The local-global conjecture for Apollonian circle packings is false
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910591836225536 |
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| author | Haag, Summer Kertzer, Clyde Rickards, James Stange, Katherine E. |
| author_facet | Haag, Summer Kertzer, Clyde Rickards, James Stange, Katherine E. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_02749 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The local-global conjecture for Apollonian circle packings is false Haag, Summer Kertzer, Clyde Rickards, James Stange, Katherine E. Number Theory 52C26, 11D99, 11-04 (Primary) 20H10, 11E12, 11A15, 11B99 (Secondary) 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. |
| title | The local-global conjecture for Apollonian circle packings is false |
| topic | Number Theory 52C26, 11D99, 11-04 (Primary) 20H10, 11E12, 11A15, 11B99 (Secondary) |
| url | https://arxiv.org/abs/2307.02749 |