Journal de Théorie des Nombres de Bordeaux 13 (2001), 1–

S-integral points on elliptic curves - Notes on a paper of B.M.M. de Weger

par Emanuel HERRMAN^{∗} et Attila PETHŐ^{∗∗}

Résumé. Nous donnons une nouvelle preuve beaucoup plus courte d’un résultat de B. M. M de Weger. Cett preuve est basée sur la théorie des formes linéaires de logarithmes complexes, p-adiques et elliptiques, pour lesquelles nous obtenons une majoration en confrontant les résultats de Hajdu et Herendi à ceux de Rémond et Urfels.

Abstract. In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and p-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

1. Introduction

In a recent paper [12] B.M.M. de Weger solved the Diophantine equation

y^{2} = x^{3} − 228x + 848 (1)

completely in rational numbers x, y such that their denominator in the lowest form is a power of 2. With other words, he solved (1) in S-integers where S = {2, ∞}. De Weger uses in the proof algebraic number theoretical considerations and lower estimates for linear forms in complex and q-adic logarithms of algebraic numbers.

In the present paper we will give a much shorter proof of a generalization of Theorem 1 of [12]. Here we use the theory of elliptic curves and linear forms in elliptic logarithms. More precisely, we are using a theorem of Rémond and Urfels [6], which can be applied for curves of rank at most 2. An alternative method which avoids lower bounds for linear forms in q-adic elliptic logarithms is given in [5]. However the bounds coming from [5] are in the actual case much larger as working directly with the Theorem of Rémond and Urfels (cf. Section 3).

Manuscrit reçu le 3 août 1999.

∗ This paper was partly written while the author was a Visiting Scholar at the School of Mathematics and Statistics at the University of Sydney. His research was supported in part by grants from the Australian Research Council and the Defense Science and Technology Organization.

∗∗ Research partially supported by Hungarian National Foundation for Scientific Research, Grant No. T25157 and T16975.

We now state our result.

Theorem 1. Let S = {2, 3, 5, 7, ∞}. Then the equation

y^{2} = x^{3} − 228x + 848

has only 65 S-integer solutions (x, ±y) listed in Table 2 at the end of this paper.

2. Notations and Auxiliary Results

Let the elliptic curve be defined by the equation

y^{2} = x^{3} + Ax + B, A, B ∈ ℤ. (2)

Let S = {q_{1}, …, q_{s−1}, q_{s} = ∞} be a set of primes including the infinite prime. To simplify the presentation we assume that the equation (2) is minimal for every finite prime q ∈ S. For the general case we refer to the paper [5].

Let P_{1}, …, P_{r} denote a basis of the Mordell-Weil group E(ℚ) and let g be the order of the torsion subgroup E_{tors}(ℚ) of E(ℚ). Let ĥ denote the Néron-Tate height on E(ℚ). Designate by λ the smallest eigenvalue of the positive definite regulator matrix (ĥ(P_{i}, P_{j}))_{1≤i,j≤r}.

Let ℘(u) be the Weierstrass ℘-function corresponding to the curve E(ℂ). Let Ω = ⟨ω_{1}, ω_{2}⟩ be its fundamental lattice and ω_{1} its real period. There exists, for any P = (x, y) ∈ E(ℂ), an element u ∈ ℂ/Ω such that (x, y) = (℘(u), \frac{1}{2}℘′(u)) . This is called the (complex) elliptic logarithm of P. In the sequel u_{i,∞} denotes the elliptic logarithm of P_{i} for i = 1, …, r. We put u′_{i,∞} = g \frac{u_{i,∞}}{ω_{1}} .

For a finite prime q ∈ S let E_{0}(ℚ_{q}) denote the points of E(ℚ_{q}) with nonsingular reduction modulo q. Then the index [E(ℚ_{q}) : E_{0}(ℚ_{q})] is finite, and equal to the Tamagawa number c_{q} because by our assumption equation (2) is minimal at q. Let further Ẽ denote the reduced curve E modulo q. Let N_{q} = #Ẽ(𝔽_{q}) be the number of rational points of Ẽ/𝔽_{q}. With the order g of the torsion group, we define the number

m = m_{q} = lcm(g, c_{q} · N_{q}).

