Semistable models for modular curves of arbitrary level

Abstract

We produce an integral model for the modular curve X(N p m ) over the ring of integers of a sufficiently ramified extension of 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(N p 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 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.

Maximal varieties and the local Langlands correspondence for 𝐺𝐿(𝑛)

“…2.13) which turns out to be representable. In §2 we review a result from [Wei12] which furnishes a linear-algebra description of M H 0 ,∞ . We begin with the formal O K -module H 0 /F q of dimension 1 and height n. Let H be any lift of H 0 to OK, whereK is the completion of the maximal unramified extension of K. We consider the "universal cover"H = lim ← − H (inverse limit with respect to multiplication by a uniformizer of K) as a K-vector space object in the category of formal schemes over OK. ThenH does not depend on the choice of lift H, and as a formal scheme we haveH ≃ −→ Spf OK T 1/q ∞ .…”

“…At present, this fact can only be proved using global methods. A program initiated by the second author in [Wei10] and [Wei12] aims to obtain a purely local proof by first constructing a sufficiently nice model of M H 0 ,m , and then computing the cohomology of M H 0 ,∞ using the nearby cycles complex on the special fiber of this model. This idea has roots in the work of T. Yoshida [Yos10], who found an open affinoid in M H 0 ,1 whose reduction turned out to be a certain Deligne-Lusztig variety for the group GL n over k. Using this affinoid, Yoshida showed by purely local methods that the local Langlands correspondence for depth zero supercuspidal representations of GL n (K) is realized in the cohomology of M H 0 ,1 .…”

“…Our work is concerned with a large class of supercuspidals of positive depth. Instead of working with any particular layer M H 0 ,m of the tower, we work directly with M H 0 ,∞ , which carries the structure of a perfectoid space (see [Wei12], or [SW12] for the case of general Rapoport-Zink spaces). The following theorem does not require any global methods.…”

“…The Lubin-Tate space at infinite level. We review some recent results from[Wei12] and[SW12] concerning moduli of p-divisible groups with infinite level structures. Suppose as usual that H 0 is a one-dimensional formal O K -module over F q of height n. If (R, R + ) is an object of CAffK ,OK , and (H, ρ) is a deformation of H 0 to (R, R + ), then (as above) (H, ρ) corresponds to a family of pairs (H i , ρ i ) defined over a covering of Spa(R, R + ).…”

Geometric approach to the local Jacquet-Langlands correspondence

“…It is known that the ring A ∞,O C with the G 1 -action has very simple description; see [Wei,§2] and [IT15a,§1].…”