Computing integral points on elliptic curves.

*(English)*Zbl 0816.11019A method is developed for computing explicitly all integral points on a Weierstrass model of an elliptic curve over \(\mathbb{Q}\), based on estimates of linear forms in elliptic logarithms.

In a few words, we can describe the notion of elliptic logarithm of an elliptic curve as follows: Let \(y^ 2= x^ 3+ ax+b\) be a short Weierstrass model of an elliptic curve \(E/\mathbb{Q}\). There exists a Weierstrass \(\wp\)-function (doubly periodic meromorphic function) such that \(x= \wp(z)\), \(y= \wp'(z)\) parametrizes the curve \(E(\mathbb{C})\). Let \(P= (x(P), y(P))\in E(\mathbb{C})\) and \(z\in \mathbb{C}\) such that \(\wp(z)= x(P)\), \(\wp'(z)= y(P)\); then, we say that \(z\) is the elliptic logarithm of the point \(P\).

The idea of using estimates of linear forms in elliptic logarithms of points on \(E(\overline{\mathbb{Q}})\), in order to find explicitly all integral points on \(E(\mathbb{Q})\) is not new [see Appendix IV of D. W. Masser, Elliptic functions and transcendence, Lect. Notes Math. 437, Springer, Berlin (1975; Zbl 0312.10023); Chapter VI, §8 of S. Lang, Elliptic curves; diophantine analysis, Grundlehren Math. Wiss. 231, Springer, Berlin (1978; Zbl 0388.10001); or Chapter IX, §5 of J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer, New York (1986; Zbl 0585.14026)]. However, there is a long way from theory to practice.

First of all, an explicit lower bound for linear forms as above should be proved. N. Hirata-Kohno gave such an effective bound [see Invent. Math. 104, 401-433 (1991; Zbl 0716.11035), Corollary 2.16], but it was S. David, who made this bound explicit [Minorations de formes linéaires de logarithmes elliptiques, Publ. Math. Univ. Pierre et Marie Curie 106, Problèmes diophantiens 1991-1992, exposé no 3].

Second, the method assumes the possibility of computing explicitly a Mordell-Weil basis for \(E(\mathbb{Q})\), and it is only recently that powerful techniques were developed for this purpose. The first and third named authors, for example, have developed such a technique [Elliptic curves and related topics, CRM Proc. Lect. Notes 4, 61-83 (1994; Zbl 0809.14024)]; other powerful tools are provided by Cremona’s mwrank, based on Cremona’s algorithm [J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press (1992; Zbl 0758.14042)] and Apecs of I. Stewart, based on Maple V (the authors of the present paper do not use either of the last two tools).

If \(P_ 1, \dots, P_ r\) is a Mordell-Weil basis for \(E(\mathbb{Q})\) and \(P\) is the typical integral point, then \(P= m_ 1 P_ 1+ \cdots+ m_ r P_ r+T\) (\(T\) a torsion point), \(m_ 1,\dots,m_ r\in \mathbb{Z}\) and the estimate of a certain linear form in elliptic logarithms provides a very large upper bound for \(\max_{1\leq i\leq r} | m_ i|\). A powerful technique is then necessary for reducing this very large upper bound to a manageable size. B. M. M. de Weger proposed such a technique, based on the LLL-basis reduction algorithm [see Chapter 3 of B. M. M. de Weger, Algorithms for diophantine equations, CWI Tract 65, Amsterdam (1989; Zbl 0687.10013), but also section 3 of the reviewer and B. M. M. de Weger, J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)].

Two more computational problems are (1) the computation of the Néron- Tate height and its “distance” from the usual naive height; work on this has been done by the third named author [Math. Z. 147, 35-51 (1976; Zbl 0311.14003)] (Apecs also, based on an algorithm by J.H. Silverman [Computing heights on elliptic curves, Math. Comput. 51, 339- 358 (1988; Zbl 0656.14016)], calculates automatically the Néron-Tate height of a given rational point) and (2) the computation of the elliptic logarithm of a point; an ingenious trick of D. Zagier [Math. Comput. 48, 425-436 (1987; Zbl 0611.10008)] makes it possible, with very high precision (hundreds of decimal digits; this is necessary for the reduction of the large upper bound mentioned above).

Finally, a synthesis of all these ingredients has to be made and this is done successfully by the authors of the present paper. It is somewhat amazing that, a little earlier, R. J. Stroeker and the reviewer had developed a completely similar method [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)]. Obviously, the authors of the paper under review worked independently from Stroeker and the reviewer; moreover, they solved, as applications of the method, three very impressive numerical examples, related to elliptic curves of rank 5 and 6.

