New York Journal of Mathematics

New York J. Math. 12 (2006) 257–268.

Small rational points on elliptic curves over number fields

Clayton Petsche

Abstract. Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of results of Hindry–Silverman, giving an upper bound for the number of k-rational torsion points, and a lower bound for the canonical height of nontorsion k-rational points, in terms of expressions depending explicitly on the degree d = [k : ℚ] of k and the Szpiro ratio σ of E/k. The bounds exhibit only polynomial dependence on both d and σ.

Contents

1. Introduction 257 2. Elliptic curves over number fields 260 3. A few preliminary lemmas 261 4. A bound on the number of small points 264 References 267

1. Introduction

Let k be a number field of degree d = [k : ℚ], and let E/k be an elliptic curve. In [12], Merel used deep facts about the arithmetic of modular curves to prove that there is a universal bound C(d) depending only on d such that |E(k)_{tor}| ≤ C(d). Quantitative refinements of Merel’s theorem due to Parent [13] and Oesterlé (cf. [7]) give bounds which depend exponentially on d, but it is still unknown whether one can give a bound C(d) whose growth is polynomial in d. Such a result—or a proof that no such bound is possible—would be of value, both for its intrinsic interest and for its implications in cryptography; cf. [2], [8].

In this paper we give an explicit polynomial bound on |E(k)_{tor}| depending on d and the Szpiro ratio σ, a certain quantity associated to the elliptic curve E/k which we will now define. Recall that the conductor 𝔣_{E/k} and the minimal discriminant

257^{ISSN 1076-9803/06}

Received August 3, 2005; revised August 26, 2006.

Mathematics Subject Classification. 11G05, 11G07, 11G50.

Key words and phrases. Elliptic curves, heights, torsion points, Szpiro ratio, Lang’s conjecture.

Δ_{E/k} are certain integral ideals of O_{k} that are supported on the primes at which E/k has bad reduction. The Szpiro ratio is given by

σ = \frac{log |ℕ_{k/ℚ}(Δ_{E/k})|}{log |ℕ_{k/ℚ}(𝔣_{E/k})|} (1)

when E/k has at least one place of bad reduction; by convention we put σ = 1 if E/k has everywhere good reduction.

Theorem 1. Let k be a number field of degree d = [k : ℚ], and let E/k be an elliptic curve with Szpiro ratio σ. Then

|E(k)_{tor}| ≤ c_{1}dσ^{2} log(c_{2}dσ^{2}), (2)

where c_{1} = 134861 and c_{2} = 104613.

Strictly speaking (2) is not a uniform bound, in that the right-hand side depends on the elliptic curve E/k. However, the result is perhaps more interesting in view of the fact that one generally expects σ to be small. More precisely, let Σ(k) denote the set (with multiplicities) of Szpiro ratios σ of all elliptic curves E/k. A wellknown conjecture of Szpiro [17] asserts that Σ(k) is bounded, and that 6 is its largest limit point. (As pointed out by Masser [10], 6 is in fact a limit point.) An analogue of Szpiro’s conjecture is known to hold in the function field case, and in the number field case it can be shown to follow from the ABC conjecture.

Note that if Szpiro’s conjecture is true, then Theorem 1 gives a uniform bound on |E(k)_{tor}| in terms of k only. The first result showing that Szpiro’s conjecture implies such a uniform bound is due to Frey; the argument was written up by Flexor–Oesterlé [4]. Also in [4], the authors treated the special case of everywhere good reduction (σ = 1), giving a bound on |E(k)_{tor}| that is exponential in d. In [7] Hindry–Silverman improved this to a bound of O(d log d); thus our bound (2) recovers theirs (with slightly different constants) in this special case. In the general case, Hindry–Silverman [5] have given a bound on |E(k)_{tor}| that is exponential in both d and σ. Thus the main interest in our bound (2) is in its explicit nature, and in the fact that it exhibits only polynomial growth in both d and σ. It is also worth mentioning here another bound of Flexor–Oesterlé from the paper [4]: they show that if there exists a place of additive reduction then |E(k)_{tor}| ≤ 48d.

Finally, it should be pointed out that our inequality (2) does not imply the full theorem of Merel, even if we allow ourselves Szpiro’s conjecture, as the resulting bound would depend on the number field k. To hope for more seems doubtful; for if we let Σ_{d} denote the union of the sets Σ(k) over all number fields k of degree d = [k : ℚ], then we are presented with the question of whether sup Σ_{d} is finite, and if so, how it depends on d. A similar question relating to the ABC conjecture has been investigated by Masser [11].

