TORSION POINTS ON ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE

A. Agboola Version of June 17, 1995

§0. Introduction and statement of results. In this paper we shall study the Galois module structure of certain Kummer orders obtained by dividing torsion points on an elliptic curve.

Let E be an elliptic curve defined over a number field F. We suppose that all endomorphisms of E are defined over F, and that E/F has everywhere good reduction. For any field L, we write O_{L} for its ring of integers and L^{c} for an algebraic closure of L; we set Ω_{L} = Gal(L^{c}/L).

Let p > 3 be a rational prime, and write G_{i} for the subgroup of elements of E(F^{c}) which are killed by the endomorphism [p^{i}] of E. Let 𝔅_{i}(F) = 𝔅_{i} denote the O_{F} -Hopf algebra which represents the O_{F} -group scheme of [p^{i}]-torsion on E, and let 𝔄_{i}(F) = 𝔄_{i} be the Cartier dual of 𝔅_{i}. A detailed description of these algebras is given in [T] (see also [ST]). There it is shown that 𝔅_{i} is an O_{F} -order in the algebra B_{i}(F) = B_{i} = Map(G_{i}, F^{c})^{Ω_{F}}, and 𝔄_{i} is an O_{F} order in the algebra A_{i}(F) = A_{i} = (F^{c}G_{i})^{Ω_{F}} (where here Ω_{F} acts upon both G_{i} and F^{c}). (Here and elsewhere we shall omit from our notation the dependence of our constructions on the underlying field F unless there is some danger of ambiguity.)

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Partially supported by an NSF Postdoctoral Research Fellowship.

Suppose that Q ∈ E(F), and write

G_{Q}(i) = {Q′ ∈ E(F^{c}) : [p^{i}]Q′ = Q}  (0.1)

Define the Kummer algebra F_{Q}(i) by

F_{Q}(i) = Map(G_{Q}(i), F^{c})^{Ω_{F}}  (0.2)

Then [F_{Q}(i) : F] = |G_{i}|, and A_{i} acts on F_{Q}(i) via

(f. ∑_{g∈G_{i}} a_{g}g)(Q′) = ∑_{g∈G_{i}} a_{g}f(Q′ + g)  (0.3)

for f ∈ F_{Q}(i) and ∑_{g∈G_{i}} a_{g}g ∈ A_{i}.

Let O_{Q}(i) denote the integral closure of O_{F} in F_{Q}(i). In general O_{Q}(i) does not admit an action of 𝔄_{i}. We define the Kummer order ℭ_{Q}(i) to be the maximal 𝔄_{i}-stable submodule of O_{Q}(i), i.e.

ℭ_{Q}(i) = {x ∈ O_{Q}(i)|x.𝔄_{i} ⊆ O_{Q}(i)}  (0.4)

It is shown in [T] that ℭ_{Q}(i) is a twisted form of 𝔅_{i}, and that ℭ_{Q}(i) is a locally free 𝔄_{i}-module. Thus ℭ_{Q}(i) defines an element (ℭ_{Q}(i)) in the locally free classgroup Cl(𝔄_{i}) of 𝔄_{i}. Hence we obtain a map

ψ_{i} : E(F) ⟶ Cl(𝔄_{i})  (0.5)

given by ψ_{i}(Q) = (ℭ_{Q}(i)). Since E/F has everywhere good reduction, the map ψ_{i} is a group homomorphism (see Theorem 1 of [T]), and ψ_{i}(Q) = 0 if and only if ℭ_{Q}(i) is a globally free 𝔄_{i}-module. (We remark that in [T] it is assumed that E has complex multiplication. This assumption is not necessary for setting up the

TORSION POINTS ON ELLIPTIC CURVES 3

basic theory. See [BT] and [A1].) For an interesting geometric description of the homomorphism ψ_{i} in terms of line bundles on E and the connection with a Galoismodule theoretic analogue of a now classical conjecture of Tate, we refer the reader to [A1].

The following result is a special case of Theorem 1 of [ST]. Theorem (Srivastav-Taylor). Suppose that E is a CM elliptic curve. Then E(F)_{torsion} ⊆ ker(ψ_{i}).

This theorem gives information regarding the Galois module structure of rings of integers of certain abelian extensions of F. It may therefore be regarded as yielding an integral version of the Krönecker Jugendtraum.

The purpose of this paper is to prove the following result. Theorem 1. Suppose that E is an elliptic curve. Then E(F)_{torsion} ⊆ ker(ψ_{i}).

The proof of Theorem 1 depends upon the use of elementary intersection theory to prove integrality results concerning certain resolvent elements that arise as special functions on E. Our methods differ from those of previous authors (see e.g. [ST], [CNS], [CNT]) in that we avoid the use of modular functions and the q-expansion principle in obtaining such integrality statements. We remark that the techniques introduced in this paper may also be used to prove an analogue of Theorem 1 in the function field case, and we refer the reader to [A2] for further details. Finally, we mention that the situation in which p ≤ 3 has been studied by several authors in the case of CM elliptic curves (see [CNS] and the references quoted therein). It seems almost certain that the intersection theoretic methods of this paper may also be used to obtain partial results similar to those contained in [CNS], although we

4 A. AGBOOLA

have not carried out this project.

I would like to express my warmest thanks to Martin Taylor for his helpful comments on an earlier version of this paper. I am also extremely grateful to the referee for his very careful reading of the manuscript and for his many suggestions which have led to improvements in the exposition.

§1. Preliminary results. In this section we recall certain results from [T], [BT], [Sh] and [ST] that we shall require.

We begin by describing the relationship between (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}) and (ℭ_{Q}(j)) ∈ Cl(𝔄_{j}) for 0 < j < i, using the methods of §2 of [ST].

The natural surjection [p^{i−j}] : G_{i} → G_{j} induces a surjective homomorphism A_{i} → A_{j} of Hopf algebras (which we shall also denote by [p^{i−j}]) given by

[p^{i−j}]( ∑_{g∈G_{i}} α_{g}g) = ∑_{g∈G_{i}} α_{g}([p^{i−j}]g).  (1.1)

Similarly, the inclusion G_{j} → G_{i} induces an inclusion A_{j} → A_{i} of Hopf algebras. Since these maps are induced by homomorphisms of group schemes, we can pass to the integral level and deduce that 𝔄_{j} may be viewed as either a quotient algebra or a subalgebra of 𝔄_{i}.

