Deformations of Hilbert Modular Galois Representations and Adjoint Selmer Groups

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to a Hilbert modular form under mild assumptions on the residual image. In particular, we do not assume the residual image satisfies the Taylor–Wiles hypothesis. Using this, we deduce that the localization and completion of the universal deformation ring for the residual representation at the characteristic zero point induced from the Hilbert modular form is formally smooth of the correct dimension. We do this by employing the Taylor–Wiles patching strategy in a way inspired by the work of Skinner–Wiles to get around the Taylor–Wiles hypothesis. Along the way we give a characterization of smooth closed points on the generic fibre of Kisin’s potentially semistable deformation in terms of their Weil–Deligne representations.