A conjecture was made by Dunfield and Calegari that certain congruence covers of an arithmetic hyperbolic 3-manifold have trivial first betti number (which corresponds to the non-existence of certain automorphic forms, conjectured based on the generalized Riemann hypothesis and the Langlands proram). This was subsequently proved by Boston and Ellenberg using methods from pro-$p$ groups.
Ian Agol
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