Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. (English) Zbl 0805.11026

Stroeker, R. J.; Tzanakis, N.

Acta Arith. 67, No. 2, 177-196 (1994).

In order to compute all integer points on a Weierstraß equation for an elliptic curve \(E/\mathbb{Q}\), one may translate the linear relation between rational points on \(E\) into a linear form of elliptic logarithms. An upper bound for this linear form can be obtained by employing the Néron-Tate height function and a lower bound is provided by a recent theorem of S. David. Combining these two bounds allows for the estimation of the integral coefficients in the group relation, once the group structure of \(E(\mathbb{Q})\) is fully known. Reducing the large bound for the coefficients so obtained to a manageable size is achieved by applying a reduction process due to de Weger. In the final section two examples of elliptic curves of rank 2 and 3 are worked out in detail.

Reviewer: R.J.Stroeker (Rotterdam)

MSC:

11D25	Cubic and quartic Diophantine equations
11G05	Elliptic curves over global fields

Keywords:

Mordell equation; cubic diophantine equation; integer points; Weierstraß equation; elliptic curve; linear form of elliptic logarithms