State of the art for integral models of PEL type Shimura varieties with deep level structure - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:12:07Z http://mathoverflow.net/feeds/question/109468 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109468/state-of-the-art-for-integral-models-of-pel-type-shimura-varieties-with-deep-leve State of the art for integral models of PEL type Shimura varieties with deep level structure Ricky 2012-10-12T16:42:39Z 2013-02-20T12:00:04Z <p>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.</p> <p>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 $E$.</p> <p>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.</p> <p><strong>Question 1</strong> 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.</p> <p>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.</p> <p><strong>Question 2</strong> 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 <a href="http://arxiv.org/abs/math/0011202" rel="nofollow">http://arxiv.org/abs/math/0011202</a>), but it seems that the general case is open (here I am always assuming that $G$ is quasi-split).</p> <p>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.</p> <p>Thank you!</p> http://mathoverflow.net/questions/109468/state-of-the-art-for-integral-models-of-pel-type-shimura-varieties-with-deep-leve/122389#122389 Answer by Ulrich Goertz for State of the art for integral models of PEL type Shimura varieties with deep level structure Ulrich Goertz 2013-02-20T12:00:04Z 2013-02-20T12:00:04Z <p>Here are some remarks:</p> <p>Rapoport's Guide paper which you mention in your comment certainly is a good starting point. There is also a <a href="http://arxiv.org/abs/1011.5551" rel="nofollow">survey paper</a> 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.</p> <p>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.)</p> <p>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.)</p> <p>For type $D$ less is known, but see the papers of Smithling for some results.</p> <p>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, <a href="http://www.its.caltech.edu/~mantovan/papers/PEL.pdf" rel="nofollow">On the cohomology of certain PEL type Shimura varieties</a>, Duke Math. J. 129 (2005), no. 3, 573--610). </p>