On non-Archimedean curves omitting few components and their arithmetic analogues

TL;DR

This paper investigates the degeneracy of non-Archimedean analytic maps avoiding certain divisors on algebraic varieties, establishing geometric conditions for their existence and exploring arithmetic analogues related to integral points.

Contribution

It provides new criteria for the non-existence of non-Archimedean maps on specific varieties and connects these results to Diophantine approximation and integral points.

Findings

Characterization of non-Archimedean map degeneracy on rational ruled surfaces.

Necessary and sufficient conditions for the absence of non-Archimedean maps.

Results on integral points over integers and imaginary quadratic fields.

Abstract

Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When X is a rational ruled surface and D_1 and D_2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into $X\setminus D_1 \cup D_2$. Using a dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over the integers or the ring of integers of an imaginary quadratic field.