A NOTE ON ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE IN GLOBAL FUNCTION FIELDS

A. Agboola Version of June 17, 1995

§0. Introduction and statement of results. The purpose of this paper is to study the Galois module structure of certain Kummer orders obtained by dividing torsion points on an elliptic curve defined over a global function field.

For any field L, we shall write L^{c} for a separable closure of L, and Ω_{L} for Gal(L^{c}/L). Let r be a prime, and let 𝔽_{r} denote the finite field containing r elements. Let k be a field such that either k ⊆ 𝔽_{r}^{c} or k ⊆ ℂ. Suppose that C is a smooth, geometrically irreducible curve defined over k. Set F = k(C), the function field of C over k. Let S = {v_{1}, …, v_{t}} be a fixed, non-empty, finite set of places of F, and let O_{F,S} = O_{F} denote the ring of functions in F which are regular away from S. O_{F} is the function field analogue of the ring of integers of a number field. Write O^{c} for the integral closure of O_{F} in F^{c}. If L/F is any finite extension of F, then we shall write O_{L} for the integral closure of O_{F} in L.

Let E/F be an abelian variety defined over F. In what follows, we shall always assume that S contains all places of bad reduction of E. We shall also suppose that all endomorphisms of E that we consider are defined over F. We write \underline{O} for the origin of the group law on E.

Typeset by AMS-TEX 2 A. AGBOOLA

Partially supported by an NSF Postdoctoral Research Fellowship.

Let p > 3 be a rational prime with p ≠ r if k ⊆ 𝔽_{r}^{c}, and write G_{i} for the subgroup of elements of E(F^{c}) which are killed by the endomorphism [p^{i}] of E. The O_{F} - group scheme of p^{i}-torsion points on E is affine and étale and is therefore equal to Spec(𝔅_{i}(F)), where 𝔅_{i}(F) = 𝔅_{i} = Map(G_{i}, O^{c})^{Ω_{F}} is the O_{F} -Hopf algebra consisting of Ω_{F} -maps from G_{i} to O^{c}. (Thus 𝔅_{i} is the unique O_{F} -maximal order in the algebra B_{i}(F) := Map(G_{i}, F^{c})^{Ω_{F}}.) It follows that the O_{F} -Cartier dual of 𝔅_{i} is 𝔄_{i}(F) = 𝔄_{i} = (O^{c}G_{i})^{Ω_{F}} (here Ω_{F} acts on both O^{c} and G_{i}). 𝔄_{i}(F) is thus the unique O_{F} -maximal order in the F -algebra A_{i}(F) = A_{i} = (F^{c}G_{i})^{Ω_{F}}.

Now 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}.

The F -algebra structure of F_{Q}(i) may be described as follows. Let Q^{′(1)}, …, Q^{′(s)} be a set of representatives of the Ω_{F} -orbits of G_{Q}(i). Then, as an F -algebra, we have

F_{Q}(i) ≃ ∏_{i=1}^{s} F [Q^{′(i)}]

where F [Q^{′(i)}] is the field obtained by adjoining the coordinates of Q^{′(i)} to F. Explicitly, the isomorphism is given by f ↦ ∏_{i=1}^{s} f(Q^{′(i)}) for f ∈ F_{Q}(i). Note also that if G_{i} ⊆ E(F), then all the fields F [Q^{′(i)}] are the same.

ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE 3

Let O_{Q}(i) denote the integral closure of O_{F} in F_{Q}(i). Then O_{Q}(i) (the Kummer order) is an 𝔄_{i}-module. As 𝔄_{i} is the maximal order of A_{i}, it follows that O_{Q}(i) is a locally free 𝔄_{i}-module (see eg [CR] Proposition 31.2). Thus, if Cl(𝔄_{i}) denotes the locally free classgroup of 𝔄_{i}, then we have a map

ψ_{i} : E(F) ⟶ Cl(𝔄_{i})  (0.4)

given by ψ_{i}(Q) = (O_{Q}(i)), where (O_{Q}(i)) is the class of O_{Q}(i) in Cl(𝔄_{i}). As E has good reduction at all places of O_{F}, it follows exactly as per Theorem 1 of [T] that ψ_{i} is a homomorphism, and so in particular that the image of ψ_{i} is annihilated by |G_{i}|. Observe that, since G_{i} is abelian, 𝔄_{i} satisfies that Eichler condition. Hence O_{Q}(i) is a globally free 𝔄_{i}-module if and only if ψ_{i}(Q) = 0.