Next, we observe that G_{i} acts on Map(G_{i}, F^{c}) via translations, i.e.

f^{g}(h) = f(g + h) ∀f ∈ Map(G_{i}, F^{c}), g, h ∈ G_{i}.  (1.2)

The isomorphism G_{i}/G_{j} ≃ G_{i−j} induces identifications

F^{c}G_{i−j} = (F^{c}G_{i})^{G_{j}}  (1.3)

and

Map(G_{i−j}, F^{c}) = Map(G_{i}, F^{c})^{G_{j}}.  (1.4)

TORSION POINTS ON ELLIPTIC CURVES 5

The identifications (1.3) and (1.4) in turn induce isomorphisms of 𝔄_{i−j} with a subalgebra of 𝔄_{i} and 𝔅_{i−j} with a subalgebra of 𝔅_{i}. Proposition 1.1. There are isomorphisms

F_{Q}(j) ≃ Σ_{i−j}.F_{Q}(i)  (1.5)

as A_{j}-modules, and

ℭ_{Q}(j) ≃ \frac{Σ_{i−j}}{p^{i−j}} .ℭ_{Q}(i)  (1.6)

as 𝔄_{j}-modules. These isomorphisms are compatible with the inclusions ℭ_{Q}(j) ↪ F_{Q}(j) and ℭ_{Q}(i) ↪ F_{Q}(i).

Here Σ_{i−j} = Σ_{g∈G_{i−j}}g viewed as an element of A_{i}, and A_{j} (resp. 𝔄_{j}) acts on the right hand side of (1.5) (resp. (1.6)) via the homomorphism [p^{i−j}]. Proof. See §2 of [ST], especially Proposition 1. (It is assumed in [ST] that E has complex multiplication, but this assumption is not necessary for this particular result.) □

The following result is an immediate consequence of Proposition 1.1. Proposition 1.2. Suppose that ℭ_{Q}(i) is a globally free 𝔄_{i}-module. Then ℭ_{Q}(j) is 𝔄_{j}-globally free for all 0 < j < i. □

Proposition 1.2 implies that, in order to prove Theorem 1, we may replace p^{i} by a higher power of p. Let l and l′ be distinct odd primes not equal to p. Then, by replacing p^{i} by a higher power of p if necessary, we shall henceforth assume that

p^{i} ≡ 1 mod(ll′).  (1.7)

We next observe that it follows from the definition of ℭ_{Q}(i) that ψ_{i}(Q) in fact depends only upon the image of Q in E(F)/p^{i}E(F). Hence, in order to prove

6 A. AGBOOLA

Theorem 1, we may assume that Q ∈ E(F) is a p-power torsion point. We shall make this assumption from now on. Observe that, with this assumption, we have

p^{i}.(ℭ_{Q}(i)) = 0  (1.8)

in Cl(𝔄_{i}).

We shall now describe certain results concerning the algebras 𝔄_{i} and 𝔅_{i}.

For each prime 𝔮 of O_{F}, let G_{i,𝔮} denote the O_{F_{𝔮}}-group scheme obtained by localising and completing the O_{F} -group scheme afforded by G_{i} at 𝔮. Then there is an exact sequence

0 ⟶ G_{i,𝔮}^{0} ⟶ G_{i,𝔮} ⟶ G′_{i,𝔮} ⟶ 0  (1.9)

of O_{F,𝔮}-group schemes, where G_{i,𝔮}^{0} is the connected component of the identity of G_{i,𝔮}, and G′_{i,𝔮} is the maximal étale quotient G_{i,𝔮}/G_{i,𝔮}^{0}. The group scheme G′_{i,𝔮} is represented by the algebra 𝔅′_{i,𝔮} := Map(G′_{i,𝔮}(F_{𝔮}^{c}), O_{F_{𝔮}^{c}})^{Ω_{F𝔮}}; we remark that 𝔅′_{i,𝔮} is the unique maximal order in the algebra B′_{i,𝔮} := Map(G′_{i,𝔮}(F_{𝔮}^{c}), F_{𝔮}^{c})^{Ω_{F𝔮}}.

Let 𝔉(X, Y) ∈ O_{F_{𝔮}}[[X, Y]] denote the formal group afforded by the kernel of reduction modulo 𝔮 on E(F_{𝔮}). If b is an endomorphism of E, we write [b](X) for the power series corresponding to the induced endomorphism b of 𝔉(X, Y). Define an O_{F 𝔮}-ideal f_{𝔮} of O_{F_{𝔮}}[[X]] by

f_{𝔮} = [p^{i}](X).𝔒_{F_{𝔮}}[[X]].  (1.10)

Then the algebra 𝔅_{i,𝔮}^{0} := O_{F_{𝔮}}[[X]]/f_{𝔮} represents the group scheme G_{i}^{0} (see e.g. II §7 of [BT]). Here 𝔅_{i,𝔮}^{0} is viewed as an order in the algebra B_{i,𝔮}^{0} := Map(G_{i,𝔮}^{0}(F_{𝔮}^{c}), F_{𝔮}^{c})^{Ω_{F𝔮}} via the rule that

[b](X)(g) = [b](\underline{g})  (1.11)

TORSION POINTS ON ELLIPTIC CURVES 7

for g ∈ G_{i,𝔮}^{0}(F_{𝔮}^{c}), where \underline{g} denotes the parameter for g on the formal group 𝔉.

