I have not been keeping this page current since 1999 and I apologise to interested readers. I should point out that Mathscinet is a very useful tool for obtaining research information in Mathematics, if your institution is a subscriber. For instance, the following link will update you on the latest publications on diophantine approximation in characteristic p. Of course, diophantine approximation in positive characteristic gives a different result.

You can also look up an old version of the paper, which I created as an experiment using LaTeXtoHTML but I am not entirely happy with it. Disclaimer: this is an old version and is NOT the published version. Some of the results are not the latest and there are a couple of mistakes.

1. For S. T. Jeong's result see his paper "Rational points on algebraic curves that change genus", J. Number Theory, 67 (1997) 170-181.
2. [Ki] has appeared in Compositio Math. 105 (1997) 43-54.
3. [W] has appeared as "The truncated second main theorem for function fields", J. Number Theory 58 (1996) 139-157.
4. For other results on diophantine approximation on abelian varieties see T. Scanlon, "The abc theorem for commutative algebraic groups in characteristic p", IMRN, (1997), 17, pp 881-898.
5. For a systematic treatment of local distance functions see J. F. Voloch "Distance functions on varieties over non-archimedean local fields", Rocky Mountain Journal of Math.,7 (1997) 635-641.
6. The following two results are proved in J. F. Voloch "The equation ax+by=1 in characteristic p", J. Number Theory, 73 (1998) 195-200. Let K be a field of positive characteristic p. If K is finitely generated over its prime field, and G is a finitely generated subgroup of (K^*)² of finite rank r then an equation ax+by=1 has either infinitely many solutions or the number of solutions is bounded in terms of p and r. This is related to theorem 3 of section 2. If K is now arbitrary of characteristic p and G is a subgroup of (K^*)² of finite dimension over the rationals then an equation ax+by=1 has finitely many solutions (x,y) in G unless (a,b)ⁿ is in G for some positive integer n. This is related to theorem 3 of section 3, it deals with multiplicative groups only but allows p-power division. See item 9. below.
7. For a nice introductory survey on the work of Hrushovski mentioned in section 3, see A. Pillay, Model theory and diophantine geometry, Bull. Amer. Math. Soc. 34 (1997), pp. 405-422. But don't miss the corrections. Here is a correction to the corrections. The paper [V2], cited as [2] in Pillay's corrections, deals only with curves.
8. M. Kim has proved that a non-isotrivial curve of genus at least two over a function field K of characteristic p has only finitely many points over the perfection of K. See M. Kim, Purely inseparable points on curves of higher genus, Math. Res. Lett. 4 (1997), no. 5, 663--666.