\(S\)-integral points on elliptic curves. (English) Zbl 0817.11031

This paper gives an algorithm to determine all “small” \(S\)-integral points on a Weierstraß model of an elliptic curve over \(\mathbb{Q}\). Here “small” has the following meaning. If \(P\) is an \(S\)-integral point (\(S\) is a finite set of \(\mathbb{Q}\)-primes, including infinity), then \(P= \sum_{i=1}^ r q_ i P_ i+T\), where \(\{P_ 1, \dots, P_ r\}\) is a complete set of generators of infinite order of the Mordell-Weil group \(E(\mathbb{Q})\) and \(T\in E(\mathbb{Q} )_{\text{tors}}\). Now only those \(S\)- integral points \(P\) are considered for which \(\max_{1\leq i\leq r} | q_ i|\) is bounded by a (large) constant \(K\). An example is given with \(K\) of size \(10^{40}\). Use is made of the theory of linear forms in real-valued elliptic logarithms. Subsequently, it is shown that this large bound \(K\) may be reduced to one of size \(\sqrt {(r+1)\ln K}\) by applying the \(L^ 3\) algorithm. Usually this reduction process can be repeated several times.


11G05 Elliptic curves over global fields
14H52 Elliptic curves


Full Text: DOI


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