ON ARITHMETIC CLASS INVARIANTS

A. AGBOOLA AND G. PAPPAS

1. Introduction

Let F be a number field with ring of integers O_{F}, and let S be a finite set of places of F. Assume that S contains the set S_{∞} of archimedean places of F, and write S_{f} for the set of finite places contained in S. Let O_{S} (or O, when there is no danger of confusion) denote the ring of S_{f}-integers of F. Write F^{c} for an algebraic closure of F.

Let Y be any scheme over Spec(O). Suppose that G is a finite, flat commutative group scheme over Y of exponent N, and let G^{D} denote the Cartier dual of G. Let π : X → Y be a G-torsor, and write π_{0} : G → Y for the trivial G-torsor. Then O_{X} is an O_{G}-comodule, and so it is also an O_{G^{D}}-module (see [12]). As an O_{G^{D}}-module, O_{X} is locally free of rank one, and it therefore gives a line bundle M_{π} over G^{D}. Set L_{π} := M_{π} ⊗ M_{π_{0}}^{−1}. Then the map

ψ : H^{1}(Y, G) → Pic(G^{D}) ; [π] ↦ [L_{π}]

is a homomorphism which is often referred to as the ‘class invariant homomorphism’.

Suppose now that Y = Spec(O), and set G = Spec(𝔅), G^{D} = Spec(𝔄), and X = Spec(ℭ). Then the algebra ℭ is a twisted form of 𝔅, and the homomorphism ψ measures the Galois module structure of this twisted form (see [8]). The class invariant homomorphism was first introduced by W. Waterhouse (see [33], but also [15], éxposé VII) and was later further developed in the context of Galois module theory by M. J. Taylor (see [31], [32]). Taylor originally considered group schemes given by torsion points on abelian varieties with complex multiplication; the corresponding torsors are obtained by dividing points in the Mordell-Weil groups of these abelian varieties. In [30], it was shown that, for elliptic curves with complex multiplication, the class invariant homomorphism vanishes on the classes of torsors obtained by dividing torsion points of order coprime to 6. This implies the existence of Galois generators for certain rings of integers of abelian extensions of imaginary quadratic fields. This vanishing result was extended to all elliptic curves in [3] and [22]. The class

Date: August 4, 2000.

2 A. AGBOOLA AND G. PAPPAS

invariant homomorphism for torsors that are obtained by dividing points on abelian varieties has subsequently been studied in greater generality (see for example [2], [5], [23]).

The main goal of this paper is to introduce and study an arithmetic (i.e. ‘Arakelov’) refinement of the class invariant homomorphism and to study its values on torsors obtained by dividing points on abelian varieties. Assume as above that we have Y = Spec(O). We shall show that the line bundle L_{π} associated to the torsor π carries canonical metrics at all of the places in S; in particular it carries canonical hermitian metrics at all archimedean places in S. As a result of this, we shall see that the homomorphism ψ : H^{1}(Spec(O), G) → Pic(G^{D}) lifts to a homomorphism

ψ̂ : H^{1}(Spec(O), G) → P̂ic(G^{D}),

where P̂ic(G^{D}) is the Picard group of isomorphism classes of line bundles on G^{D} endowed with metrics at all places in S. In particular, if S = S_{∞}, then P̂ic(G^{D}) is just the Arakelov Picard group of the one-dimensional scheme G^{D}.

Now let G̃^{D} denote the normalisation of G^{D}. By composing ψ and ψ̂ with the natural pullback maps Pic(G^{D}) → Pic(G̃^{D}) and P̂ic(G^{D}) → P̂ic(G̃^{D}), we obtain homomorphisms

φ : H^{1}(Spec(O), G) → Pic(G̃^{D}) and φ̂ : H^{1}(Spec(O), G) → P̂ic(G̃^{D}).

We show that if the generic fibre of G^{D} is a constant group scheme (i.e. G^{D} is ‘generically constant’), then the kernel of φ̂ is very small (see Theorem 4.1).

In the remainder of the paper, we apply our results to study torsors that are obtained by dividing points on abelian varieties. Let E be an abelian scheme of dimension d over O, and write E^{D} for its dual. For each integer N, let [N] : E → E denote the multiplication-by-N map on E, and write E[N] for the O-group scheme of N-torsion on E. The Cartier dual of E[N] may be identified with the O-group scheme E^{D}[N] of N-torsion on E^{D}. We write { \atop E^{D}[N]} for the normalisation of E^{D}[N].

Suppose that P : Spec(O) → E is an O-valued point of E. Then we may form the following fibre product:

[N]^{−1}(P) := Spec(O) ×_{E,[N]} E ⟶ E
↓  ↓^{[N]}
Spec(O)  ⟶^{P} E.

Since [N] : E → E is an E[N]-torsor, [N]^{−1}(P) is also an E[N]-torsor and it is determined up to isomorphism by the image of P modulo [N] · E(O). In fact, [N]^{−1}(P) is just the

ON ARITHMETIC CLASS INVARIANTS 3

E[N]-torsor given by the image of P under the natural injection

\frac{E(O)}{[N] · E(O)} ↪ H^{1}(Spec(O), E[N])

afforded by Kummer theory on E. Let L_{[N]^{−1}(P)} denote the line bundle on E^{D}[N] that is associated to [N]^{−1}(P). By our earlier results, L_{[N]^{−1}(P)} carries a canonical metric at all places in S. Let \overline{L}_{[N]^{−1}(P)} denote the corresponding metrised line bundle. Then since E(O) ≃ E(F), we obtain a map

ψ̂_{N} : \frac{E(F)}{[N] · E(F)} → P̂ic(E^{D}[N]) ; [P] ↦ [\overline{L}_{[N]^{−1}(P)}]

which is a group homomorphism. By composing ψ̂_{N} with the natural pullback homomorphism P̂ic(E^{D}[N]) → P̂ic({ \atop E^{D}[N]}), we obtain a homomorphism

φ̂_{N} : \frac{E(F)}{[N] · E(F)} → P̂ic({ \atop E^{D}[N]}).

The class ψ̂(P) may be described in terms of the restriction of metrised line bundles on E^{D} to subgroup schemes of torsion points. Let L(P) denote the rigidified line bundle on E^{D} that corresponds to the point P via the duality between E and E^{D}. Then, at each place in S, L(P) carries a natural ‘Néron metric’ which arises via a canonical splitting of the extension of E^{D} by G_{m} which corresponds to P (see [17], [19], [20]). Write \overline{L}(P) for the line bundle L(P) endowed with these metrics. We show the following result.

Theorem 1.1. With the above notation, we have ψ̂_{N}(P) = (\overline{L}(P) |_{E^{D}[N]}) in P̂ic(E^{D}[N]).

In order to describe our main result on the arithmetic class invariants for abelian varieties, we have to introduce some further notation. Let l be a prime number. Then, for each positive integer n, the inclusion map E^{D}[l^{n−1}] → E^{D}[l^{n}] induces pullback homomorphisms P̂ic(E^{D}[l^{n}]) → P̂ic(E^{D}[l^{n−1}]) and P̂ic({ \atop E^{D}[l^{n}]}) → P̂ic({ \atop E^{D}[l^{n−1}]}). These homomorphisms are compatible with the corresponding maps ψ̂ and φ̂. Hence, we may pass to inverse limits to obtain homomorphisms

Ψ̂_{l} := \underleftarrow{lim} ψ̂_{l^{n}} : E(F) ⊗_{Z} Z_{l} → \underleftarrow{lim} P̂ic(E^{D}[l^{n}])

and

Φ̂_{l} := \underleftarrow{lim} φ̂_{l^{n}} : E(F) ⊗_{Z} Z_{l} → \underleftarrow{lim} P̂ic({ \atop E^{D}[l^{n}]}).

Let disc(F/Q) denote the discriminant of F/Q. The following result shows that, for almost all primes l, the class invariants φ̂_{l^{n}}(P) (and therefore also ψ̂_{l^{n}}(P)) determine P when P is not a torsion point of order coprime to l.

Theorem 1.2. Assume that none of the places of F that lie above l are contained in S. Suppose also that l does not divide 6 · disc(F/Q). Then Φ̂_{l} is injective.

The main ingredients of the proof of this theorem are results of Serre and Bogomolov concerning the l-adic representation attached to E (see [7], [25], [26], [27]), and a result of Fontaine which essentially gives an integral comparison theorem between the crystalline and étale cohomology of the l-divisible group scheme of E (see [14]).

We remark that Theorem 1.2 is currently the only known general injectivity result about class invariants that applies to all abelian varieties defined over number fields. For example, apart from certain cases involving elliptic curves, it is not known whether the invariants ψ_{l^{n}}(P) (as n varies) determine P if P is a point of infinite order (see the remarks at the end of Section 5 of this paper).

In the final section of the paper we show that a natural pairing < , >_{arith} constructed using the arithmetic class invariant homomorphism on Mordell-Weil groups coincides with a certain height pairing constructed by Mazur and Tate (see [19]). This enables us to give a new interpretation of the Mazur-Tate circle and ideal class pairings in terms of Galois module structure invariants attached to torsors of finite group schemes.

