R=T theorems for weight one modular forms

We prove modularity of certain residually reducible ordinary 2- dimensional p-adic Galois representations with determinant a finite order odd character χ. For certain non-quadratic χ we prove an R = T result for T the weight 1 specialisation of the cuspidal Hida Hecke algebra acting on nonclassical weight 1 forms. Under the additional assumption that no two cuspidal Hida families congruent to an Eisenstein series cross in weight 1 we show that T is reduced. For quadratic χ we prove that the quotient of R corresponding to deformations split at p is isomorphic to the Hecke algebra acting on classical CM weight 1 modular forms.