We fix a prime p. It is a conjecture of Serre, cf. [S], that every odd, continuous, irreducible representation
As notation, for a number field F, we denote by G F its absolute Galois group. We fix embeddings @: Q / ร C and @ p : Q / ร Q p , and denote by p the place above p fixed by @ p . If S is a finite set of places of F, we denote by G F, S the Galois group Gal(F S รF ), where F S is the maximal algebraic extension of F in C which is unramified outside S. We will always assume that S includes all the places of F above p and also all the infinite places of F. For any finite extension M of F, we denote by M S the maximal algebraic extension of M in C which is unramified outside the places of M above those in S, and denote by G M, S the Galois group Gal(M S รM).
We point out a preliminary implication of Serre's conjecture of which nothing, as yet, seems to be known. Let \ be a representation of G Q, S as above which is odd, continuous, and irreducible for some fixed finite set S. We say that such a representation is of Serre type. It is not known if such a \ has a p-adic lift, i.e., if there exists a finite extension K of Q p , inside Q p , such that there is a continuous representation
whose reduction modulo the maximal ideal p K of the ring of integers O K of K, is isomorphic to . If such is the case, we will say that \ has a lift to GL 2 (K). The existence of such a lift, for some K, is certainly implied by the conjectures in [S]. As we are fixing the set S, in order to see this implication, we need the refinements of the basic conjecture in [S], stated at the beginning of the note. These refinements are stated in [S], and now proven to be implied by the basic conjecture, by the work of Ribet et al.