Favre, Charles; Kiwi, Jan; Trucco, Eugenio A non-Archimedean Montel’s theorem. (English) Zbl 1267.32016 Compos. Math. 148, No. 3, 966-990 (2012). The goal of the paper is to prove a version of Montel’s theorem for analytic functions over non-Archimedean complete valued fields. Theorem A. Suppose \(k\) is a non-Archimedean complete non-trivially valued field and let \(X\) be a connected open subset of the projective line \(\mathbb{P}^{1,\text{an}}_k\). Let \(f_n: X\to \mathbb{P}^{1,\text{an}}_k\setminus\{0, \infty\}\) be a sequence of analytic maps. Then there exists a subsequence \(\{f_{n_j}\}\) which converges pointwise to a map \(f: X\to\mathbb{P}^{1,\text{an}}_k\). After that the authors give some sufficient conditions guaranteeing that the limit map is continuous. As one of the final outcomes they obtain the following Corollary D. Any family of meromorphic functions on an open subset \(X\) of \(\mathbb{P}^{1,\text{an}}_k\) such that, for all \(x\in X\), the local unseparable degrees at \(x\) are bounded, and which avoids three points in \(\mathbb{P}^{1,\text{an}}_k\), is both normal and equicontinuous at any rigid point. Reviewer: Sergey M. Ivashkovich (Villeneuve d’Ascq) Cited in 6 Documents MSC: 32P05 Non-Archimedean analysis 37P50 Dynamical systems on Berkovich spaces 32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) Keywords:Montel’s theorem; normal family; non-Archimedean analysis PDFBibTeX XMLCite \textit{C. Favre} et al., Compos. Math. 148, No. 3, 966--990 (2012; Zbl 1267.32016) Full Text: DOI arXiv References: [2] doi:10.1090/S0273-0979-98-00755-1 · Zbl 1037.30021 · doi:10.1090/S0273-0979-98-00755-1 [3] doi:10.5802/aif.1869 · Zbl 1137.37319 · doi:10.5802/aif.1869 [4] doi:10.5802/afst.367 · Zbl 0004.11802 · doi:10.5802/afst.367 [6] doi:10.1017/CBO9780511735233.012 · doi:10.1017/CBO9780511735233.012 [7] doi:10.1007/b100262 · Zbl 1064.14024 · doi:10.1007/b100262 [9] doi:10.1007/BF01111543 · doi:10.1007/BF01111543 [10] doi:10.1112/S0024610700001447 · Zbl 1022.11060 · doi:10.1112/S0024610700001447 [13] doi:10.1112/plms/pdp022 · Zbl 1254.37064 · doi:10.1112/plms/pdp022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.