Modularity lifting theorems for ordinary Galois representations

We generalize results of Clozel, Harris and Taylor by proving modularity lifting theorems for ordinary l -adic Galois representations of any dimension of an imaginary CM or totally real number field. The main theorems are obtained by establishing an R red = T theorem over a Hida family. A key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing l and to determine their irreducible components.