INDEPENDENCE OF POINTS ON ELLIPTIC CURVES ARISING FROM

SPECIAL POINTS ON MODULAR AND SHIMURA CURVES, I: GLOBAL RESULTS

ALEXANDRU BUIUM AND BJORN POONEN

to Ken Ribet on his 60^{th} birthday

Abstract. Given a correspondence between a modular curve S and an elliptic curve A, we prove that the intersection of any finite-rank subgroup of A with the set of points on A corresponding to CM-points on S is finite. We prove also a version in which S is replaced by a Shimura curve and A is replaced by a higher-dimensional abelian variety.

1. Introduction

1.1. CM-points mapping into a finite rank subgroup. Let N ∈ Z satisfy N > 3. Let X_{1}(N) over \overline{Q} be the complete modular curve attached to the group Γ_{1}(N). If Y_{1}(N) ⊂ X_{1}(N) is the non-cuspidal locus then Y_{1}(N)(\overline{Q}) is in bijection with the set of isomorphism classes of pairs (E, α) where E is an elliptic curve over \overline{Q} and α: Z/NZ ↪ E(\overline{Q}) is an injection. A CM-point on the curve X_{1}(N) is a point in Y_{1}(N)(\overline{Q}) represented by a pair (E, α) such that E has complex multiplication, i.e., End(E) ≠ Z. Let CM ⊂ X_{1}(N)(\overline{Q}) be the set of CM-points. Say that an abelian group Γ is of finite rank if dim_{Q}(Γ ⊗_{Z} Q) < ∞. We prove that the images of CM-points in any elliptic curve A are mostly independent in A(\overline{Q}):

Theorem 1.1. Let A be an elliptic curve over \overline{Q}. Let Φ: X_{1}(N) → A be a non-constant morphism. Let Γ ≤ A(\overline{Q}) be a finite rank subgroup. Then Φ(CM) ∩ Γ is finite.

In fact, we extend Theorem 1.1 in various directions. First, we can replace the morphism Φ by a correspondence between X_{1}(N) and A. Second, we can replace A by a higherdimensional abelian variety. Third, we can “fatten” Γ in the style of [20] by replacing Γ by Γ + B_{ϵ} where B_{ϵ} is a set of points of small Néron-Tate height. Fourth, we can prove variants where X_{1}(N) is replaced by a Shimura curve. For all of these, see Section 2.

1.2. Previous work. Most previous work on problems related to this paper concerned Heegner points, which are certain special points in Φ(CM) for a morphism Φ: X_{1}(N) → A. The study of the linear dependences among Heegner points (and their traces) plays an important role in work on the Birch and Swinnerton-Dyer conjecture, especially in the breakthroughs by Gross-Zagier [10] and Kolyvagin [13]. See [7] for an exposition of this circle of ideas. See also [18, 24, 6] for more recent advances, especially in relation to Mazur’s conjectures in [14]. In particular, [18] proved that there are only finitely many torsion Heegner points on any elliptic curve over Q. Along slightly different lines it was recently proved in [21] that if Q_{1}, …, Q_{s} are Heegner points associated to distinct quadratic imaginary fields and if the odd parts of the class numbers of these fields are sufficiently large then Q_{1}, …, Q_{s} are linearly independent. Recall that, by the classical theory of complex multiplication, the set of all points in CM defined over a given number field is finite; this, together with the Hermite-Minkowski theorem, implies that Φ(CM) ∩ Γ is finite for any finitely generated Γ ≤ A(\overline{Q}). In contrast, our finiteness results allow Γ to have points of unbounded degree (by Northcott’s theorem, this is always the case if Γ contains an infinite set of bounded height, for instance an infinite set of torsion points).

Date: October 5, 2008.

2000 Mathematics Subject Classification. 11G18, 14G20.

Key words and phrases. modular curve, Shimura curve, CM point.

Pink has formulated and proved some results toward a conjecture combining the Mordell-Lang and André-Oort conjectures [19]. Our Theorem 2.1 may also be seen as combining the Mordell-Lang conjecture with the André-Oort conjecture (the latter appears only in a trivial case, however, since we consider only 1-dimensional Shimura varieties). But as far as we can tell, Pink’s conjecture, even if fully proved, would not imply any of our results: his conjecture concerns subvarieties of mixed Shimura varieties such as the universal family of abelian varieties over some Shimura variety, whereas our Theorem 2.1 concerns subvarieties of the product of a Shimura curve (a modular curve) with an abelian variety where the abelian variety is not required to bear any relationship to the Shimura curve.

