\(S\)-integer points on elliptic curves
Let \(K\) be a number field and let \(E/K\) be an elliptic curve defined over \(K\) given by a quasi-minimal Weierstraß equation \[ E: y^ 2= x^ 3+ Ax+ B, \qquad A,B\in {\mathcal O}_ K, \tag{1} \] where \({\mathcal O}_ K\) is the ring of integers of \(K\). This means, denoting the discriminant of (1) by \(\Delta\), that \(| N_{K/ \mathbb{Q}} (\Delta) |\) is minimal, subject to \(A,B\in {\mathcal O}_ K\). Further, let \(S\) be a finite set of places of \(K\) containing all the archimedean ones. In this paper an explicit upper bound is given for all \(S\)-integral points on \(E\) in terms of the number of places at which the \(j\)-invariant is non-integral, the absolute degree of \(K\), and the number of places in \(S\). This completes \textit{J. Silverman}'s [J. Reine Angew. Math. 378, 60- 100 (1987; Zbl 0608.14021)] in which it was proved that such an effectively computable bound exists without doing the actual computing of the constants involved.
scientific article; zbMATH DE number 769661