Lang-Vojta Conjecture over function fields for surfaces dominating $\mathbb{G}_m^2$

TL;DR

This paper proves a significant case of the Lang-Vojta conjecture over function fields for certain surfaces, extending previous results and explicitly calculating all involved constants.

Contribution

It extends the Lang-Vojta conjecture proof to the nonsplit case for surfaces over function fields, building on prior split case results and making all constants explicit.

Findings

Proves the nonsplit case of the Lang-Vojta conjecture for certain surfaces.

Extends previous results from split cases and complements of divisors in projective space.

Provides explicit constants in the proof.

Abstract

We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $\mathbb{G}_m^2$. This extends results of Corvaja and Zannier, who proved the conjecture in the split case, and results of Corvaja and Zannier and the second author that were obtained in the case of the complement of a degree four and three component divisor in $\mathbb{P}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.