# Reduction of dynatomic curves

## TL;DR

This paper investigates the reduction properties of dynatomic curves associated with polynomial maps, identifying primes where these curves retain smoothness and irreducibility after reduction, and providing new examples of good reduction.

## Contribution

It extends Morton’s work by partially characterizing primes for which dynatomic curves remain smooth and irreducible modulo p, with applications to specific cases n=7,8,11.

## Key findings

- Identification of primes with good reduction properties
- New examples of primes with smooth and irreducible reductions
- Application to dynatomic curves for specific n values

## Abstract

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.