Abstract
We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.
Highlights
The problem of the reduction modulo p of a Shimura variety has a long and complicated history, perhaps beginning with Kronecker
The case of the modular curve is essentially solved after the work of Igusa, Deligne, Drinfeld and Katz-Mazur
If the level structure is of Γ0(p)-type, the modular curve has semi-stable reduction
Summary
The problem of the reduction modulo p of a Shimura variety has a long and complicated history, perhaps beginning with Kronecker. The second case, which is a new observation of the current paper, is that of the local model associated to an even ramified quasi-split orthogonal group G, the cocharacter {μ} that corresponds to the orthogonal Grassmannian of isotropic subspaces of maximal dimension, and the parahoric K given by the stabilizer of an almost selfdual lattice. The hypothesis that each factor Gad,i be absolutely simple is essential to our method It implies that the translation element associated to {μ} in the extended affine Weyl group for Gad,i is not too large and this limits drastically the number of possibilities of LM triples with associated local models of good reduction. The list of Theorem 5.6 contains two more cases of LM triples with semi-stable associated local models, both for orthogonal groups, which seem to be new.
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