Semistable models for modular curves of arbitrary level

Abstract

We produce an integral model for the modular curve $$X(Np^m)$$ over the ring of integers of a sufficiently ramified extension of $$\mathbf {Z}_p$$ 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 $$X(Np^m)$$ , which is a union of copies of a Lubin–Tate curve. In doing so we tie together non-abelian 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 $$p^m$$ but rather at infinite level. We show that the infinite-level 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.