Effective estimation of integer points of a family of algebraic curves
If \(X\), \(Y\), \(Z\) are affine algebraic curves defined over a number field, with \(Y,Z\) covers of \(X\) and \(Z/X\) unramified, then integral points on \(Y\) map to integral points on \(X\) and those lift to integral points on \(Z\) perhaps after base field extension. The author has a simple geometric criterion on a plane curve \(Y\), which allows him to find \(X\) and \(Z\) as above, in such a way that \(Z\) is a curve where it is known how to find the integral points (for example an elliptic curve). In this way he obtains an effective bound for the integral points on the curves that satisfy his criterion. The main result improves ealier work of \textit{Yu. F. Belotserkovskij} [Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1988, No. 6, 34-39 (1988; Zbl 0669.10037)].
scientific article; zbMATH DE number 1051571
scientific article