The Mordell Conjecture, proved by Faltings, states that a curve of genus at least 2 over a number field k has only finitely many k-rational points.
Unfortunately, the proof of this conjecture is neither explicit nor effective, in the sense that it does not provide a bound on the 'size' of the points, nor a method to find such a bound. Effective methods in this context are rare and often difficult to apply to obtain explicit results.
In this talk I will present a joint work with F. Veneziano and E. Viada in which we prove, in particular, the explicit Mordell Conjecture for curves in $E^N$, where $E$ is an elliptic curve without CM having Mordell-Weil group of rank 1. For certain families of curves in $E\times E$, of quite general shape and increasing genus, we give a method of easy application to bound the height of their rational points. In many examples (more than 10^4) the bounds we obtain are so small that a computer search for the rational points is faisable. This last is made with PARI-GP using an algorithm by K. Belabas.