A note on supersingular abelian varieties

TL;DR

This paper demonstrates that supersingular abelian varieties are isogenous to superspecial ones without field extension increase, constructs examples not defined over finite fields, and clarifies endomorphism algebra properties.

Contribution

It shows the isogeny relation between supersingular and superspecial abelian varieties without extending fields and constructs counterexamples related to endomorphism algebras.

Findings

Supersingular abelian varieties are isogenous to superspecial ones without increasing field extensions.

Constructs superspecial abelian varieties not directly defined over finite fields.

Provides corrections and clarifications on endomorphism algebras of supersingular elliptic curves.

Abstract

In this note we show that any supersingular abelian variety is isogenous to a superspecial abelian variety without increasing field extensions. The proof uses minimal isogenies and the Galois descent. We then construct a superspecial abelian variety which not directly defined over a finite field. This answers negatively to a question of the author [J. Pure Appl. Alg., 2013] concerning of endomorphism algebras occurring in Shimura curves. Endomorphism algebras of supersingular elliptic curves over an arbitrary field are also investigated. We correct a main result of the author's paper [Math. Res. Let., 2010].