Let 𝔅_{i,𝔮} denote the local completion of the algebra 𝔅_{i} at 𝔮, and set B_{i,𝔮}(F) = B_{i,𝔮} = 𝔅_{i,𝔮} ⊗_{O_{F𝔮}} F_{𝔮}. The exact sequence (1.9) splits when restricted to the closed fibre of Spec(O_{F_{𝔮}}) (cf. e.g. [Sh] p.68). Hence there are isomorphisms of algebras

𝔅_{i,𝔮} ≃ 𝔅′_{i,𝔮} ⊗ 𝔅_{i,𝔮}^{0}  (1.12)

and

B_{i,𝔮} ≃ B′_{i,𝔮} ⊗ B_{i,𝔮}^{0}.  (1.13)

Write j : B_{i,𝔮} → B_{i,𝔮}^{0} for the obvious projection map. We record the following result for future use. Proposition 1.3. Let 𝔐_{i,𝔮} denote the maximal order in the algebra B_{i,𝔮}. Suppose that η ∈ 𝔐_{i,𝔮} is such that j(η) ∈ 𝔅_{i,𝔮}^{0}. Then η ∈ 𝔅_{i,𝔮}. Proof. This follows easily from the immediately preceding discusion together with the fact that 𝔅′_{i,𝔮} is the maximal order in B′_{i,𝔮}. □

It follows from the argument given in §2, Proposition 1 of [T] that the algebra 𝔄_{i} is described explicitly by

𝔄_{i} = {p^{−i} ∑_{g∈G_{i}} f(g)g : f ∈ 𝔅_{i}}.  (1.14)

If M/F is a finite extension, then, since E/F has everywhere good reduction, we have that

𝔅_{i}(M) = 𝔅_{i}(F) ⊗_{O_{F}} O_{M}, 𝔄_{i}(M) = 𝔄_{i}(F) ⊗_{O_{F}} O_{M}  (1.15)

and

𝔅_{i}(F) = 𝔅_{i}(M)^{Ω_{F}} , 𝔄_{i}(F) = 𝔄_{i}(M)^{Ω_{F}} .  (1.16)

8 A. AGBOOLA

We shall now give an alternative description of ℭ_{Q}(i) as a Q-twist of the algebra 𝔅_{i} (see §4 of [T]).

Let N/F be a finite extension containing the coordinates of G_{i} and G_{Q}(i). Then there is an isomorphism of N-algebras (and A_{i}-modules) B_{i}(N) ≃ N_{Q} induced by translation by any Q′ ∈ G_{Q}(i). So there is an isomorphism of N-algebras and A_{i}-modules given by

ξ : B_{i} ⊗_{F} N ⟶ F_{Q}(i) ⊗_{F} N  (1.17)

where ξ(b ⊗ n)(Q′ + g) = b(g)n for b ∈ B_{i}, n ∈ N, and g ∈ G_{i}. (Here Ω_{F} acts on both terms of (1.17) via the second factor.) Then we have

ℭ_{Q}(i) = [ξ(𝔅_{i} ⊗_{O_{F}} O_{N})]^{Ω_{F}} .  (1.18)

For any finite extension M/F, it follows that

ℭ_{Q}(i)(M) = ℭ_{Q}(i)(F) ⊗_{O_{F}} O_{M}, ℭ_{Q}(i)(F) = ℭ_{Q}(i)(M)^{Ω_{F}} .  (1.19)

We conclude this section by recalling the following result from [T] regarding a change of basefield. For any finite extension M/F, there is a commutative diagram

\begin{matrix}E(M) & ⟶^{ψ_{i,M}} & Cl(𝔄_{i}(M)) \\ ^{T r_{M/F}} ↓ & & ↓^{Res} \\ E(F) & ⟶^{ψ_{i,F}} & Cl(𝔄_{i}(F)) \end{matrix}  (1.20)

where T r_{M/F} is the trace map, and Res is the restriction map on classgroups defined in §4 of [T].

§2. Néron symbols and intersection multiplicities. In this section we shall recall some elementary facts concerning intersection theory on curves and arithmetic surfaces. For full details, we refer the reader to Chapter III of [L] and to [G].

TORSION POINTS ON ELLIPTIC CURVES 9

Let N be a number field, and let v denote a non-archimedean absolute value on N. Write π_{v} for a uniformising parameter in O_{N_{v}}, and suppose that v is normalised so that v(π_{v}) = 1. We write q_{v} for the cardinality of the residue field of N_{v}, and for each α ∈ N_{v}, we set |α|_{v} = q_{v}^{−v(α)}.

Suppose that C/N_{v} is a complete, non-singular, geometrically irreducible curve, and assume that C(N_{v}) is non-empty. Write Div^{0}(C(N_{v})) for the set of elements of degree zero in the free abelian group on C(N_{v}), and view this as a subgroup of Div^{0}(C/N_{v}).

If Z ∈ Div^{0}(C/N_{v}) is the divisor of a rational function f, and Z′ = ∑ n_{i}(P_{i}) is a 0-cycle in Div^{0}(C/N_{v}) whose support is disjoint from that of Z, then we set

f(Z′) = ∏_{i} f(P_{i})^{n_{i}}.  (2.1)

(It is easy to see that this is well-defined.)

The following theorem is due to Néron (see §3 of [G] for a proof). Theorem 2.1. Let Z ∈ Div^{0}(C(N_{v})), Z′ ∈ Div^{0}(C/N_{v}), and suppose that Z, Z′ have disjoint supports. Then one can define (in precisely one way) a real number < Z, Z′ >_{v} satisfying the following properties:

(1) The pairing < Z, Z′ >_{v} is bilinear.

(2) If Z′ = (f) is principal, then < Z, Z′ >_{v}= log|f(Z)|_{v}.

(3) The pairing is symmetric, i.e. < Z, Z′ >_{v}=< Z′, Z >_{v} if Z′ ∈ Div^{0}(C(N_{v})).

(4) Write supp(Z′) for the support of Z′, and fix P_{0} ∈ C(N_{v})\supp(Z′). Then the map

C(N_{v})\supp(Z′) ⟶ ℝ

defined by P ↦< (P) − (P_{0}), Z′ >_{v} is continuous. □

10 A. AGBOOLA

(We remark that the above pairing may also be defined when v is archimedean, but we shall not require this fact.)

Since we are assuming that v is non-archimedean, the pairing < Z, Z′ >_{v} may be described in terms of intersection multiplicities. In order to explain this, we shall first of all recall some basic facts concerning intersection theory on an arithmetic surface. (We refer the reader to Chapter III of [L] for further details.)

Set Y = Spec(O_{N}), and let X be a regular arithmetic surface over Y, with π : X → Y. Write 𝔒 for the structure sheaf of X. Suppose that W, W′ are effective divisors on X without common component, and let x be a closed point of X. The intersection number i_{x}(W, W′) is defined by

i_{x}(W, W′) = length(𝔒_{x}/(f, g)),  (2.1)

where length is that of an 𝔒_{x}-module, and W (resp. W′) is represented locally by a function f (resp. g) in a neighbourhood of x.