We are now able to state the main result of this paper. Theorem 1. Suppose that E is an elliptic curve. Then E(F)_{torsion} ⊆ ker(ψ_{i}).

.

Theorem 1 is the function field analogue of a conjecture first stated by M. J. Taylor for CM elliptic curves over number fields (see [T]). A large part of this conjecture (for CM elliptic curves) was proved in [ST]. The main technique of proof in [ST] was the use of modular functions and the q-expansion principle to prove integrality statements concerning certain resolvent elements that arise as special functions on E. A different proof relying upon elementary intersection theory rather than modular functions, and valid for all elliptic curves with everywhere good reduction was given in [A1]. The techniques used in proving Theorem 1 of the present paper are very similar to (albeit somewhat easier than) those used in treating the corresponding result over number fields as described in [A1]. For further results on the

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class invariant homomorphism over function fields, we refer the reader to [A2].

§1 Preliminary results. In this section we recall certain preliminary results concerning Kummer orders that we shall require.

We first of all give an alternative description of O_{Q}(i) as a Q-twist of the algebra 𝔅_{i} (cf. §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}(i) 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  (1.1)

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 (1.1) via the second factor.) Then we have

O_{Q}(i) = [ξ(𝔅_{i} ⊗_{O_{F}} O_{N})]^{Ω_{F}} .  (1.2)

For any finite extension M of F, it follows that

O_{Q}(i)(M) = O_{Q}(i)(F) ⊗_{O_{F}} O_{M}, O_{Q}(i)(F) = O_{Q}(i)(M)^{Ω_{F}} .  (1.3)

We shall now describe the relationship between (O_{Q}(i)) ∈ Cl(𝔄_{i}) and (O_{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).  (1.4)

ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE 5

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 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})andg, h ∈ G_{i}.  (1.5)

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

F^{c}G_{i−j} = (F^{c}G_{i})^{G_{j}}  (1.6)

and

Map(G_{i−j}, F^{c}) = Map(G_{i}, F^{c})^{G_{j}}.  (1.7)

The identifications (1.6) and (1.7) 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)  (1.8)

as A_{j}-modules, and

O_{Q}(j) ≃ Σ_{i−j}.O_{Q}(i)  (1.9)

as 𝔄_{j}-modules. These isomorphisms are compatible with the inclusions O_{Q}(j) → F_{Q}(j) and O_{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 (1.8) (resp. (1.9)) via the surjective homomorphism [p^{i−j}]. Proof. Via (1.3), together with the fact that ⊗_{O_{F}} O_{N} is faithfully flat, we may assume that the field F contains the coordinates of G_{i} and G_{Q}(i). Next, we observe

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that (1.2) allows us to assume in addition that P = \underline{O}, i.e. that O_{Q}(i) = 𝔅_{i} and O_{Q}(j) = 𝔅_{j}. The result now follows via the discussion immediately preceding the statement of Proposition 1.1. □

Proposition 1.1 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. Suppose further that (r, ll′) = 1 if k ⊆ 𝔽_{r}. Then, by replacing p^{i} by a higher power of p if necessary, we shall henceforth assume that

p^{i} ≡ 1 mod(ll′).  (1.10)

We next observe that it follows from the definition of O_{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 Theorem 1, we may in fact 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  (1.11)

in Cl(𝔄_{i}).

