ON P -ADIC HEIGHT PAIRINGS AND LOCALLY FREE CLASSGROUPS OF HOPF ORDERS

A. Agboola Version of April 30th, 1996

Let E be an elliptic curve with complex multiplication by the ring of integers 𝔒 of an imaginary quadratic field K. The purpose of this paper is to describe certain connections between the arithmetic of E on the one hand and the Galois module structure of certain arithmetic principal homogeneous spaces arising from E on the other. The present paper should be regarded as a complement to [AT]: we assume that the reader is equipped with a copy of the latter paper and that he is not averse to referring to it from time to time.

An outline of the contents of this paper is as follows. After setting up a certain amount of notation in §0, we give an account of the algebraic p-adic height pairing on E in §1. The pairing that we describe was first introduced by Bernadette Perrin-Riou (see [PR1], [PR2]). In §2 we describe the class invariant homomorphism. This homomorphism measures information regarding the module structure of certain principal homogeneous spaces of torsion sub-group schemes of E. In the next two sections, we describe what we call the modified classgroup of the Hopf algebra representing a certain torsion subgroup scheme of E, and we explain precisely how this group is related to the height pairing on E (see Theorem 4.2). This gives a

Typeset by AMS-TEX 2 A. AGBOOLA

The author was partially supported by an NSF Postdoctoral Research Fellowship while the research described in this paper was carried out.

very direct and explicit link between the Galois module structure of the principal homogeneous spaces that we consider and the infinite descent on E described by Perrin-Riou in [PR1], [PR2]. Theorem 4.2 of the present paper therefore goes some way towards addressing the point raised in the introduction of [AT] concerning the precise nature of the connection between the algebraic p-adic height pairing and the class invariant homomorphism in the present setting. It seems quite likely to us that a similar relationship exists in the more general context of the p-adic height pairings described in [PR3]. It would be of some interest to obtain a better understanding of the relationship between class invariants and p-adic height pairings in general. (See [A3] for some further remarks on this.)

The final two sections of this paper are logically independent of the previous sections, although they involve a similar circle of ideas. Here we develop the bare beginings of an Iwasawa theory of classgroups of Hopf orders, using techniques of Kervaire-Murthy and Ullom (cf. [KM], [U1], [U2]). In particular, we prove an elliptic analogue of a result of Ullom (see Theorem 6.5).

§0. Notation.

We begin by fixing a certain amount of notation that will be in force for the rest of this paper. Let F/K be a finite extension over which E acquires everywhere good reduction. Set Δ = Gal(F/K), and assume that all endomorphisms of E are defined over F. If α ∈ 𝔒 we shall sometimes write [α] for the corresponding endomorphism of E. For any field L, we write L^{c} for an algebraic closure of L, and we set Ω_{L} = Gal(L^{c}/L).

Let p be an odd rational prime which splits in 𝔒 with p𝔒 = 𝔭𝔭^{∗}. Choose π ∈ 𝔭 with 𝔭^{h} = π𝔒 for some h ≥ 1, and write π^{∗} for the complex conjugate of π. Set

ON p-ADIC HEIGHT PAIRINGS 3

q = ππ^{∗}.

Let E_{π^{n}} (resp. E_{π^{∗n}}) denote the subgroup of elements of E(ℚ^{c}) which are killed by π^{n} (resp. π^{∗n}). (We shall also often write G_{n} (resp. G_{n}^{∗}) instead of E_{π^{n}} (resp. E_{π^{∗n}}).) Let

w_{n} : E_{π^{n}} × E_{π^{∗n}} ⟶ μ_{q^{n}}  (0.1)

denote the Weil pairing on E. For each R ∈ E_{π^{n}}, we define a character χ_{R}^{(n)} ∈ Ĝ_{n} by

χ_{R}^{(n)} = w_{n}(g, R)  ∀g ∈ E_{π^{∗n}}  (0.2)

This identifies Ĝ_{n} with E_{π^{∗n}}. If ω ∈ Ω_{F} and χ ∈ Ĝ_{n}, then ω acts on χ via χ^{ω}(g) = χ(g^{ω^{−1}})^{ω}. The identification (2) preserves this action since the Weil pairing is Ω_{F} -equivariant.

For each R ∈ E_{π^{n}} (resp. R^{∗} ∈ E_{π^{∗n}}), we let F [R] (resp. F [R^{∗}]) denote the field obtained by adjoining the coordinates of R (resp. R^{∗}) to F. We write F_{n} = ∪_{R∈E_{π^{n}}} F [R] (resp. F_{n}^{∗} = ∪_{R∈E_{π^{∗n}}} F [R]). Set F_{∞} = ∪_{n}F_{n} (resp. F_{∞}^{∗} = ∪_{n}F_{n}^{∗}), and write Γ_{1} = Gal(F_{∞}/F). Let N_{∞}/F be the unique ℤ_{p} extension contained in F_{∞}/F, and set Γ = Gal(N_{∞}/F).

§1. The algebraic p-adic height pairing.

In this section we shall describe the algebraic p-adic height pairing on E/F. The reader may refer to §3 of [PR1] or to Chapter IV of [PR2] for full details of the results we use.

