M-Bigness in compatible systems

Abstract

Taylor–Wiles type lifting theorems allow one to deduce that if ρ is a “sufficiently nice” l-adic representation of the absolute Galois group of a number field whose semi-simplified reduction modulo l, denoted ρ ¯ , comes from an automorphic representation then so does ρ. The recent lifting theorems of Barnet-Lamb–Gee–Geraghty–Taylor impose a technical condition, called m-big, upon the residual representation ρ ¯ . Snowden–Wiles proved that for a sufficiently irreducible compatible system of Galois representations, the residual images are big at a set of places of Dirichlet density 1. We demonstrate the analogous result in the m-big setting using a mild generalization of their argument.