$m$-bigness in compatible systems

TL;DR

This paper extends the concept of 'big' residual images in Galois representations to the 'm-big' setting, showing that compatible systems have 'm-big' residual images at almost all places, which is crucial for automorphy lifting theorems.

Contribution

It generalizes Snowden-Wiles' result to the 'm-big' case, establishing that compatible systems have 'm-big' residual images at a density one set of places.

Findings

Residual images are 'm-big' at a set of places with Dirichlet density 1.

The proof adapts Snowden-Wiles' argument to the 'm-big' context.

Supports the application of Taylor-Wiles type lifting theorems under 'm-big' conditions.

Abstract

Taylor-Wiles type lifting theorems allow one to deduce that for $\rho$ a "sufficiently nice" $l$-adic representation of the absolute Galois group of a number field whose semi-simplified reduction modulo $l$, denoted $\overline{\rho}$, comes from an automorphic representation then so does $\rho$. The recent lifting theorems of Barnet-Lamb-Gee-Geraghty-Taylor impose a technical condition, called \emph{$m$-big}, upon the residual representation $\overline{\rho}$. 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.