Computing S-Integral Points on Elliptic Curves

1. Introduction. By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demyanenko (see [L3], p. 140) states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Silverman [Si1] for elliptic curves with integral modular invariant j over K and by Hindry and Silverman [HSi] for algebraic function fields K. On the other hand, beginning with Baker [B], effective bounds for the size of the coefficients of integral points on E have been found by various authors (see [L4]). The most recent bound was established by W. Schmidt [Sch, Th. 2]. However, the bounds are rather large and therefore can be used only for solving some particular equations (see [TdW1], [St]) or for treating a special model of elliptic curves, namely Thue curves of degree 3 (see [GSch]). The Siegel–Baker method (see [L3]) for the calculation of integer points on elliptic curves over K = Q requires some detailed information about certain quartic number fields. Computing these fields often represents a hard problem and, moreover, this approach does not seem to be appropriate. That is why in general all the results mentioned above cannot be used for the actual calculation of all integral points on an elliptic curve E over Q. However, there is another method suggested by Lang [L1], [L3] and further developed by Zagier [Za]. We shall work out the Lang–Zagier method and turn it into an algorithm for determining all integral points on elliptic curves E over Q using elliptic logarithms. The algorithm requires the knowledge of a basis of the Mordell–Weil group E(Q) and of an explicit lower bound for linear forms in elliptic logarithms. Compared to the Siegel–Baker