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Companion forms and weight one forms

Kevin Buzzard·Richard Taylor
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Abstract

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 ρ:G_\Q -> GL_2(O_L) 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(Frob_p) has eigenvalues αand βwith distinct reductions modulo λ. 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(ρ(\Frob_q)). In particular ρhas finite image and for any embedding i of L in \C, the Artin L-function L(i o ρ, s) is entire.

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Kevin Buzzard, Richard Taylor. 1999-05-01. Companion forms and weight one forms. https://arxiv.org/abs/math/9905207

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