Semistable models for modular curves of arbitrary level

By Jared Weinstein. 2010

57 citations recorded by the source corpus.

We produce an integral model for the modular curve $$X(Np^m)$$X(Npm) over the ring of integers of a sufficiently ramified extension of $$\mathbf {Z}_p$$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 $$X(Np^m)$$X(Npm), 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 $$p^m$$pm 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d8cfa3fd235197940d58c897303e71891e48b8c0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Peter Scholze
2. Jared Weinstein
3. Ruochuan Liu
4. Kiran S. Kedlaya
5. B. Bhatt
6. Laurent Fargues
7. Lue Pan
8. W. Gan

In Mathematical Physics

1. Peter Scholze
2. Jared Weinstein
3. Ruochuan Liu
4. Kiran S. Kedlaya
5. B. Bhatt
6. Laurent Fargues
7. W. Gan
8. Wiesława Nizioł

Citations in the corpus

• Moduli of p-divisible groups (2012, result): “L. This suggests defining an integral model Mˆh ∞ as the fibre product of H˜h η and Mˆ1 ∞ over ˜µp∞. On the other hand we have the integral models Mˆh n of finite level, due to Drinfeld. It is proved in [Wei12] that Mˆ is the inverse limit lim ←− Mˆ n in the category of formal schemes over Spf Zp. 6.5.”
• p-ADIC GEOMETRY (2017, background): “1-torsor. To restore full GL 2(Q p)-equivarience, we consider the perfectoid space M LT;1= Ue x GL 2(Q p) 1 GL 2(Q p) ˘= G Z Ue x; which is known as the Lubin-Tate tower at innite level, cf. [SW13], [Wei16]. On the other hand, Ue x! 2 turns out to be an O D -torsor, where D=Q pis the quaternion alge”