Semistable models for modular curves of arbitrary level

We produce an integral model for the modular curve X(Np) over the ring of integers of a sufficiently ramified extension of Zp whose special fiber is a semistable curve in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of X(Np), which is a union of copies of a Lubin-Tate curve. In doing so we tie together nonabelian Lubin-Tate theory to the representation-theoretic point of view afforded by Bushnell-Kutzko types. For our analysis it was essential to work with the Lubin-Tate curve not at level p but rather at infinite level. We show that the infinitelevel Lubin-Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin-Tate spaces of finite level.