The intersection W.W′ is defined by

W.W′ = ∑_{x} i_{x}(W.W′)[x];  (2.2)

this is a finite sum and should be viewed as a 0-cycle on X.

Let y be a closed point of Y, with associated valuation v. Then the number i_{y}(W.W′) (which we shall also sometimes write i_{v}(W.W′)) is defined by

i_{y}(W.W′) = ∑_{x} i_{x}(W.W′)[k(x) : k(y)]  (2.3)

where the sum is taken over all x ∈ supp(W) ∩ supp(W′) lying above y. (Here k(x) (resp. k(y)) denotes the residue field at x (resp. y).)

TORSION POINTS ON ELLIPTIC CURVES 11

Now write X_{N} for the generic fibre of X, and let Div^{0}(X_{N}) denote the group of divisors on X_{N} of degree zero. For any irreducible divisor Z ∈ Div(X_{N}), its Zariski closure Z in X is an irreducible horizontal divisor on X. Conversely, every irreducible horizontal divisor on X arises in this way. Extend the association Z ↦ Z by linearity to all of Div(X_{N}); if Z = ∑_{i} n_{i}(P_{i}) is a divisor on X_{N}, we write Z = ∑_{i} n_{i}(P_{i}) for the associated irreducible horizontal divisor on X.

Suppose that Z, Z′ ∈ Div^{0}(X_{N}) and that all the components of Z, Z′ are rational over N. Write X_{N_{v}} for the generic fibre of X ×_{Y} Spec(O_{N_{v}}). Then Z, Z′ may be regarded as elements of Div^{0}(X_{N_{v}}) with components rational over N_{v}. Hence, if in addition Z, Z′ have disjoint supports, then we may refer to < Z, Z′ >_{v} as in Theorem 2.1 (cf. [L], Chapter III, Proposition 4.4).

Let X_{y} := π^{−1}(y) denote the fibre of π above y. We are now in a position to describe the Néron pairing in terms of intersection multiplicities (see §3 (especially (3.7)) of [G]). Theorem 2.2. With the above notation, suppose that Z, Z′ ∈ Div^{0}(X_{N}) have disjoint supports and that all the components of Z, Z′ are rational over N. Suppose further that X_{y} is smooth. Then,

< Z, Z′ >_{v}= −i_{y}(Z.Z′)log(q_{v}).  (2.4 □)

Remark. If X_{y} is not smooth, then the term i_{y}(Z.Z′) in (2.4) must be modified (cf.Chapter III, Theorem 5.2 of [L]).

§3. Intersection multiplicities on elliptic curves. We retain the notation of the previous section, and we suppose in addition that X is an elliptic surface over Y with X_{y} smooth for each closed point y of Y. (We continue to write v for

12 A. AGBOOLA

the valuation attached to y.)

Suppose that P, Q_{1}, Q_{2} are distinct points on X_{N}. We assume furthermore that these points are rational over N, and we set P′ = P + Q_{2}, Q′ = Q_{1} + Q_{2}. Since the map X → X given by T ↦ T + Q_{2} is an automorphism of the surface X, it follows that we have

i_{y}(P.Q_{1}) = i_{y}(P′.Q′).  (3.1)

Hence

i_{y}(P.Q_{1}) = i_{y}(\underline{O}.S),

where S = P − Q_{1} and \underline{O} is the origin of the group law on X.

This enables us to apply the following technique (see [S] p.85-91). Let ω ≠ 0 be a differential of the first kind on X_{N}, and choose a parameter z such that ω = dz + (higher order terms) about \underline{O}. (So we may take z = s/t, in terms of Weierstrass coordinates s, t on X_{N}.) Write P̃ for the reduction of P ∈ X_{N}. Then

i_{y}(\underline{O}.P) = 0 if P̃ ≠ ˜ \underline{O}

= v(z(P)) otherwise.

So the local intersection multiplicity i_{y}(\underline{O}.P) is determined by the parameter z(P) of P on the formal group about the origin. The following result is an immediate consequence of the preceding discussion. Proposition 3.1. Suppose that Z = (P), Z′ = (P′) are divisors on X_{N} which are rational over N. Then i_{y}(Z.Z′) is non-zero only if the points P, P′ coincide when reduced modulo v. □

§4. Special functions. We now return to the setup and notation established in §0 and §1. Thus, E/F is an elliptic curve with everywhere good reduction, l and

TORSION POINTS ON ELLIPTIC CURVES 13

l′ are distinct odd primes and Q ∈ E(F) is a p-power torsion point. (Recall that p > 3 is a prime satisfying p^{i} ≡ 1(modulo ll′).) The purpose of this section is to describe two special functions on E that will play a crucial role in the proof of Theorem 1.

Let E_{l} (resp.E_{l′}) denote the group of l (resp. l′) torsion points of E, and write F (E_{l}) for the field obtained by adjoining the coordinates of the points in E_{l} to F. Let θ and ϕ be two independent l-torsion points. Choose a function D_{θ,ϕ}, rational over F (E_{l}) such that the divisor of D_{θ,ϕ} is given by

(D_{θ,ϕ}) = ∑_{k=0}^{l−1}(kθ) − ∑_{k=0}^{l−1}(ϕ + kθ).  (4.1)

(In what follows, we shall write D for D_{θ,ϕ}.) D(z) and D(z + θ) have the same divisor, and so,

D(z + θ) = ω.D(z) ω ∈ F (E_{l}).  (4.2)

Since l.θ = \underline{O}, it follows that ω^{l} = 1.

We write

w : G_{i} × G_{i} ⟶ μ_{p^{i}}  (4.3)

for the Weil pairing on G_{i} × G_{i}. Suppose that ν ∈ G_{i}. Then we define a homomorphism χ_{ν} : G_{i} → μ_{p^{i}} by

χ_{ν}(γ) = w(l.γ, ν), γ ∈ G_{i}.  (4.4)

It follows that the characters of G_{i} are precisely the χ_{ν}’s.

Next, consider the function H(z) = D(p^{i}z)/D(z). H(z) has neither a zero nor a pole at z = \underline{O} and H(\underline{O}) = p^{i}. The following result is immediate.

