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# Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

University of Toronto, Department of Mathematics, 40 St. George St., Toronto, ON M5S 2E4, Canada
Source: arXiv

ABSTRACT Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.

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##### Article:On the flatness of models of certain Shimura varieties of PEL-type
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ABSTRACT: Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of . Rapoport and Zink have defined a model of the Shimura variety over the ring of integers of the completion of the reflex field at a place lying over p, with parahoric level structures at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced.
Mathematische Annalen 10/2001; 321(3):689-727. · 1.30 Impact Factor
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##### Article:Local models for Ramified unitary groups
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 01/2003; 73(1):67-80. · 0.22 Impact Factor
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##### Article:Local models in the ramified case. II. Splitting models
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ABSTRACT: This paper is a continuation of our paper math.AG/0006222. We study the reduction of certain PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" \$p\$-adic integral models of these Shimura varieties and study their 'etale local structure. In particular, we exhibit a stratification of their (singular) special fibers and give a partial calculation of the sheaf of nearby cycles.
06/2002;

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### Keywords

conjecture

conjectured

flat scheme

give étale-local neighborhoods

integral models

linear-algebraic moduli problems

Local models

localizations

new local models

new models

original local models

quasi-split GUn

Shimura models

Shimura variety ramifies

so-called wedge

unitary similitude groups

Zink