We begin by recalling various elementary facts about Selmer groups. If 𝔮 is a prime of F, we write k_{𝔮} for the residue field of F at 𝔮, and we let Ẽ(k_{𝔮}) denote the reduction of F at 𝔮. Write E_{1,𝔮}(F) for the kernel of reduction of E(F) at 𝔮, and

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define E_{1,𝔭}(F) = E_{1}(F) via exactness of the sequence

0 ⟶ E_{1}(F) ⟶ E(F) ⟶ ∏_{𝔮|𝔭} Ẽ(k_{𝔮}).  (1.1)

Let L be any extension of K over which E is defined. The Selmer group S(L)^{π^{∗n}} is defined to be the kernel of the natural homomorphism

H^{1}(Ω_{L}, E_{π^{∗n}}) ⟶ ∏_{𝔮} H^{1}(Ω_{L_{𝔮}}, E).  (1.2)

(Here L_{𝔮} denotes the local completion of L at the prime 𝔮 of L.) The Selmer group S(L)^{π^{n}} is defined similarly. We set S(L) = \underrightarrow{lim} S(L)^{π^{n}}, and we write Y (L) for the Pontryagin dual of S(L).

Define Σ(F)^{π^{∗n}} to be the subgroup of S(F)^{π^{∗n}} which makes the sequence

0 ⟶ Σ(F)^{π^{∗n}} ⟶ S(F)^{π^{∗n}} ⟶ ∏_{𝔮|𝔭} H^{1}(Ω_{F_{𝔮}}, E_{π^{∗n}})

exact. Set Σ(F) = \underleftarrow{lim} Σ(F)^{π^{∗n}}. Then there is a natural injection

E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} → Σ(F)

afforded by Kummer theory on E.

Let J(F_{n}) denote the group of ideles of F_{n}, and write U_{n}^{(𝔭)} for the subgroup of J(F_{n}) consisting of ideles which are equal to 1 at all places above 𝔭 and which are units elsewhere. Set C_{n} = J(F_{n})/U_{n}^{(𝔭)}F_{n}^{x}, and let W_{n} = ∏_{𝔮|𝔭} μ_{q^{n}}(F_{n,𝔮}).

The first step in the construction of the p-adic height pairing is the following result (see Proposition 3.13 of [PR1]). Proposition 1.1. There is a natural exact sequence

Hom(E_{π^{n}}, W_{n})^{Ω_{F}} ⟶ Hom(E_{π^{n}}, C_{n}(p))^{Ω_{F}} ⟶^{η_{n}} Σ(F)^{π^{∗n}} ⟶ 0.

ON p-ADIC HEIGHT PAIRINGS 5

(Here C_{n}(p) denotes the p-primary part of C_{n}.)

We refer the reader to [PR1] for a proof of this result. We shall give the definition of the homomorphism η_{n} in a moment. Before doing so, however, let us first explain how Proposition 1.1 is used to construct the p-adic height pairing.

We first observe that Proposition 1.1 yields an isomorphism

η_{n}^{−1} : Σ(F)^{π^{∗n}} ⟶ \frac{Hom(E_{π^{n}}, C_{n}(p))^{Ω_{F}}}{Hom(E_{π^{n}}, W_{n})^{Ω_{F}}} .

Define

Ξ_{n} : Σ(F)^{π^{∗n}} ⟶ \frac{Hom(E_{π^{n}}, C_{n}(p))^{Ω_{F}}}{Hom(E_{π^{n}}, W_{n})^{Ω_{F}}}

by Ξ_{n}(s)(R) = η_{n}^{−1}(π^{∗−n}R). For each n, let X_{n}/F_{n} be the maximal abelian pro-p extension of F_{n} which is unramified away from primes dividing 𝔭, and set X_{n} = Gal(X_{n}/F_{n}), X_{∞} = \underrightarrow{lim} X_{n}. The global Artin map yields a surjection

(−, X_{n}/F_{n}) : C_{n}(p) ⟶ X_{n},

and this induces a homomorphism

Ξ′_{n} : Σ(F)^{(π^{∗n})} ⟶ Hom(E_{π^{n}}, X_{n})^{Γ_{1}}  (1.3)

which is defined by Ξ′_{n} = (−, X_{n}/F_{n}) ∘ Ξ_{n}(s).

It is shown in §3.2 of [PR1] that we may take inverse limits of (1.3) to obtain a homorphism

Φ_{F} : Σ(F) ⟶ Hom(T_{π}, X_{∞})^{Γ_{1}}.

(Here T_{π} denotes the π-adic Tate module of E.) Φ_{F} is an isomorphism if the weak 𝔭-adic Leopoldt conjecture holds for F.

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It is an immediate consequence of a theorem of Coates (see Theorem 12 of [C]) that there is an isomorphism

f : Hom(T_{π}, X_{∞})^{Γ_{1}} ⟶ Y (F_{∞})^{Γ_{1}},

and it may easily be shown via Pontryagin duality (see [PR1] p. 378-379) that there is a natural homomorphism

α_{F} : Y (F_{∞})^{Γ_{1}} ⟶ Hom(E(F) ⊗ 𝔒_{𝔭}, 𝔒_{𝔭}).

Hence we obtain a pairing

{, } : E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} × E(F) ⊗_{𝔒} 𝔒_{𝔭} ⟶ 𝔒_{𝔭}

given by {x, y} = (α_{F} ∘f ∘Φ_{F} (x))(y); this extends to a pairing (which we shall also denote by {, })

{, } : E_{(}F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} × E(F) ⊗_{𝔒} 𝔒_{𝔭} ⟶ K_{𝔭}.

This pairing {, } is the algebraic p-adic height pairing on E.

We shall now give a description of the homomorphism η_{n}.

Suppose that f ∈ Hom(E_{π^{n}}, C_{n}(p))^{Ω_{F}}. Let f̄ ∈ Map(E_{π^{n}}, J(F_{n})) be a representative of f. Write d_{n} : U_{n}^{(𝔭)}F_{n}^{x} → F_{n}^{x} for the natural projection map.