14 A. AGBOOLA

Lemma 4.1. The divisor of the function H(z) is given by

(H(z)) = ∑_{g∈G_{i}\\underline{O}}[∑_{k=0}^{l−1}(kθ + g) − ∑_{k=0}^{l−1}(ϕ + kθ + g)]. □

Define the resolvent function R_{ν}(z) by

R_{ν}(z) = \frac{1}{p^{i}} ∑_{γ∈G_{i}} \frac{D(p^{i}z)}{D(z + γ)}χ_{ν}(−γ).  (4.5)

Note that R_{ν}(z) is well-defined independently of the choice of D, and that R_{ν}(\underline{O}) = 1. Also we have (using the fact that p^{i} ≡ 1(ll′)):

D(p^{i}(z + θ)) = D(p^{i}z + θ) = ω.D(p^{i}θ),  (4.6)

and

D(z + γ + θ) = ωD(z + γ).  (4.7)

Hence

R_{ν}(z + θ) = R_{ν}(z).  (4.8)

We shall now determine the divisor of R_{ν}(z). Proposition 4.2.

(a) If ν = \underline{O}, then R_{ν}(z) = 1.

(b) If ν ≠ \underline{O}, then

(R_{ν}(z)) = ∑_{g∈G_{i}}[∑_{k=0}^{l−1}(ν′ + kθ + g + ϕ) − ∑_{k=0}^{l−1}(kθ + g + ϕ)]

where ν′ is any point in E(F^{c}) satisfying [p^{i}]ν′ = ν.

Remark. This is Theorem 3 of [ST]. We give an argument below (which is somewhat different from that in [ST]) for the convenience of the reader and because

TORSION POINTS ON ELLIPTIC CURVES 15

we believe that the more algebraic techniques used here may be of use in treating the case of higher dimensional abelian varieties.

Proof.

(a) We have that

(D(p^{i}z)) = ∑_{g∈G_{i}}[∑_{k=0}^{l−1}(kθ + g) − ∑_{k=0}^{l−1}(ϕ + kθ + g)]  (4.9)

and

(D(z + γ)) = ∑_{k=0}^{l−1}(kθ − γ) − ∑_{k=0}^{l−1}(ϕ + kθ − γ).  (4.10)

Hence D(p^{i}z)^{−1} and ∑_{γ∈G_{i}} D(z + γ)^{−1} have the same polar divisor. It follows from (4.8) that R\underline{O}(z) gives a well-defined function on the elliptic curve E/ < θ >, with at worst a simple pole. This implies, via the Riemann-Roch theorem, that R\underline{O}(z) is constant. Thus we have

R\underline{O}(z) = R\underline{O}(\underline{O}) = 1.  (4.10a)

(b) Let f_{ν}(z) be a function on E whose divisor is given by

(f_{ν}(z)) = ∑_{g∈G_{i}}(ν′ + g) − ∑_{g∈G_{i}}(g)  (4.11)

where ν′ is any point in E(F^{c}) satisfying [p^{i}]ν′ = ν. Then we have

\frac{f_{ν}(z + λ)}{f_{ν}(z)} = w(λ, ν)  (4.12)

for all λ ∈ G_{i} (see e.g. [Si] Chapter III §8). Define

F_{ν}(z) = ∏_{k=0}^{l−1} f_{ν}(z + kθ).  (4.13)

16 A. AGBOOLA

Then

(F_{ν}(z)) = ∑_{k=0}^{l−1}[ ∑_{g∈G_{i}}(ν′ + g + kθ) − ∑_{g∈G_{i}}(g + kθ)]  (4.14)

and furthermore we have that

\frac{F_{ν}(z + γ)}{F_{ν}(z)} = w(l.γ, ν) = χ_{ν}(γ)  (4.15)

for all γ ∈ G_{i}. It follows from the definition of F_{ν}(z) that

F_{ν}(z + θ) = F_{ν}(z).  (4.16)

We also have that

R_{ν}(z + γ) = R_{ν}(z).χ_{ν}(γ)  (4.17)

for all γ ∈ G_{i}.

Next, we observe that

(D(p^{i}z)^{−1}) = ∑_{g∈G_{i}}[∑_{k=0}^{l−1}(ϕ + kθ + g) − ∑_{k=0}^{l−1}(kθ + g)]  (4.18)

and

(F_{ν}(z − ϕ)) = ∑_{g∈G_{i}}[∑_{k=0}^{l−1}(ν′ + ϕ + kθ + g) − ∑_{k=0}^{l−1}(kθ + g + ϕ)].  (4.19)

Hence

Polar divisor of \frac{F_{ν}(z − ϕ)}{D(p^{i}z)} = ∑_{g∈G_{i}} ∑_{k=0}^{l−1}(kθ + g).  (4.20)

Now

Polar divisor of \frac{R_{ν}(z)}{D(p^{i}z)} = Polar divisor of ∑_{γ∈G_{i}} \frac{1}{D(z + γ)}.χ_{ν}(−γ)
= ∑_{g∈G_{i}} ∑_{k=0}^{l−1}(kθ + g).  (4.21)

TORSION POINTS ON ELLIPTIC CURVES 17

Thus, it follows from (4.15)-(4.16) that the quotient F_{ν}(z − ϕ)/R_{ν}(z) yields a well-defined function on the elliptic curve E/ < G_{i}, θ > with at worst a simple pole. This implies, via the Riemann-Roch theorem, that F_{ν}(z −ϕ)/R_{ν}(z) is constant. So,

F_{ν}(z − ϕ) = c_{1}.R_{ν}(z),  (4.22)

and since R_{ν}(\underline{O}) = 1, it follows that c_{1} = F_{ν}(−ϕ). Thus we have

R_{ν}(z) = F_{ν}(−ϕ)^{−1}.F_{ν}(z − ϕ),  (4.23)

and now part (b) of the proposition follows. □

§5. Integrality results. We shall now use the results in §2 and §3 to obtain integrality statements concerning special values of the functions R_{ν}(z) and H(z).

For any finite extension N/F, let E/O_{N} denote the Néron minimal model of E/N. If Z = ∑_{i} n_{i}(P_{i}) is a divisor on the generic fibre E_{N}, then we write Z = ∑_{i} n_{i}(P_{i}) for the Zariski closure of Z on E. Our principal tool will be the following proposition. Proposition 5.1. Let f be a function on E with divisor Z, and suppose that Z′ ∈ Div^{0}(E). Assume also that all components of Z, Z′ are rational over N. Then for each finite place v of N, we have that

ord_{v}(f(Z′)) = i_{v}(Z.Z′).