A problem somewhat related to counting rational torsion points is that of giving a lower bound on the Néron–Tate canonical height ĥ(P), for nontorsion rational points P, depending explicitly on the relevant data associated to the elliptic curve E/k. In particular, a conjecture of Lang asserts that if P ∈ E(k) is not a torsion point, then

ĥ(P) ≥ c log |ℕ_{k/ℚ}(Δ_{E/k})|, (3) where c = c(k) > 0 is a constant depending only on k. We show that one can take for c a certain expression depending explicitly on d and σ.

Theorem 2. Let k be a number field of degree d = [k : ℚ], and let E/k be an elliptic curve with minimal discriminant Δ_{E/k} and Szpiro ratio σ. Then

ĥ(P) ≥ c(d, σ) log |ℕ_{k/ℚ}(Δ_{E/k})| (4)

for all nontorsion points P ∈ E(k), where

c(d, σ) = \frac{1}{10^{15}d^{3}σ^{6} log^{2}(c_{2}dσ^{2})}, (5)

and c_{2} = 104613.

A consequence of Theorem 2 is that Szpiro’s conjecture implies Lang’s conjecture; this fact was originally proved by Hindry–Silverman [5], who showed that (3) holds with a value of c depending exponentially on d and σ. Thus again the main interest in (4) is in the fact that c(d, σ) exhibits only polynomial decay in d and σ. Compare with the results of David [3], who uses methods from transcendence theory to obtain a similar bound.

Let us now briefly summarize our approach, in which we extend the methods of Hindry–Silverman [7] to include a treatment of the places of bad reduction. Denote by M_{k} the set of all places of k, and given v ∈ M_{k}, let k_{v} be the completion of k at v. Let ĥ : E(k) → [0, +∞) denote the Néron–Tate canonical height, and recall that given a point P ∈ E(k) \ {O}, we have the local decomposition ĥ(P) = ∑_{v∈M_{k}} \frac{d_{v}}{d} λ_{v}(P), where d_{v} = [k_{v} : ℚ_{v}] is the local degree and λ_{v} : E(k_{v})\\{O\} → ℝ is the appropriately normalized Néron local height function (cf. [16], §VI.1). Given a set Z = {P_{1}, …, P_{N}} ⊂ E(k) of N distinct small rational points, we estimate the height-discriminant sum

Λ(Z) = \frac{1}{N^{2}} ∑_{1≤i,j≤N} ĥ(P_{i} − P_{j}), (6)

from above globally via the parallelogram law, and from below locally using the decomposition Λ(Z) = ∑_{v∈M_{k}} \frac{d_{v}}{d} Λ_{v}(Z), where

Λ_{v}(Z) = \frac{1}{N^{2}} ∑_{i≠j^{1≤i,j≤N}} λ_{v}(P_{i} − P_{j}). (7)

In order to obtain the necessary archimedean estimates we follow [7], using the pigeonhole principle to pass to a subset of Z of positive density, all of whose points are close to each other in E(k_{v}) at a particular archimedean place v. At the nonarchimedean places we give an analytic lower bound on Λ_{v}(Z) in terms of the valuation of the minimal discriminant (cf. Lemma 4 below). Although we will not specifically require this interpretation in the present paper, this inequality can be viewed as a quantitative form of a local equidistribution principle for small points, as developed in [1]. Finally, it is worth noting that we do not decompose the minimal discriminant into “small power” and “large power” parts as Hindry–Silverman do in [5]. Instead, we assemble the local information at the different places of bad reduction into global information by a simple application of Jensen’s inequality.

Acknowledgements. The author would like to express his gratitude to Matt Baker for his many valuable suggestions, and to the anonymous referee, who offered several insightful comments on the content and the exposition of this paper.

2. Elliptic curves over number fields

In this section we fix some notation, and we review some of the relevant facts concerning elliptic curves over number fields.

Let O_{k} be the ring of integers of a number field k, and denote by M_{k}^{∞} and M_{k}^{0} the set of all archimedean and nonarchimedean places of k respectively. For each place v ∈ M_{k} we will select the normalized absolute value | |_{v} that, when restricted to ℚ, coincides with one of the usual archimedean or p-adic absolute values. Given a place v ∈ M_{k}^{0} lying over the rational prime p, let O_{v} and M_{v} be the ring of integers and maximal ideal in k_{v} respectively. Let 𝔭_{v} = O_{k} ∩ M_{v} denote the prime ideal of O_{k} corresponding to the place v, and let π_{v} ∈ M_{v} be a uniformizer. Thus |ℕ_{k/ℚ}(𝔭_{v})| = |O_{v}/M_{v}| = p^{f_{v}} and |π_{v}|_{v} = p^{−1/e_{v}}, where f_{v} and e_{v} are the residual degree and ramification index, respectively. Recall that d_{v} = [k_{v} : ℚ_{p}] = e_{v}f_{v}.