Acknowledgements. We are very grateful to D. Bertrand, Ph. Cassou-Nogués, A. J. de Jong, B. Erez, M. Larsen, K. Ribet and M. J. Taylor for helpful conversations. The first author was partially supported by NSF grant no. DMS-9700937. He would like to thank the Université de Bordeaux I and Harvard University for their hospitality while part of this work was carried out. The second author was partially supported by NSF grant no. DMS-9970378 and by a Sloan Research Fellowship.

Notation. If K is any field, then K^{c} denotes an algebraic closure of K.

If K is a number field, and v is any place of K, then we write K_{v} for the local completion of K at v and O_{K,v} (or O_{v} when there is no danger of confusion) for the ring of integers of K_{v}. For any O_{K}-module M, we set M_{v} := M ⊗_{O_{K}} O_{K,v}.

If v is a non-archimedean place of K and ϖ_{v} denotes a uniformising parameter of O_{K,v}, then we suppose that v is normalised so that v(ϖ_{v}) = 1. If q_{v} is the cardinality of the residue field of K_{v}, then for each α ∈ K_{v} we set |α|_{v} = q_{v}^{−v(α)}.

ON ARITHMETIC CLASS INVARIANTS 5

2. Torsors and canonical metrics

Let π : X → Spec(O) be a G-torsor, and let L_{π} denote its associated G^{D}-line bundle. In this section, we shall explain how to endow L_{π} with canonical metrics at all places contained in S. (In fact, the construction will show that L_{π} carries a canonical metric at all places of F.) This will then enable us to construct the homomorphism ψ̂ and the pairing < , >_{arith}.

For each place v ∈ S, we fix an algebraic closure F_{v}^{c} of the field F_{v} and we let σ_{v} : F ↪ F_{v}^{c} denote the corresponding embedding. We also write | · |_{v} for the unique extension to F_{v}^{c} of the v-adic absolute value | · |_{v} on F_{v}.

Suppose that V is any scheme over Spec(O). A metrised line bundle \overline{L} on V is a pair (\overline{L}, || · ||) which consists of a line bundle L on V together with a family || · || = \{|| · ||_{v}\}_{v∈S} of metrics on the line bundles L ⊗_{F,σ_{v}} F_{v}^{c}. (Here L denotes the restriction of L to the generic fibre of V.) We require that each metric || · ||_{v} take values in R_{≥0} and satisfy

||a · x||_{v} = |a|_{v} · ||x||_{v}

for all a ∈ F_{v}^{c} and x ∈ L ⊗_{F,σ_{v}} F_{v}^{c}. We also require that each || · ||_{v} is invariant under the action of Gal(F_{v}^{c}/F_{v}).

The set of isomorphism classes of metrised line bundles on V forms a group (with the group operation being given by tensor product) which we denote by P̂ic(V). The identity element of this group is the isomorphism class of the structure sheaf O_{V} of V endowed with the trivial metric | · |_{v} at all places v in S.

We next recall (see [33], [15], éxposé VII, or [23]) that there is a canonical isomorphism

H^{1}(Spec(O), G) ≃ Ext^{1}(G^{D}, G_{m}).  (2.1)

In particular one can associate a canonical central extension

1 → G_{m} → G(π) → G^{D} → 1  (2.2)

to the torsor π which is such that the corresponding line bundle over G^{D} is equal to L_{π}. (This construction is explained in detail in [33].) Let L_{π} denote the restriction of L_{π} to the generic fibre of G^{D}.

Now suppose that v is a place in S, and consider the extension

1 → G_{m}(F_{v}^{c}) → G(π)(F_{v}^{c}) → G^{D}(F_{v}^{c}) → 1

over F_{v}^{c}. Since G(π) is the complement of the zero section in L_{π}, it follows that each metric on the fibres of G(π)(F_{v}^{c}) above G^{D}(F_{v}^{c}) uniquely determines a metric on the line bundle

6 A. AGBOOLA AND G. PAPPAS

L_{π} ⊗_{F,σ_{v}} F_{v}^{c}. In order to specify a canonical metric on G(π)(F_{v}^{c}), we appeal to the following result.

Proposition 2.1. Let K be any locally compact field. Suppose that H_{1} is a commutative, locally compact group over K, and let H_{2} be a closed subgroup of H_{1} which is such that H_{1}/H_{2} is compact. Then any continuous homomorphism λ : H_{2}(K) → R has a unique extension to a continuous homomorphism λ̃ : H_{1}(K) → R.

Proof. A proof of this result is given in [17], Chapter 11, Lemma 6.1.

Proposition 2.1 implies that, for each finite extension K of F_{v}, there is a unique continuous homomorphism v_{π}^{K} : G(π)(K) → R which is such that v_{π}^{K}(z) = log |z|_{v} for all z ∈ K^{∗} ⊂ G(π)(K). If F_{v} ⊆ K ⊆ L with [L : F_{v}] < ∞, then the uniqueness assertion of Proposition 2.1 implies that v_{π}^{L}|_{G(π)(K)} = v_{π}^{K}. Hence, by passing to the direct limit over all finite extensions of F_{v}, we see that there is a unique homomorphism

v_{π} : G(π)(F_{v}^{c}) → R

which is such that v_{π}(z) = log |z|_{v} for all z ∈ (F_{v}^{c})^{∗} ⊂ G(π)(F_{v}^{c}). We define

|| · ||_{v} : G(π)(F_{v}^{c}) → R_{>0}  (2.3)

by setting ||x||_{v} = exp(v_{π}(x)) for all x ∈ G(π)(F_{v}^{c}). Then, if z ∈ F_{v}^{c}, we have that ||z.x||_{v} = |z|_{v}||x||_{v}, and so it follows that || · ||_{v} defines a metric on each fibre of G(π)(F_{v}^{c}) lying above G^{D}(F_{v}^{c}). It is easy to see that || · ||_{v} is invariant under the action of Gal(F_{v}^{c}/F_{v}). We use the same symbol || · ||_{v} for the induced metric on the line bundle L_{π} ⊗_{F,σ_{v}} F_{v}^{c}. The metrics \{||·||_{v}\}_{v∈S} are the canonical metrics associated to L_{π}, and we write \overline{L}_{π} for the corresponding metrised line bundle on G^{D}.

Now suppose that π′ : X′ → Spec(O) is another G-torsor, and write π′′ for the compositum of π and π′. The product G(π) · G(π′) of the extensions G(π) and G(π′) is given by the following pullback diagram:

G(π) · G(π′) ⟶ G(π) × G(π′)
↓  ↓
G^{D}  ⟶^{Δ} G^{D} × G^{D},

where Δ denotes the diagonal map. It follows from the construction of the extension (2.2) (see [33]) that there are canonical isomorphisms

G(π′′) ≃ G(π) · G(π′) and L_{π′′} ≃ L_{π} ⊗ L_{π′}.  (2.4)

ON ARITHMETIC CLASS INVARIANTS 7

The uniqueness assertion of Proposition 2.1 implies that for each v ∈ S, the homomorphism v_{π′′} : G(π′′)(F_{v}^{c}) → R is induced by the homomorphism v_{π} ⊗v_{π′} : G(π)(F_{v}^{c})×G(π′)(F_{v}^{c}) → R which is defined by (v_{π} ⊗ v_{π′})(x, x′) = v_{π}(x) + v_{π′}(x′) for x ∈ G(π)(F_{v}^{c}) and x′ ∈ G(π′)(F_{v}^{c}). This in turn implies that (2.4) induces an isometry

\overline{L}_{π′′} ≃ \overline{L}_{π} ⊗ \overline{L}_{π′}

of metrised line bundles on G^{D}. We therefore obtain the following result.

Theorem 2.2. With the above notation, the map

ψ̂ : H^{1}(Spec(O), G) → P̂ic(G^{D}) ; [π] ↦ [\overline{L}_{π}]

is a group homomorphism which lifts the homomorphism ψ.

Remark 2.3. An alternative (and somewhat more explicit) description of the canonical metrics {|| · ||_{v}}_{v∈S} on L_{π} may be given as follows. Recall that π_{0} : G → Spec(O) denotes the trivial G-torsor over O, and that N is the exponent of G. We observe that there are canonical isomorphisms

G(π)^{N} ≃ G(π^{N}) ≃ G(π_{0}) ≃ G^{D} × G_{m}

of extensions. It follows via functoriality that these isomorphisms induce a canonical isometry

ξ_{π} : \overline{L}_{π}^{⊗N} ⟶^{∼} \overline{O}_{G^{D}}  (2.5)

of metrised line bundles on G^{D}. Hence, if v ∈ S and s is a section of L_{π} ⊗_{F,σ_{v}} F_{v}^{c}, then

||s||_{v} = |ξ_{π}(s^{⊗N})|_{v}^{1/N}

We therefore see that the metric || · ||_{v} is the pullback via ξ_{π} of the Nth root of the trivial metric on O_{G^{D}}.

