Introduction to Dieudonn\'e modules and supersingular abelian varieties revisited

TL;DR

This paper provides an accessible overview of Dieudonné modules and supersingular abelian varieties, offering simplified proofs of key theorems related to their structure and uniqueness properties.

Contribution

It presents simplified proofs of important theorems on supersingular elliptic curves and superspecial abelian varieties, enhancing understanding of their classification.

Findings

Proof of uniqueness of products of supersingular elliptic curves

Simplified proof of Oort's theorem for superspecial abelian varieties

Clarification of the structure of Dieudonné modules in this context

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.