In particular, f(Z′) is an algebraic integer if and only if i_{v}(Z.Z′) ≥ 0 for all finite v. Proof. This follows immediately from Theorem 2.1(2) and Theorem 2.2. □

Fix a choice of ν ∈ Ĝ_{i} with ν ≠ \underline{O} and set Z_{1} = (R_{ν}(z)), Z_{2} = (H(z)) (these are divisors on E); so

Z_{1} := ∑_{g∈G_{i}}[∑_{k=0}^{l−1}(ν′ + kθ + ϕ + g) − ∑_{k=0}^{l−1}(g + kθ + ϕ)]  (5.1)

18 A. AGBOOLA

and

Z_{2} := ∑_{g∈G_{i}\\underline{O}}[∑_{k=0}^{l−1}(kθ + g) − ∑_{k=0}^{l−1}(g + kθ + ϕ)].  (5.2)

(Note that the divisor Z_{1} depends upon our choice of ν, although we omit this dependence from our notation.)

Let ψ be a primitive l′-torsion point of E, and let β be any p-power torsion point of E. Define a divisor Z_{3} on E by

Z_{3} := (β + ψ) − (\underline{O}).  (5.3)

Now choose N/F to be a sufficiently large extension that all components of the Z_{i}’s are rational over N, and regard each Z_{i} as being a divisor on E_{N}. We observe that

(a) supp(Z_{3}) is disjoint from supp(Z_{1}) ∪ supp(Z_{2}).

(b) Z_{3} and Z_{i} (i = 1, 2) can only intersect on vertical fibres of E/O_{N} lying above places v dividing p, l or l′. This follows from Proposition 3.1 together with the fact that ∪_{j=1}^{3}supp(Z_{j}) contains only points that are killed by l, l′ or a power of p.

In this section, we shall evaluate i_{v}(Z_{3}.Z_{i}) (i = 1, 2) for all such finite places v of O_{N}. Proposition 5.2. If v|l, then

i_{v}(Z_{1}.Z_{3}) = ∑_{k=0}^{l−1} i_{v}((kθθθ + ϕ ϕ ϕ).(\underline{O})).

In all other cases, i_{v}(Z_{1}.Z_{3}) = 0. Proof. Suppose that v ∤ l. Then supp(Z_{1}) and supp(Z_{3}) are disjoint when reduced modulo v, and so we deduce from Proposition 3.1 that i_{v}(Z_{1}.Z_{3}) = 0 in this case.

If v|l, then it follows from the fact that the local intersection multiplicity is both bilinear and translation invariant (see §2 and §3) that we have

i_{v}(Z_{1}.Z_{3}) = ∑_{k=0}^{l−1} i_{v}((kθθθ + ϕ ϕ ϕ).(\underline{O})).

as asserted. □

We observe from Proposition 5.2 that i_{v}(Z_{1}.Z_{3}) ≥ 0 for all finite v, and that i_{v}(Z_{1}.Z_{3}) is independent of the choice of β and of ν ≠ \underline{O}. For any two algebraic numbers a and b, write a ∼ b if a/b is a unit. The following result is an immediate consequence of Propositions 5.1 and 5.2 together with equation (4.10)(a).

Proposition 5.3.

(a) R_{ν}(β + ψ) is an algebraic integer for all p-power torsion points β of E and all ν ∈ Ĝ_{i}. Furthermore, R_{ν}(β + ψ) is a unit at all primes not dividing l.

(b) Let β_{1}, β_{2} be any p-power torsion points of E. Then

R_{ν}(β_{1} + ψ) ∼ R_{ν}(β_{2} + ψ)

for all ν ∈ Ĝ_{i}.

Proof. This follows from the immediately preceding discussion together with the fact that R\underline{O}(z) = 1 for all z. □

We now turn our attention to the function H(z).

Proposition 5.4. If v|p, then

i_{v}(Z_{2}.Z_{3}) = − ∑_{g∈G_{i}\\underline{O}} i_{v}((g).(\underline{O})).

20 A. AGBOOLA

In all other cases, i_{v}(Z_{2}.Z_{3}) = 0. Proof. Suppose that v ∤ p. Then supp(Z_{2}) and supp(Z_{3}) are disjoint when reduced modulo v. Hence it follows from Proposition 3.1 that i_{v}(Z_{2}.Z_{3}) = 0 in this case.

If on the other hand v|p, then again it follows from Proposition 3.1 together with the fact that the local intersection multiplicity is both bilinear and translation invariant that

i_{v}(Z_{2}.Z_{3}) = − ∑_{g∈G_{i}\\underline{O}} i_{v}((g).(\underline{O})). □

Proposition 5.5. H(β + ψ) is a global unit for all p-power torsion points β. Proof. Recall that H(\underline{O}) = p^{i}. From Proposition 5.1 applied to Z_{2} and Z_{3}, we have, for each finite place v of N

ord_{v}(\frac{H(β + ψ)}{H(\underline{O})}) = i_{v}(Z_{2}.Z_{3})

i.e

ord_{v}(\frac{H(β + ψ)}{p^{i}}) = − ∑_{g∈G_{i}\\underline{O}} i_{v}((g).(\underline{O})) if v|p
= 0 otherwise.  (5.4)

for all p-power torsion points β.

Set β = \underline{O}. Then H(ψ) = D(p^{i}ψ)/D(ψ) = 1 (since p^{i} ≡ 1(ll′)), and so we have

ord_{v}(p^{−i}) = − ∑_{g∈G_{i}\\underline{O}} i_{v}((g).(\underline{O})) if v|p
= 0 otherwise.  (5.5)

From (5.4) and (5.5) we deduce that ord_{v}(H(β + ψ)) = 0 for all finite v and all p-power torsion points β. Hence H(β + ψ) is a global unit, as asserted. □

TORSION POINTS ON ELLIPTIC CURVES 21

§6. Proof of Theorem 1. In this section we shall use our earlier results to give a proof of Theorem 1. Let us begin by recalling the salient facts concerning the situation with which we are dealing.

We assume that E/F is an elliptic curve with everywhere good reduction, and Q ∈ E(F) is a p-power torsion point. The numbers l and l′ are distinct odd primes, and we suppose further that p^{i} ≡ 1(ll′). We have that θ and ϕ are independent l-torsion points, while ψ is a primitive l′-torsion point of E.

