Resolution of non-singularities for Mumford curves

Abstract. We will study the following question : given a hyperbolic curve X over the algebraic closure of Q_p, a semistable stable model of this curve and a closed point x of this semistable model, is there a finite étale cover of the curve Y -> X such that the minimal semistable model of Y above the given semistable model of X has a vertical component above x ? A. Tamagawa answered positively for the stable model. It is true for any semistable model when X is a Mumford curve. We will give applications to the tempered fundamental group.