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...
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2021
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| Online Access: | https://hdl.handle.net/1721.1/136298 |
| Summary: | © 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. |
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