Let M be the field F (E_{ll′}), and define a function h on E by

h(z) = \frac{D(p^{i}z + ψ)}{D(z + ψ)}.  (6.1)

Then the functions D(z) and h(z) both lie in the function field M(E). Observe that h(z) = H(z + ψ), since ψ is an l′-division point and p^{i} ≡ 1(ll′). Lemma 6.1. For the field M as above, we have

[M : F]|[l(l + 1)(l − 1)^{2}][l′(l′ + 1)(l′ − 1)^{2}]

Proof. The group Gal(M/F) is a subgroup of GL_{2}(𝔽_{l}) × GL_{2}(𝔽_{l′}). The result now follows from the fact that for any prime q, the group GL_{2}(𝔽_{q}) is of order q(q + 1)(q − 1)^{2}. □ Lemma 6.2. Let w = HCF {l(l + 1)(l − 1)^{2}|l ≠ p is an odd rational prime}. Suppose that q > 3 is a prime. Then q ∤ w. Proof. Choose a prime l_{1} such that l_{1} ≡ 3(q) and l_{1} ≠ p.. (This may be done, via Dirichlet’s theorem on primes in an arithmetic progression.) Then q ∤ l_{1}(l_{1} +1)(l_{1} − 1)^{2}, and the result follows. □

22 A. AGBOOLA

Next, we observe that if Q is any p-power torsion point in E(F), the function h defines an element h_{Q} of M_{Q}(i) by the rule

h_{Q}(Q′) = h(Q′), Q′ ∈ G_{Q}(i).  (6.2)

Note that h_{Q}(Q′) is always finite. If we take Q = \underline{O}, then we have that G\underline{O}(i) = G_{i}, and (6.2) defines a function h\underline{O} on G_{i}. We define a resolvend element ρ ∈ A_{i}(M) by

ρ = \frac{1}{p^{i}} ∑_{g∈G_{i}} h\underline{O}(g)g^{−1}.  (6.3)

Proposition 6.3. Let Q ∈ E(F) be a p-power torsion point. Then h_{Q} ∈ ℭ_{Q}(i)(M). Proof. Let N be some large extension of M containing the coordinates of all points of G_{i} and G_{Q}(i). From §1 (see (1.17)-(1.19)) it follows that ℭ_{Q}(i)(M) = ξ(𝔅_{i}(N))∩ M_{Q}(i). Hence, since h_{Q} ∈ M_{Q}(i), the result will follow if we show that h_{Q} ∈ ξ(𝔅_{i}(N)).

Let M_{i}(N) denote the maximal order in the algebra B_{i}(N). Proposition 5.5 implies that ξ^{−1}(h_{Q}) ∈ M_{i}(N)^{x} (cf. (1.17)). We shall show that ξ^{−1}(h_{Q}) ∈ 𝔅_{i}(N)_{𝔮} for each prime 𝔮 of O_{N}.

Set η = ξ^{−1}(h_{Q}) and suppose that 𝔮 ∤ p. Then M_{i}(N)_{𝔮} = 𝔅_{i}(N)_{𝔮}, and so we have that η ∈ 𝔅_{i}(N)_{𝔮}, as required.

Suppose now that 𝔮|p. It follows from Proposition 1.3 that, in order to show that η ∈ 𝔅_{i}(N)_{𝔮}, it suffices to show that j(η) ∈ 𝔅_{i,𝔮}^{0}(N). For this we use an argument motivated by that given in Proposition 5 of [ST]. I am grateful to the referee for supplying some of the details given below.

Set Z_{4} = (h(z)), a divisor on E_{N}, and write Z_{4} for the associated divisor on the Néron model E/O_{N}. We claim that the fibre E_{𝔮} of E over 𝔮 does not occur in Z_{4}.

TORSION POINTS ON ELLIPTIC CURVES 23

For observe that this is the same as saying that ord_{w(𝔮)}(h(z)) = 0, where ord_{w(𝔮)} is the discrete valuation of the function field N(E) corresponding to the generic point of E_{𝔮}. Now

h(z) = H(z + ψ) = \frac{D(p^{i}(z + ψ))}{D(z + ψ)} = \frac{[p^{i}]^{∗}(D(z + ψ))}{D(z + ψ)},

where [p^{i}]^{∗} : N(E) → N(E) is the algebra homomorphism induced by the morphism [p^{i}] : E → E. Since [p^{i}]^{∗} fixes a uniformiser for O_{N,𝔮}, which is also a uniformiser in O_{E,w(𝔮)} (where O_{E} denotes the structure sheaf of E), we see that ord_{w(𝔮)}([p^{i}]^{∗}f) = ord_{w(𝔮)}(f) for all f ∈ N(E). Hence

ord_{w(𝔮)}(h(z)) = ord_{w(𝔮)}([p^{i}]^{∗}D(z + ψ)) − ord_{w(𝔮)}(D(z + ψ)) = 0,

as asserted.

Now let Ẽ denote the reduction of E modulo 𝔮, and write k for the residue field of O_{N} at 𝔮. Since h(z) ∈ O_{E,w(𝔮)}^{x}, we may define h̄(z) ∈ k^{c}(Ẽ) to be the (non-zero) image of h(z) in the residue field of w(𝔮) (noting that this latter field is naturally contained in k^{c}(Ẽ)). The divisor of h̄(z) on Ẽ is contained in the intersection of the fibre E_{𝔮} with the divisor Z_{4}. All points appearing in the divisor of h̄(z) have order dividing ll′, and so cannot be p-power torsion points.

