Residual Irreducibility of Compatible Systems

Abstract

We show that if |$\{\rho_{\ell}\}$| is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation |$\overline{\rho}_{\ell}$| is absolutely irreducible for |$\ell$| in a density 1 set of primes. The key technical result is the following theorem: the image of |$\rho_{\ell}$| is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as |$\ell$| varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF
• Oxford University Research Archive (ORA) (University of Oxford) - View - PDF