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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.

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• Last edited by Kiran S. Kedlaya on 2021-03-07 13:23:47
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