Consider the sequence of homomorphisms

Hom(E_{π^{n}}, C_{n}(p))^{Ω_{F}} ⟶ Ext^{1}(E_{π^{n}}, U_{n}^{(𝔭)}F_{n}^{x})^{Ω_{F}} ⟶^{d_{n}} Ext^{1}(E_{π^{n}}, F_{n}^{x})^{Ω_{F}} . (1.4)

Let f_{1} ∈ Ext^{1}(E_{π^{n}}, F_{n}^{x})^{Ω_{F}} denote the image of f under this sequence of homomorphisms. It follows via standard theory (see e.g. [R] p.211) that f_{1} is represented by the map

(R, S) ⟼ d_{n}(\frac{f̄(R + S)}{f̄(R)f̄(S)}), R, S ∈ E_{π^{n}}.

ON p-ADIC HEIGHT PAIRINGS 7

Next, consider the exact sequence

0 ⟶ F_{n}^{x} ⟶ F̄^{x} ⟶ F̄^{x}/F_{n}^{x} ⟶ 0;

this gives the Ω_{F} -equivariant exact sequence

Hom(E_{π^{n}}, F̄^{x}) ⟶ Hom(E_{π^{n}}, F̄^{x}/F_{n}^{x}) ⟶ Ext^{1}(E_{π^{n}}, F_{n}^{x}) ⟶ Ext^{1}(E_{π^{n}}, F̄_{n}^{x}).

Now Ext^{1}(E_{π^{n}}, F̄_{n}^{x}) = 0 since F̄^{x} is divisible. Hence there exists β ∈ Hom(E_{π^{n}}, F̄^{x}/F_{n}^{x}) which maps onto f_{1} ∈ Ext^{1}(E_{π^{n}}, F_{n}^{x}). Any two different choices of β differ by an element of Hom(E_{π^{n}}, F̄^{x}) = Hom(E_{π^{n}}, μ_{q^{n}}) ≃ E_{π^{∗n}}. Since f_{1} is fixed by Ω_{F}, it follows that for each σ ∈ Ω_{F}, we have β^{σ} = h_{σ}β, where h_{σ} ∈ Hom(E_{π^{n}}, μ_{q^{n}}) ≃ E_{π^{∗n}}. The element η_{n}(f) ∈ H^{1}(Ω_{F} , E_{π^{n}}) is defined to be the cohomology class represented by the cocycle {σ ↦ h_{σ}}. It may be easily checked from the description that η_{n}(f) is trivial at all places 𝔮 with 𝔮|𝔭. Hence η_{n}(f) ∈ Σ(F)^{(π^{∗n})}, as claimed.

We conclude this section by remarking that it is the homomorphism η_{n} that lies at the heart of the construction of the p-adic height pairing. In §4, we shall relate η_{n} to the Galois module structure of principal homogeneous spaces arising from points on E.

§2. The class invariant homomorphism.

The purpose of this section is to recall various facts that we shall require concerning the class invariant homomorphism. For full details, we refer the reader to [A1], [AT], [BT] and [T1].

Let 𝔅_{n} denote the 𝔒_{F} -Hopf algebra which represents the 𝔒_{F} -group scheme of [π^{∗n}]-torsion on E, and let 𝔄_{n} denote the Cartier dual of 𝔅_{n}. Then 𝔅_{n} is an 𝔒_{F} -order in the algebra B_{n} = Map_{Ω_{F}} (G_{n}, ℚ^{c}), and 𝔄_{n} is an order in the algebra

8 A. AGBOOLA

A_{n} = (F^{c}G_{n})^{Ω_{F}} (where Ω_{F} acts upon both G_{n} and F^{c}). Let E_{π^{n}}\Ω_{F} denote a set of representatives of Ω_{F} orbits of E_{π^{n}}. Then the Wedderburn decomposition of A_{n} is given by

A_{n} = ∏_{R∈E_{π^{n}} \Ω_{F}} F [R]

(see e.g. [A1] lemma 3.3). It is shown in [T1] that

A_{n} = { ∑_{g∈G_{n}} f(g)g^{−1}|f ∈ B_{n}}

and

𝔄_{n} = {(π^{∗})^{−n} ∑_{g∈G_{n}} f(g)g^{−1}|f ∈ 𝔅_{n}}.

Let E/𝔒_{F} denote the Néron model of E/F. The endomorphism π^{∗n} of E yields the exact Kummer sequence of commutative group schemes

0 ⟶ E_{π^{∗n}} ⟶ E ⟶^{π^{∗n}} E ⟶ 0.

This in turn yields the following exact sequence of flat cohomology:

0 ⟶ E_{π^{∗n}} ⟶ E(𝔒_{F}) ⟶^{π^{∗n}} E(𝔒_{F}) ⟶ H^{1}(𝔒_{F} , E_{π^{∗n}}).

Since E(F) ≃ E(𝔒_{F}) (via the universal property of the Néron model), we obtain a homomorphism

\frac{E(F)}{π^{∗n}E(F)} ≃ \frac{E(𝔒_{F})}{π^{∗n}E(𝔒_{F})} ⟶ H^{1}(𝔒_{F} , E_{π^{∗n}}).

Thus each point Q ∈ E(F) yields an element of H^{1}(𝔒_{F} , E_{π^{∗n}}); this element is represented by an algebra ℭ_{Q}(n) which is a principal homogeneous space of 𝔅_{n}. It may be shown that ℭ_{Q}(n) is a locally free 𝔄_{n}-module, and so determines an element (ℭ_{Q}(n)) in the locally free classgroup Cl(𝔄_{n}) of 𝔄_{n}. Hence we obtain a map

ψ_{n} : E(F) ⟶ Cl(𝔄_{n})

ON p-ADIC HEIGHT PAIRINGS 9

defined by ψ_{n}(Q) = (ℭ_{Q}(n)). It is shown in [T1] that if E/F has everywhere good reduction, then ψ_{n} is a homomorphism. (For a discussion of ψ_{n} in terms of the geometry of E, we refer the reader to [A2].)

