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

E

MANUEL

H

ERRMANN

A

TTILA

P

ETHÖ

S-integral points on elliptic curves - Notes on a

paper of B. M. M. de Weger

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{2 (2001),}

p. 443-451

<http://www.numdam.org/item?id=JTNB_2001__13_2_443_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

S-integral points

on

elliptic

curves -

Notes

on a

paper

of B. M.

M.

de

Weger

par EMANUEL HERRMANN * et ATTILA

PETHO

**

RÉSUMÉ. Nous donnons une nouvelle preuve

beaucoup plus

courte

d’un résultat de B. M. M de

_Weger.

Cette 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

re-sult 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}

(1)

y 2 = x 3 -

228x + 848

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, oo}.

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

Remond 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

Manuscrit reçu le 3 aout 1999.

*

This _paper was _partly written while the author was a _Visiting Scholar at the School of Math-ematics 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,

We now state our result.

Theorem 1. Let S =

{2, 3, 5, ?, oo}.

Then the

_equation

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

Let S = =

00}

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 E} S. For the

_general

case we refer to the

paper

[5].

Let

_{Pl, ... , Pr}

denote a basis of the Mordell-Weil _group and

let g

be the order of the torsion

_subgroup

of

_{E(Q) .}

Let h

denote the N6ron-Tate

_height

on

E(~).

_{Designate by A}

the smallest

_eigenvalue

of the

positive

definite

_regulator

matrix

Let

_p(u)

be the Weierstrass

_{p-function corresponding}

to the curve

E((C).

Let S2 =

(WI, (2)

be its fundamental lattice and _wl its real

period.

There

exists,

for _{any P} =

(~, y)

_E

E(C),

an element u E such that

_{(x, y) =}

ip’(u)) .

This is called the

_(complex)

_{elliptic logarithm}

of P. In the

sequel

Ui,oo denotes the

elliptic logarithm

of Pi

for i =

1, ... ,

_r. We

_put

i _ui-oo

u

W1

For a finite

_{prime q}

E S let denote the

_points

of

_{E(Q )}

with

non-singular

reduction modulo _q. Then the index

_{~E(~ ) : Eo (Q )]}

is

_finite,

and

equal

to the

_Tamagawa

_{number cq because}

_by

our

assumption equation

(2)

is minimal at _q. Let further E denote the reduced curve E modulo _q. Let

NQ

= be the number of rational

points

of

E /JFq.

With the order _g

of the _{torsion group,} we define the number m =

mQ =

lcm(g,

eq .

Nq).

Finally

for the finite

_places

q E

S,

let denote the

q-adic elliptic

logarithm

of

_mPi

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.

Theorem A. Let the

_elliptic

curve

E(Q)

be

_{defined by equation}

_(2),

which

P =

(x,

y) E E(Zs)

has the

_{representation}

with ni E

Z, I

=

1, ... ,

_r, and T a torsion

_point

_of

_E(Q).

For

_{N(P) =}

=

_{1, ... ,}

r},

_we have

with

where

with

_Ao

=

4A 3+

27B2. Moreover,

there exists _a

place

q E S such that

zuith E Z

_if

_{q =} oo and _{nr+1 =} 0

otherwise,

and with

ks = f3¡

if

q = oo and

_k5

= 1 otherwise.

0 -

3wl

Theorem A

_together

with numerical

_Diophantine

_{approximation}

tech-niques

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 Remond 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

For a finite

place q

E S’ let

Theorem B.

_Assuming

that r 2 and

_using

the notations introduced in Theorem A and above we have

where

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,

Peth6

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, oo}.

It is easy to

check,

that

(1)

is minimal for every finite

prime q

E S.

_Actually,

it

is a

global

minimal model of E. The discriminant of E is A

= -160o

with

Ao

= -27993600. We have

where the

_only

non-trivial torsion

_point

is

_{(4, 0)}

and a basis of the infinite

part

of the Mordell-Weil _group is

PI

=

(-2,36),P2

=

(-11,45).

(See

Tzanakis

_[10],

or one of of the _{programs apecs}

[13],

y

Magma’

[1],

mwrank

[14]

or Simath

[15].)

Now we can

_compute

the fundamental

_{parallelogram}

of the associated

Weierstrass

_p-function

and

_get

The

_regulator

matrix of E is

hence its smallest

_eigenvalue

is

_{given by}

A = 0.375922.

Using

Tate’s

_algorithm

_[9]

we

_compute

the

Tamagawa

numbers

The curve E has additive reduction at the

_primes

2 and

_3,

_{multiplicative}

reduction at 5 and

_good

reduction at 7.

_Hence,

-Using

these data we can

_compute

the numbers mq and obtain

3.2.

_Upper

Bounds for

_N(P).

