Computing all \(S\)-integral points on elliptic curves. (English) Zbl 0949.11033

Pethő, Attila; Zimmer, Horst G.; Gebel, Josef; Herrmann, Emanuel

Math. Proc. Camb. Philos. Soc. 127, No. 3, 383-402 (1999).

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) and let \(E(\mathbb{Q})\) be the group of rational points on \(E\). Further, let \(S\) be a given finite set of places of \(\mathbb{Q}\) including the infinite place. The authors describe a method of computing all \(S\)-integral points in \(E(\mathbb{Q})\), i.e. those rational points on \(E\) that are integral outside \(S\). In 1933 K. Mahler [J. Reine Angew. Math. 170, 168-178 (1933; Zbl 0008.20002)], generalizing C. L. Siegel’s result of 1929 [Abh. Preuss. Akad. Wiss. 1929, 1-70 (1929; JFM 56.0180.05)], proved that there can only be finitely many such points.
Until quite recently all existing approaches were either ineffective or, if effective, not of much practical use. Based on ideas of S. Lang and D. Zagier, R. J. Stroeker and N. Tzanakis [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)] and independently the first three named authors of the present paper [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] developed a method in which both the group structure of \(E(\mathbb{Q})\) and estimates of linear forms in elliptic logarithms play a decisive role. This method which initially aims at finding all integral points was made suitable for \(S\)-integral points by N. Smart [Math. Proc. Camb. Philos. Soc. 116, 391-399 (1994; Zbl 0817.11031)]. However, his work depends on an unproven lower bound for linear forms in \(q\)-adic elliptic logarithms. The present authors overcome this difficulty by combining the diophantine approximation approach of Siegel-Baker-Coates with the Lang-Zagier elliptic logarithm method, which is made possible by a result of L. Hajdu and T. Herendi [J. Symb. Comput. 25, 361-366 (1998; Zbl 0923.11048)].
Finally the authors illustrate their method by means of the rank 4 curve given by the equation \(y^2= x^3- 172x+ 505\) [A. Wiman, 12te Skand. Matematikerkongressen, Lund, 317-323 (1953; Zbl 0055.27106)] with \(S= \{3, 5, 7, \infty\}\); they find 144 \(S\)-integral points.

Reviewer: Roelof J.Stroeker (Rotterdam)

MSC:

11G05	Elliptic curves over global fields
14H52	Elliptic curves
11Y50	Computer solution of Diophantine equations

Keywords:

elliptic curve; group of rational points; \(S\)-integral points