Dedicated to Professor Pit-Mann Wong on the occasion of his sixtieth birthday 1. Introduction. Beginning with the work of Osgood, Vojta, and Lang, it has been observed that many statements in Nevanlinna theory closely resemble statements in Diophantine approximation. Qualitatively, in the simplest case, holomorphic curves in a variety X should correspond to infinite sets of integral points on X. A detailed dictionary between Nevanlinna theory and Diophantine approximation has been developed by Vojta [13]. This correspondence has been influential, inspiring conjectures and results in both subjects. Relatively recently, a p-adic analogue of Nevanlinna theory and value distribution theory has been developed, and many analogous results proven. Similar to the correspondence between classical Nevanlinna theory and Diophantine approximation, we discuss here a correspondence between p-adic Nevanlinna theory and certain Diophantine statements over the integers Z or the rational numbers Q. Roughly speaking, at least for certain classes of varieties, a nonconstant p-adic analytic map into a variety X should correspond to an infinite set of Z-integral points on X. We discuss this in the next section, making some observations towards a precise formulation. While we lack such a precise formulation in general, this correspondence already appears to be useful in suggesting both results and proofs of statements concerning p-adic analytic maps and integral points on varieties. We illustrate this in the last section, giving several examples of parallel p-adic and arithmetic results. Aside from their illustrative purpose, some of these results may be of independent interest.

Abstract. Let k be an algebraically closed eld complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D , . . . , D n be e ective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into X ∖ ∪ n i= D i under various geometric conditions. When X is a rational ruled surface and D and D are ample, we obtain a necessary and su cient condition such that there is no nonArchimedean analytic map from k into X∖D ∪D . Using the dictionary between nonArchimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over Z or the ring of integers of an imaginary quadratic eld.

Abstract. We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is nonsingular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.