Forcing ramification in the relative deformation method

TL;DR

This paper extends the forcing ramification technique within the relative deformation framework, ensuring ramification at chosen primes and linking local smoothness to the vanishing of certain Selmer groups, advancing deformation theory and Galois representations.

Contribution

It generalizes the forcing ramification argument to the setting of Fakhruddin-Khare-Patrikis's lifting results, connecting local smoothness with Selmer group vanishing.

Findings

Finite set of primes can be chosen for ramification.

Local lifts correspond to smooth points in universal deformation rings.

Vanishing of Bloch-Kato Selmer group is established.

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.