Regular models of certain Shimura varieties (Q700535)

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 \textit{P. Deligne} and \textit{M. Rapoport} [Lect. Notes Math. 349, 143-316 (1973; Zbl 0281.14010)] for modular curves, and it extends the techniques of \textit{R. J. Taylor} and \textit{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 \(\Gamma_1 (q)\).