Suppose that G^{D} is generically constant, and recall that G̃^{D} denotes the normalisation of G^{D}. Then it follows that

OG̃^{D} = Map(G^{D}(F), O),  (2.6)

and so we have

P̂ic(G̃^{D}) ≃ Map(G^{D}(F), P̂ic(O)),
Pic(G̃^{D}) ≃ Map(G^{D}(F), Pic(O)).

We define

< , >_{arith}: H^{1}(Spec(O), G) × G^{D}(F) → P̂ic(O)  (2.7)

to be the pairing which is induced by the homomorphism

φ̂ : H^{1}(Spec(O), G) → P̂ic(G̃^{D}) ≃ Map(G^{D}(F), P̂ic(O)).  (2.8)

Thus, if R ∈ G(F), then we extend R to an O-valued point R : Spec(O) → G^{D} of G^{D}, and we have

< [π], R >_{arith}= [R^{∗}\overline{L}_{π}].  (2.9)

We write

< , >_{class}: H^{1}(Spec(O), G) × G^{D}(F) → Pic(O)  (2.10)

for the pairing that is induced by composing < , >_{arith} with the natural map

P̂ic(G̃^{D}) → Pic(G̃^{D})  (2.11)

given by forgetting metrics. It follows immediately from the definitions that this is the same as the pairing induced by the homomorphism

φ : H^{1}(Spec(O), G) → Pic(G̃^{D}) ≃ Map(G^{D}(F), Pic(O)).  (2.12)

We refer to < , >_{class} as the ‘ideal class pairing’.

Let C(G̃^{D}) denote the kernel of the natural map (2.11), and write C(O) for the subgroup of P̂ic(O) which is similarly defined. We refer to C(G̃^{D}) and C(O) as the ‘circle groups’ of G̃^{D} and O respectively. Then φ̂ induces a homomorphism

Ker(φ) → C(G̃^{D}) ≃ Map(G^{D}(F), C(O)).  (2.13)

We write

< , >_{circ}: Ker(φ) × G^{D}(F) → C(O)  (2.14)

for the pairing induced by (2.13), and we refer to this as the ‘circle pairing’.

In Section 7 we shall show that the pairings defined above are related to certain pairings defined by Mazur and Tate (see [19]) in the case in which G is a torsion subgroup scheme of an abelian scheme.

ON ARITHMETIC CLASS INVARIANTS 9

Remark 2.4. An explicit description of the groups P̂ic(O) and C(O) may be given as follows.

Let J(F) denote the group of ideles of F, and define

J_{/S}(F) := {α ∈ J(F) | α_{v} = 1 for all places v of F lying above S }.

Write δ_{1} : F^{∗} → J_{/S}(F) for the obvious natural map, and define

δ_{2} : F^{∗} → ∏_{v∈S} R_{>0}^{∗}; a ↦ ∏_{v∈S} |a|_{v}^{−1}.

Let

δ : F^{∗} → \frac{J_{/S}(F)}{∏_{v∉S} O_{v}^{∗}} × ∏_{v∈S} R_{>0}^{∗}

be the homomorphism induced by the map a ↦ (δ_{1}(a), δ_{2}(a)). Then it is not hard to show (cf. [9], Chapter 1, Cor. 5.5, for example) that

P̂ic(O) ≃ (\frac{J_{/S}(F)}{∏_{v∉S} O_{v}^{∗}} × ∏_{v∈S} R_{>0}^{∗})/ δ(F^{∗})  (2.15)

and

C(O) ≃ \frac{∏_{v∈S} R_{>0}^{∗}}{δ_{2}(O^{∗})} .  (2.16)

It may also be shown (see e.g. no. 3.5.2 of [19], or [9]) that there is a (non-canonical) isomorphism C(O) ≃ R × (R/Z)^{|S|−1}, i.e. C(O) is isomorphic to a direct product of a real line and |S| − 1 circles. This is the motivation for our choice of terminology with regard to the circle group.

3. Kummer Theory

Our main aim in this section is to apply Kummer theory in order to study torsors over Spec(F). Throughout this section, we will assume that our base scheme Y is equal to Spec(F).

Suppose therefore that G is a group scheme over Spec(F), and let π : X → Spec(F) be a G-torsor. Write G^{D} = Spec(A), and let L_{π} denote the line bundle on G^{D} associated to π. Since Pic(G^{D}) = 0, we may choose a trivialisation ϕ_{π} : A → L_{π}, and this induces a trivialisation

ϕ_{π}^{N} : A ⟶^{∼} L_{π}^{⊗N}.

10 A. AGBOOLA AND G. PAPPAS

Now since L_{π} is associated to a G^{D}-torsor, there is a canonical trivialisation

ξ_{π} : L_{π}^{⊗N} ⟶^{∼} A  (3.1)

(cf. (2.5)). Then the isomorphism ξ_{π} ∘ ϕ_{π}^{N} : A ⟶^{∼} A is multiplication by an element a_{π} ∈ A^{∗}. It is not hard to check that changing ϕ_{π} or replacing X by an isomorphic G-torsor alters a_{π} by multiplying it by an element of A^{∗N}. Also, if π′ : X′ → Spec(F) is another G-torsor, then since there is a canonical isomorphism L_{π·π′} ≃ L_{π} ⊗ L_{π′}, it follows that we have the equality \overline{a}_{π·π′} = \overline{a}_{π} · \overline{a}_{π′} in A^{∗}/(A^{∗})^{N}. We thus have the following result.

Proposition 3.1. The map η_{F} : H^{1}(Spec(F), G) → A^{∗}/(A^{∗})^{N} given by [π] → \overline{a}_{π} is a homomorphism.

Now suppose that G^{D} is a constant group scheme over Spec(F). Then

A^{∗}/(A^{∗})^{N} ≃ Map(G^{D}(F), F^{∗}/(F^{∗})^{N}).

For each element R : Spec(F) → G^{D} in G^{D}(F), write χ_{R} : G → μ_{N} for the corresponding character of G. Then χ_{R} induces a homomorphism (which we denote by the same symbol)

χ_{R} : H^{1}(Spec(F), G) → H^{1}(Spec(F), μ_{N}) ; [π] → [π(χ_{R})]

We write

ev_{R} : A^{∗}/(A^{∗})^{N} ≃ Map(G^{D}(F), F^{∗}/(F^{∗})^{N}) → F^{∗}/(F^{∗})^{N}

for the map a ↦ a(R) given by ‘evaluation at R’. The following result shows that the homomorphism η_{F} may be described in terms of Kummer theory.

Proposition 3.2. Let the hypotheses and notation be as above. Then the following diagram is commutative:

H^{1}(Spec(F), G) ⟶^{χ_{R}} H^{1}(Spec(F), μ_{N})
η_{F} ↓ ↑^{Kummer}
A^{∗}/(A^{∗})^{N}  ⟶^{ev_{R}} F^{∗}/(F^{∗})^{N}.

(3.2)

(Here the right-hand vertical arrow is the natural isomorphism afforded by Kummer theory.)

Proof. Let χ_{R}^{D} : μ_{N}^{D} = Z/NZ → G^{D} denote the homomorphism induced by χ_{R} via Cartier duality, and write 1 : Spec(F) → Z/NZ for the F -valued point of Z/NZ corresponding to the character of μ_{N} given by the identity map Id : μ_{N} → μ_{N}. Then we have that R = χ_{R}^{D} ∘1. Now the extension of μ_{N}^{D} = Z/NZ by G_{m} corresponding to the μ_{N}-torsor π(χ_{R})

ON ARITHMETIC CLASS INVARIANTS 11

is canonically isomorphic to the pullback via χ_{R}^{D} of the extension of G^{D} by G_{m} corresponding to the G-torsor π. Hence if L_{π(χ_{R})} denotes the line bundle on μ_{N}^{D} associated to π(χ_{R}), then there is a natural isomorphism

(χ_{R}^{D})^{∗}L_{π} ≃ L_{π(χ_{R})}.

We therefore deduce that there is a canonical isomorphism R^{∗}L_{π} ≃ 1^{∗}L_{π(χ_{R})} of line bundles on Spec(F). Hence it follows from the definitions of a_{π} and a_{π(χ_{R})} that we have the equality \overline{a}_{π}(R) = \overline{a}_{π(χ_{R})}(1) in F^{∗}/(F^{∗})^{N}. Thus, to prove the proposition, it suffices to show that if τ : V → Spec(F) is any μ_{N}-torsor, then τ is represented in H^{1}(Spec(F), μ_{N}) ≃ F^{∗}/(F^{∗})^{N} by the element a_{τ}(1) ∈ F^{∗}. This follows from the standard proof of the main statement of Kummer theory.

Corollary 3.3. Suppose that G^{D} is a constant group scheme over Spec(F). Then the homomorphism η_{F} is injective.

Proof. Since G^{D} is a direct sum of cyclic constant group schemes, we may assume without loss of generality that G^{D} ≃ Z/NZ and G ≃ μ_{N}. Suppose that π is a G-torsor which is such that η_{F} (π) = 0. Then Proposition 3.2 implies that χ_{R}(π) = 0 for all R ∈ G^{D}(F). Hence π is a trivial torsor.