Finally for the finite places q ∈ S, let u′_{i,q} denote the q-adic elliptic logarithm of mP_{i} for i = 1, …, r. For the definition and basic properties of q-adic elliptic logarithms we refer to Silverman [7] and to [5]. Now we state the main result of [5] in the special case considered, i.e. for curves given in short Weierstrass form.

S-integral points on elliptic curves 3

Theorem A. Let the elliptic curve E(ℚ) be defined by equation ( 2), which is minimal for every finite prime q ∈ S. Assume that the S-integral point P = (x, y) ∈ E(ℤ_{S}) has the representation

P = ∑_{i=1}^{r} n_{i}P_{i} + T (3)

with n_{i} ∈ ℤ, i = 1, …, r, and T a torsion point of E(ℚ). For N(P) = max{|n_{i}|, i = 1, …, r}, we have

N(P) ≤ N_{0} = \sqrt{\frac{1}{λ}(\frac{k_{1}}{2} + k_{2})} (4)

with k_{2} = log max{|2A|^{1/2}, |4B|^{1/3}},

k′_{1} =7 · 10^{38s+49}s^{20s+15}Q^{24}(log^{∗} Q)^{4s−2}k_{3}(log k_{3})^{2}((20s − 19)k_{3} + log(ek_{4})),
k_{1} =k′_{1} + 2 log 6,

where log^{∗} Q = max{log Q, 1} for Q = max{q_{1}, …, q_{s−1}}, s=#S,

k_{3} =\frac{32}{3} \sqrt{|Δ_{0}|} (8 + \frac{1}{2} log |Δ_{0}|)^{4} ,
k_{4} =10^{4} max\{16A^{2}, 256\sqrt{|Δ_{0}|}^{3}\}

with Δ_{0} = 4A^{3} + 27B^{2}. Moreover, there exists a place q ∈ S such that

|∑_{i=1}^{r} n_{i}u′_{i,q} + n_{r+1}|_{q} ≤ k_{5} exp\{−\frac{λ}{s}N(P)^{2} + \frac{k_{2}}{s} \} (5)

with n_{r+1} ∈ ℤ if q = ∞ and n_{r+1} = 0 otherwise, and with k_{5} = \frac{2g}{3ω_{1}} if q = ∞ and k_{5} = 1 otherwise.

Theorem A together with numerical Diophantine approximation techniques is sufficient to prove our Theorem 1. However it was pointed out already in [5] that combining the method of Smart [8] with results of David [2] and of Rémond and Urfels [6] one can obtain a much better estimate for N(P) as by the one implied by Theorem A. In the sequel we assume r ≤ 2. To formulate the next theorem we have to introduce further notations. Let j = \frac{j_{1}}{j_{2}} with j_{1}, j_{2} ∈ ℤ and gcd(j_{1}, j_{2}) = 1 be the j−invariant of E(ℚ). Put

h = log max{4|Aj_{2}|, 4|Bj_{2}|, |j_{1}|},
log V_{i} = max \{ĥ(P_{i}), h, \frac{3π|u′_{i,∞}|_{∞}^{2}}{Imτ} \} , i = 1, 2,

log V_{0} = max \{h, \frac{3π}{Imτ} \} ,
k_{6,∞} = \frac{k_{2} + s log k_{5}}{λ} ,
k_{7,∞} = \frac{2 · 10^{68} · s · h^{5}}{λ} ∏_{i=0}^{2} log V_{i}.

