Non-big subgroups for l large

TL;DR

This paper characterizes non-big residual Galois representations in dimension n>2, showing they are either reducible, induced, or tensor products involving Artin representations, thus clarifying the 'big' condition in lifting theorems.

Contribution

It provides a classification of non-big residual Galois representations for large primes, simplifying the understanding of the 'big' hypothesis in automorphic lifting theorems.

Findings

Non-big representations are either reducible, induced, or tensor products with Artin representations.

The classification applies for primes larger than a constant depending on n.

This work clarifies the technical 'big' condition in the context of automorphic forms.

Abstract

Lifting theorems form an important collection of tools in showing that Galois representations are associated to automorphic forms. (Key examples in dimension n>2 are the lifting theorems of Clozel, Harris and Taylor and of Geraghty.) All present lifting theorems for n>2 dimensional representations have a certain rather technical hypothesis---the residual image must be `big'. The aim of this paper is to demystify this condition somewhat. For a fixed integer n, and a prime l larger than a constant depending on n, we show that n dimensional mod l representations which fail to be big must be of one of three kinds: they either fail to be absolutely irreducible, are induced from representations of larger fields, or can be written as a tensor product including a factor which is the reduction of an Artin representation in characteristic zero. Hopefully this characterization will make the bigness condition more comprehensible, at least for large l.