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Local models in the ramified case. I: The EL-case. (English) Zbl 1063.14029
The subject of the paper under review is to investigate some simply-defined schemes (\(M=M(\Lambda, \text{\textbf{r}}\), \(\widetilde M\), \(N\) etc.) in order to get information about more complicated objects, namely models (denoted by \(X\)) of Shimura varieties of endomorphism and level type over Spec \({\mathcal O}_E\), where \(E\) is the completion of the reflex field of \(X\) at a finite prime. Particularly, information about the phenomenon of possible non-flatness of \(X\) over Spec \({\mathcal O}_E\) is obtained. An example of such \(X\) is a Shimura variety associated to group \(G=\text{Res}_{E/F}\text{GL}(n)\) where \(E/F\) is a totally ramified extension of local fields.
\(M=M(\Lambda, \text{\textbf{r}})\) is called a naive local model for the moduli problem corresponding to \(X\). \(M\) is closely related to \(X\), namely, it contains some closed subschemes (local models) which are locally for the étale topology isomorphic to \(X\).
Schemes \(M(\Lambda, \text{\textbf{r}})\) are defined in terms of linear algebra. Namely, fix a totally ramified extension \(F/F_0\) of local fields, a vector space \(V\) over \(F\), an \({\mathcal O}_F\)-lattice \(\Lambda \subset V\) and a multidimension \(\text{\textbf{r}}=\{r_{\varphi}\}\) (here the \(\varphi\)’s run over all inclusions of \(F/F_0\) in \(\bar F_0\)) such that the reflex field of \(\text{\textbf{r}}\) is \(E\). \(M\) represents a functor that (roughly speaking) associates to an extension \(S/F_0\) the set of sublattices \(\mathcal F\) of \(\Lambda \otimes_{F_0}S\) such that the multidimension of \(\mathcal F\) is \(\text{\textbf{r}}\). Similarly, \(\widetilde M\) represents a functor associating to \(S\) the pair (\(\mathcal F\), a base of \(\mathcal F\)), and \(N(S)\) is a set of some \(r\times r\)-matrices over \(S\) (\(r=\sum_{\varphi}r_{\varphi}\)). There is a diagram \(M\overset{\pi}{\leftarrow} \widetilde M \overset{\phi}{\to}N\). Further, let \(M^{\text{loc}}\) be the closure of \(M\otimes_{{\mathcal O}_E}E\) in \(M\). The following theorems are proved:
Theorem A. The morphism \(\phi\) is smooth.
Theorem B. \(M^{\text{loc}}\) is normal, Cohen-Macaulay, its special fiber is reduced, normal with rational singularities.
The existence of an inclusion of the special fiber of \(M\) to the affine Grassmannian permits to prove a theorem on its Schubert subvariety. The description of the resolution of the singularities of \(N\) over the Galois closure of \(F/F_0\) permits to calculate the complex of nearby cycles on \(M\) (together with the Gal(\(F_0\))-action) in terms of the constant sheaves on some varieties related to \(M\); the statement of this result is too large to be given here.
The final result of the discussion about relations between \(X\) and \(M\) is the following: while \(X\) is not always flat over \({\mathcal O}_E\), it is conjectured that \(M^{\text{loc}}\) is always flat over \({\mathcal O}_E\).

MSC:
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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