2010/06/06 by Thomas Barnet-Lamb, Barnet-Lamb, Thomas
Mathematics · #11R39 (primary) #20D99 (secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT #msc:11R39 #msc:20D99
paper · pdf · doi:10.48550/arxiv.1006.1110
30 pages
arxiv created 2010/06/06 · openalex publication_date 2010/06/06 · arxiv updated 2010/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.