Our next result deals with the valuations of certain Lagrange resolvents. Suppose that a ∈ F_{Q}(i) and χ is a character of G. The resolvent of a at χ is defined by

(a|χ) = ∑_{g∈G_{i}} χ(g^{−1})a^{g} ∈ F_{Q}(i)

For each place v of O_{F}, let O_{F,v} (resp. O_{Q,v}(i), resp. 𝔄_{i,v}(F)) denote the semilocal completion of O_{F} (resp. O_{Q}(i), resp. 𝔄_{i}(F)) at v. Choose a_{v} ∈ O_{Q,v}(i) such that O_{Q,v}(i) = a_{v}.𝔄_{i,v}(F). As E/F has good reduction at all places of O_{F}, it follows from the criterion of Néron-Ogg-Shafarevitch and the description of F_{Q}(i)

ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE 7

given in §0 that F_{Q}(i)/F is unramified at all places of O_{F}. The following result is a simple extension of Proposition 4.3 in Chapter 1 of [F] from fields to Galois algebras. Proposition 1.3. Let a_{v} be as above. Then for all χ ∈ Ĝ_{i}, we have that (a_{v}|χ) ∈ O_{Q,v}(i)^{∗}. □

We conclude this section by recalling the following result from [T] regarding a change of basefields. (The result in [T] is proved for number fields, but it is easy to see that the proof given there carries over to our present situation.) 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}  (1.12)

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

§2. Intersection multiplicities and an integrality principle. In this section we shall describe a method for proving integrality statements about special values of functions defined on E.

Write Div^{0}(E) for the group of divisors of degree zero on E. If Z ∈ Div^{0}(E) is the divisor of a rational function f on E, and Z′ = ∑ n_{i}(P_{i}) is a 0-cycle on E 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.

Let E/O_{F} denote the Nèron minimal model of E/F over O_{F}. If Z = ∑_{i} n_{i}(P_{i}) is any divisor on the generic fibre E_{F} of E, then we write Z = ∑_{i} n_{i}(P_{i}) for the Zariski

8 A. AGBOOLA

closure of Z on E. If v is a place of O_{F}, and D_{1}, D_{2} are horizontal divisors on E which intersect properly, then we write i_{v}(D_{i}.D_{2}) for the intersection multiplicity of D_{1} and D_{2} at v (see e.g. Chapter III of [L] for definitions and further details regarding intersection multiplicities). Our principal tool for proving integrality results will be the following proposition. Proposition 2.1. Let f be a function on E with divisor Z, and suppose that Z′ ∈ Div^{0}(E) with supp(Z) disjoint from supp(Z′). (Here supp(Z) denotes the support of Z, with similar notation for Z′.) Assume that all components of Z, Z′ are rational over F. Then for each place v of O_{F}, we have that

ord_{v}(f(Z′)) = i_{v}(Z.Z′)  (2.2)

In particular, f(Z′) is integral at v if and only if i_{v}(Z.Z′) ≥ 0. Proof. This proposition simply summarises certain elementary facts concerning intersection multiplicities. See e.g. Chapter III of [L] (especially Theorems 5.1 and 5.2) for full details. □

Let us now explain how we use this proposition. Suppose that P_{1}, P_{2} are distinct torsion points on E(F), with each of order prime to the characteristic of F. Let v be a place of O_{F}. Then P_{1}, P_{2} remain disjoint when reduced modulo v, and so we have that i_{v}(P_{1}.P_{2}) = 0. The following result is now immediate. Proposition 2.2. Let the notation be as in Proposition 2.1. Suppose in addition that supp(Z) and supp(Z′) consist of disjoint sets of torsion points of E(F) of order prime to the characteristic of F. Then

ord_{v}(f(Z′)) = i_{v}(Z.Z′) = 0  (2.3)

ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE 9

for each place v of O_{F}. Hence f(Z′) is integral over O_{F}. □

We conclude this section by introducing a piece of notation. Suppose that a, b ∈ O^{c}. Then we write a ∼ b if a/b ∈ O^{c∗}.

§3 Special functions The purpose of this section is to describe two special functions that will play a major role in the proof of Theorem 1. These functions are the same as those used to prove the corresponding result in the number field case (cf. §4 of [A1]).

