# The smooth locus in infinite-level Rapoport-Zink spaces

## TL;DR

This paper proves that in infinite-level Rapoport-Zink spaces, the dense open subset of points with no extra endomorphisms is cohomologically smooth, revealing a deep geometric structure related to supersingular elliptic curves.

## Contribution

It establishes the existence of a dense cohomologically smooth locus in infinite-level Rapoport-Zink spaces, characterized by the absence of extra endomorphisms, extending understanding of their geometric properties.

## Key findings

- The cohomologically smooth locus is dense and open in the infinite-level Rapoport-Zink space.
- This locus corresponds to $p$-divisible groups without extra endomorphisms.
- In the modular curve case, it matches supersingular elliptic curves with no extra endomorphisms.

## Abstract

Rapoport-Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let $\mathscr{M}_{\infty}$ be an infinite-level Rapoport-Zink space of EL type, and let $\mathscr{M}_{\infty}^\circ$ be one geometrically connected component of it. We show that $\mathscr{M}_{\infty}^{\circ}$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^\infty)^{\circ}$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.