Lifting of nearly extraordinary Galois representations

Let (rho) over bar be a continuous, 2-dimensional and absolutely irreducible mod p representation of the absolute Galois group of a number field K/Q. In this work, we study the existence of lifts of (rho) over bar to GL(2)(Z(p)), for which the restrictions to the decomposition groups above p are abelian. The tools and philosophy come from the Taylor-Ramakrishna method. As an application we produce finitely ramified extensions over Q(mu(p infinity)) with Galois group SL2(Z(p)), for some p. These extensions are unramified at p. (C) 2012 Elsevier Inc. All rights reserved.