Andrew Snowden, Andrew Wiles
Published 2009-08-13, updated 2010-04-21Version 3
Clozel, Harris and Taylor have recently proved a modularity lifting theorem of the following general form: if rho is an l-adic representation of the absolute Galois group of a number field for which the residual representation rho-bar comes from a modular form then so does rho. This theorem has numerous hypotheses; a crucial one is that the image of rho-bar must be "big," a technical condition on subgroups of GL(n). In this paper we investigate this condition in compatible systems. Our main result is that in a sufficiently irreducible compatible system the residual images are big at a density one set of primes. This result should make some of the work of Clozel, Harris and Taylor easier to apply in the setting of compatible systems.
$m$-bigness in compatible systems
Relatively projective groups as absolute Galois groups
Galois action on knots I: Action of the absolute Galois group
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