, a_{2}) ( \begin{matrix}Mτ \\ 1 \end{matrix} ) |{Mτ \atop 1} ) = ε(M)^{2}φ ((a_{1}, a_{2})M \binom{τ}{1} |{τ \atop 1} )  (56)

for a_{1}, a_{2} ∈ ℚ, Im(τ) > 0, M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl_{2}(ℤ). Herein ε(M) is the 24-th

root of unity arising from the transformation formula of the Dedekind η-function (see e.g. [KL, pp.27/28]). Furthermore for z = a_{1}τ + a_{2}, Im(τ) > 0, a_{1}, a_{2} ∈ ℚ, we have the q-expansion

φ (z |{τ \atop 1} )

= −q_{τ}^{\frac{1}{2} B_{2}(a_{1})}e^{2πia_{2}(a_{1}−1)/2}(1 − q_{z}) ∏_{n=1}^{∞}(1 − q_{τ}^{n}q_{z})(1 − q_{τ}^{n}q_{z}^{−1}), (57)

where B_{2}(X) = X^{2} − X + \frac{1}{6} is the second Bernoulli polynomial and q_{z} = e^{2πiz} (see [KL, p.29]).

Note that φ is homogenous of degree 0, i.e. φ (αz |{αω_{1} \atop αω_{2}} ) = φ (z |{ω_{1} \atop ω_{2}} ) for all α ∈ ℂ^{∗}. It is not a function of lattices, but depends on the choice of a basis ω_{1}, ω_{2}. Nevertheless we will write φ (z |𝔴) in the sequel. Moreover φ is non-meromorphic as a function of z. We therefore introduce the “analytic substitute”

ϑ (z |{ω_{1} \atop ω_{2}} ) = 2πi exp(−\frac{1}{2} \frac{η_{2}}{ω_{2}} z^{2})σ (z |𝔴) η^{2}(\frac{ω_{1}}{ω_{2}} )ω_{2}^{−1}, z ∈ ℂ.

This ϑ-function is homogenous of degree 0 and holomorphic for z ∈ ℂ with simple zeroes in z ∈ 𝔴. By abuse of notation we write, as for the φ-function, ϑ (z |𝔴) instead of ϑ (z |{ω_{1} \atop ω_{2}} ).

From the transformation formula of the Weierstrass σ-function we deduce

ϑ (z + ω_{1} |{ω_{1} \atop ω_{2}} ) = −e^{− \frac{2πi}{ω_{2}} (z+ \frac{ω_{1}}{2} )}ϑ (z |{ω_{1} \atop ω_{2}} ) ,
ϑ (z + ω_{2} |{ω_{1} \atop ω_{2}} ) = −ϑ (z |{ω_{1} \atop ω_{2}} )

(see [BBC, Prop. 3.3]). ϑ is very closely related to the φ-function. The Legendre relation η_{2}ω_{1} − η_{1}ω_{2} = 2πi implies

ϑ (z |{ω_{1} \atop ω_{2}} ) = e^{−πia_{1} \frac{z}{ω_{2}}} φ (z |{ω_{1} \atop ω_{2}} ) .  (58)

Finally we introduce for z = a_{1}ω_{1} + a_{2}ω_{2}, a_{1}, a_{2} ∈ ℝ, the short hand

z^{∗} = a_{1}η_{1} + a_{2}η_{2}

and set

h_{𝔴}(u, v) = uv^{∗} − u^{∗}v = 2πi(u_{1}v_{2} − u_{2}v_{1})  (59)

for u = u_{1}ω_{1} + u_{2}ω_{2}, v = v_{1}ω_{1} + v_{2}ω_{2}. The second equality follows immediately from the Legendre relation.

Proposition 5.11 [Sch2, (3.19)] Let 𝔴 ⊆ 𝔴̂ be two complex lattices,

𝔴 = ℤω_{1} + ℤω_{2}, 𝔴̂ = ℤ\frac{ω_{1}}{n_{1}} + ℤ\frac{ω_{2}}{n_{2}} , Im(\frac{ω_{1}}{ω_{2}} ) > 0, n_{1}, n_{2} ∈ ℕ.

(i) Let z, γ ∈ ℂ. Then

exp (\frac{1}{2}h_{𝔴}(ξ, γ)) \frac{φ (z + γ + ξ |𝔴)}{φ (z + ξ |𝔴)}

only depends on ξ modulo 𝔴.

(ii) Let χ be an abelian character of 𝔴̂/𝔴, uniquely determined by

χ (\frac{ω_{1}}{n_{1}} ) = e^{2πiμ_{1}/n_{1}}, χ (\frac{ω_{2}}{n_{2}} ) = e^{2πiμ_{2}/n_{2}}, μ_{1}, μ_{2} ∈ ℤ,

and set δ_{χ} = \frac{1}{[𝔴̂:𝔴]} (γ + μ_{2}ω_{1} − μ_{1}ω_{2}). Then

∑_{ξ∈𝔴̂/𝔴} e^{\frac{1}{2} h_{𝔴}(ξ,γ)} \frac{φ (z + γ + ξ |𝔴)}{φ (z + ξ |𝔴)}χ̄(ξ) =

= e^{− \frac{1}{2} h_{𝔴}(z,μ_{2}ω_{1}−μ_{1}ω_{2})} · \sqrt[12]{\frac{Δ(𝔴̂)}{Δ(𝔴)}} · \frac{φ (γ |𝔴) φ (z + δ_{χ} |𝔴̂)}{φ (δ_{χ} |𝔴̂) φ (z |𝔴̂)}.

5.4 Special values of the φ-function 61

Proof Part (i) follows easily from (55) and (59). The second part is just a reformulation of [Sch2, (3.19)]. For the convenience of the reader we give the details. We put

𝔴′ = \frac{1}{ω_{2}} 𝔴 = ℤτ + ℤ,
𝔴̂′ = \frac{1}{ω_{2}} 𝔴̂ = ℤ\frac{τ}{n_{1}} + ℤ\frac{1}{n_{2}}

with τ = \frac{ω_{1}}{ω_{2}} and view χ as an abelian character of 𝔴̂′/𝔴′. We write z′ = \frac{z}{ω_{2}} , ξ′ = \frac{ξ}{ω_{2}} and γ′ = \frac{γ}{ω_{2}} . Then [Sch2, (3.19)] reads as follows

∑_{ξ′∈𝔴̂′/𝔴′} e^{2πiγ′a′_{1}(ξ′)} \frac{ϑ (z′ + ξ′ + γ′ |{τ \atop 1} )}{ϑ (z′ + ξ′ |{τ \atop 1} )}χ̄(ξ′)

= e^{2πiμ_{2}z′} · n_{2} · (\frac{η(n_{2}τ/n_{1})}{η(τ)})^{2} · \frac{ϑ (γ′ |{τ \atop 1} ) ϑ (n_{2}z′ + δ′ |{n_{2}τ/n_{1} \atop 1} )}{ϑ (δ′ |{n_{2}τ/n_{1} \atop 1} ) ϑ (n_{2}z′ |{n_{2}τ/n_{1} \atop 1} )}

with δ′ = \frac{1}{n_{1}} (γ′ + μ_{2}τ − μ_{1}) and ξ′ = a′_{1}(ξ′)τ + a′_{2}(ξ′). Since ϑ is homogenous of degree 0 this implies

∑_{ξ∈𝔴̂/𝔴} e^{2πi \frac{γ}{ω_{2}} a_{1}(ξ)} \frac{ϑ (z + ξ + γ |𝔴)}{ϑ (z + ξ |𝔴)}χ̄(ξ)

= e^{2πiμ_{2} \frac{z}{ω_{2}}} n_{2} (\frac{η(n_{2}τ/n_{1})}{η(τ)})^{2} \frac{ϑ (γ |𝔴) ϑ (z + δ_{χ} |𝔴̂)}{ϑ (δ_{χ} |𝔴̂) ϑ (z |𝔴̂)},

where we have set ξ = a_{1}(ξ)ω_{1} + a_{2}(ξ)ω_{2}. From the equality

Δ \binom{ω_{1}}{ω_{2}} = (\frac{2πi}{ω_{2}})^{12} η (\frac{ω_{1}}{ω_{2}} )^{24}

we derive

n_{2} (\frac{η(n_{2}τ/n_{1})}{η(τ)})^{2} = \sqrt[12]{\frac{Δ(𝔴̂)}{Δ(𝔴)}}.

Using (58) we now replace the ϑ-functions by the φ-functions. Then a long, but straight-forward computation proves (ii).

5.4 Special values of the φ-function

In the following we will investigate the arithmetical nature of certain singular values of the φ-function. To that end we will apply the Shimura reciprocity law as it is presented in [St, §4]. We follow very closely the exposition of Schertz in [Sch2, §4].

Let (m_{1}, m_{2}), (n_{1}, n_{2}) ∈ ℤ^{2} and f ∈ ℕ be such that \frac{1}{f} (n_{1}, n_{2}) ∉ ℤ^{2} and consider the function

g(τ) = \frac{φ (\frac{1}{f} (m_{1}, m_{2}) \binom{τ}{1} |{τ \atop 1} )}{φ (\frac{1}{f} (n_{1}, n_{2}) \binom{τ}{1} |{τ \atop 1} )}, Im(τ) > 0.

From (55) and (56) we deduce that g(τ) is a modular function for the congruence subgroup

Γ(2f^{2}) = \{M ∈ Sl_{2}(ℤ) | M ≡ ± ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) (mod 2f^{2})\} .

By (57) the q-expansion of g(τ) has coefficients in ℚ(ζ_{2f^{2}}).

Let 𝔞 = ℤω_{1} + ℤω_{2}, Im(\frac{ω_{1}}{ω_{2}} ) > 0, be a fractional ideal of k and put ω = \frac{ω_{1}}{ω_{2}} . From [St, Th. 3] we deduce

g(ω) ∈ k(2f^{2}).  (60)

Let p be a rational prime relatively prime to 2f^{2}d_{k/ℚ} such that p = 𝔭𝔭̄ decomposes in k/ℚ. Let B be an integral 2 × 2 matrix of determinant p such that B \binom{ω_{1}}{ω_{2}} is a ℤ-basis of 𝔭̄𝔞. Then the action of the Frobenius automorphism σ(𝔭) on g(ω) is explicitly described by [St, Th. 3]. Namely

g(ω)^{σ(𝔭)} = [g ∘ pB^{−1}] (Bω).  (61)

The recipe for computing g ∘ pB^{−1} is explained in [St, pp.210/211]. We have to decompose the matrix pB^{−1} in the form

pB^{−1} = M_{1} ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) M_{2} with M_{1}, M_{2} ∈ Sl_{2}(ℤ)

and may then use the associativity of the symbol ∘. For (a_{1}, a_{2}) ∈ ℚ^{2} such that f(a_{1}, a_{2}) ∈ ℤ^{2} and M ∈ Sl_{2}(ℤ) we obtain from (56)

φ ((a_{1}, a_{2}) \binom{τ}{1} |{τ \atop 1} ) ∘ M = ε(M)^{2}φ ((a_{1}, a_{2})M \binom{τ}{1} |{τ \atop 1} ) ,  (62)

and (57) implies

φ ((a_{1}, a_{2}) \binom{τ}{1} |{τ \atop 1} ) ∘ ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) = φ ((a_{1}, a_{2}) ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) \binom{τ}{1} |{τ \atop 1} ) . (63)

Using (62) and (63) we derive

[g ∘ pB^{−1}] (τ) = \frac{φ (\frac{1}{f} (m_{1}, m_{2})pB^{−1} \binom{τ}{1} |{τ \atop 1} )}{φ (\frac{1}{f} (n_{1}, n_{2})pB^{−1} \binom{τ}{1} |{τ \atop 1} )}.  (64)

We are particularly interested in the action of elements σ ∈ Gal(k(2f^{2})/k(1)) on g(ω). Let λ ∈ O_{k} be relatively prime to 2f^{2}. By the Tchebotarev density theorem we can find λ_{1} ∈ O_{k} such that λ ≡ λ_{1} (mod 2f^{2}), (λ_{1}, 2fd_{k/ℚ}) = 1 and (λ_{1}) = 𝔭 is a prime of k which decomposes in k/ℚ. Obviously σ(λ) = σ(λ_{1}).

As a basis of 𝔭̄𝔞 we take λ̄_{1}ω_{1}, λ̄_{1}ω_{2} and define B by

λ̄_{1} ( \begin{matrix}ω_{1} \\ ω_{2} \end{matrix} ) = B ( \begin{matrix}ω_{1} \\ ω_{2} \end{matrix} ) .

It follows that p = λ_{1}λ̄_{1} and Bω = ω. Therefore we obtain from (61) and (64)

g(ω)^{σ(λ)} = \frac{φ (\frac{λ_{1}}{f} (m_{1}, m_{2}) \binom{ω}{1} |{ω \atop 1} )}{φ (\frac{λ_{1}}{f} (n_{1}, n_{2}) \binom{ω}{1} |{ω \atop 1} )}.

Since λ_{1} ≡ λmod 2f^{2} we deduce from (55) the final formula

g(ω)^{σ(λ)} = \frac{φ (\frac{λ}{f}(m_{1}, m_{2}) \binom{ω}{1} |{ω \atop 1} )}{φ (\frac{λ}{f}(n_{1}, n_{2}) \binom{ω}{1} |{ω \atop 1} )}, λ ∈ O_{k}, (λ, 2f^{2}) = 1.  (65)

Let now 𝔠, 𝔠^{∗} be two integral ideals of k such that (∏_{𝔭|𝔠} 𝔭) | 𝔠^{∗} | 𝔠 and let 𝔩 be an integral ideal such that (𝔩, 𝔠𝔠̄) = 1. We set l = min(ℕ ∩ 𝔩) and put

𝔤 = 𝔩𝔠, 𝔤^{∗} = 𝔩𝔠^{∗}.

We decompose all occuring ideals into a rational and primitive part such that

\begin{aligned}𝔩 & = & l_{1}(ℤα + ℤl_{2}), \\ 𝔠 & = & c_{1}(ℤα + ℤc_{2}), \\ 𝔠^{∗} & = & c_{1}^{∗}(ℤα + ℤc_{2}^{∗}), \\ 𝔤 & = & g_{1}(ℤα + ℤg_{2}), g_{1} = l_{1}c_{1}, g_{2} = l_{2}c_{2}, \\ 𝔤^{∗} & = & g_{1}^{∗}(ℤα + ℤg_{2}^{∗}), g_{1}^{∗} = l_{1}c_{1}^{∗}, g_{2}^{∗} = l_{2}c_{2}^{∗}. \end{aligned}  (66)

We write β = \frac{α}{g_{2}} and β^{∗} = \frac{α}{g_{2}^{∗}} and define

φ_{𝔤} = \frac{φ (\frac{l}{g} + \frac{ν_{0}}{l} |{β \atop 1} )}{φ (\frac{l}{g} |{β \atop 1} )},

φ_{𝔤^{∗}} = \frac{φ (\frac{l}{g^{∗}} + \frac{ν_{0}}{l} |{β^{∗} \atop 1} )}{φ (\frac{l}{g^{∗}} |{β^{∗} \atop 1} )},  (67)

where we choose ν_{0} ∈ ℤ such that

\begin{aligned}(ν_{0}, l) = 1, & \frac{g}{l} | ν_{0}, if 2 | l, \\ & 2\frac{g}{l} | ν_{0}, if 2 ∤ l. \end{aligned}  (68)

Proposition 5.12 [Sch2, Satz 4.3] (i) With the above notation one has

φ_{𝔤} ∈ \begin{cases}k(2l𝔤), & if 2 | l, \\ k(l𝔤), & if 2 ∤ l, \end{cases}

φ_{𝔤^{∗}} ∈ \begin{cases}k(2l𝔤^{∗}), & if 2 | l, \\ k(l𝔤^{∗}), & if 2 ∤ l, \end{cases}

(ii) For ξ ∈ \frac{l^{4}}{g} 𝔤^{∗} one obtains

φ_{𝔤}^{σ(1+ \frac{g}{l} ξ)} = \frac{φ (\frac{l}{g} + \frac{ν_{0}}{l} + ξ |{β \atop 1} )}{φ (\frac{l}{g} + ξ |{β \atop 1} )}.

(iii) There exists μ ∈ ℤ such that φ_{𝔤}/φ_{𝔤^{∗}}^{μ} ∈ k(𝔤). In addition, if 𝔠 is the power of a prime ideal, then μ can be chosen such that

μ ≡ 1mod Φ(𝔠^{∗}),

where Φ denotes the Euler totient function for the ring O_{k}.

Proof The proof is very similar to the proof of [Sch2, Satz 4.3]. But since we do not assume that 𝔩 is a rational prime, which decomposes in k/ℚ, some minor modifications are necessary.

From (60) we know that φ_{𝔤} ∈ k(2g^{2}). Any Galois automorphism σ ∈ Gal(k(2g^{2})/k(𝔤)) can be represented as σ = σ(1 + gξ) with ξ ∈ \frac{1}{g}𝔤 = ℤβ + ℤ. We write ξ = xβ + y with x, y ∈ ℤ and set

a = l + ν_{0} \frac{g}{l} , b = l,
a^{∗} = l + ν_{0} \frac{g^{∗}}{l} , b^{∗} = l,

where g = g_{1}g_{2}, g^{∗} = g_{1}^{∗}g_{2}^{∗}. From (65), together with (55), we deduce

φ_{𝔤}^{σ(1+gξ)}/φ_{𝔤} = (−1)^{xy(a^{2}−b^{2})+(x+y)(a−b)}e^{− \frac{2πi}{2} ν_{0}^{2} \frac{g}{l^{2}} x}.  (69)

Note that β = \frac{g_{2}^{∗}}{g_{2}} β^{∗} and set

x^{∗} = \frac{g_{1}}{g_{1}^{∗}} x, y^{∗} = \frac{g}{g^{∗}} y.

Then x^{∗}, y^{∗} ∈ ℤ. Again from (55) and (65) we obtain

φ_{𝔤^{∗}}^{σ(1+gξ)−1} = (−1)^{x^{∗}y^{∗}(a^{∗2}−b^{∗2})+(x^{∗}+y^{∗})(a^{∗}−b^{∗})}e^{− \frac{2πi}{2} ν_{0}^{2} \frac{g^{∗}}{l^{2}} x^{∗}}.  (70)

If we assume in addition that σ(1 + gξ) ∈ Gal(k(2g^{2})/k(l𝔤)) (resp. σ(1 + gξ) ∈ Gal(k(2g^{2})/k(2l𝔤)), if 2 | l), then we have l | x, y (resp. 2l | x, y). Considering (68) it follows immediately from (69) that σ(1 + gξ) fixes φ_{𝔤}, and hence φ_{𝔤} ∈ k(l𝔤) (resp. φ_{𝔤} ∈ k(2l𝔤), if 2 | l). Analogously we show φ_{𝔤^{∗}} ∈ k(l𝔤^{∗}) (resp. φ_{𝔤^{∗}} ∈ k(2l𝔤^{∗}), if 2 | l). This proves (i).

For the proof of (ii) let ξ ∈ \frac{l^{4}}{g} 𝔤^{∗}. The decompositions in (66) imply that

\frac{g}{l} ξ = xβ + y with x, y ∈ ℤ, l^{3} | x, y.

The formula (65) leads to

φ_{𝔤}^{σ(1+ \frac{g}{l} ξ)} = \frac{φ (\frac{a}{g}(1 + \frac{g}{l} ξ) |{β \atop 1} )}{φ (\frac{b}{g} (1 + \frac{g}{l} ξ) |{β \atop 1} )}

= \frac{φ (\frac{l}{g} + \frac{ν_{0}}{l} + ξ + \frac{ν_{0}}{l} (xβ + y) |{β \atop 1} )}{φ (\frac{l}{g} + ξ |{β \atop 1} )}.

The statement of the proposition is now a consequence of the transformation formula (55).

Finally we have to show that there exists μ ∈ ℤ such that φ_{𝔤}/φ_{𝔤^{∗}}^{μ} ∈ k(𝔤). From (69) and (70) we deduce that μ has to be chosen such that

(−1)^{xy(a^{2}−b^{2})+(x+y)(a−b)−μ(x^{∗}y^{∗}(a^{∗2}−b^{∗2})+(x^{∗}+y^{∗})(a^{∗}−b^{∗}))}e^{− \frac{2πi}{2} (ν_{0}^{2} \frac{g}{l^{2}} x−μν_{0}^{2} \frac{g^{∗}}{l^{2}} x^{∗})} = 1. (71)

We claim that it suffices to take μ ∈ ℤ such that

ν_{0}^{2}(\frac{g}{l^{2}} x − μ\frac{g^{∗}}{l^{2}} x^{∗}) = ν_{0}^{2} \frac{g_{1}}{l^{2}} x(g_{2} − μg_{2}^{∗}) ≡ 0 (mod 2)  (72)

holds. We have to prove that if (72) holds, then the first factor in (71) is equal to 1, too. Since 2 divides ν_{0} for 2 ∤ l by (68) and

a − b = ν_{0} \frac{g}{l} , a^{∗} − b^{∗} = ν_{0} \frac{g^{∗}}{l} ,

this is obvious in this case. For 2 | l we first note that μ must be odd by (72). Moreover

xy(a^{2} − b^{2}) + (x + y)(a − b) − μ(x^{∗}y^{∗}(a^{∗2} − b^{∗2}) + (x^{∗} + y^{∗})(a^{∗} − b^{∗}))

≡ ν_{0}^{2}xy \frac{1}{l^{2}} gg_{1}(g_{2} − g_{2}^{∗}) + xν_{0} \frac{g_{1}}{l} (g_{2} − g_{2}^{∗})
= ν_{0}^{2}xycc_{1}(c_{2} − c_{2}^{∗}) + xν_{0}c_{1}(c_{2} − c_{2}^{∗}) (mod 2).

Since c_{2} and c_{2}^{∗} are both odd the above claim follows.

It remains to prove the existence of μ ∈ ℤ such that (72) holds. (72) is equivalent to

ν_{0}^{2} \frac{c_{1}}{l} x(c_{2} − μc_{2}^{∗}) ≡ 0 (mod 2).

If 2 ∤ l, then 2 | ν_{0} and it suffices to solve μc_{2}^{∗} ≡ c_{2}mod l. This is possible since (c_{2}^{∗}, l) = 1. If 2 | l, then (2l, c_{2}^{∗}) = 1 and we solve c_{2}^{∗}μ ≡ c_{2} (mod 2l).

Finally we have to check the congruence condition of (iii). To that end let

𝔠 = 𝔭^{r+m}, 𝔠^{∗} = 𝔭^{r} with r, m ≥ 1.

If 𝔭 is inert, then c_{2} = c_{2}^{∗} and we may take μ = 1. For 𝔭 ramified or decomposed we have

Φ(𝔭^{r}) = (p − 1)p^{r−1} with p = min(ℕ ∩ 𝔭).

If 𝔭 decomposes, then c_{2}^{∗} = p^{r} and c_{2} = p^{r+m}. Hence we have to solve

p^{r}μ ≡ p^{r+m} (mod l) (resp. (mod 2l))
μ ≡ 1 (mod p^{r−1}(p − 1)). (73)

We will show at the end that this can be done.

Suppose now that 𝔭 is ramified. If 2 | r and 2 | m, we obtain c_{2} = c_{2}^{∗} = 1 and we take μ = 1. In the case 2 | r and 2 ∤ m we get c_{2}^{∗} = 1 and c_{2} = p. Thus we need to solve

μ ≡ p (mod l) (resp. (mod 2l))
μ ≡ 1 (mod p^{r−1}(p − 1)). (74)

Suppose that 2 ∤ r and 2 | m. Then c_{2}^{∗} = c_{2} = p and we choose μ = 1. Finally, if 2 ∤ rm, then c_{2}^{∗} = p and c_{2} = 1 and we have to find μ ∈ ℤ such that

pμ ≡ 1 (mod l) (resp. (mod 2l))
μ ≡ 1mod p^{r−1}(p − 1). (75)

In order to solve (73) or (74) it suffices to show that

μ ≡ p^{s} (mod l) (resp. (mod 2l)), s > 0
μ ≡ 1 (mod p^{r−1}(p − 1)).

has a solution. We set μ = 1 + νp^{r−1}(p − 1) and wish to solve

ν · p^{r−1}(p − 1) ≡ p^{s} − 1 (mod l) (resp. (mod 2l)).

We put d = (l, p − 1) (resp. d = (2l, p − 1)) and solve

ν · \frac{p^{r−1}(p − 1)}{d} ≡ \frac{p^{s} − 1}{d} (mod \frac{l}{d}) (resp. (mod \frac{2l}{d})).

To find a solution for (75) we choose q ∈ ℤ such that pq ≡ 1 (mod l) (resp. (mod 2l)) and have to find ν ∈ ℤ such that

μ = 1 + νp^{r−1}(p − 1) ≡ q (mod l) (resp. (mod 2l)).

Since q(p − 1) + (q − 1) = pq − 1 ≡ 0 (mod l) (resp. (mod 2l)) we have d | q − 1. Therefore it is possible to solve

νp^{r−1} \frac{p − 1}{d} ≡ \frac{q − 1}{d} (mod \frac{l}{d}) (resp. (mod \frac{2l}{d})).

In order to complete our collection of properties of the φ-function we consider the prime ideal factorization of its singular values. For a fractional ideal 𝔞 of k and ξ ∈ k ∖ 𝔞 we denote by o (ξ |𝔞) the denominator of ξ𝔞^{−1}. The following proposition is originally due to Ramachandra [Ra]. Its present formulation is easily derived from [Sch1, Satz 3].