Corollary 3.4. Suppose that G^{D} is a constant group scheme over Spec(F). Then the image of η_{F} is equal to

Hom(G^{D}(F), F^{∗}/(F^{∗})^{N}) ⊂ Map(G^{D}(F), F^{∗}/(F^{∗})^{N}) ≃ A^{∗}/(A^{∗})^{N}.

Proof. In order to prove this result, it suffices to consider the case in which G^{D} ≃ Z/NZ and G ≃ μ_{N}. Suppose that R and R′ are elements of G^{D}(F). Then (using additive notation for the group law in G^{D}(F))

χ_{R+R′} = χ_{R} · χ_{R′},

and this implies that η_{F} (π) ∈ Hom(G^{D}(F), F^{∗}/(F^{∗})^{N}).

Since

Hom(G^{D}(F), F^{∗}/(F^{∗})^{N}) ≃ F^{∗}/(F^{∗})^{N} ≃ H^{1}(Spec(F), μ_{N}),

we see that the image of η_{F} is equal to Hom(G^{D}(F), F^{∗}/(F^{∗})^{N}) by applying Proposition 3.2 with any R which is such that χ_{R} is an isomorphism.

12 A. AGBOOLA AND G. PAPPAS

4. Cyclotomic Extensions

In this section we shall apply Proposition 3.2 to give a criterion that will enable us to detect elements lying in the kernel of φ̂.

Let G be a group scheme over Spec(O) which is such that G^{D} is generically constant. We write G_{/F} and G_{/F}^{D} for the generic fibres of G and G^{D} respectively. Set G^{D} = Spec(𝔄) and G_{/F}^{D} = Spec(A). Then

A = Spec(𝔄) ⊗_{O} F ≃ Map(G^{D}(F), F).

If G̃^{D} denotes the normalisation of G^{D}, then we have OG̃^{D} = Map(G^{D}(F), O).

Let π : X → Spec(O) be a G-torsor, and let π_{F} : X_{/F} → Spec(F) be its generic fibre. We shall view π_{F} as a G_{/F} -torsor. Suppose that μ(F) is the group of all roots of unity in F, and write L := F (μ(F)^{1/N}) for the field obtained by adjoining all Nth roots of all elements in μ(F) to F. We have the following result.

Theorem 4.1. Let the hypotheses and notation be as above, and suppose in addition that φ̂(π) = 0. Then the torsor π_{F} becomes trivial over the field L.

Proof. Write e : G̃^{D} → G^{D} for the obvious natural map. Since φ̂(π) = 0, we may choose an isometry ϕ̃_{π} : \overline{O}G̃^{D} ⟶^{∼} e^{∗}\overline{L}_{π}, and this induces an isometry

ϕ̃_{π}^{N} : \overline{O}G̃^{D} ⟶^{∼} e^{∗}\overline{L}_{π}^{⊗N}.

Next we note that the canonical isometry (2.5) of metrised line bundles on G^{D} induces an isometry

ξ̃_{π} : e^{∗}\overline{L}_{π}^{⊗N} ⟶^{∼} \overline{O}G̃^{D}.

The composition ξ̃_{π} ∘ϕ̃_{π}^{N} : \overline{O}G̃^{D} ⟶^{∼} \overline{O}G̃^{D} is multiplication by an element a_{π} ∈ O∗ ; G̃^{D}. Since this composition is an isometry, it follows that a_{π}(R) ∈ O^{∗} and |a_{π}(R)|_{v} = 1 for all R ∈ G^{D}(F) and all places v ∈ S. As S contains all infinite places of F, this implies that a_{π}(R) is a root of unity for all R ∈ G^{D}(F).

Now consider the G_{/F} -torsor π_{F}. It follows from the definition of the map η_{F} given at the begining of Section 3 that we have

η_{F} (π_{F}) = \overline{a}_{π} ∈ A^{∗}/(A^{∗})^{N}.

Hence Proposition 3.2 implies that for each R ∈ G^{D}(F), the element a_{π}(R) ∈ F^{∗} is a Kummer representative in H^{1}(Spec(F), μ_{N}) of the torsor π_{F} (χ_{R}). Therefore π_{F} (χ_{R}) becomes trivial over the field F (a_{π}(R)^{1/N}) ⊆ L. We therefore deduce that if

Res : H^{1}(Spec(F), G_{/F}^{D}) → H^{1}(Spec(L), G_{/L}^{D})

denotes the restriction homomorphism on cohomology, then η_{L}(Res(π_{F})) = 0. Hence Corollary 3.3 implies that Res(π_{F}) = 0, as claimed.

5. Arithmetic Class Invariants attached to abelian varieties

In this section we shall discuss arithmetic class invariants arising via the division of points on abelian varieties O.

Let E be an abelian scheme of dimension d over Spec(O), and write E^{D} for the dual abelian scheme. Let P denote the Poincaré line bundle on E ×_{O} E^{D}. For each O-valued point P : Spec(O) → E of E, we set

L(P) := (P ×_{O} Id)^{∗}(P).

Then L(P) is a rigidified line bundle on E^{D}, and it corresponds to the point P under the duality between E and E^{D}. For each place v in S, we may endow L(P) ⊗_{F,σ_{v}} F_{v}^{c} with a metric in the following way.

Since the G_{m}-torsor associated to P supports the structure of a biextension on E ×_{O} E^{D}, it follows that there exists a commutative extension

1 → G_{m} → G(P) → E^{D} → 1  (5.1)

which is such that the total space G(P) is equal to the complement of the zero section in L(P). Then, just as in the case of G-torsors described in Section 2, it follows from Proposition 2.1 that there exists a unique homomorphism

v_{P} : G(P)(F_{v}^{c}) → R

which is such that v_{P} (z) = log |z|_{v} for all z ∈ (F_{v})^{c∗} ⊂ G(P)(F_{v}^{c}). Hence the map

|| · ||_{v} := exp ∘v_{P} : G(P)(F_{v}^{c}) → R_{>0}

induces a metric || · ||_{v} on L(P) ⊗_{F,σ_{v}} F_{v}^{c}, and this metric is invariant under the action of Gal(F_{v}^{c}/F_{v}). We refer to || · ||_{v} as the Néron metric at v on L(P). We write \overline{L}(P) for the metrised line bundle on E^{D} which is obtained by endowing L with the Néron metric at all places v ∈ S.

14 A. AGBOOLA AND G. PAPPAS

Now let

1 → G_{m} → G_{N}(P) → E^{D}[N] → 1

denote the extension of E^{D}[N] by G_{m} that is obtained by pulling back (5.1) along the inclusion map E^{D}[N] → E^{D}. This corresponds to an E[N]-torsor π_{N}(P) : X_{N}(P) → Spec(O) via the isomorphism (2.1). Let \overline{L}_{π_{N} (P)} be the metrised line bundle on E^{D}[N] associated to π_{N}(P), as described in Section 2. Then it follows from the definitions that there is a natural isometry

\overline{L}_{π_{N} (P)} ≃ \overline{L}(P)|_{E^{D}[N]}.

Recall from the introduction that there is an E[N]-torsor [N]^{−1}(P) which is obtained by dividing the point P by N. We have the following result.

Theorem 5.1. There is an isometry

\overline{L}_{[N]^{−1}(P)} ≃ \overline{L}(P)|_{E^{D}[N]}  (5.2)

of metrised line bundles on E^{D}[N].

Proof. As \overline{L}_{[N]^{−1}(P)} and \overline{L}(P)|_{E^{D}[N]} are the metrised line bundles associated to the E[N]- torsors [N]^{−1}(P) and X_{N}(P) respectively, in order to prove the result it suffices to show that

[N]^{−1}(P) ≃ X_{N}(P).  (5.3)

Write [N]^{−1}(P)_{/F} (respectively X_{N}(P)_{/F}) for the generic fibre of [N]^{−1}(P) (respectively X_{N}(P)). Since any E[N]-torsor is determined by its generic fibre, (5.3) will follow if we show that

[N]^{−1}(P)_{/F} ≃ X_{N}(P)_{/F} .  (5.4)

This isomorphism (5.4) is explained in [1] (see especially (10) in the proof of Theorem 1. See also Proposition 3.1 of [24] for a detailed proof of this isomorphism.)

Remark 5.2. Note that Theorem 1.1 is an immediate consequence of Theorem 5.1. This stregnthens results of [1] and [22], and it shows that the arithmetic class invariant homomorphism ψ̂_{N} may be interpreted in terms of restricting the metrised line bundle \overline{L}(P) on E^{D} to the torsion subgroup scheme E^{D}[N]. The reader may consult [4] for a further discussion concerning the restriction of metrised line bundles on arithmetic varieties to horizontal subschemes.

