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Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-$\\ell$ fundamental group of $P^1 - \\{0,1,\\infty\\}$.\n  Under GRH, we demonstrate the set of classes is finite for any fixed $K$ and any fixed dimension. Without GRH, we prove a semistable version of the result. 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