Crystalline Representations for GL(2) over Quadratic Imaginary Fields

Let K be a quadratic imaginary field and π an irreducible regular algebraic cuspidal automorphic representation of GL(2,AK). Under the assumption that the central character χπ is isomorphic to its complex conjugate, Taylor et al. associated p-adic Galois representations ρπ,p : GK → GL(2,Qp) which are unramified except at finitely many places, and such that the truncated L-function of the Galois representation equals the truncated L-function of π. We extend this result to include crystallinity of the Galois representation at p, under some restrictions on π (we require distinct Satake parameters). We first follow Kisin and Lai in constructing geometric families of finite slope overconvergent Siegel modular forms. This is achieved by defining overconvergent Siegel modular forms geometrically, and then showing that an Atkin-Lehner operator acts completely continuously on the space of such forms. Geometric families are then defined using eigenvarieties. We exhibit, in a rigid neighborhood of a theta lift of the representation π, a dense set of classical Siegel modular forms whose associated Galois representations are crystalline at p. Using this dense set of classical points we construct an analytic Galois representation in the chosen neighborhood of the theta lift. Finally, we appeal to a theorem of Kisin to show that crystalline periods at the dense set of classical points extend to the theta lift of π.