Consider the formal group afforded by E_{1}(N_{𝔮}), the kernel of reduction of E modulo 𝔮. Let t = t(z) denote the parameter of a point z ∈ E_{1}(N_{𝔮}) on this formal group, and write α′ for the parameter of any p-power torsion point in E_{1}(N_{𝔮}). For any point z ∈ E_{1}(N_{𝔮}), we have the following expansion of h̄ as a Laurent series in k^{c}[[t]]:

h̄(z + α′) = ∑_{i=0}^{∞} b̄_{i}t^{i}

24 A. AGBOOLA

with b̄_{0} ≠ 0, since the divisor of h̄ contains no p-power torsion points. Hence we have that h(z + α′) is a unit in O_{N,𝔮}[[t]], i.e.

h(z + α′) = ∑_{i=0}^{∞} b_{i}t^{i},

with b_{0} ∈ O_{N,𝔮}^{x} and b_{i} ∈ O_{N,𝔮}. It therefore follows that j(η) ∈ 𝔅_{i,𝔮}^{0}(N), as asserted. This completes the proof. □

The following corollary is now an immediate consequence of (1.14). Corollary 6.4. The resolvend element ρ lies in 𝔄_{i}(M). □

We now prove a special case of Theorem 1. Theorem 6.5. Let Q ∈ E(F) be a p-power torsion point. Then

ℭ_{Q}(i)(M).ρ = h_{Q}.𝔄_{i}(M),

and so ℭ_{Q}(i)(M) is 𝔄_{i}(M)-free. Proof. The proof of this assertion is similar to that of Theorem 5 of [ST].

We shall show that the equality holds everywhere locally; this will imply the desired result. Let 𝔮 be a prime of O_{M}. Then we may write ℭ_{Q}(i)(M)_{𝔮} = x_{𝔮}.𝔄_{i}(M)_{𝔮}. For some λ_{𝔮} ∈ A_{i}(M_{𝔮}), we have

x_{𝔮}.ρλ_{𝔮} = h_{Q}.  (6.4)

We shall show that in fact λ_{𝔮} ∈ 𝔄_{i}(M)_{𝔮}^{x}; this will establish the result.

We begin by observing that, since h_{Q} ∈ ℭ_{Q}(i)(M) and 𝔄_{i}(M)_{𝔮} = {x ∈ A_{i}(M_{𝔮}) : ℭ_{Q}(i)(M)_{𝔮}.x ⊆ ℭ_{Q}(i)(M)_{𝔮}}, we have that

ρλ_{𝔮} ∈ 𝔄_{i}(M)_{𝔮}.  (6.5)

TORSION POINTS ON ELLIPTIC CURVES 25

Next, recall that if x ∈ M_{Q} and ν ∈ G_{i}, then we have the Lagrange resolvent

(x|χ_{ν}) = ∑_{g∈G_{i}} x^{g}χ_{ν}(g^{−1}).  (6.6)

If g′ ∈ G_{i}, then (x^{g′}|χ_{ν}) = (x|χ_{ν}).χ_{ν}(g′), and so for each λ ∈ ℚ^{c}G_{i}, we have

(xλ|χ_{ν}) = (x|χ_{ν}).χ_{ν}(λ).  (6.7)

Hence (6.4) implies that

(x_{𝔮}|χ_{ν}).χ_{ν}(ρ).χ_{ν}(λ_{𝔮}) = (h_{Q}|χ_{ν}).  (6.8)

Now the proof of Theorem 3 of [T] implies that if Q′ ∈ G_{Q}(i), then (x_{𝔮}|χ_{ν})(Q′) ∼ p^{i}. Also, we have that χ_{ν}(ρ) = R_{ν}(ψ), and (h_{Q}|χ_{ν})(Q′) = p^{i}R_{ν}(Q′ + ψ). Hence, evaluating (6.8) at Q′ ∈ G_{Q}(i), we obtain

p^{i}R_{ν}(ψ).χ_{ν}(λ_{𝔮}) ∼ p^{i}R_{ν}(Q′ + ψ).  (6.9)

Next, we recall (see Proposition 5.3) that

R_{ν}(ψ) ∼ R_{ν}(Q′ + ψ) ∼ \{ \begin{matrix}1 if ν = \underline{O} \\ an l − unit if ν ≠ \underline{O}. \end{matrix}  (6.10)

This implies that χ_{ν}(λ_{𝔮}) is always a unit, and that χ_{ν}(ρ) is also a unit at 𝔮 unless ν ≠ \underline{O} and 𝔮|l. There are now two cases to consider.

Case 1. Suppose that 𝔮 ∤ l. Corollary 6.4 implies that ρ ∈ 𝔄_{i}(M)_{𝔮}. Since also χ_{ν}(ρ) is a unit for all ν in this case, it follows that in fact ρ ∈ 𝔄_{i}(M)_{𝔮}^{x}. Thus, from (6.4), it follows that λ_{𝔮} ∈ 𝔄_{i}(M)_{𝔮}, and so we have that λ_{𝔮} ∈ 𝔄_{i}(M)_{𝔮}^{x} since χ_{ν}(λ_{𝔮}) is a unit for all ν ∈ G_{i}.

Case 2. Suppose that 𝔮|l. Then, since l ∤ |G_{i}|, it follows that 𝔄_{i}(M)_{𝔮} is the maximal order. Now we deduce that λ_{𝔮} ∈ 𝔄_{i}(M)_{𝔮}^{x} since the χ_{ν}(λ_{𝔮}) are units for all ν ∈ G_{i}. □

We are now in a position to prove Theorem 1.

Consider the trace-restriction square (1.20):

\begin{matrix}E(M) & ⟶^{ψ_{i,M}} & Cl(𝔄_{i}(M)) \\ ^{T r_{M/F}} ↓ & & ↓^{Res} \\ E(F) & ⟶^{ψ_{i,F}} & Cl(𝔄_{i}(F)) \end{matrix}  (6.11)

If Q ∈ E(F) is a p-power torsion point, we may regard Q as lying in E(M), and Theorem 6.5 implies that

ψ_{i,F} (T r(Q)) = Res(ψ_{i,M}(Q)) = 0  (6.12)

Next, we observe that we also have

ψ_{i,F} (T r(Q)) = ψ_{i,F} ([M : F]Q) = [M : F].ψ_{i,F} (Q)  (6.13)

Hence we obtain that

[M : F]ψ_{i,F} (Q) = 0  (6.14)

i.e.

[M : F](ℭ_{Q}(i)) = 0.  (6.15)

Now let l and l′ vary among all odd primes not equal to p. Then it follows from Lemmas 6.1 and 6.2 that w^{2}ψ_{i,F} (Q) = 0. Now recall that |G_{i}|ψ_{i,F} (Q) = 0 (see (1.8)). Since p > 3, we have that (|G_{i}|, w) = 1 by Lemma 6.2, and so finally we deduce that (ℭ_{Q}(i)(F)) = 0 in Cl(𝔄_{i}(F)). This completes the proof of Theorem 1. □

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

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