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Residual Irreducibility of Compatible Systems
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 …