$m$-bigness in compatible systems

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

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