ON ARITHMETIC CLASS INVARIANTS 15

We shall now discuss certain inverse limits. Let l be a prime number. For each positive integer n, let

p_{n} : P̂ic(E^{D}[l^{n}]) → P̂ic(E^{D}[l^{n−1}])

be the pullback homomorphism induced by the inclusion map E^{D}[l^{n−1}] → E^{D}[l^{n}]. Write

red : \frac{E(F)}{[l^{n}] · E(F)} → \frac{E(F)}{[l^{n−1}] · E(F)}

for the homomorphism given by reduction modulo l^{n−1}. The following result follows directly from Theorems 5.1 and 1.1.

Proposition 5.3. The following diagram is commutative:

E(F)/([l^{n}] · E(F)) ⟶^{ψ̂_{l^{n}}}  P̂ic(E^{D}[l^{n}])
red↓  ↓^{p_{n}}
E(F)/([l^{n−1}] · E(F)) ⟶^{ψ̂_{l^{n−1}}} P̂ic(E^{D}[l^{n−1}]).

(5.5)

Taking inverse limits using the diagram (5.5) yields a homomorphism

Ψ̂_{l} := \underleftarrow{lim} ψ̂_{l^{n}} : E(F) ⊗_{Z} Z_{l} → \underleftarrow{lim} P̂ic(E^{D}[l^{n}])).

By composing Ψ̂_{l} with the natural maps

\underleftarrow{lim} P̂ic(E^{D}[l^{n}]) → \underleftarrow{lim} Pic(E^{D}[l^{n}]),
\underleftarrow{lim} P̂ic(E^{D}[l^{n}]) → \underleftarrow{lim} P̂ic({ \atop E^{D}[l^{n}]}),

and

\underleftarrow{lim} P̂ic(E^{D}[l^{n}]) → \underleftarrow{lim} Pic({ \atop E^{D}[l^{n}]}),

respectively, we obtain homomorphisms

Ψ_{l} : E(F) ⊗_{Z} Z_{l} → \underleftarrow{lim} Pic(E^{D}[l^{n}]),  (5.6)
Φ̂_{l} : E(F) ⊗_{Z} Z_{l} → \underleftarrow{lim} P̂ic({ \atop E^{D}[l^{n}]}),  (5.7)

and

Φ_{l} : E(F) ⊗_{Z} Z_{l} → \underleftarrow{lim} Pic({ \atop E^{D}[l^{n}]}).  (5.8)

We shall now make a few remarks concerning these maps.

16 A. AGBOOLA AND G. PAPPAS

(1) If E is an elliptic curve, and l > 3, then it is shown in [30], [2], and [22] that all l-power torsion points in E(F) lie in the kernel of Ψ_{l}. If l ≤ 3, then this is no longer true in general (see [6], [11]).

(2) Suppose that E is an elliptic curve with complex multiplication, and that l is a prime of ordinary reduction. Then it is shown in [5] that (subject to certain technical hypotheses) the map Ψ_{l} is injective modulo torsion. On the other hand, the kernel of Φ_{l} is infinite in general (see [1], [5]), and it may be described in terms of the l-adic height pairing on E.

(3) It follows from Theorem 6.4 below that Φ̂_{l} and Ψ̂_{l} are injective modulo torsion for all abelian schemes E and all primes l. It seems reasonable to expect that the same is true for Ψ_{l} (but not for Φ_{l}).

6. Proof of Theorem 1.2

In this section, we shall prove Theorem 1.2.

Throughout this section (unless explicitly stated otherwise), we assume that l is a prime number satisfying the hypotheses of Theorem 1.2. We shall write E_{l^{n}} and E_{l^{n}}^{D} for the groups E[l^{n}](F^{c}) and E^{D}[l^{n}](F^{c}) respectively.

Fix compatible identifications E_{l^{n}} ≃ (Z/l^{n}Z)^{2d}, n > 0 which give

E_{l^{∞}} := \underrightarrow{lim}_{n}E_{l^{n}} ≃ (Q_{l}/Z_{l})^{2d}, T_{l}(E) := \underleftarrow{lim}_{n}E_{l^{n}} ≃ Z_{l}^{2d}.

We shall use the l-adic representation

ρ : Gal(F^{c}/F) → Aut_{Z_{l}}(T_{l}(E)) ≃ GL(2d, Z_{l}).

The composition det ·ρ is equal to ϵ^{d}, where

ϵ : Gal(F^{c}/F) → Z_{l}^{∗}

is the l-adic cyclotomic character. We shall also use the Galois representations

ρ_{n} : Gal(F^{c}/F) → Aut_{Z_{l}}(E_{l^{n}}) ≃ GL(2d, Z/l^{n}Z).

Let K_{n} denote the fixed field of Ker(ρ_{n}); this is the extension of F that is obtained by adjoining the coordinates of points in E_{l^{n}} to F. Write F_{n} := K_{n}(μ_{l^{n}}). Then it follows from the existence of the Weil pairing on E that the points of E_{l^{n}}^{D} are rational over F_{n}. We set K_{∞} := ∪_{n}K_{n} and F_{∞} := ∪_{n}F_{n}.

Our main strategy for proving Theorem 1.2 may be described as follows. Suppose that P ∈ E(F) ⊗_{Z} Z_{l} lies in the kernel of Φ̂_{l}. For each integer n > 0, choose P_{n} ∈ E(F) so that

P_{n} ⊗ 1 ≡ P mod l^{n}[E(F) ⊗_{Z} Z_{l}]

Then Theorem 4.1 implies that for every integer n > 0, the extension L_{n}(P) := F_{n}(\frac{1}{l^{n}} P_{n}) of F_{n} is obtained by adjoining to F_{n} an l^{n}th root of a root of unity in F_{n}. The field L_{n}(P) is independent of the choices of P_{n} and \frac{1}{l^{n}} P_{n}. Thus, for all n > 0, we have

F ⊆ F_{n} ⊆ L_{n}(P) ⊆ F_{2n}.  (6.1)

We shall prove the Theorem 1.2 by establishing the following result, which is itself of some independent interest.

Theorem 6.1. Suppose that P ∈ E(F) ⊗_{Z} Z_{l}. Then, for all sufficiently large n, the extension L_{n}(P)/F_{n} is not a cyclotomic extension.

We shall require the following lemma.

Lemma 6.2. K_{∞} = F_{∞}.

Proof. We first observe that plainly K_{∞} ⊂ F_{∞}.

Let β : E → E^{D} be a polarisation of E which is defined over F. The l-primary part of the kernel of β is a finite, flat, commutative group scheme H. Suppose that H is of exponent l^{N}. Then, for n >> 0, β induces an injection E[l^{n}]/H → E^{D}[l^{n}], and so we may view E[l^{n}]/H as being a subscheme of E^{D}[l^{n}].

The points of the group scheme E[l^{n}]/H are rational over the field K_{n}. Since H is of exponent l^{N}, not all points of E[l^{n}]/H are killed by l^{n−N−1}, and so it follows that there is some point of E[l^{n}]/H which is of exact order l^{n−N}.

Now composing the Weil pairing

E[l^{n−N}] × E^{D}[l^{n}] → μ_{l^{n−N}}

with the natural map E[l^{n}] → E[l^{n}]/H ⊂ E^{D}[l^{n}] gives a pairing

E[l^{n−N}] × E[l^{n}] → E[l^{n−N}] × E^{D}[l^{n}] → μ_{l^{n−N}} .

We therefore deduce that for n >> 0, we have μ_{l^{n−N}} ⊂ K_{n}. This implies that F_{∞} ⊂ K_{∞}, and so it follows that F_{∞} = K_{∞} as asserted.

18 A. AGBOOLA AND G. PAPPAS

Let μ(F_{n}) denote the group of all roots of unity in F_{n}, and write M_{n} := F_{n}(μ(F_{n})^{1/l^{n}}) for the field obtained by adjoining all l^{n}th roots of all elements in μ(F_{n}) to F_{n}. Set M_{∞} := ∪_{n}M_{n}. Then, since μ_{l^{n}} ⊂ F_{n}, it follows that M_{∞} = F_{∞} and so Lemma 6.2 implies that M_{∞} = K_{∞}. Consider the restriction homomorphism

Res : H^{1}(F, T_{l}(E)) → H^{1}(K_{∞}, T_{l}(E)) = H^{1}(M_{∞}, T_{l}(E)).

The restriction-inflation sequence implies that the kernel of this homomorphism is equal to H^{1}(Gal(K_{∞}/F), T_{l}(E)). This is a finite group (see [26], Corollary to Theorem 2).

Proposition 6.3. Let l be any prime, and suppose that P ∈ E(F)⊗_{Z} Z_{l} is of infinite order. Then Theorem 6.1 holds for the element P, and so Φ̂_{l}(P) ≠ 0.

Proof. We shall argue via contradiction. Suppose that Theorem 6.1 does not hold for P, and hence that Φ̂_{l}(P) = 0. Let [P] ∈ H^{1}(F, T_{l}(E)) denote the image of P under the natural injective map

E(F) ⊗_{Z} Z_{l} → H^{1}(F, T_{l}(E))

arising from Kummer theory on E. For each integer n > 0, let [P_{n}] denote the image of [P] in H^{1}(F, E_{l^{n}}).

