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On the arithmetic moduli schemes of PEL Shimura varieties. (English) Zbl 0978.14023
The aim of this paper is to study the moduli space for PEL Shimura varieties constructed by M. Rapoport and Th. Zink [“Period spaces for \(p\)-divisible groups”, Ann. Math. Stud. 141 (1996; Zbl 0873.14039)]. A detailed study of the Shimura varieties associated to unitary groups \(U(r,s)\) over an imaginary quadratic field \(K\) is done. It turns out that, when \(|r-s|>1\), the models given by Rapoport-Zink’s construction are not flat over those primes \(\mathcal P\) of \(K\) that divide the discriminant of \(K\). This fact contradicts the flatness conjecture stated in the above-mentioned work of Rapoport and Zink. However, the author proves that the moduli scheme for the modified moduli problem is flat in the case of unitary groups of type \(U(r,1)\).

MSC:
14G35 Modular and Shimura varieties
14D20 Algebraic moduli problems, moduli of vector bundles
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14L05 Formal groups, \(p\)-divisible groups
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