p-rank (reviewed)

Let $A/\mathbb{F}_q$ be an abelian variety where $q=p^r$. The $p$-rank of an abelian variety is the dimension of the geometric $p$-torsion as a $\mathbb{F}_p$-vector space: $$p\operatorname{-rank}(A) = \dim_{\mathbb{F}_p}( A(\overline{\mathbb{F}}_p)[p] ).$$ The $p$-rank is at most the dimension of $A$, with equality if and only if $A$ is ordinary; the difference between the two is the $p$-rank deficit of $A$.