Lang-Vojta's conjecture over functions fields for surfaces dominating tori

The celebrated Lang-Vojta Conjecture predicts degeneracy of $S$-integral points on varieties of log general type defined over number fields. It admits a geometric analogue over function fields, where stronger results have been obtained applying a method developed by Corvaja and Zannier. In this talk, we present a recent result for non-isotrivial surfaces over function fields dominating a two-dimensional torus. This extends Corvaja and Zannier’s result in the isotrivial case and it is based on a refinement of gcd estimates for polynomials evaluated at $S$-units. This is a joint work with A. Turchet.