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 $A$.

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- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2021-07-31 11:03:06

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- 2022-03-26 16:23:23 by Bjorn Poonen
- 2021-07-31 11:03:06 by Kiran S. Kedlaya (Reviewed)
- 2021-03-07 13:23:47 by Kiran S. Kedlaya (Reviewed)
- 2021-03-07 13:22:25 by Kiran S. Kedlaya
- 2016-10-29 19:22:32 by Christelle Vincent (Reviewed)

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