Year: 2010
Type: preprint
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 {\em 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 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 $p^m$ 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.
Source: arXiv (Cornell University)
Author: Weinstein, Jared
Institution: Boston University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2952812803