Finally we mention [22, Theorem 1.1], which gives an estimate for the average height of the points in the image of a Hecke correspondence T_{m} applied to a point of a modular curve; see also [1] for a stronger version of this result. The estimate grows with m, so at least one of the points in the image must be non-torsion; if moreover these points are Galois conjugates (as is true for large prime m if they are not CM, as pointed out in the proof of [22, Corollary 0.3]), then all are non-torsion.

1.3. Structure of the paper. Section 2 states all our variants of Theorem 1.1, and Section 3 proves them. We use methods quite different from those used in the papers mentioned in Section 1.2: our proofs rely on the fact that the limit measure promised by equidistribution results for Galois orbits in abelian varieties is incompatible with the limit measure in equidistribution results for modular curves and Shimura curves.

1.4. The sequel to this paper. In [4], we prove “local” versions of these results, in which \overline{Q} is replaced by the completion of the maximal unramified extension of the ring Z_{p} of padic integers, and CM is replaced by the subset CL of canonical lift points. Although these substitutions would appear to give a weaker statement, there are also several advantages:

(1) We obtain effective bounds for the size of the intersection.

(2) The results remain valid for certain Γ of infinite rank.

(3) The method applies also to prove results where CM is replaced by a (partial) isogeny

class of points in the modular or Shimura curve. The proofs in [4] use arguments very different from those in this paper: they involve the theory of arithmetic differential equations in the sense of [3].

Acknowledgments. While writing this paper the authors were partially supported by NSF grants: the first author by DMS-0552314, and the second author by DMS-0301280 and DMS-0841321. We are indebted to M. Christ, M. Kim, J. H. Silverman, P. Vojta, and J. F. Voloch for their remarks and suggestions. We thank also W. Duke, P. Michel, and S. Zhang for discussing equidistribution of CM-points. Finally we thank the referees for several helpful comments.

2. Variants

2.1. Modular curves and higher-dimensional abelian varieties. By variety, we mean a separated scheme of finite type over a field. By a coset in an abelian variety A over an algebraically closed field, we mean a translate of an abelian subvariety of A.

Theorem 2.1. Let S = X_{1}(N) over \overline{Q} for some N ≥ 1. Let A be an abelian variety over \overline{Q}. Let X be a closed irreducible subvariety of S × A. Let Γ ≤ A(\overline{Q}) be a finite-rank subgroup. If X(\overline{Q}) ∩ (CM ×Γ) is Zariski dense in X, then X = S′ × A′ where S′ is a subvariety of S and A′ is a coset in A.

If in Theorem 2.1 we drop the assumption that X(\overline{Q}) ∩ (CM ×Γ) is Zariski dense in X, we can apply Theorem 2.1 to the irreducible components of the Zariski closure of X(\overline{Q}) ∩ (CM ×Γ) to deduce the following equivalent form of Theorem 2.1.

Theorem 2.2. Let S, A, Γ be as in Theorem 2.1. Let X be a closed subvariety of S × A. Then the intersection X(\overline{Q}) ∩ (CM ×Γ) is contained in a subvariety Z ⊆ X that is a finite union of products S′ × A′ where each S′ is a subvariety of S and each A′ is a coset in A.

We can actually strengthen Theorem 2.1, as [20] strengthened the Mordell-Lang conjecture, by fattening Γ as follows. Let h: A(\overline{Q}) → R_{≥0} be a canonical height function attached to some symmetric ample line bundle on A. For Γ ≤ A(\overline{Q}) and ϵ ≥ 0, let

Γ_{ϵ} := \{ γ + a | γ ∈ Γ, a ∈ A(\overline{Q}), h(a) ≤ ϵ \}.

Theorem 2.3. Assume that S, A, X, Γ are as in Theorem 2.1. If X(\overline{Q})∩(CM ×Γ_{ϵ}) is Zariski dense in X for every ϵ > 0, then X = S′ × A′ where S′ is a subvariety of S and A′ is a coset in A.

Just as Theorem 2.1 implied Theorem 2.2, Theorem 2.3 implies the following more general (but equivalent) version of itself:

Theorem 2.4. Assume that S, A, Γ are as in Theorem 2.3. Let X be a closed subvariety of S × A defined over \overline{Q}. Then for some ϵ > 0, the intersection X(\overline{Q}) ∩ (CM ×Γ_{ϵ}) is contained in a subvariety Z ⊆ X that is a finite union of products S′ ×A′ where each S′ is a subvariety of S and each A′ is a coset in A.

