Preperiodic points for quadratic polynomials over cyclotomic quadratic fields
Volume 196 / 2020
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.
