I describe two projects, one with Clozel and Taylor, the other with Shepherd-Barron and Taylor, that develop two of three steps needed to extend the methods initiated by Wiles to a certain class of Galois representations of arbitrary dimension. Missing is a treatment of level raising, which we reduce to a technical condition ("Ihara's Lemma") on automorphic forms with values in finite fields. The principal novelty is the use of Diophantine properties of moduli spaces of Calabi-Yau hypersurfaces to prove that certain n-dimensional Galois representations, including the odd symmetric powers of Tate modules of elliptic curves with rational non-integral j-invariant, become residually modular over unspecified totally real fields.