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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 \(\Gamma_1 (q)\).

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