In a few words, we can describe the notion of elliptic logarithm of an elliptic curve as follows: Let \(y^ 2= x^ 3+ ax+b\) be a short Weierstrass model of an elliptic curve \(E/\mathbb{Q}\). There exists a Weierstrass \(\wp\)-function (doubly periodic meromorphic function) such that \(x= \wp(z)\), \(y= \wp'(z)\) parametrizes the curve \(E(\mathbb{C})\). Let \(P= (x(P), y(P))\in E(\mathbb{C})\) and \(z\in \mathbb{C}\) such that \(\wp(z)= x(P)\), \(\wp'(z)= y(P)\); then, we say that \(z\) is the elliptic logarithm of the point \(P\).

The idea of using estimates of linear forms in elliptic logarithms of points on \(E(\overline{\mathbb{Q}})\), in order to find explicitly all integral points on \(E(\mathbb{Q})\) is not new [see Appendix IV of D. W. Masser, Elliptic functions and transcendence, Lect. Notes Math. 437, Springer, Berlin (1975; Zbl 0312.10023); Chapter VI, §8 of S. Lang, Elliptic curves; diophantine analysis, Grundlehren Math. Wiss. 231, Springer, Berlin (1978; Zbl 0388.10001); or Chapter IX, §5 of J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer, New York (1986; Zbl 0585.14026)]. However, there is a long way from theory to practice.

First of all, an explicit lower bound for linear forms as above should be proved. N. Hirata-Kohno gave such an effective bound [see Invent. Math. 104, 401-433 (1991; Zbl 0716.11035), Corollary 2.16], but it was S. David, who made this bound explicit [Minorations de formes linéaires de logarithmes elliptiques, Publ. Math. Univ. Pierre et Marie Curie 106, Problèmes diophantiens 1991-1992, exposé no 3].

Second, the method assumes the possibility of computing explicitly a Mordell-Weil basis for \(E(\mathbb{Q})\), and it is only recently that powerful techniques were developed for this purpose. The first and third named authors, for example, have developed such a technique [Elliptic curves and related topics, CRM Proc. Lect. Notes 4, 61-83 (1994; Zbl 0809.14024)]; other powerful tools are provided by Cremona’s mwrank, based on Cremona’s algorithm [J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press (1992; Zbl 0758.14042)] and Apecs of I. Stewart, based on Maple V (the authors of the present paper do not use either of the last two tools).

If \(P_ 1, \dots, P_ r\) is a Mordell-Weil basis for \(E(\mathbb{Q})\) and \(P\) is the typical integral point, then \(P= m_ 1 P_ 1+ \cdots+ m_ r P_ r+T\) (\(T\) a torsion point), \(m_ 1,\dots,m_ r\in \mathbb{Z}\) and the estimate of a certain linear form in elliptic logarithms provides a very large upper bound for \(\max_{1\leq i\leq r} | m_ i|\). A powerful technique is then necessary for reducing this very large upper bound to a manageable size. B. M. M. de Weger proposed such a technique, based on the LLL-basis reduction algorithm [see Chapter 3 of B. M. M. de Weger, Algorithms for diophantine equations, CWI Tract 65, Amsterdam (1989; Zbl 0687.10013), but also section 3 of the reviewer and B. M. M. de Weger, J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)].

Two more computational problems are (1) the computation of the Néron- Tate height and its “distance” from the usual naive height; work on this has been done by the third named author [Math. Z. 147, 35-51 (1976; Zbl 0311.14003)] (Apecs also, based on an algorithm by J.H. Silverman [Computing heights on elliptic curves, Math. Comput. 51, 339- 358 (1988; Zbl 0656.14016)], calculates automatically the Néron-Tate height of a given rational point) and (2) the computation of the elliptic logarithm of a point; an ingenious trick of D. Zagier [Math. Comput. 48, 425-436 (1987; Zbl 0611.10008)] makes it possible, with very high precision (hundreds of decimal digits; this is necessary for the reduction of the large upper bound mentioned above).

Finally, a synthesis of all these ingredients has to be made and this is done successfully by the authors of the present paper. It is somewhat amazing that, a little earlier, R. J. Stroeker and the reviewer had developed a completely similar method [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)]. Obviously, the authors of the paper under review worked independently from Stroeker and the reviewer; moreover, they solved, as applications of the method, three very impressive numerical examples, related to elliptic curves of rank 5 and 6.

Reviewer: N.Tzanakis (Iraklion)

##### MSC:

11D25 | Cubic and quartic Diophantine equations |

11G05 | Elliptic curves over global fields |

11J86 | Linear forms in logarithms; Baker’s method |