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

Abstract

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.

Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points

“…On the other hand, if each geometric point of ∩ n j=1 D j is k-rational and that the divisors D j are nef and big, the condition vol(D j ) > d becomes D n j > 1. In this way we recover large part of the main arithmetic results of [16].…”

“…Before Siegel's theorem on integral points of curves, Runge [18] proved some special cases over Q under the assumption that the divisor D on the curve consisted of at least 2 components defined over Q. Runge's method has been extended to the higher dimensional setting by Levin [13,15] (see also Le Fourn's extension [12]), and over Q the main technical assumption is that the divisor D on the variety consists of several components D j defined over Q having empty total intersection ∩ j D j . This last assumption has been relaxed by Levin and Wang [16], by requiring that ∩ j D j is finite and its geometric points are in fact rational over Q, among other technical assumptions on the geometry of the divisors D j .…”

Non-archimedean hyperbolicity and applications

“…The notion of K-analytic hyperbolicity introduced above (Definition 2.3) has not appeared before in the literature. However, the following weaker notion was studied in the work of Cherry [17]; see also [2,16,18,50,51]. Definition 2.7.…”

“…Although the notion of K-analytic Brody hyperbolicity introduced above has not appeared before in the literature, we were first led to investigate this notion by the work of Cherry; see [2,16,17,18,50,51].…”

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

“…To explain how to do this, we introduce a new notion of hyperbolicity for rigid analytic varieties (and also adic spaces) over a non-archimedean field K of characteristic zero; see [53, §2]. This notion is inspired by the earlier work of Cherry [19] (see also [6,20,21,64,65]).…”