A GEOMETRIC DESCRIPTION OF THE CLASS INVARIANT HOMOMORPHISM

A. Agboola Version of May 27, 1994

In recent years, a certain amount of work has been done on the Galois structure of principal homogeneous spaces of finite group schemes that are constructed via dividing points on abelian varieties. This study was begun by M. J. Taylor in [T1] and was originally motivated by the fact that such principal homogeneous spaces are very closely connected with certain rings of integers. The starting point of the theory is the so-called class invariant homomorphism which was first introduced by Waterhouse in [W]. The purpose of this note is to point out a simple geometric description of this homomorphism (as considered in [T1]) in terms of the restriction of certain line bundles to torsion subgroup schemes of the abelian variety in question. We shall also make a number of remarks concerning both old results and new questions that arise in light of this description.

In what follows, we shall confine ourselves to the case of abelian varieties defined over number fields. It should however be noted that there is no difficulty in carrying out a similar analysis of the analogous situation over global function fields (cf. [A3]).

I would like to thank K. Ribet for drawing my attention to [R]. I am also grateful to T. Chinburg, M. J. Taylor, and W. Messing for interesting conversations.

Typeset by AMS-TEX 2 A. AGBOOLA

Partially supported by an NSF Postdoctoral Research Fellowship

§1. In this section we shall recall the definition of the class invariant homomorphism for abelian varieties. We refer the reader to [W], [T1] and [BT] for further details.

Let F be a number field with ring of integers 𝔒_{F}. Suppose that A/F is an abelian variety with everywhere good reduction, and let A/𝔒_{F} denote the Néron model of A/F. Write Â (resp. Â) for the dual abelian variety of A (resp. A). (We shall frequently omit the dependence upon F from our notation when there is no danger of confusion.) It follows from the universal property of the Néron model that there are natural isomorphisms A(𝔒_{F}) ≃ A(F) (resp. Â(𝔒_{F}) ≃ Â(F)); we shall feel free to make such identifications without further notice.

Let A_{p^{n}} denote the 𝔒_{F} -group scheme of p^{n}-torsion on A. Then A_{p^{n}} is affine with Cartier dual Â_{p^{n}}, and both A_{p^{n}} and Â_{p^{n}} are twisted constant groups. Hence we may apply Theorem 2 of [W] and deduce that there is an isomorphism

ν_{1} : Ext(A_{p^{n}}, 𝔾_{m}) ⟶ H^{1}(𝔒_{F} , Â_{p^{n}}).

(Here Ext(A_{p^{n}}, 𝔾_{m}) is computed in the category of fpqc group schemes, and we take cohomology with respect to the fpqc site.)

This isomorphism may be described as follows. Let V be an extension of A_{p^{n}} by 𝔾_{m}; then there is an exact sequence of commutative group schemes

1 ⟶ 𝔾_{m} ⟶ V ⟶ A_{p^{n}} ⟶ 0.  (1)

Applying the functor Hom(A_{p^{n}}, −) to (1) yields the exact sequence

0 ⟶ Hom(A_{p^{n}}, 𝔾_{m}) ⟶ Hom(A_{p^{n}}, V) ⟶ Hom(A_{p^{n}}, A_{p^{n}})  (2)

THE CLASS INVARIANT HOMOMORPHISM 3

i.e.

0 ⟶ Â_{p^{n}} ⟶ Hom(A_{p^{n}}, V) ⟶ Hom(A_{p^{n}}, A_{p^{n}}).  (3)

Let ι ∈ Hom(A_{p^{n}}, A_{p^{n}}) denote the identity map, and define Y to be the inverse image of ι in (3) above (i.e. Y is the sheaf of group-theoretic sections of (1)). Then Y is a principal homogeneous space of Â_{p^{n}} (see e.g. Theorem 2’ of [W]) and so determines an element ν_{1}(V) of H^{1}(𝔒_{F} , Â_{p^{n}}).

