On the Modularity of Certain 2-adic Galois Representations

We prove some results of the form “r residually irreducible and residually modular implies r is modular,” where r is a suitable continuous odd 2-dimensional 2-adic representation of the absolute Galois group of Q. These results are analogous to those obtained by A. Wiles, R. Taylor, F. Diamond, and others for p-adic representations in the case when p is odd; some extra work is required to overcome the technical difficulties present in their methods when p = 2. The results are subject to the assumption that any choice of complex conjugation element acts nontrivially on the residual representation, and the results are also subject to an ordinariness hypothesis on the restriction of r to a decomposition group at 2. Our main theorem (Theorem 4) plays a major role in a programme initiated by Taylor to give a proof of Artin’s conjecture on the holomorphicity of L-functions for 2-dimensional icosahedral odd representations of the absolute Galois group of Q; some results of this programme are described in a paper that appears in this issue, jointly authored with K. Buzzard, N. Shepherd-Barron, and Taylor. Introduction Let ` be a rational prime, and let GQ = Gal(Q̄/Q) be an absolute Galois group of Q. Conjectures of J.-M. Fontaine and B. Mazur [FM] and of R. Langlands predict that any irreducible, continuous, odd representation ρ : GQ → GL2(Q̄`) which is unramified almost everywhere and potentially semistable at ` arises from a modular form, while a conjecture of J.-P. Serre [S2] predicts that every irreducible, continuous, odd representation ρ̄ : GQ → GL2(F̄`) also arises from a modular form. In both these cases, the term “odd” means that the image of a complex conjugation element of GQ should have determinant −1. A key step of the proof that every semistable elliptic curve defined over Q is modular (see [Wi2], [TW]) relates these two conjectures by showing that if ` is odd, then for certain `-adic representations ρ : GQ → GL2(Q̄`) with irreducible mod-` reduction ρ̄ : GQ → GL2(F̄`) the modularity of ρ can be DUKE MATHEMATICAL JOURNAL Vol. 109, No. 2, c © 2001 Received 22 December 1999. 2000 Mathematics Subject Classification. Primary 11F80.