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arXiv · 1605.03936

Residual irreducibility of compatible systems

Stefan Patrikis·Andrew Snowden·Andrew Wiles
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Abstract

We show that if $\{ρ_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overlineρ_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of primes. The key technical result is the following theorem: the image of $ρ_{\ell}$ 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.

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Stefan Patrikis, Andrew Snowden, Andrew Wiles. 2016-06-06. Residual irreducibility of compatible systems. https://arxiv.org/abs/1605.03936

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