# Resolution of Nonsingularities for Mumford Curves

### Emmanuel Lepage

Université Pierre et Marie Curie VI, Paris, France

## Abstract

Given a Mumford curve $X$ over $\overline{\mathbf Q}_p$, 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 $Y$ over $\mathcal X$ 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 $\overline{\mathbf Q}_p$ with isomorphic tempered fundamental groups are isomorphic over $\mathbf Q_p$.

## Cite this article

Emmanuel Lepage, Resolution of Nonsingularities for Mumford Curves. Publ. Res. Inst. Math. Sci. 49 (2013), no. 4, pp. 861–891

DOI 10.4171/PRIMS/122