Let E/k be an elliptic curve, and let j_{E} denote its j-invariant. The conductor and minimal discriminant of E/k are the integral ideals

𝔣_{E/k} = ∏_{v∈M_{k}^{0}} 𝔭_{v}^{η_{v}}, and Δ_{E/k} = ∏_{v∈M_{k}^{0}} 𝔭_{v}^{δ_{v}} (8)

of O_{k}, respectively, where the exponents η_{v} and δ_{v} are given as follows. Fix a place v ∈ M_{k}^{0}, and define δ_{v} = ord_{v}(Δ_{v}), where Δ_{v} ∈ O_{v} denotes the discriminant of a minimal Weierstrass equation for E/k_{v}; thus |Δ_{v}|_{v} = |π_{v}|_{v}^{δ_{v}}. The exponent η_{v} of the conductor has a rather complicated Galois-theoretic definition (cf. [16], §IV.10); alternatively, it can be characterized by Ogg’s formula

δ_{v} = η_{v} + m_{v} − 1, (9)

where m_{v} is the number of components on the special fiber of the minimal proper regular model of E/k_{v} (cf. [16], §IV.11). An immediate consequence of (9) is that η_{v} ≤ δ_{v}, from which we deduce the lower bound σ ≥ 1 on the Szpiro ratio.

Let E_{0}(k_{v}) denote the subgroup of E(k_{v}) consisting of those points whose reduction (with respect to a minimal Weierstrass equation for E/k_{v}) is nonsingular, and let

c_{v} = |E(k_{v})/E_{0}(k_{v})| (10)

be the cardinality of the quotient. It is well-known that the set E_{0}(k_{v}), and thus also the number c_{v}, do not depend on the choice of the minimal Weierstrass equation used to define them.

Finally, we recall that the various data discussed above are governed by the reduction type of E/k_{v}. To be precise, if E/k_{v} has good reduction then δ_{v} = η_{v} = 0 and c_{v} = 1; if E/k_{v} has split multiplicative reduction then η_{v} = 1 and c_{v} = δ_{v}; and finally, in all other cases we have c_{v} ≤ 4. For proofs of these assertions see [16], §IV.9.

Following Rumely (cf. [14] §2.4), it will be useful to decompose the nonarchimedean Néron local height function λ_{v} : E(k_{v}) \ {O} → ℝ into a sum

λ_{v}(P − Q) = i_{v}(P, Q) + j_{v}(P, Q), (11) where i_{v} is a nonnegative arithmetic intersection term, and j_{v} is bounded (cf. also [1]). These functions are most naturally described by first passing to a finite extension K_{v} of k_{v} over which E achieves either good or split multiplicative reduction (cf. [15], Prop. VII.5.4). Recall that the absolute value | |_{v} extends uniquely to K_{v}, and that the Néron local height function is invariant under this finite extension (cf. [16], Thm. VI.1.1). We now consider the cases of integral and nonintegral j-invariant separately:

Case 1: |j_{E}|_{v} ≤ 1. Then j_{v} is identically zero and λ_{v}(P − Q) = i_{v}(P, Q).

Case 2: |j_{E}|_{v} > 1. We have maps

E(k_{v}) ↪ E(K_{v}) ⟶^{∼} K_{v}^{×}/q^{ℤ} → ℝ/ℤ, (12)

where q ∈ K_{v}^{×} with |q|_{v} = |1/j_{E}|_{v} < 1; here the first map is inclusion, the second map is the isomorphism afforded by Tate’s uniformization theory [18], and the third map is given by u ↦ log |u|_{v}/ log |q|_{v}. Let r denote the composition of the three maps in (12). Then

j_{v}(P, Q) = \frac{1}{2}B_{2}(r(P − Q)) log |j_{E}|_{v},

where

B_{2}(t) = (t − [t])^{2} − \frac{1}{2}(t − [t]) + \frac{1}{6} = \frac{1}{2π^{2}} ∑_{m∈ℤ\\{0\}} \frac{1}{m^{2}} e^{2πimt}

is the periodic second Bernoulli polynomial. If r(P) ≠ r(Q) then i_{v}(P, Q) = 0. On the other hand if r(P) = r(Q), then we can select representatives u(P), u(Q) ∈ K_{v}^{×}/q^{ℤ} for P and Q under the Tate isomorphism such that |u(P)|_{v} = |u(Q)|_{v}; in this case

i_{v}(P, Q) = − log |1 − u(P)/u(Q)|_{v}.

Finally, we note for future reference that E_{0}(k_{v}) = ker(r); in other words, a point P ∈ E(k_{v}) has nonsingular reduction if and only if r(P) = 0.

3. A few preliminary lemmas

We begin with a purely group theoretical counting lemma.

