Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

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.