Moduli of p-divisible groups

TL;DR

This paper establishes that Rapoport-Zink spaces at infinite level are perfectoid spaces, describes them via p-adic Hodge theory, and proves duality and classification results for p-divisible groups, advancing the understanding of their geometric and arithmetic structures.

Contribution

It proves that Rapoport-Zink spaces at infinite level are perfectoid and provides a p-adic Hodge theoretic description, including duality isomorphisms and classification of p-divisible groups.

Findings

Rapoport-Zink spaces at infinite level are perfectoid spaces.

Duality isomorphisms between basic Rapoport-Zink spaces at infinite level.

Classification of p-divisible groups over algebraically closed fields.

Abstract

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p-adic Hodge theory. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of p-divisible groups over the ring of integers of a complete algebraically closed field in the spirit of Riemann's classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonn\'e module functor for p-divisible groups over semiperfect rings (meaning rings on which the Frobenius map is surjective).