Lemma 3. Let G_{0} be a subgroup of a group G with finite index c = [G : G_{0}], and let Z be a finite subset of G with cardinality N. Then

|\{(x, y) ∈ Z × Z | xy^{−1} ∈ G_{0}\}| ≥ \frac{N^{2}}{c} . (13)

Proof. Denote by G/G_{0} the set of right-cosets of G_{0}, and for each C ∈ G/G_{0} let N_{C} = |Z ∩ C|. Note that xy^{−1} ∈ G_{0} if and only if both x and y lie in the same right-coset. It follows that the number on the left-hand side of (13) is

\begin{aligned}∑_{xy^{−1}∈G_{0}^{(x,y)∈Z×Z}} 1 & = ∑_{C∈G/G_{0}} N_{C}^{2} \\ & = ∑_{C∈G/G_{0}} \{(N_{C} − \frac{N}{c} )^{2} + \frac{2N_{C}N}{c} − \frac{N^{2}}{c^{2}} \} \\ & ≥ \frac{2N}{c} ∑_{C∈G/G_{0}} N_{C} − \frac{N^{2}}{c^{2}} ∑_{C∈G/G_{0}} 1 \\ & = \frac{N^{2}}{c} . \end{aligned}  □

The following lower bound on the nonarchimedean local sum (7) is a variant of [6], Proposition 1.2. To be precise, our bound (14) is given in terms of the valuation of the minimal discriminant, rather than the valuation of the j-invariant as in [6], a distinction that is only relevant at the places of additive reduction.

Lemma 4. Let E/k be an elliptic curve, let v be a nonarchimedean place of k, and let Δ_{v} ∈ O_{v} be the discriminant of a minimal Weierstrass equation for E/k_{v}. If Z = {P_{1}, …, P_{N}} ⊂ E(k_{v}) is a set of N distinct k_{v}-rational points, then

Λ_{v}(Z) ≥ (\frac{1}{c_{v}^{2}} − \frac{1}{N} ) \frac{1}{12} log |1/Δ_{v}|_{v}. (14)

Proof. First, note that

log^{+} |j_{E}|_{v} ≤ log |1/Δ_{v}|_{v}, (15)

with equality if and only if E/k_{v} has good or multiplicative reduction (cf. [15], Prop. VII.5.1). Also, the Néron local height function satisfies the lower bound λ_{v}(P) ≥ \frac{1}{12} log |1/Δ_{v}|_{v} for points P ∈ E_{0}(k_{v}) \ \{O\} (cf. [16], Thm. VI.4.1). It follows from this and the decomposition (11) that

i_{v}(P, Q) ≥ \frac{1}{12}(log |1/Δ_{v}|_{v} − log^{+} |j_{E}|_{v}) ≥ 0

for P − Q ∈ E_{0}(k_{v}) \ {O}. As i_{v} is nonnegative, it follows that

∑_{i≠j^{1≤i,j≤N}} i_{v}(P_{i}, P_{j}) ≥ (M − N)\frac{1}{12}(log |1/Δ_{v}|_{v} − log^{+} |j_{E}|_{v}), (16)

where M is the number of ordered pairs (P_{i}, P_{j}) ∈ Z × Z with P_{i} − P_{j} ∈ E_{0}(k_{v}). By Lemma 3 we have the lower bound M ≥ N^{2}/c_{v}, which in combination with (16) yeilds

\frac{1}{N^{2}} ∑_{i≠j^{1≤i,j≤N}} i_{v}(P_{i}, P_{j}) ≥ (\frac{1}{c_{v}} − \frac{1}{N} ) \frac{1}{12}(log |1/Δ_{v}|_{v} − log^{+} |j_{E}|_{v}). (17)

We will now show that

\frac{1}{N^{2}} ∑_{i≠j^{1≤i,j≤N}} j_{v}(P_{i}, P_{j}) ≥ (\frac{1}{c_{v}^{2}} − \frac{1}{N} ) \frac{1}{12} log^{+} |j_{E}|_{v}. (18)

For if |j_{E}|_{v} ≤ 1 then both sides of (18) are zero. On the other hand, assume that |j_{E}|_{v} > 1. Since the map r is trivial on the identity component E_{0}(k_{v}), it is well-defined on the quotient E(k_{v})/E_{0}(k_{v}), which has cardinality c_{v}. Therefore r(E(k_{v})) ⊆ ⟨1/c_{v}⟩ ⊂ ℝ/ℤ, and we conclude that e^{2πimr(P_{j})} = 1 for all P_{j} ∈ Z ⊂ E(k_{v}), whenever c_{v} | m. It follows that

(19) ∑_{i≠j^{1≤i,j≤N}} j_{v}(P_{i}, P_{j}) = ∑_{1≤i,j≤N} j_{v}(P_{i}, P_{j}) − \frac{N}{12} log |j_{E}|_{v}

= \frac{log |j_{E}|_{v}}{4π^{2}} ∑_{m∈ℤ\\{0\}} \frac{1}{m^{2}} | ∑_{1≤j≤N} e^{2πimr(P_{j})}|^{2} − \frac{N}{12} log |j_{E}|_{v}

≥ \frac{N^{2} log |j_{E}|_{v}}{4π^{2}} ∑_{m∈c_{v}ℤ\\{0\}} \frac{1}{m^{2}} − \frac{N}{12} log |j_{E}|_{v}

= \frac{N^{2} log |j_{E}|_{v}}{12c_{v}^{2}} − \frac{N}{12} log |j_{E}|_{v},

which implies (18).

