Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
© The Authors 2020. Fix d ≥ 2 and a field k such that char k - d. Assume that k contains the dth roots of 1. Then the irreducible components of the curves over k parameterizing preperiodic points of polynomials of the form zd+c are geometrically irreducible and have gonality tending to 1. This impli...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Wiley
2021
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| Online Access: | https://hdl.handle.net/1721.1/136298 |
| _version_ | 1826211803090124800 |
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| author | Doyle, John R Poonen, Bjorn |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Doyle, John R Poonen, Bjorn |
| author_sort | Doyle, John R |
| collection | MIT |
| description | © The Authors 2020. Fix d ≥ 2 and a field k such that char k - d. Assume that k contains the dth roots of 1. Then the irreducible components of the curves over k parameterizing preperiodic points of polynomials of the form zd+c are geometrically irreducible and have gonality tending to 1. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of zd+c. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo{Severi inequality. |
| first_indexed | 2024-09-23T15:11:39Z |
| format | Article |
| id | mit-1721.1/136298 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:11:39Z |
| publishDate | 2021 |
| publisher | Wiley |
| record_format | dspace |
| spelling | mit-1721.1/1362982023-02-17T21:08:16Z Gonality of dynatomic curves and strong uniform boundedness of preperiodic points Doyle, John R Poonen, Bjorn Massachusetts Institute of Technology. Department of Mathematics © The Authors 2020. Fix d ≥ 2 and a field k such that char k - d. Assume that k contains the dth roots of 1. Then the irreducible components of the curves over k parameterizing preperiodic points of polynomials of the form zd+c are geometrically irreducible and have gonality tending to 1. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of zd+c. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo{Severi inequality. 2021-10-27T20:34:46Z 2021-10-27T20:34:46Z 2020 2021-05-25T18:42:52Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/136298 en 10.1112/S0010437X20007022 Compositio Mathematica Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Wiley MIT web domain |
| spellingShingle | Doyle, John R Poonen, Bjorn Gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| title | Gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| title_full | Gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| title_fullStr | Gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| title_full_unstemmed | Gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| title_short | Gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| title_sort | gonality of dynatomic curves and strong uniform boundedness of preperiodic points |
| url | https://hdl.handle.net/1721.1/136298 |
| work_keys_str_mv | AT doylejohnr gonalityofdynatomiccurvesandstronguniformboundednessofpreperiodicpoints AT poonenbjorn gonalityofdynatomiccurvesandstronguniformboundednessofpreperiodicpoints |