A p-adic Nevanlinna–Diophantine correspondence

Abstract

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.

On Non-Archimedean Curves Omitting Few Components and their Arithmetic Analogues

“…Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in [2], 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 field.Z or the ring of integers of an imaginary quadratic field. Thus, a second objective is to prove an appropriate arithmetic analogue of all of our results on non-Archimedean analytic curves, further illustrating and justifying the correspondence proposed in [2]. Before discussing our main results, we briefly…”

“…Z or the ring of integers of an imaginary quadratic field. Thus, a second objective is to prove an appropriate arithmetic analogue of all of our results on non-Archimedean analytic curves, further illustrating and justifying the correspondence proposed in [2]. Before discussing our main results, we briefly…”

“…Our primary object of study is the degeneracy of non-Archimedean analytic maps from k to an ndimensional projective variety X omitting an effective divisor with at least n irreducible components. As argued in [2], results on non-Archimedean analytic curves in this context should have arithmetic counterparts in Diophantine geometry; they should correspond to results on integral points over recall some aspects of the correspondence in [2] as well as several examples of parallel non-Archimedean and arithmetic results.…”

“…A generalization of Theorem 1.2A to higher dimensions was obtained by Lin and Wang [10], and a refinement was given by An, Levin and Wang in [2] as follows. Z ⊂ X such that the image of any non-constant analytic map f :…”

“…In view of the aforementioned analogies, this suggests that to obtain a Diophantine analogue of non-Archimedean analytic curves, we should consider only rings of integers with a finite unit group, i.e., Z or the ring of integers of an imaginary quadratic field. This observation was the starting point for work in [2], where more generally it was argued that, at least for certain classes of varieties, a non-constant non-Archimedean analytic map into a variety X should correspond to an infinite set of O k -integral points on X, where O k = Z or the ring of integers of an imaginary quadratic field. For completeness, we also mention that this correspondence can frequently be made quantitative (see [2]), resulting in parallel statements in non-Archimedean Nevanlinna theory and Diophantine approximation.…”