Isogenies of abelian varieties over finite fields

TL;DR

This paper investigates conditions for when abelian varieties over finite fields are isogenous over the base field, and when isogenies are defined over the base field, enhancing understanding of their algebraic structure.

Contribution

It provides new criteria for determining when isogenies and isogeny relations are defined over the base finite field, improving the classification of abelian varieties over finite fields.

Findings

Conditions for abelian varieties to be F-isogenous when isogenous over an extension field.

Criteria for when a given isogeny is defined over the base finite field.

Enhanced understanding of the Galois action on abelian varieties and their isogenies.

Abstract

In this paper we give conditions under which two abelian varieties that are defined over a finite field $F$, and are isogenous over some larger field, are $F$-isogenous. Further, we give conditions under which a given isogeny is defined over $F$.