We now observe that L_{n}(P) is the smallest extension of F_{n} which trivialises the image of [P_{n}] in H^{1}(F_{n}, E_{l^{n}}). Since, by our initial assumption, L_{n}(P) ⊆ F_{2n} (see (6.1)), it follows that the image of [P_{n}] in H^{1}(F_{∞}, E_{l^{n}}) is trivial for all n > 0. This in turn implies that Res([P]) = 0, which is a contradiction, since Res has finite kernel.

Thus L_{n}(P)/F_{n} is not a cyclotomic extension if n is sufficiently large, and hence Φ̂_{l}(P) ≠ 0.

The following result is an immediate corollary of the proof of Proposition 6.3.

Theorem 6.4. The map Φ̂_{l} (and therefore also Ψ̂_{l}) is injective modulo torsion for all primes l.

We now consider the case in which P ∈ E(F) ⊗_{Z} Z_{l} is a non-trivial torsion point. Since P is torsion, it follows that in fact P ∈ E(F). It is sufficient to show that if l · P = 0, then Φ̂_{l}(P) ≠ 0.

Assume therefore P ∈ E(F) with l · P = 0 and P ≠ 0. We shall again argue via contradiction. Suppose that Φ̂_{l}(P) = 0 and hence that Theorem 6.1 does not hold for the point P.

Set F′ := F (μ_{l^{∞}}). Then it follows from (6.1) that for all n > 0,

L_{n}(P) ⊂ M_{n} ⊂ F_{n}F′.

Let ρ′ and ρ′_{n} denote the restrictions of ρ and ρ_{n} respectively to the subgroup Gal(F^{c}/F′) of Gal(F^{c}/F). Write F′_{n} for the extension of F′ which is fixed by Γ′_{n} := Ker(ρ′_{n}). Consider the field

L′_{n}(P) := L_{n}(P)F′_{n}.

Our assumption that Theorem 6.1 does not hold for the point P implies that

L′_{n}(P) = F′_{n}  (6.2)

for all n > 0. This will lead to the desired contradiction.

We shall use the following facts about the Galois representation ρ (see [7]):

A. Let 𝔊 denote the algebraic group over Q_{l} which is given by the Zariski closure of Im(ρ) in GL(2d, Q_{l}). Then Im(ρ) is open with respect to the l-adic topology on 𝔊(Q_{l}).

B. The algebraic group 𝔊/Q_{l} contains the diagonal torus of homotheties G_{m}.

For simplicity of notation, we shall often write T instead of T_{l}(E) in what follows.

Let 𝔊′ denote the algebraic subgroup of 𝔊 consisting of elements with determinant 1.

Proposition 6.5. For n >> 0, we have

ρ(Γ′_{n}) = ρ(Ker(ρ′_{n})) = Ker(SL(T) → SL(T/l^{n}T)) ∩ 𝔊′(Q_{l}).

Proof. Notice that since det(ρ) = ϵ^{d}, it follows that, for all n > 0, det(ρ) is trivial on the subgroup Γ′_{n} corresponding to F′_{n}. Hence it is clear that

ρ(Γ′_{n}) ⊆ Ker(SL(T) → SL(T/l^{n}T)) ∩ 𝔊′(Q_{l}).

By (A) above, Im(ρ) is open in the l-adic topology of 𝔊(Q_{l}), and therefore Im(ρ)∩𝔊′(Q_{l}) is open in the l-adic topology of 𝔊′(Q_{l}). We have that Im(ρ′) ⊂ 𝔊′(Q_{l}). We will show that Im(ρ′) is open in the l-adic topology of 𝔊′(Q_{l}).

The subgroup Im(ρ′) of 𝔊′(Q_{l}) is equal to the subgroup of Im(ρ) ∩ 𝔊′(Q_{l}) which consists of images ρ(σ) of σ ∈ Gal(F^{c}/F) for which ϵ(σ) = 1. Since every element of Im(ρ) ∩ 𝔊′(Q_{l}) is the image ρ(σ) of some σ ∈ Gal(F^{c}/F) for which ϵ^{d}(σ) = 1, it follows that there are a finite number of cosets (parametrised by a subset of the d-th roots of unity in Z_{l}^{∗}) of Im(ρ′)

20 A. AGBOOLA AND G. PAPPAS

in Im(ρ) ∩ 𝔊′(Q_{l}). Therefore Im(ρ′) is l-adically open in Im(ρ) ∩ 𝔊′(Q_{l}), and so it is also l-adically open in 𝔊′(Q_{l}). Now since

Ker(SL(T) → SL(T/l^{n}T)) ∩ 𝔊′(Q_{l})

for n >> 0 gives a fundamental system of neighbourhoods of the identity in 𝔊′(Q_{l}), we have that

Ker(SL(T) → SL(T/l^{n}T)) ∩ 𝔊′(Q_{l}) = Ker(SL(T) → SL(T/l^{n}T)) ∩ Im(ρ′)

whenever n is sufficiently large.

The result now follows since

Ker(SL(T) → SL(T/l^{n}T)) ∩ Im(ρ′) = ρ(Ker(ρ′_{n})) = ρ(Γ′_{n}).

We now observe that (6.2) implies that the natural action of the group ρ(Γ′_{n}) on T/l^{n+1}T fixes the element \frac{1}{l^{n}} P. (Note that there is no well-defined choice of \frac{1}{l^{n}} P. However, any two choices differ by an l^{n}-torsion point, and so, since ρ(Γ′_{n}) fixes all l^{n}-torsion points, it makes sense to say that ρ(Γ′_{n}) fixes \frac{1}{l^{n}} P.) Hence, Proposition 6.5 implies that for all sufficiently large n, the natural action of the group

Ker(SL(T) → SL(T/l^{n}T)) ∩ 𝔊′(Q_{l})

on T/l^{n+1}T also fixes the element \frac{1}{l^{n}} P.

Let λ be any prime ideal of F which divides lO_{F}. Let F_{λ} denote the local completion of F at λ. We write C_{l} for the completion of the algebraic closure F_{λ}^{c} of F_{λ}, and we let ℛ denote the completion of the ring of integers of F_{λ}^{c}. Then ℛ is a flat (non-noetherian) Z_{l}-algebra which is a valuation ring with valuation v : ℛ − {0} → Q_{≥0}, say. The field C_{l} is the fraction field of ℛ. We set T_{ℛ} := T ⊗_{Z_{l}} ℛ and

Δ_{n} := Ker(SL(T_{ℛ}) → SL(T_{ℛ} /l^{n}T_{ℛ})) ∩ 𝔊′(C_{l}).

Then Δ_{n} acts on T_{ℛ} /l^{n+1}T_{ℛ} = (T/l^{n+1}T) ⊗_{Z_{l}} ℛ.

Proposition 6.6. Assume that Theorem 6.1 does not hold for the point P, and hence that (6.2) is true for all n > 0. Then the element

\frac{1}{l^{n}} P ⊗ 1 ∈ (T/l^{n+1}T) ⊗_{Z_{l}} ℛ

is fixed by Δ_{n} for all sufficiently large n.

ON ARITHMETIC CLASS INVARIANTS 21

Proof. Let 𝔤′ denote the Lie algebra of 𝔊′(Q_{l}). Then 𝔤′ is a Q_{l}-vector space which is a Q_{l}-subspace of End_{Q_{l}}(T ⊗_{Z_{l}} Q_{l}). Set

γ′ := 𝔤′ ∩ End_{Z_{l}}(T).

There is an exact sequence

0 → γ′ → End_{Z_{l}}(T) ⊕ 𝔤′ → End_{Q_{l}}(T ⊗_{Z_{l}} Q_{l})  (6.3)

of Z_{l}-modules, and γ′ is a Z_{l}-lattice in 𝔤′. Since ℛ is flat over Z_{l}, and Q_{l} ⊗_{Z_{l}} ℛ = C_{l}, (6.3) gives an exact sequence

0 → γ′ ⊗_{Z_{l}} ℛ → End_{ℛ} (T_{ℛ}) ⊕ (𝔤′ ⊗_{Q_{l}} C_{l}) → End_{C_{l}}(T ⊗_{Z_{l}} C_{l}).  (6.4)

Hence it follows that

(𝔤′ ⊗_{Q_{l}} C_{l}) ∩ End_{ℛ} (T_{ℛ}) = γ′ ⊗_{Z_{l}} ℛ.

The natural map End_{Z_{l}}(T) → End_{Z_{l}}(T/l^{n}T) induces a Lie homomorphism

τ_{n} : γ′ → End_{Z_{l}}(T/l^{n}T)

whose kernel we denote by γ′_{n}. It follows from Proposition 6.5 together with standard properties of the exponential on l-adic Lie groups (see for example [16] or [18]) that, for all n >> 0, the exponential map gives a bijection between γ′_{n} and the l-adic Lie group ρ(Γ′_{n}). We now assume that n is large enough for the exponential map to satisfy this property.

Now consider the homomorphism

τ_{n} ⊗_{Z_{l}} ℛ : γ′ ⊗_{Z_{l}} ℛ → End_{ℛ} (T_{ℛ} /l^{n}T_{ℛ}).

