Reduction of dynatomic curves

Abstract

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families F of polynomial maps, such as the family fc(x) = xm + c, where m ≥ 2. We do this by making use of the dynatomic modular curves Y1(n) (respectively Y0(n)) which parametrize maps f in F together with a point (respectively orbit) of period n for f . The key point in our strategy is to study the set of primes p for which the reduction of Y1(n) modulo p fails to be smooth or irreducible. Morton gave an algorithm to construct, for each n, a discriminant Dn whose list of prime factors contains all the primes of bad reduction for Y1(n). In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime p dividing Dn: one guarantees that p is in fact a prime of bad reduction for Y1(n), yet this same criterion implies that Y0(n) is geometrically irreducible. The other guarantees that the reduction of Y1(n) modulo p is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of Y1(n) for several primes dividing Dn when n = 7, 8, 11, and fc(x) = x2 + c. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.