Newly reducible iterates in families of quadratic polynomials

Abstract

We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f. For each choice of critical point \gamma of f(x), we consider the family g_{\gamma,m}(x)= (x - \gamma)^2 + m + \gamma, m \in K. For fixed n \geq 3 and nearly all values of \gamma, we show that there are only finitely many m such that g_{\gamma,m} has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of \gamma. These results complement those obtained by Danielson and Fein in the higher-degree case. Our method involves translating the problem to one of finding rational points on certain hyperelliptic curves, determining the genus of these curves, and applying Faltings' theorem.