Newly reducible iterates in families of quadratic polynomials

Abstract

We examine the question of when a quadratic polynomial $f\left(x\right)$ defined over a number field $K$ can have a newly reducible $n$-th iterate, that is, $f^n\left(x\right)$ irreducible over $K$ but $f^{n+1}\left(x\right)$ reducible over $K$, where $f^n$ denotes the $n$-th iterate of $f$. For each choice of critical point $\gamma$, we consider the family

$$g_{\gamma ,m}\left(x\right)={\left(x-\gamma \right)}^2+m+\gamma ,m\in K.$$

For fixed $n\ge 3$ and nearly all values of $\gamma$, we show that there are only finitely many $m$ such that $g_{\gamma ,m}$ has a newly reducible $n$-th 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 (Proc. Amer. Math. Soc. 130:6 (2002), 1589–1596) 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.