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Regular models of certain Shimura varieties. (English) Zbl 1008.11022
The authors develop a technique for studying the bad reduction of Shimura varieties attached to twisted unitary groups, which are higher dimensional analogues of the modular curves $$X_1 (p)$$. The method presented in this paper is analogous to the well-known theory of P. Deligne and M. Rapoport [Lect. Notes Math. 349, 143-316 (1973; Zbl 0281.14010)] for modular curves, and it extends the techniques of R. J. Taylor and A. Wiles [Ann. Math. 141, 553-572 (1995; Zbl 0823.11030)] to deformations of mod $$\ell$$ Galois representations of dimension greater than two. The Taylor-Wiles method requires information about the tame ramification at $$q$$ of the Galois representations associated to modular forms of level $$\Gamma_1 (q)$$ for some primes $$q$$ highly congruent to 1 modulo $$\ell$$, and the extension of this to higher dimension is based on the detailed study of the singularities of the special fiber at $$q$$ for level subgroups generalizing $$\Gamma_1 (q)$$.

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