Recall that E/F is an elliptic curve with good reduction at all places of F not in S. The numbers l and l′ are distinct odd primes not equal to p or r, and Q ∈ E(F) is a p-power torsion point. We suppose further that p satisfies p^{i} ≡ 1(ll′).

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).  (3.1)

(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}).  (3.2)

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

Write

w : G_{i} × G_{i} ⟶ μ_{p^{i}}  (3.3)

for the Weil pairing on G_{i} × G_{i}. Suppose that ν ∈ G_{i}. Then we define a homomor-

10 A. AGBOOLA

phism χ_{ν} : G_{i} → μ_{p^{i}} by

χ_{ν}(γ) = w(l.γ, ν), γ ∈ G_{i}.  (3.4)

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. Lemma 3.1. 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).  (3.5)

Note that R_{ν}(z) is well-defined independently of the choice of D, and that R_{ν}(\underline{O}) = 1. Our next result tells us about the divisor of R_{ν}(z). Proposition 3.2.

(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. Proof. This may be proved exactly as in the number field case. We refer the reader to Proposition 4.2 of [A1] for details. □

§4. 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).

We retain the notation of the previous sections. Fix a choice of ν ∈ G_{i} with ν ≠ 0, and set Z_{1} = (R_{ν}(x)), Z_{2} = (H(z)) (these are divisors on E_{F}); so,

Z_{1} := ∑_{g∈G_{i}}[∑_{k=0}^{l−1}(ν′ + kθ + ϕ + g) − ∑_{k=0}^{l−1}(g + kθ + ϕ)]  (4.1)

and

Z_{2} := ∑_{g∈G_{i}\\underline{O}}[∑_{k=0}^{l−1}(kθ + g) − ∑_{k=0}^{l−1}(g + kθ + ϕ)].  (4.2)

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

Now 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_{F} by

Z_{3} = (β + ψ) − (\underline{O}).  (4.3)

Choose N/F to be a sufficiently large extension so that all components of Z_{1}, Z_{2}, Z_{3} 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) do not intersect on any vertical fibre of E. This is because, for each place v of O_{L}, the divisors Z_{3} and Z_{i} (i = 1, 2) remain distinct when reduced modulo v.

The following result is an immediate consequence of these observations. Proposition 4.1. Let v be a place of O_{N}. Then i_{v}(Z_{i}.Z_{2}) = 0 for 1 = 1, 2. □

By combining Proposition 4.1 with Proposition 2.2, we obtain the following result.

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Proposition 4.2. (a) R_{ν}(β + ψ) ∈ O^{c∗} for all p-power torsion points β of E and all ν ∈ G_{i}. Thus, if β_{1}, β_{2} are any p-power torsion points of E, then we have in particular that

R_{ν}(p^{i}(β_{1} + ψ)) ∼ R_{ν}(p^{i}(β_{2} + ψ)) ∼ 1

for all ν ∈ G_{i}.

(b) H(β + ψ) ∈ O^{c∗} for all p-power torsion points β of E. □

§5. Proof of Theorem 1. In this section we shall use our earlier results to give a proof of Theorem 1. The method used is the same as in the number field case.

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)}.  (5.1)

Then the functions D(z) and h(z) both lie in the function field M(E).

Lemma 5.1. 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 5.2. Let 𝔖 be the set of odd rational primes satisfying l ≠ p and (if k ⊆ 𝔽_{r}) l ≠ r. (If k ⊆ ℂ then we simply ignore this latter condition.) Let w = HCF {l(l + 1)(l − 1)^{2}|l ∈ 𝔖}. Suppose that q > 3 is a prime. Then q ∤ w.

