Companion forms and weight one forms
Kevin Buzzard, Richard Taylor
Abstract
In this paper we prove the following theorem. Let L/p be a finite extension with ring of integers OL and maximal ideal lambda. Theorem 1. Suppose that p >= 5. Suppose also that ρ:G -> GL2(OL) is a continuous representation satisfying the following conditions. 1. ρramifies at only finitely many primes. 2. ρmod λis modular and absolutely irreducible. 3. ρis unramified at p and ρ(Frobp) has eigenvalues αand βwith distinct reductions modulo λ. Then there exists a classical weight one eigenform f = Σn=1∞ am(f) qm and an embedding of (am(f)) into L such that for almost all primes q, aq(f)=tr(ρ(q)). In particular ρhas finite image and for any embedding i of L in , the Artin L-function L(i o ρ, s) is entire.
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