2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157886We 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. These can be applied to the study of exceptional zero of p-adic L-function.Number Theory11F80text