Let $K$ be a quadratic extension of $\mathbb Q$ and let $p \neq 2$ be a prime that is **inert** in $K$. Let $X$ be the Shimura variety associated to the unitary group $\operatorname{U}(2,1)$ over $K$ (after a choice of a suitable integral PEL datum). We have an integral model $\mathcal X$ of $X$ defined over $\mathcal O\_E \otimes \mathbb Z\_p$, where $E$ is the reflex field. Let $A$ be a abelian variety corresponding to a $k$-point of $\mathcal X$, where $k$ is a field $k$ of characteristic $p$. Its $p$-torsion $A[p]$ has rank $6$ and it is equipped with an action of $\mathbb F\_{p^2}$ (this is true regardless the assumption on $k$ of course).



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> Question: What can be said about the $p$-rank of $A$? It must be $0$ or $2$ (since $A[p^\infty]$ is principally polarized), but I do not know whether both these cases really appear (I believe so) or not.
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p-rank stratification in unitary Shimura variety
Yes, both these cases appear. This follows from the Remark, page 92, of my [PhD thesis](http://people.brandeis.edu/~jbellaic/preprint/these.pdf).
This is in the unpublished chapter III, devoted to the study of the Shimura variety for U(2,1)
(a.k.a Picard modular surface) over $W(k)$, $k$ a field of char $p$, with level structures either spherical or Iwahori at $p$. In this chapter 
there are more precise results about the different type of abelian varieties that 
can appear, and in which dimensions.


