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The smooth locus in infinite-level Rapoport–Zink spaces

Part of: Algebraic groups Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  03 November 2020

Alexander B. Ivanov  and
Jared Weinstein
Alexander B. Ivanov
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115Bonn, [email protected]
Jared Weinstein
Affiliation:
Department of Mathematics and Statistics, Boston University, Boston, MA02215, [email protected]
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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 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.

Keywords

p-divisible groupsRapoport–Zink spacesperfectoid spacesvector bundles on the Fargues–Fontaine curvecohomological smoothness

MSC classification

Primary: 14G22: Rigid analytic geometry 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 9 , September 2020 , pp. 1846 - 1872
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Copyright
Copyright © The Author(s) 2020

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