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

Published:2015-09-07
Printed: Feb 2017
• Aaron Levin,
Department of Mathematics , Michigan State University , East Lansing, MI 48824
• Julie Tzu-Yueh Wang,
Institute of Mathematics , Academia Sinica , No. 1, Sec. 4, Roosevelt Road , Taipei 10617 , TAIWAN
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## Abstract

Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let $D_1,\dots, D_n$ be effective nef divisors intersecting transversally in an $n$-dimensional nonsingular projective variety $X$. We study the degeneracy of non-Archimedean analytic maps from $\mathbf{k}$ 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 $\mathbf{k}$ into $X\setminus D_1 \cup D_2$. Using the dictionary between non-Archimedean 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 $\mathbb{Z}$ or the ring of integers of an imaginary quadratic field.
 Keywords: non-Archimedean Picard theorem, non-Archimedean analytic curves, integral points
 MSC Classifications: 11J97 - Analogues of methods in Nevanlinna theory (work of Vojta et al.) 32P05 - Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 32H25 - Picard-type theorems and generalizations {For function-theoretic properties, see 32A22}

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