Bigness in Compatible Systems

Abstract

Clozel, Harris and Taylor have recently proved a modularity lifting theorem of the following general form: if $$\rho $$ is an $$\ell $$ -adic representation of the absolute Galois group of a number field for which the residual representation $$\overline{\rho }$$ comes from a modular form then so does $$\rho $$ . This theorem has numerous hypotheses; a crucial one is that the image of $$\overline{\rho }$$ must be “big,” a technical condition on subgroups of $$\mathrm {GL}_n$$ . 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.

Locations

• Springer proceedings in mathematics & statistics - View
• arXiv (Cornell University) - View - PDF