The local-global conjecture for Apollonian circle packings is false
成果类型：
Article
署名作者：
Haag, Summer; Kertzer, Clyde; Rickards, James; Stange, Katherine E.
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2024.200.2.6
发表日期：
2024
页码：
749-770
关键词：
摘要：
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 BrauerManin obstruction. Based on computational evidence, we formulate a new conjecture.