We next observe that there is an obvious natural homomorphism

ν_{2} : Ext(A_{p^{n}}, 𝔾_{m}) ⟶ H^{1}(A_{p^{n}}, 𝔾_{m}) = Pic(A_{p^{n}})

and that Kummer theory on A affords us a natural map

ν_{3} : Â(F) ≃ Â(𝔒_{F}) ⟶ Â(𝔒_{F})/p^{n}Â(𝔒_{F}) ⟶ H^{1}(𝔒_{F} , Â_{p^{n}}).

The class invariant homomorphism

ψ_{n} : Â(F) ⟶ Pic(A_{p^{n}})

is defined by ψ_{n} = ν_{2} ∘ ν_{1}^{−1} ∘ ν_{3}.

This homomorphism may be described somewhat more concretely as follows. Let F^{c} be an algebraic closure of F, and write Ω_{F} for the absolute Galois group of F. Let G denote the group of F^{c}-valued points of Â_{p^{n}}, and let ℌ (resp. 𝔅) be an 𝔒_{F} -Hopf algebra that represents the group scheme Â_{p^{n}}/𝔒_{F} (resp. A_{p^{n}}/𝔒_{F}). It is shown in [T1] that ℌ (resp. 𝔅) is an 𝔒_{F} -order in the algebra H = (F^{c}G)^{Ω_{F}} (resp. Map(G, F^{c})^{Ω_{F}}), where here Ω_{F} acts on both G and F^{c}.

Let Q ∈ Â(F), and set

G_{Q} = {Q′ ∈ Â(F^{c})|p^{n}Q′ = Q}.

4 A. AGBOOLA

Define the Kummer algebra F_{Q} by F_{Q} = Map(G_{Q}, F^{c})^{Ω_{F}}, and write 𝔒_{Q} for the integral closure of 𝔒_{F} in F_{Q}. The algebra H acts on F_{Q} by

(f.α)(Q′) = ∑_{g∈G} α_{g}f(Q′ + g)

for f ∈ F_{Q} and α = ∑_{g∈G} α_{g}g ∈ H.

In general 𝔒_{Q} is not acted upon by ℌ. Define the Kummer order ℭ_{Q} to be the largest ℌ-module contained in 𝔒_{Q}. It is shown in §3 of [T1] (see also p.186 of [BT]) that ℭ_{Q} is a principal homogeneous space of the algebra 𝔅, and that ℭ_{Q} is a locally free ℌ-module. Then (ibid.) ψ_{n}(Q) = (ℭ_{Q}) ∈ Pic(A_{p^{n}}).

§2. Recall that, via standard theory, there are canonical isomorphisms

Â ≃ Pic^{0}(A) ≃ Ext(A, 𝔾_{m}).  (4)

Thus each point Q ∈ Â(F) determines a (rigidified) line bundle L_{Q} on A, and an extension

1 ⟶ 𝔾_{m} ⟶ G(Q) ⟶ A ⟶ 0  (5)

of A by 𝔾_{m} (cf. [Mu, §13 and §23] or [Mi, §11]).

The inclusion map A_{p^{n}} ↪ A induces a natural homomorphism

ν_{4} : Ext(A, 𝔾_{m}) ⟶ Ext(A_{p^{n}}, 𝔾_{m}).

Hence, from (5), we obtain an extension G(Q)^{(n)} := ν_{4}(G(Q))

1 ⟶ 𝔾_{m} ⟶ G(Q)^{(n)} ⟶ A_{p^{n}} ⟶ 0  (6)

of A_{p^{n}} by 𝔾_{m}.

We thus have a homomorphism

ψ′_{n} : Â(F) ⟶ Pic(A_{p^{n}}/𝔒_{F})  (7)

THE CLASS INVARIANT HOMOMORPHISM 5

defined by ψ′_{n} = ν_{2} ∘ν_{4}. Note that the map ψ′_{n} is simply that induced by restricting the line bundle L_{Q} to the subgroup scheme A_{p^{n}} of A.

