Moving Targets on Curves

Recent papers of Vojta ([V 3], [V 4]) and Ru and Vojta ([R-V]) show that in Roth’s and Wirsing’s theorems the points being approximated may vary provided that their height remains small relative to the height of the points with which they are being approximated. In [R-V] and [V 4] the technique of proof is a variation of a technique of Steinmetz from Nevanlinna theory which consists of embedding the points in a certain sense into a projective space and then invoking the Schmidt subspace theorem. It should be noted that, to our knowledge, the first “moving targets” result was proven in the context of Nevanlinna theory (see [Os]). For these proofs, it does not appear to be possible to bound the ratio of the heights of the approximations to the heights of the points being approximated (which we refer to as “targets”). That is to say, these proofs require that h(Q) = o(h(P )), where Q denotes a “target” and P a point approximating Q. In [V 3], however, the technique of proof is a modification of Roth’s original proof ([Ro]) with the added ingredients of Dyson’s lemma (in place of Roth’s lemma) and metrized line sheaves (in place of polynomials). This proof yields an explicit constant; indeed the constant yielded is quite small compared with what might be proven without the precision of arithmetic intersections (compare with [B-P], which implicitly contains a moving targets result). In this note, we generalize the Wirsing’s theorem with moving targets of Ru and Vojta to higher genus curves. Of course, for curves of genus greater than 1 there are only finitely many rational points, so the result follows trivially from the fixed target theorem. So for practical purposes we have merely extended the result to elliptic curves. We prove the result without explicit constants, for explicit constants seem to be difficult to control, depending in particular on such things as negativity of Green functions, arithmetic degrees of line bundles, and bounds on arithmetic discriminants (none of which appear in the genus 0 case, more or less because of the fact that degree determines a divisor up to linear rather than algebraic equivalence), as well as dimension counting arguments on line bundles more complicated than