2.2. Shimura curves. Let D be a non-split indefinite quaternion algebra over Q. Fix a maximal order O_{D} once and for all. Let X^{D}(U) be the Shimura curve attached to the pair (D, U), where U is a sufficiently small compact subgroup of (O_{D} ⊗ (\underleftarrow{lim} Z/mZ))^{×} such that X^{D}(U) is connected: see [5, 27]. A fake elliptic curve^{1} is a pair (E, i) consisting of an abelian surface E over \overline{Q} and an embedding i: O_{D} → End(E). The set X^{D}(U)(\overline{Q}) is in bijection with the set of isomorphism classes of fake elliptic curves equipped with a level U structure in the sense of [5, 27].

The classification of endomorphism algebras [16, p. 202] shows that for any fake elliptic curve (E, i), the algebra (End E) ⊗ Q is isomorphic to either D or D ⊗ K ≃ M_{2}(K) for some imaginary quadratic field K embeddable in D. In the latter case, (E, i) is called CM; then E is isogenous to the square of an elliptic curve with CM by an order in K. A CM-point of S(\overline{Q}) is a point whose associated (E, i) is CM. Let CM ⊂ S(\overline{Q}) be the set of CM-points on S.

1In the literature this is sometimes called a “false elliptic curve”.

Theorem 2.5. Let S = X^{D}(U) over \overline{Q}, let A be an abelian variety over \overline{Q}, and let Φ: S → A be a morphism. Let Γ ≤ A(\overline{Q}) be a finite-rank subgroup. Then Φ(CM) ∩ Γ is finite.

Finally, we can again fatten Γ:

Theorem 2.6. In the notation of Theorem 2.5, there exists ϵ > 0 such that Φ(CM) ∩ Γ_{ϵ} is finite.

Theorems 2.3 and 2.6 imply Theorem 1.1 and all the other results in this section, so we focus on them.

3. Proofs

We begin with Theorem 2.3; for its proof we need some measure-theoretic prerequisites.

Lemma 3.1. Let S be a smooth projective curve over C. Let X be a (possibly singular) closed N-dimensional subvariety of P_{C}^{n}. Let π: X → S be a morphism. Let s ∈ S(C). Equip S(C) and P^{n}(C) with real analytic Riemannian metrics. Let B_{r} be the open disk in S with center s and radius r, and let B′_{r} = B_{r} − {s}. Then there exists δ > 0 such that the N-dimensional volume of π^{−1}(B′_{r}) with respect to the metric on P^{n}(C) is O(r^{δ}) as r → 0.

Proof. Irreducible components of X having dimension less than N have zero N-dimensional volume, so we may reduce to the case in which X is irreducible of dimension N.

Define Δ := {z ∈ C : |z| < 1} and ▲ := {z ∈ C : |z| ≤ 1/2}. Let g_{P} and g_{S} be the given metrics on P^{n}(C) and S(C). Let μ be Lebesgue measure on C^{N}. Fix a holomorphic chart ι_{S}: Δ → S(C) mapping 0 to s.

We may assume that dim π^{−1}(s) < N. By work of Hironaka, there exists a desingularization p: Y → X (see [12, Corollary 3.22]) and we may assume that the fiber of the map f := π ∘ p: Y → S above s is a simple normal crossing divisor (see [12, Theorem 3.21]). Then for each y ∈ f^{−1}(s), there exists a holomorphic chart ι_{Y} : Δ^{N} → Y (C) mapping 0 to y such that f is given with respect to ι_{Y} and ι_{S} by

h: Δ^{N} → Δ
z = (z_{1}, …, z_{N}) ↦ u(z)z_{1}^{e_{1}} ⋯ z_{N}^{e_{N}}

for some nonvanishing holomorphic function u: Δ^{N} → C and e_{i} ∈ Z_{≥0}. By compactness, there exist ε > 0 and finitely many ι_{Y} such that the sets ι_{Y} (▲^{N}) cover f^{−1}(B_{ε}).

Since ▲ is compact, (ι_{S}^{∗}g_{S})|_{▲} is bounded above and below by positive constants times the standard metric. Similarly, the pullback of g_{P} to ▲^{N} is bounded above in terms of the standard metric. Thus we reduce to showing that for each ι_{Y}, there exists δ > 0 such that

μ ({z ∈ ▲^{N} : |h(z)| < r}) = O(r^{δ})

as r → 0.