We are now in a position to state the main result of this note. Theorem 1. ψ_{n} = ψ′_{n}. Proof. To prove the result it suffices to show that

ν_{1} ∘ ν_{4}(G(Q)) = ν_{3}(Q).  (8)

We first of all note that the natural homomorphism

η : H^{1}(𝔒_{F} , Â_{p^{n}}) ⟶ H^{1}(F, G)

is an injection (see e.g. [BT, Lemma 2.1]), and so (8) will follow if we show that

η ∘ ν_{1} ∘ ν_{4}(G(Q)) = η ∘ ν_{3}(Q).  (9)

We next observe that by reasoning exactly as above in the definition of ν_{i}’s, but this time working over Spec(F) rather than Spec(𝔒_{F}), we may define natural homomorphisms ν′_{i}:

ν′_{1} : Ext(A_{p^{n}}, 𝔾_{m}) ⟶ H^{1}(F, Â_{p^{n}})
ν′_{2} : Ext(A_{p^{n}}, 𝔾_{m}) ⟶ H^{1}(A_{p^{n}}, 𝔾_{m}) = Pic(A_{p^{n}})
ν′_{3} : Â(F) ⟶ H^{1}(F, Â_{p^{n}})
ν′_{4} : Ext(A, 𝔾_{m}) ⟶ Ext(A_{p^{n}}, 𝔾_{m})

(It is perhaps worth remarking that Pic(A_{p^{n}}) = 0, and so the map ν′_{2} is not particularly interesting!)

6 A. AGBOOLA

Let G(Q) (resp. G(Q)^{(n)}) denote the generic fibre of G(Q)/𝔒_{F} (resp. G(Q)^{(n)}/𝔒_{F}), and set θ = ν′_{1} ∘ ν′_{4}. Then G(Q) is the extension of A/F by 𝔾_{m} afforded by Q ∈ A(F), and so by functoriality we have

η ∘ ν_{1} ∘ ν_{4}(G(Q)) = θ(G(Q))

and

η ∘ ν_{3}(Q) = ν′_{3}(Q)

However, Proposition 3.1 of [R] asserts that

θ(G(Q)) = ν′_{3}(Q)  (10)

(note that Ribet’s ξ is our ν′_{3}, and his P is our Q), and this implies the result.

For the sake of completeness, we shall now give a brief sketch of the proof of (10). The reader may refer to §3 of [R] for full details.

Let D be the divisor on A determined by the point Q (cf. (4) above). Let F^{c}(A) denote the function field of A/F^{c}, and for each a ∈ A(F^{c}), write T_{a} : A → A for the translation-by-a map. Ribet shows that G(Q)^{(n)}(F^{c}) is isomorphic to the group of pairs (a, f) ∈ A_{p^{n}}(F^{c}) × F^{c}(A)^{x} satisfying:

(f) = T_{a}^{∗}D − D  (11)
f(x)f(x + a) … f(x + (p^{n} − 1)a) = 1  (12)

under the multiplication

(a_{1}, f_{1})(a_{2}, f_{2}) = (a_{1} + a_{2}, g)

where g(x) = f_{2}(x + a_{1})f(x).

THE CLASS INVARIANT HOMOMORPHISM 7

Now choose a p^{n}th root Q′, say, of Q on Â(F^{c}), and let E be the corresponding divisor on A/F^{c}. Then

[p^{n}]^{∗}(E) = D + (h)

for some function h ∈ F^{c}(A), and it may be shown that the map s : A_{p^{n}}(F^{c}) → G(Q)^{(n)}(F^{c}) defined by

a ⟼ (a, h(x)/h(x + a))

is a group-theoretic section of the natural surjection G(Q)^{(n)}(F^{c}) → A_{p^{n}}(F^{c}). It follows that θ(G(Q)) ∈ H^{1}(F, Â_{p^{n}}) is represented by the cocycle δ_{σ} = σ.s−s (where σ ∈ Ω_{F}). By explicitly computing δ_{σ}, it may be deduced that

δ_{σ}(a) = e_{n}(a, (σ − 1)Q′),

where e_{n} denotes the Weil pairing. This establishes (10), as required. □

It follows from Theorem 1 that there are certain constraints on the image of ψ_{n}. Let W be any flat, commutative 𝔒_{F} -group scheme, and write m : W × W → W for the multiplication map. Let p_{i} : W × W → W (i = 1, 2) denote projection onto the ith factor. Recall (see eg [Se, Chapter VII, §3]) that a class c ∈ Pic(W) is said to be primitive if m^{∗}(c) = p_{1}^{∗}(c) + p_{2}^{∗}(c). Let PPic(W) denote the subgroup of Pic(W) consisting of the primitive elements of Pic(W). Then (op. cit.) PPic(W) is an additive functor of W, and if W is an abelian variety, then PPic(W) = Pic^{0}(W). Hence we deduce that the image of ψ_{n} is contained in PPic(A_{p^{n}}), i.e. that we in fact have

ψ_{n} : Â(F) ⟶ PPic(A_{p^{n}}/𝔒_{F}).

