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Local models of Shimura varieties and a conjecture of Kottwitz. (English) Zbl 1294.14012
Invent. Math. 194, No. 1, 147-254 (2013); erratum ibid. 194, No. 1, 255 (2013).
This paper addresses a problem in arithmetic algebraic geometry. Local models are schemes over a discrete valuation ring, defined as closed subschemes of products of Grassmann schemes by “linear algebra conditions”. They are related to various topics in arithmetic geometry. One of them – maybe the most prominent one – is the theory of integral models of Shimura varieties. This connection, relying on Grothendieck-Messing theory, allows one to define such local models which étale-locally model the singularities of certain integral models of Shimura varieties of PEL-type with parahoric level structure. This theory was developed by de Jong, Deligne, Pappas, Rapoport, Zink and others.
For example, the computation of the trace of Frobenius on the sheaf of nearby cycles, which is required for the standard strategy of computing the local Hasse-Weil zeta function, can be carried out on the local model. In this way, T. Haines and B. C. Ngô [Compos. Math. 133, No. 2, 117–150 (2002; Zbl 1009.11042)] proved for \(\mathrm{GL}_n\) and \(\mathrm{GSp}_{2g}\) a conjecture of Kottwitz describing this trace as a central element in the corresponding parahoric Hecke algebras, partly following previous work of D. Gaitsgory [Invent. Math. 144, No. 2, 253–280 (2001; Zbl 1072.14055)] in the equal characteristic case.
Pappas and Zhu give a group-theoretic definition of “local models” in the context of an arbitrary reductive group. Having a unified approach which is independent of the theory of Shimura varieties is an important step forward in the theory.
Furthermore, they prove the conjecture of Kottwitz in this case, and show that their local model coincides with the one studied previously in most cases of PEL Shimura varieties. Conjecturally, this new construction of local models should also yield étale-local models of suitable integral models of Shimura varieties of Hodge type.
An essential step is the construction of a group scheme \(\mathcal G\) over the \(2\)-dimensional \(\mathbb A^1_{\mathcal O}=\mathrm{Spec} \mathcal O[u]\), where \(\mathcal O\) is a discrete valuation ring, whose base change to \(\text{Quot}(\mathcal O)[[u]]\), and to \((\mathcal O/\varpi)[u]\) respectively, gives Bruhat-Tits schemes for the underlying parahoric group over these discrete valuation rings. If the situation arises from a Shimura variety, then \(\mathcal O\) is of mixed characteristic, and one obtains a tool to “interpolate” between the mixed characteristic \(\mathcal O\) and the equal characteristic \((\mathcal O/\varpi)[[u]]\). It is then possible to define “local” and “global” variants of the affine Grassmannian, à la Beilinson and Drinfeld, and eventually to define the local model as the schematic closure of a certain Schubert cell in the generic fiber, the closure being taken inside a degeneration of a parahoric affine flag variety to the affine Grassmannian. While this idea is already in the work of Gaitsgory and of Haines and Ngô, in the situation at hand it is considerably more complicated to carry it through.
(The erratum only corrects the misprint of two names in the bibliography.)

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