Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

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.

10 pages; v2 has some improvements in exposition

References in corpus (1)

• Reduction of dynatomic curves

Cited by in corpus (4)

• Reduction of dynatomic curves
• Moduli Spaces for Dynamical Systems with Portraits
• New families satisfying the Dynamical Uniform Boundedness Principle over function fields
• Uniform bounds on periodic points of polynomials with good reduction