The local-global conjecture for Apollonian circle packings is false

TL;DR

This paper disproves the local-global conjecture for many Apollonian circle packings by identifying new obstructions related to quadratic and quartic reciprocity, showing certain curvatures are missed.

Contribution

It demonstrates that the local-global conjecture fails for many packings due to obstructions from the thin Apollonian group, introducing new reciprocity-based obstructions.

Findings

The local-global conjecture is false for many packings.

Certain quadratic and quartic families of curvatures are systematically missed.

New reciprocity-based obstructions explain these misses.

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 will appear 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.