Isogeny classes and endomorphisms algebras of abelian varieties over finite fields

TL;DR

This paper constructs examples of simple ordinary abelian varieties over algebraic closures of finite fields that, despite not being isogenous, share the same endomorphism algebra, highlighting new phenomena in their classification.

Contribution

It demonstrates the existence of non-isogenous simple ordinary abelian varieties with isomorphic endomorphism algebras over algebraic closures of finite fields.

Findings

Existence of such abelian varieties with shared endomorphism algebras

Construction method for non-isogenous varieties with isomorphic endomorphism algebras

Implications for classification of abelian varieties over finite fields

Abstract

We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.