Apollonian packings: The rise and fall of the local to global conjecture

Nearly twenty years ago, a paper of Graham, Lagarias, Mallows, Wilks, and Yan on the number theory of Apollonian circle packings sparked an interest in the number theory community, which was just developing tools to handle arithmetic problems involving so-called thin groups. At the time, these packings were the only naturally occurring example of such an arithmetic problem and, naturally, number theorists sprang upon the opportunity to discover all their rich properties, thinness notwithstanding. In 2010, building upon conjectures of Graham et al., the author and Katherine Sanden gave evidence toward the local to global conjecture for Apollonian circle packings, stating that in any integral packing, any large enough integer that satisfied certain congruence conditions modulo 24 must appear as a curvature in the packing. For thirteen years, most everyone believed this conjecture to be true. In this article, we will explore the history of this conjecture and its fascinating downfall after Haag, Kertzer, Rickards, and Stange proved that, in fact, infinitely many integral Apollonian packings fail to abide by the local to global principle, and come with extra quadratic and quartic obstructions.