Finally, by the decomposition (11) and the lower bounds (17) and (18), and using the inequality (15), we have

Λ_{v}(Z) = \frac{1}{N^{2}} ∑_{i≠j^{1≤i,j≤N}}(i_{v}(P_{i}, P_{j}) + j_{v}(P_{i}, P_{j}))

≥ (\frac{1}{c_{v}^{2}} − \frac{1}{c_{v}} ) \frac{1}{12} log^{+} |j_{E}|_{v} + (\frac{1}{c_{v}} − \frac{1}{N} ) \frac{1}{12} log |1/Δ_{v}|_{v}

≥ (\frac{1}{c_{v}^{2}} − \frac{1}{c_{v}} ) \frac{1}{12} log |1/Δ_{v}|_{v} + (\frac{1}{c_{v}} − \frac{1}{N} ) \frac{1}{12} log |1/Δ_{v}|_{v}

= (\frac{1}{c_{v}^{2}} − \frac{1}{N} ) \frac{1}{12} log |1/Δ_{v}|_{v},

which completes the proof. □

The following archimedean analogue of Lemma 4 is a quantitative refinement of a result due to Elkies (cf. [9], §VI.); for a detailed proof see [1], Appendix A.

Lemma 5. Let E/k be an elliptic curve with j-invariant j_{E}, and let Z ⊂ E(k) be a set of N distinct k-rational points. If v is an archimedean place of k, then

Λ_{v}(Z) ≥ −\frac{log N}{2N} − \frac{1}{12N} log^{+} |j_{E}|_{v} − \frac{16}{5N} . (20)

In order to obtain the necessary nonarchimedean estimates we will require the following lemma due to Hindry–Silverman (cf. Lemma 1 of [7] and Proposition 2.3 of [5]). Let j : ℍ = {τ ∈ ℂ | ℑ(τ) > 0} → ℂ denote the modular j-function, and let L = ℤ + τℤ be a normalized lattice with τ ∈ ℍ; thus j(τ) is the j-invariant of the elliptic curve ℂ/L. Let λ : (ℂ/L) \ {0} → ℝ denote the Néron function, as given in [16], §VI.3.

Lemma 6 (Hindry–Silverman). If z = r_{1} + r_{2}τ ∈ ℂ \ {0}, where r_{1}, r_{2} ∈ ℝ and max{|r_{1}|, |r_{2}|} ≤ 1/24, then

λ(z) ≥ \frac{1}{288} max\{1, log |j(τ)|\}.

Finally, we will require the following inequality, the proof of which is elementary and given in [1], §6.

Lemma 7. Let N ≥ 1 satisfy the bound N ≤ A log N +B for constants A > 0, B ≥ 0. Then N ≤ (\frac{e}{e−1})(A log A + B).

4. A bound on the number of small points

In this section we will prove the results stated in the introduction, which are both consequences of the following bound on the number of small rational points.

Proposition 8. Let k be a number field of degree d = [k : ℚ], and let E/k be an elliptic curve with Szpiro ratio σ. Then

|\{P ∈ E(k) | ĥ(P) ≤ \frac{log |ℕ_{k/ℚ}(Δ_{E/k})|}{2^{13}3dσ^{2}}\}| ≤ c_{1}dσ^{2} log(c_{2}dσ^{2}), (21)

where c_{1} = 134861 and c_{2} = 104613.

Proof. To ease notation we will henceforth suppress the subscripts on the notations ℕ_{k/ℚ}, Δ_{E/k}, and 𝔣_{E/k}. Let S denote the set on the left-hand side of (21), and let N denote the largest integer satisfying |S| ≥ 24^{2}(N − 1) + 1; thus |S| ≤ 24^{2}N. We will show that

N ≤ (148dσ^{2}) log N + 971dσ^{2}. (22)

Assuming for now that this holds, it follows by Lemma 7 that

\begin{aligned}N & ≤ (\frac{e}{e − 1})(148dσ^{2} log(148dσ^{2}) + 971dσ^{2}) \\ & = (\frac{e}{e − 1})148dσ^{2} log(148e^{971/148}dσ^{2}). \end{aligned}

In view of this and the fact that |S| ≤ 24^{2}N, the bound (21) follows immediately.