Let us now make a remark on Galois action. Write C_{Q}(n) = ℭ_{Q}(n) ⊗_{𝔒_{F}} F; then C_{Q}(n) is a principal homogeneous space of B_{n}, and so determines an element s_{Q}(n) ∈ H^{1}(F, E_{π^{∗n}}). In general, C_{Q}(n) is not stable under the action of G_{n}. However, C_{Q}(n) ⊗ F_{n}/F is a Galois algebra with Galois group isomorphic to Gal(F_{n}/F) ⋉ G_{n}, where for g ∈ G_{n} and γ ∈ Gal(F_{n}/F), we have

γ^{−1}gγ = γ(g)

(i.e. Galois action of γ on G_{n}). If c ∈ C_{Q}(n) ⊗ F^{c} and σ ∈ Ω_{F}, then it follows from Galois descent (cf. [BT] pages 181-183) that

c^{σ} = c^{s_{Q}(n)(σ)}.

We shall write PH(𝔅_{n}) (resp. PH(B_{n})) for the group of principal homogeneous spaces of 𝔅_{n} (resp. B_{n}), and we let PH_{𝔭}(𝔅_{n}) (resp. PH_{𝔭}(B_{n})) denote the group of principal homogeneous spaces of 𝔅_{n} (resp. B_{n}) that are locally trivial at all places dividing 𝔭. There are natural injections PH(𝔅_{n}) ↪ PH(B_{n}) and PH_{𝔭}(𝔅_{n}) ↪ PH_{𝔭}(B_{n}).

We now turn our attention to locally free classgroups. Let J(A_{n}) denote the group of finite ideles of A_{n}. The theory of classgroups (see e.g. Chapter VI of [CR]) furnishes us with isomorphisms

Cl(𝔄_{n}) ≃ \frac{J(A_{n})}{(∏_{𝔮} 𝔄_{n,𝔮}^{x}).A_{n}^{x}} ≃ \frac{Map(E_{π^{n}}, J(ℚ^{c}))^{Ω_{F}}}{∏_{𝔮}(Det(𝔄_{n,𝔮}^{x})). Map(E_{π^{n}}, (F^{c})^{x})^{Ω_{F}}} .  (2.1)

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This notation may be explained as follows. Suppose that u ∈ J(A_{n}). Then u determines a map Det(u) ∈ Map(E_{π^{n}}, J(ℚ^{c}))^{Ω_{F}} which is defined by

Det(u)(R)_{𝔮} = χ_{R}^{(n)}(u_{𝔮}).

The second isomorphism in (2.1) is induced by the map u ↦ Det(u) from J(A_{n}) to Map(E_{π^{n}}, J(ℚ^{c}))^{Ω_{F}}.

We shall now describe the construction of an idele in J(A_{n}) representing the class (ℭ_{Q}(n)) ∈ Cl(𝔄_{n}). Define the resolvend map

r : C_{Q}(n) ⟶ (C_{Q}(n) ⊗ F_{n})[G_{n}]

by

r(c) = ∑_{g∈G_{n}} c^{g}g^{−1}.

Then r is a homomorphism of A_{n}-modules (but not of algebras, since it does not preserve multiplication). It is a standard result in Galois theory that r(c) is invertible if and only if C_{Q}(n) = c.A_{n}.

For each prime 𝔮 of F, choose a local generator c_{𝔮} ∈ ℭ_{Q}(n)_{𝔮} such that ℭ_{Q}(n)_{𝔮} = c_{𝔮}.𝔄_{n,𝔮}, and let c ∈ C_{Q}(n) satisfy C_{Q}(n) = c.A_{n}. Then we have that r(c) = r(c_{𝔮}).u_{𝔮}, with u_{𝔮} ∈ A_{n,𝔮}^{x}. The idele (u_{𝔮}) ∈ J(A_{n}) represents the class (ℭ_{Q}(n)) ∈ Cl(𝔄_{n}).

§2. The modified class invariant homomorphism.

In this section we shall describe a map that we call the modified class invariant homomorphism. It is this map that gives the link between the module structure of principal homogeneous spaces and the algebraic p-adic height pairing.

Let J_{𝔭}(𝔄_{n}) denote the subgroup of ∏_{𝔮} Det(𝔄_{n,𝔮}^{x}) consisting of those elements that are equal to 1 at all places dividing 𝔭. If 𝔮 is a prime of F, we write G_{n}(F_{𝔮}) for

ON p-ADIC HEIGHT PAIRINGS 11

the subgroup of elements of G_{n} that are rational over F_{𝔮}. The modified classgroup Cl′(𝔄_{n}) of 𝔄_{n} is defined as follows:

Cl′(𝔄_{n}) = \frac{J(A_{n})}{J_{𝔭}(𝔄_{n}).(∏_{𝔮|𝔭} G_{n}(F_{𝔮})).A_{n}^{x}} .

We remark that (unlike Cl(𝔄_{n})) Cl′(𝔄_{n}) is an infinite group, and there is a natural surjection

e_{n} : Cl′(𝔄_{n}) ⟶ Cl(𝔄_{n}).

