Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
Name
dyn_gonality.pdf
Description
Accepted version
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426.5 KB
Format
Adobe PDF
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Author(s)
Doyle, John R
Poonen, Bjorn
Date Issued
2020
Journal
Compositio Mathematica
Publisher
Wiley
Version
Author's final manuscript
Abstract
© 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1112/S0010437X20007022
