DEFORMATIONS OF GQp AND GL2(Qp) REPRESENTATIONS

We show that Colmez's functor from GL(2)(Q(p)) representations to G(Qp) representation produces essentially all two dimensional representations of G(Qp). The method compares the deformation theory for the two kinds of representations: An Ext group calculation of Colmez implies that the deformation space for GL(2)(Q(p),) representations is closed in that for G(Qp)-representations. A local version of the Gouvea-Mazur "infinite fern" argument shows that this closed subspace is also dense.