Now suppose that ℭ ∈ PH_{𝔭}(𝔅_{n}), with C := ℭ ⊗ F. For each 𝔮|𝔭, there is a natural isomorphism of F_{𝔮}-algebras and A_{n,𝔮}-modules

i_{𝔮} : C_{𝔮} ⟶ B_{n,𝔮};

i_{𝔮} is only defined up to an element of Aut_{F_{𝔮}}(B_{n,𝔮}) = G_{n}(F_{𝔮}). There is also an isomorphism

j_{n,𝔮} : B_{n,𝔮} ⟶ A_{n,𝔮}

of A_{n,𝔮}-modules (but not of algebras) defined by

j_{𝔮}(f) = ∑_{g∈G_{n}} f(g).g^{−1}.

Suppose that C = c.A_{n}, and set

ν_{n,𝔮}(c) = j_{𝔮} ∘ i_{𝔮}(c).

Then ν_{n,𝔮}(c) gives a well-defined element of A_{n,𝔮}^{x}/G_{n}(F_{𝔮}).A_{n}^{x}. We define the modified class invariant homomorphism

ψ′_{n} : PH_{𝔭}(𝔅_{n}) ⟶ Cl′(𝔄_{n})

12 A. AGBOOLA

by explaining how to construct an idele in J(A_{n}) representing ψ′_{n}(ℭ). Suppose that ℭ_{𝔮} = c_{𝔮}.𝔄_{n,𝔮} for each prime 𝔮 of F. Then ψ′_{n}(ℭ) is represented by the idele whose components are given by

\{ \begin{matrix}r(c).r(c_{𝔮})^{−1} & 𝔮 ∤ 𝔭 \\ ν_{n,𝔮}(c) & 𝔮|𝔭 \end{matrix}

It may be easily checked that this gives a well-defined homomorphism independent of the choices of c and c_{𝔮}. We shall abuse notation and write ψ′_{n}(Q) for ψ′_{n}(ℭ_{Q}(n)) if Q ∈ Σ(F)^{(π^{∗n})}.

Proposition 3.1. Suppose that Q ∈ Σ(F)^{(π^{∗n})}.Then e_{n} ∘ ψ′_{n}(Q) = ψ_{n}(Q).

Proof. Choose a generator c ∈ C_{Q}(n) such that C_{Q}(n) = c.A_{n} and ℭ_{Q}(n)_{𝔮} = c.𝔄_{n,𝔮} for all places 𝔮|𝔭. Then ψ_{n}(Q) is represented by the idele

\{ \begin{matrix}r(c).r(c_{𝔮})^{−1} & 𝔮 ∤ 𝔭 \\ 1 & 𝔮|𝔭 \end{matrix}

Hence the class (e_{n} ∘ ψ′_{n}(Q)).ψ_{n}(Q)^{−1}. is represented by the idele (v_{𝔮}) that is equal to ν_{n,𝔮}(c) for 𝔮|𝔭 and equal to 1 elsewhere.

Now since ℭ_{Q}(n)_{𝔮} = c.𝔄_{n,𝔮} for all places 𝔮|𝔭, it follows that ν_{n,𝔮}(c) is a generator of 𝔄_{n,𝔮} as a module over itself. Hence for each 𝔮 with 𝔮|𝔭), we have ν_{n,𝔮}(c) ∈ 𝔄_{n,𝔮}^{x}, and now the result follows, since the idele (v_{𝔮}) lies in the denominator of the classgroup. □

Let m be a positive integer. We shall now give the definition of the mth Adams operation [m] on Cl′(𝔄_{n}). (There is also a similar operation on Cl(𝔄_{n}) which is defined in an identical manner.)

For any commutative F -algebra S, say, there is a natural map [m] : S[G_{n}] →

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S[G_{n}] defined by

[m] ∑_{g∈G_{n}} s_{g}g = ∑_{g∈G_{n}} s_{g}g^{m}.

Now suppose that α ∈ Cl′(𝔄_{n}) is represented by the idele (v_{𝔮}) ∈ J(A_{n}). The mth Adams operation [m] : Cl′(𝔄_{n}) → Cl′(𝔄_{n}) is defined as follows: [m]α is the class represented by the idele ([m]v_{𝔮}) ∈ J(A_{n}). It may easily be checked that this gives a well-defined operation on Cl′(𝔄_{n}). Proposition 3.2. Suppose that Q ∈ Σ(F)^{(π^{∗n})}. Then ([m]ψ′_{n}(Q))(ψ′_{n}(Q))^{−m} = 0 for all positive integers m. Proof. It is shown in Lemma 4.15 and Proposition 7.5 of [AT] that ([m]r(c))(r(c))^{−m} ∈ A_{n}^{x} and that ([m]r(c_{𝔮}))(r(c_{𝔮}))^{−m} ∈ 𝔄_{n,𝔮}^{x}. This implies the result. □

§4. The class invariant and the p-adic height.

In this section, we shall describe the link between the homomorphisms ψ′_{n} of §3 and η_{n} of §1. We thus obtain a direct relationship between the class invariant of Q on the one hand, and the p-adic height of Q on the other.

We begin by observing that there is a natural homomorphism

ξ_{n} : Cl′(𝔄_{n}) ⟶ \frac{Map(E_{π^{n}}, C_{n}(p))^{Ω_{F}}}{∏_{𝔮|𝔭} Hom(E_{π^{n}}, μ_{q^{n}}(F_{𝔮}))^{Ω_{F}}}

defined as follows. Suppose that α ∈ Cl′(𝔄_{n}) is represented by the idele (v_{𝔮}) of J(A_{n}). Then ξ_{n}(α) is represented by the map R ↦ (χ_{R}(v_{𝔮})) of Map(E_{π^{n}}, J(F^{c}))^{Ω_{F}}.

