p-rank stratification in unitary Shimura variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:02:33Z http://mathoverflow.net/feeds/question/119963 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119963/p-rank-stratification-in-unitary-shimura-variety p-rank stratification in unitary Shimura variety Ricky 2013-01-26T20:34:20Z 2013-01-26T23:44:43Z <p>Let $K$ be a quadratic extension of $\mathbb Q$ and let $p \neq 2$ be a prime that is <strong>inert</strong> 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).</p> <blockquote> <p>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.</p> </blockquote> http://mathoverflow.net/questions/119963/p-rank-stratification-in-unitary-shimura-variety/119974#119974 Answer by Joël for p-rank stratification in unitary Shimura variety Joël 2013-01-26T23:44:43Z 2013-01-26T23:44:43Z <p>Yes, both these cases appear. This follows from the Remark, page 92, of my <a href="http://people.brandeis.edu/~jbellaic/preprint/these.pdf" rel="nofollow">PhD thesis</a>. This is in the unpublished chapter III, devoted to the study of the Shimura variety for U(2,1) (a.k.a Picard modular surface) over $W(k)$, $k$ a field of char $p$, with level structures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.</p>