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On the Hilbert-Blumenthal moduli problem 1

In this article we compare different conditions on abelian schemes with real multiplication which occur in the integral models of the Hilbert-Blumenthal Shimura variety considered by Rapoport, Deligne, Pappas and Kottwitz. We show that the models studied by Deligne/Pappas and Kottwitz are isomorphic over Spec Z(p). We also examine the associated local models and prove that they are equal.