The following result is an immediate consequence of Proposition 3.2. Proposition 4.1. Suppose that Q ∈ Σ(F)^{(π^{∗n})}. Then

ξ_{n} ∘ ψ′_{n}(Q) ∈ \frac{Hom(E_{π^{n}}, C_{n}(p))^{Ω_{F}}}{∏_{𝔮|𝔭} Hom(E_{π^{n}}, μ_{q^{n}}(F_{𝔮}))^{Ω_{F}}} .

We are now able to describe the relationship between ψ′_{n} and η_{n}.

14 A. AGBOOLA

Theorem 4.2. Suppose that Q ∈ Σ(F)^{(π^{∗n})}. Then η_{n} ∘ ξ_{n} ∘ ψ′_{n}(Q) = Q. Proof. This follows from carefully unwinding the definitions of the homomorphisms involved. We begin by observing that ξ_{n}∘ψ′_{n}(Q) is represented by f̄ ∈ Map(E_{π^{n}}, J(F^{c}))^{Ω_{F}} defined by

\begin{aligned}f̄(R) & = χ_{R}(u_{𝔮})  𝔮 ∤ 𝔭 \\ & = χ_{R}(ν_{n,𝔮}(c))  𝔮|𝔭 \end{aligned}

Let f_{1} ∈ Ext^{1}(E_{π^{n}}, F_{n}^{x})^{Ω_{F}} denote the image of ξ_{n} ∘ ψ′_{n}(Q) under the sequence of homomorphisms (1.4). Then f_{1} is represented by the map

(R, S) ⟼ d_{n}(\frac{f̄(R + S)}{f̄(R)f̄(S)})

i.e.

(R, S) ⟼ ∏_{𝔮|𝔭} \frac{χ_{R+S}(ν_{n,𝔮}(c))}{χ_{R}(ν_{n,𝔮}(c))χ_{S}(ν_{n,𝔮}(c))}.

Next we note that since C_{n}(Q) is a principal homogeneous space of B_{n}, there is an isomorphism (of algebras and A_{n} ⊗ F^{c}-modules)

i′ : C_{n}(Q) ⊗ F^{c} ⟶ B_{n} ⊗ F^{c}.

There is also an isomorphism

j′ : B_{n} ⊗ F^{c} ⟶ A_{n} ⊗ F^{c}

of A_{n} ⊗ F^{c}-modules (but not of algebras) defined by j′(f) = ∑_{g∈G_{n}} f(g)g^{−1}. Set ν′_{n} = j′ ∘ i′.

Recall from §1 that there is a natural surjection

Hom(E_{π^{n}}, (F^{c})^{x}/F_{n}^{x}) ⟶ Ext^{1}(E_{π^{n}}, F_{n}^{x}).

ON p-ADIC HEIGHT PAIRINGS 15

Define β ∈ Hom(E_{π^{n}}, (F^{c})^{x}/F_{n}^{x}) by β(R) = χ_{R}(ν′_{n}(c)) (mod F_{n}^{x}). Then β ↦ f_{1} under the above surjection.

The fact that η_{n} ∘ ξ_{n} ∘ ψ′_{n}(Q) = Q now follows from the definition of η_{n} together with the remarks on Galois action on principal homogeneous spaces in §1. □

§5. Further remarks on classgroups.

The remainder of this paper is devoted to an analysis of a certain piece of Cl(𝔄_{n}) via the techniques of Ullom and Kervaire-Murty (see [KM], [U1], [U2]). In order to carry out this analysis, we shall require certain results proved in [AT]. We therefore make the following assumptions from now on:

(i) We assume that F/K is abelian, and that F and K(E_{p^{∞}}) are linearly disjoint over K. (ii) We assume that the prime 𝔭 is completely split in F/K.

The assumption (i) implies that the Wedderburn decompositions of A_{n} and B_{n} are given by

A_{n} ≃ ⊕_{i=0}^{n}F_{i}, B_{n} ≃ ⊕_{i=0}^{n}F_{i}^{∗}.  (5.1)

Let {χ_{i}}_{i=0}^{n} denote a sequence of characters of G_{n}^{∗} such that χ_{i} has order q^{i} and χ_{i+1}^{q} = χ_{i}. We assume that the above decomposition of A_{n} is induced by ⊕_{i=0}^{n}χ_{i}.

We shall now describe two different homomorphisms from Cl(𝔄_{n}) to Cl(𝔄_{n−1}). It is most important that the reader keep in mind the distinction between these maps.

Let α ∈ Cl(𝔄_{n}), and let m_{n} ∈ J(A_{n}) be a representative of α in (2.1). Then q_{n}(α) ∈ Cl(𝔄_{n−1}) is defined to be the class represented by the element

R ⟼ χ_{R}(m_{n}), R ∈ E_{π^{n−1}}

16 A. AGBOOLA

of Map(E_{π^{n−1}}, J(ℚ^{c}))^{Ω_{F}}. This gives us a homomorphism (‘passage to quotient’; see chapter 1, §4 of [T2])

q_{n} : Cl(𝔄_{n}) ⟶ Cl(𝔄_{n−1}).

The other map that we wish to consider is the restriction map on classgroups. Suppose that α ∈ Cl(𝔄_{n}) and m_{n} ∈ J(A_{n}) are as above. Then res_{n/n−1}(α) ∈ Cl(𝔄_{n−1}) is defined to be the class represented by the element

R ⟼ (Ind_{G_{n−1}}^{G_{n}} χ_{R})(m_{n}), R ∈ E_{π^{n−1}}

of Map(E_{π^{n−1}}, J(ℚ^{c}))^{Ω_{F}}. (Here Ind_{G_{n−1}}^{G_{n}} χ_{R} denotes the character of G_{n} induced from G_{n−1} by χ_{R}.) Now a straightforward computation shows that we have

(Ind_{G_{n−1}}^{G_{n}} χ_{R})(m_{n}) = ∏_{{R′∈G_{n}|[q]R′=R}} χ_{R′}(m_{n})
= N_{F_{n}/F_{n−1}}(χ_{R′})(m_{n})

for any choice of R′ ∈ G_{n} with [q]R′ = R. Hence a representing map for res_{n/n−1}(α) ∈ Cl(𝔄_{n−1}) is given by

R ⟼ N_{F_{n}/F_{n−1}}(χ_{R})(m_{n}) = ∏_{{R′∈G_{n}|[q]R′=R}} χ_{R′}(m_{n}).

