Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb{Z}[\pi ,q\mathrm{/}\pi ]$-ideal and a fractional $W{\otimes }_{{\mathbb{Z}}_p}{\mathbb{Z}}_p[\pi ,q\mathrm{/}\pi ]$-ideal, with $\pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified extension of ${\mathbb{Q}}_p$ of degree $a$. The latter ideal should be compatible at $p$ with the former and stable under the action of a semilinear Frobenius (and Verschiebung) operator; it will be the Dieudonné module of the corresponding abelian variety. Using this categorical description we create effective algorithms to compute isomorphism classes of these objects and we produce many new examples exhibiting exotic patterns.