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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).

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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up vote 4 down vote accepted

Yes, both these cases appear. This follows from the Remark, page 92, of my PhD thesis. This is in the unpublished chapter III, devoted to the study of the Shimura variety for U(2,1) (a.k.a Picard modular surface) over $W(k)$, $k$ a field of char $p$, with level structures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.

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Thank you very much Joël! –  Ricky Jan 27 '13 at 9:22
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