  We say that a two dimensional p-adic Galois representation 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 rho is a sufficiently irreducible
odd weight two p-adic representation of a totally real field F and either F has
odd degree or rho is indecomposable at some finite place away from p then rho
occurs as the Tate module of a GL_2-type abelian variety. This establishes some
new cases of the Fontaine-Mazur conjecture.