Let u_{min} := inf{|u(z)| : z ∈ ▲^{N}} > 0. We may assume that r < u_{min}. Fix E > ∑ e_{i}. Let ρ := (r/u_{min})^{1/E} < 1. If |z_{i}| ≥ ρ for all i, then |h(z)| ≥ u_{min}ρ^{E} = r. Thus

{z ∈ ▲^{N} : |h(z)| < r} ⊆ {z ∈ ▲^{N} : |z_{i}| < ρ for some i}.

The volume of the latter is O(ρ^{2}) = O(r^{2/E}) as r → 0. □

Lemma 3.2. Let Y, H be varieties over C, with Y proper. Let X be a closed subvariety of Y × H. Let (h_{i}) be a sequence in H(C) converging to h_{∞}. For i ≤ ∞, let X_{i} be the fiber of X → H above h_{i}. View X_{i} as a subvariety of Y. Then any open neighborhood N of X_{∞}(C) in the complex topology contains X_{i}(C) for all sufficiently large i.

Proof. The open set ((Y × H) − X)(C) ∪ (N × H(C)) contains Y (C) × {h_{∞}}, so it contains also Y (C) × U for some open neighborhood U of h_{∞} in H, by the “tube lemma for compact spaces” (Lemma 5.8 on p. 169 of [17]). For large i, we have h_{i} ∈ U, and then X_{i}(C) ⊆ N. □

From now on, we assume that S = X_{1}(N) as in Theorem 2.3. Let H := {τ ∈ C : Im τ > 0}, and let H^{∗} := H ∪ P^{1}(Q), so S(C) = Γ_{1}(N)\H^{∗}. Let ∞_{S} ∈ S(C) be the image of the cusp ∞ ∈ P^{1}(Q) ⊂ H^{∗}. Choose a real analytic Riemannian metric on S(C). Define B_{r} to be the open disk in S(C) with center ∞_{S} and radius r with respect to the metric. Let μ_{H} be the probability measure on S(C) whose pullback to H equals a multiple of the hyperbolic measure \frac{dx dy}{y^{2}}.

We next show that μ_{H} blows up relative to the Riemannian metric near the cusp ∞_{S}. (The Riemannian volume of B_{r} is only O(r^{2}) as r → 0.)

Lemma 3.3. There exists u > 0 such that for all sufficiently small r > 0, we have μ_{H}(B_{r}) > u/ log(1/r).

Proof. Let τ be the usual parameter on H. Then q := e^{2πiτ} is an analytic uniformizer at ∞_{S} ∈ S(C) (defined on some neighborhood of ∞_{S}) because the subgroup of Γ_{1}(N) ⊆ PSL_{2}(Z) stabilizing ∞ ∈ H^{∗} is generated by ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) . So there exists c > 0 such that for all sufficiently small r, in the fundamental domain in H, the part corresponding to B_{r} contains the part where |q| < cr. The inequality |q| < cr is equivalent to Im(τ) > \frac{1}{2π} log(1/(cr)), so μ_{H}(B_{r}) is at least a constant times 1/ log(1/(cr)) for sufficiently small r. If u is small enough, this exceeds u/ log(1/r) for all sufficiently small r. □

We will need also an equidistribution result for CM points. The first such equidistribution result was proved in [8], and this has been generalized in several directions by several authors: see Section 5.4 of the survey paper [15], for instance. The version we use is a special case of a result in [28].

Lemma 3.4. Let k be a finite extension of Q. Fix an embedding \overline{k} ↪ C. Let S be a modular curve X_{1}(N) or a Shimura curve X^{D}(U) over \overline{k}. Let (x_{i}) be an infinite sequence of distinct CM-points in S(\overline{k}). The uniform probability measure on the Gal(\overline{k}/k)-orbit of x_{i} converges weakly as i → ∞ to the measure μ_{H} on S(C).

Proof. This follows from Corollary 3.3 of [28]. Namely, we choose δ < 1/2 as on p. 3663 of [28], choose ϵ > 0 so that δ/2 + 1/4 + ϵ < 1/2, and define the “CM-suborbit” O(x_{i}) as the Gal(\overline{k}/k)-orbit of x_{i}. The hypothesis of Corollary 3.3 of [28] is satisfied, by the Brauer-Siegel theorem (see the first remark following Corollary 3.3 of [28]). □

Proof of Theorem 2.3. Let A′ be the image of X → A. By Corollary 9 of [20] (also proved partially independently as Theorem 1.2 of [26]) applied to A′ ⊆ A, we have that A′ is a coset. We may translate to assume that A′ is an abelian subvariety, and hence reduce to the case where X → A is surjective. We may assume also that X → S is surjective, since dim S = 1. We want X = S × A. Suppose not. Then X → A is generically finite, say of degree d.

