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Introduction to Dieudonné modules and supersingular abelian varieties revisited

Chia-Fu Yu

Abstract

In this Note we present an expository account for \dieu modules and revisit supersingular abelian varieties. We give a simple proof of the uniqueness of products of two or more supersingular elliptic curves (a theorem due to Deligne, Ogus and Shioda), and of Oort's theorem for superspecial abelian varieties.

Introduction to Dieudonné modules and supersingular abelian varieties revisited

Abstract

In this Note we present an expository account for \dieu modules and revisit supersingular abelian varieties. We give a simple proof of the uniqueness of products of two or more supersingular elliptic curves (a theorem due to Deligne, Ogus and Shioda), and of Oort's theorem for superspecial abelian varieties.
Paper Structure (10 sections, 24 theorems, 80 equations)

This paper contains 10 sections, 24 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Dieudonné modules
  3. Newton polygons and $a$-numbers
  4. Cartier-Dieudonné theory
  5. The construction of deformations
  6. Proof of the Manin-Dieudonné theorem
  7. $F$-spaces
  8. Endomorphism algebras of $F$-spaces
  9. The Manin-Dieudonné Theorem
  10. Supersingular abelian varieties

Key Result

Theorem 3

There is an anti-equivalence of categories between the category of $p$-divisible groups over $k$ and the category of $W$-free Dieudonné modules over $k$.

Theorems & Definitions (48)

  • Definition 1
  • Example 2
  • Theorem 3: cf. demazure
  • Lemma 4
  • proof
  • Theorem 5: Manin-Dieudonné manin:thesis
  • Remark 6
  • Definition 7
  • Definition 8
  • Theorem 9: zink:cartier, chai:cartier
  • ...and 38 more