The kernel of τ_{n} ⊗_{Z_{l}} ℛ is equal to γ′_{n} ⊗_{Z_{l}} ℛ. The Lie algebra of 𝔊′(C_{l}) is equal to

𝔤′ ⊗_{Q_{l}} C_{l} ⊂ End_{C_{l}}(T_{C_{l}}).

The exponential is defined on elements of End_{ℛ} (T_{ℛ}) ⊂ End_{C_{l}}(T_{C_{l}}) which reduce to zero modulo lℛ, and so it is defined on Ker(τ_{n} ⊗_{Z_{l}} ℛ). The values of the exponential map on elements of Ker(τ_{n} ⊗_{Z_{l}} ℛ) belong on the one hand to 𝔊′(C_{l}) because

Ker(τ_{n} ⊗_{Z_{l}} ℛ) ⊂ 𝔤′ ⊗_{Q_{l}} C_{l},

and on the other hand to Ker(SL(T_{ℛ}) → SL(T_{ℛ} /l^{n}T_{ℛ})). Hence we obtain a map

Exp : Ker(τ_{n} ⊗_{Z_{l}} ℛ) → Δ_{n}.

22 A. AGBOOLA AND G. PAPPAS

A similar argument shows that the logarithm defines a map in the other direction which is inverse to Exp. We conclude that Exp gives a bijection between

Ker(τ_{n} ⊗_{Z_{l}} ℛ) = γ′_{n} ⊗_{Z_{l}} ℛ

and Δ_{n}. Now, by our initial assumption, ρ(Γ′_{n}) fixes \frac{1}{l^{n}} P ∈ T/l^{n+1}T and therefore, since we have chosen n to be large, γ′_{n} annihilates \frac{1}{l^{n}} P. We conclude that γ′_{n} ⊗_{Z_{l}} ℛ annihilates \frac{1}{l^{n}} P ⊗ 1, and therefore Δ_{n} fixes \frac{1}{l^{n}} P ⊗ 1, as asserted.

This completes the proof of the Proposition.

We now continue with the proof of Theorem 1.2. Consider the canonical Hodge-Tate decomposition

T ⊗_{Z_{l}} C_{l} := V_{C_{l}} = V (0) ⊕ V (1),

with each V (i) (i = 0, 1) a C_{l}-subspace of V_{C_{l}} of dimension d. By [14] (Proposition 11 on p. 406; see also 5.10 and 5.11), there are ℛ-lattices Λ(i) (i = 0, 1) in V (i), and an element α ∈ ℛ of valuation v(α) = (l − 1)^{−1} such that

α(Λ(0) ⊕ Λ(1)) ⊂ T ⊗_{Z_{l}} ℛ ⊂ Λ(0) ⊕ Λ(1).

(Using the terminology of [14], Λ(0) = t_{H′}^{∗}(O_{C_{l}}) and Λ(1) = t_{H}(T_{l}(Ω)), where H stands for the l-divisible group of the abelian scheme E over O_{F_{λ}}. The stated property of α follows from [14], Corollary to Theorem 3 (see 4.10). Note that here we are using our assumption that F/Q is unramified at l.)

Set

Λ := Λ(0) ⊕ Λ(1).  (6.5)

This grading defines a cocharacter

Υ : G_{m/ℛ} → GL(Λ)

which when basechanged to C_{l} produces the Hodge cocharacter

Υ_{C_{l}} : C_{l}^{∗} → GL(V_{C_{l}}) = GL(T ⊗_{Z_{l}} C_{l}).

It follows from the canonicity of the Hodge-Tate decomposition and Tannakian equivalence that the image of the Hodge cocharacter is contained in 𝔊(C_{l}) (cf. [27], 1.4). We therefore see that

Υ(G_{m/ℛ})) = Υ_{C_{l}}(ℛ^{∗}) ⊂ GL(Λ) ∩ 𝔊(C_{l}).

ON ARITHMETIC CLASS INVARIANTS 23

On the other hand, (B) above implies that we have diag(C_{l}^{∗}) ⊂ 𝔊(C_{l}). This implies that

diag(ℛ^{∗}) ⊂ GL(Λ) ∩ 𝔊(C_{l}).

Hence there is a homomorphism

ℛ^{∗} × ℛ^{∗} → GL(Λ) ∩ 𝔊(C_{l})

defined by (r_{1}, r_{2}) ↦ Υ(r_{1}) · diag(r_{2}). Since det(Υ(r_{1}) · diag(r_{2})) = r_{1}^{d}r_{2}^{2d}, it follows that there exists an injective homomorphism

κ : ℛ^{∗} → SL(Λ) ∩ 𝔊′(C_{l})

which is defined by r ↦ Υ(r)^{2} · diag(r^{−1}).

Let ℛ_{l^{n}α}^{∗} denote the subgroup of ℛ^{∗} consisting of those elements r which satisfy r ≡ 1 mod αl^{n}ℛ.

Lemma 6.7. We have that

κ(ℛ_{l^{n}α}^{∗}) ⊂ Δ_{n} = Ker(SL(T_{ℛ}) → SL(T_{ℛ} /l^{n}T_{ℛ})) ∩ 𝔊′(C_{l}).  (6.6)

Proof. Write r = 1+αl^{n}r′ and r^{−1} = 1+αl^{n}r′′, with r′, r′′ ∈ ℛ. It is easy to see that v(r′) = v(r′′). Now choose a basis e_{i}(0), e_{i}(1) (i = 1, ..., g) of Λ which respects the decomposition (6.5). With respect to this basis, we can write

κ(r) = I + αl^{n}M(r),  (6.7)

where M(r) = diag(r′′, …, r′′, r′, …, r′) is a diagonal matrix with coefficients in ℛ. We have that

αl^{n}T_{ℛ} ⊂ αl^{n}Λ = l^{n}(αΛ) ⊂ l^{n}T_{ℛ} .

Therefore κ(r) preserves T_{ℛ} and is congruent to the identity modulo l^{n}T_{ℛ}. This shows that κ(r) ∈ Δ_{n}, and so (6.6) holds as asserted.

Proposition 6.8. Suppose that r ∈ ℛ_{l^{n}α}^{∗} is such that

κ(r)(\frac{1}{l^{n}} P ⊗ 1) = \frac{1}{l^{n}} P ⊗ 1  (6.8)

in T_{ℛ} /l^{n+1}T_{ℛ}. Then

r ≡ 1 mod l^{n+1}α^{−1}ℛ.  (6.9)

24 A. AGBOOLA AND G. PAPPAS

Proof. Lift \frac{1}{l^{n}} P ∈ T/l^{n+1}T to an element P̃_{n} of T. Then (6.7) and (6.8) imply that

αl^{n}M(r)(P̃_{n} ⊗ 1) ∈ l^{n+1}T_{ℛ} ,

which gives

M(r)(P̃_{n} ⊗ 1) ∈ lα^{−1}T_{ℛ} .

Since P ≠ 0, we have that \frac{1}{l^{n}} P ⊗ 1 is an element of an ℛ/l^{n+1}ℛ-basis of T_{ℛ} /l^{n+1}T_{ℛ}, and P̃_{n} ⊗ 1 is an element of an ℛ-basis of T_{ℛ}. Hence, since

αΛ ⊂ T_{ℛ} ⊂ Λ,

it follows that P̃_{n} ⊗ 1 is an element of an ℛ-basis of Λ multiplied by an element β ∈ ℛ with v(α) ≥ v(β) ≥ 0. Therefore, with respect to the basis e_{i}(0), e_{i}(1), we may write

P̃_{n} ⊗ 1 = β · (x_{1}, …, x_{d}, x_{d+1}, …, x_{2d}),

where at least one of the coordinates x_{i} is a unit (i.e. satisfies v(x_{i}) = 0). From

diag(r′′, …, r′′, r′, …, r′)(P̃_{n} ⊗ 1) ∈ lα^{−1}T_{ℛ} ⊂ lα^{−1}Λ,

we now obtain that

(r′′x_{1}, …, r′′x_{d}, r′x_{d+1}, …, r′x_{2d}) ∈ lα^{−1}β^{−1}Λ ∈ lα^{−2}Λ.

As v(r′) = v(r′′), we can write r′′ = r′u, where u is a unit in ℛ. Setting x′_{i} = x_{i}u for i = 1, …, d, we obtain

(x′_{i}, …, x′_{d}, x_{d+1}, …, x_{2d}) ∈ lα^{−2}r′^{−1}Λ.

Since at least one of the x_{i} or x′_{i} is a unit, we conclude that v(lα^{−2}r′^{−1}) ≤ 0, and so

v(r′) = v(r′′) ≥ v(lα^{−2}).

Hence it follows that r ≡ 1 mod l^{n+1}α^{−1}ℛ as claimed.