Proposition 5.13 Let 𝔞 denote a fractional ideal of k and let ξ ∈ k ∖ 𝔞. Then one has

φ (ξ |𝔞) ∼ \begin{cases}1, & if o (ξ |𝔞) is composite, \\ 𝔭^{1/Φ(𝔭^{n})}, & if o (ξ |𝔞) = 𝔭^{n} is a prime ideal power. \end{cases}

5.5 Resolvents

Let us recall the notation of Sections 5.1 and 5.2. F is a finite abelian extension of k and E/F is an elliptic curve with complex multiplication by O_{k} and the additional property that F (E_{tor})/k is abelian. Associated to E/F there is a Groessencharacter φ of k and we have set 𝔣 = lcm(𝔣_{φ}, 𝔣_{F/k}). Recall that 𝔣 only depends on E/F.

Let ξ( , 𝔴) be the corresponding Weierstrass isomorphism as in (40) . Let r, m ≥ 1 be rational integers and fix an prime ideal 𝔭 of k such that (𝔭, 𝔣) = 1. We have set G = E[𝔭^{m}] and for a fixed primitive 𝔭^{r+m}-torsion point Q_{0} = ξ(ρ_{0}, 𝔴) we defined Γ by {Q_{0}+_{E} P | P ∈ G}. In Section 5.2 we further wrote K = k(𝔣𝔭^{r}), L = k(𝔣𝔭^{r+m}), Ω = Ω_{K} and defined

A = (K̄G)^{Ω}, B = Map(G, K̄)^{Ω}, C = Map(Γ, K̄)^{Ω}.

Via the K−algebra isomorphism τ : C → L of (52) we endowed L with the structure of an A-module. This A-action is explicitly described by (54).

For θ ∈ L we set h_{θ} = τ^{−1}(θ). By the definition of τ this implies h_{θ}(Q_{0}) = θ. Recall the definition of the resolvent in (12). For any abelian character χ of G we obtain from (54)

\begin{aligned}(h_{θ}, χ)_{Q_{0}} & = & ∑_{g∈G} h_{θ}(Q_{0} +_{E} g)χ̄(g) \\ & = & ∑_{g∈G} h_{θ}(Q_{0})^{σ(1+ \frac{βg}{ρ_{0}} )}χ̄(g) \\ & = & ∑_{g∈G} θ^{σ(1+ \frac{βg}{ρ_{0}} )}χ̄(g), \end{aligned}  (76)

where β_{g} ∈ 𝔭^{−m}𝔣𝔴 is such that g = ξ(β_{g}, 𝔴).

Our aim in the following will be to construct elements θ ∈ L such that the ideal factorization of (h_{θ}, χ)_{Q_{0}} can be computed.

Theorem 5.14 Assume that 𝔣 is composite and (𝔣, 𝔭𝔭̄) = 1. Then there exists an element θ ∈ L such that

(h_{θ}, χ)_{Q_{0}} ∼ 𝔭^{m}  (77)

for all abelian characters χ of G. Moreover

θ ∼ 𝔭^{\frac{1}{Φ(𝔭^{r})} − \frac{1}{Φ(𝔭^{r+m})}} .  (78)

Remark 5.15 Up to minor modifications this is [Sch2, Satz 2.2].

Proof We set

𝔤 = 𝔣𝔭^{r+m}, 𝔤^{∗} = 𝔣𝔭^{r}

and define φ_{𝔤} and φ_{𝔤^{∗}} as in (67) with 𝔩 = 𝔣, 𝔠 = 𝔭^{r+m}, 𝔠^{∗} = 𝔭^{r}. We choose μ ∈ ℤ as in part (iii) of Proposition 5.12. Then (1 − μ)/Φ(𝔭^{r}) ∈ ℤ and we can find κ ∈ k(1) such that

κ ∼ 𝔭^{(1−μ)/Φ(𝔭^{r})}.

We set

θ = \frac{φ_{𝔤}}{φ_{𝔤^{∗}}^{μ}} · κ.

Then θ ∈ L by Proposition 5.12. Furthermore it follows from Proposition (5.13) that

φ_{𝔤} ∼ \frac{φ (l + g \frac{ν_{0}}{l} |𝔩𝔭^{r+m} )}{φ (l |𝔩𝔭^{r+m})} ∼ 𝔭^{− \frac{1}{Φ(𝔭^{r+m})}} ,
φ_{𝔤^{∗}} ∼ \frac{φ (l + g^{∗} \frac{ν_{0}}{l} |𝔩𝔭^{r} )}{φ (l |𝔩𝔭^{r})} ∼ 𝔭^{− \frac{1}{Φ(𝔭^{r})}}

Hence θ ∼ 𝔭^{− \frac{1}{Φ(𝔭^{r+m})} + \frac{μ}{Φ(𝔭^{r})} + \frac{1−μ}{Φ(𝔭^{r})}} = 𝔭^{\frac{1}{Φ(𝔭^{r})} − \frac{1}{Φ(𝔭^{r+m})}} , which proves (78).

In order to verify (77) we choose a set R of representatives of 𝔭^{−m}l^{3}𝔣𝔴 modulo l^{3}𝔣𝔴. Then

G = {ξ(β, 𝔴) | β ∈ R}

and from (76) we deduce

(h_{θ}, χ)_{Q_{0}} = ∑_{β∈R} θ^{σ(1+ \frac{β}{ρ_{0}} )}χ̄(ξ(β, 𝔴)).

From β ∈ 𝔭^{−m}l^{3}𝔣𝔴 and ρ_{0} ∈ 𝔭^{−r−m}𝔴 we easily deduce

\frac{l}{g} · \frac{β}{ρ_{0}} ∈ \frac{l^{4}}{g} 𝔣𝔭^{r}.

Considering the isomorphism

𝔭^{−m}l^{3}𝔣𝔴/l^{3}𝔣𝔴 ⟼ \frac{l^{4}}{g} 𝔣𝔭^{r}/\frac{l^{4}}{g} 𝔣𝔭^{r+m},
β ⟼ \frac{l}{g} · \frac{β}{ρ_{0}} ,

we obtain

(h_{θ}, χ)_{Q_{0}} = ∑_{ξ} θ^{σ(1+ \frac{g}{l} ξ)}χ̄(ξ),  (79)

where ξ runs over a set of representatives of \frac{l^{4}}{g} 𝔣𝔭^{r}/\frac{l^{4}}{g} 𝔣𝔭^{r+m} and χ is interpreted as a character of this group in the obvious way. From (79) and Proposition 5.12 we deduce

(h_{θ}, χ)_{Q_{0}} = (∑_{ξ} \frac{φ (\frac{l}{g} + \frac{ν_{0}}{l} + ξ |{β \atop 1} )}{φ (\frac{l}{g} + ξ |{β \atop 1} )}χ̄(ξ)) · φ_{𝔤^{∗}}^{−μ} · κ,

where we have to take into account that σ(1 + \frac{g}{l} ξ) acts trivially on φ_{𝔤^{∗}}, since \frac{g}{l} ξ ∈ l^{3}𝔣𝔭^{r} and φ_{𝔤^{∗}} ∈ k(l𝔣𝔭^{r}), if 2 ∤ l (resp. φ_{𝔤^{∗}} ∈ k(2l𝔣𝔭^{r}) , if 2 | l).

By the homogenity of φ (and the usual misuse of notation) we can write

(h_{θ}, χ)_{Q_{0}} = (∑_{ξ} \frac{φ (l + \frac{ν_{0}g}{l} + ξ |𝔣𝔭^{r+m} )}{φ (l + ξ |𝔣𝔭^{r+m})}χ̄(ξ)) · φ_{𝔤^{∗}}^{−μ} · κ,

where now ξ runs over a set of representatives of l^{4}𝔣𝔭^{r}/l^{4}𝔣𝔭^{r+m}. Recall now the definition of the function h_{𝔴} in (59). Since ξ ∈ l^{4}𝔣𝔭^{r} and 2 | ν_{0}, if 2 ∤ l, we easily find

\frac{1}{2}h_{𝔣𝔭^{r+m}}(ξ, ν_{0} \frac{g}{l}) ∈ 2πiℤ.

We apply Proposition 5.11 and therefore adopt the notation used there. Write 𝔣𝔭^{r+m} = ℤω_{1} + ℤω_{2}, 𝔣𝔭^{r} = ℤ\frac{ω_{1}}{n_{1}} + ℤ\frac{ω_{2}}{n_{2}} , χ(\frac{ω_{1}}{n_{1}} ) = e^{2πiμ_{1}/n_{1}}, χ(\frac{ω_{2}}{n_{2}} ) = e^{2πiμ_{2}/n_{2}} and δ_{χ} = \frac{1}{N_{k/ℚ}(𝔭^{m})}(ν_{0} \frac{g}{l} + μ_{2}ω_{1} − μ_{1}ω_{2}). Then Proposition 5.11 yields

(h_{θ}, χ)_{Q_{0}} = ζ · \sqrt[12]{\frac{Δ(𝔣𝔭^{r})}{Δ(𝔣𝔭^{r+m})}} · φ_{𝔤^{∗}}^{−μ} · κ · \underbrace{\frac{φ (g \frac{ν_{0}}{l} |𝔣𝔭^{r+m} ) φ (l + δ_{χ} |𝔣𝔭^{r})}{φ (δ_{χ} |𝔣𝔭^{r}) φ (l |𝔣𝔭^{r})}}_{=:A}

with some root of unity ζ. We easily prove that 𝔣 divides o (g \frac{ν_{0}}{l} |𝔣𝔭^{r+m} ), o (l + δ_{χ} |𝔣𝔭^{r}), o (δ_{χ} |𝔣𝔭^{r}) and that o (l |𝔣𝔭^{r}) = 𝔭^{r}. Since 𝔣 is composite we derive from Proposition 5.13 A ∼ 𝔭^{− \frac{1}{Φ(𝔭^{r})}} . [KL, Ch. 11,Th. 3.1] implies that \sqrt[12]{Δ(𝔣𝔭^{r})/Δ(𝔣𝔭^{r+m})} ∼ 𝔭^{m}. Collecting everything we get

(h_{θ}, χ)_{Q_{0}} ∼ 𝔭^{m}𝔭^{− \frac{μ}{Φ(𝔭^{r})} + \frac{1−μ}{Φ(𝔭^{r})} − \frac{1}{Φ(𝔭^{r})}} = 𝔭^{m}

5.6 Elliptic curves and Lubin-Tate theory

In this Section we keep all the notations of the previous sections.

In the sequel we will work out the connection between the relative Lubin-Tate theory presented in Section 4 and the global situation described in this section. Since we are mainly concerned with the relative extension k(𝔣𝔭^{r+m})/k(𝔣𝔭^{r}), we may without loss of generality assume that F = k(𝔣). Let 𝔓 be a prime ideal of F above 𝔭. Then 𝔓 is totally ramified in k(𝔣𝔭^{s})/F for all s ≥ 0 and, for any intermediate field N in L/F, we write 𝔓_{N} for the unique prime ideal of N above 𝔓. Let ı : ℚ̄ ↪ ℚ̄_{p} be a field embedding defining the prime 𝔓_{L}. Via ı we will view elements of ℚ̄ as elements of ℚ̄_{p} and usually omit the ı in our notation. Furthermore, for any finite extension N/F we write Ñ for the completion of ı(N).

We fix a minimal Weierstrass equation of E over the localization of O_{F} at 𝔓 and via ı we view E as an elliptic curve defined over F̃ = F_{𝔓}, denoted by Ẽ. Let Ê denote the formal group of Ẽ with respect to the parameter z = −\frac{x}{y}. From Lemma 5.8 we know that Ê is a relative Lubin-Tate extension with respect to the unramified extension F_{𝔓}/k_{𝔭}. More precisely, Ê is the unique group law corresponding to

f(z) = λ̂(𝔭)(z) = Λ(𝔭)z + ….  (80)

Let N be a finite extension of F. From [Sil1, Ch. VII, Prop. 2.2] we know that the map

E_{1}(Ñ) ⟶ Ê(𝔭Ñ), P = (x, y) ⟼ z(P) = −\frac{x}{y}  (81)

is a isomorphism, where E_{1} denotes the kernel of reduction. Let θ : O_{k} → End(E) be normalized. Via ı we can identify the rings End(E) and End(Ẽ) and thus obtain a ring isomorphism θ̃: O_{k} → End(Ẽ). By general theory we know that θ̃(α)(E_{1}(L̃)) ⊆ E_{1}(L̃), α ∈ O_{k}, (see [CT, diagram on p. 49]), hence E_{1}(L̃) is an O_{k}-module. Furthermore Ê(𝔭L̃) is also an O_{k}-module by [dS, Prop. 1.5, p.9]. Namely, there is a ring isomorphism Ok̃ → End(Ê), α ↦ [α]_{f}, and hence a ring embedding O_{k} ↪ End(Ê).

Lemma 5.16 The map E_{1}(Ñ) → Ê(𝔭Ñ), P ↦ z(P), is an isomorphism of O_{k}-modules for any finite extension Ñ/F̃.

Proof Let P ∈ E_{1}(Ñ) and α ∈ O_{k}. Then we have to show that

z(θ̃(α)(P)) = [α]_{f} (z(P)) .

The isogeny θ̃(α) induces an endomorphism ˆ θ̃(α) ∈ End(Ê) such that

E_{1}(Ñ) ⟶^{θ̃(α)} E_{1}(Ñ)
↓  ↓
Ê(𝔭Ñ) \underrightarrow{ˆθ̃(α)} Ê(𝔭Ñ)

commutes. It remains to show that ˆ θ̃(α) = [α]_{f}. To that end, by [dS, Prop. 1.5, p.9], it is enough to prove

ˆθ̃(α)(z) ≡ αz (mod deg 2).

This follows as in the proof of Lemma (5.8) from (ω ∘ θ̃(α))(z) = αω(z).

Corollary 5.17 (i) Let π ∈ Ok̃ be any uniformizing parameter for 𝔭 and let ω ∈ ℚ̄_{p} be a primitive π^{n}-torsion point of Ê. Then P = z^{−1}(ω) is a primitive 𝔭^{n}-torsion point of E(ℚ̄).

(ii) Conversely, if P ∈ E(ℚ̄) is a primitive 𝔭^{n}-torsion point, then z(P) is a primitive π^{n}-torsion point of Ê for any uniformizing parameter π ∈ Ok̃.

Proof We first prove (i). We have to show that

θ̃(α)(P) = 0Ẽ for all α ∈ 𝔭^{n},  (82)
θ̃(α)(P) ≠ 0Ẽ, if 𝔭^{n} ∤ α.  (83)

If w_{𝔭}(α) = s, then α/π^{s} ∈ OF̃^{∗}. By Lemma 5.16 the following equivalence holds:

θ̃(α)(P) = 0_{E} ⟺ [α]_{f}(ω) = [\frac{α}{π^{s}}]_{f} ∘ [π^{s}]_{f}(ω) = 0.

Since [\frac{α}{π^{s}}]_{f} is an isomorphism, this implies both (82) and (83).

In order to prove the converse in (ii) we first note that E[𝔭^{n}] ⊆ E_{1}(F̃_{n}), where we denote by F̃_{n} the field generated by the 𝔭^{n}-torsion points of Ê over F̃. This follows from Ê(𝔭F̃_{n}) ≃ E_{1}(F̃_{n}) and |G_{f,n}| = N_{k/ℚ}(𝔭^{n}) = |E[𝔭^{n}]| (see [dS, Ch. I, Prop. 1.7]), where G_{f,n} is the set of all 𝔭^{n}-torsion points of Ê.

We have to prove that ω = z(P) satisfies

[π^{n}]_{f}(ω) = 0 and [π^{s}]_{f}(ω) ≠ 0, 0 ≤ s < n.  (84)

To that end we choose π_{1} ∈ O_{k} such that w_{𝔭}(π_{1}) = 1. Then π = π_{1} · ε with a unit ε ∈ O∗ ; k̃ and since [ε]_{f} is an isomorphism we obtain [π^{s}]_{f}(ω) = 0 ⟺ [π_{1}^{s}]_{f}(ω) = 0 for 0 ≤ s ≤ n. Now (ii) follows from Lemma 5.16.

Corollary 5.17, together with [dS, Ch. I, Prop. 1.8] and Corollary 5.6, implies that for s ≥ 0 the completion of F (E[𝔭^{s}]) = k(𝔣𝔭^{s}) is equal to the field F̃_{n} = F̃(G_{f,n}), where, as in Section 4 ,

G_{f,n} = {ω ∈ Ê(ℚ̄_{p}) | [π^{n}]_{f}(ω) = 0}, π ∈ O_{k} such that w_{𝔭}(π) = 1,

is the set of all 𝔭^{n}-division points of Ê. In particular we obtain

L̃ = F̃_{r+m}, K̃ = F̃_{r}.  (85)

We fix an uniformizing parameter π ∈ O_{k} for 𝔭 and set

G̃ = G_{f,m} = {ω ∈ Ê(ℚ̄_{p}) | [π^{m}]_{f}(ω) = 0},
Γ̃ = {z(Q_{0}) +_{f} ω | ω ∈ G̃}.

Then we derive from Lemma (5.17) a group isomorphism

G ⟼ G̃, g ⟼ z(g)  (86)

and a bijection

Γ ⟼ Γ̃, Q ⟼ z(Q)  (87)

Let

A = (K̄G)^{Ω_{K}} ,  Ã = ( ¯K̃G̃)^{ΩK̃}
= (LG)^{Gal(L/K)},  = (L̃G̃)^{Gal(L̃/K̃)},
C = Map(Γ, K̄)^{Ω_{K}} , C̃ = Map(Γ̃, ¯ K̃)^{ΩK̃}
= Map(Γ, L)^{Gal(L/K)}  = Map(Γ̃, L̃)^{Gal(L̃/K̃)}.

(88)

Then

Φ : K̃ ⊗_{K} A ⟼ Ã, α ⊗ ∑_{g∈G} a_{g}g ⟼ ∑_{g∈G} αa_{g}z(g)  (89)

is an isomorphism of K̃-algebras and we will identify both algebras in the sequel.

Lemma 5.18 The map

Ψ : K̃ ⊗_{K} C ⟶ C̃,
α ⊗ h ⟼ (z(Q) ↦ αh(Q)) , Q ∈ Γ

is an isomorphism of Ã-algebras.

Proof Let Q_{1}, …, Q_{t} be a set of orbit representatives of Γ modulo the action of Gal(L/K). For each m ∈ {1, …, t} let K̃_{m} denote the fixed field of stab_{Ω_{K}} (Q_{m}) ≤ Ω_{K}, and let z_{m,1}, …, z_{m,t_{m}} be an K-basis of K̃_{m}.(The notation K̃_{m} does not mean completion for a short moment, but adapts the notation of Lemma 2.11.) For 1 ≤ m ≤ t and 1 ≤ n ≤ t_{m}, let c_{m,n} ∈ C be defined by

c_{m,n}(Q) := \begin{cases}ω(z_{m,n}), & if Q = ^{ω}Q_{m} for some ω ∈ Ω_{L}, \\ 0, & otherwise, \end{cases}

for Q ∈ Γ. Then, by Lemma 2.11(ii), the elements c_{m,n}, 1 ⩽ m ⩽ t, 1 ⩽ n ⩽ t_{m}, form an K-basis of C. Obviously this basis is mapped to a K̃-basis of C̃. Hence Ψ is an isomorphism of K̃-algebras.

Finally we have to show

Ψ ((α ⊗ h) (β ⊗ ∑_{g∈G} a_{g}g)) = Ψ(α ⊗ h) · ∑_{g∈G} βa_{g}z(g)

for h ∈ C, α, β ∈ K̃ and ∑_{g∈G} a_{g}g ∈ A. This is a straight forward consequence from the definitions: for the left hand side we compute

(Ψ ((α ⊗ h)(β ⊗ ∑_{g∈G} a_{g}g))) (z(Q)) = ∑_{g∈G} αβa_{g}h(Q +_{E} g)

for the value at z(Q), Q ∈ Γ, and the right hand side yields

(Ψ(α ⊗ h) · ∑_{g∈G} βa_{g}z(g)) (z(Q)) = ∑_{g∈G} βa_{g}Ψ(α ⊗ h) (z(g) +_{f} z(Q))

= ∑_{g∈G} βa_{g}Ψ(α ⊗ h) (z(g +_{E} Q))
= ∑_{g∈G} αβa_{g}h(g +_{E} Q).

The preceeding discussion may be summarized in the following commutative diagram of Ã-modules

K̃ ⊗_{K} C ⟶^{1⊗τ} K̃ ⊗_{K} L = L̃
Ψ↓  ↓
C̃  ⟶^{τ̃}  L̃

where τ(h) = h(Q_{0}) and τ̃(h̃) = h̃(z(Q_{0})) for h ∈ C and h′ ∈ C′.

Finally we have to take care of our resolvents.

Lemma 5.19 Let h ∈ C and put h̃ = Ψ(h) ∈ C̃. Then

(h̃, χ̃)_{z(Q_{0})} = (h, χ)_{Q_{0}},

where χ and χ̃ are related by χ = χ̃ ∘ z.

Proof This is just a straight forward computation:

(h̃, χ̃)_{z(Q_{0})} = ∑_{g∈G} h̃(z(Q_{0}) +_{f} z(g))χ̃(z(g))^{−1}
= ∑_{g∈G} h̃(z(Q_{0} +_{E} g))χ̄(g)
= ∑_{g∈G} h(Q_{0} +_{E} g)χ̄(g)
= (h, χ)_{Q_{0}}.

5.7 Module structure

Recall the situation and notation as it is presented in Section 5.2. In addition, let C denote the integral closure of O_{K} in C and let A^{ass} be its associated order in A.

Our aim is to study the A^{ass}-module structure of C. The case r = 0 is already settled by Corollary 2.15. Namely, define a map l ∈ C by

l(Q) = \begin{cases}1, & if Q = 0_{E}, \\ 0, & if Q ≠ 0_{E}, \end{cases}

and recall that A^{∘} = (OK̄G)^{Ω}. Then we have the following

Theorem 5.20 Let r = 0. Then A^{ass} = A^{∘} and C is free of rank one over A^{ass}. An explicit generator is given by the map l.

Proof This is the assertion of Corollary 2.15.

In the sequel we concentrate on the case r ≥ 1. Recall that L = k(𝔣𝔭^{r+m}) and K = k(𝔣𝔭^{r}). In order to control the ramication of primes outside 𝔭 we assume that the elliptic curve E/F and K satisfy the following hypothesis:

E attains everywhere good reduction over K  (90)

This is in fact a mild condition, which, for a fixed elliptic curve E/F, excludes only finitely many extensions K = k(𝔣𝔭^{r}). In Lemma 5.26 we will precisely describe those cases that do not satisfy (90).

Lemma 5.21 Let K = k(𝔣𝔭^{r}), r ≥ 1, and let 𝔔 be a prime ideal of O_{K} not dividing 𝔭. Then 𝔔 is unramified in L/K, L = k(𝔣𝔭^{r+m}), for all m ≥ 1 if and only if 𝔔 is a prime of good reduction for E/K.

Proof The criterion of Ogg-Néron-Shafarevich (see [dS, Ch. II, Th. 1.8]) implies that 𝔔 is a prime of good reduction if and only if K(E[𝔭^{∞}])/K is unramified at 𝔔. Since K(E[𝔭^{∞}]) = k(𝔣𝔭^{∞}) by Proposition 5.3 the Lemma follows.

We will completely determine A^{ass} by its local components and thereby prove that it is an Hopf order, more precisely, the Cartier dual of the O_{K}-Hopf order which represents the O_{K}-group scheme of 𝔭^{m}-torsion points on E. Moreover we will show that C is locally free of rank 1 over A^{ass}. Finally we will prove that, under certain restrictions on 𝔣 and 𝔭, C is also globally free over A^{ass}.

Definition 5.22 We define an O_{K}-module Ã ⊆ A by describing its localizations at each prime 𝔔 of K. If 𝔔 ∤ 𝔭 let

Ã_{𝔔} = (O_{K_{𝔔}}G)^{Ω_{K}} .

For 𝔔 | 𝔭 we choose a uniformizing parameter π ∈ O_{k} for 𝔭 and a minimal Weierstrass equation for E defined over the localization O_{K,𝔔}. For i ≥ 0 we define σ_{i} ∈ A by

σ_{i} = \frac{1}{π^{m}} ∑_{g∈G} z(g)^{i}(g − g_{0}),

where

z(g) = \begin{cases}−\frac{x}{y}, & if g = (x, y) ≠ 0_{E}, \\ 0, & if g = 0_{E}. \end{cases}

Then

Ã_{𝔔} = O_{K_{𝔔}} · g_{0} + ∑_{i=0}^{q^{m}−2} O_{K} · σ_{i},

where q = N_{k/ℚ}(𝔭).

Proposition 5.23 Ã is a O_{K}-Hopf order in A. More precisely, it is the Cartier dual of the O_{K}-Hopf order which represents the O_{K}-group scheme of 𝔭^{m}-torsion points on E.

Proof Let B^{gs} denote the O_{K}-Hopf order which represents the O_{K}-group scheme of 𝔭^{m}-torsion points on E. An explicit description of B^{gs} is given in [T2, § 2]. For each prime 𝔔 of O_{K} not dividing 𝔭, its localization B_{𝔔}^{gs} is equal to Map(G, OK̄)_{𝔔}^{Ω_{K}}. For a prime 𝔔 | 𝔭 we have

B_{𝔔}^{gs} = O_{K_{𝔔}}[[X]]/(f^{(m)}(X)),

where f(z) is given by (80) and f^{(m)} is defined in (25). Here we view, as in Section 3, B_{𝔔}^{gs} as an O_{K_{𝔔}}-order in Map(G, K̄)_{𝔔}^{Ω_{K}} via the rule b(X)(g) = b(z(g)) for b(X) ∈ O_{K_{𝔔}}[[X]] and g ∈ G.

