Resolution of non-singularities for Mumford curves

Abstract

Given a Mumford curve $X$ over $\overline{{\mathbb{Q}}_p}$, we show that for every semistable model $\mathcal{X}$ of $X$ and every closed point $x$ of this semistable model, there exists a finite étale cover $Y$ of $X$ such that every semistable model of $Y$ has a vertical component above $X$. We then give applications of this to the tempered fundamental group. In particular, we prove that two punctured Tate curves $\overline{{\mathbb{Q}}_p}$ with isomorphic tempered fundamental groups are isomorphic over ${\mathbb{Q}}_p$.