Gonality of Dynatomic Curves and Strong Uniform Boundedness of Preperiodic Points

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 z + c are geometrically irreducible and have gonality tending to ∞. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of z + 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.