Forcing ramification in the relative deformation method

Abstract

In this paper we generalize the forcing ramification argument of Khare-Ramakrishna to the setting of the lifting result by Fakhruddin-Khare-Patrikis. In particular, we show that in the relative deformation method the finite set of primes can be chosen so that the geometric characteristic 0 lift will be ramified at each of those primes. Moreover, we show that the restrictions of this lift to the local absolute Galois groups will correspond to formally smooth points in the generic fibers of the local universal lifting rings at each prime. Eventually, we prove that the local formal smoothness at each of the primes implies the vanishing of the geometric Bloch-Kato Selmer group associated to the adjoint representation of that characteristic 0 lift.