According to Lemma 2.3, we have a decomposition

B ≃ ∏_{i=1}^{r} K_{i},

where in this situation each of the fields K_{i} is a subfield of L/K, and hence by (90) and Lemma 5.21 unramified at primes 𝔔 not dividing 𝔭. Thus the assertion of the Theorem follows for 𝔔 ∤ 𝔭 from Lemma 2.4 and Corollary 2.5. For 𝔔 | 𝔭 it is an immediate consequence of Lemma 5.16 and Proposition 4.5.

In the following we write again A^{gs} for Ã, reflecting the assertion of the previous proposition.

Theorem 5.24 Let r ≥ 1 and suppose that E/F and K satisfy (90). Then: (i) The associated order A^{ass} is equal to A^{gs}. (ii) C is a locally free A^{ass}-module of rank 1.

Proof Let first 𝔔 be a prime ideal dividing 𝔭. By Lemma 5.18 the isomorphism Φ in (89) maps A_{𝔔}^{ass} to the associated order of the integral closure C̃ of O_{K_{𝔔}} in C̃. Hence the assertion of the Theorem follows from Proposition 5.23 and Corollary 4.10.

Next we will prove that C_{𝔔} is locally free over A_{𝔔}^{gs} for primes 𝔔 such that 𝔔 ∤ 𝔭. To achieve this we apply [CH, Th. 5.4].Let

I = {λ ∈ A_{𝔔}^{gs} | μλ = ε(μ)λ, ∀μ ∈ A_{𝔔}^{gs}}

denote the ideal of integrals in A_{𝔔}^{gs}. It is an easy exercise, using the basis described in Lemma 2.1, to show that I = O_{K_{𝔔}} ∑_{g∈G} g. Since L/K is unramified at 𝔔 by Lemma 5.21, it follows that C_{𝔔} · I = O_{K_{𝔔}}, and this, in turn, proves local freeness by [CH, Prop. (5.4)].

To sum up, C is a locally free A^{gs}-module, hence A^{ass} = A^{gs}.

As in the Lubin-Tate theory we consider the natural map

ξ′ : B ⟶ C,
f ⟼ (Q ↦ f(Q −_{E} Q_{0})).

From the explicit local description of B^{gs} we derive

B_{L}^{gs} = O_{L} ⊗_{O_{K}} B^{gs}, A_{L}^{gs} = O_{L} ⊗_{O_{K}} A^{gs}.

In the terminology of [T2] or [BT] the next proposition shows that C is a principal homogeneous space for B^{gs}.

Proposition 5.25 The map ξ′ induces an isomorphism

ξ : O_{L} ⊗_{O_{K}} B^{gs} ⟶ O_{L} ⊗_{O_{K}} C

of A_{L}^{gs}-modules.

Proof We show that ξ induces an isomorphism for the localizations at each prime 𝔔 of O_{L}. For a prime 𝔔 dividing 𝔭 this follows from the corresponding result of the Lubin-Tate theory, namely Proposition 4.14. For primes 𝔔 not dividing 𝔭 we set 𝔮 = 𝔔 ∩ O_{K}. Then B^{gs} is the unique maximal O_{K,𝔮}-order in B. By hypothesis (90) the extension L/K is unramified at 𝔮 and therefore O_{L,𝔮} ⊗ B^{gs} (resp. O_{L,𝔮} ⊗ C) is the maximal O_{L,𝔮}-order in B (resp. C). Now the result follows, since ξ is obviously an isomorphism on the field level.

Before we discuss the question of global freeness we explicitly describe the triples 𝔣, 𝔭, r for which the hypothesis (90) is not satisfied.

Lemma 5.26 If k ≠ ℚ(\sqrt{−3}), then E/F and K do not satisfy (90) if and only if one of the following conditions holds:

(a) 2 = 𝔭_{2}𝔭̄_{2} is decomposed in k/ℚ, 𝔭 = 𝔭_{2}, r = 1, 𝔣 = 𝔮^{s}, s ≥ 1.

(b) 2 = 𝔭_{2}𝔭̄_{2} is decomposed in k/ℚ, 𝔭 = 𝔭_{2}, r = 1, 𝔣 = 𝔮^{s}𝔭̄_{2}, s ≥ 1 and (𝔮, 𝔭̄) = 1.

(c) 2 = 𝔭_{2} is inert in k/ℚ, 𝔭 = 𝔭_{2} and r = 1, 𝔣 = 𝔮^{s}, s ≥ 1.

(d) 2 = 𝔭_{2}^{2} is ramified in k/ℚ, 𝔭 = 𝔭_{2}, r ∈ {1, 2} and 𝔣 = 𝔮^{s}, s ≥ 1.

If k = ℚ(\sqrt{−3}), then we have the following exclusions:

(e) 𝔭 = (2), r = 1 and 𝔣 = 𝔮^{s}, s ≥ 1.

(f) 𝔭 = 𝔭_{3} (where (3) = 𝔭_{3}^{2}), r = 1 and 𝔣 = 𝔮^{s}, s ≥ 1.

Here 𝔮 denotes always a prime of k with (𝔮, 𝔭) = 1.

Proof Since 𝔭 ∤ 𝔣 the elliptic curve E/K has good reduction for all primes 𝔔 of O_{K} dividing 𝔭. Let 𝔔 be a prime of O_{K} not dividing 𝔭. By Lemma 5.21 it suffices to show that k(𝔣𝔭^{r+m})/k(𝔣𝔭^{r}) is unramified at 𝔔 for m ≥ 1, except in the cases mentioned in the Lemma.

Let 𝔮 = 𝔔 ∩ O_{k}. As before we write L = k(𝔣𝔭^{r+m}) and K = k(𝔣𝔭^{r}). Then 𝔮 ramifies in L if and only if 𝔮 | 𝔣. Hence we may assume that

𝔣 = 𝔮^{s}𝔣′, (𝔣′, 𝔮) = 1, s ≥ 1.

We denote by e_{L/k}, e_{K/k}, e_{L/K} the ramification indices of primes above 𝔮 in L/k, K/k, L/K, respectively. By class field theory (e.g., [Neu, Kap. VI, §7, Aufgabe 1] or [Ha2, Th. IVa]) we derive

e_{L/k} = [k(𝔣𝔭^{r+m}) : k(𝔣′𝔭^{r+m})] = \frac{w(𝔣𝔭^{r+m})}{w(𝔣′𝔭^{r+m})}Φ(𝔮^{s}),
e_{K/k} = [k(𝔣𝔭^{r}) : k(𝔣′𝔭^{r})] = \frac{w(𝔣𝔭^{r})}{w(𝔣′𝔭^{r})}Φ(𝔮^{s}),

where Φ denotes the Euler function of the ring O_{k} and w(𝔞) the order of {ζ ∈ O_{k}^{∗} | ζ ≡ 1(mod 𝔞)} for any integral ideal 𝔞 of O_{k}. Hence we obtain

e_{L/K} = \frac{w(𝔣𝔭^{r+m})w(𝔣′𝔭^{r})}{w(𝔣′𝔭^{r+m})w(𝔣𝔭^{r})}.  (91)

If k ≠ ℚ(\sqrt{−1}), ℚ(\sqrt{−3}), then we have

w(𝔞) = \begin{cases}2, & if 𝔞 | (2), \\ 1, & otherwise. \end{cases}  (92)

For k = ℚ(\sqrt{−1}) we set (2) = 𝔭_{2}^{2}. Then we have

w(𝔞) = \begin{cases}4, & if 𝔞 | 𝔭_{2}, \\ 2, & if 𝔞 = (2), \\ 1, & otherwise. \end{cases}  (93)

Finally, for k = ℚ(\sqrt{−3}) we set (3) = 𝔭_{3}^{2}. Then

w(𝔞) = \begin{cases}6, & if 𝔞 = (1), \\ 3, & if 𝔞 = 𝔭_{3}, \\ 2, & if 𝔞 = (2), \\ 1, & otherwise. \end{cases}  (94)

Now the assertion of the Lemma follows easiy from (91), (92), (93) and (94). For the convenience of the reader we record the ramification degrees e_{L/K}:

(a), (b) (c)  (d)  (e) (f)
k ≠ ℚ(\sqrt{−1}), ℚ(\sqrt{−3})  2  2  2  − −
k = ℚ(\sqrt{−1})  −  − \begin{matrix}4, & if r = 1, m ≥ 2 \\ 2, & if r = m = 1 \\ 2, & if r = 2 \end{matrix} − −
k = ℚ(\sqrt{−3})  −  −  −  2 3

Due to the lack of appropriate resolvents we are not able to prove a complete result for the question of global freeness of C over A^{ass}. As a first global result we obtain

Theorem 5.27 Assume that r ≥ 1. If 𝔣 is composite and (𝔣, 𝔭𝔭̄) = 1, then C is free of rank one over its associated order A^{ass}. An explicit generator is given by the map h_{θ} constructed in the proof of Theorem 5.14.

Remark 5.28 By Lemma 5.26 the assumptions of the theorem guarantee that hypothesis (90) is satisfied.

Proof Let h_{θ} ∈ C be the map constructed in Theorem 5.14. Then

(h_{θ}, χ)_{Q_{0}} ∼ 𝔭^{m}  (95)

for all abelian characters χ of G. We shall show that the global element h_{θ} generates C_{𝔔} at each prime 𝔔 of O_{K}.

Let L′ be a splitting field for A and B. By Theorem 2.14 we have

[C : h_{θ} · A^{∘}]_{O_{K}}^{2} O_{L′} = d_{B/K} · d_{C/K}^{−1} ∏_{χ∈G}(h_{θ}, χ)_{Q_{0}}^{2}.  (96)

Since B and C are unramified outside 𝔭 by Lemma 5.21 equation (95) implies that C_{𝔔} = h_{θ} · A_{𝔔}^{ass} for each prime 𝔔 ∤ 𝔭.

For primes 𝔔 | 𝔭 it follows from Lemma 5.19 and (95) that for h̃_{θ} = Ψ(h_{θ}) we have

(h̃_{θ}, χ̃)_{z(Q_{0})} ∼ π^{m}.

Thus, by Theorem 4.12 (a), h̃_{θ} generates C̃_{𝔔} over its associated order. But then Lemma 5.18 implies that C_{𝔔} is generated by h_{θ} over A_{𝔔}^{ass}.

We now use the fact that the associated orders behave well under a change of the base field to deduce a more complete result from Theorem 5.27. Let w_{k} denote the number of roots of unity in k.

Theorem 5.29 Assume that r ≥ 1. If (𝔣, 𝔭𝔭̄) = 1 and (𝔭, w_{k}) = 1, then C is free of rank one over its associated order.

Proof Let 𝔮 be an O_{k}-prime ideal with (𝔣𝔭𝔭̄, 𝔮) = 1 and set K′ = k(𝔣𝔮𝔭^{r}), L′ = k(𝔣𝔮𝔭^{r+m}). Since the discriminants d_{K′/K} and d_{C} are relatively prime, we have

O_{K′} ⊗_{O_{K}} C ≃ C_{K′}.

From this and Theorem 5.24 (or from the explicit local description of A^{ass}) we derive

A_{K′}^{ass} ≃ O_{K′} ⊗_{O_{K}} A^{ass}.  (97)

Let cl(A^{ass}) (resp. cl(A_{K′}^{ass})) denote the classgroup of locally free A^{ass}-modules (resp. A_{K′}^{ass}-modules). Because of (97) we have a restriction homomorphism

res : cl(A_{K′}^{ass}) ⟶ cl(A^{ass}).

From Theorem 5.27 we know that the class (C_{K′})_{A_{K′}^{ass}} of C_{K′} in cl(A_{K′}^{ass}) is trivial. On the other hand res((C_{K′})_{A_{K′}^{ass}}) = [K′ : K](C_{K})_{A^{ass}}. Varying the extension K′/K we conclude that

d · (C_{K})_{A^{ass}} = 0  (98)

with d = gcd{[K′ : K] | K′/K as above }. It follows easily from global class field theory that d | w_{k} (see e.g. [ST, Lemma 3]). Hence we get w_{k} · (C_{K})_{A}^{ass} = 0 in cl(A^{ass}).

We now consider the base change induced by applying O_{L} ⊗_{O_{K}} . From the explicit local desription of the O_{K}-order B^{gs} we derive

O_{L} ⊗_{O_{K}} A^{gs} ≃ A_{L}^{gs}.

Hence we again have a restriction homomorphism

res : cl(A_{L}^{gs}) ⟶ cl(A^{gs}).

We claim that B_{L}^{gs} is a free A_{L}^{gs}-module with generator π^{m}l, l defined as in (23). Indeed, localizing at a prime 𝔔 of O_{L} dividing 𝔭 this follows from Corollary 4.6. For primes 𝔔 not dividing 𝔭 we have B_{L,𝔔}^{gs} = Map(G, O_{L,𝔔}) and A_{L,𝔔}^{gs} = O_{L,𝔔}G. Thus the claim is immediate, since π is a unit at 𝔔.

By Proposition 5.25 the A_{L}^{gs}-module O_{L} ⊗_{O_{k}} C is globally free and we therefore conclude

res ((O_{L} ⊗_{O_{k}} C)_{A_{L}^{gs}}) = N_{k/ℚ}(𝔭^{m}) · (C_{K})_{A}^{gs} = 0.

Together with (98) this implies (C_{K})_{A^{gs}} = 0. Since A^{gs} is commutative, the Eichler condition holds, and thus C is a free A^{gs} = A^{ass}-module.

The next corollary describes consequences of Theorem 5.29 for the classical Galois module structure of rings of integers in abelian extensions of a quadratic imaginary number field.

Corollary 5.30 Let r, m be integers with 1 ≤ m ≤ r. Let 𝔣 be an integral O_{k}- ideal with w(𝔣) = 1. Let 𝔭 be an O_{k}-prime ideal with (𝔣, 𝔭𝔭̄) = 1 and (𝔭, w_{k}) = 1 and set K = k(𝔣𝔭^{r}), L = k(𝔣𝔭^{r+m}). Then O_{L} is free over its associated order in KGal(L/K).

Proof By [dS, Ch. II, Lemma 1.4] there exists an elliptic curve E defined over F = k(𝔣) with complex multiplication by O_{k} such that the associated Groessencharacter ψ_{E/F} is of the form ψ_{E/F} = φ∘N_{F/k} with a Groessencharacter φ of k of infinity type (1, 0) and conductor 𝔣_{φ} = 𝔣. Thus the assertion of the corollary follows from Theorem 5.29 and Lemma 5.10.

Remark 5.31 The extensions k(𝔣𝔭^{r+m})/k(𝔣𝔭^{r}) covered by Corollary 5.30 are not completely contained in the results of [CT] and [Sch2]. On the other hand our results do not completely include the results of [Sch2].

For the construction of the associated orders Schertz uses exactly our approach applied to the analytic Fueter model or the Deuring model of an elliptic curve. His results extending those of Corollary 5.30 are obtained by really tricky constructions of resolvent elements ([Sch2, Satz 2.1 and 2.2]) and some descent argument. However, it is clear from this work, that these methods can not be extended to treat the case m > r for classical Galois module structure.

In the last part of this section we will use the results of Theorems 5.20, 5.24 and 5.27 to derive results in the composite case. Namely, let 𝔠 and 𝔪^{∗} be integral ideals of k and set 𝔪 = 𝔠𝔪^{∗}. We assume that (𝔣, 𝔠) = 1 and write

𝔠 = ∏_{i=1}^{s} 𝔭_{i}^{m_{i}}, 𝔪^{∗} = ∏_{i=1}^{s} 𝔭_{i}^{r_{i}}, 𝔪 = ∏_{i=1}^{s} 𝔭_{i}^{r_{i}+m_{i}}, 0 ≤ r_{i}, m_{i}, r_{i} + m_{i} ≥ 1.

We let G = E[𝔠] be the group of 𝔠-torsion points and choose primitive 𝔭_{i}^{r_{i}+m_{i}}- torsion points Q_{i,0} for i = 1, …, s. If we put

Q_{0} = Q_{1,0} +_{E} … +_{E} Q_{s,0},

then Q_{0} is a primitive 𝔪-torsion point of E and we easily derive

G = ∏_{i=1}^{s} G_{i},  (99)

Γ := {Q_{0} +_{E} g | g ∈ G} = ∏_{i=1}^{s} Γ_{i},  (100)

where

G_{i} := E[𝔭_{i}^{m_{i}}], Γ_{i} := {Q_{i,0} +_{E} g | g ∈ G_{i}}, i = 1, … s.

Finally we write K = F (E[𝔪^{∗}]) = k(𝔣𝔪^{∗}), L = F (E[𝔪]) = k(𝔣𝔪) and define

A = (K̄G)^{Ω_{K}} , , B = Map(G, K̄)^{Ω_{K}} , C = Map(Γ, K̄)^{Ω_{K}} .

For i = 1, …, s set 𝔣_{i} = 𝔣𝔪^{∗}/𝔭_{i}^{r_{i}} and assume that either

r_{i} = 0
or  m_{i} = 0
or  m_{i}, r_{i} ≥ 1, (𝔣_{i}, 𝔭_{i}𝔭̄_{i}) = 1 and (𝔭_{i}, w_{k}) = 1
or  m_{i}, r_{i} ≥ 1, 𝔣_{i} is composite and (𝔣_{i}, 𝔭_{i}𝔭̄_{i}) = 1.

Let B^{gs} ⊆ B denote the O_{K}-Hopf order which represents the affine O_{K}-group scheme of 𝔠-torsion on E. As before we write A^{gs} for its Cartier dual in A.

Theorem 5.32 Assume the notation and conditions as above. Then C is free over its associated order A^{ass} in A. If, in addition, r_{i} ≥ 1 for i = 1, …, s, then A^{ass} = A^{gs}.

Remark 5.33 A^{ass} and a generating element are explicitly given in the proof.

Proof Writing A_{i} = (K̄G_{i})^{Ω_{K}}, C_{i} = Map(Γ_{i}, K̄)^{Ω_{K}}, it is easily verified that firstly

A ≃ A_{1} ⊗_{K} … ⊗_{K} A_{s},

and secondly that

\begin{matrix}C_{1} ⊗_{K} … ⊗_{K} C_{s} & ⟶ & C, \\ h_{1} ⊗_{K} … ⊗_{K} h_{s} & ⟼ & (Q_{1} +_{E} … +_{E} Q_{s} ↦ h_{1}(Q_{1}) ⋯ h_{s}(Q_{s})) \end{matrix} (101)

is an isomorphism of A-modules.

By Theorem 5.20 or 5.27 there exists a map h_{i} ∈ C_{i} such that C_{i} = h_{i} · A_{i}^{ass} for each i ∈ {1, …, s}, where C_{i} denotes the integral closure of O_{K} in C_{i} and A_{i}^{ass} its associated order in A_{i}. Since the discriminants d_{C_{1}/K}, …, d_{C_{s}/K} are mutually coprime, the isomorphism in (101) restricts to an isomorphism

C_{1} ⊗_{O_{K}} … ⊗_{O_{K}} C_{s} ⟶ C

of A_{1}^{ass} ⊗_{O_{K}} … ⊗_{O_{K}} A_{s}^{ass}-modules. Therefore

A^{ass} ≃ A_{1}^{ass} ⊗_{O_{K}} … ⊗_{O_{K}} A_{s}^{ass}  (102)

and a generating function is given by the image of h_{1} ⊗ … ⊗ h_{s} under the isomorphism in (101).

If r_{i} ≥ 1 for i = 1, …, s, then (102) and Theorem 5.24 imply A^{ass} = A^{gs}.

6 Kummer orders and Taylor’s conjecture

In [T2], M.J. Taylor introduced the notion of a Kummer order with respect to the group law of an abelian variety. In this section we shall study the Galois module structure of these Kummer orders in the case when the variety is an elliptic curve.

We begin by describing the situation considered by Taylor. For more details the reader is referred to [T2] and [ST].

As always, let k denote a quadratic imaginary number field. Let M/k be a finite extension and E/M an elliptic curve with everywhere good reduction and admitting complex multiplication by O_{k}. Let 𝔞 be a non-zero integral ideal and set

G(𝔞) = E[𝔞], A(𝔞) = A_{M}(𝔞) = (M̄G(𝔞))^{Ω_{M}} ,
B(𝔞) = B_{M}(𝔞) = Map(G(𝔞), M̄)^{Ω_{M}} .

In the following we will define an M-algebra M_{P} for each P ∈ E(M). If 𝔞 = (a) is principal, then M_{P} is given by

M_{P} = Map(G_{P} (𝔞), M̄)^{Ω_{M}} ,

with G_{P} (𝔞) = {P′ ∈ E(M̄) | [a](P′) = P }. In the general case the definition is more involved. Since we will need it later we recall some details from [ST].

For integers i ≥ j ≥ 0 we fix an isomorphism of groups

τ : G(𝔞^{i})/G(𝔞^{j}) ⟶ G(𝔞^{i−j}).  (103)

We always assume that τ is Ω_{M}-equivariant. The group G(𝔞^{j}) acts on M̄G(𝔞^{i}) by translation and (103) induces an isomorphism

\begin{matrix}τ̃ : (M̄G(𝔞^{i}))^{G(𝔞^{j})} & ⟶ & M̄G(𝔞^{i−j}), \\ ∑_{g∈G(𝔞^{i})} a_{g}g & ⟼ & ∑_{g∈G(𝔞^{i})} a_{g}τ(g). \end{matrix}  (104)

Note that τ̃ is Ω_{M}-equivariant if τ has this property.

Next we choose a positive integer h such that 𝔞^{h} = (a) is principal and set

M_{P} (𝔞) = (Map(G_{P} ((a)), M̄)^{G(𝔞^{h−1})})^{Ω_{M}} .  (105)

In the following we write G = G(𝔞), A = A(𝔞), B = B(𝔞) and M_{P} = M_{P} (𝔞). Then A acts on M_{P} via the rule

(f · ∑_{g∈G((a))} a_{g}g) (P′) = ∑_{g∈G((a))} a_{g}f(P′ +_{E} g).

Let B^{gs} ⊆ B be the O_{M}-group scheme of 𝔞-torsion on E and denote by A^{gs} its Cartier dual in A. Recall that B^{gs}, A^{gs} are Hopf orders in B, A, respectively.

Let O_{P} = O_{P} (M) be the integral closure of O_{M} in M_{P} and define the Kummer algebra Õ_{P} to be the largest A^{gs}-module contained in O_{P}, i.e.,

Õ_{P} = Õ_{P} (M) = {x ∈ O_{P} | xA^{gs} ⊆ O_{P} }.

In [T2] it is proved that Õ_{P} is a locally free A^{gs}-module and moreover that

ψ : E(M) ⟶ cl(A^{gs}), P ⟼ \binom{}{Õ_{P}}_{A^{gs}}

is a group homomorphism. Herein cl(A^{gs}) denotes the classgroup of locally free A^{gs}-modules and \binom{}{Õ_{P}}_{A^{gs}} the class of Õ_{P} in cl(A^{gs}).

In [T2] the following conjecture was stated:

Conjecture 6.1 For any non-zero O_{k}-ideal (a), a ∈ O_{k},

E(M)_{tor} ⊆ ker ψ.

In [ST] Conjecture 6.1 was generalized for arbitrary integral ideals and furthermore they proved

Theorem Let (𝔞, w_{k}) = 1, where w_{k} denotes the number of roots of unity in k. Then

E(M)_{tor} ⊆ ker ψ.

Recently, A. Agboola proved an analogue of the Theorem of Srivastav and Taylor for elliptic curves without complex multiplication. Namely, let p > 3 be a rational prime and consider the conjecture for 𝔞 = (p^{i}), i ≥ 1. Then, by [Ag, Theorem 1],

E(M)_{tor} ⊆ ker ψ.

See also [Pa], where among other things, a group scheme theoretic proof of Agboola’s Theorem is given. Finally we mention that Agboola’s Theorem does not hold for p = 2 as the infinite series of counter-examples in [BK] shows.

In the sequel we will apply the results of Section 5.7 in order to derive results concerning Conjecture 6.1.

Let F be a finite abelian extension of k and let E/F be an elliptic curve with complex multiplication by O_{k}. Assume further that F (E_{tor})/k is abelian. Let 𝔣 be defined as in (44). Let 𝔭 be a prime ideal of O_{k} such that (𝔣, 𝔭) = 1. Let P be a primitive 𝔭^{r}-torsion point, r ≥ 1, and let 𝔞 = 𝔭^{m}, m ≥ 1. Set K = F (E(𝔣𝔭^{r}]) = k(𝔣𝔭^{r}).

Theorem 6.2 Assume the above notation and suppose that E/F and K satisfy the hypothesis (90). Then, for M = K,

A^{gs} = A^{ass}, and therefore O_{P} = Õ_{P} .

Remarks 6.3 (i) The hypothesis (90) is almost always satisfied, except in the cases listed in Lemma 5.26.

(ii) During the proof of Theorem 6.2 we will show that there exists an isomorphism of A-modules

M_{P} ≃ C = Map(Γ, M̄)^{Ω_{M}} ,

where Γ = {Q_{0} +_{E} g | g ∈ E[𝔭^{m}]} with an appropriate primitive 𝔭^{r+m}-torsion point Q_{0}. A^{ass} can therefore be viewed as the associated order of the integral closure C of O_{M} in C or as the associated order of O_{P}.

