0 On the flatness of local models for the symplectic group

1 Introduction In order to study arithmetic properties of a variety over an algebraic number field, it is desirable to have a model over the ring of integers. We are interested in the particular case of a Shimura variety (of PEL type), and ask for a model over O E , where E is the completion of the reflex field at a prime of residue characteristic p > 0. Since a Shimura variety of PEL type is a moduli space of abelian varieties with certain additional structure, it is natural to define a model by posing the moduli problem over O E ; in the case of a parahoric level structure at p, which is the case we are interested in, such models have been defined by Rapoport and Zink [RZ]. These models are almost never smooth, and it is interesting to study the singularities occuring in the special fibre. It even is not obvious whether the models are flat over O E , although flatness certainly belongs to the minimum requirements for a reasonable model. Rapoport and Zink have conjectured that their models are indeed flat. In the case of a unitary group that splits over an unramified extension of Q p , flatness has been proved in [G]. In this article we will extend this result to the case of the symplectic group, i.e. we will show the flatness for groups of the form Res F/Qp GSp 2r , where F is a finite unramified extension of Q p. In the case where the underlying group does not split over an unramified extension, the flatness conjecture is not true as it stands, as has been pointed out by Pappas [P]; see also [PR] for ideas on how to proceed in this case. Several special cases of our main theorem have been known before, by the work of Chai and Norman [CN], de Jong [dJ] and Deligne and Pappas [DP].