The not-so-local-global conjecture

I will describe new obstructions found for Apollonian circle packings, which contradicts the local-global conjecture for many configurations. These obstructions come from quadratic and quartic reciprocity, and we expect similar obstructions to exist in other thin group orbits. Based on computational evidence, a new conjecture is formulated. This is joint work with Summer Haag, Clyde Kertzer, and Katherine E. Stange.