The group Γ is contained in the division hull of a finitely generated group Γ_{0}. Choose a number field k ⊂ \overline{Q} such that A, S, X are all defined over k and Γ_{0} ≤ A(k).

Since X(\overline{Q}) ∩ (CM ×Γ_{ϵ}) is Zariski dense in X for every ϵ > 0, and since X has only countably many subvarieties, we may choose a generic infinite sequence of points x_{i} = (s_{i}, a_{i}) ∈ X(\overline{Q}) with s_{i} ∈ CM and a_{i} ∈ Γ_{ϵ_{i}} where ϵ_{i} → 0. (“Generic” means that each proper subvariety of X contains at most finitely many x_{i}.) In particular, each s_{i} appears only finitely often. Since class numbers of imaginary quadratic fields tend to infinity, we have [k(s_{i}) : k] → ∞. So [k(x_{i}) : k] → ∞. For all but finitely many i, the a_{i} lie in the open locus above which the fibers of X → A have size d, and then [k(x_{i}) : k] ≤ d[k(a_{i}) : k]. Thus [k(a_{i}) : k] → ∞.

The a_{i} form a sequence of almost division points relative to k in the sense of [26]. By passing to a subsequence we may assume that they have a coherent limit (C, b + T) in the sense of [26], where C is an abelian subvariety of A, and b ∈ A(C)/C(C), and T is a finite set of torsion points of A/C. Since [k(a_{i}) : k] → ∞, we have dim C > 0 by definition of

coherent limit. By replacing X by its image under S × A ⟶^{(id,ϕ)} S × Ã for a suitable isogeny ϕ: A → Ã, we may reduce to the case where T = {0} and A ≃ B × C for some abelian subvariety B of A. Identify A/C with B. Write a_{i} = (b_{i}, c_{i}) with b_{i} ∈ B(\overline{Q}) and c_{i} ∈ C(\overline{Q}). By definition of T, we have b_{i} ∈ B(k). By Theorem 1.1 of [26], the uniform probability measure on the orbit Gal(\overline{k}/k)a_{i} (supported on \{b_{i}\} × C(C)) converges weakly as i → ∞ to the C(C)-invariant probability measure on {b} × C(C). So the uniform probability measure on Gal(\overline{k}/k)c_{i} converges to Haar measure μ_{C} on C(C).

For each i, let X_{b_{i}} be the fiber of the projection X → B above b_{i}, viewed as a subvariety of S × C. Since the b_{i} are generic in B, we may pass to a subsequence to assume that the X_{b_{i}} have the same Hilbert polynomial (with respect to some embedding S × C ↪ P^{N}) and that the corresponding points of the Hilbert scheme H converge in the complex topology; let X_{b_{∞}} ⊆ S × C be the closed subscheme corresponding to the limit. We have dim X_{b_{i}} < dim(S × C) for all finite i (and hence also for i = ∞), since otherwise by genericity of the b_{i}, we would have X = S × B × C = S × A.

Let π_{C}: S × B × C → C be the projection. Also, for i ≤ ∞, let π_{S,i}: X_{b_{i}} → S be the projection.

Choose a real analytic Riemannian metric on C(C) whose associated volume form equals μ_{C}. Let g = dim C. Let B′_{r} = B_{r} − {∞}. By Lemma 3.3, there exists u > 0 such that μ_{H}(B′_{r}) = μ_{H}(B_{r}) > u/ log(1/r) for all sufficiently small r. On the other hand, Lemma 3.1 implies that for some δ > 0, the g-dimensional volume of π_{S,∞}^{−1}(B′_{r}) is O(r^{δ}) as r → 0. Let L_{r} := π_{C}(π_{S,∞}^{−1}(B′_{r})). Projection onto C can only decrease g-dimensional volume, so μ_{C}(L_{r}) = O(r^{δ}). Thus we may fix r > 0 such that μ_{H}(B′_{r}) > μ_{C}(L_{r}). Let L = L_{r}. Fix a compact annulus K ⊆ B′_{r} large enough so that μ_{H}(K) > μ_{C}(L).

