On a rigidity of some modular Galois deformations

Abstract

Let F be a totally real field and ρ = (ρ_λ)_λ be a compatible system of two dimensional λ-adic representations of the Galois group of F. We assume that ρ has a residually modular λ-adic realization for some λ. In this paper, we consider local behaviors of modular deformations of λ-adic realizations of ρ at unramified primes. In order to control local deformations at specified unramified primes, we construct certain Hecke modules. Applying Kisin's Taylor-Wiles system, we obtain an R = T type result supplemented with local conditions at specified unramified primes. As a consequence, we shall show a potential rigidity of some modular deformations of infinitely many λ-adic realizations of ρ.