For a finite place q ∈ S let

α_{q} = \{ \begin{matrix}3, & if q = 2 \\ \frac{1}{q−1}, & otherwise \end{matrix}
σ_{q} = (q^{α_{q}} max{|u′_{1,q}|_{q}, |u′_{2,q}|_{q}})^{−1},
d_{q} = max{1, 1/ log σ_{q}},
a_{i} = max{1, ĥ(P_{i})}, i = 1, 2,
β = max{log N(P), log |A|_{∞}, log |B|_{∞}, a_{1}, a_{2}, d_{q}},
γ = max{log |A|_{∞}, log |B|_{∞}, log β},
k_{6,q} = \frac{k_{2}}{λ} ,
k_{7,q} ≥ (3.6 · 10^{25}s · a_{1}a_{2}d_{q}^{6} log σ_{q})/λ.

Theorem B. Assuming that r ≤ 2 and using the notations introduced in Theorem A and above we have

N(P) ≤ N_{1} := max{N_{q} : q ∈ S},

where

N_{q} = \{ \begin{matrix}2^{5}\sqrt{k_{6,∞}k_{7,∞}}(log 5^{5}k_{7,∞})^{5/2}, & if q = ∞, \\ 2^{4}\sqrt{k_{6,q}k_{7,q}}(log 4^{4}k_{7,q})^{2}, & if q ∈ S \ \{∞\}. \end{matrix}

Proof. Combining inequality (5) with the lower bounds for linear forms in elliptic logarithms due to David [2] and for linear forms in at most two q−adic elliptic logarithms due to Rémond and Urfels [6] one obtains the upper bound for N(P) analogously as described for example in Gebel, Pethő and Zimmer [3, 4]. Therefore we omit the details.

3. Proof of Theorem 1

3.1. Basic data of the elliptic curve. In the sequel we denote by E the elliptic curve over ℚ defined by equation (1). Let S = {2, 3, 5, 7, ∞}. It is easy to check, that (1) is minimal for every finite prime q ∈ S. Actually, it is a global minimal model of E. The discriminant of E is Δ = −16Δ_{0} with Δ_{0} = −27993600. We have

E(ℚ) ≃ ℤ/2ℤ ⊕ ℤ^{2},

where the only non-trivial torsion point is (4, 0) and a basis of the infinite part of the Mordell-Weil group is P_{1} = (−2, 36), P_{2} = (−11, 45). (See Tzanakis [10], or one of of the programs apecs [13], Magma^{1} [1], mwrank [14] or Simath [15].)

Now we can compute the fundamental parallelogram of the associated Weierstrass ℘-function and get

ω_{1} = 0.767848, ω_{2} = −0.631356 · i and τ = \frac{ω_{1}}{ω_{2}} = 1.216188 · i.

The regulator matrix of E is

R = (\begin{matrix}0.423441 & −0.158771 \\ −0.158771 & 0.906408 \end{matrix}) ,

hence its smallest eigenvalue is given by λ = 0.375922.

Using Tate’s algorithm [9] we compute the Tamagawa numbers

c_{2} = 4, c_{3} = 4, c_{5} = 2 and c_{7} = 1.

The curve E has additive reduction at the primes 2 and 3, multiplicative reduction at 5 and good reduction at 7. Hence,

N_{2} = 2, N_{3} = 3, N_{5} = 6 and N_{7} = 12.

Using these data we can compute the numbers m_{q} and obtain

m_{2} = 8, m_{3} = 12, m_{5} = 12 and m_{7} = 12.

3.2. Upper Bounds for N(P).

(i) The first way to obtain an upper bound for N(P) is to calculate N_{0} of Theorem A. We have actually Q = 7, s = 5,

k_{2} = log max{456^{1/2}, 3392^{1/3}} = 3.061246,
k_{3} =\frac{32}{3} \sqrt{|Δ_{0}|} (8 + \frac{1}{2} log |Δ_{0}|)^{4} = 4.258342 · 10^{9},
(6)  k_{4} =10^{4} max{16 · 228^{2}, 256 · |Δ_{0}|^{3/2}} = 3.791649 · 10^{17}

and k_{1} = 3.730724 · 10^{369}, hence N(P) ≤ N_{0} = 7.044216 · 10^{184}.

