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arXiv · 1711.04233

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

John R. Doyle·Bjorn Poonen
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Abstract

Fix $d \ge 2$ and a field $k$ such that $\mathrm{char}~k \nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^d+c$ are geometrically irreducible and have gonality tending to $\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^d+c$. 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.

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BibTeXRIS

John R. Doyle, Bjorn Poonen. 2017-11-12. Gonality of dynatomic curves and strong uniform boundedness of preperiodic points. https://doi.org/10.1112/s0010437x20007022

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