The local-global conjecture for Apollonian circle packings is false

Abstract

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes appears as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.

Authors

Summer Haag

University of Colorado Boulder, Boulder, Colorado, USA

Clyde Kertzer

University of Colorado Boulder, Boulder, Colorado, USA

James Rickards

Saint Mary's University, Halifax, Nova Scotia, Canada

Katherine E. Stange

University of Colorado Boulder, Boulder, Colorado, USA