Residual irreducibility of compatible systems

2016/05/12 by Patrikis, Stefan, Snowden, Andrew, Wiles, Andrew
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1605.03936

Abstract

We show that if \ρ_ℓ\ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation ρ_ℓ is absolutely irreducible for ℓ in a density 1 set of primes. The key technical result is the following theorem: the image of ρ_ℓ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as ℓ 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.

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