1Magma version 2.6 will have an implementation of the algorithm described in [5].

(ii) Another, a bit more complicated, way to find an upper bound for N(P) is to compute N_{1} = max{N_{q} : q ∈ S} as defined in Theorem B. Consider first the case q = ∞. Then we have

h = log max{4 · 228 · 75, 4 · 848 · 75, 2^{5} · 19^{3}} = 12.446663,
log V_{0} = max\{h, \frac{3π}{Imτ} \} = 12.446663,
log V_{1} = max\{ĥ(P_{1}), h, \frac{3πg^{2}|u_{1,∞}|_{∞}^{2}}{ω_{1}^{2}Imτ} \} = 21.645104,
log V_{2} = max\{ĥ(P_{2}), h, \frac{3πg^{2}|u_{2,∞}|_{∞}^{2}}{ω_{1}^{2}Imτ} \} = 28.279603,
k_{5,∞} =\frac{4}{3ω_{1}} = 1.736455,
k_{6,∞} =15.483196,
k_{7,∞} =6.054145 · 10^{78}.

Thus we obtain N_{∞} ≤ 1.530526 · 10^{47} after a simple computation.

Next we have to consider the cases q = 2, 3, 5 and 7. In Table 1 below you find the actual values of α_{q}, σ_{q} and d_{q}.

Table 1

\begin{matrix}q & 2 & 3 & 5 & 7 \\ α_{q} & 3 & 1/2 & 1/4 & 1/6 \\ σ_{q} & 2 & 3^{1/2} & 5^{3/4} & 7^{5/6} \\ d_{q} & 1/ log 2 & 2/ log 3 & 1 & 1 \\ k_{7,q} & 2.992592 · 10^{27} & 9.5742 · 10^{27} & 5.779766 · 10^{26} & 7.76455 · 10^{26} \end{matrix}

The following values are independent of q ∈ {2, 3, 5, 7}

a_{1} = max{1, ĥ((−2, 36))} = max{1, 0.423441} = 1,
a_{2} = max{1, ĥ((−11, 45))} = max{1, 0.906408} = 1,
k_{6,q} = k_{2}/λ = 8.143301.

Choosing the worst cases from Table 1 we see that we can take

k_{7,q} = k_{7,3} = 9.5742 · 10^{27}, q = 2, 3, 5, 7,

thus

N_{q} = N_{3} = 2.187487 · 10^{19}, q = 2, 3, 5, 7.

These inequalities imply

N(P) ≤ N_{1} = max{N_{q} : q ∈ S} = 1.530526 · 10^{47}

by Theorem B. Since N_{1} is much smaller than N_{0} we use this value in the sequel.

3.3. Reduction of the large upper bound for N(P). By Theorem 1, and by the last section we have to solve the Diophantine approximation problem

|n_{1}u′_{1,q} + n_{2}u′_{1,q} + n_{3}|_{q} ≤ k_{5}exp{0.075184 · N(P)^{2} + 0.6122492},
N(P) ≤ N_{1} = 1.530526 · 10^{47}

for each q ∈ S.

To solve these systems we use the well known reduction procedure of de Weger [11]. (See also Smart [8].) For details about the high precision computation of q-adic elliptic logarithms we refer to Pethő et al. [5]. We shall also use the notations introduced there.

We first take q = ∞ and perform a de Weger reduction with C = 10^{142}. We obtain the new upper bound N(P) ≤ M_{∞} = 67 in the case q = ∞. Comparing this bound with N_{q}, q = 2, 3, 5, 7 we obtain

N(P) ≤ N_{3} = 2.187487 · 10^{19},

i.e. we may perform the q−adic reduction steps with this value.

To do this we compute for each q ∈ S\{∞}, the q-adic elliptic logarithms of m_{q}P_{i}, i = 1, 2, with precision at least

n_{2} = 129, n_{3} = 82, n_{5} = 56, n_{7} = 46.

