# Reduction of dynatomic curves

07 Mar 2018

The dynatomic modular curves parametrize polynomial maps together with a point of period \$n\$. It is known that the dynatomic curves \$Y_1(n)\$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form \$f_c(z) = z^m +c\$ where \$m\geq 2\$. In the present paper, we build on the work of Morton to partially characterize the primes \$p\$ for which the reduction modulo \$p\$ of \$Y_1(n)\$ remains smooth and/or irreducible. As an application, we give new examples of good reduction of \$Y_1(n)\$ for several primes dividing the ramification discriminant when \$n=7,8,11\$. The proofs involve arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.