Effective analysis of integral points on algebraic curves

LetK be an algebraic number field,S?S _t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)| _v >1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if

1. x:C→P ¹ is a Galois covering andg(C)≥1;
2. the integral closure of $bar Q$ [x] in $bar Q$ (C) has at least two units multiplicatively independent mod $bar Q$ *.

This generalizes famous results of A. Baker and other authors on the effective solution of Diophantine equations.