(iii) Theorem 6.2 generalizes [T2, Remark 2, p.431].

Proof of Theorem 6.2 The Theorem follows immediatedly from Theorem 5.23, if we can show that there exists an isomorphism of A-modules

M_{P} ≃ C = Map(Γ, M̄)^{Ω_{M}}

for an appropriate set Γ = {Q′_{0}+_{E} g′ | g′ ∈ E[𝔭^{m}]} with a primitive 𝔭^{r+m}-torsion point Q′_{0}.

Let h ∈ ℕ be such that (𝔭^{m})^{h} = (a) is principal and recall the definition of M_{P} in (103) – (105). We choose b ∈ O_{k} such that (b) = (𝔭^{m})^{h−1} · 𝔟 with an integral O_{k}-ideal 𝔟 such that (𝔟, 𝔭) = 1 and specify an isomorphism τ as in

(103) by

τ : G(𝔞^{h})/G(𝔞^{h−1}) ⟶ G(𝔞),
g + G(𝔞^{h−1}) ⟼ [b](g).

Then the induced isomorphism

τ̃ : (M̄G(𝔞^{h}))^{G(𝔞^{h−1})} ⟼ M̄G(𝔞)  (106)

is Ω_{M}-equivariant and we may therefore identify the Ω_{M}-invariants.

The set G_{P} ((a)) is of the form

G_{P} ((a)) = {Q_{0} +_{E} g | g ∈ G((a))},

where Q_{0} is a primitive 𝔭^{r+mh}-torsion point such that [a](Q_{0}) = P. We set Q′_{0} = [b](Q_{0}) and note that Q′_{0} is a primitive 𝔭^{r+m}-torsion point. Let Γ = {Q′_{0} +_{E} g′ | g′ ∈ G(𝔞)} and consider the natural map

Ψ : Map(G_{P} ((a)), M̄)^{G(𝔞^{h−1})} ⟶ Map(Γ, M̄),
f ⟼ f′,

where f′(γ′) = f(γ) with γ ∈ G_{P} ((a)) such that [b](γ) = γ′. Ψ is well-defined since [b](γ_{1}) = [b](γ_{2}) implies γ_{1} −_{E} γ_{2} ∈ G(𝔞^{h−1}). The map Ψ is obviously bijective and also Ω_{M}-equivariant:

(Ψ(f^{ω})) (Q′_{0} +_{E} [b](g)) = f^{ω}(Q_{0} +_{E} g) = ω (f(Q_{0}^{ω^{−1}} +_{E} g^{ω^{−1}}))
(Ψ(f)^{ω}) (Q′_{0} +_{E} [b](g)) = ω (Ψ(f)(Q′_{0}^{ω^{−1}} +_{E} [b](g)^{ω^{−1}})) = ω (f(Q_{0}^{ω^{−1}} +_{E} g^{ω^{−1}})) .

Hence it remains to show that Ψ is an A-module homomorphism where we

identify ((M̄G(𝔞^{h}))^{G(𝔞^{h−1})})^{Ω_{M}} and (M̄G(𝔞))^{Ω_{M}} via (106). To that end we

have to establish the equality

Ψ (f ( ∑_{g∈G(𝔞^{h})} a_{g}g)) = Ψ(f) · ∑_{g∈G(𝔞^{h})} a_{g}[b](g),

where ∑_{g∈G(𝔞^{h})} a_{g}g ∈ (M̄G(𝔞^{h}))^{G(𝔞^{h−1})}. Since a_{g+_{E}g_{1}} = a_{g} for g_{1} ∈ G(𝔞^{h−1}) and f(γ +_{E} g_{1}) = f(γ) we easily compute

(Ψ(f( ∑_{g∈G(𝔞^{h})} a_{g}g))) (γ′) = ∑_{g∈G(𝔞^{h})} a_{g}f(γ +_{E} g)

= |G(𝔞^{h−1})| ∑_{g∈G(𝔞^{h})/G(𝔞^{h−1})} a_{g}f(γ +_{E} g)

and, since [b](g_{1}) = 0_{E} for g_{1} ∈ G(𝔞^{h−1}),

(Ψ(f) · ∑_{g∈G(𝔞^{h})} a_{g}[b](g)) (γ′) = ∑_{g∈G(𝔞^{h})} a_{g}Ψ(f)(γ′ +_{E} [b](g))

= ∑_{g∈G(𝔞^{h})/G(𝔞^{h−1})} |G(𝔞^{h−1})|a_{g}f(γ +_{E} g).

Finally we apply Theorem 5.27 to exhibit an infinite class of elliptic curves E/M with everywhere good reduction for which Taylor’s conjecture holds in full generality. To that end let 𝔣 be a composite O_{k}-ideal such that w(𝔣) = 1. By [dS, Ch. II, Lemma 1.4] there exists an elliptic curve E defined over F = k(𝔣) with complex multiplication by O_{k} such that the associated Groessencharacter ψ_{E/F} is of the form ψ_{E/F} = φ∘N_{F/k} with a Groessencharacter φ of k of infinity type (1, 0) and conductor 𝔣_{φ} = 𝔣.

Theorem 6.4 Let 𝔣 be a composite integral O_{k}-ideal with w(𝔣) = 1 and (𝔣, w_{k}) = 1. Let E/k(𝔣) be an elliptic curve as above and let M be any extension of k(𝔣) such that E/M has everywhere good reduction. Then Taylor’s conjecture holds in full generality, i.e.,

E(M)_{tor} ⊆ ker(ψ)

for all integral ideals 𝔞 of O_{k}.

Proof From [ST, Theorem 1] we already know that the theorem holds for 𝔞 relatively prime with w_{k}. As it is shown in [ST, §2] it therefore suffices to prove P ∈ ker ψ for 𝔞 = 𝔭^{m}, 𝔭 | w_{k}, m ≥ 1 and P a torsion point with 𝔭-power O_{k}-annihilator.

Suppose that P ∈ E(M) is a primitive 𝔭^{r}-trosion point for some r ≥ 0. By [T2, Proposition 2] we may assume that r ≥ 1. Hence M contains the ray class field K = F (E[𝔭^{r}]) = k(𝔣𝔭^{r}) by Proposition 5.3. By Lemma 5.26 E/K has everywhere good reduction. Thus Theorem 5.27 and Theorem 6.2 imply that C_{K} ≃ Õ_{P} is free over its associated order A_{K}^{ass} = A_{K}^{gs}. The result follows now by applying O_{M} ⊗_{O_{K}} , since both A_{K}^{gs} and Õ_{P} behave well with respect to a change of fields (see [ST, (2.2a), (2.5), (2.6)]).

Part II

The conjecture of Chinburg-Stark for abelian extensions of a quadratic imaginary field

1 Introduction

This part is concerned with a strong version of Stark’s conjecture [St] for abelian extensions of an imaginary quadratic number field.

Let N/K be a finite Galois extension of number fields with Galois group G. Let S be a finite G-stable set of places of N containing the set S_{∞} of archimedian places. We write E_{S} for the ℤG-module of S-units of N. For S = S_{∞} we also set E_{N} = E_{S_{∞}}.

Let ℤS denote the free ℤ-module on S, with G acting by permuting the primes in S. Let ΔS be the kernel of the augmentation map ℤS → ℤ which sends every place 𝔭 in S to 1. Then the Dirichlet unit theorem implies that ℚ⊗_{ℤ}E_{S} ≃ ℚ⊗_{ℤ}ΔS as ℚG-modules. Therefore we may fix (but not canonically) a G-embedding φ : ΔS → E_{S}.

For any complex valued character χ of G we write R_{φ}(χ) for the Tate regulator associated to φ and χ in [Ta2] (see also Definition 2.4).

Let L(s, χ) denote the Artin L-function associated to χ with the Euler factors at places of S omitted. Let c(χ) denote the first non-zero coefficient in the Taylor expansion of L(s, χ) at s = 0. Thus R_{φ}(χ) and c(χ) are non-zero, possibly transcendental complex numbers and the idea of Stark’s conjecture is that R_{φ}(χ) accounts precisely for the transcendental part of c(χ). Setting

A_{φ}(χ) = \frac{R_{φ}(χ)}{c(χ)}

Tate’s formulation of Stark’s conjecture is as follows:

Stark’s conjecture

A_{φ}(χ)^{σ} = A_{φ}(χ^{σ}) for all σ ∈ Aut_{ℚ}(ℂ).

In particular, it is conjectured that A_{φ}(χ) is contained in the field ℚ(χ) of character values of χ. Note that Stark’s conjecture is known to be true for abelian extensions of ℚ and for abelian extensions of an imaginary quadratic number field (both cases are a consequence of [Ta2, Ch.IV, Prop.3.9]).

We fix a subfield F of ℂ such that F/ℚ is a finite Galois extension and so that every representation of G is realizable over F. The field F is just an auxiliary field which is needed for housing the invariants that we will attach to N/K. For any number field M we write O_{M} for the ring of algebraic integers of M.

Let now S be large in the following sense: assume that S contains S_{∞} and the set of all ramified primes in N/K, and that the S-class group cl_{S} is trivial. Then a strong version of Stark’s conjecture, due to Chinburg [Ch1], asserts that the fractional O_{F} -ideal generated by A_{φ}(χ) is an Euler characteristic O_{F} -ideal q_{φ}(χ) (see Definition 2.2) constructed from the Galois cohomologies of ΔS and E_{S}. This was first proved by Tate [Ta2] for ℚ-valued characters and recently by Ritter and Weiss (up to difficulties with the prime 2) for extensions N/K with N/ℚ abelian and all characters χ of G = Gal(N/K). We will refer to this strong form of Stark’s conjecture as the conjecture of Chinburg-Stark.

If M is a number field let h_{M} denote the class number of M. For two fractional O_{M}-ideals 𝔞 and 𝔟 and a rational prime l we write 𝔞 ∼_{l} 𝔟 if and only if (𝔞/𝔟, l) = 1. Then the first main result of this paper reads as follows

Theorem 1.1 Let N/k be an abelian extension of an imaginary quadratic number field k. Let l be a rational prime such that l ∤ [N : k]h_{k}. Let N/K be a subextension of N/k. Then

q_{φ}(χ) ∼_{l} A_{φ}(χ̌)O_{F}

for all characters χ of G = Gal(N/K). Here χ̌ denotes the contragredient of χ.

Since q_{φ} and A_{φ} are both functorial with respect to induction of characters it suffices to prove Theorem 1.1 for the extension N/k. The proof of Theorem 1.1 follows closely the strategy of Ritter and Weiss for their proof of q_{φ}(χ) = A_{φ}(χ) in the cyclotomic case [RW2, Theorem A]. The starting point is a generalized Tate sequence of [RW1] associated to S

0 ⟶ E_{S} ⟶ A ⟶ B ⟶ ∇ ⟶ 0  (1)

with A cohomologically trivial, B stably free and where ∇ originates from a unique extension

0 ⟶ cl_{S} ⟶ ∇ ⟶ ¯∇ ⟶ 0,

where ¯∇ is a known ℤG-lattice. For large sets S we have ∇ = ΔS and thus (1) generalizes the Tate sequence of [Ta1] which was originally used in the definition of q_{φ} (see [Ta2, Ch.II, Def.6.2]).

For arbitrary sets S (always assuming S_{∞} ⊆ S) the invariant q_{φ} is attached to G-homomorphisms

φ : ∇ ⟶ E_{S} ⊕ ℤG^{r},  (2)

with finite kernel and cokernel. Here r denotes the number of G-orbits of ramified primes which are not in S. If we adapt A_{φ} to these φ (see Definition 2.5) then [RW2, Theorem B] asserts that the ratio

𝔞(χ) = \frac{A_{φ}(χ̌)}{q_{φ}(χ)}

is independent of S and φ.

This result allows to work with the set S = S_{∞}. To prove Theorem 1.1 it is necessary to construct an explicit G-homomorphism φ as in (2). In practice it is sometimes possible to define G-embeddings \underline{φ} : ΔS → E_{N} by means of explicitly known units such as the cyclotomic and elliptic ones. In [RW2, Section 8] a recipe is given how to derive a G-map φ from a given \underline{φ}. Using this explicit construction of φ, Ritter and Weiss also compute an expression for 𝔞(χ) in terms of \underline{φ}, up to fractional O_{F} -ideal factors which divide |G|. Therefore a main step towards a proof of the Chinburg-Stark conjecture is the construction of a G-embedding \underline{φ}. This will be done in Section 5.

In order to be able to approach the conjecture of Chinburg-Stark prime by prime we shall interprete the ideal 𝔞(χ) l-adically. Let ℚ̄ denote a fixed algebraic closure of ℚ. Fix a rational prime l and denote by ℂ_{l} a fixed completion of an algebraic closure of the l-adic rationals ℚ_{l}. Choose an embedding i_{l} : ℚ̄ → ℂ_{l} and write F_{l} for the completion of i_{l}(F) in ℂ_{l}. Let O_{l} denote the valuation ring of F_{l}. If χ_{l} is a ℂ_{l}-character of G, then χ_{l} = i_{l} ∘ χ for a unique F -character χ and we define 𝔞^{(l)}(χ_{l}) = i_{l}(𝔞(χ))O_{l}.

Our aim is to show that 𝔞(χ) is prime to l for all characters χ of G. By a variant of [We, Chapter 14, Prop. 10] it is enough to prove that 𝔞^{(l)}(ψ)^{−1} is integral whenever ψ is a non-trivial ℚ_{l}-irreducible character of a subextension N′/K′ of N/k with ([N′ : K′], l) = 1 and [K′ : k] = l^{t}, t ≥ 0. We may therefore assume the following situation

[figure: k; K; K_{0}; N; l^{t}; l^{t}; l′; l′]

with (l, l′) = 1, t ≥ 0 and ψ a non-trivial ℚ_{l}-irreducible character of G = Gal(N/K).

Corollary 1.2 Let N/k be an abelian p-extension of an imaginary quadratic number field k. Let l be a rational prime such that l = p or l ∤ h_{k}. Let N/K be a subextension of N/k. Then

q_{φ}(χ) ∼_{l} A_{φ}(χ̌)O_{F}

for all characters χ of G = Gal(N/K). In particular, q_{φ}(χ) = A_{φ}(χ̌)O_{F}, if h_{k} = 1.

Proof For primes l ≠ p the assertion follows from Theorem 1.1. For l = p the result is immediate from the above discussion.

Let ψ be a non-trivial ℚ_{l}-character of G = Gal(N/K). The computations of Section 5 and 6 reduce the problem of proving the integrality of 𝔞^{(l)}(ψ)^{−1} essentially to the problem of establishing a divisibility result of the form

ℓ_{ℤ_{l}} ((ℤ_{l}⊗_{ℤ}cl_{N})^{ψ}) | ℓ_{ℤ_{l}} ((ℤ_{l}⊗_{ℤ}(E_{N}/C_{N}))^{ψ}) .  (3)

Here, for a torsion ℤ_{l}-module X, ℓ_{ℤ_{l}}(X) denotes the usual ℤ_{l}-order ideal of X (see e.g. [CR, page 86]). Writing e_{ψ} for the idempotent associated to ψ, the ℤ_{l}G-submodule X^{ψ} = e_{ψ}X is the ψ-eigenspace of the ℤ_{l}G-module X. Finally, C_{N} denotes the group of elliptic units defined by Rubin in [Ru2]. In [Ru2], C_{N} is only defined for extensions N containing the Hilbert class field k(1). In fact it suffices to require that N contains the Hilbert-l-class field k(1)_{l}. This is the reason for the restriction to primes l ∤ h_{k} in Theorem 1.1.

Using Kolyvagin’s Euler system method [Ko], the divisibility assertion (3) is proved in [Ru2, Section 3] under the additional assumption l ∤ [N : k]. This assumption is responsible for the restriction to primes l ∤ [N : k] in Theorem 1.1.

At this point we should compare Theorem 1.1 to the result of Ritter and Weiss. The essential reason why they can prove an almost complete result for abelian extensions N of ℚ is that the main conjecture of Iwasawa theory for totally real fields is known to be true by famous work of A.Wiles [Wi]. Indeed, their problems with the prime 2 are only due to the present state of the main conjecture. It should also be remarked that for the proof of their main theorem it is not enough to know the main conjecture over ℚ, but also for certain totally real fields.

Using Rubin’s work [Ru2] on the “main conjectures” for imaginary quadratic fields we can prove a kind of a complementary result to Theorem 1.1.

Theorem 1.3 Let k be an imaginary quadratic number field with class number h_{k} = 1. Let l be a prime and let K_{0}/k be abelian with l ∤ [K_{0} : k]. Let K_{∞}/K_{0} be a ℤ_{l}-extension such that K_{∞}/k is abelian. Assume that every O_{k}-prime ideal 𝔭 which divides l and ramifies in K_{∞}/K_{0} does not split in K_{0}/k. Let N/K_{0} be a finite subextension of K_{∞}/K_{0} and assume that l does not divide the number of roots of unity in N. Then

q_{φ}(χ) ∼_{l} A_{φ}(χ̌)

for all characters χ of G = Gal(N/k).

To illustrate Theorem 1.3 we consider an example.

Example Let k be an imaginary quadratic number field with class number h_{k} = 1. Let l ≠ 2 be a rational prime that splits in k/ℚ and write l = 𝔭𝔭̄, 𝔭 ≠ 𝔭̄. For n ≥ 0 we set K_{n} = k(𝔭^{n+1}). Then K_{∞}/K_{0} is a ℤ_{l}-extension, which is obviously abelian over k. If we consider N = K_{n}, n ≥ 1, then q_{φ}(χ) ∼_{l} A_{φ}(χ̌) for all characters χ of Gal(N/k).

We conclude this introductory Section with a brief outline of the structure of this part. In Sections 2 and 3 we recall the necessary definitions and results from [RW2]. Section 5 is devoted to the construction of an embedding \underline{φ} : ΔS → E_{N}. The construction of \underline{φ} is motivated by the construction of Ramachandra units (see [Wa, Ch.8]) with cyclotomic units replaced by elliptic units. The relevant results on elliptic units are collected in Section 4. In Section 6 we construct another group C_{Ha} of explicit units and compute the index of im(\underline{φ}) in C_{Ha}. The group C_{Ha} is independent of \underline{φ} and contained in C_{N}. The knowledge of the index [C_{Ha} : im(\underline{φ})] allows therefore to reduce the proof of Theorem 1.1 to the divisibility assertion (3). This is achieved in Section 7, where the main theorems are proved. Finally, in Section 8, we discuss the implications of our results for the Galois module structure of units.

2 Preliminary results

Let N/K be a finite Galois extension of number fields and write G = Gal(N/K). Let S be a finite G-stable set of primes of N containing the set S_{∞} of archimedian primes.

In this section we briefly recall the construction of Ritter and Weiss [RW2] of a q-index and an A-number for such a set S. Originally these invariants were only defined when S is large, i.e., when S contains S_{∞} and all ramified primes and the S-class group cl_{S} is trivial (see [Ta2] and [Ch1]). The definitions were based on a so-called Tate sequence [Ta1] for large sets S which gives a cohomological connection between E and ΔS. The new definitions use the generalized Tate sequence of [RW1] which we shall describe first.

Let 𝔓 be a prime of N and put 𝔭 = 𝔓 ∩ O_{K}. We write N_{𝔓}/K_{𝔭} for the localization of N/K. Let G_{𝔓} denote the decomposition group, I_{𝔓} the inertia group and ϕ_{𝔓} the Frobenius automorphism of the corresponding residue class extension. In addition, set Ḡ_{𝔓} = G_{𝔓}/I_{𝔓}, e_{𝔓} = |I_{𝔓}| and f_{𝔓} = |Ḡ_{𝔓}|. For any group H we write ΔH for the kernel of the augmentation map.

Definition 2.1 [GW] The inertial lattice W_{𝔓} attached to N_{𝔓}/K_{𝔭} is the free ℤ-module on the basis

w_{g} = (g − 1, 1 + ϕ_{𝔓} + … + ϕ_{𝔓}^{a(g)−1}) ∈ ΔG_{𝔓} ⊕ ℤḠ_{𝔓}, g ∈ G_{𝔓},

where ḡ = (g mod I_{𝔓}) = ϕ_{𝔓}^{a(g)} and 1 ≤ a(g) ≤ f_{𝔓}. The G_{𝔓}-action is induced by the natural action of G_{𝔓} on ΔG_{𝔓} ⊕ ℤḠ_{𝔓}.

The projection on the first summand induces a short exact sequence

0 ⟶ ℤ ⟶ W_{𝔓} ⟶ ΔG_{𝔓} ⟶ 0

with 1 ↦ w_{1}, which implies ℚ⊗_{ℤ}W_{𝔓} = ℚG_{𝔓}. Dualizing this sequence yields

0 ⟶ ΔG_{𝔓}^{∘} ⟶ W_{𝔓}^{∘} ⟶^{ı_{𝔓}} ℤ ⟶ 0

where ı_{𝔓}(f) = f(w_{1}) for f ∈ W_{𝔓}^{∘} = Hom_{ℤ}(W_{𝔓}, ℤ). For later reference we define a map ρ_{𝔓} ∈ W_{𝔓}^{∘} by

ρ_{𝔓}(g) = \begin{cases}1, & if g = 1, \\ 1 + e_{𝔓}, & if 1 ≠ g ∈ I_{𝔓}, \\ e_{𝔓}, & if g ∉ I_{𝔓}. \end{cases}  (4)

We fix a choice ∗ of G-orbit representatives in the set of all primes of N and for a G-stable subset S we write S_{∗} for the intersection of ∗ and S. Furthermore we set

S^{ram} = {𝔓 ramified in N/K, 𝔓 ∉ S},
∘W = ^{∘}W_{S} = ⨁_{𝔓∈S_{∗}^{ram}} ind_{G_{𝔓}}^{G}W_{𝔓}^{∘},

and finally

¯∇ = ¯∇_{S} = ker(ℤS ⊕ ^{∘}W ⟶^{ı′} ℤ),

where ı′ is the augmentation on ℤS and ind_{G_{𝔓}}^{G}ı_{𝔓} followed by the augmentation map ind_{G_{𝔓}}^{G}ℤ ⟶ ℤ on ind_{G_{𝔓}}^{G}W_{𝔓}^{∘} for 𝔓 ∈ S_{∗}^{ram}.

The main result of [RW1] is a Tate sequence associated to S

0 ⟶ E_{S} ⟶ A ⟶ B ⟶ ∇ ⟶ 0

with A cohomologically trivial, B stably free and where ∇ originates from a unique extension

0 ⟶ cl_{S} ⟶ ∇ ⟶ ¯∇ ⟶ 0

which is explicitly described in [RW1]. Note that the torsion submodule of ∇ is the S-class group cl_{S} and that for a large set S we have ∇ = ΔS.

We are now in position to give the definitions of the q-index and the Anumber. Let r = r_{S} be the number of G-orbits of ramified primes of N/K which are not in S, i.e., r = |S_{∗}^{ram}|. Then the invariants are associated to G-homomorphisms

φ : ∇ ⟶ Ẽ = E_{S} ⊕ ℤG^{r}

with finite kernel and cokernel (in [RW2] such maps are called isogenies).

We fix a subfield F of ℂ such that F/ℚ is finite Galois with group Γ = Gal(F/ℚ) and large enough that every representation of G can be realized over F. For a character χ of G we choose a F G-module V = V_{χ} affording χ and an O_{F} G-lattice M such that F ⊗_{O_{F}} M = V. We write Ĝ for the trace element ∑_{g∈G} g. Motivated by Tate’s definition of the q-index we consider the composite map

φ_{M} : Hom_{O_{F}} (M, ∇)_{G} ⟶^{Ĝ} Hom_{O_{F}} (M, ∇)^{G} ⟶^{φ} Hom_{O_{F}} (M, Ẽ)^{G}.

Here and in all what follows we adopt the convention that whenever it is clear from the context any ℤG-lattice is to be replaced by the appropriate base change which makes the notations meaningful. Above this simply means that ∇ and Ẽ have to be read as O_{F} ⊗_{ℤ}∇ and O_{F} ⊗_{ℤ}Ẽ.

Recall that for a finitely generated torsion O_{F} -module X the O_{F} -order ideal ℓ_{O_{F}} (X) is defined as the product 𝔭_{1} ⋯ 𝔭_{t} if O_{F} /𝔭_{1}, …, O_{F} /𝔭_{t} are the composition factors of X.

Definition 2.2 [RW2]

q_{φ}(χ) = \frac{ℓ_{O_{F}} (coker(φ_{M}))}{ℓ_{O_{F}} (ker(φ_{M}))}

For reasons of completeness we include the following

Lemma 2.3 [RW2, Lemma 3] a) q_{φ}(χ) does not depend on the choices of F, V and M.

b) If χ is irreducible, then q_{φ}(χ) is a fractional O_{F} -ideal which is generated by a fractional ideal of the field ℚ(χ) of character values of χ.

c) q_{φ}(χ)^{γ} = q_{φ}(χ^{γ}) for γ ∈ Γ.

Next we come to the definition of the generalized A-number of [RW2], which involves two main ingredients. The first one will be denoted by c(χ) and is defined to be the leading coefficient in the Taylor expansion of the Artin Lfunction for χ with the Euler factors at 𝔓 ∈ S removed. The second one is a generalized Stark-Tate regulator R_{φ}(χ) which we shall introduce next. We write | · |_{𝔓} for the absolute values attached to places 𝔓 of N normalized in the usual way. Let

λ : ℝ⊗_{ℤ}E_{S} ⟶ ℝ⊗_{ℤ}ΔS,
u ⟼ ∑_{𝔓∈S} log |u|_{𝔓}𝔓

denote the usual Dirichlet isomorphism. Our aim is to define a generalized Dirichlet map

λ̃ : ℝ⊗_{ℤ}Ẽ ⟶ ℝ⊗_{ℤ}∇.