It now remains only to prove (22). Let v_{0} be an archimedean place of k, chosen so that |j_{E}|_{v_{0}} = max_{v|∞}(|j_{E}|_{v}). We then have a corresponding embedding σ : E(k) ↪ ℂ/L, where L = ℤ + τℤ is a normalized lattice, and |j(τ)| = |j_{E}|_{v_{0}}. If λ : (ℂ/L) \ {0} → ℝ denotes the Néron function on the complex torus, then λ_{v_{0}} = λ ∘ σ.

Divide the torus ℂ/L into the 24^{2} parallelograms

P_{m_{1},m_{2}} = \{z = r_{1} + r_{2}τ | \frac{m_{1} − 1}{24} ≤ r_{1} ≤ \frac{m_{1}}{24} and \frac{m_{2} − 1}{24} ≤ r_{2} ≤ \frac{m_{2}}{24} \},

where 1 ≤ m_{1}, m_{2} ≤ 24. By the pigeonhole principle there exists a set Z = {P_{1}, …, P_{N}} ⊆ S of N distinct points such that σ(Z) is contained in one of the 24^{2} parallelograms. In particular, it follows that the difference σ(P_{i}) − σ(P_{j}) between any two points in σ(Z) must lie in one of the four parallelograms P_{m_{1},m_{2}}, m_{1}, m_{2} ∈ \{1, 24\}. Therefore, by Lemma 6 we have λ_{v_{0}}(P_{i} − P_{j}) ≥ \frac{1}{288} max\{1, log |j_{E}|_{v_{0}}\} for all such pairs.

It follows from the above considerations that

Λ_{v_{0}}(Z) ≥ \frac{(N − 1)}{288N} max\{1, log |j_{E}|_{v_{0}}\}; (23)

and by Lemma 5 and the maximality of |j_{E}|_{v_{0}}, we have

(24)  ∑_{v≠v_{0}^{v|∞}} \frac{d_{v}}{d} Λ_{v}(Z) ≥ − ∑_{v≠v_{0}^{v|∞}} \frac{d_{v}}{d} \{\frac{log N}{2N} + \frac{1}{12N} log^{+} |j_{E}|_{v} + \frac{16}{5N} \}

≥ −\frac{log N}{2N} − \frac{1}{12N} log^{+} |j_{E}|_{v_{0}} − \frac{16}{5N} .

If N ≤ 24d, then (22) plainly holds; so we may henceforth assume that N ≥ 24d+1. Then combining the estimates (23) and (24) we have

(25)  ∑_{v∈M_{k}^{∞}} \frac{d_{v}}{d} Λ_{v}(Z) ≥ −\frac{log N}{2N} − \frac{16}{5N}

+ \frac{1}{N} \{\frac{(N − 1)d_{v_{0}}}{288d} − \frac{1}{12}\} max\{1, log |j_{E}|_{v_{0}}\}

≥ −\frac{log N}{2N} − \frac{16}{5N} + \frac{1}{N} \{\frac{N − 1}{288d} − \frac{1}{12}\}

= −\frac{log N}{2N} − \frac{197}{60N} + \frac{(N − 1)}{288dN}.

We now turn to the lower bounds on Λ_{v}(Z) at the nonarchimedean places; in particular we will show that

∑_{v∈M_{k}^{0}} \frac{d_{v}}{d} Λ_{v}(Z) ≥ \frac{1}{12d}(\frac{1}{16σ^{2}} − \frac{1}{N} ) log |ℕ(Δ)|. (26)

First, if E/k has everywhere good reduction then (26) plainly holds, since in that case log |ℕ(Δ)| = 0 and the left-hand side is nonnegative by Lemma 4. So we may assume that the set M_{k}^{br} = {v ∈ M_{k}^{0} | E/k_{v} has bad reduction} is nonempty.

At this point we will require Jensen’s inequality: if {w_{v}} is a finite set of positive weights with ∑_{v} w_{v} = W, if x_{v} > 0 for all v, and if ϕ(x) is a convex function for x > 0, then

∑_{v} w_{v}ϕ(x_{v}) ≥ W ϕ(\frac{1}{W} ∑_{v} w_{v}x_{v}). (27)

Applying this with ϕ(x) = 1/x, with {w_{v} = η_{v} log |ℕ(𝔭_{v})| | v ∈ M_{k}^{br}} as our weights (thus W = log |ℕ(𝔣)|), and with x_{v} = η_{v}c_{v}^{2}/δ_{v}, we have

(28)  ∑_{v∈M_{k}^{br}} \frac{δ_{v}}{c_{v}^{2}} log |ℕ(𝔭_{v})| ≥ W (\frac{1}{W} ∑_{v∈M_{k}^{br}} \frac{η_{v}^{2}c_{v}^{2}}{δ_{v}} log |ℕ(𝔭_{v})|)^{−1}

≥ W^{2}( ∑_{v∈M_{k}^{br}} 16δ_{v} log |ℕ(𝔭_{v})|)^{−1}

= \frac{(log ℕ(𝔣))^{2}}{16 log |ℕ(Δ)|}.