Lemma 6.7 and Proposition 6.8 imply that if r is any element of ℛ_{l^{n}α}^{∗} (i.e. r ≡ 1 mod l^{n}αℛ) which is not congruent to 1 modulo l^{n+1}α^{−1}ℛ, then κ(r) is a non-trivial element of Δ_{n} which does not fix the element \frac{1}{l^{n}} P ⊗ 1 ∈ T_{ℛ} /l^{n+1}T_{ℛ}. Since v(α) = (l − 1)^{−1} and l > 3, such elements r do exist. This contradicts Proposition 6.6. We therefore deduce that in fact Theorem 6.1 holds for the point P. Hence Φ̂_{l}(P) ≠ 0.

This completes the proof of Theorem 1.2.

ON ARITHMETIC CLASS INVARIANTS 25

7. Class Invariants and Mazur-Tate Pairings

In this section we shall explain the relationship between the pairings constructed at the end of Section 2 and a refinement of the canonical height pairing on abelian varieties constructed by Mazur and Tate in [19]. A good reference for background concerning this material is [9] (see also [10]).

We retain the notation established at the begining of Section 5. Throughout this section we shall assume that S = S_{∞}, and so O is the ring of integers of F. If X is any scheme over Spec(O), then we shall write X_{/F} for the generic fibre of X.

Write Div^{0}(E_{/F}^{D}) for the group of divisors on E_{/F}^{D} that are rational over F and algebraically equivalent to zero. Let Z_{0}(E_{/F}^{D}) denote the group of zero-cycles on E_{/F}^{D} of degree zero which are of the form

Z = ∑_{i} n_{i}(Q_{i})

with each Q_{i} rational over F. If \overline{L} is any metrised line bundle on E^{D}, then the metrised line bundle Z^{∗}\overline{L} on Spec(O) is defined by

Z^{∗}\overline{L} := ⨂_{i} Q_{i}^{∗}\overline{L}^{⊗n_{i}}.

(Here we have identified each point Q_{i} on E_{/F}^{D} with its Zariski closure on E^{D}.)

Suppose that Z ∈ Z_{0}(E_{/F}^{D}) and D_{1} ∈ Div^{0}(E_{/F}^{D}) have disjoint supports, and let v be any place of F. Then the Néron symbol < Z, D_{1} >_{v} may be described as follows (see [17], Chapter 11, Theorem 6.2). Let P : Spec(F) → E_{/F} be the point on E_{/F} corresponding to D_{1} under the duality between E and E^{D}. We may identify P with the corresponding Spec(O)-valued point of E. Choose any rational section

s_{D_{1}} : E^{D} → G(P)

of (5.1) such that the generic fibre of the divisor of s_{D_{1}} is equal to D_{1}. Then s_{D_{1}} induces a map (which we denote by the same symbol)

s_{D_{1}} : E^{D}(F_{v}^{c}) → G(P)(F_{v}^{c}).

The Néron symbol < Z, D_{1} >_{v} is then equal to

< Z, D_{1} >_{v}= −v_{P} ∘ s_{D_{1}}(Z),  (7.1)

where we extend s_{D_{1}} to Z_{0}(E_{/F}^{D}) via linearity. It may be shown that this is independent of the choice of s_{D_{1}}.

If v is a non-archimedean place of F and ϖ_{v} is a local uniformiser of F at v, then it may be shown that

< Z, D_{1} >_{v}= i_{v}(Z, D_{1}) log |ϖ_{v}|_{v},  (7.2)

where i_{v}(Z, D_{1}) denotes the intersection multiplicity of D_{1} and Z at v (see [17], Chapter 11, §5).

For each place v of F, write

λ_{v}(Z, D_{1}) = \begin{cases}ϖ_{v}^{i_{v}(Z,D_{1})}, & if v ∤ ∞; \\ exp(< Z, D_{1} >_{v}), & if v | ∞. \end{cases}  (7.3)

Then we may view ∏_{v} λ_{v}(Z, D_{1}) as being an element of J_{/S}(F)×∏_{v∈S}(F_{v}^{c})^{∗}. Hence, via the isomorphism (2.15), we obtain an element [∏_{v} λ_{v}(Z, D_{1})] ∈ P̂ic(O). By using the section s_{D_{1}} to compute Z^{∗}\overline{L}(P), we see from (7.2), (7.3) and the definition of the Néron metrics on \overline{L}(P) (see Section 5) that we have

[Z^{∗}\overline{L}(P)] = [∏_{v} λ_{v}(Z, D_{1})]  (7.4)

in P̂ic(O).

We shall now recall the definition of the Mazur-Tate pairing

b_{MT} : E(F) × E^{D}(F) → P̂ic(O)  (7.5)

(cf. [9], [10], or [19]).

Suppose that P ∈ E(F) and Q ∈ E^{D}(F). It may be shown that we may choose D_{P} ∈ Div^{0}(E_{/F}^{D}) and Z_{Q} ∈ Z_{0}(E_{/F}^{D}) satisfying the following properties:

(1) The divisor D_{P} corresponds to the point P under the duality between E and E^{D};

(2) If Z_{Q} = ∑_{i} n_{i}(Q_{i}), then ∑_{i}[n_{i}]Q_{i} = Q;

(3) D_{P} and Z_{Q} have disjoint supports.

Then we define

b_{MT}(P, Q) = [∏_{v} λ_{v}(Z_{Q}, D_{P})] ∈ P̂ic(O).

It may be shown that b_{MT}(P, Q) is independent of all choices made in its definition.

ON ARITHMETIC CLASS INVARIANTS 27

It will be helpful to express the definition of the pairing b_{MT} in terms of metrised line bundles as follows. Let Z_{Q} = (Q) − (\underline{O}). We may then choose D_{P} to satisfy conditions (1) and (3) above. As \underline{O}^{∗}\overline{L}(P) is the trivial line bundle, it follows from (7.4) that we have

b_{MT}(P, Q) = [Q^{∗}\overline{L}(P)][\underline{O}^{∗}\overline{L}(P)]^{−1} = [Q^{∗}\overline{L}(P)].  (7.6)

Now suppose that N is a positive integer, and that E^{D}[N] is generically constant. Write E_{N}^{D} = E^{D}[N](F). Then we have a pairing

< , >_{arith}: H^{1}(Spec(O), E[N]) × E_{N}^{D} → P̂ic(O)

as defined in (2.7). The following result shows that the pairing

b_{MT} : E(F) × E_{N}^{D} → P̂ic(O)

induced by (7.5) has an interpretation in terms of the Galois module structure of E[N]- torsors.

Theorem 7.1. Suppose that P ∈ E(F) and R ∈ E_{N}^{D}. Then

b_{MT}(P, R) =< [N]^{−1}(P), R >_{arith} .

Proof. This just follows from unwinding the definitions, using (7.6), Theorem 5.1, and (2.9). We have

b_{MT}(P, R) = [R^{∗}\overline{L}(P)] = [R^{∗}\overline{L}([N]^{−1}(P))] =< [N]^{−1}(P), R >_{arith} .

Mazur and Tate have used (7.5) to construct two further pairings that are of interest. These may be described as follows.

By composing b_{MT} with the natural surjection P̂ic(O) → Pic(O), we obtain a pairing

b_{class} : E(F) × E^{D}(F) → Pic(O).  (7.7)

The pairing b_{class} is called the Mazur-Tate ideal class pairing. It has been extensively studied by G. Call (see [9], [10]).

To define the second pairing, we set

E(F) ×_{∘} E^{D}(F) := {(P, Q) ∈ E(F) × E^{D}(F) | b_{class}(P, Q) = 0}.

Then if (P, Q) ∈ E(F) ×_{∘} E^{D}(F), we have that b_{MT}(P, Q) ∈ C(O). We write

b_{circ} : E(F) ×_{∘} E^{D}(F) → C(O)

28 A. AGBOOLA AND G. PAPPAS

for the restriction of b_{MT} to E(F) ×_{∘} E^{D}(F), and we refer to b_{circ} as the Mazur-Tate circle pairing.

Set

E(F) ×_{∘} E_{N}^{D} := (E(F) × E_{N}^{D}) ∩ (E(F) ×_{∘} E^{D}(F)).

The following result is an immediate corollary of Theorem 7.1. It gives a new interpretation of the Mazur-Tate pairings in terms of the Galois structure of E[N]-torsors.

Corollary 7.2. Suppose that R ∈ E_{N}^{D} and P ∈ E(F). Then

b_{class}(P, R) =< [N]^{−1}(P), R >_{class} .

If φ([N]^{−1}(P)) = 0, then (P, R) ∈ E(F) ×_{∘} E_{N}^{D} for all R ∈ E_{N}^{D}, and we have

b_{circ}(P, R) =< [N]^{−1}(P), R >_{circ} .

ON ARITHMETIC CLASS INVARIANTS 29

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(A. Agboola) Department of Mathematics, Harvard University, Cambridge, MA 02138.

Permanent Address: Department of Mathematics, University of California, Santa Barbara, CA 93106.

  E-mail address: agboola@math.ucsb.edu

(G. Pappas) Department of Mathematics, Michigan State University, East Lansing, MI 48824.

  E-mail address: pappas@math.msu.edu