The local-global conjecture for Apollonian circle packings is false
| Autor: | Haag, Summer, Kertzer, Clyde, Rickards, James, Stange, Katherine E. |
|---|---|
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Annals of Mathematics, Vol. 200, No. 2, pp. 749-770 (September 2024) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4007/annals.2024.200.2.6 |
| Popis: | 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. Comment: 19 pages, 2 figures. Slightly expanded from the published version |
| Databáze: | arXiv |
| Externí odkaz: |
|