Some cases of Vojta’s conjecture on integral points over function fields

In the present paper we solve, in particular, the function field version of a special case of Vojta’s conjecture for integral points, namely for the variety obtained by removing a conic and two lines from the projective plane. This will follow from a bound for the degree of a curve on such a surface in terms of its Euler characteristic.

This case is special, but significant, because it lies “at the boundary”, in the sense that it represents the simplest case of the conjecture which is still open. Also, it was already studied in the context of Nevanlinna Theory by M. Green in the seventies.

Our general results immediately imply the degeneracy of solutions of Fermat type equations $z^d=P(x^m,y^n)$ for all $d\ge 2$ and large enough $m,n$, also in the case of non-constant coefficients. Such equations fall apparently out of all known treatments.

The methods used here refer to derivations, as is usual in function fields, but contain fundamental new points. One of the tools concerns an estimation for the $\gcd (1-u,1-v)$ for $S$-units $u,v$; this had been developed also in the arithmetic case, but for function fields we may obtain a much more uniform quantitative version.

In the Appendix we shall finally point out some other implications of the methods to the problem of torsion-points on curves and related known questions.