We shall now focus our attention on certain natural automorphisms of Cl(𝔄_{n}). Write κ : Ω_{F} → ℤ_{p}^{x} for the character giving the action of Ω_{F} on the π-adic Tate module T_{π} of E. Let Aut(G_{n}) denote the group of automorphisms of G_{n}. Then there is a canonical isomorphism Aut(G_{n}) → Gal(F_{n}/F) given by τ ↦ σ_{τ}. This isomorphism induces an action of Aut(G_{n}) on A_{n} via Galois action on each factor occuring in the Wedderburn decomposition (4.1) of A_{n}. We now examine this

ON p-ADIC HEIGHT PAIRINGS 17

action. Suppose that m = ∑_{g} a_{g}g is an element of A_{n}, and that R ∈ E_{π^{n}}. For each τ ∈ Aut(G_{n}), we have

[χ_{R}(m)]^{σ_{τ}} = ∑_{g∈G_{n}} a_{g}^{σ_{τ}} w_{n}(R, g)^{σ_{τ}}

= ∑_{g∈G_{n}} a_{g}^{σ_{τ}} w_{n}(R^{σ_{τ}} , g^{σ_{τ}})

= ∑_{g∈G_{n}} a_{g}w_{n}(R^{σ_{τ}} , g)

= ∑_{g∈G_{n}} a_{g}w_{n}(κ(σ_{τ})R, g)

= ∑_{g∈G_{n}} a_{g}w_{n}(R, κ(σ_{τ})g)

= χ_{R}([κ(σ_{τ})]m).

Now [κ(σ_{τ})] : A_{n} → A_{n} is a homomorphism of algebras. Hence Aut(G_{n}) preserves the denominator of (2.1), whence it follows that Aut(G_{n}) acts on Cl(𝔄_{n}) via its action on A_{n}. Thus, Cl(𝔄_{n}) is an Aut(G_{n})-module.

We next observe that Aut(G_{n}) ≃ Δ × Γ_{n}, where |Δ| = p − 1, and Γ_{n} is cyclic of order q^{n}. Set Γ = \underleftarrow{lim} Γ_{n} (where here the inverse limit is taken with respect to the natural quotient map Γ_{n} → Γ_{n−1}). Then \underleftarrow{lim} Cl(𝔄_{n}) (inverse limit taken with respect to the restriction maps res_{n/n−1}) carries the structure of a Γ-module in the obvious manner.

Let Cl(𝔄_{n})^{(p)} denote the p-primary part of Cl(𝔄_{n}), and set Q(F) = \underleftarrow{lim} Cl(𝔄_{n})^{(p)} (where here again the inverse limit is taken with respect to the restriction maps res_{n/n−1}). Then Q(F) is a Λ := ℤ_{p}[[Γ]]-module.

§6. The kernel group.

Write M_{n} for the 𝔒_{F} -maximal order of A_{n}. The kernel group D(𝔄_{n}) of 𝔄_{n} is defined to be the kernel of the natural homomorphism Cl(𝔄_{n}) → Cl(M_{n}). Since

18 A. AGBOOLA

𝔄_{n,𝔭} = M_{n,𝔮} for 𝔮 ∤ 𝔭 (see Proposition 2.1 of [AT]), it follows from standard theory that

D(𝔄_{n}) ≃ \frac{M_{n,𝔭}^{x}}{𝔄_{n,𝔭}^{x}.A_{n}^{x}} .  (6.1)

The following result is proved in §9 (see especially (9.7)) of [AT]. Proposition 6.1. There is a natural homomorphism

δ_{n} : D(𝔄_{n}) ⟶ \frac{𝔒_{F_{n−1},𝔭}^{x} mod 𝔭}{Im(𝔒_{F_{n−1}}^{x})}  (6.2)

where Im(𝔒_{F_{n−1}}^{x}) denotes the image of 𝔒_{F_{n−1}}^{x} mod 𝔭. This homomorphism is defined as follows. Set L_{n} = F_{n}.F_{n}^{∗}; then via the existence of the Weil pairing, we see that μ_{q^{n}} ⊂ L_{n}. Let f ∈ Gal(L_{n}/F (μ_{q^{n}})) denote the Frobenius automorphism for the primes of F (μ_{q^{n}}) above 𝔭. Suppose that α ∈ D(𝔄_{n}) is represented in (6.1) by an element m ∈ M_{n,𝔭}^{x}. Then

δ_{n}(α) = N_{F_{n}/F_{n−1}}(χ_{n}(m)).χ_{n−1}(m)^{−f} Im(𝔒_{F_{n−1}}^{x}) mod 𝔭. □

Now set V_{n} = ker(q_{n}|D(𝔄_{n})). We remark that classes in V_{n} may be represented by elements m ∈ M_{n,𝔭}^{x} satisfying χ_{i}(m) = 1 for i < n. It is easily seen that res_{n/n−1}(D(𝔄_{n})) ⊂ D(𝔄_{n−1}) and that res_{n/n−1}(V_{n}) ⊂ V_{n−1}. In the remainder of this section, we shall determine the structure of \underleftarrow{lim} V_{n} (where the inverse limit is taken with respect to the restriction map on classgroups) as a Λ-module. We begin by examining \underleftarrow{lim} Ker(δ_{n}|V_{n}). Lemma 6.3. \underleftarrow{lim} Ker(δ_{n}|V_{n}) = 0. Proof. Suppose that (s_{n}) ∈ \underleftarrow{lim} Ker(δ_{n}|V_{n}). Then each class s_{n} ∈ V_{n} may be represented (via the isomorphism (6.1) by an element m_{n} ∈ M_{n,𝔭}^{x} satisfying χ_{i}(m_{n}) = 1 for i < n.

