In this paper we prove the following theorem. Let L/\Q_p be a finite
extension with ring of integers O_L and maximal ideal lambda.
Theorem 1. Suppose that p >= 5. Suppose also that \rho:G_\Q -> GL_2(O_L) is a
continuous representation satisfying the following conditions.
1. \rho ramifies at only finitely many primes.
2. \rho mod \lambda is modular and absolutely irreducible.
3. \rho is unramified at p and \rho(Frob_p) has eigenvalues \alpha and \beta
with distinct reductions modulo \lambda.
Then there exists a classical weight one eigenform
f = \sum_{n=1}^\infty a_m(f) q^m
and an embedding of \Q(a_m(f)) into L such that for almost all primes q,
a_q(f)=tr(\rho(\Frob_q)). In particular \rho has finite image and for any
embedding i of L in \C, the Artin L-function L(i o \rho, s) is entire.Comment: 15 pages, published version, abstract added in migratio