To that end let λ̃ be any ℝG-homomorphism such that the diagram

\begin{matrix}0 ⟶ ℝ⊗_{ℤ}E_{S} & ⟶ & ℝ⊗_{ℤ}Ẽ & ⟶ & ℝG^{r} ⟶ 0 \\ ^{λ}↓ & & λ̃↓ & & ρ̃↓ \\ 0 ⟶ ℝ⊗_{ℤ}ΔS & ⟶ & ℝ⊗_{ℤ} ¯∇ & ⟶ & ℝ⊗_{ℤ} ^{∘}W ⟶ 0 \end{matrix}  (5)

commutes. Here, for the definition of ρ̃, we view ℤG^{r} as ⨁_{𝔓∈S_{∗}^{ram}} ind_{G_{𝔓}}^{G}ℤG_{𝔓} and

set ρ̃(1_{𝔓}) = − log(N𝔓)⊗ρ_{𝔓} with ρ_{𝔓} as in (4).

Let V̌ be a ℂG-module affording the contragredient χ̌ of χ and denote by [λ̃φ | Hom_{ℂG}(V̌ , ∇)] the endomorphism of Hom_{ℂG}(V̌ , ∇) which maps a homomorphism f to the composite λ̃ ∘ φ ∘ f.

By [RW2, Lemma 4] the determinant of [λ̃φ | Hom_{ℂG}(V̌ , ∇)] does not depend on the choice of λ̃ making diagram (5) commute.

Definition 2.4 [RW2]

R_{φ}(χ) = det[λ̃φ | Hom_{ℂG}(V̌ , ∇)]

Finally we define the generalized A-number of [RW2].

Definition 2.5 [RW2]

A_{φ}(χ) = \frac{R_{φ}(χ)}{c_{S}(χ) ( ∏_{𝔓∈S_{∗}^{ram}} e_{𝔓} log(N𝔓))^{χ(1)}} .

When S is large Chinburg [Ch1] conjectured that

q_{φ}(χ) = A_{φ}(χ̌)O_{F}  (6)

for all characters χ of G. This was first proved by Tate [Ta2] for ℚ-valued characters and by Ritter and Weiss [RW2, Theorem A] (up to certain problems with the prime 2) for extensions N/K with N/ℚ abelian and all characters χ of G = Gal(N/K).

In [RW2] the proof of the above conjecture is based on the following

Theorem 2.6 [RW2, Theorem B] The ratio 𝔞(χ) = \frac{A_{φ}(χ̌)O_{F}}{q_{φ}(χ)} is independent of φ and S.

This Theorem allows to work with small sets S; in particular we can take S = S_{∞}.

3 Some reductions

We write 𝔞_{N/K}(χ) instead of 𝔞(χ) if we want to emphasize the dependence on N/K. The next proposition, which lists some basic properties of the ratio 𝔞(χ), is a direct consequence of work of Tate [Ta2] and Chinburg [Ch1] and Theorem 2.6.

Proposition 3.1 [RW2, Proposition 9]

a) 𝔞_{N/K} is additive, i.e., 𝔞_{N/K}(χ_{1} + χ_{2}) = 𝔞_{N/K}(χ_{1})𝔞_{N/K}(χ_{2}) for all characters χ_{1}, χ_{2} of G.

b) 𝔞_{N/K}(1) = O_{F}.

c) Let H be a subgroup of G. Then for any character ψ of H we have

𝔞_{N/N^{H}} (ψ) = 𝔞_{N/K}(ind_{H}^{G}ψ).

d) Let H be a normal subgroup of G and χ̄ a character of Ḡ = G/H. Then we have

𝔞_{N^{H}/K}(χ̄) = 𝔞_{N/K}(inflG ; Ḡχ̄).

In practice it is in special cases possible to exhibit a G-embedding \underline{φ} : ΔS ⟶ E_{N} by means of explicitly known units such as the cyclotomic and elliptic units (e.g. [RW2, Section 10] or Section 5). It is therefore of great importance to give an expression for 𝔞(χ) in terms of \underline{φ}. In [RW2, Section 8] a canonical homomorphism φ : ∇ → Ẽ with finite kernel and cokernel is constructed, starting out from a given G-embedding \underline{φ} : ΔS → E_{N}. This map is then used to prove another result of Ritter and Weiss, which we will state after introducing the necessary notation.

Define O′ = O_{F} [\frac{1}{|G|}]. For G abelian and an irreducible character χ of G we write e_{χ} for the χ-idempotent

e_{χ} = \frac{1}{|G|} ∑_{g∈G} χ(g)g^{−1} ∈ O′G.

If X is a O′G-module we let X^{χ} = e_{χ}X denote its χ-isotypic component. Recall also that cl_{S} is the S-class group of N.

Proposition 3.2 [RW2, Proposition 10] Let G be abelian and let χ be an irreducible character of G. Let V = V_{χ} be a ℂG-module affording χ. Then

𝔞(χ)O′ = \frac{det (λ\underline{φ} | Hom_{ℂG}(V, ΔS))}{c_{S}(χ̌)} · \frac{ℓ_{O′} (cl_{S}^{χ})}{ℓ_{O′} (coker(\underline{φ})^{χ})}.

In the final part of this section we are concerned with certain l-adic reductions that will allow to approach Conjecture (6) prime by prime. To that end we assume that Stark’s conjecture holds, so that by Lemma 2.3 𝔞(χ) is a fractional O_{F} -ideal satisfying 𝔞(χ)^{γ} = 𝔞(χ^{γ}) for γ ∈ Γ.

We view all our number fields as subfields of a fixed algebraic closure ℚ̄ of the rationals. For a rational prime l denote by ℂ_{l} a fixed completion of an algebraic closure of the l-adic rationals ℚ_{l} and choose once and for all an embedding i_{l} : ℚ̄ → ℂ_{l}. Write F_{l} for the completion of i_{l}(F) in ℂ_{l}. If χ_{l} is a ℂ_{l}-character of G, then χ_{l} = i_{l} ∘ χ for some F -character χ and we define 𝔞^{(l)}(χ_{l}) = i_{l}(𝔞(χ))O_{l}, where O_{l} is the valuation ring in F_{l}.

Proposition 3.3 [We, Proposition 10] a) 𝔞^{(l)} is independent of the choice of i_{l}.

(b) Let N/K be abelian. Suppose that 𝔞_{N′/K′}^{(l)}(χ_{l}) (resp. 𝔞_{N′/K′}^{(l)}(χ_{l})^{−1}) is integral, whenever χ_{l} is a non-trivial ℚ_{l}-irreducible character of a cyclic subextension N′/K′ of N/K which has degree prime to l and so that K′/K is a cyclic l-extension. Then 𝔞_{N/K}(χ) is relatively prime to l for all χ of G = Gal(N/K).

Proof The proof is similar to the proof of [We, Prop. 10]. Therefore we only give a sketch of a proof, referring the reader to the details in [We].

Since 𝔞_{N/K} is functorial with respect to inflation we may assume that N/K is cyclic. For any finite group H we write R_{ℚ_{l}}(H) for the ring of virtual ℚ_{l}- characters.

The proof of [We, Lemma 19] shows the surjectivity of the map

⨁_{N′/K′} R_{ℚ_{l}}(Gal(N′/K′)) ⟶^{ind∘inf} R_{ℚ_{l}}(G),

N′/K′ as in the statement of the Proposition. By Proposition 3.1 this implies that 𝔞_{N/K}(ψ) (resp. 𝔞_{N/K}(ψ)^{−1}) is integral for all ℚ_{l}-irreducible characters ψ of G. The regular representation is the sum of the irreducibles. From 𝔞_{N/K}(ind_{1}^{G}1) = (1) (Proposition 3.1) we therefore conclude 𝔞_{N/K}(ψ) = (1) for all ℚ_{l}-irreducible characters of G.

The rest of the proof is the same as the proof of [We, Prop.10].

4 Elliptic units

The aim of this section is to define the elliptic units in abelian extensions of an imaginary quadratic number field k and to summarice the relevant arithmetical properties. Our main references are [Co1, Appendix], [Ru1, Appendix] and [dS, II,2].

Most of the properties that we shall need have long been known for the 12th powers. In order to deal with the 12th roots we first recall some definitions and results of [Co1, Appendix].

Let F be a field of characteristic 0 and let E be an elliptic curve defined over F. Let E′/F be another curve and let ϕ : E ⟶ E′ be an isogeny of elliptic curves which is also defined over F. We shall always assume that (| ker(ϕ)|, 6) = 1.

We choose a generalized Weierstrass equation for E over F,

E : y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, a_{i} ∈ F,  (7)

and let x_{E} be the x-coordinate function given by this model.

Following Coates we set

ζ_{ϕ}(P_{E}) = ∏_{R}(x_{E}(P_{E}) − x_{E}(R))^{−1},  (8)

where R runs over any set of representatives of ker(ϕ) \ {0_{E}} modulo {±1}. This is a well-defined F -rational function on E, for which the following holds:

Proposition 4.1 [Co1, Appendix, Prop.1] There exists a unique c_{E}(ϕ) ∈ F^{∗} such that, for all β ∈ End_{F} (E) with ker(β) ∩ ker(ϕ) = {0_{E}}, we have

\frac{ζ_{ϕ}(β(P_{E}))}{∏_{R∈ker(β)} ζ_{ϕ}(P_{E} +_{E} R)} = c_{E}(ϕ)^{deg(β)−1}.

Next we define a F -rational function

γ_{ϕ}(P_{E}) = c_{E}(ϕ)ζ_{ϕ}(P_{E}).

Note that γ_{ϕ}(P_{E}) does not depend on the choice of a generalized Weierstrass equation for E over F.

Lemma 4.2 Suppose that F is an algebraic extension of a field k and let σ ∈ Gal(k̄/k). Then

γ_{ϕ}(P_{E})^{σ} = γ_{ϕ^{σ}}(P_{E}^{σ}).

Proof By the definition of ζ_{ϕ} we have

ζ_{ϕ}^{σ}(P_{E}^{σ}) = ∏_{R∈(ker(ϕ)\{0})/{±1}} (x_{E}^{σ}(P_{E}^{σ}) − (x_{E}(R))^{σ})^{−1}
= ∏_{R∈(ker(ϕ)\{0})/{±1}} (x_{E}^{σ}(P_{E}^{σ}) − x_{E}^{σ}(R^{σ}))^{−1}  (9)
= ∏_{R∈(ker(ϕ^{σ})\{0})/{±1}} (x_{E}^{σ}(P_{E}^{σ}) − (x_{E}^{σ}(R)))^{−1} = ζ_{ϕ^{σ}}(P_{E}^{σ}).

Next we claim that

c_{E}(ϕ)^{σ} = c_{E^{σ}}(ϕ^{σ}).

To prove the claim we have to show that for all β ∈ End_{F} (E)

(c_{E}(ϕ)^{σ})^{deg(β^{σ})−1} = \frac{ζ_{ϕ^{σ}} (β^{σ}(P_{E}^{σ}))}{∏_{R∈ker(ϕ^{σ})} ζ_{ϕ^{σ}} (P_{E}^{σ} +_{E^{σ}} R)}.

But this follows immediately from (9) and the defining equation for c_{E}(ϕ), since the right hand side equals

(\frac{ζ_{ϕ} (β(P_{E}))}{∏_{R∈ker(ϕ)} ζ_{ϕ} (P_{E} +_{E} R)})^{σ}

The lemma is now a consequence of the claim and (9).

Let λ : E → A be an isogeny of elliptic curves defined over F such that ker(λ) ∩ ker(ϕ) = {0_{E}}. To any such λ we associate a new isogeny

ψ : A ⟶ A′ = A/λ(ker(ϕ))  (10)

by taking the canonical projection map (see e.g. [Sil1, Ch. III, Prop. 4.12]).

Proposition 4.3 [Co1, Appendix, Theorem 3 and Theorem 4] (a) For all isogenies λ : E → A as above, we have

γ_{ψ}(λ(P_{E})) = ∏_{R∈ker(λ)} γ_{ϕ}(P_{E} +_{E} R).

(b) For all β ∈ End_{F} (E) with ker(β) ∩ ker(ϕ) = {0_{E}} we have

γ_{ϕ}(β(P_{E})) = ∏_{R∈ker(β)} γ_{ϕ}(P_{E} +_{E} R).

Corollary 4.4 Let λ : E → A be an F -isogeny such that ker(λ) ∩ ker(ϕ) = {0_{E}} and let ψ be the associated isogeny as in (10). Let t_{A} = −x_{A}/y_{A} and t_{E} = −x_{E}/y_{E} be local parameters at zero and write

t_{A} (λ(P_{E})) = Λt_{E} + higher powers of t_{E}

with Λ ∈ F^{∗}. Then we have

∏_{R∈ker(λ)\\{0_{E}\}} γ_{ϕ}(R) = \frac{c_{A}(ψ)}{c_{E}(ϕ)} · Λ^{deg(ϕ)−1}.

Remark By [Sil1, III, Th. 4.10(c)] the isogeny λ is unramified. Therefore the element Λ ∈ F^{∗} satisfying the hypothesis of the Corollary always exists.

Proof Proposition 4.3 (a) implies

∏_{R∈ker(λ)\\{0_{E}\}} γ_{ϕ}(R) = \frac{γ_{ψ}(λ(P_{E}))}{γ_{ϕ}(P_{E})}|_{P_{E}=0_{E}}

= \frac{c_{A}(ψ)}{c_{E}(ϕ)} · \frac{ζ_{ψ}(λ(P_{E}))}{ζ_{ϕ}(P_{E})}|_{P_{E}=0_{E}} .

If we expand ζ_{ψ} and ζ_{ϕ} in terms of the local parameters at zero we obtain

ζ_{ψ}(P_{A}) = t_{A}^{deg(ψ)−1} + higher powers of t_{A},
ζ_{ϕ}(P_{E}) = t_{E}^{deg(ϕ)−1} + higher powers of t_{E}.

Since ker(λ) ∩ ker(ϕ) = {0_{E}}, we have deg(ψ) = deg(ϕ), and thus the expansion of ζ_{ψ}(λ(P_{E}))/ζ_{ϕ}(P_{E}) at zero is given by

\frac{ζ_{ψ}(λ(P_{E}))}{ζ_{ϕ}(P_{E})} = Λ^{deg(ϕ)−1} + higher powers of t_{E}.

Hence \frac{ζ_{ψ}(λ(P_{E}))}{ζ_{ϕ}(P_{E})} |_{P_{E}=0_{E}} = Λ^{deg(ϕ)−1} and the proof is complete.

We shall now give an expression of the 12th powers of c_{E}(ϕ) in terms of the discriminants of E and E′. We fix a generalized Weierstrass model

E′ : y^{2} + a′_{1}xy + a′_{3}y = x^{3} + a′_{2}x^{2} + a′_{4}x + a′_{6}, a′_{i} ∈ F,  (11)

and let Δ(E′) (resp. Δ(E)) be the discriminant of this equation (resp. of (7)). Let

ω_{E} = \frac{dx_{E}}{2y_{E} + a_{1}x_{E} + a_{3}} , ω_{E′} = \frac{dx_{E′}}{2y_{E′} + a′_{1}x_{E′} + a′_{3}}

be the standard differentials associated with (7) and (11). Since ω_{E} is a F -basis of the one-dimensional space of holomorphic F -rational differentials, we may define a number δ(E, E′) ∈ F^{∗} by

ω_{E′} ∘ ϕ = δ(E, E′)ω_{E}.

Proposition 4.5 [Co1, Appendix, Theorem 8]

c_{E}(ϕ)^{12} = \frac{Δ(E)^{deg(ϕ)}}{Δ(E′)δ(E, E′)^{12}} .

In the second part of this section we shall rephrase the above results in terms of analytic functions. For that purpose we introduce a special class of elliptic curves defined over an abelian extension F of our imaginary quadratic number field k. For a detailed discussion of these curves the reader is referred to [dS, II, 1.4] or [GS].

In the following we assume that the elliptic curve E satisfies the hypothesis

(i) E has complex multiplication by O_{k},
(ii) F (E_{tor})/k is abelian.

(12)

The condition (ii) is equivalent to the existence of a Groessencharacter φ of k of type (1, 0) such that

ψ_{E/F} = φ ∘ N_{F/k},

where ψ_{E/F} denotes the unique Groessencharacter of F associated to E. Denoting the conductors of φ and F/k by 𝔣_{φ} and 𝔣_{F/k}, respectively, we define an integral O_{k}-ideal 𝔣 = lcm(𝔣_{φ}, 𝔣_{F/k}). This ideal depends only on E/F and not on the choice of φ.

If 𝔞 is an integral ideal of k we let w(𝔞) be the number of roots of unity in k congruent to 1 modulo 𝔞. We write k(𝔞) for the ray class field of conductor 𝔞. If N/k is an abelian extension and 𝔞 an integral ideal relatively prime to 𝔣_{N/k} we write σ(𝔞) or (𝔞, N/k) for the corresponding Artin automorphism. Furthermore we denote by E[𝔞] the finite subgroup of E(F̄) consisting of all points P which are annihilated by all elements α ∈ 𝔞.

Let now 𝔞 be an integral O_{k}-ideal with (𝔞, 𝔣) = 1. By [dS, II, Proposition 1.5] there exists a unique isogeny

λ(𝔞) : E ⟶ E^{σ(𝔞)},  (13)

defined over F, of degree N𝔞 and characterized by

P^{σ(𝔞)} = λ(𝔞)(P)

for all P ∈ E[𝔠], where 𝔠 is any integral ideal of k relatively prime to 𝔞.

Let ω = ω_{E} be a F -rational holomorphic differential on E and define Λ(𝔞) ∈ F^{∗} by

ω^{σ(𝔞)} ∘ λ(𝔞) = Λ(𝔞)ω.  (14)

Whenever E is given by a generalized Weierstrass equation we assume that Λ is associated with the standard differential.

Let [ ] : O_{k} → End_{F} (E) be normalized such that ω ∘[α] = αω for all α ∈ O_{k}. We recall from [dS, II, 1.5] that if (𝔞, F/k) = 1, then

Λ(𝔞) ∈ k^{∗}, λ(𝔞) = [Λ(𝔞)], Λ(𝔞) = φ(𝔞).

We view all our number fields as subfields of the complex number field ℂ. Let L be the period lattice of ω and

ξ(·, L) : ℂ/L ⟶ E(ℂ),
z + L ⟼ \begin{cases}[0, 1, 0], & if z ∈ L, \\ [℘(z) − \frac{a_{1}^{2}+4a_{2}}{12} , \frac{℘′(z)−a_{1}℘(z)}{2} + \frac{a_{1}^{3}+4a_{1}a_{3}−12a_{3}}{24} , 1] , & if z ∉ L. \end{cases}

the corresponding analytic uniformization. Here, as usual, ℘ denotes the Weierstrass function associated to the complex lattice L and ℘′ its derivative.

For an integral ideal 𝔞 with (𝔞, 𝔣) = 1 the period lattice of ω^{σ(𝔞)} on E^{σ(𝔞)} is given by L_{𝔞} = Λ(𝔞)𝔞^{−1}L and we have the commuting diagram

\begin{matrix}0 ⟶ 𝔞^{−1}L/L & ⟶ & ℂ/L & ⟶^{·Λ(𝔞)} & ℂ/L_{𝔞} ⟶ 0 \\ ↓ & ^{ξ(·,L)}↓ & ^{ξ(·,L_{𝔞})}↓ \\ 0 ⟶ ker λ(𝔞) & ⟶ & E(ℂ) & ⟶^{λ(𝔞)} & E^{σ(𝔞)}(ℂ) ⟶ 0 \end{matrix}  (15)

To conclude the collection of facts about this special class of elliptic curves we finally recall

Proposition 4.6 [dS, II, Prop. 1.6] Let 𝔤 be an integral O_{k} ideal such that 𝔣 | 𝔤. Then k(𝔤) = F (E[𝔤]).

We now express the elliptic units obtained from special values of the function γ_{ϕ} by means of complex analytic functions and derive their basic arithmetical properties. For that purpose let 𝔟 be an integral O_{k}-ideal such that w(𝔟) = 1. By [dS, II, Lemma 1.4] there exists a Groessencharacter φ of k of type (1, 0) and conductor 𝔟 and an elliptic curve E defined over F = k(𝔟) satisfying (12) and such that ψ_{E/F} = φ ∘ N_{F/k}. We fix a generalized Weierstrass equation for E and let L denote the period lattice associated to the standard invariant differential. Let 𝔞 be an integral O_{k}-ideal such that (𝔞, 6𝔟) = 1 and consider the isogeny

ϕ = λ(𝔞) : E ⟶ E^{σ(𝔞)}.

Then, for any z ∈ ℂ, we define

θ_{0}(z, 𝔞) := γ_{ϕ}(ξ(z, L))
= c_{E}(ϕ) · ∏_{u} \frac{1}{℘(z | L) − ℘(u | L)},  (16)

where u runs over a set of non-zero representatives of 𝔞^{−1}L/L modulo {±1}. The 12th power of θ_{0}(z, 𝔞) is the function θ(z, L, 𝔞) considered in [dS, II, 2.3]. Indeed, by Proposition 4.5 and the definition of Λ(𝔞),

θ_{0}(z, 𝔞)^{12} = \frac{Δ(L)^{N𝔞}}{Δ(L_{𝔞})Λ(𝔞)^{12}} ∏_{u} \frac{1}{(℘(z | L) − ℘(u | L))^{12}}

= \frac{Δ(L)^{N𝔞}}{Δ(𝔞^{−1}L)} ∏_{u} \frac{1}{(℘(z | L) − ℘(u | L))^{12}} ,

where now Δ also denotes the discriminant function for complex lattices (see e.g. [Sil1, VI, Prop.3.6]). Note also that θ_{0}(z, 𝔞)^{12} can be expressed in terms of an important normalization of the Weierstrass σ-function which we shall define next. We choose a ℤ-basis ω_{1}, ω_{2} of L such that Im(ω_{1}/ω_{2}) > 0 and write η(τ), Im(τ) > 0, for the Dedekind η-function. Let η_{1}, η_{2} denote the basic quasiperiods of the Weierstrass ζ-function and for any z = a_{1} + a_{2} ∈ ℂ, a_{1}, a_{2} ∈ ℝ, put z^{∗} = a_{1}η_{1} + a_{2}η_{2}. Writing σ (z |L) for the Weierstrass σ-function attached to the lattice L we define (see e.g. [Sch2] or [KL, page 29])

φ (z |{ω_{1} \atop ω_{2}} ) = 2πie^{−zz^{∗}/2}σ (z |L) η^{2}(\frac{ω_{1}}{ω_{2}} )ω_{2}^{−1}.

This function is non-meromorphic as a function of z and it is not a function of lattices but depends on the choice of a basis ω_{1}, ω_{2}. Its 12th power is often called the fundamental theta function associated to L [dS, II,2.3]

θ(z, L) = Δ(L)e^{−6zz^{∗}}σ (z |L)^{12} = φ (z |{ω_{1} \atop ω_{2}} )^{12} .

Indeed, the 12-th power is a function of lattices and we will also write φ (z |L)^{12}.

The following relation between φ and θ_{0} will be of great importance since certain sums of values of φ arise in the Kronecker limit formula and its applications to L-series. We have [dS, II, 2.3]

θ_{0}(z, 𝔞)^{12} = \frac{φ^{12N(𝔞)}(z | L)}{φ^{12}(z | 𝔞^{−1}L)}.  (17)

For later reference we recall some fundamental properties of the function φ.

Lemma 4.7 (a) The function φ (z |{ω_{1} \atop ω_{2}} ) is homogeneous of degree 0, i.e.,

φ (λz |{λω_{1} \atop λω_{2}} ) = φ (z |{ω_{1} \atop ω_{2}} ) for all λ ∈ ℂ^{∗}.

(b) Let L be a complex lattice and z ∈ ℂ \ L. Let N ∈ ℕ be minimal with Nz ∈ L. Then φ^{12N} does not depend on the choice of a basis of L and furthermore

φ^{12N}(z + ω | L) = φ^{12N}(z | L)

for all ω ∈ L.

(c) Let L be a complex lattice with complex multiplication by O_{k}. Let 𝔤 ≠ (1) be an integral O_{k}-ideal and let τ be a primitive 𝔤-torsion point of ℂ/L. Then

φ^{12N(𝔤)}(τ | L) ∈ k(𝔤).

(d) Let L, 𝔤 and τ be as in (c). Let 𝔠 be an integral O_{k}-ideal with (𝔤, 𝔠) = 1. Then

φ^{12N(𝔤)}(τ | L)^{σ(𝔠)} = φ^{12N(𝔤)}(τ | 𝔠^{−1}L).

Proof (a) This is almost immediate from the definition (see e.g. [KL, pages 27–29]).

(b) This is a consequence of [KL, page 28, K 2].

(c) The lattice L can be written in the form L = Ω𝔞 with an integral O_{k}-ideal 𝔞 and Ω ∈ ℂ^{∗}. Since τ is a primitive 𝔤-torsion point there exists an integral O_{k}-ideal 𝔟 such that 𝔤τ = 𝔟𝔞Ω, (𝔟, 𝔤) = 1. By (a) we derive φ^{12N}(τ | L) = φ^{12N}(1 | 𝔤𝔟^{−1}). Now (c) follows from [KL, Ch.11, Th.1.1].

