On Two Dimensional Weight Two Odd Representations of Totally Real Fields

We say that a two dimensional p-adic Galois representation GF → GL2(Qp) of a number field F is weight two if it is de Rham with Hodge-Tate weights 0 and −1 equally distributed at each place above p; for example, the Tate module of an elliptic curve has this property. The purpose of this paper is to establish a variety of results concerning odd weight two representations of totally real fields in as great a generality as we are able. Most of these results are improvements upon existing results. Three of our main results are as follows. (1) We prove a modularity lifting theorem for odd weight two representations, extending a theorem of Kisin to include representations which are not potentially crystalline. (2) We show that essentially any odd weight two representation is potentially modular, following the ideas of Taylor. (3) We show that one can lift essentially any odd residual representation to a minimally ramified weight two p-adic representation, using some ideas of Khare-Wintenberger. As an application of these results we show that if ρ is a sufficiently irreducible odd weight two p-adic representation of a totally real field F and either F has odd degree or ρ is indecomposable at some finite place v p then ρ occurs as the Tate module of a GL2-type abelian variety. This establishes some new cases of the Fontaine-Mazur conjecture.