Preperiodic points for quadratic polynomials with small cycles over quadratic fields

Abstract

Given a number field $K$ and a polynomial $f(z) \in K[z]$, one can naturally construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points of $f$, with an edge $α\to β$ if and only if $f(α) = β$. The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers $n \ge 1$ and $d \ge 2$, there are only finitely many isomorphism classes of directed graphs $G(f,K)$ as one ranges over all number fields $K$ of degree $n$ and polynomials $f(z) \in K[z]$ of degree $d$. In the case $(n,d) = (1,2)$, Poonen has given a complete classification of all directed graphs which may be realized as $G(f,\mathbb{Q})$ for some quadratic polynomial $f(z) \in \mathbb{Q}[z]$, under the assumption that $f$ does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case $(n,d) = (2,2)$. By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen's -- with a similar assumption on points of large period -- but over all quadratic extensions of $\mathbb{Q}$.