(d) This is also a consequence of [KL, Ch.11, Th.1.1].

The next proposition summarizes important arithmetical properties of special values of the function θ_{0}. In particular, it implies that these elements can be used to construct an Euler system in the sense of [Ru2].

Proposition 4.8 (a) Let 𝔤 be an integral O_{k}-ideal with 𝔟 | 𝔤 and let τ be a primitive 𝔤-torsion point of ℂ/L. Then θ_{0}(τ, 𝔞) ∈ k(𝔤).

(b) Let 𝔤 and τ be as in (a) and let 𝔠 be an integral O_{k}-ideal prime to 𝔤. Then

θ_{0}(τ, 𝔞)^{σ(𝔠)} = γ_{ϕ^{σ(𝔠)}}(ξ(Λ(𝔠)τ, L_{𝔠})).

If, in addition, one has (𝔠, F/k) = 1, then

θ_{0}(τ, 𝔞)^{σ(𝔠)} = θ_{0}(Λ(𝔠)τ, 𝔞).

(c) Let 𝔤 and τ be as in (a). If 𝔤 is composite (i.e., divisible by at least two distinct prime O_{k}-ideals), then θ_{0}(τ, 𝔞) is a unit. If 𝔤 = 𝔭^{n} is the power of a prime O_{k}-ideal 𝔭, then

θ_{0}(τ, 𝔞)O_{k(𝔤)} = (𝔭O_{k(𝔤)})^{(N𝔞−1)/Φ(𝔭^{n})} ,

where Φ is the Euler function of the ring O_{k}. Thus θ_{0}(τ, 𝔞)^{σ−1} is a unit for all σ ∈ Gal(k(𝔤)/k).

(d) Let 𝔤 and τ be as in (a). Let 𝔭 be a prime O_{k}-ideal such that 𝔭 ∤ 𝔤. Let τ_{1} be a primitive 𝔭-torsion point of ℂ/L. Then

N_{k(𝔭𝔤)/k(𝔤)}θ_{0}(τ + τ_{1}, 𝔞) = θ_{0}(τ, 𝔞)^{σ(𝔭)−1}.

(e) Let 𝔭, 𝔤, τ, τ_{1} be as in (d). Then

θ_{0}(τ + τ_{1}, 𝔞) ≡ θ_{0}(τ, 𝔞)

modulo all primes of k(𝔭𝔤) above 𝔭.

Proof

(a) Since γ_{ϕ} is defined over F, the special value θ_{0}(τ, 𝔞) = γ_{ϕ}(ξ(τ, L)) is contained in F (E[𝔤]), which equals k(𝔤) by Proposition 4.6.

(b) The first assertion is an immediate consequence of Lemma 4.2 and the commutative diagram (15). Indeed, writing σ = σ(𝔠) we have

θ_{0}(τ, 𝔞)^{σ} = γ_{ϕ}(ξ(τ, L))^{σ}
= γ_{ϕ}^{σ} (ξ(τ, L)^{σ})
= γ_{ϕ^{σ}} (ξ(Λ(𝔠)τ, L_{𝔠})) .

The second statement follows then from ϕ^{σ} = ϕ and L_{𝔠} = L.

(c) The prime factorization of special values of φ (z |L)^{12} is originally due to Ramachandra [Ra]. The assertion in (c) follows immediately from (17) and [Sch1, Satz 3].

(d) By class field theory and part (b) we obtain

N_{k(𝔭𝔤)/k(𝔤)}θ_{0}(τ + τ_{1}, 𝔞) = ∏_{α} θ_{0}(τ + τ_{1}, 𝔞)^{σ(α)} = ∏_{α} θ_{0}(ατ + ατ_{1}, 𝔞),

where α runs through a set of representatives of (O_{k}/𝔭𝔤)^{∗} with α ≡ 1 (mod 𝔤). Thus

N_{k(𝔭𝔤)/k(𝔤)}θ_{0}(τ + τ_{1}, 𝔞) = ∏_{β∈𝔭^{−1}L/L,β≢0} θ_{0}(τ + β, 𝔞)

= \frac{∏_{β∈𝔭^{−1}L/L} γ_{ϕ}(ξ(τ + β, L))}{γ_{ϕ}(ξ(τ, L))}.

We now apply Proposition 4.3(a) for the isogeny

λ = λ(𝔭) : E ⟶ E^{σ(𝔭)}.

Note that we can take ϕ^{σ(𝔭)} for the associated isogeny ψ : E^{σ(𝔭)} → E^{σ(𝔭)}/λ(ker(ϕ)). Hence we find

N_{k(𝔭𝔤)/k(𝔤)}θ_{0}(τ + τ_{1}, 𝔞) = \frac{γ_{ϕ^{σ(𝔭)}}(ξ(Λ(𝔭)τ, L_{𝔭}))}{γ_{ϕ}(ξ(τ, L))}
=^{(b)} θ_{0}(τ, 𝔞)^{σ(𝔭)−1}.

(e) Since θ_{0}(τ + τ_{1}, 𝔞) is a unit it suffices to show that θ_{0}(τ, 𝔞)/θ_{0}(τ + τ_{1}, 𝔞) ≡ 1. Using the equality [La1, Ch.18, §1, Th.2]

℘ (z_{1} |L) − ℘ (z_{2} |L) = −\frac{σ (z_{1} + z_{2} |L) σ (z_{1} − z_{2} |L)}{σ^{2}(z_{1} | L)σ^{2}(z_{2} | L)}, z_{1}, z_{2} ∈ ℂ,

we obtain

\frac{℘(τ + τ_{1}) − ℘(τ)}{℘(τ) − ℘(u)} = \frac{φ(2τ + τ_{1})φ(τ_{1})φ(u)^{2}}{φ^{2}(τ + τ_{1})φ(τ + u)φ(τ − u)},  (18)

where we omit the lattice L in our notation. From the definition (16) of θ_{0} we derive

\frac{θ_{0}(τ, 𝔞)}{θ_{0}(τ + τ_{1}, 𝔞)} = ∏_{u} \frac{℘(τ + τ_{1}) − ℘(u)}{℘(τ) − ℘(u)}

= ∏_{u} (1 + \frac{℘(τ + τ_{1}) − ℘(τ)}{℘(τ) − ℘(u)})

By (18) and [Sch1, Satz 3] each of the factors is congruent 1 modulo all primes above 𝔭. This concludes the proof.

5 The Ramachandra unit lattice

Let k be an imaginary quadratic number field and let N be an abelian extension of k. Our ultimate goal is to prove Conjecture (6) (or at least some approximation to it) for all abelian extensions N/k and all characters χ of Gal(N/k). By Proposition 3.1 we may assume firstly, that N/k is cyclic, and secondly, that χ is a faithful linear character of Gal(N/k). Moreover, Stark’s conjecture holds for abelian extensions of an imaginary quadratic number field and we can therefore apply the results of Proposition 3.3.

Fix a rational prime l. The idea is to prove (6) prime by prime by applying Proposition 3.3(b). Thus it suffices to consider the following situation

[figure: k; K; K_{0}; N; l^{t}; l^{t}; l′; l′]  (19)

where G = Gal(N/K) has order l′ prime to l and [K : k] = l^{t} with t ≥ 0. We also write H for the group Gal(N/k).

Let 𝔣 be any multiple of the conductor 𝔣_{N/k} of N. Write

𝔣 = ∏_{i=1}^{s} 𝔭_{i}^{e_{i}}

and assume that Φ(𝔭^{e_{i}}) > 2 for i = 1, …, s. We choose an O_{k}-prime ideal 𝔟 such that

w(𝔟) = 1, (𝔟, 6𝔣) = 1,

ψ(σ(𝔟)) ≠ 1 for all non-trivial linear characters ψ of H.

(20)

Since N/k is cyclic, there are infinitely many primes 𝔟 satisfying (20) by the Chebotarev density theorem.

Let 𝔞 be another auxiliary integral O_{k}-ideal such that (𝔞, 6𝔟𝔣) = 1. Let φ be a Groessencharacter of k of type (1, 0) and of conductor 𝔟 and let E be an elliptic curve defined over F = k(𝔟) satisfying (12) and such that ψ_{E/F} = φ ∘ N_{K/k}. Fix a Weierstrass model for E and a period lattice L associated to the standard invariant differential. If necessary we replace E by one of its conjugates so that we can assume without loss of generality that L = ΩO_{k} with Ω ∈ ℂ^{∗}. Recall also the definition of the function θ_{0}(z, 𝔞) in (16).

Motivated by the cyclotomic case (see e.g. [Wa, page 147]) we write 𝔫_{I} = ∏_{i∈I} 𝔭_{i}^{e_{i}} for each subset I of {1, …, s}. We fix a primitive 𝔟-torsion point τ of ℂ/L and for each j ∈ {1, …, s} a primitive 𝔭_{j}^{e_{j}}-torsion point τ_{j}. Finally, for a subset J ⊆ {1, …, s} we set τ_{J} = τ + ∑_{j∈J} τ_{j} and

θ_{J} = θ_{0}(τ_{J}, 𝔞).

From Proposition 4.8(a) we deduce θ_{0}(τ_{J}, 𝔞) ∈ k(𝔟𝔫_{J}).

Definition 5.1

θ_{N} = N_{k(𝔣𝔟)/N} ( ∏_{J⊆{1,... ,s}} θ_{0}(τ_{J}, 𝔞)) .

Note that in contrast to the cyclotomic case (see [RW2, §10]) it is here necessary to include the set J = ∅ in order to include the so-called unramified units.

Henceforth we assume that S = S_{∞}. As before we view all our number fields as subfields of ℂ. This defines an archemedian place 𝔭_{∞} of S and for all α ∈ N^{∗} we have |α|_{𝔭_{∞}} = |α|^{2} with | · | denoting the usual complex value. Note that ℤS = ℤH𝔭_{∞}. Hence there is a unique H-homomorphism ℤS → N^{∗} mapping 𝔭_{∞} to θ_{N}. By Proposition 4.8(c) this homomorphism restricts to a map

\underline{φ} : ΔS ⟶ E_{N}, (h − 1)𝔭_{∞} ⟼ θ_{N}^{h−1}, h ∈ H.  (21)

Actually, we should now proof that im(\underline{φ}) is a unit lattice of full rank. However, this will follow immediately from the expression we shall compute for det(λ\underline{φ} | Hom_{ℂG}(V, ΔS))/c_{S}(χ̌) (see Proposition 5.3), where χ is any non-trivial linear character of G.

Definition 5.2 For any non-trivial linear character ψ of H we set

g(ψ) = ∏_{𝔭_{i}|𝔣,𝔭_{i}∤𝔣_{ψ}} (Φ(𝔭_{i}^{e_{i}}) + ψ(𝔭_{i})^{e_{i}−1}(ψ(𝔭_{i}) − 1)) ,

where ψ is interpreted as a Dirichlet character via the Artin map.

By our choice of 𝔣 we obtain g(ψ) ≠ 0 for all non-trivial linear characters ψ of H.

Proposition 5.3 Let χ be a non-trivial linear character of G with representation space V. Then

\frac{det(λ\underline{φ} | Hom_{ℂG}(V, ΔS))}{c_{S}(χ̌)} = ζ_{χ} ∏_{ψ|χ} (−g(ψ)(ψ(𝔟) − 1)(N(𝔞) − ψ(𝔞))) ,

where ψ runs through all characters of H extending χ and ζ_{χ} is a root of unity.

The proof of Proposition 5.3 will occupy the rest of this section. We start by computing det(λ\underline{φ} | Hom_{ℂG}(V, Δ_{S})). We take V = ℂGe_{χ} and derive

Hom_{ℂG}(V, ΔS) ≃ e_{χ}ℂS = ∑_{ψ|χ} e_{ψ}ℂS
= ∑_{ψ|χ} e_{ψ}ℂH𝔭 = ∑_{ψ|χ} e_{ψ}ℂ𝔭_{∞}.

Thus the set {e_{ψ}𝔭_{∞} | ψ extends χ} can be viewed as a ℂ-basis of Hom_{ℂG}(V, ΔS). Computing the matrix representation of λ\underline{φ} with respect to this basis we obtain

λ\underline{φ}(e_{ψ}𝔭_{∞}) = λ\underline{φ} (\frac{1}{|H|} ∑_{h∈H} ψ(h^{−1})(h𝔭_{∞} − 𝔭_{∞})) , since ψ ≠ 1

= λ (\frac{1}{|H|} ∑_{h∈H} ψ(h^{−1})θ_{N}^{h−1})

= \frac{1}{|H|} ∑_{h∈H} ψ(h^{−1}) ∑_{h′∈H} log |θ_{N}^{h−1}|_{h′·𝔭_{∞}}h′ · 𝔭_{∞}

= \frac{1}{|H|} ∑_{h∈H} ψ(h^{−1}) ∑_{h′∈H} log |θ_{N}^{(h−1)h′^{−1}}|_{𝔭_{∞}}h′ · 𝔭_{∞}

= \frac{1}{|H|} ∑_{h∈H} ψ(h^{−1}) ∑_{h′∈H} log |θ_{N}^{hh′^{−1}}|_{𝔭_{∞}}h′ · 𝔭_{∞}, since ψ ≠ 1

= \frac{1}{|H|} ∑_{h′∈H} ∑_{l∈H} ψ(h′^{−1}l^{−1}) log |θ_{N}^{l}|_{𝔭_{∞}}h′ · 𝔭_{∞}, l = hh′^{−1}

= ( ∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}|_{𝔭_{∞}}) e_{ψ}𝔭_{∞}.

Therefore λ\underline{φ} is represented by a diagonal matrix with determinant

det(λ\underline{φ} | Hom_{ℂG}(V, ΔS)) = ∏_{ψ|χ} 2 ∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}|.  (22)

We will express the sums ∑_{h∈H} ψ(h^{−1} log |θ_{N}^{h}| in terms of the Dirichlet Lseries L(s, ψ) at s = 1. By the definition of θ_{N} we obtain

∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}| = ∑_{g∈Gal(k(𝔣𝔟)/k)} ψ(g^{−1}) ∑_{J⊆{1,... ,s}} log |θ_{0}(τ_{J}, 𝔞)|,  (23)

where we view ψ as a character of Gal(k(𝔣𝔟)/k) by inflation. By (17) the above expression is equal to

\frac{1}{12} ∑_{g∈Gal(k(𝔣𝔟)/k)} ψ(g^{−1}) ∑_{J⊆\{1,... ,s\}} log |(\frac{φ^{12N(𝔞)}(τ_{J} | L)}{φ^{12}(τ_{J} | 𝔞^{−1}L)})^{g}|.  (24)

Since φ^{12N(𝔫_{J} 𝔟)}(τ_{J} | 𝔞^{−1}L) = φ^{12N(𝔫_{J} 𝔟)}(τ_{J} | L)^{σ(𝔞)} by Lemma 4.7(d) the expression (24) becomes

\frac{N(𝔞) − ψ(𝔞)}{12} ∑_{J⊆\{1,... ,s\}} \frac{1}{N(𝔫_{J}𝔟)} ×

∑_{g∈Gal(k(𝔣𝔟)/k)} ψ(g^{−1}) log |(φ^{12N(𝔟𝔫_{J})}(τ_{J} | L))^{g}|.  (25)

Since τ_{J} is a primitive 𝔫_{J}𝔟-torsion point of ℂ/L, there exists an integral O_{k}-ideal 𝔠_{J} such that

𝔫_{J}𝔟τ_{J} = 𝔠_{J}Ω, (𝔠_{J}, 𝔫_{J}𝔟) = 1.

By Lemma 4.7(a) and (d) we derive

φ^{12N(𝔫_{J} 𝔟)}(τ_{J} | L) = φ^{12N(𝔫_{J} 𝔟)}(1 | 𝔫_{J}𝔟𝔠_{J}^{−1})
= φ^{12N(𝔫_{J} 𝔟)}(1 | 𝔫_{J}𝔟)^{σ(𝔠_{J})}.

Therefore the expression in (25) becomes

\frac{N(𝔞) − ψ(𝔞)}{12} ∑_{J⊆\{1,... ,s\}} \frac{ψ(𝔠_{J})}{N(𝔫_{J}𝔟)} ×

∑_{g∈Gal(k(𝔣𝔟)/k)} ψ(g^{−1}) log |(φ^{12N(𝔫_{J} 𝔟)}(1 | 𝔫_{J}𝔟))^{g}|.  (26)

Following [Ro] we define for integral O_{k}-ideals 𝔤, 𝔤_{1} with 𝔤 | 𝔤_{1} and each abelian character ψ of Gal(k(𝔤)/k)

S_{𝔤}(ψ, 𝔤_{1}) = ∑_{C∈cl(𝔤_{1})} ψ(C^{−1}) log |φ_{𝔤}(C)| ,  (27)

where cl(𝔤_{1}) denotes the ray class group modulo 𝔤_{1} and ψ is viewed as a character of cl(𝔤_{1}) by inflation. For the definition of the invariant φ_{𝔤}(C) we choose an integral ideal 𝔠 in the class C and set

φ_{𝔤}(C) = \begin{cases}φ^{12N(𝔤)}(1 | 𝔤𝔠^{−1}), & if 𝔤 ≠ (1), \\ |\frac{N(𝔠^{−1})^{6}Δ(𝔠^{−1})}{(2π)^{12}} | , & if 𝔤 = (1). \end{cases}

Note that this definition does not depend on the choice of 𝔠 ∈ C ([Ro, pp. 15/16]).

108 5 THE RAMACHANDRA UNIT LATTICE

By Lemma 4.7(d) and (26) we can therefore write

∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}|

= \frac{N(𝔞) − ψ(𝔞)}{12} ∑_{J⊆\{1,... ,s\}} \frac{ψ(𝔠_{J})}{N(𝔫_{J}𝔟)}S_{𝔫_{J} 𝔟}(ψ, 𝔣𝔟).

Lemma 5.4 [Wa, §8, Lemma 8.4] If 𝔣_{ψ} ∤ 𝔫_{J}𝔟, then S_{𝔫_{J} 𝔟}(ψ, 𝔣𝔟) = 0. Lemma 5.5 [Wa, §8, Lemma 8.5] If 𝔣_{ψ} | 𝔫_{J}𝔟, then

S_{𝔫_{J} 𝔟}(ψ, 𝔣𝔟) = Φ(𝔫_{I})S_{𝔫_{J} 𝔟}(ψ, 𝔫_{J}𝔟),

where I = {1, …, s}. Proofs The proofs are analogous to the proofs of [Wa, §8, Lemma 8.4 and 8.5] using the fact that φ^{12N(𝔫_{J} 𝔟)}(1 | 𝔫_{J}𝔟) ∈ k(𝔫_{J}𝔟) by Lemma 4.7(c).

By the above Lemmas we obtain

∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}|

= \frac{N(𝔞) − ψ(𝔞)}{12} ∑_{{J,𝔣_{ψ} |𝔫_{J} \atop I=\{1,... ,s\}\J}} \frac{ψ(𝔠_{J})}{N(𝔫_{J}𝔟)}Φ(𝔫_{I})S_{𝔫_{J} 𝔟}(ψ, 𝔫_{J}𝔟).  (28)

Lemma 5.6 [Ro, Corollaire 2] If 𝔣_{ψ} | 𝔫_{J}𝔟, then

S_{𝔫_{J} 𝔟}(ψ, 𝔫_{J}𝔟) = \frac{N(𝔫_{J}𝔟)w(𝔫_{J}𝔟)}{N(𝔣_{ψ})w(𝔣_{ψ})}(1 − ψ(𝔟)^{−1}) ( ∏_{{j∈J \atop 𝔭_{j} ∤𝔣_{ψ}}} (1 − ψ(𝔭_{j})^{−1})) S_{𝔣_{ψ}}(ψ, 𝔣_{ψ}).

Substituting the equality of Lemma 5.6 into (28) we have

∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}|

= \frac{N(𝔞) − ψ(𝔞)}{12N(𝔣_{ψ})w(𝔣_{ψ})} ∑_{{J,𝔣_{ψ} |𝔫_{J} \atop I=\{1,... ,s\}\J}} [ψ(𝔠_{J}) ∏_{{j∈J \atop 𝔭_{j} ∤𝔣_{ψ}}} ψ(𝔭)^{−e_{j}} ] (1 − ψ(𝔟)^{−1}) ×
Φ(𝔫_{I}) ( ∏_{{j∈J \atop 𝔭_{j} ∤𝔣_{ψ}}} ψ(𝔭)^{e_{j}−1}(ψ(𝔭_{j}) − 1)) S_{𝔣_{ψ}}(ψ, 𝔣_{ψ}).  (29)

Lemma 5.7 The value

ζ_{ψ} = ψ(𝔠_{J}) ∏_{{j∈J \atop 𝔭_{j} ∤𝔣_{ψ}}} ψ(𝔭_{j})^{−e_{j}}

does not depend on J as long as 𝔣_{ψ} | 𝔫_{J}.

Proof Let J_{1} ⊆ {1, …, s} be minimal with 𝔣_{ψ} | 𝔫_{J}. Then τ_{J} = τ_{J_{1}} + ν with a torsion point ν of order prime to 𝔣_{ψ}.From the definition of 𝔠_{J} and 𝔠_{J_{1}} we derive

ψ(𝔠_{J}) ∏_{{j∈J \atop 𝔭_{j} ∤𝔣_{ψ}}} ψ(𝔭_{j})^{−e_{j}} = ψ(\frac{τ_{J}}{Ω} 𝔟 ∏_{𝔭_{j}|𝔣_{ψ}} 𝔭_{j}^{e_{j}}),

ψ(𝔠_{J_{1}}) ∏_{{j∈J_{1} \atop 𝔭_{j} ∤𝔣_{ψ}}} ψ(𝔭_{j})^{−e_{j}} = ψ(\frac{τ_{J_{1}}}{Ω} 𝔟 ∏_{𝔭_{j}|𝔣_{ψ}} 𝔭_{j}^{e_{j}}).

Hence it suffices to show that τ_{J}/τ_{J_{1}} belongs to the principle ray class modulo 𝔣_{ψ}. But this follows from \frac{τ_{J}}{τ_{J_{1}}} − 1 = \frac{ν}{τ_{J_{1}}} , since ν has order prime to 𝔣_{ψ}.

Lemma 5.8

∑_{{J,𝔣_{ψ} |𝔫_{J} 𝔟 \atop I=\{1,... ,s\}\J}} Φ(𝔫_{I}) ∏_{{j∈J \atop 𝔭_{j} ∤𝔣_{ψ}}} ψ(𝔭_{j})^{e_{j}−1}(ψ(𝔭_{j}) − 1) = g(ψ).

Proof For fixed ψ let K = {i ∈ {1, …, s} | 𝔭_{i} ∤ 𝔣_{ψ}}. By expanding the product for g(ψ) given in Definition 5.2 we find

g(ψ) = ∑_{I∈K} Φ(𝔫_{I}) ∏_{j∈K\I} ψ(𝔭_{j})^{e_{j}−1}(ψ(𝔭_{j}) − 1).

A subset I ⊆ {1, …, s} occurs in the sum if and only if 𝔣_{ψ} | 𝔫_{J}𝔟 with J = {1, …, s} \ I. Thus the Lemma follows easily from

j ∈ J and 𝔭_{j} ∤ 𝔣_{ψ} ⟺ j ∈ K \ I.

Applying the results of Lemma 5.7 and 5.8 to the expression in (29) we get

∑_{h∈H} ψ(h^{−1}) log |θ_{N}^{h}|

= \frac{N(𝔞) − ψ(𝔞)}{12N(𝔣_{ψ})w(𝔣_{ψ})}ζ_{ψ}(1 − ψ(𝔟)^{−1})g(ψ)S_{𝔣_{ψ}}(ψ, 𝔣_{ψ}).  (30)

Our next aim is to compute c_{S}(χ̌) in terms of the S_{𝔣_{ψ}}(ψ, 𝔣_{ψ}) with ψ extending χ. By the formal properties of Artin L-series [Ta2, Ch.0, 4.2] we have

L(s, χ) = L(s, ind_{G}^{H}χ) = ∏_{ψ|χ} L(s, ψ).

Since ψ is abelian and S = S_{∞} the Artin L-function L(s, ψ) is equal to the primitive Dirichlet L-function attached to ψ

In the following we write d_{k} for the discriminant of k/ℚ, 𝔡_{k} for the different ideal and Tr for the trace from k to ℚ.

Definition 5.9 [Ro, 2.4] Let 𝔤 be an integral O_{k}-ideal and ψ a character of cl(𝔤). Let γ be an element of k^{∗} such that γ𝔡_{k}𝔤 is integral and prime to 𝔤. Then we define a Gauss sum τ(ψ, 𝔤) by

τ(ψ, 𝔤) = ∑_{λ∈O_{k}/𝔤} ψ̄(λγ𝔡_{k}𝔤)e^{2πiTr(λγ)}.

If ψ is primitive, we simply write τ(ψ).

Note that τ(ψ, 𝔤) does not depend on the choice of γ.