This precision is necessary to carry out the q-adic de Weger reduction. For this purpose we use the method of [5].

u′_{1,2} =134584334573222732131510464853384888320 + O(2^{128})
u′_{2,2} =224603122385055121905025779589746548856 + O(2^{128})
u′_{1,3} =35130898366670225251067310603381664587 + O(3^{81})
u′_{2,3} =32674326287561878726624624078558984866 + O(3^{81})
u′_{1,5} =118414103305724592543524002578287458095 + O(5^{55})
u′_{2,5} =193714651202697832194263283063279750580 + O(5^{55})
u′_{1,7} =49086609441793589144883973076015987885 + O(7^{46})
u′_{2,7} =723939447229120403790851561285560713079 + O(7^{46})

Now we perform the q-adic de Weger reduction with the values C_{2} = 2^{128}, C_{3} = 3^{81}, C_{5} = 5^{55} and C_{7} = 7^{46} and obtain the new bound

N(P) ≤ max{M_{∞} = 67, M_{2} = 12, M_{3} = 13, M_{5} = 13, M_{7} = 13}.

This new upper bound for N(P) can be further reduced. On repeating this reduction process 3-times, we eventually get N(P) ≤ 13, which cannot be reduced any further.

8 Emanuel Herrmann, Attila Pethő

Table 2

S-integral points P = (x, y) = (\frac{ξ}{ζ^{2}} , \frac{η}{ζ^{3}} ) = ∑_{i=1}^{2} n_{i}P_{i} + T_{j}, j = 0, 1

on E : y^{2} = x^{3} − 228x + 848 for S = {2, 3, 5, 7, ∞}
rank 2
basis P_{1} = (−2, 36), P_{2} = (−11, 45)
torsion T_{0} = O, T_{1} = (4, 0)
#  ξ  η ζ  F (n_{1}, n_{2}, j)
1  4  0 1  (0, 0, 1)
2  −11  45 1  (0, 1, 0)
3  16  36 1  (0, 1, 1)
4  94  −900 1  (1, −1, 0)
5  2  −20 1  (1, −1, 1)
6  −2  36 1  (1, 0, 0)
7  34  180 1  (1, 0, 1)
8  14  −20 1  (1, 1, 0)
9  −14  −36 1  (1, 1, 1)
10  754  −20700 1  (1, 2, 1)
11  196  2736 1  (2, −1, 1)
12  13  9 1  (2, 0, 0)
13  −16  20 1  (2, 0, 1)
14  52  −360 1  (2, 1, 1)
15  53  371 1  (2, 2, 0)
16  814  23220 1  (3, 1, 0)
17 534256 −390502764 1  (4, 3, 1)
18  97  −783 2  2 (0, −2, 0)
19  1  −225 2  2 (2, 1, 0)
20  857  −25027 2  2 (4, 0, 0)
21  49  855 4  2^{2} (2, −1, 0)
22 −16439  −631035 32  2^{5} (2, 3, 0)
23  −44  −1160 3  3 (0, −2, 1)
24  34  172 3  3 (3, 1, 1)
25  1534  42020 9  3^{2} (3, −1, 0)
26  94  −828 5  5 (1, 2, 0)
27  629  −13133 5  5 (2, −2, 0)
28 −194  −5796 5  5 (3, 0, 0)
29  6361  −282141 20 2^{2} × 5 (4, 2, 0)
30 −818  −468 7  7 (1, −2, 0)
31  16  9540 7  7 (2, 2, 1)
32  946  −20700 7  7 (3, 0, 1)
33  8516 1163623840 343  7^{3} (4, −2, 1)

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[14] mwrank, a package to compute ranks of elliptic curves over the rationals. http://www.maths.nott.ac.uk/personal/jec/ftp/progs.

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Emanuel Herrmann FR 6.1 Mathematik Universität des Saarlandes Postfach 151150 D-66041 Saarbrücken Germany

E-mail : herrmann@math.uni-sb.de

Attila Pethő Institute of Mathematics and Informatics Kossuth Lajos University H-4010 Debrecen, P.O.Box 12 Hungary