Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

TL;DR

This paper proves that certain algebraic curves related to preperiodic points of specific polynomials have increasing gonality, leading to a function field analogue of the strong uniform boundedness conjecture and implications for number fields.

Contribution

It establishes the geometric irreducibility and unbounded gonality of dynatomic curves for polynomials of the form z^d+c, advancing the understanding of preperiodic points.

Findings

Gonality of dynatomic curves tends to infinity.

Function field analogue of strong uniform boundedness is proved.

Implications for uniform boundedness over number fields.

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.