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The smooth locus in infinite-level Rapoport–Zink spaces
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 9 / September 2020
- Published online by Cambridge University Press:
- 03 November 2020, pp. 1846-1872
- Print publication:
- September 2020
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- Article
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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 connected component of its geometric fiber. 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.
