Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

TL;DR

This paper classifies the possible preperiodic point graphs for quadratic polynomials over the cyclotomic quadratic fields () and (), extending previous classifications over  and quadratic extensions.

Contribution

It provides a conjecturally complete classification of preperiodic point graphs for quadratic polynomials over () and (), using dynamical modular curves and quadratic points on curves.

Findings

Classification over () and () completed

Identification of possible graph structures for preperiodic points

Extension of prior classifications to new quadratic fields

Abstract

Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha) = \beta$. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $\mathbb{Q}$, while recent work of the author, Faber, and Krumm has provided a detailed study of this question for all quadratic extensions of $\mathbb{Q}$. In this article, we give a conjecturally complete classification like Poonen's, but over the cyclotomic quadratic fields $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. The main tools we use are dynamical modular curves and results concerning quadratic points on curves.