Abstract:We studied framed deformations of two dimensional Galois representation of which the residue representation restrict to decomposition groups are scalars, and established a modular lifting theorem for certain cases. We then proved a family version of the result, and used it to determine the structure of deformation rings over characteristic zero fields. As a corollary, we obtain the L-invariant of adjoint square representation associated to a Hilbert Hecke eigenform.
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