# State of the art for integral models of PEL type Shimura varieties with deep level structure

The theory of PEL type Shimura varieties is nowadays well developed, but it is not easy to be updated with the latest results. Here I am particularly interested in integrals models. Let me describe what I understand.

Let $B$ be a (semi)simple algebra over $\mathbb Q$, of finite dimension. We suppose that $B$ is endowed with a positive involution $\ast$. Let $\mathcal O_B$ be an order of $B$, preserved by $\ast$. Let $(V,\Psi)$ be a finitely generated symplectic left $(B, \ast)$-module. Let $h \colon \mathbb C \to End_{B_{\mathbb R}}(V_{\mathbb R})$ be an $\mathbb R$-algebras homomorphism that gives an Hode strucure of type $(1,0),(0,1)$ on $V_{\mathbb R}$. We can now define a Shimura datum $(G,X)$ in the usual way. We obtain in particular a family of complex varieties $S_K$ parametrized by compact open subgroup $K \subseteq G(\mathbb A_f)$. We will assume that $K$ is 'small enough', in particular these varieties are moduli spaces of abelian varieties with additional (PEL) structure. It turns out that there is a number field $E$, called the reflex field, such that $S_K$ admits a canonical model defined over $E$.

In arithmetic it is very interesting to consider integral model of $S_K$. We fix a rational prime $p$. We assume that there is a lattice $\Lambda \subseteq V_{\mathbb Q_p}$ that is self-dual for $\Psi$ and we fix a compact open, small enough, subgroup $K^p \subseteq G(\mathbb A_f^p)$. Assuming that $B$ splits over an unramified extension of $\mathbb Q_p$, we have that $G(\mathbb Q_p)$ admits an hyperspecial subgroup, that we denote $K_{0,p}$. We assume that $B$ is of type A or C (another question is what can be done in the case D). It is well known that $S_{K^pK_{0,p}}$ admits a canonical integral model over $\mathcal O_E$, that is smooth over $\mathcal O_E \otimes Z_p$ and solves a very reasonable moduli problem. This goes back to Kottwitz.

Question 1 What can be done without the unramifiedness assumption? Of course in this case we do not have an hyperspcial subroup of $G(\mathbb Q_p)$, so we need a level strucure also at $p$. Rapoport and Zink have defined some integral models that satisfy a moduli problem, but it seems that their models are not even flat over the base.

Let me go back to the unramified case. For different reasons, it is interesting to consider level structures at $p$ (for example of type $\Gamma_1(Np^m)$ or $\Gamma_1(N) \cap \Gamma_0(p^m)$ in the case of modular curves). Now there is no hope for a smooth model, but of course one wants a good integral models. I am in particular interested in Iwahoric (or, more generally, parahoric) level structure. In the Siegel case, for example, we have good integral models.

Question 2 Under which assumptions we have good (flat and with a moduli interpretation) model of Shimura varieties with Iwahoric level structure at $p$? We have the models of Rapoport and Zink, but I do not if they are flat in general. Some cases are studied by Görtz (http://arxiv.org/abs/math/9912064 and http://arxiv.org/abs/math/0011202), but it seems that the general case is open (here I am always assuming that $G$ is quasi-split).

In general, I am interested in various condition 'at $p$' one have to put in order to obtain good integral models of PEL type Shimura varieties.

Thank you!

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It seems that the papers by Görtz answer my second question completely in case A of C if algebra becomes isomorphic to a product of matrix algebras over an unramified extension of $\mathbb Q_p$. Rapoport says this in his 'A guide to the reduction modulo $p$ of Shimura varieties', at the very end of section 1. This is not clear to me, as Görtz has more restrictive assumptions (for example he has $B=F$, where $F$ s the center and $V=F^n$). Can someone explain this? –  Ricky Oct 13 '12 at 8:24

Here are some remarks:

Rapoport's Guide paper which you mention in your comment certainly is a good starting point. There is also a survey paper by Pappas, Rapoport and Smithling with many more recent results; it focusses on the "local model side", but the understanding of the local model more or less is enough to understand the Shimura variety side.

Generally speaking, when the group splits over an unramified extension of $\mathbb Q_p$, the original Rapoport-Zink model is flat (types $A$ and $C$). (Re your comment: The general case can be reduced to the case $B=F$ by a Morita equivalence argument.)

If the group does not split over an unramified extension, the original (now called: naive) local model is typically not flat. Pappas and Rapoport have studied this case in a series of papers. (Of course one can pass to the flat closure in order to obtain a flat model; then the question becomes giving a moduli description of the flat closure, and maybe describe its geometry.)

For type $D$ less is known, but see the papers of Smithling for some results.

For level structures deeper than Iwahori, much less is known in general: In special cases (such as the case considered by Harris and Taylor, or, of course, for modular curves) one can obtain a nice model with a good geometric description. Haines and Rapoport have a paper on the $\Gamma_1(p)$ case. Sometimes it is also enough to work with some "abstractly defined" models (e.g. Drinfeld level structures and flat closure as in Mantovan, On the cohomology of certain PEL type Shimura varieties, Duke Math. J. 129 (2005), no. 3, 573--610).

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