ON p-ADIC HEIGHT PAIRINGS 19

Fix n, and choose M > n. Suppose that s_{M} ∈ V_{M} is represented by m_{M} with χ_{i}(m_{M}) = 1 for i < M. Since s_{n} = res_{M/n}(s_{M}), it follows that there is a representative m_{n}(M) ∈ M_{n,𝔭}^{x} of s_{n}, say, which is such that χ_{n}(m_{n}(M)) = N_{F_{M} /F_{n}}(χ_{M}(m_{M})), and χ_{i}(m_{n}(M)) = 1 for i < n.

Now s_{M} ∈ ker(δ_{M}), and so there is a global unit u_{M−1} ∈ 𝔒_{F_{M−1}}^{x} such that N_{F_{M} /F_{M−1}}(χ_{M}(m_{M})) ≡ u_{M−1}(mod 𝔭). This implies that χ_{n}(m_{n}(M)) = u_{n}.γ_{n}, where u_{n} ∈ 𝔒_{F_{n}}^{x} is a global unit, and γ_{n} ∈ 𝔒_{F_{n},𝔭}^{x}. The local unit γ_{n} may be made as 𝔭-adically close to 1 as we please by choosing M to be sufficiently large (as may be seen by an argument very similar to that given in Lemma 3.2 of [U2]). Hence m_{n}(M) lies in the denominator of the RHS of (6.1) for all sufficiently large M, and so it follows that s_{n} = 0. This establishes the result. □

Hence we deduce that the natural map

δ = \underleftarrow{lim} δ_{n} : \underleftarrow{lim} V_{n} ⟶ \underleftarrow{lim} \frac{𝔒_{F_{n−1},𝔭}^{x} mod 𝔭}{Im(𝔒_{F_{n−1}}^{x})}

(where the right-hand inverse limit is taken with respect to the norm map) is an injection. We remark that δ is clearly a Γ-homomorphism.

We shall now show that δ_{n}|V_{n} is surjective, whence it follows that δ_{n} is an isomorphism. This is an immediate consequence of the following lemma. Lemma 6.4.. Let 𝔮 be any prime of F lying above 𝔭. Then

(1 + p𝔒_{F,𝔮})N_{F_{n+1}/F_{n}}(𝔒_{F_{n+1},𝔮}^{x}) = 𝔒_{F_{n},𝔮}^{x}.

Proof. The proof is via classfield theory. We begin by observing that [𝔒_{F_{n},𝔮}^{x} : N_{F_{n+1}/F_{n}}(𝔒_{F_{n+1},𝔮}^{x})] = p, since F_{n+1,𝔮}/F_{n,𝔮} is totally ramified. Now 1 + p𝔒_{F,𝔮} is

20 A. AGBOOLA

topologically generated by 1 + p, and so to prove our result it suffices to show that the Artin symbol (1 + p, F_{n+1,𝔮}/F_{n,𝔮}) is non-trivial.

We have the following sequence of maps:

𝔒_{F,𝔮}^{x} ⟶^{Artin} Gal(F_{n+1,𝔮}/F_{𝔮}) ⟶^{transfer} Gal(F_{n+1,𝔮}/F_{n,𝔮}).

Under this sequence of maps, 1+p𝔒_{F,𝔮} maps surjectively onto the p-Sylow subgroup of Gal(F_{n+1,𝔮}/F_{n,𝔮}) since [𝔒_{F,𝔮}^{x} : 1 + p𝔒_{F,𝔮}] is of order prime to p. The result now follows from the fact that Gal(F_{n+1,𝔮}/F_{n,𝔮}) is a p-group. □

Thus, we have now shown that

\underleftarrow{lim} V_{n} ≃ \underleftarrow{lim} \frac{𝔒_{F_{n−1},𝔭}^{x} mod 𝔭}{Im(𝔒_{F_{n−1}}^{x})},

where the left-hand inverse limit is taken with respect to the restriction map on classgroups and the right-hand inverse limit is taken with respect to the norm map.

Now let H_{∞} (resp. X_{∞}) be the maximal abelian pro-p extension of F_{∞} which is everywhere unramified (resp. unramified away from 𝔭), and set Y_{∞} = Gal(X_{∞}/H_{∞}). It follows from classfield theory (cf. the argument given in §3 of [U2]) that

\underleftarrow{lim} \frac{𝔒_{F_{n−1},𝔭}^{x} mod 𝔭}{Im(𝔒_{F_{n−1}}^{x})} ≃ Y_{∞}

as Γ-modules. The following result, which may be regarded as an elliptic analogue of Theorem 3.6(a) of [U2], is now immediate. Theorem 6.5. There is a canonical isomorphism of Λ-modules

\underleftarrow{lim} V_{n} ≃ Y_{∞}. □

ON p-ADIC HEIGHT PAIRINGS 21

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School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA. (Address from September 1, 1996: Department of Mathematics, University of California, Santa Barbara, CA 93106, USA. Email: agboola@math.ucsb.edu)

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