Local models of Shimura varieties and a conjecture of Kottwitz

By Georgios Pappas, Xinwen Zhu. 2011

147 citations recorded by the source corpus.

We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1c7b369aa96a1a6d96f6ccb39dc37d81e26fda77

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Citations in the corpus

Towards a theory of local Shimura varieties (2014, methodology): “For instance, we have now a purely group-theoretical definition of local models of Shimura varieties through the work of Pappas and Zhu [76].”
Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties (2014, background): “In [26] and [21], Pappas and Zhu give a group-theoretic definition of “local models” of Shimura varieties and show that the subset of WKW̃/WK for the local model is the admissible set AdmK(μ) (defined in §1.”
On the p-adic cohomology of the Lubin-Tate tower (2015, background): “tric domains. Funkcional. Anal. i Priloˇzen., 10(2):29–40, 1976. [14] R. Elkik. Solutions d’´equations a coeﬃcients dans un anneau hens´elien. Ann. Sci. Ecole Norm. Sup.´ (4), 6:553–603 (1974), 1973. [15] M. Emerton. Ordinary parts of admissible representations of p-adic reductive groups I. Deﬁnitio”
Integral models of Shimura varieties with parahoric level structure (2015, methodology): “1By definition [56], this means that x is a special vertex in B(G,Qp) and is also special in B(G,Q p ).”
Stratifications in the reduction of Shimura varieties (2015, background): “Pappas and Zhu [40] have constructed such local models under a tameness assumption on G.”
The Geometric Satake Correspondence for Ramified Groups (2011, background): “In [24], we will show that the same characterization holds for arbitrary (tamely ramified) quasi-split groups with special parahoric level structure.”