The first inequality in (28) is Jensen’s, and the second follows from the inequality

\frac{η_{v}^{2}c_{v}^{2}}{δ_{v}} ≤ 16δ_{v} (29)

for v ∈ M_{k}^{br}. (To see this, recall that if E/k_{v} has split multiplicative reduction then δ_{v} = c_{v} and η_{v} = 1, and (29) holds; while otherwise η_{v} ≤ δ_{v} and c_{v} ≤ 4, and (29) follows in this case as well.)

Finally, combining (14) and (28) and noting that d_{v} log |1/Δ_{v}|_{v} = δ_{v} log |ℕ(𝔭_{v})|, we have

∑_{v∈M_{k}^{0}} \frac{d_{v}}{d} Λ_{v}(Z) ≥ ∑_{v∈M_{k}^{br}} \frac{d_{v}}{d} (\frac{1}{c_{v}^{2}} − \frac{1}{N} ) \frac{1}{12} log |1/Δ_{v}|_{v}

= \frac{1}{12d} ∑_{v∈M_{k}^{br}} (\frac{1}{c_{v}^{2}} − \frac{1}{N} ) δ_{v} log |ℕ(𝔭_{v})|

≥ \frac{1}{12d} (\frac{(log |ℕ(𝔣)|)^{2}}{16 log |ℕ(Δ)|} − \frac{1}{N} log |ℕ(Δ)|)

= \frac{1}{12d} (\frac{1}{16σ^{2}} − \frac{1}{N} ) log |ℕ(Δ)|,

which is (26).

We are now ready to combine these local estimates. By the parallelogram law we have the global upper bound

(30)  Λ(Z) ≤ \frac{4}{N} ∑_{j=1}^{N} ĥ(P_{j})

≤ \frac{log |ℕ(Δ)|}{2^{11}3dσ^{2}}

on the sum defined in (6), by the upper bound on ĥ(P) for P ∈ S. Therefore by (25) and (26) we have

(31) \frac{log |ℕ(Δ)|}{2^{11}3dσ^{2}} ≥ Λ(Z)

= ∑_{v∈M_{k}} \frac{d_{v}}{d} Λ_{v}(Z)

≥ −\frac{log N}{2N} − \frac{197}{60N} + \frac{(N − 1)}{288dN} + \frac{1}{12d} (\frac{1}{16σ^{2}} − \frac{1}{N} ) log |ℕ(Δ)|.

If E/k has everywhere good reduction then, then σ = 1 and log |ℕ(Δ)| = 0, and thus (31) becomes

0 ≥ −\frac{log N}{2N} − \frac{197}{60N} + \frac{(N − 1)}{288dN}. (32)

It follows that N ≤ 144d log N + 945.6d + 1 ≤ 144d log N + 946.6d, and (22) holds in this case. On the other hand, suppose that M_{k}^{br} is nonempty. If N ≤ 2^{9}σ^{2}, then (22) holds, so we may assume that N > 2^{9}σ^{2}. The bound (31) implies that

(33)  0 ≥ −\frac{log N}{2N} − \frac{197}{60N} + \frac{1}{12d} (\frac{1}{16σ^{2}} − \frac{1}{N} ) log |ℕ(Δ)| − \frac{log |ℕ(Δ)|}{2^{11}3dσ^{2}}

≥ −\frac{log N}{2N} − \frac{197}{60N} + \frac{1}{12d} (\frac{1}{2^{4}σ^{2}} − \frac{1}{2^{9}σ^{2}} − \frac{1}{2^{9}σ^{2}} ) log |ℕ(Δ)|

= −\frac{log N}{2N} − \frac{197}{60N} + \frac{5}{2^{10}dσ^{2}} log |ℕ(Δ)|

≥ −\frac{log N}{2N} − \frac{197}{60N} + \frac{5 log 2}{2^{10}dσ^{2}} ,

since log |ℕ(Δ)| ≥ log 2. We deduce that

(34)  N ≤ \{\frac{2^{9}dσ^{2}}{5 log 2}\} log N + \frac{197 · 2^{10}dσ^{2}}{60 · 5 log 2}
< (148dσ^{2}) log N + 971dσ^{2}.