We remark that this fact also follows from the results contained in §1 of [CM].

8 A. AGBOOLA

§3. Theorem 1 implies that the study of the homomorphism ψ_{n} is really the study of the behaviour of line bundles on A under restriction to certain torsion subgroup schemes of A. It is interesting to reformulate the results of [A], [AT] and [ST] in light of this fact. For example, combining Proposition 1 above with Theorem 1 of [ST] immediately yields the following result. Theorem 2 (Srivastav-Taylor). Let A/F above be a CM elliptic curve. Suppose that p > 3, and let L ∈ Pic^{0}(A)(𝔒_{F}) be a torsion line bundle. Then L|A_{p^{n}} is trivial for all n ≥ 1.

It is conjectured that a similar result holds for an arbitrary CM abelian variety with everywhere good reduction. (Note added in proof: The author has recently shown that Theorem 2 also holds in the non-CM case. See his forthcoming paper ‘Torsion points on elliptic curves and Galois module structure’.)

We shall now introduce some further notation. Let A′_{p^{n}}/𝔒_{F} denote the normalisation of A_{p^{n}}/𝔒_{F}. Composing ψ_{n} with the natural map Pic(A_{p^{n}}) → Pic(A′_{p^{n}}) gives us a homomorphism

φ_{n} : A ⟶ PPic(A′_{p^{n}}/𝔒_{F}).

The map φ_{n} may be described in terms of Kummer orders (cf. §1) as follows. By definition, A′_{p^{n}}/𝔒_{F} = Spec(M), where M is the unique maximal 𝔒_{F} -order contained in H. Suppose that Q ∈ Â(F). Then φ_{n}(Q) = (ℭ_{Q} ⊗_{ℌ} M) ∈ PPic(A′_{p^{n}}).

Next, we recall that ψ_{n} (and hence also φ_{n}) factors through the projection Â(F) → Â(F)/p^{n}Â(F). Also, the inclusion map A_{p^{n}} ↪ A_{p^{n+1}} induces natural homomorphisms H^{1}(𝔒_{F} , A_{p^{n+1}}) → H^{1}(𝔒_{F} , A_{p^{n}}) and H^{1}(A_{p^{n+1}}, 𝔾_{m}) → H^{1}(A_{p^{n}}, 𝔾_{m}). Taking inverse limits of ψ_{n} and φ_{n} with respect to these maps, and identifying Â with Pic^{0}(A) as per (4) above yields homomorphisms

ψ_{∞} : Â(𝔒_{F}) ⊗_{ℤ} ℤ_{p} ≃ Pic^{0}(A)(𝔒_{F}) ⊗_{ℤ} ℤ_{p} ⟶ \underleftarrow{lim} PPic(A_{p^{n}})

and

φ_{∞} : Â(𝔒_{F}) ⊗_{ℤ} ℤ_{p} ≃ Pic^{0}(A)(𝔒_{F}) ⊗_{ℤ} ℤ_{p} ⟶ \underleftarrow{lim} PPic(A′_{p^{n}}).

We remark that a line bundle L ∈ Pic^{0}(A)(𝔒_{F}) lies in the kernel of ψ_{∞} if and only if L|A_{p^{n}} is trivial for all n ≥ 1. We may now pose the following question. Question 1. (i) Is the kernel of ψ_{∞} finite ? (ii) Is ψ_{∞} surjective (possibly only up to finite index)?

