| A key ingredient of the proof that every elliptic curve over Q is modular is a set of results of the form ‘ r&d1; irreducible and modular implies r modular’, where r is a continuous two-dimensional ℓ-adic representation of the absolute Galois group of Q (for example that associated to an elliptic curve over Q) and r&d1; is its mod-ℓ reduction. For various technical reasons, existing results of this form require that the prime ℓ be odd. We prove some analogous results in the case ℓ = 2. When combined with work of Buzzard, Shepherd-Barron and Taylor, these results yield a proof of Artin's conjecture on the holomorphicity of L-functions for a large class of two-dimensional icosahedral odd representations of the absolute Galois group of Q. |