By [Ro, Théorème 3] the values L(1, ψ) and S_{𝔣_{ψ}}(ψ, 𝔣_{ψ}) are closely related, namely

L(1, ψ) = −\frac{2π}{6N(𝔣_{ψ})w(𝔣_{ψ})τ(ψ)\sqrt{|d_{k}|}}S_{𝔣_{ψ}}(ψ, 𝔣_{ψ}).  (31)

We will now use the functional equation for L(s, ψ) to compute the value c_{S}(ψ̌). The completed Artin L-function is given by

Λ(s, ψ) = \frac{(|d_{k}|N(𝔣_{ψ}))^{s/2} · 2}{(2π)^{s}}Γ(s)L(s, ψ)

(see e.g. [Ta2, Ch.0, §6]). The function Λ(s, ψ) satisfies the functional equation

Λ(1 − s, ψ) = W (ψ)Λ(s, ψ̌)  (32)

with the Artin root number W (ψ). Computing the limit for s → 0 on both sides of (32) we obtain

|d_{k}|^{1/2}N(𝔣_{ψ})^{1/2}π^{−1}L(1, ψ) = 2W (ψ)c_{S}(ψ̌),

since L(s, ψ) has a zero of order 1 at s = 0, Γ(1) = 1 and lim_{s→0} sΓ(s) = 1. Together with (31) and W (ψ) = N(𝔣_{ψ})^{1/2}/τ(ψ) ([Ta2, page 19]) this implies

c_{S}(ψ̌) = −\frac{S_{𝔣_{ψ}}(ψ, 𝔣_{ψ})}{6N(𝔣_{ψ})w(𝔣_{ψ})}.  (33)

Now, from (22), (30) and (33), we derive

\frac{det(λ\underline{φ} | Hom_{ℂG}(V, ΔS))}{c_{S}(χ̌)} = ∏_{ψ|χ} (−(N(𝔞) − ψ(𝔞))(ψ(𝔟) − 1)g(ψ)ζ_{ψ}) ,

which concludes the proof of Proposition 5.3.

6 A unit lattice à la Hasse

In this section we will define another G-stable unit group, denoted by C_{Ha}, and compute the index of im(\underline{φ})^{χ} in C_{Ha}^{χ} for each abelian faithful character χ of G.

Recall the situation and notation summarized at the beginning of Section 5. In addition, let σ be a generator of G = Gal(N/K).

For each abelian character ϕ of Gal(N/K_{0}) we denote by ℚ(ϕ) the field of character values of ϕ. We define the division of ϕ by

[ϕ] := {ϕ^{g} | g ∈ Gal(ℚ(ϕ)/ℚ)},

thus dividing the group Ĝal(N/K_{0}) of abelian characters of Gal(N/K_{0}) into equivalence classes. Obviously this equivalence relation does not depend on the choice of ϕ within a class.

In the sequel we write

Gal(N/K_{0}) =< ρ > with ρ^{l^{t}} = 1

and define a generator ϕ_{0} of Ĝal(N/K_{0}) by

ϕ_{0}(ρ) = ζ_{l^{t}},

where ζ_{l^{t}} is a fixed primitive l^{t}-th root of unity. Then the divisions of Ĝal(N/K_{0}) are given by

D_{i} = [ϕ_{0}^{l^{t−i}}] = {ϕ_{0}^{l^{t−i}j} | j = 1, …, l^{i}, (j, l) = 1}, i = 0, …, t,

and we easily compute

|D_{i}| = \begin{cases}l^{i−1}(l − 1), & if 1 ≤ i ≤ t, \\ 1, & if i = 0. \end{cases}

In the following let χ be a linear faithful character of G. The characters ψ of H extending χ are given by χϕ, ϕ ∈ Ĝal(N/K_{0}). We set

ker(D_{i}) = ker(ϕ), 𝔣_{i} = 𝔣_{χϕ}, N_{i} = N^{ker(D_{i})}

for any ϕ ∈ D_{i}, i = 0, …, t, and note again that these definitions do not depend on the choice of ϕ ∈ D_{i}. Moreover we have 𝔣_{N_{i}} = 𝔣_{i}. Finally we fix a primitive 𝔣_{i}𝔟-torsion point ν_{i} of ℂ/L for each i ∈ {0, …, t}.

The next definition is motivated by [Ha1, §13 and 18].

Definition 6.1 (a) For i ∈ {0, …, t} we define

θ_{i} = N_{k(𝔣_{i}𝔟)/N_{i}}(θ_{0}(ν_{i}, 𝔞)).

(b) Let C_{Ha} be the subgroup of E_{N} generated by the elements

θ_{i}^{(σ−1)σ^{v}ρ^{w}}, i = 0, …, t, v ∈ ℤ, w = 0, …, |D_{i}| − 1.

We remark that the elements θ_{i}^{σ−1} are units by Proposition 4.8(c). Of course, C_{Ha} is not a unit group of full rank, since χ is fixed and does not run over all non-trivial irreducible characters of G. But the result of Proposition 6.2 will show that the “χ-part of C_{Ha}” is a full sublattice of the “χ-part of E_{N}”.

From now on we view χ as an l-adic character. Let F be a Galois extension of ℚ such that each representation of G can be realized over F. As in Section 3 we fix an embedding i_{l} : ℚ̄ → ℂ_{l} and write F_{l} for the completion of i_{l}(F) in ℂ_{l}. The valuation ring of F_{l} will be denoted by O_{l}.

In order to state the next proposition we need the notion of a generalized index ideal. Let R be a Dedekind domain with quotient field M. Let X, Y be two full R-lattices on a M-vector space V. For each prime ideal 𝔭 of R we define [X : Y]_{𝔭} := (det(l_{𝔭}))R_{𝔭}, where l_{𝔭} is any automorphism of M_{𝔭} ⊗_{M} V with l_{𝔭}(R_{𝔭} ⊗_{R} Y) = R_{𝔭} ⊗_{R} X. Then [X : Y]_{𝔭} = R_{𝔭} for almost all primes 𝔭 and we can therefore define an R-ideal [X : Y]_{R} by requiring ([X : Y]_{R})_{𝔭} = [X : Y]_{𝔭} for all primes 𝔭 of R. We also recall the following relation

[X : Y]_{R} = \frac{ℓ_{R}(X/X ∩ Y)}{ℓ_{R}(Y/X ∩ Y)}.

Proposition 6.2 Let χ be a faithful abelian character of G. Then

[(O_{l}⊗_{ℤ}C_{Ha})^{χ} : (O_{l}⊗_{ℤ}im(\underline{φ}))^{χ}]_{O_{l}} = (∏_{ψ} g(ψ)) O_{l},

where ψ runs through all characters of H extending χ.

The rest of this section will be devoted to the proof of Proposition 6.2.

Let ζ be a primitive |H|-th root of unity and set R = O_{l}[ζ]. Let M be the field of fractions of R.

We will show that

[(R⊗_{ℤ}C_{Ha})^{χ} : (R⊗_{ℤ}im(\underline{φ}))^{χ}]_{R} = (∏_{ψ|χ} g(ψ)) R,  (34)

which immediately implies Proposition 6.2 since ∏_{ψ|χ} g(ψ) ∈ O_{l}. The computations leading to (34) will take place in M⊗_{ℤ}(k^{ab})^{∗}.

In analogy to (27) we define for integral O_{k}-ideals 𝔤, 𝔤_{1} with 𝔤 | 𝔤_{1} and an abelian character ψ of Gal(k(𝔤)/k)

S′_{𝔤}(ψ, 𝔤_{1}) = ∑_{C∈cl(𝔤_{1})} ψ(C^{−1}) ⊗ φ′_{𝔤}(C),

where

φ′_{𝔤} = \begin{cases}φ_{𝔤}(C), & if 𝔤 ≠ (1), \\ \frac{Δ(𝔠^{−1})}{Δ(𝔟𝔠^{−1})}, & if 𝔤 = (1) and 𝔠 ∈ C. \end{cases}

Note that we have to adapt the definition of φ′_{(1)} since N(𝔠)^{−1}Δ(𝔠^{−1})/(2π)^{12} (without the absolut value) is not a class invariant. However, we have the relation

∑_{g∈Gal(k(1)/k)} ψ(g^{−1}) log |(\frac{Δ(O_{k})}{Δ(𝔟)})^{g}| = (1 − ψ(𝔟)^{−1})S_{(1)}(ψ, (1)).  (35)

The strategy for our proof of (34) is as follows: we will exhibit an R-basis α_{1}, …, α_{l^{t}} of (R⊗_{ℤ}im(\underline{φ}))^{χ} and an R-basis β_{1}, …, β_{l^{t}} of (R⊗_{ℤ}C_{Ha})^{χ} and express both basis in terms of the S′_{𝔣_{ψ}}(ψ, 𝔣_{ψ}), ψ | χ,

( α_{1} ; ⋮ ; α_{l^{t}} ) = A_{1} ( ⋮ ; S′_{𝔣_{ψ}}(ψ, 𝔣_{ψ}) ; ⋮ )_{ψ|χ} ,  ( β_{1} ; ⋮ ; β_{l^{t}} ) = A_{2} ( ⋮ ; S′_{𝔣_{ψ}}(ψ, 𝔣_{ψ}) ; ⋮ )_{ψ|χ}

with matrices A_{1}, A_{2} ∈ Gl_{l^{t}}(M). Then

[(R⊗_{ℤ}C_{Ha})^{χ} : (R⊗_{ℤ}im(\underline{φ}))^{χ}]_{R} = det(A_{1}A_{2}^{−1})R.  (36)

Since (R⊗_{ℤ}im(\underline{φ}))^{χ} ≃ (R⊗_{ℤ}ΔS)^{χ} = e_{χ}RH𝔭_{∞}, an R-basis of (R⊗_{ℤ}im(\underline{φ}))^{χ} is given by the elements e_{χ}(1⊗θ_{N}^{h}) with h running through a set of representatives of H/G.

From e_{χ} = ∑_{ψ|χ} e_{ψ} and the definition of θ_{N} we obtain

e_{χ}(1 ⊗ θ_{N}^{h}) = ∑_{ψ|χ} ψ(h) \underbrace{e_{ψ} ∑_{g∈Gal(k(𝔣𝔟)/N)} ∑_{J⊆\{1,... ,s\}} 1 ⊗ θ_{0}(τ_{J}, 𝔞)^{g}}_{(∗)} .

For (*) we compute

\frac{1}{|H|} ∑_{g∈Gal(k(𝔣𝔟)/k)} ψ(g^{−1}) ∑_{J⊆\{1,... ,s\}} 1 ⊗ θ_{0}(τ_{J}, 𝔞)^{g}.

This expression is analogous to the right hand side of (23). We define

a_{1}(ψ) = \begin{cases}\frac{1}{|H|} \frac{N(𝔞)−ψ(𝔞)}{12N(𝔣_{ψ})w(𝔣_{ψ})}(1 − ψ(𝔟)^{−1})ζ_{ψ}g(ψ), & if 𝔣_{ψ} ≠ (1), \\ \frac{1}{|H|} \frac{N(𝔞)−ψ(𝔞)}{12w(1)}ζ_{ψ}g(ψ), & if 𝔣_{ψ} = (1), \end{cases}

with g(ψ) defined in Defintion 5.2 and ζ_{ψ} as in Lemma 5.6.

Note that the Lemmas 5.4,5.5 and 5.7 also hold for S replaced by S′, up to a slight difference in the case 𝔣_{ψ} = (1) which reflects the definition of φ′_{(1)}. As in Section 5 we can prove

e_{χ}(1 ⊗ θ_{N}^{h}) = ∑_{ψ|χ} ψ(h)a_{1}(ψ)S′_{𝔣_{ψ}}(ψ, 𝔣_{ψ}).  (37)

We will now describe an R-basis of (R⊗_{ℤ}C_{Ha})^{χ} and compute the matrix A_{2}. Since e_{χ}(1⊗θ_{i}^{(σ−1)σ^{v}ρ^{w}}) = (χ(σ)−1)χ(σ)^{v}e_{χ}(1⊗θ_{i}^{ρ^{w}}) and (χ(σ)−1)χ(σ)^{v} ∈ O_{l}^{∗}, the set

{e_{χ}(1 ⊗ θ_{i}^{ρ^{w}}) | i = 0, …, t, w = 0, …, |D_{i}| − 1}

constitutes an R-basis of (R⊗_{ℤ}C_{Ha})^{χ}. The set of characters ψ extending χ is given by {χϕ | ϕ ∈ Ĝal(N/K_{0})}. From the definition of θ_{i} and Ĝal(N/K_{0}) = ∪_{j=0}^{t}D_{j} we conclude

e_{χ}(1 ⊗ θ_{i}^{ρ^{w}}) = ∑_{j=0}^{t} ∑_{ϕ∈D_{j}} ϕ(ρ^{w})\frac{1}{|H|} ∑_{h∈H}(χϕ)(h^{−1}) ⊗ θ_{i}^{h}

= ∑_{j=0}^{t} ∑_{ϕ∈D_{j}} ϕ(ρ^{w})\frac{1}{|H|} ∑_{h∈H/ ker(D_{i})} ∑_{g∈ker(D_{i})}(χϕ)((hg)^{−1}) ⊗ θ_{i}^{hg}.

Since Gal(N/K_{0}) is cyclic, we have a filtration

1 = ker(D_{t}) ⊂ ker(D_{t−1}) ⊂ … ⊂ ker(D_{1}) ⊂ ker(D_{0}) = Gal(N/K_{0}).

By definition θ_{i} is contained in N_{i} = N^{ker(D_{i})}. Hence we compute further

∑_{h∈H/ ker(D_{i})} ∑_{g∈ker(D_{i})}(χϕ)((hg)^{−1}) ⊗ θ_{i}^{hg}

= ∑_{h∈H/ ker(D_{i})}(χϕ)(h^{−1}) ( ∑_{g∈ker(D_{i})}(χϕ)(g^{−1})) ⊗ θ_{i}^{h}

= \begin{cases}0, & if i < j, \\ l^{t−i} ∑_{h∈H/ ker(D_{i})}(χϕ)(h^{−1}) ⊗ θ_{i}^{h}, & if i ≥ j. \end{cases}  (38)

From (38) and the definition of θ_{i} we derive

e_{χ}(1 ⊗ θ_{i}^{ρ^{w}}) = ∑_{j=0}^{i} ∑_{ϕ∈D_{j}} ϕ(ρ^{w})\frac{l^{t−i}}{|H|} ∑_{g∈Gal(k(𝔣_{i}𝔟)/k)}(χϕ)(g^{−1}) ⊗ θ_{0}(ν_{i}, 𝔞)^{g}. (39)

The equation (17) implies

∑_{g∈Gal(k(𝔣_{i}𝔟)/k)}(χϕ)(g^{−1}) ⊗ θ_{0}(ν_{i}, 𝔞)^{g}

= \frac{N(𝔞) − (χϕ)(𝔞)}{12N(𝔣_{i}𝔟)} ∑_{g∈Gal(k(𝔣_{i}𝔟)/k)}(χϕ)(g^{−1}) ⊗ (φ^{12N(𝔣_{i}𝔟)}(ν_{i} | L))^{g} .

Since ν_{i} is a primitive 𝔣_{i}𝔟-torsion point of ℂ/L there exists an integral O_{k}-ideal 𝔠_{i} such that

(\frac{ν_{i}}{Ω} ) = 𝔠_{i}(𝔣_{i}𝔟)^{−1}, (𝔠_{i}, 𝔣_{i}𝔟) = 1.

By the homogeneity of φ^{12N(𝔣_{i}𝔟)} and Lemma 4.7(d) we conclude

φ^{12N(𝔣_{i}𝔟)}(ν_{i} | L) = φ^{12N(𝔣_{i}𝔟)}(1 | 𝔣_{i}𝔟𝔠_{i}^{−1})
= φ^{12N(𝔣_{i}𝔟)}(1 | 𝔣_{i}𝔟)^{σ(𝔠_{i})}.

Hence we obtain for 0 ≤ j ≤ i and ϕ ∈ D_{j}

∑_{g∈Gal(k(𝔣_{i}𝔟)/k)}(χϕ)(g^{−1}) ⊗ θ_{0}(ν_{i}, 𝔞)

= \frac{N(𝔞) − (χϕ)(𝔞)}{12N(𝔣_{i}𝔟)}(χϕ)(𝔠_{i}) ×

∑_{g∈Gal(k(𝔣_{i}𝔟)/k)}(χϕ)(g^{−1}) ⊗ φ^{12N(𝔣_{i}𝔟)}(1 | 𝔣_{i}𝔟)^{g}

= \frac{N(𝔞) − (χϕ)(𝔞)}{12N(𝔣_{i}𝔟)}(χϕ)(𝔠_{i})S′_{𝔣_{i}𝔟}(χϕ, 𝔣_{i}𝔟)  (40)

By the analogon of Lemma 5.7 (adapted to the definition of φ′_{(1)} we obtain for ϕ ∈ D_{j} with 0 ≤ j ≤ i

\frac{1}{|H|} ∑_{g∈Gal(k(𝔣_{i}𝔟)/k)}(χϕ)(g^{−1}) ⊗ θ_{0}(ν_{i}, 𝔞) = a_{2}^{(i)}(χϕ)S′_{𝔣_{χϕ}}(χϕ, 𝔣_{χϕ}),  (41)

where

a_{2}^{(i)}(χϕ)

= \begin{cases}\frac{1}{|H|} \frac{N(𝔞)−(χϕ)(𝔞)}{12N(𝔣_{j})w(𝔣_{j})}(χϕ)(𝔠_{i})(1 − (χϕ)(𝔟)^{−1}) ∏_{𝔭|𝔣_{i},𝔭∤𝔣_{j}}(1 − (χϕ)(𝔭)^{−1}), \\ if 𝔣_{j} ≠ (1), \\ \frac{1}{|H|} \frac{N(𝔞)−(χϕ)(𝔞)}{12w(1)}(χϕ)(𝔠_{i}) ∏_{𝔭|𝔣_{i}}(1 − (χϕ)(𝔭)^{−1}), \\ if 𝔣_{j} = (1). \end{cases}

Hence we obtain from (39) and (41) for i = 0, …, t and w = 0, …, |D_{i}| − 1

e_{χ}(1 ⊗ θ_{i}^{ρ^{w}}) = ∑_{j=0}^{i} ∑_{ϕ∈D_{j}} ϕ(ρ^{w})l^{t−i}a_{2}^{(i)}(χϕ)S′_{𝔣_{χϕ}}(χϕ, 𝔣_{χϕ}).  (42)

We are now in position to compute the matrices A_{1} and A_{2}. On the one hand we derive from (37)

( ⋮ ; e_{χ}(1 ⊗ θ_{N}^{h}) ; ⋮ )_{h∈Gal(N/K_{0})} = A_{1} ( ⋮ ; S′_{𝔣_{χϕ}}(ψ, 𝔣_{ψ}) ; ⋮ )_{ψ|χ} ,

where

A_{1} = (ψ(h))_{{h∈Gal(N/K_{0}) \atop ψ|χ}} · diag (a_{1}(ψ))_{ψ|χ} .

On the other hand (42) implies

( ⋮ ; e_{χ}(1 ⊗ θ_{i}^{ρ^{w}}) ; ⋮ )_{i=0,... ,t,w=0,... ,|D_{i}|−1} = A_{2} ( ⋮ ; S′_{𝔣_{χϕ}}(χϕ, 𝔣_{χϕ}) ; ⋮ )_{j=0,... ,t,ϕ∈D_{j}} ,

where A_{2} is a block matrix of the form

A_{2} = ( \begin{matrix}M_{00} & 0 & 0 & … & 0 \\ M_{10} & M_{11} & 0 & … & 0 \\ ⋮ & & ... \\ M_{t0} & & … & & M_{tt} \end{matrix} ) .

Each block M_{ij} is a |D_{i}| × |D_{j}|-matrix given by

M_{ij} = l^{t−i} (a_{2}^{(i)}(χϕ)ϕ(ρ^{w}))_{{w=0,... ,|D_{i}|−1 \atop ϕ∈D_{j}}}
= l^{t−i} (ϕ(ρ^{w}))_{{w=0,... ,|D_{i}|−1 \atop ϕ∈D_{j}}} · diag (a_{2}^{(i)}(χϕ))_{ϕ∈D_{j}} .

The determinant of A_{1} is easily computed. We have

det(A_{1}) = l^{\frac{1}{2} tl^{t}} · ∏_{ψ|χ} a_{1}(ψ).  (43)

Lemma 6.3

∏_{i=0}^{t} det (l^{t−i}ϕ(ρ^{w}))_{{w=0,... ,|D_{i}|−1 \atop ϕ∈D_{i}}} = ±l^{\frac{1}{2} tl^{t}}.

Proof Recall that D_{i} = {ϕ_{0}^{l^{t−i}j} | j = 1, …, l^{i}, (j, l) = 1}. It follows that (det(ϕ(ρ^{w})_{w,ϕ})^{2} equals the field discriminant of ℚ(ζ_{l^{i}})/ℚ. By [Wa, Ch.2, Prop. 2.1] we therefore derive

∏_{i=0}^{t} det (l^{t−i}ϕ(ρ^{w}))_{{w=0,... ,|D_{i}|−1 \atop ϕ∈D_{i}}}

= ∏_{i=0}^{t} l^{(t−i)|D_{i}|} · d_{ℚ(ζ_{l^{i}})/ℚ}^{1/2}

= ±l^{t} ∏_{i=1}^{t} l^{(t−i)l^{i−1}(l−1)} · l^{l^{i−1}(li−i−1)/2}.

An elementary computation shows that this is equal to ±l^{\frac{1}{2} tl^{t}}.

Lemma 6.3 implies that

det(A_{2}) = ±l^{\frac{1}{2} tl^{t}} · ∏_{i=0}^{t} ∏_{ϕ∈D_{i}} a_{2}^{(i)}(χϕ).  (44)

Now (34) (and thus also Proposition 6.2) is an immediate consequence of (36), (43), (44) and the definition of a_{1}(ψ) and a_{2}^{(i)}(χϕ).

7 Proofs of the main results

We assume the situation described at the beginning of Section 5. From Proposition 3.2, Proposition 5.3 and Proposition 6.2 we conclude

𝔞^{(l)}(χ) = (∏_{ψ|χ}(N(𝔞) − ψ(𝔞))(ψ(𝔟) − 1)) \frac{ℓ_{O_{l}}(cl_{N}^{χ})}{ℓ_{O_{l}}((E_{N}/C_{Ha})^{χ})}.  (45)

Since χ is a faithful abelian character of G the hypothesis (20) implies that ψ(𝔟) is a non-trivial root of unity of order dividing l′. Therefore ψ(𝔟) − 1 ∼_{l} 1.

Lemma 7.1 (a) Suppose that either l ∤ w_{N} or K_{0} ≠ k(ζ_{l}). Then there exists an integral O_{k}-ideal 𝔞 such that

∏_{ψ|χ}(N(𝔞) − ψ(𝔞)) ∼_{l} ℓ_{O_{l}}(μ_{N}^{χ}) ∼_{l} 1.

(b) If K_{0} = k(ζ_{l}), then

ℓ_{O_{l}}(μ_{N}^{χ}) | ∏_{ψ|χ}(N(𝔞) − ψ(𝔞))

for all integral O_{k}-ideals 𝔞.

Proof (a) If l ∤ w_{N} then obviously ℓ_{O_{l}}(μ_{N}^{χ}) = 1. For the choice of the integral ideal 𝔞 we consider the tower of fields k ⊆ N ⊆ N(ζ_{l}) and take 𝔞 such that (𝔞, N(ζ_{l})/k) ≠ 1 and (𝔞, N/k) = 1. Then ψ(𝔞) = 1 and since ζ_{l}^{(𝔞,N(ζ_{l})/k)} = ζ_{l}^{N(𝔞)} we obtain N(𝔞) − ψ(𝔞) = N(𝔞) − 1 ≢ 0(mod l).

If l | w_{N}, but K_{0} ≠ k(ζ_{l}), then we have the following diagram of fields

[figure: k; K; K_{0}; N; K(ζ_{l}); k(ζ_{l})]

with N ≠ K(ζ_{l}). We choose 𝔞 such that (𝔞, N/k) ∈ Gal(N/K(ζ_{l})) \ {1}. Then N(𝔞) − ψ(𝔞) = N(𝔞) − χ(𝔞) = N(𝔞) − 1 + 1 − χ(𝔞). Since (𝔞, N/k) acts trivially on ζ_{l} we have N(𝔞) ≡ 1(mod l). The character value χ(𝔞) is a root of unity of order prime to l and therefore (1 − χ(𝔞), l) = 1, which implies the assertion.

(b) The set of characters ψ | χ is given by χϕ, ϕ ∈ Gal(N/K_{0}). On writing σ(𝔞) = σ_{1}σ_{2} with σ_{1} ∈ G, σ_{2} ∈ Gal(N/K_{0}) we compute

∏_{ψ|χ}(N(𝔞) − ψ(𝔞)) = ∏_{ϕ∈Gal(N/K_{0})}(N(𝔞) − χ(σ_{1})ϕ(σ_{2})ϕ(𝔞))

= ∏_{j=0}^{l^{t}−1}(N(𝔞) − χ(σ_{1})ζ_{l^{t}}^{j}) = N(𝔞)^{l^{t}} − χ(σ_{1})^{l^{t}}.