ELLIPTIC CURVES AND GALOIS MODULE STRUCTURE 13

Proof. Choose a prime l_{1} ∈ 𝔖 such that l_{1} ≡ 3(q). (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. □

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).  (5.2)

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 (5.2) 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}.  (5.3)

Proposition 5.3. Let Q ∈ E(F) be a p-power torsion point. Then h_{Q} ∈ O_{Q}(i)(M). Proof. Let N be some finite extension of F containing the coordinates of all points of G_{i} and G_{Q}(i). From §1 (see (1.1)-(1.3)) it follows that O_{Q}(i)(M) = ξ(𝔅_{i}(N))∩M_{Q}. Hence, since h_{Q} ∈ M_{Q}, the result will follow if we show that h_{Q} ∈ ξ(𝔅_{i}(N)). But this is certainly the case, because 𝔅_{i}(N) is the unique O_{N}-maximal order in B_{i}(N), and so Proposition 4.2(b) implies that ξ^{−1}(h_{Q}) ∈ 𝔅_{i}(N)^{∗}. □

Recall that 𝔄_{i}(M) is the unique O_{M}-maximal order in A_{i}(M). The following corollary is an immediate consequence of this fact. Corollary 5.4. The resolvend element ρ lies in 𝔄_{i}(M). □

We now prove a special case of Theorem 1. Theorem 5.5. Let Q ∈ E(F) be a p-power torsion point. Then

O_{Q}(i)(M).ρ = h_{Q}.𝔄_{i}(M),

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and so O_{Q}(i)(M) is 𝔄_{i}(M)-free. Proof. We shall show that the equality holds everywhere locally; this will imply the desired result.

Let v be a place of O_{M}. Then we may write O_{Q,v}(i)(M) = x_{v}𝔄_{i,v}(M). For some λ_{v} ∈ A_{i}(M_{v}), we have

x_{v}.ρλ_{v} = h_{Q}  (5.4)

We shall show that in fact λ_{v} ∈ 𝔄_{i,v}(M)^{∗}; this will establish the result. 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}).  (5.5)

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

(xλ|χ_{ν}) = (x|χ_{ν}).χ_{ν}(λ).  (5.6)

Hence, (5.4) implies that

(x_{𝔮}|χ_{ν}).χ_{ν}(ρ).χ_{ν}(λ_{𝔮}) = (h_{Q}|χ_{ν}).  (5.7)

Now Proposition 1.3 implies that if Q′ ∈ G_{Q}(i), then (x_{ν}|χ_{ν})(Q′) ∼ 1. Also, we have that χ_{ν}(ρ) = R_{ν}(ψ) ∼ 1 and (h_{Q}|χ_{ν})(Q′) = p^{i}R_{ν}(Q′ +ψ) ∼ 1 (cf. Proposition 4.2). Hence, evaluating (5.7) at Q′ ∈ G_{Q}(i), we obtain

χ_{ν}(λ_{ν}) ∼ 1.  (5.8)

Therefore λ_{ν} ∈ 𝔄_{i,v}(M)^{∗}, and this implies the desired result. □

We now prove Theorem 1.

Consider the trace-restriction square (1.12):

\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 5.5 implies that

ψ_{i,F} (T r(Q)) = Res(ψ_{i,M}(Q)) = 0  (5.10)

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)  (5.11)

Hence we obtain that

[M : F]ψ_{i,F} (Q) = 0  (5.12)

i.e.

[M : F](O_{Q}(i)) = 0.  (5.13)

Now let l and l′ vary among all odd primes not equal to p or r. Then it follows

from Lemma 5.1 that w^{2}ψ_{i,F} (Q) = 0. Now recall that p^{i}ψ_{i,F} (Q) = 0 (see (1.11)).

Since p > 3, we have that (p, w) = 1 by Lemma 5.2, and so finally we deduce that

(O_{Q}(i)(F)) = 0 in Cl(𝔄_{i}(F)). This completes the proof of Theorem 1. □

References

[A1] A. Agboola, Torsion points on elliptic curves and Galois module structure, preprint.

[A2] A. Agboola, Abelian varieties and Galois module structure in global function fields, Math. Zeit. 271 (1994), 407-419.

[CR] C. W. Curtis and I. Reiner, Methods of representation theory Vol. 1, Wiley, 1981.

[F] A. Fröhlich, Galois module structure of algebraic integers, Springer, 1983.

[L] S. Lang, Introduction to Arakelov theory, Springer, 1988.

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

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

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

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