It seems possible to regard an affirmative answer to these questions as a loose analogue of the Tate conjecture for abelian varieties (now a theorem of Faltings (see [F] or [CS])). This states that the natural map

End_{F} (A) ⊗_{ℤ} ℤ_{p} ⟶ End_{F} (T_{p}(A))

is an isomorphism. (Here T_{p}(A) is the p-adic Tate module of A, and End_{F} (A) (resp. End_{F} (T_{p}(A))) is the ring of endomorphisms of A (resp. T_{p}(A)) that are defined over F.) It is shown in [AT] that Question 1(i) has an affirmative answer (subject to certain technical hypotheses) in the case that A/F is a CM elliptic curve and p is a prime of good ordinary reduction.

We may also ask similar questions about the kernel of φ_{∞}. When A/F is a CM elliptic curve and p is an ordinary prime, it is shown in [A1] and [AT] that the kernel of φ_{∞} is equal to a canonical subgroup of A(F) ⊗_{ℤ} ℤ_{p} first defined by R. Greenberg using Iwasawa theory (see [G]). This subgroup may be described in terms of the p-adic height pairing on A (see [P]) and is in general of infinite order. (See also

[A2] for a discussion of φ_{∞} in the context of p-adic representations.) It would be

of some interest to obtain a better understanding of the relationship between class

invariants and p-adic heights. (See [A4] for further remarks on this in the case of

CM elliptic curves.) We conclude with the following question.

Question 2. Suppose that the kernel of ψ_{∞} is of infinite order. Does this imply

that that the p-adic height pairing on A (cf. [PR] or [Sc]) is degenerate ?

References

[A1] A. Agboola, Iwasawa theory of elliptic curves and Galois module structure, Duke Math. J. 71 (1973), 441-462.

[A2] A. Agboola, p-adic representations and Galois module structure, in preparation.

[A3] A. Agboola, Abelian varieties and Galois module structure in global function fields, Math. Z. (to appear).

[A4] A. Agboola, On p-adic height pairings and locally free classgroups of Hopf orders, preprint.

[AT] A. Agboola, M. J. Taylor, Class invariants of Mordell-Weil groups, Crelle 447 (1994), 23-61.

[BT] N .P. Byott, M. J. Taylor, Hopf orders and Galois module structure, In: Group rings and classgroups, K. W. Roggenkamp, M. J. Taylor (eds.), Birkhauser, 1992.

[CM] L. Childs, A. Magid, The Picard invariant of a principal homogeneous space, J. Pure and Appl. Alg. 4 (1974), 273-286.

[CS] G. Cornell, J. Silverman (eds.), Arithmetic Geometry, Springer, 1986.

[F] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpen, Invent. Math. 73 (1983), 349-366.

[G] R. Greenberg, Iwasawa theory for p-adic representations, Advanced Studies in Pure Mathematics 17 (1989), Academic Press, 97-137.

[Mi] J. Milne, Abelian varieties, In: Arithmetic Geometry, G. Cornell, J. Silverman (eds.), Springer, 1986.

[Mu] D. Mumford, Abelian Varieties, OUP, 1970.

[PR] B. Perrin-Riou, Théorie d’Iwasawa et hauteurs p-adique, Invent. Math. 109 (1992), 137– 185.

[P] A. Plater, Height Pairings on Elliptic Curves, Ph.D. Thesis, Cambridge University, 1991.

[R] K. Ribet, Kummer theory on extensions of abelian varieties by tori, Duke Math. J. 49 (1979), 745–761.

[Sc] P. Schneider, Iwasawa theory for abelian varieties—a first approach, Invent. Math. 71 (1983), 251–293.

[Se] J.-P. Serre, Algebraic Groups and Class Fields, Springer, 1988.

[ST] A. Srivastav, M. J. Taylor, Elliptic curves with complex multiplication and Galois module structure, Invent. Math. 99 (1990), 165–184.

[T1] M. J. Taylor, Mordell-Weil groups and the Galois module structure of rings of integers, Ill. J. Math. 32 (1988), 428–452.

[W] W. Waterhouse, Principal homogeneous spaces and group scheme extensions, AMS Transactions 153 (1971), 181–189.

Department of Mathematics, UC Berkeley Berkeley, CA 94720, USA.

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