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Moduli of p-divisible groups

Peter Scholze, Jared Weinstein

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 Dieudonné module functor for p-divisible groups over semiperfect rings (meaning rings on which the Frobenius map is surjective).

Moduli of p-divisible groups

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 Dieudonné module functor for p-divisible groups over semiperfect rings (meaning rings on which the Frobenius map is surjective).

Paper Structure

This paper contains 30 sections, 89 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Adic spaces
  3. Adic spaces
  4. Formal schemes and their generic fibres
  5. Perfectoid spaces
  6. Inverse limits in the category of adic spaces
  7. Preparations on $p$-divisible groups
  8. The universal cover of a $p$-divisible group
  9. The universal vector extension, and the universal cover
  10. The Tate module of a $p$-divisible group
  11. The logarithm
  12. Explicit Dieudonné theory
  13. Dieudonné theory over semiperfect rings
  14. Statement of result
  15. The case $\mathbb{Q}_p/\mathbb{Z}_p\to \mu_{p^\infty}$
  16. ...and 15 more sections

Key Result

Proposition 2.1.6

Theorems & Definitions (131)

  • Remark 1.0.1
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Proposition 2.1.6
  • Proposition 2.2.1
  • Proposition 2.2.2
  • Definition 2.3.1
  • ...and 121 more