Mod $p$ Galois representations of solvable image

Abstract. It is proved that, for a number field $K$ and a prime number $p$, there exist only finitely many isomorphism classes of continuous semisimple Galois representations of $K$ into $\GL_d(\Fpbar)$ of fixed dimension $d$ and bounded Artin conductor outside $p$ which have solvable images. Some auxiliary results are also proved.


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