For a compact subset M′ of a metric space M, let N_{ρ}M′ be the set of points in M whose distance to M′ is less than ρ. Fix ρ > 0 such that N_{ρ}K ⊆ B′_{r}. By Lemma 3.2 with Y = S × C, with H the Hilbert scheme, and X the universal family in Y × H, we have X_{b_{i}}(C) ⊆ N_{ρ}X_{b_{∞}}(C) after discarding finitely many i. In particular, every point of π_{S,i}^{−1}(K) is within ρ of a point of X_{b_{∞}}(C). The S-projections of the points of X_{b_{∞}}(C) so used are then within ρ of K, so

π_{S,i}^{−1}(K) ⊆ π_{S,∞}^{−1}(N_{ρ}K) ⊆ π_{S,∞}^{−1}(B′_{r}).

Projecting to C, we obtain

(3.5)  π_{C}(π_{S,i}^{−1}(K)) ⊆ L.

Now as i → ∞, the fraction of points of Gal(\overline{k}/k)x_{i} whose S-projection lies in K tends to μ_{H}(K) by Lemma 3.4, and the fraction of points of Gal(\overline{k}/k)x_{i} whose C-projection lies in L tends to μ_{C}(L). But (3.5) implies that the first set of points is contained in the second set of points, so μ_{H}(K) ≤ μ_{C}(L), contradicting the choice of K. □

For the proof of Theorem 2.6, we will need the following:

Lemma 3.6. Let Φ: S → A be a morphism from a Shimura curve to an elliptic curve A over C. Let μ_{H} be the hyperbolic probability measure on S(C). Let μ_{A} be the Haar probability measure on A(C). Then Φ_{∗}μ_{H} ≠ μ_{A}.

Proof. By replacing S with a finite cover, we may assume that H → S(C) is unramified. The universal cover of A(C) is not biholomorphic to H, so the composition H → S(C) → A(C) cannot be unramified. Hence Φ is ramified. Pick s ∈ S(C) at which the ramification index e is > 1. Let a = Φ(s). Choose a Riemannian metric on A(C) inducing the Haar probability measure μ_{A}. Let B_{r} be the disk of radius r centered at a. With respect to suitable uniformizing parameters, Φ near s is equivalent to z ↦ z^{e}, so there exists c > 0 such that μ_{H}(Φ^{−1}(B_{r})) > cμ_{A}(B_{r})^{1/e} for all sufficiently small r. In particular, for sufficiently small r, we have (Φ_{∗}μ_{H}) (B_{r}) = μ_{H}(Φ^{−1}(B_{r})) > μ_{A}(B_{r}). □

Proof of Theorem 2.6. As in the first three sentences of the proof of Theorem 2.3, we may use Corollary 9 of [20] to reduce to the case that Φ is surjective. If A = 0, there is nothing to show, so we may assume that A is an elliptic curve.

Choose a number field k ⊂ \overline{Q} such that A, S, X are all defined over k and Γ is contained in the division hull of A(k).

If the conclusion fails, then there is an infinite sequence (s_{i}) in CM with Φ(s_{i}) ∈ Γ_{ϵ_{i}} for some ϵ_{i} → 0. Let a_{i} = Φ(s_{i}). By Lemma 3.4, the uniform probability measure on Gal(\overline{k}/k)s_{i} converges weakly to μ_{H} on S(C). It follows that the uniform probability measure on Gal(\overline{k}/k)a_{i} converges weakly to Φ_{∗}μ_{H} on A(C).

On the other hand, (a_{i}) is a sequence of almost division points. By the previous paragraph, [k(s_{i}) : k] → ∞, so [k(a_{i}) : k] → ∞. Passing to a subsequence, we may assume that (a_{i}) has a coherent limit, which can only be (A, {0}), since [k(a_{i}) : k] → ∞. By Theorem 1.1 of [26], the uniform probability measure on Gal(\overline{k}/k)a_{i} converges weakly to the Haar measure μ_{A} on A(C).

The previous two paragraphs imply that Φ_{∗}μ_{H} = μ_{A}, contradicting Lemma 3.6. □

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University of New Mexico, Albuquerque, NM 87131

  E-mail address: buium@math.unm.edu
  URL: http://math.unm.edu/~buium

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA Current address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA

02139-4307, USA

  E-mail address: poonen@math.mit.edu
  URL: http://math.mit.edu/~poonen