A non-Archimedean Montel's theorem (Q2894212)

The goal of the paper is to prove a version of Montel's theorem for analytic functions over non-Archimedean complete valued fields.NEWLINENEWLINE 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 \(f: X\to\mathbb{P}^{1,\text{an}}_k\).NEWLINENEWLINE After that the authors give some sufficient conditions guaranteeing that the limit map is continuous. As one of the final outcomes they obtain the followingNEWLINENEWLINE 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.