Computing \(S\)-integral points on elliptic curves. (English) Zbl 0899.11012

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

Cohen, Henri (ed.), Algorithmic number theory. Second international symposium, ANTS-II, Talence, France, May 18-23, 1996. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1122, 157-171 (1996).

In a previous paper [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] J. Gebel, A. Pethő and H. G. Zimmer developed a method, due to Lang and Zagier, for computing all the integral points on an elliptic curve \(E\) over the field \(\mathbb{Q}\) of rational numbers. This method requires knowledge of the Mordell-Weil group \(E(\mathbb{Q})\) and relies on a method of S. David [Mém. Soc. Math. Fr., Nouv. Sér. 62 (1995; Zbl 0859.11048)] for computing lower bounds for linear forms in elliptic logarithms. In the present paper the authors develop a \(p\)-adic method for finding \(S\)-integral points on elliptic curves and list some tables concerning \(S\)-integral points on some curves of the form \(y^2= x^3+k\). Their method relies on a new explicit bound for linear forms in two \(p\)-adic elliptic logarithms which was recently established by G. Rémond and F. Urfels [J. Number Theory 57, 133-169 (1996; Zbl 0853.11055)].
For the entire collection see [Zbl 0852.00023].

Reviewer: R.P.Steiner (Bowling Green)

MSC:

11D25	Cubic and quartic Diophantine equations
11G05	Elliptic curves over global fields
14H52	Elliptic curves
11-04	Software, source code, etc. for problems pertaining to number theory
11J86	Linear forms in logarithms; Baker’s method

Keywords:

\(p\)-adic method; \(S\)-integral points on elliptic curves; tables