(i)

The first _way to obtain an _upper bound for

N(P)

is to calculate

No

of Theorem A. We have

_{actually Q}

=

7, s

=

5,

and

_{kl -}

3.730724

_.10~,

hence

_N(P)

_No

= 7.044216 -

10184.

(ii)

Another,

a bit more

complicated,

_way to find an _upper bound for

N(P)

is to

_compute

_Nl

=

max~Nq : q

E

_S}

as defined in Theorem B.

1

Thus we obtain

_Noo

1.530526 .

1047

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 aq, o-q and

dq.

*

Table 1

The

_following

values are

independent

of _q E

_{2,3,5,7}

Choosing

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

thus

These

_inequalities

_imply

by

Theorem B. Since

_Nl

is much smaller than

_No

we use this value in the

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 approximatioi}

problem

for each _q E 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 Peth6 et al.

_[5].

We shall also use the notations introduced there.

We first take _q = oo and

_perform

a de

_Weger

reduction with C =

1014z.

We obtain the new _upper bound

N(P)

_MO

= 67 in the case _q = _oo.

Comparing

this bound with =

2, 3, 5, 7

_we obtain

i.e. we _may

_perform

the

_q-adic

reduction

_steps

with this value.

To do this we

_compute

for

each q

E the

_{q-adic elliptic logarithms}

of i =

1, 2,

with

precision

_at least

This

_precision

_{is necessary} to _carry out the

_q-adic

de

_Weger

reduction. For

this purpose we use the method of

[5].

Now we

perform

the

_q-adic

de

_Weger

reduction with the values

_C2

=

2128,

,

C3

=

381,

C5

=

555

and

C7

=

746

and obtain the new bound

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.

References

[1] W. _BOSMA, J. _CANNON, C. _PLAYOUST, The Magma algebra system I: The user lan-guage. J. Symb. Comp., 24, 3/4 (1997), 235-265. _(See also the Magma home page at

http://www.maths.usyd.edu.au:8000/u/magma/)

[2] S. DAVID, Minorations de _formes linéaires de _{logarithmes elliptiques.} Mém. Soc. Math. France _(N.S.) 62 _(1995).

[3] J. _GEBEL, A. H. G. _{ZIMMER, Computing integral points} on _elliptic curves. Acta Arith. 68 _(1994), 171-192.

[4] J. GEBEL, A. H. G. _{ZIMMER, Computing S-integral points} on _elliptic curves.

Al-gorithmic number theory (Talence, 1996), 157-171, Lecture Notes in Comput. Sci. _1122,

Springer, Berlin, 1996.

[5] A. H. G. _ZIMMER, J. GEBEL, E. HERRMANN, Computing all S-integral points on

elliptic curves. Math. Proc. Cambr. Phil. Soc. 127 _(1999), 383-402.

[6] G. RÉMOND, F. URFELS, Approximation diophantienne de logarithmes elliptiques p-adiques.

J. Numb. Th. 57 _(1996), 133-169.

[7] J. H. SILVERMAN, The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106,

Springer-Verlag, New York, 1986.

[8] N. P. SMART, S-integral Points on _elliptic curves. Math. Proc. Cambr. Phil. Soc. 116

(1994), 391-399.

[9] J. T. TATE, Algorithm for determining the _type of a _{singular fibre} in an _{elliptic pencil.}

Modular functions of one _variable, IV _(Proc. Internat. Summer _School, Univ. Antwerp, Antwerp, 1972), 33-52, Lecture Notes in Math. _{476, Springer, Berlin,} 1975.

[10] N. TZANAKIS, Solving elliptic Diophantine equations by estimating linear _forms in _elliptic

logarithms. The case _{of quartic equations.} Acta Arith. 75 (1996), 165-190.

[11] B. M. M. DE _{WEGER, Algorithms for Diophantine equations.} PhD Thesis, Centr. for Wiskunde en _{Informatica, Amsterdam,} 1987.

[12] B. M. M. DE _{WEGER, S-integral} solutions to a Weierstrass _equation, J. Théor. Nombres

Bordeaux 9 _(1997), 281-301.

[13] Apecs, Arithmetic _{of plane elliptic} curves, ftp://ftp.math.mcgill.ca/pub/apecs.

[14] mwrank, a _package to _compute ranks _{of elliptic} curves over the rationals.

http://www.maths.nott.ac.uk/personal/jec/ftp/progs.

[15] Simath, a _{computer algebra system for algorithmic} number _theory.

http://simath.math.uni-sb.de.

Emanuel HERRMANN FR 6.1 Mathematik Universitat des Saarlandes Postfach 151150

D-66041 Saarbrucken

Germany

E-mail : hernaanncQmath . uni-sb . de

Attila PFTH6

Institute of Mathematics and Informatics Kossuth Lajos University

H-4010 Debrecen, P.O.Box 12

Hungary