Companion forms and weight one forms

We have three motivations for looking at this theorem. Firstly it can be seen as partial confirmation of a conjecture of Fontaine and Mazur which asserts that if ρ : GQ → GLn(Qp) is a continuous irreducible representation ramified at only finitely many primes and such that the image of the inertia group at p is finite, then the image of ρ is finite (see [FM]). Our theorem verifies this conjecture in the case that n = 2, p ≥ 5 and the reduction (ρ mod λ) is modular, irreducible and takes Frobp to an element with distinct eigenvalues. In our opinion the most serious assumption here is that (ρ mod λ) should be modular, but we remind the reader that if (ρmodλ) is odd then a conjecture of Serre (see [S2]) predicts that it is necessarily modular.