Lang–Vojta conjecture over function fields for surfaces dominating Gm2

We prove the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of Gm2. This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in P2. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.