We write w_{N} = l^{s} · k with (l, k) = 1, t ≥ s − 1. Since μ_{N}^{χ} = e_{χ}(O_{l}⊗_{ℤ}μ_{N}) is cyclic generated by e_{χ}(1⊗ζ_{l^{s}}) it suffices to show that N(𝔞)^{l^{t}} −χ(σ_{1})^{l^{t}} annihilates e_{χ}(1 ⊗ ζ_{l^{s}}). On the one hand we have σ_{1}^{l^{t}}e_{χ}(1 ⊗ ζ_{l^{s}}) = χ(σ_{1})^{l^{t}}e_{χ}(1 ⊗ ζ_{l^{s}}). On

the other hand σ_{1}^{l^{t}}e_{χ}(1 ⊗ ζ_{l^{s}}) = \frac{1}{|G|} ∑_{g∈G} χ(g^{−1}) ⊗ ζ_{l^{s}}^{σ_{1}^{l^{t}} g} = \frac{1}{|G|} ∑_{g∈G} χ(g^{−1}) ⊗

(ζ_{l^{s}}^{N(𝔞)^{l^{t}}} )^{g} = N(𝔞)^{l^{t}}e_{χ}(1 ⊗ ζ_{l^{s}}). Hence N(𝔞)^{l^{t}} − χ(𝔞)^{l^{t}} annihilates e_{χ}(1 ⊗ ζ_{l^{s}}).

In the sequel we assume that l is fixed and satisfies either

l ∤ w_{N} or (l | w_{N} and K_{0} ≠ k(ζ_{l})) .  (46)

Then Lemma 7.1(a) implies that

𝔞^{(l)}(χ) = \frac{ℓ_{O_{l}}(cl_{N}^{χ})}{ℓ_{O_{l}}((E_{N}/(μ_{N} × C_{Ha}))^{χ})}.  (47)

In order to show that this quotient equals (1) (or to derive at least some divisibility result) we will employ the Euler system method as presented by Rubin in [Ru2, §1-3].

We fix a power M of l and unfortunately have to impose the additional hypothesis

N contains the Hilbert-l-class field k(1)_{l}.  (48)

We define L = L_{N,M} to be the set of primes 𝔩 of k not dividing w_{k} and satisfying

(i)  𝔩 splits completely in N/k
(ii)  N(𝔩) ≡ 1(mod M).

Lemma 7.2 For 𝔩 ∈ L there exists a unique extension N[𝔩] of N of degree M in Nk(𝔩). Further N[𝔩]/N is cyclic, totally ramified at all primes above 𝔩 and unramified at all primes not dividing 𝔩.

Proof The proof is just an adaption of the proof of [Ru2, Lemma 1.1] (see also [BH, page ???]).

For each 𝔩 ∈ L there exists a unique cyclic extension C/k(1) such that M = [C : k(1)] and C ⊆ k(𝔩). Let C′ be the unique cyclic subextension of C/k(1)_{l} with [C′ : k(1)_{l}] = M. If we set N[𝔩] = NC′, then the assertions of the Lemma follow from global class field theory.

We write S = S_{N,M} for the set of squarefree integral ideals of k which are divisible only by primes 𝔩 ∈ L. For 𝔤 ∈ S with 𝔤 = ∏_{i=1}^{n} 𝔩_{i} we set N[𝔤] = N[𝔩_{1}] ⋯ N[𝔩_{n}]. Note that for 𝔩_{1}, 𝔩_{2} ∈ L, 𝔩_{1} ≠ 𝔩_{2}, the extensions N[𝔩_{1}]/N and N[𝔩_{2}]/N are linearly disjoint.

For an integral O_{k}-ideal 𝔠 we write S(𝔠) for the subset of S consisting of ideals 𝔤 ∈ S with (𝔠, 𝔤) = 1. Furthermore we let U_{M}(𝔠) = U(𝔠) denote the set of functions α : S(𝔠) → N̄^{∗} such that the following axioms hold:

ES(i)  α(𝔤) ∈ N[𝔤]^{∗} for all 𝔤 ∈ S(𝔠).
ES(ii)  α(𝔤) is a global unit if 𝔤 ≠ (1).
ES(iii)  N_{N[𝔤]/N[𝔤/𝔩]}(α(𝔤)) = α(𝔤/𝔩)^{σ(𝔩)−1}, if 𝔩 | 𝔤.
ES(iv)  α(𝔤) ≡ α(𝔤/𝔩)^{(N(𝔩)−1)/M} modulo all primes above 𝔩, if 𝔩 | 𝔤.

We define U_{N} = U_{N,M} to be the disjoint union of the sets U(𝔠) over all integral ideals 𝔠 of O_{k}. Then U_{N} is closed under multiplication, inverses and the action of Gal(N̄/k).

Recall the definition (16) of the function θ_{0} which was based on a fixed elliptic curve E defined over F = k(𝔟). The period lattice of E was denoted by L. Following [Ru2] we will now define a group of elliptic units. Let 𝔠 be an integral O_{k}-ideal such that 𝔟 | 𝔠. Let τ ∈ ℂ/L be a primitive 𝔠-torsion point of ℂ/L and let 𝔞 be an integral O_{k}-ideal with (𝔞, 6𝔠) = 1. For every 𝔩 ∈ L we fix a primitive 𝔩-torsion point τ_{𝔩} of ℂ/L. Then, for every 𝔤 ∈ S(𝔞𝔠), we set

α_{τ,𝔞}(𝔤) = N_{k(𝔠𝔤)N/N[𝔤]}(θ_{0}(τ + ∑_{𝔩|𝔤} τ_{𝔩}, 𝔞)).

From Proposition 4.8 we conclude that α_{τ,𝔞} ∈ U(𝔞𝔠). As in [Ru2] we write C_{N} for the group of global units generated by

{α_{τ,𝔞}(1)^{σ−1} : τ, α as above , σ ∈ Gal(N/k)}.

Then C_{N} is a ℤGal(N/k)-module and we define the group of elliptic units by

C_{N} = μ_{N}C_{N}.  (49)

Theorem 7.3 [Ru2, Theorem 3.2] Suppose that l ∤ [N : k] and k(1)_{l} ⊆ N. Let ψ be an irreducible ℚ_{l}-character of G whose restriction to Gal(N/N ∩ k(ζ_{l})) is non-trivial. Then

ℓ_{ℤ_{l}}(cl_{N}^{ψ}) | ℓ_{ℤ_{l}}((E_{N}/C_{N})^{ψ}).

Remarks 7.4 (a) Unlike Rubin in [Ru2, §1] we do not assume that l does not divide the number w_{k(1)} of roots of unity in k(1). For that reason we have to replace the condition “ψ ≠ 1” by “ψ|_{Gal(N/N∩k(ζ_{l}))} ≠ 1” (see [Ru3, Theorem 3.2] or [Ho, Theorem 5.9]).

(b) In our applications we can assume that ψ is a faithful character. Therefore the restriction of ψ to Gal(N/N ∩ k(ζ_{l})) is non-trivial unless N ⊆ k(ζ_{l}). But in this case N/ℚ is abelian, l ≠ 2 and we deduce from [RW2, Theorem A] and the induction property for 𝔞^{(l)}(ψ) that 𝔞_{N/k}^{(l)}(ψ) = (1).

Recall the definition of the elements θ_{N} (Definition 5.1) and θ_{i}, i = 0, …, t (Definition 6.1). The next lemma shows that the unit groups im(\underline{φ}) and C_{Ha} are contained in C_{N}.

Lemma 7.5 (a) θ_{N}^{h−1} ∈ C_{N} for h ∈ H \ {1}.

(b) θ_{i}^{σ−1} ∈ C_{N} for i ∈ {0, …, t} and σ ∈ G = Gal(N/K).

Proof (a) For each subset J ⊆ {0, …, t} we define a map α_{J} = α_{τ_{J} ,𝔞} : S(𝔫_{J}𝔟𝔞) → N̄^{∗} by

α_{J}(𝔤) = N_{k(𝔫_{J} 𝔟𝔤)N/N[𝔤]} (θ_{0}(τ_{J} + ∑_{𝔩|𝔤} τ_{𝔩}, 𝔞)) .

Considering the diagram

[figure: k; N; k(𝔣𝔟; k(𝔫_{J}𝔟); N_{i}]

we derive from Proposition 4.8(c)

N_{k(𝔣𝔟)/N}(θ_{0}(τ_{J}, 𝔞)) = N_{k(𝔫_{J} 𝔟)N/N}N_{k(𝔣𝔟)/k(𝔫_{J} 𝔟)N}(θ_{0}(τ_{J}, 𝔞))
= N_{k(𝔫_{J} 𝔟)N/N}(θ_{0}(τ_{J}, 𝔞))^{[k(𝔣𝔟):k(𝔫_{J} 𝔟)N]} = α_{J}(1)^{[k(𝔣𝔟):k(𝔫_{J} 𝔟)N]}.

Hence θ_{N}^{h−1} ∈ C_{N} for all h ∈ H \ {1}.

(b) For each i ∈ {0, …, t} we define a map α_{i} = α_{ν_{i},𝔞} : S(𝔣_{i}𝔟𝔞) → N̄^{∗} by

α_{i}(𝔤) = N_{k(𝔣_{i}𝔟𝔤)N/N[𝔤]} (θ_{0}(ν_{i} + ∑_{𝔩|𝔤} τ_{𝔩}, 𝔞))

Since θ_{0}(ν_{i}, 𝔞) ∈ k(𝔣_{i}𝔟) by Proposition 4.8 the diagram of fields

[figure: k; N; Nk(𝔣_{i}𝔟); k(𝔣_{i}𝔟); N_{i}]

implies that θ_{i} = N_{k(𝔣_{i}𝔟)/N_{i}}(θ_{0}(ν_{i}, 𝔞)) = N_{k(𝔣_{i}𝔟)∩N/N_{i}}(α_{i}(1)). Hence θ_{i}^{σ−1} ∈ C_{N}.

Proof of Theorem 1.1 By the inflation property (Proposition 3.1(d)) we may assume that N/k is cyclic. By Proposition 3.3(c) it suffices to show that 𝔞^{(l)}(ψ)^{−1} is integral whenever ψ is a non-trivial ℚ_{l}-irreducible character of N/k. By the inflation property we may also assume that ψ is faithful. Write

ψ = ∑_{α∈Gal(ℚ_{l}(χ)/ℚ_{l})} χ^{α},

where χ is a non-trivial linear character of Gal(N/k). Without loss of generality we may assume that hypothesis (46) is satisfied. In the exceptional case, N = K_{0} = k(ζ_{l}), the absolut extension N/ℚ is abelian and l ≠ 2. Thus 𝔞^{(l)}(ψ) = (1) follows from [RW2, Theorem A] (see also Remark 7.4(b)).

From (47) and the l-adic analogue of Propositon 3.1(a) we obtain

𝔞^{(l)}(ψ) = \frac{ℓ_{ℤ_{l}}(cl^{ψ})}{ℓ_{ℤ_{l}}((E_{N}/(μ_{N} × C_{Ha}))^{ψ})}.

By Lemma 7.5 (μ_{N} × C_{Ha})^{ψ} ⊆ C_{N}^{ψ}. Since l ∤ h_{k} we have k(1)_{l} = k and assumption (48) is trivially satisfied. Therefore we can apply Theorem 7.3. If the restriction of ψ to Gal(N/N ∩ k(ζ_{l})) is non-trivial, then 𝔞^{(l)}(ψ)^{−1} is integral. Otherwise N ⊆ k(ζ_{l}), l ≠ 2 and we obtain 𝔞^{(l)}(ψ) = (1) from [RW2, Theorem A].

Certainly it is unsatisfactory that we have to exclude prime divisors l of [N : k] in Theorem 1.1. To deal with these primes we have to consider a diagram of fields like in (19). In this situation we need a divisibility result of the form

ℓ_{ℤ_{l}}(cl_{N}^{ψ}) | ℓ_{ℤ_{l}}((E_{N}/C_{N})^{ψ})  (50)

for all ℚ_{l}-irreducible characters ψ of Gal(N/K).

A statement like this is given in [Ho, Theorem 5.9]. Unfortunately there is an error in the proof. To point out this mistake we use the notation of [Ho]. On page 334 of [Ho] a tower of fields F [λ]/F is exhibited satisfying the axioms EF(i)–(iv). But the fields constructed in the last but one paragraph do not satisfy EF(ii). On the contrary, each finite place above λ ∩ K_{0} (and not only λ) is totally ramified in F [λ]/F. Despite strong effort this mistake has not yet been fixed. Note however, that an analogous assertion for abelian extensions N/ℚ holds true. This can be derived from [RW2, §11 and Theorem A] together with a result analogous to Proposition 6.2. We also mention that the proof of [RW2, Theorem A] heavily relies on the results concerning the main conjecture of Iwasawa theory for totally real fields of A.Wiles [Wi].

Assuming (50) we can prove a stronger result.

Theorem 7.6 Let k be an imaginary quadratic number field with class number h_{k} = 1. Let L be an abelian number field extension of k. Suppose that for each prime l and all subextensions N/k of L/k the divisibility assertion of (50) holds. Suppose further that w_{L} = 3^{a}w_{k}^{b} with a ≥ 0, b ≥ 1. Then 𝔞(χ) = (1) for all characters χ of Gal(L/k).

Proof We fix a prime l and show (l, 𝔞(χ)) = 1. By Proposition 3.1(d) it suffices to show that 𝔞^{(l)}(ψ)^{−1} is integral for all cyclic subextensions N/k of L/k and all ℚ_{l}-irreducible faithful characters ψ of Gal(N/K) with K as in diagram (19).

Suppose first l ≠ 3. Then w_{L} = 3^{a}w_{k}^{b} implies that either (46) is satisfied or that k = K_{0}. If k = K_{0} we are done since then N = K. If (46) holds then

𝔞^{(l)}(ψ) = \frac{ℓ_{ℤ_{l}}(cl_{N}^{ψ})}{ℓ_{ℤ_{l}}((E_{N}/(μ_{N} × C_{Ha}))^{ψ})}.

By Lemma 7.5 we have (μ_{N} × C_{Ha})^{ψ} ⊆ C_{N}^{ψ}. Hence (50) implies that 𝔞^{(l)}(ψ) is integral.

If l = 3 then either (46) is satisfied, in which case we procedd as before, or K_{0} = k(ζ_{3}). But then N/K is of degree 2 and we obtain 𝔞^{(l)}(ψ) = (1) from 𝔞^{(l)}(ind_{1}^{Gal(N/K)}1) = (1) = 𝔞^{(l)}(1).

Remarks (a) Theorem 7.6 should be compared to [BH, Theorem 1].

(b) The assumption h_{k} = 1 is needed to define the subgroup C_{N} for all primes l. Without the hypothesis h_{k} = 1 we can derive a result of the form

(𝔞(χ), l) = 1 for all primes l with l ∤ h_{k}

and all characters χ of Gal(L/k).

In order to prove Theorem 1.3 we need some more notation. For the convenience of the reader we adopt most of the notation of [Ru2]. We write

Γ = Gal(K_{∞}/K_{0}), Δ = Gal(K_{0}/k), G = Δ × Γ = Gal(K_{∞}/k).

As always we assume l ∤ |Δ|. For every extension F of k in K_{∞} write A_{F} for the l-part of the ideal class group cl_{F}, E_{F} for the group of global units of F and C_{F} for the group of elliptic units as defined in (49). Fix an O_{k}-prime ideal 𝔭 and write U_{F} for the group of local units of F ⊗_{ℤ}k_{𝔭} which are congruent to 1 modulo all primes above 𝔭. Let Ē_{F} and C̄_{F} be the closures of E_{F} ∩ U_{F} and C_{F} ∩ U_{F}, respectively, in U_{F}.

Finally we define

A_{∞} = lim_{←} A_{F} , Ē_{∞} = lim_{←} Ē_{F} , C̄_{∞} = lim_{←} C̄_{F} ,

where the inverse limits are taken over all finite extensions F of k in K_{∞} with respect to the norm maps.

We define the Iwasawa algebra

Λ = ℤ_{l}[[G]] = lim_{←} ℤ_{l}[Gal(F/k)]

with the inverse limit taken over all finite extensions F of k in K_{∞}. For a closed subgroup H of G we define J (H) to be the Λ-ideal generated by {γ −1 : γ ∈ H}. We write D_{𝔭} for the decomposition group of an O_{k}-ideal 𝔭 in K_{∞}/k. Finally let T ⊆ {𝔭 : 𝔭 | l} be the set of primes of k which ramify in K_{∞}/K_{0} and define J_{T} = ∏_{𝔭∈T} J (D_{𝔭}).

For every irreducible ℚ_{l}-character ψ of Δ we have

Λ^{ψ} = e_{ψ}ℤ_{l}[[Δ × Γ]] ≃ R_{ψ}[[Γ]],

where R_{ψ} is the valuation ring in the unramified extension of ℚ_{l} of degree dim(ψ). Furthermore each Λ^{ψ} is (non-canonically) isomorphic to a power series ring over R_{ψ},

Λ^{ψ} ≃ R_{ψ}[[T]].

A Λ-module Y is called a torsion Λ-module if and only if Y^{ψ} is a torsion Λ^{ψ}- module for every ψ. By the well-known classification theorem for Λ^{ψ}-modules we know that for every finitely generated torsion Λ^{ψ}-module Y^{ψ} we can find power series f_{i,ψ} ∈ Λ^{ψ} and a pseudo-isomorphism

Y^{ψ} ⟶ ⊕Λ^{ψ}/f_{i,ψ}Λ^{ψ}.

Then char(Y^{ψ}) = (∏ f_{i,ψ}) Λ^{ψ} is a well-defined invariant of the Λ^{ψ}-module Y^{ψ} and is usually called the characteristic ideal.

In [Ru2, §5] it is proved that A_{∞} and Ē_{∞}/C̄_{∞} are finitely generated torsion Λ-modules. Moreover it is shown in the proof of [Ru2, Theorem 4.1] that

char(A_{∞}^{ψ}) | char((Ē_{∞}/C̄_{∞})^{ψ})  (51)

for all ℚ_{l}-irreducible characters ψ of Δ (see in particular the table at the end of [Ru2, §10]).

Proof of Theorem 1.3 Note first that every subfield N′ of N is again contained in some ℤ_{l}- extension K′_{∞}/K′_{0}:

[figure: k; K; K_{0}; N; K′_{0}; N′; K_{n}; K_{∞}; K′_{n}; K′_{∞}]

By the inflation property of Proposition 3.1(d) we we may therefore assume that N/k is cyclic. By Proposition 3.3(c) it is enough to show that 𝔞^{(l)}(ψ)^{−1} is integral for all non-trivial ℚ_{l}- irreducible characters of Gal(N/K). By the inflation property we may also assume that ψ is faithful. By assumption the hypothesis (46) is satisfied. This implies

𝔞^{(l)}(ψ) = \frac{ℓ_{ℤ_{l}}(cl^{ψ})}{ℓ_{ℤ_{l}}((E_{N}/(μ_{N} × C_{Ha})^{ψ})}

By Lemma 7.5 we have (μ_{N} × C_{Ha})^{ψ} ⊆ C_{N}^{ψ} and therefore it suffices to prove

ℓ_{ℤ_{l}}(cl_{N}^{ψ}) | ℓ_{ℤ_{l}}((E_{N}/C_{N})^{ψ}).  (52)

This will be a consequence of (51). Consider the canonical embedding α : E_{N} → ∏_{𝔓|𝔭} O_{𝔓}^{∗}, where 𝔭 is the fixed prime of k above l. Since the index of U_{N} in ∏_{𝔓|𝔭} O_{𝔓}^{∗} is prime to l the map α induces a map

α̃ : ℤ_{l}⊗_{ℤ}E_{N} = ℤ_{l}⊗_{ℤ}α^{−1}(U_{N}) ⟶ U_{N},
s ⊗ β ⟼ β^{s}.

It follows from [Br, Theorem 2’] that α̃ is injective. (A proof of this is achieved along the lines of proof of Leopoldt’s conjecture for abelian extensions K/ℚ [Wa, Theorem 5.31]).

Note that

α̃(ℤ_{l}⊗_{ℤ}E_{N}) = Ē_{N}, α̃(ℤ_{l}⊗_{ℤ}C_{N}) = C̄_{N}.  (53)

By hypothesis 𝔭 ∈ T does not split in K_{0}/k. Therefore we have Δ ⊆ D_{𝔭} for all 𝔭 ∈ T. This implies J_{T}^{ψ} = Λ^{ψ} for all non-trivial ℚ_{l}-irreducible characters ψ of Δ. Hence we derive from [Ru2, Corollary 7.9(ii)] and (51)

ℓ_{ℤ_{l}}(cl_{N}^{ψ}) | ℓ_{ℤ_{l}}((Ē_{N}/C̄_{N})^{ψ}).

Together with (53) this implies (52).

8 Galois module structure

In this Section we discuss the implications of Theorem 1.1 for the Galois module structure of global units.

Let N/K be a Galois extension of number fields with group G and let S be a large set of places of N. One of the most important invariants attached to the group E_{S} of S-units of N is the so-called Chinburg class Ω_{m}. We briefly recall its definition.

For S large we have a Tate sequence

0 ⟶ E_{S} ⟶ A ⟶ B ⟶ ΔS ⟶ 0  (54)

with A and B cohomologically trivial and whose extension class in Ext_{ℤG}(ΔS, E_{S}) is the canonical class α_{3} defined in [Ta1].

Let cl(ℤG) denote the locally free class group. Every cohomologically trivial ℤG-module has a resolution of length at most two by locally free ℤG-modules. Therefore, by Schanuel’s Lemma, every cohomologically trivial ℤG-module C defines a class (C) in cl(ℤG).

The Chinburg class is then defined by

Ω_{m} = (A) − (B) ∈ cl(ℤG).

By [Ch1, Theorem 3.1] the class Ω_{m} is independent of S and the chosen Tate sequence as long as S is large and the extension class of (54) is the canonical class. The analogy with the tame additive theory of Fröhlich-Taylor [Ch1] suggest the Conjecture of Chinburg Ω_{m} equals the root number class W in cl(ℤG).

For the definition of the root number class W see for example [Fr1]. We only recall the W = 0 if the extension N/K is trivial.

Let M ⊆ ℚG be a maximal ℤ-order. Tensoring with M over ℤG from the left induces an epimorphism cl(ℤG) → cl(M) whose kernel is independent of the choice of M. As usual we denote the kernel group by D(ℤG).

We briefly recall the Hom-description of cl(ℤG)/D(ℤG) ≃ cl(M) in its ideal theoretic form. Let F/ℚ be a finite Galois extension with group Γ such that every representation of G is realizable over F. Let R(G) be the character ring of G. Write I_{F} for the group of fractional O_{F} -ideals. Let I_{ℤ}(G) be the group of f ∈ Hom_{Γ}(R(G), I_{F}) such that for all irreducible characters χ the ideal f(χ) is an ideal of ℚ(χ). Define P_{ℤ}(G) to be the subgroup of I_{ℤ}(G) whose elements are homomorphisms of the form χ ↦ b(χ)O_{F} for some Γ-homomorphism b : R(G) → F^{∗} which satisfies: b(χ) is totally positive, if χ is a simplectic irreducible character. Then Fröhlich’s Hom-description of cl(ℤG) induces an isomorphism

cl(ℤG)/D(ℤG) ≃ I_{ℤ}(G)/P_{ℤ}(G).

Let φ : ΔS → E_{S} be a G-embedding. On the one hand, by [Ch1, Proposition 3.1] the homomorphism χ ↦ q_{φ}(χ) is an element of I_{ℤ}(G) and represents −Ω_{m} in cl(ℤG)/D(ℤG). On the other hand, if Stark’s conjecture is true for N/K, then χ ↦ A_{φ}(χ)O_{F} represents the root number class W in cl(ℤG)/D(ℤG). Thus the conjecture of Chinburg-Stark would impliy

W = Ω_{m} in cl(ℤG)/D(ℤG).

Let now k be an imaginary quadratic field and let N/k be a finite abelian extension. Let N/K be a subextension of N/k and as before write G = Gal(N/K). Then Chinburg’s conjecture simply reads as Ω_{m} = 0.

Let M ⊆ ℚG be the unique maximal ℤ-order. Let D_{1}, …, D_{n} be the divisions of Ĝ. For i = 1, …, n we choose a character χ_{i} ∈ D_{i} and thus fix an isomorphism

M ⟶ ⨁_{i=1}^{n} O_{ℚ(χ_{i})}, χ ⟼ (χ_{i}(λ))_{i} .

This isomorphism induces an isomorphism

cl(M) ≃ cl(ℤG)/D(ℤG) ⟶^{ψ} ⨁_{i=1}^{n} cl(O_{ℚ(χ_{i})}).

For a number field L and a natural number N we set

cl(O_{L}, \frac{1}{N}) := cl(O_{L})/ < (𝔭) : 𝔭 | N > .

Finally we define cl(M, 1/N) such that ψ induces an isomorphism

cl(M, \frac{1}{N}) ≃ ⨁_{i=1}^{n} cl(O_{ℚ(χ_{i})}, \frac{1}{N}).

Theorem 8.1 Let N/K be a subextension of an abelian extension N/k. Then

Ω_{m}(N/K) = 0 in cl (M, \frac{1}{[N : k]h_{k}} ) .

Proof The proof is immediate from Theorem 1.1 and the above discussion.

If N/k is a p-extension, then each of the fields ℚ(χ_{i}) is of the from ℚ(ζ_{p^{r}}) with r ≥ 0. It follows that cl(M, 1/[N : k]h_{k}) = cl(M, 1/h_{k}), since every prime divisor 𝔭 of p is principal in ℚ(ζ_{p^{r}}). Therefore Corollary 1.2 does not give any additional information.

From Theorem 1.3 we obtain

Theorem 8.2 Assume the situation of Theorem 1.3. Let N/K be a subextension of N/k. Then

Ω_{m}(N/K) = 0 in cl (M, \frac{1}{w_{N}[K_{0} : k]})

.

This Theorem should be compared to [BH, Theorem 1]. Note however, hat there might be a gap in their proof because it is based on the Euler system method as presented in [Ho] (see the discussion preceeding Theorem 7.6).

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