𝑅=𝑇 theorems for weight one modular forms

Abstract

We prove modularity of certain residually reducible ordinary 2-dimensional p p -adic Galois representations with determinant a finite order odd character χ \chi . For certain non-quadratic χ \chi we prove an R = T R=T result for T T the weight 1 specialisation of the cuspidal Hida Hecke algebra acting on non-classical 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 T is reduced. For quadratic χ \chi we prove that the quotient of R R corresponding to deformations split at p p is isomorphic to the Hecke algebra acting on classical CM weight 1 modular forms.