Thus (22) holds in this case as well, and the proof of Proposition 8 is complete. □

Proof of Theorems 1 and 2. Again, let S denote the set on the left-hand side of (21). Theorem 1 follows trivially from Proposition 8, since E(k)_{tor} ⊆ S. To see that Theorem 2 follows, let P ∈ E(k) be a nontorsion point, and let M be the largest integer such that ĥ((M −1)P) ≤ (log |ℕ_{k/ℚ}(Δ_{E/k})|)/2^{13}3dσ^{2}. Then the first M multiples O, P, 2P, …, (M − 1)P of P are contained in the set S, and therefore M ≤ c_{1}dσ^{2} log(c_{2}dσ^{2}) by Proposition 8. On the other hand, by the maximality of M we have M^{2}ĥ(P) = ĥ(MP) > (log |ℕ_{k/ℚ}(Δ_{E/k})|)/2^{13}3dσ^{2}. Therefore

ĥ(P) > M^{−2} \frac{log |ℕ_{k/ℚ}(Δ_{E/k})|}{2^{13}3dσ^{2}}
≥ c(d, σ) log |ℕ_{k/ℚ}(Δ_{E/k})|,

where c(d, σ) is given by (5). □

References

[1] Baker, Matthew; Petsche, Clayton. Global discrepancy and small points on elliptic curves. Int. Math. Res. Not. 61 (2005), 3791–3834. MR2205235.

[2] Cheng, Qi; Uchiyama, Shigenori. Nonuniform polynomial time algorithm to solve decisional Diffie–Hellman problem in finite fields under conjecture. Topics in cryptology—CT-RSA 2002, 290–299, Lecture Notes in Comput. Sci., 2271, Springer, Berlin, 2002. MR2048688, Zbl 1048.94009.

[3] David, Sinnou. Points de petite hauteur sur les courbes elliptiques. J. Number Theory. 64 (1997), no. 1, 104–129. MR1450488 (98k:11067), Zbl 0873.11035.

[4] Flexor, Marguerite; Oesterlé, Joseph. Sur les points de torsion des courbes elliptiques. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183, (1990), 25–36. MR1065153 (91g:11057), Zbl 0737.14004.

268 Clayton Petsche

[5] Hindry, Marc; Silverman, Joseph H. The canonical height and integral points on elliptic curves. Invent. Math. 93 (1988), no. 2, 419–450. MR0948108 (89k:11044), Zbl 0657.14018.

[6] Hindry, Marc; Silverman, Joseph H. On Lehmer’s conjecture for elliptic curves. Séminaire de Théorie des Nombres, Paris 1988–1989, 103–116, Progr. Math., 91, Birkhäuser Boston, Boston, MA, 1990. MR1104702 (92e:11062), Zbl 0741.14013.

[7] Hindry, Marc; Silverman, Joseph H. Sur le nombre de points de torsion rationnels sur une courbe elliptique. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 2, 97–100. MR1710502 (2000g:11047), Zbl 0993.11028.

[8] Koblitz, Neal; Menezes, Alfred J. Obstacles to the torsion-subgroup attack on the decision Diffie–Hellman problem. Math. Comp. 73 (2004), no. 248, 2027–2041 (electronic). MR2059749 (2006i:94055), Zbl 1064.94013.

[9] Lang, Serge. Introduction to Arakelov Theory. With an appendix by Paul Vojta: Diophantine inequalities and Arakelov theory. Springer-Verlag, New York, 1988. MR0969124 (89m:11059), Zbl 0667.14001.

[10] Masser, David W. Note on a conjecture of Szpiro. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183, (1990), 19–23. MR1065152 (91g:11058), Zbl 0742.14027.

[11] Masser, David W. On abc and discriminants. Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150 (electronic). MR1912990 (2003d:11050), Zbl 1030.11011.

[12] Merel, Loic. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), no. 1-3, 437–449. MR1369424 (96i:11057), Zbl 0936.11037.

[13] Parent, Pierre. Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. Reine Angew. Math. 506 (1999), 85–116. MR1665681 (99k:11080), Zbl 0919.11040.

[14] Rumely, Robert. Capacity theory on algebraic curves. Lecture Notes in Mathematics, 1378. Springer-Verlag, Berlin, 1989. MR1009368 (91b:14018), Zbl 0679.14012.

[15] Silverman, Joseph H. The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986. MR0817210 (87g:11070), Zbl 0585.14026.

[16] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994. MR1312368 (96b:11074), Zbl 0911.14015.

[17] Szpiro, Lucien. Discriminant et conducteur des courbes elliptiques. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183, (1990), 7–18. MR1065151 (91g:11059), Zbl 0742.14026.

[18] Tate, John. A review of non-Archimedean elliptic functions. Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong, 1993), 162–184, Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995. MR1363501 (97d:11096), Zbl 1071.11508.

Department of Mathematics, The University of Georgia, Athens, GA 30602-7403 clayton@math.uga.edu

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