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

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.


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