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We let $B$ be a simple algebra over $\mathbb Q$, with the usual notations for PEL type Shimura varieties. In his paper "Ordinariness in good reductions of Shimura varieties of PEL-type" (available here, Wedhorn proved, under the assumption that $p$ is unramified in $B$, that the ordinary locus in (the reduction) a PEL type Shimura variety is non-empty if and only if it is dense if and only if $p$ splits completely in the reflex field.

So my question is the following:

Are there similar results, even partial, without the unramifiedness assumption?

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2 Answers 2

There is a recent work of Marc-Hubert Nicole - for a summary - - just click on the title of his talk.

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Here's the paper I recently talked to him that he hopes that the condition for non-emptiness of ordinary locus is a technical condition and can be replaced by \mu ordinary locus which they show is always non-empty. – Arijit Feb 20 '13 at 15:49

In the case of Iwahori level structure, there is the result of Stamm (cited in Wedhorn's paper) who gives an example where the ordinary locaus fails to be dense (Hilbert-Blumenthal case).

Stamm's result is reproved and generalized in

Philipp Hartwig, Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type,

who shows that, still in the Iwahori case (loc.cit. Theorem 1.1.1):

Assume that $\mathcal B$ is the symplectic PEL datum associated with a totally real extension $F/\mathbb Q$. Assume that there is only a single prime of $\mathcal O_F$ dividing $p$. Then the ordinary locus lies dense in $\mathcal A_F$ if and only if $p$ is totally ramified in $\mathcal O_F$.

Of course, if the ordinary locaus is dense in the Iwahori case, then the same is true for all parahoric level structures.

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Thank you very much for pointing out Hartwig's paper! – Ricky Feb 20 '13 at 13:17

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