Existence of geometrically simple ordinary abelian varieties over a fixed finite field

Hui Zhu

Given a prime number $p$ and a positive integer $d$, we prove that there exist geometrically simple $d$-dimensional ordinary abelian varieties defined over the finite field of $p$ elements. We present an algorithm to determine whether an abelian variety over a finite field $k$, given with the characteristic polynomial of its Frobenius endomorphism relative to $k$, is geometrically simple.