Deligne module (awaiting review)

A Deligne module is a finitely generated free $\mathbb{Z}$-module $M$ together with a $\mathbb{Z}$-linear endomorphism $F$ inducing an action of the Frobenius order $\mathbb{Z}[F,V]$ of some isogeny class determined by a Weil polynomial $h(x)$ such that the action of $F$ on $M$ has characteristic polynomial $h(x)$. If the characteristic polynomial $h(x)$ is squarefree, than $M$ can be identified with a fractional $\mathbb{Z}[F,V]$-ideal in in the étale $\mathbb{Q}$-algebra $\mathbb{Q}[F]$. The whole category of ordinary abelian variety over any finite field $\mathbb{F}_q$ can be described using Deligne modules, by a result of Deligne [MR:254059]. The same holds for the category of abelian varieties over a prime finite field $\mathbb{F}_p$ whose characteristic polynomials don't have real roots, thanks to a result by Centeleghe-Stix [MR:3317765, arXiv:1501.02446v1].